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.
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W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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)
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W. Winkler and P. Nehter
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)
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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)
2 Thermodynamics of Fuel Cells
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)
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W. Winkler and P. Nehter
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)
2 Thermodynamics of Fuel Cells
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)
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W. Winkler and P. Nehter
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.
2 Thermodynamics of Fuel Cells
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
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W. Winkler and P. Nehter
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-
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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:
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
× 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.
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
˝ 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)
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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
W. Winkler and P. Nehter
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
2 Thermodynamics of Fuel Cells
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:
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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-
2 Thermodynamics of Fuel Cells
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
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[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-
2 Thermodynamics of Fuel Cells
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
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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:
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∂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:
3 Mathematical Models: A General Overview
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)
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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)
3 Mathematical Models: A General Overview
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.
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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
3 Mathematical Models: A General Overview
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)
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S. Ubertini and R. Bove
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-
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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.
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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:
3 Mathematical Models: A General Overview
∇ 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.
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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;
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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:
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S. Ubertini and R. Bove
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.
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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].
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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)
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S. Ubertini and R. Bove
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.
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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.
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S. Ubertini and R. Bove
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)
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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”).
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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];
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• 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
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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)
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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.:
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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.
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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
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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
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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.
References 1. Bove R., Lunghi P., Sammes N.M., 2005. SOFC mathematic model for systems simulations. Part One: From a micro-detailed to macro-black-box model. International Journal of Hydrogen Energy, 30(2), 181–187.
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2. Campanari S., 2001. Thermodynamic model and parametric analysis of a tubular SOFC module. Journal of Power Sources 92, 26–34. 3. Costamagna P., Magistri L., Massardo A.F., 2001. Design and part-load performance of a hybrid system based on a solid oxide fuel cell reactor and a micro gas turbine. Journal of Power Sources 96, 352–368. 4. Chan S.H., Khor K.A., Xia Z.T., 2001. A complete polarization model of a solid oxide fuel cell and its sensitivity to the change of cell components thickness. Journal of Power Sources 93, 130–140. 5. Lin Z., Stevenson J.W., Khaleel M.A., 2003. The effect of interconnect rib size on the fuel cell concentration polarization in planar SOFCs. Journal of Power Sources 117, 92–97. 6. Bove R., Sammes N.M, 2005. The effect of current collectors configuration on the performance of a tubular SOFC. In Proceedings of the Ninth International Symposium on Solid Oxide Fuel Cells (SOFC IX), May 15–20, Quebec City, Canada, S.C. Singhal and J. Mizusaki (Eds.), Electrochemical Society, Vol. 1, pp. 780–781. 7. Bieberle A., Gauckler L.J., 2002. State-space modeling of the anodic SPFC system Ni H2 H2 O|YSZ. Solid State Ionics 146, 23–41. 8. Bessler W., 2005. New computational approach for SOFC impedance from detailed electrochemical reaction-diffusion models. Solid State Ionics 176, 997–1011. 9. Mitterdorfer A., Gauckler L.J., 1999. Reaction kinetics of the Pt, O2 (g)|c-ZrO2 system: Precursor-mediated adsorption. Solid State Ionics 120(1/4), 211–225. 10. Mitterdorfer A., Gauckler L.J., 1999. Identification of the reaction mechanism of the Pt, O2 (g)|yttria-stabilized zirconia system: Part I: General framework, modelling, and structural investigation. Solid State Ionics 117(3/4), 187–202. 11. Mitterdorfer A., Gauckler L.J., 1999. Identification of the reaction mechanism of the Pt, O2 (g)|yttria-stabilized zirconia system: Part II: Model implementation, parameter estimation, and validation. Solid State Ionics 117(3/4), 203–217. 12. Sunde, S., 2000. Simulations of composite electrodes in fuel cells. Journal of Electroceramics 5(2), 153–182. 13. Adler S.B., Lane, J.A., Steele B.C.H., 1996. Electrode kinetics of porous mixed-conducting oxygen electrodes. Journal of the Electrochemical Society 143(11), 3554–3564. 14. Ioselevich A.S. and Kornyshev A.A., 2001. Phenomenological theory of Solid Oxide Fuel Cell Anode. Fuel Cells 1(1), 40–65. 15. Costamagna P., Costa P., Antonucci V., 1998. Micro-modeling of solid oxide fuel cell electrodes. Electrochemica Acta 43(3/4), 375–394. 16. Whitaker S., 1986. Flow in porous media I: A theoretical derivation of Darcy’s law. Transport in Porous Media 1(1), 3–25. 17. Nabae Y., Yamanaka I., Hatano M., Otsuka K., 2006. Catalytic behavior of Pd–Ni/composite anode for direct oxidation of methane in SOFCs. Journal of the Electrochemical Society 153(1), A140–A145. 18. Weber A., Sauer B., Müller A.C., Herbstritt D., Ivers-Tiffée E., 2002. Oxidation of H2, CO and methane in SOFCs with Ni/YSZ-cermet anodes. Solid State Ionics Volumes 152-153, pp. 543-550. 19. Perry Murray E., Tsai T. and Barnett S.A., 1999. A direct-methane fuel cell with a ceria-based anode. Nature 400, 649–651. 20. Andreassi L., Toro C., Ubertini S., 2007. Modeling carbon monoxide direct oxidation in solid oxide fuel cells. In Proceedings ASME European Fuel Cell Technology and Applications Conference, EFC2007-39057. 21. Campanari S., Iora P., 2004. Definition and sensitivity analysis of a finite volume SOFC model for a tubular cell geometry. Journal of Power Sources 132(1/2), 113–126. 22. Achenbach E., 1994. Three-dimensional and time dependent simulation of a planar solid oxide fuel cell stack. Journal of Power Sources 49, 333–348. 23. Agnew G.D., Bernardi D., Collins R.D., Cunninghama R.H., 2006. An internal reformer for a pressurised SOFC system. Journal of Power Sources 157(2/3), 832–836. 24. Dicks A.L., Pointon K.D., Siddle A., 2000. Intrinsic reaction kinetics of methane steam reforming on a nickel/zirconia anode. Journal of Power Sources 86, 523–530.
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25. Hirsch, C. 1992. Numerical Computation of Internal and External Flows. John Wiley & Sons. 26. Abbott M.B., Basco D.R., 1989. Computational Fluid Dynamics: An Introduction for Engineers. Longman Scientific & Technical/Wiley, Harlow, Essex/New York. 27. Patankar S.V., 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation. 28. Ferziger J.H., Peric M., 1995. Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin. 29. Bird R.B., Steward W.E., Lightfoot E.N., 1960. Transport Phenomena. J. Wiley & Sons, New York. 30. Chase M.W., Jr., 1998. NIST-JANAF Themochemical Tables, Fourth Edition, J. Phys. Chem. Ref. Data, Monograph 9, pp. 1-1951. 31. Todd, B., Young, J.B., 2002. Thermodynamic and transport properties of gases for use in SOFC modeling. Journal of Power Sources 110, 186–200. 32. Khaleel M.M.A., Lin Z., Singh P., Surdoval W., Collin D., 2004. A finite element analysis modeling tool for solid oxide fuel cell development: Coupled electrochemistry, thermal and flow analysis in MARC. Journal of Power Sources 130, 136–148. 33. Ferguson J.R., 1992, SOFC two dimensional “unit cell” modeling. SOFC Stack Design Tool, International Energy Agency Final Report. 34. Ferguson J.R., Fiard J.M., Herbin, R., 1996. Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. Journal of Power Sources 58, 109–122. 35. Gemmen R.S., Johnson C.D., 2005, Effect of load transient on SOFC operation – Current reversal on loss of load. Journal of Power Sources 144, 152–164. 36. Zhao F., Virkar A.V., 2005. Dependence of polarization in anode-supported solid oxide fuel cells on various cell parameters. Journal of Power Sources 141(1), 79–95. 37. Gileadi E., 1993. Electrode Kinetics for Chemists, Chemical Engineers, and Material Scientists. Wiley-VCH. 38. Costamagna P., Honegger K., 1998. Modeling of solid oxide heat exchanger integrated stacks and simulation at high fuel utilization. Journal of the Electrochemical Society 145, 3995–4007. 39. Andreassi L., Bove R., Rubeo G., Ubertini S., Lunghi P., 2007. Experimental and numerical analysis of a radial flow Solid Oxide Fuel Cell. International Journal of Hydrogen Energy 32(17), 4559–4574. 40. Costamagna P., Selimovic A., Del Borghi M., Agnewc G., 2004. Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC). Chemical Engineering Journal 102, 61–69. 41. Gurau V., Liu H., Kakac S., 1998. Two-dimensional model for proton exchange membrane fuel cells. AIChE Journal 44(11), 2410–2422. 42. Daun K.J., Beale S.B., Liu F., Smallwood G.J., 2006. Radiation heat transfer in planar SOFC electrolytes. Journal of Power Sources 157, 302–310. 43. Kanamura K., Yoshioka S., Takehara Z., 1991. Dependence of entropy change of single electrodes on partial pressure in Solid Oxide Fuel Cells. Journal of the Electrochemical Society 138(7), 2165–2167. 44. Ito Y., Kaiya H., Yoshizawa S., Ratkje S.K., Førland T., 1984. Electrode Heat balances of electrochemical cells. Journal of the Electrochemical Society 131(11), 2504–2509. 45. Ito Y., Foulkes F.R., Yoshizawa S., 1982. Energy analysis of a steady-state electrochemical reactor. Journal of the Electrochemical Society 129(9), 1936–1943. 46. Froment G.F., Bishoff, K.B., 1990. Chemical Reactors Analysis and Design. J. Wiley & Sons, New York. 47. Bear J., Buchlin J.M., 1991. Modeling and Application of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Boston, MA. 48. Bear J., 1972. Dynamics of Fluids in Porous Media. Dover Publications, New York. 49. Crank J., 1956. The Mathematics of Diffusion. Oxford University Press, New York, pp. 12–15. 50. Beale S.B., 2004. Calculation procedure for mass transfer in fuel cells. Journal of Power Sources 128, 185–192. 51. Kim J.W., Virkar A.V., Fung K.Z., Metha K., Singhal S.C., 1999. Polarization effects in intermediate temperature, anode-supported Solid Oxide Fuel Cells. Journal of the Electrochemical Society 146(1), 69–78.
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52. Hwang J.J., Chen C.K., Lai D.Y., 2005. Computational analysis of species transport and electrochemical characteristics of a MOLB-type SOFC. Journal of Power Sources 140, 235–242. 53. Autissier N., Larrain D., Van Herle J., Favrat D., 2004. CFD simulation tool for solid oxide fuel cells. Journal of Power Sources 131, 313–319. 54. Iwata M., Hikosaka T., Morita M., Iwanari T., Ito K., Onda K., Esaki Y., Sakaki Y., Nagata S., 2000. Performance analysis of planar-type unit SOFC considering current and temperature distribution. Journal of Power Sources 132, 297–308. 55. Roos M., Batawi E., Harnisch U., Hocker Th., 2003. Efficient simulation of fuel cell stacks with the volume averaging method. Journal of Power Sources 118, 86–95. 56. Veldsink J.W., van Damme R.M.J., Versteeg G.F., van Swaaij W.P.M., 1995. The use of the dusty-gas model for the description of transport with chemical reaction in porous media. Chemical Engineering and the Biomedical Engineering Journal 57(2), 115–125. 57. Yakabe H., Hishinuma M., Uratani M., Matsuzaki Y., Yasuda I., 2000. Evaluation and modeling of performance of anode-supported solid oxide fuel cell. Journal of Power Sources 86, 423–431. 58. Lehnert W., Meusinger J., Thom F., 2000. Modelling of gas transport phenomena in SOFC anodes. Journal of Power Sources 87, 57–63. 59. Suwanwarangkul R., Croiset E., Fowler M.W., Douglas P.L., Entchev E., Douglas M.A., 2003. Performance comparison of Fick’s, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode. Journal of Power Sources 122, 9–18. 60. Kestin J., Wakeham W.A., 1988. Transport Properties of Fluids; Thermal conductivity, Viscosity, and Diffusion Coefficient. Hemisphere Publishing, New York. 61. Cussler E.L., 1984. Diffusion-Mass Transfer in Fluid Systems. Cambridge University Press, Cambridge, MA. 62. Reid R.C, Prausnitz J.M., Sherwood T.K., 1977. The Properties of Gases and Liquids. McGraw-Hill, New York. 63. Jiang Y., Virkar A.V., 2003. Fuel composition and diluent effect on gas transport and performance of anode-supported SOFCs. Journal of the Electrochemical Society 150(7), A942–A951. 64. Burt A.C., Celik I.B., Gemmen R.S., Smirnov A.V., 2003. Influence of radiative heat transfer on variation of cell voltage within a stack. In Proceedings of the 1st International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, April 21–23, 2003. 65. Murthy S., Fedorov G, 2003. Radiation heat transfer analysis of the monolith type solid oxide fuel cell. Journal of Power Sources 124(2), 453–458. 66. VanderSteen J.D.J., Pharoah J.G., 2006. Modeling radiation heat transfer with participating media in Solid Oxide Fuel Cells. Journal of Fuel Cell Science and Technology 3(1), 62–67. 67. Norman F., Bessette, N.F., Wepfer W.J., 1996. Prediction of on-design and off-design performance for a solid oxide fuel cell power module. Energy Conversion and Management 37(3), 281–293. 68. Smith G.D., 1985. Numerical Solution of Partial Differential Equations: Finite Difference Methods, third edition. Claredon Press, Oxford. 69. Versteeg H.K., Malalasekera W., 1995. An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Addison Wesley Longman, Harlow, England. 70. Eymard R., Gallouet T., Herbin R., 2000. Finite volume methods. In Handbook of Numerical Analysis, Vol. VII, P.G. Ciarlet, J.L. Lions (Eds.). North Holland, Amsterdam, pp. 713–1020. 71. Zienkiewicz O.C., Taylor R., 2000. The Finite Element Method: Volumes 1, 2 & 3, fifth edition. Elsevier, Butterworth–Heinemann. 72. Reddy J.N., 1993. An Introduction to the Finite Element Method. McGraw-Hill. 73. Bathe K.J., 1996. Finite Element Procedures. Prentice Hall, Englewood Cliffs, NJ. 74. Thompson J.F., Warsi Z.U.A., Mastin C.W., 1985. Numerical Grid Generation, Foundations and Applications. North Holland, Amsterdam. 75. Thompson J.F., Soni B., Weatherill N. (Eds.), 1998. Handbook of Grid Generation. CRC Press. 76. Liseikin V.D., 1999. Grid Generation Methods. Springer. 77. Owen S.J., Shephard M.S., 2003. Trends in unstructured mesh generation. Special Issue, International Journal for Numerical Methods in Engineering 58(2).
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78. Singhal S.C., Kendall K., 2003. High-Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications. Elsevier Science & Technology. 79. Ciano C., Calì M., Melhus O., Verda V., 2006. A model for the configuration design of a tubular Solid Oxide Fuel Cell Stack. ASME Paper IMECE2006-16141, Chicago, IL, November 5–10, 2006. 80. Bossel U., 1992. Facts & figures. Final Report on SOFC data. International Energy Agency, Programme on R & D & D on Advanced Fuel Cells, Annex II. 81. Repetto L., Agnew G., Del Borghi A., Di Benedetto F., Costamagna P., 2007. Detailed simulation of the ohmic resistance of SOFCs. Journal of Fuel Cell Science and Technology 4(4), in press. 82. Incropera F.P., DeWitt D.P., 1990. Fundamentals of Heat and Mass Transfer. Wiley, New York. 83. Haynes C., Wepfer W.J., 2000. ‘Design for power’ of a commercial grade tubular solid oxide fuel cell. Energy Conversion & Management 41, 1123–1139. 84. Fuel Cell Handbook, sixth edition, 2002. US Department of Energy, National Energy Technology Laboratory, Morgantown/Pittsburgh.
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
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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
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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),
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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
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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
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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-
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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
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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)
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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
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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-
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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
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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.
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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.
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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
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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.
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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.
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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
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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-
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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-
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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
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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.)
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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.
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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
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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.
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(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
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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-
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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
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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.
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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. <5%) of exit hydrogen concentration, at least 80 grid nodes are required for this problem. Table A.1 shows the values of exit concentration and temperature predicted by the different grids. Using the solutions with 40, 20 and 10 grid points for Richardson extrapolation with Equation (A.16) we have the following system of equations in terms of extrapolated value and extrapolation constants of exit mass fraction of hydrogen.
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Fig. A.5 Percentage grid convergence error in the predictions of exit temperature and hydrogen mass fraction as a function of number of grid points used: the solution with 640 grid points is taken as converged (grid independent) solution and the grid convergence error is calculated as abs[(φconverged − φh )/φconverged ].
⎡
⎤⎡ ⎤ ⎡ ⎤ 1 −2.50 × 10−3 −6.25 × 10−6 3.7253 × 10−2 φext ⎣ 1 −5.00 × 10−3 −2.50 × 10−5 ⎦ ⎣ c1 ⎦ = ⎣ 3.9945 × 10−2 ⎦ . c2 1 −1.00 × 10−2 −1.00 × 10−4 4.5168 × 10−2 Solving the system we have, for hydrogen mass fraction at exit φext = 3.4507 × 10−2 ,
c1 = −1.1093,
c2 = 4.3135.
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Table A.1 Convergence of hydrogen mass fraction and temperature at the exit of the fuel channel. No. of nodes
Grid size (m)
H2 mass fraction at exit
Temperature (K) at exit
10 20 40 80 160 320 640
1.0000E-02 5.0000E-03 2.5000E-03 1.2500E-03 6.2500E-04 3.1250E-04 1.5625E-04
4.5168E-02 3.9945E-02 3.7253E-02 3.5841E-02 3.5114E-02 3.4744E-02 3.4558E-02
1373.4977 1382.3140 1387.1079 1389.6956 1391.0502 1391.7464 1392.0986
Similarly for the exit temperature, the system of equations and its solution are ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 −2.50 × 10−3 −6.25 × 10−6 φext 1387.1079 ⎣ 1 −5.00 × 10−3 −2.50 × 10−5 ⎦ ⎣ c1 ⎦ = ⎣ 1382.3140 ⎦ c2 1373.4977 1 −1.00 × 10−2 −1.00 × 10−4 and φext = 1392.1591,
c1 = 2071.9113,
c2 = 20577.6966.
Thus, in the calculation with 40 grid nodes, the estimated discretization error is 3.4507 × 10−2 − 3.7253 × 10−2 δdis = abs 3.4507 × 10−2 = 7.96% in exit hydrogen mass fraction and
δdis = abs
1392.1591 − 1387.1079 = 0.36% in exit temperature. 1392.1591
Now for using the second method for extrapolation, given by Equation (A.17), we solve the following non-linear equation for the apparent order p of the method and the extrapolated valued, φext . These equations are obtained by substituting the grid sizes and their respective predicted exit concentration values from Table 5.1 into Equations (A.20–A.24) −0.0052228 , 2p = −0.0026923 −0.0026923 , φext = 0.037253 + (2p − 1) which give p = 0.956 and φext = 0.034389. Similarly for the exit temperature 2p =
4.79394 , 8.81623
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φext = 1387.11 +
4.79394 , (2p − 1)
which give p = 0.879 and φext = 1392.82 K. Note that the calculated apparent order is close to the theoretical order of the first order upwind scheme used in the present calculations. In order to verify the estimated apparent order of the method we may extrapolate the solution for the exit temperature and hydrogen concentration for grid size of 80 nodes using the solution from 40 node grid and the extrapolation constants found above. These values are found to be 1389.71 K and 0.035865, respectively, which are close to the actual values given in Table A.1 thus confirming our uncertainty analysis.
Indirect Model Validation In the absence of elaborate experimental data to validate the multi-dimensional fuel cell models, an indirect validation may be performed by comparing the predictions of two or more independent models for an identical test case. The models used in indirect validation may differ from each other in dimensionality, simplifying approximations that were used etc, and hence it may not be possible to identically match all the input parameters to the model. However, care must be taken to match the test cases as closely as possible. This approach was used in Achenbach (1994a) where eight independent models from different authors were compared against each other for a set of pre-defined benchmark test cases. An example from their results is given in Table A.2 where three of the integral output parameters from the models viz. maximum solid temperature, minimum solid temperature, and the air exit temperature are compared. This table also shows the predictions from DREAM-SOFC code (Pakalapati, 2006; Pakalapati et al., 2006) for the benchmark case for the purpose of validation against the other eight codes which were validated against each other. The predictions of DREAM-SOFC are not included in the calculation of the mean and standard deviation values shown in Table A.2. Deviation values given in Table A.2 are the absolute differences between the dream predictions and the average of the other eight predictions for each parameter. It can be seen that the DREAM SOFC results fall more or less within the standard deviation range from the mean of eight other predictions. To make a more formal assessment of the possible errors (a combination of numerical modeling errors) in the calculations of DREAM-SOFC, here we treat the other eight results as outcomes from eight repetitions of a virtual experiment, the differences being a result of experimental uncertainty. With this assumption, we can calculate the 90% confidence interval for the ‘true error’ in the DREAM results using the t-distribution as: S0 90% C.I. = (φ¯ 0 − φ˜D ) ± √ t0.05 (A.26) n0
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Table A.2 Predictions from nine different independent models with same material properties and operating conditions for a cross SOFC. Author I II III IV V VI VII VIII DREAM Mean SD Deviation
Max. solid T (◦ C)
Min. solid T (◦ C)
Air Exit T (◦ C)
1170 1220 1153 1182 1157 1121 1170 1162 1155 1167 28 12
912 911 910 915 907 909 918 913 914 912 3 2
1063 1066 1079 1040 1067 1057 1078 1067 1080 1065 12 15
where φ¯0 , S0 and n0 are the average, standard deviation and number of cases for the results other than DREAM-SOFC’s, φ˜ D is the single result of DREAM-SOFC and t0.05 is from the t-distribution tables with n0 − 1 degrees of freedom. From Equation (A.26), 90% confidence intervals for ‘true error’ in predictions of DREAMSOFC are: Maximum Solid Temperature: (−8.05◦C, 32.05◦C) Minimum Solid Temperature: (−4.15◦C, 0.15◦C) Air Exit Temperature: (−23.59◦C, −6.41◦C) Since these estimated error bounds are small compared to the actual value of these parameters, it may be said that DREAM-SOFC is indirectly validated against other models.
References Achenbach, E. (1994a) Status of the IEA Benchmark Test I on Stack Modeling, IEA Workshop, Rome, February 22–24, 1994. Achenbach, E. (1994b) Three-dimensional and time-dependant simulation of a planar solid oxide fuel cell stack, Journal of Power Sources 49, 333–348. Aguiar, P., Chadwick, D. and Kershenbaum, L. (2002) Modelling of an indirect internal reforming solid oxide fuel cell, Chemical Engineering Science 57, 1665–1677. Beil, A. and Seume, J. (2006) Unsteady performance of a PEMFC system including autothermal methane reforming, in Proceedings of the 4th International ASME Conference on Fuel Cell Science, Engineering and Technology, June 19–21, Irvine, CA. Billingham, J., King, A.C., Copcutt, R.C. and Kendall, K. (2000) Analysis of a model for a loaded, planar, solid oxide fuel cell, Siam Journal of Applied Mathematics 60(2), 574–601. Burt, A.C. (2005) Refinement of numerical models and parametric study of SOFC stack performance, Dissertation, West Virginia University, Morgantown.
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Burt, A., Celik, I., Gemmen, R. and Smirnov, A. (2003a) Cell-to-cell performance variations within a stack, in ECS SOFC-VIII Proceedings Vol. 2003-07, April 27–May 2, Paris, France, pp. 1487–1500. Burt, A., Celik, I.,Gemmen, R. and Smirnov, A. (2003b) Influence of radiative heat transfer on variation of cell voltage within a stack, in Proceedings of First International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, April 21–23, 2003. Burt, A., Celik, I., Gemmen, R. and Smirnov, A. (2004a) A numerical study of cell-to-cell variations in a SOFC stack, Journal of Power Sources 126, 76–87. Burt, A.C., Celik, I.B., Gemmen, R.S., Smirnov, A.V. and Rogers, W.A. (2004b) Cell-to-cell variations with increasing SOFC stack size, in Proceedings of Second International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, June 14–16, 2004, pp. 31–38. Calise, F., D’Accadia, M., Vanoli, L. and Palombo, A. (2005) One-dimensional model of a tubular solid oxide fuel cell, in Proceedings of the 1st European Fuel Cell Technology and Applications Conference, Rome, Italy, December 14–16. Celik, I. (2006) Introduction to fuel cell technology, Lecture Notes, Department of Mechanical and Aerospace Engienering, West Virginia University, Morgantown, WV, USA. Celik, I. and Elizalde-Blancas, F. (2006) Verification of various measures for numerical uncertainty using manufactured solutions: An iteration and grid convergence study, Report No. MAE:IC06TR01, Departmental Report, Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown, WV, USA. Celik, I.B. and Pakalapati, S.R. (2003) A modular approach to SOFC modeling, Final Report submitted to Oak Ridge Associated Universities (ORAU) Oak Ridge, Tennessee. Celik, I., Pakalapati, S. and Salazar-Villalpando, M. (2005) Theoretical calculation of the electrical potential at the electrode/electrolyte interfaces of solid oxide fuel cells, Journal of Fuel Cell Science and Technology 2(4), 238–245. Costamagna, P. and Honegger, K. (1998) Modeling of solid oxide heat exchanger integrated stacks and simulation at high fuel utilization, Journal of the Electrochemical Society 145(11), 3995– 4007. Costamagna, P., Arato, E., Achenbach, E. and Reus, U. (1994) Fluid dynamic study of fuel cell devices: simulation and experimental validation, Journal of Power Sources 52, 243–249. Elizalde-Blancas, F., Pakalapati, S.R., Cayan, F. and Celik, I. (2007a) One-dimensional transient lumped model for fast fuel cell stack analysis, in proceedings of Renewable Energy Symposium, Frostburg, MD, USA, September 14–15. Elizalde-Blancas, F., Pakalapati, S.R. and Celik, I. (2007b) One-dimensional transient lumped model for fast fuel cell stack analysis, Report submitted to Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown, WV. 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, 109–122. Ferziger, J.H. and Peric, M. (1996) Further discussion of Numerical Errors in CFD, International Journal for Numerical Methods in Fluids 23, 1263–1274. FLUENT 6.2 (2005) Fuel Cell Modules Manual, FLUENT Inc., Lebanon, NH. Gemmen, R. and Trembly, J. (2006) On the mechanisms and behavior of coal syngas transport and reaction within the anode of a solid oxide fuel cell, Journal of Power Sources 161, 1084–1095. Gemmen, R. and Johnson, C. (2004) Dynamics of solid oxide fuel cell operation, in Pproceedings of the Second International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, June 14–16, pp. 229–238. Gemmen, R.S., Liese, E., Rivera, J., Jabbari, F. and Brouwer, J. (2000a) Development of dynamic modeling tools for solid oxide and molten carbonate hybrid fuel cell gas turbine systems, International Gas Turbine Institute Meeting of the ASME, May 8–12, 2000. Gemmen, R., Rogers, W. and Prinkey, M. (2000b) Application of a computational fluid dynamics code to fuel cells – Integrated SOFC fuel cell and post xxidizer, American Flame Research Committee (AFRC) International Symposium, Newport Beach, CA, USA, September 2000. Ghezel-Ayagh, H., Lukas, M. and Junker, S. (2004) Dynamic modeling and simulation of a hybrid fuel cell/gas turbine power plant for control system development, in Proceedings of the Second
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International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, June 14–16, pp. 325–329. Grens, E.A. and Tobias, C.W. (1964) Analysis of dynamic behavior of flooded porous electrodes, Zeitschrift für Elektrochemie 68(3), 236–249. Gubner, A., Froning, D., Haart, B. and Stolten, D. (2003) Complete modeling of kW-range SOFC stacks, ECS Proceedings 2003-17, 1436–1441. Hirata, H. and Hori, M. (1996) Gas-flow uniformity and cell performance in a molten carbonate fuel cell stack, Journal of Power Sources 63, 115–120. Incropera, F. and DeWitt, D.P. (2002) Fundamentals of Heat and Mass Transfer, 5th Edition, Wiley, Hoboken, NJ. Jiang, W., Fang, R., Khan, J. and Dougal, R. (2006) Parameter setting and analysis of a dynamic tubular SOFC model, Journal of Power Sources 162(1), 316–326. Koh, J., Seo, H., Yoo, Y. and Lim, H. (2002) Consideration of numerical simulation parameters and heat transfer models for a molten carbonate fuel cell stack, Chemical Engineering Journal 87, 367–379. Kolavennu, P., Telotte, John C. and Palanki, S. (2004) Modeling of a fuel cell power system for automotive applications, in Proceedings of the Second International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, June 14–16, pp. 177–181. Krotz, D. (2003) Almost there: A commercially viable fuel cell, http://enews.lbl.gov/ScienceArticles/Archive/MSD-fuel-cells.html. Li, P. and Suzuki, K. (2004) Numerical modeling and performance study of a tubular SOFC, Journal of the Electrochemical Society 151, A548–A557. Lin, Z., Khaleel, M., Surdoval, W. and Collins, D. (2003) Finite element analysis of solid oxide fuel cells: Coupled electrochemistry, thermal and flow analysis in MARC, in Proceedings SECA Modeling and Simulation Training Session, Morgantown, WV, August 29, 2003. Liu, Z., Mao, Z., Wang, C., Zhuge, W. and Zhang, Y. (2006) Numerical simulation of a mini PEMFC stack, Journal of Power Sources 160, 1111–1121. Lu, Y., Schaefer, L. and Li, P. (2005) Numerical study of a flat tube high power density solid oxide fuel cell. Part I: Heat/mass transfer and fluid flow, Journal of Power Sources 140, 331–339. Ma, Z. (2000) A combined differential and integral model for high temperature fuel cells, Ph.D. Thesis Georgia Institute of Technology. Maggio, G., Recupero, V. and Mantegazza, C. (1996) Modelling of temperature distribution in a solid polymer electrolyte fuel cell stack, Journal of Power Sources 62, 167–174. Masuda, H., Ito, K., Kakimoto, Y., Miyazaki, T., Ashikaga, K. and Sasaki, K., (2006) Numerical simulation of two-phase flow and transient response in polymer electrolyte fuel cell, in Proceedings of FUELCELL2006, The 4th International Conference on Fuel Cell Science Engineering and Technology, Irvine, CA, June 19–21. Mendenhall, W. and Sincich, T. (2007) Statistics for Engineering and Sciences, 5th Edition, Prentice Hall, Englewood Cliffs, NJ. Minh, N.Q. (1995) Science and Technology of Solid Oxide Fuel Cells, Elsevier Sciences, Amsterdam. Murthy, S. and Fedorov, A. (2003) Radiation heat transfer analysis of the monolith type solid oxide fuel cell, Journal of Power Sources 124, 453–458. Nishino, T., Iwai, H. and Suzuki, K. (2006) Comprehensive numerical modeling and analysis of a cell-based indirect internal reforming tubular SOFC, Journal of Fuel Cell Science and Technology 3, 33–44. O’Hayre, R., Cha, S-W., Colella, W. and Prinz, F.B. (2006) Fuel Cell Fundamentals, John Wiley & Sons, Hoboken, NJ. Pakalapati, S.R. (2003) A numerical study of current distribution inside the cathode and electrolyte of a solid oxide fuel cell, MS Thesis, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV. Pakalapati, S.R. (2006) A new reduced order model for solid oxide fuel cell modeling, PhD Thesis, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV.
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Pakalapati, S., Yavuz, I., Elizalde-Blancas, F. and Celik, I. (2006) Comparison of a multidimensional model with a reduced order pseudo three-dimensional model for simulation of solid oxide fuel ells, in Proceedings of the 4th International ASME Conference on Fuel Cell Science, Engineering and Technology, Irvine, CA, June 19–21. Pakalapati, S.R., Elizalde-Blancas, F. and Celik, I. (2007) Numerical simulation of solid oxide fuel cell stacks: Comparison between a reduced order pseudo three-dimensional model and a multidimensional model, Proceedings of 5th ASME International Fuel Cell Science, Engineering and Technology Conference, New York, USA, June 18–20. Richardson, L.F. (1910) The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Transactions of the Royal Society of London, Ser. A 210, 307–357. Roache, P. (1998) Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM. Sanchez, D., Chacartegui, R., Munoz, A. and Sanchez, T. (2006) Thermal and electrochemical model of internal reforming solid oxide fuel cells with tubular geometry, Journal of Power Sources 160(2), 1074–1087. Suzuki, M., Shikazono, N., Fukagata, K. and Kasagi, N. (2006) Numerical analysis of heat/mass transfer and electrochemical reaction in an anode supported flat-tube solid oxide fuel cell, in Proceedings of FUELCELL2006, The 4th International Conference on Fuel Cell Science Engineering and Technology, Irvine, CA, June 19–21. Suwanwarangkul, R., Croiset, E., Entchev, E., Charojrochkul, S., Pritzker, M.D., Fowler, M.W., Douglas, P.L., Chewathanakup, S. and Mahaudom H. (2006) Experimental and modeling study of solid oxide fuel cell operating with syngas fuel, Journal of Power Sources 161, 308–322. Valluru, S. (2005) Steady state thermal analyses of two-dimensional and three-dimensional solid oxide fuel cells, MS Thesis, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV. VanderSteen, J. and Pharoah, J. (2004) The role of radiative heat transfer with participating gases on the temperature distribution in solid oxide fuel cells, in Proceedings of Fuel Cell Science, Engineering, and Technology, Rochester, NY, June 14–16, 2004, pp. 483–490. Virkar, A., Chen, J., Tanner, C. and Kim, J. (2000) The role of electrode microstructure on activation and concentration polarizations in solid oxide fuel cells, Solid State Ionics 131, 189–198. Yakabe, H., Hishinuma, M., Uratani, M., Matsuzaki, Y. and Yasuda, I. (2000) Evaluation and modeling of performance of anode-supported solid oxide fuel cell, Journal of Power Sources 86, 423–431. Yuan, J., Rokni, M. and Sundén, B. (2003) Three-dimensional computational analysis of gas and heat transport phenomena in ducts relevant for anode-supported solid oxide fuel cells, International Journal of Heat and Mass Transfer 46, 809–821. Zitney, S., Prinkey, M., Shahnam, M. and Rogers, W. (2004) Coupled CFD and process simulation of a fuel cell auxiliary power unit, in Pproceedings of the Second International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY, June 14–16, pp. 339–345.
Chapter 6
IP-SOFC Model Simone Grosso, Laura Repetto and Paola Costamagna DICHEP, University of Genoa, Italy
6.1 Introduction to the IP-SOFC Geometry The Integrated Planar-Solid Oxide Fuel Cell (IP-SOFC) [1] is a proprietary solid oxide fuel cell design currently developed by Rolls-Royce Fuel Cell Systems Ltd (RRFCS) for integration into an SOFC-gas turbine hybrid system. The IP-SOFC is substantially a cross between tubular and planar geometries, seeking to combine the thermal compliance qualities of the former and the low-cost component fabrication and short current paths of the latter. The various elements of the stack architecture are displayed in Figure 6.1, starting from a cross-section of the single cell, showing the sequence of the various components (Figure 6.1a). The main electrochemical reactions occurring within the IP-SOFC are: 1 O2 + 2e− → O−− cathode, 2 H2 + O−− → H2 O + 2e− anode,
(6.2)
CO + O−− → CO2 + 2e−
(6.3)
anode.
(6.1)
However, other chemical reactions occur at the anodic side when CH4 , CO and H2 O are present in the fuel flow: CH4 + H2 O ↔ CO + 3H2 CO + H2 O ↔ CO2 + H2
reforming,
(6.4)
shifting.
(6.5)
In order to obtain high voltages, a number of single cells are electrically connected in series by means of conductive interconnections. This arrangement is achieved by printing a number of stripe-shaped single cells on each side of a porous ceramic support, called the tube, as shown in Figure 6.1d. The anodic side of the cell is in contact with the ceramic support; the feeding fuel is fed into a number of vertical channels present inside the tube, and reaches the anodic side of the cells by diffusing through R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 183–205. © Springer Science+Business Media B.V. 2008
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Fig. 6.1 Scheme of the IP-SOFC single cell (a), manufacturing process (b), ceramic support (c), tube (d) and bundle (e).
Fig. 6.2 Scheme of the IP-SOFC stack.
the porous support. The oxidant flows horizontally, external to the tube itself, skimming over the cathodic side of the cells. A number of IP-SOFC tubes are assembled and connected forming a bundle (Figure 6.1e). The connection is in-series from the point of view of the anodic fuel flow, since the exit of the anodic channels of one tube is connected to the inlet of the anodic channels of the subsequent tube. On the cathodic side, the oxidant feed is in parallel through the different tubes of the same bundle, in the horizontal direction. The IP-SOFC bundle is then the basic unit of the IP-SOFC stack (Figure 6.2), where several bundles are coupled to a reformer in which the fresh fuel (CH4 ) reacts with a part of the water-rich SOFC anode off-gas, in order to produce an H2 and COrich gas mixture. The fuel produced by the reformer is fed in parallel to the various electrochemical bundles. The oxidant stream crosses all the bundles of the stack in series. The hot oxidant stream exiting the last electrochemical bundle flows through the reformer as well, in order to provide the heat necessary for the endothermic reforming reaction.
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Finally, the IP-SOFC stack is an essential part of a hybrid plant, which combines this technology together with a gas turbine in order to provide a high efficiency power generation system. This chapter is divided into three main sections, focusing respectively on the simulation of the single cell (Section 6.2), of the tube (Section 6.3) and of the bundle (Section 6.4). In each section, the results obtained from the simulation model are reported, and Sections 6.2 and 6.3 also discuss a comparison to experimental data provided by RRFCS.
6.2 Single Cell Simulation The simulation of the single cell is based on the fundamental assumption that temperatures and gas compositions are uniform all over the cell. This assumption may be not fully reliable in the case of fuel cells deposited onto a tube and operating into a system, since the air flowing in cross-flow with the fuel can cause non uniformities of temperature across the cell. This topic will be discussed in more depth in Section 6.3. Nevertheless, the assumption of uniformity of all the chemical and physical variables throughout the cell is usually justified in the case of single cells (usually assembled into three-cell short-tube arrangements) operated at the experimental scale in laboratory ovens. In this case, the flux of air and fuel fed into the cells largely exceeds the stoichiometric value, so that reactant concentrations can be considered as uniform, as well as the temperatures (which can also be considered equal to the oven temperature, since the oven radiation is by far the dominating heating effect). With the assumption of uniformity of the values of the chemical and physical variables throughout the cell, the cell model is reduced to an electrochemical model, based on the evaluation of the theoretical reversible voltage (VNernst) through the Nernst equation (see also Chapter 3): −G0 RT −G νi VNernst = (6.6) = − ln (pi ) . nF nF nF i
In Equation (6.6) pi is the partial pressure of the species i, while νi is the stoichiometric coefficient of the specie i inside the overall electrochemical reaction, positive for reactants and negative for products. Since in SOFCs the rate of the electrochemical reaction of CO is often negligible compared to that of H2 , only the electrochemical reaction of H2 is taken into consideration in this chapter, and thus the overall electrochemical reaction reads as follows: 1 H2 + O2 → H2 O. 2
(6.7)
Usually, in SOFCs, the cell voltage under open circuit conditions equals the theoretical reversible voltage, while under operation the cell voltage is obtained by subtracting the various internal losses from the theoretical reversible voltage [2] (see
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Anode Electrolyte Cathode Interconnect
Thickness [µm]
Conductivity [S cm−1 ]
50 10 50 10
304 129 6.11 × 10−2 22 × 103
also Chapter 3): Vcell = VNernst − ηOhm − ηact − ηcon .
(6.8)
The way to evaluate ohmic losses (ηOhm ), activation losses (ηact ) and concentration losses (ηcon ) will be described in Section 6.2.1 and 6.2.2.
6.2.1 Ohmic Resistance: FEM Approach Ohmic losses are evaluated by multiplying the cell internal ohmic resistance by the electrical current. Internal ohmic resistances account for a large proportion of the losses occurring in IP-SOFCs and significantly affect the cell performance. However, a detailed evaluation of these is not an elementary task as the complexity of the geometry makes it difficult to apply simple laws. Complex geometries require the solution of a PDE equation, which can be performed analytically or numerically. The first method has already been applied in other works [3, 4], but unfortunately its applicability is rather limited since it does not permit the simulation of several of the fine details of the geometry that might quite significantly influence the overall ohmic resistance. Thus, for a more detailed simulation, the numerical approach is the most suitable. Due to their higher flexibility and accuracy, Finite Elements Methods (FEMs) [5] are often preferred to numerical methods; their basic concept consists first of all in establishing a “weak” variational formulation of the mathematical problem; the second step is to introduce the concept of shape functions that are defined into small sub-regions of the domain (see also Chapter 3). Finally, the variational equations are discretised and form a linear system where the unknowns are the coefficients in the linear combination. Due to difficulty in implementing FEMs, commercial software packages like Comsol Multiphysics are often used, since they allow the application of these methods without implementing an “in-house” finite element code. Specific details of the working and potentialities of Comsol Multiphysics can be found in the programme manuals [6, 7]. Concerning the cell geometry adopted for the simulation, the lengths of the various components are reported in Figure 6.3 (in particular, the active cell length equals 0.5 cm, the active cell being the region where the anode, cathode and electrolyte overlap). Table 6.1 shows the geometrical data and conductivities used for the sim-
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Fig. 6.3 Schematic and dimensions of the IP-SOFC single cell.
ulation of the IP-SOFC repetitive unit at 900◦C; all the layers conductivities are literature values [3]. In the following, some results are reported concerning the simulation of an IP-SOFC operating with a current density of 0.32 Acm−2 flowing through the active part of the cell. It must be noted that the Comsol Multiphysics simulation does not take into account the electrochemical reactions occurring at the cathode-electrolyte and electrolyte-anode interfaces, with corresponding activation losses (which will be treated separately in Section 6.2.2). Simulation results are reported in Figures 6.4 to 6.6. The streamlines in Figure 6.4a indicate the current density path. The vector that represents the current density that enters and exits the cathodic electrode is directed in the x direction (see Figure 6.3). Figure 6.4b shows an enlargement of the left side of the cell where the anode, the cathode and the electrolyte interface, while Figure 6.4c shows an extension of the contact region between the electrolyte and the interconnection. It can be noted that the streamlines plotted in Figure 6.4 are evidently interrupted, but this is merely a graphical problem that does not affect the calculation of the area specific ohmic resistance (ASRo ) [8]. Another remark about Figure 6.4 is that the spatial density of the current lines is uniform and is not linked to the norm of the current density vectors; the current lines are displayed both in the regions where there is a significant current density and in the regions where there is no current density. Indeed, current lines indicate the direction of the current density vectors, but do not give information on the electrical current concentration; this should always be considered when interpreting the results given by Comsol Multiphysics. To gain information about the distribution of the current crossing the electrolyte, the values of the current density norm along a horizontal line in the electrolyte are plotted versus cell length in Figure 6.5. Considering the distributions of the current density norm reported in Figure 6.5, it can be noted that (with this choice of geometry and conductivities) there is no current density flowing in the electrolyte in the regions outside the active part of the cell (i.e. the region where anode, electrolyte and cathode overlap). In the active part, the current density is visibly non-uniform, but nevertheless there are no regions where the current density is zero, i.e. there are no regions which are unutilised.
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Fig. 6.4 Simulation results of distribution of electrical current density through the IP-SOFC: (a) general view; (b) enlargement of the left hand side; (c) enlargement of the right hand side.
Finally, Figure 6.6 shows the colour map of the voltage obtained with Comsol Multiphysics; the potential value is higher where the current enters the structure, on the left hand side, while it is lower where the current exits the cell, on the right hand side. With this choice of geometry and conductivities the area specific resistance is ASRo ≈ 0.35 ohm cm2 at T = 900◦ C. The ASRo is quite sensitive to temperature, mainly due to the change of the electrolyte conductivity with temperature [3]. Once the ASRo is known, the ohmic losses can be evaluated using Ohm’s law:
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Fig. 6.5 Comsol simulation of the current density norm within the electrolyte.
Fig. 6.6 Comsol simulation: colour map of the electrical potential.
ηOhm = ASRo · J.
(6.9)
6.2.2 Concentration and Activation Losses Activation polarizations can be simulated through the typical equations generally used for SOFCs, i.e. the Butler–Volmer equation (see also Chapter 3): nF nF ηact − exp −(1 − β) ηact , (6.10) J = J0 exp β RT RT where β is the apparent charge transfer-coefficient. Equation (6.10) applies to both the anode and the cathode; under open circuit conditions, direct and reverse electrochemical reactions occur simultaneously at each electrode, both equal to the exchange current density J0 , which can be expressed as a function of the Arrhenius law and of the composition of the reacting gases [9]:
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J0,c = γc J0,a = γa
0.25
PO2 pref
PH2 pref
Eact,c , RT Eact,a PH2 O , exp − pref RT exp −
(6.11)
(6.12)
where γ is a phenomenological coefficient, pref is a reference pressure (1 atm) and Eact is the activation energy of the process. Concerning concentration losses, also in that case the treatment is not specific to the IP-SOFC geometry, but on the contrary a general treatment applicable to all types of SOFCs is applied (Chapter 3, Appendix). In proximity to the reacting sites in the electrodes, the concentration of reactants and products of the electrochemical reaction is different from the concentration in the bulk flow of the gases, due to the occurrence of the electrochemical reaction. This concentration gradient is the driving force of a diffusion process. At each electrode, diffusion occurs in two different ways: (i) external diffusion, in the gas channel, in the boundary layer next to the electrode, and (ii) within the porous structure of the electrode, to/from the reaction sites which are usually located at the interface between electrode and electrolyte in the case of the cathode (LSM electrode), or randomly within the electrode in the case of the anode (Ni/YSZ electrode). As a consequence of this effect, the “true” theoretical reversible voltage of the cell must be calculated through an equation analogous to Equation (6.6), but taking into account the reactant/product concentration occurring in proximity to the reaction sites: −G0 RT −G VNernst = (6.13) = − ln (pi∗ )νi , nF nF nF i
where pi∗ denotes the reactant and product partial pressure at the reaction sites. Thus, the difference between Equation (6.6) and Equation (6.13) gives the departure from the theoretical thermodynamic voltage caused by the diffusion effect, i.e. the concentration loss (see also Equation (3.77)): ηcon
∗ 0 ∗ 1/2 pH2 pH2 O pO2 RT RT − ln ln =− , 0 0 ∗ nF nF pH2 pH2 O pO 2
(6.14)
or, expressed in terms of the molar fraction of the gases Xi : ηcon
∗ ∗ 1/2 0 XH2 XH XO2 RT RT O 2 − ln ln =− , 0 X∗ 0 nF nF XH XO H2 O 2 2
(6.15)
where the superscript “0” indicates the feed values. The first term of formulae (6.14) and (6.15) represents the anodic concentration overpotential while the second term the cathodic one. It can be shown [10] that by (i) neglecting the external diffusion; (ii) considering the electrochemical reaction occurring only at the interface between the electrode and the electrolyte; (iii) applying the treatment of diffusion over a
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stagnant film, based on Fick’s first law of diffusion; (iv) relating the diffusive flows of hydrogen, oxygen and water to the electrical current density J through Faraday’s law, and (v) assimilating multicomponent diffusion to binary diffusion where necessary, the molar fractions of oxygen, hydrogen and water at the reaction sites can be calculated from the following equations (see also Equations (A3.12), (A3.13) and (A3.16)): J RT lc ∗ 0 XO , (6.16) = 1 + (X − 1) exp O2 2 4nF DO2 ,eff P ∗ 0 XH = XH − 2 2 ∗ 0 XH = XH + 2O 2O
J RT la , 2nF DH2 ,eff P
(6.17)
J RT la . 2nF DH2 O,eff P
(6.18)
In the equations, lc and la represent the thickness of the cathode and of the anode respectively; P is the total pressure, and Di,eff represents the effective diffusion coefficient of the species i (either at the anode or at the cathode). To evaluate the effective diffusion coefficient, the Bosanquet formula [11] has been used: 1 Di,eff
=
1 Dij,eff
+
1 Dik,eff
(6.19)
.
To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures Dij,eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11]. Finally, combining Equations (6.15–6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)): ⎛ ⎞ J RT la 1− 0 nF DH2 ,eff P XH ⎟ RT ⎜ 2 ln ⎝ (6.20) ηcon,a = − ⎠, J RT la nF 1+ 0 nF DH2 O,eff P XH
ηcon,c
2O
RT 1 1 J RT lc ln . =− − − 1 exp 0 0 2nF 2nF DO2 ,eff P XO XO 2 2
(6.21)
Concentration overpotentials can be very high in low temperature fuel cells, where they often cause the occurrence of limiting currents (such as in PEMFCs, which operate at 80◦ C). However, they are less critical in SOFCs, due to the high operating temperatures (about 1000◦C).
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Fig. 6.7 Comparison between simulation and experimental results for the 3-cell tube fed with H2 /H2 O at the anode and air at the cathode.
6.2.3 V-I Performance of Single Cells and Comparison with Experimental Data In this section, the simulation results obtained from an electrochemical model based on the equations reported in Sections 6.2, 6.2.1 and 6.2.2 are reported, together with experimental data. The experimental data were obtained at RRFCS, and they were used to calibrate and validate the electrochemical part of the model [10]. The single cells used in the experiments were assembled into 3-cell short tubes; all the experimental data have been obtained by keeping the cells in a furnace at uniform and controlled temperature, which is the temperature set in the simulations. An overall value of 6 Nl min−1 has been assumed as the feeding flow rate at the anodic side (97% H2 , 3% H2 O), and 14 Nl min−1 of air at the cathodic side. Figure 6.7 shows the results obtained in terms of characteristic current-voltage curves. The voltage reported in the ordinate is the average among the voltages of the three cells (which, anyway, are very close to each other), while the current density reported in the abscissa, is exactly the same for each of the three cells, since the cells are electrically connected in series. To generate the set of curves in the figure, the less-known parameters (coefficients of the activation polarisation kinetics) were adjusted, in the range indicated by previous literature results, using the 950◦C data; the data used for the electrochemical parameters are reported in Table 6.2. Then only the temperature was changed to generate the curves for the rest of the operational temperatures. Since the electrochemical model is extremely sensitive to temperature, the good agreement between simulation and experimental data obtained in a wide temperature range (from 689◦C to 989◦C) allows us to assume that the equations and the coefficients used to set the model are physically correct. The average error is about 2.5%, which is of the same order of magnitude, or even lower, than the experimental error of the thermocouples used to measure the operating temperature for every set of experimental data. The analysis of Figure 6.7 leads to some considerations: (i) the open circuit potential increases for decreasing operating temperature. Indeed, this is
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Fig. 6.8 Comparison between the different sources of overpotential at different temperatures and at current density 290 mA cm−2 .
a widely known result for SOFCs, which is due to the fact that for SOFCs (unlike other fuel cells, like PEMFCs) the open circuit potential is in good agreement with the thermodynamic reversible potential (Equation (6.6)), which in turn is directly proportional to the G of the electrochemical reaction, which decreases with increasing temperature. (ii) The slope of the characteristic curves increases with decreasing temperature, which is related to the increase of both ohmic and activation effects. (iii) The characteristic curves are not perfectly straight lines and this becomes more pronounced at low temperatures. In order to explain this effect, we must consider that the relative importance of the different sources of overpotentials is different at different operating temperatures. The model results reported in Figure 6.8 show that (i) concentration polarizations can be considered of minor importance at all temperatures in the case study, (ii) at high temperature ohmic overpotentials are the prevailing source of loss (being about twice as large as the sum of anodic and cathodic activation overpotentials), (iii) at low temperatures activation overpotentials are the prevailing contribution. While ohmic losses are a linear function of the electrical current at all temperatures (Equation (6.9)), activation overpotentials tend to be linear at the typical operating conditions of SOFCs (above 900◦ C) while at lower temperatures the activation overpotential is a logarithmic function of the electrical current (see the Butler–Volmer equation, Equation (6.10)). Thus, at high temperature both ohmic and activation polarizations show a linear V-I relationship, and this explains why both the experimental and theoretical V-I curves appear rather linear, while at low temperature, where the activation polarization is a logarithmic function of the current and accounts for the largest part of the overall polarization, both the experimental and theoretical V-I curves show a marked bending. Another remark is that anodic activation losses are higher than the cathodic ones in the case study. Under other operating conditions (with higher content of water at the anodic side) the anodic activation is usually lower than the cathodic one, but in the present case the anodic activation is rather high due to the fact that, as mentioned before, the water content of the feeding fuel is only 3% (even if water is produced by the electrochemical reaction, the fuel flow is in huge excess and thus
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S. Grosso et al. Table 6.2 Parameters of the electrochemical model. βa βc γa γc Eact,a Eact,c pref
0.5 0.3 5.5 × 108 [A m−2 ] 7 × 108 [A m−2 ] 100 × 103 [J mol−1 ] 120 × 103 [J mol−1 ] 1 [atm]
the water content at the anodic side in practice is not influenced by the amount of water produced electrochemically). Due to the small water content of the fuel, the exchange current density tends to be small (see Equation (6.12)), and thus the anodic polarization tends to be anomalously high.
6.3 Single Tube Simulation 6.3.1 One-Dimensional Model: Mass and Energy Balances Several models have been proposed in the literature for the steady-state and transient simulation of various SOFC concepts (for example, see [9, 13–17]). Here, a onedimensional (1-D), steady state model of an IP-SOFC single tube and bundle has been developed [18], under the hypothesis reported below: • One-dimensional model along the x coordinate displayed in Figure 6.1 (here, the origin of the x axis is located on the bottom edge of the tube); • Fuel: pre-reformed methane (i.e. a mixture of CH4 , CO2 , CO, H2 , H2 O); oxidant: air; • Shifting and reforming reactions are assumed to be at equilibrium within the anodic flow; • Same electrical current for all the cells of the tube; • No heat exchange by radiation between gases and solid; • No heat exchange between different tubes of the same bundle and between the tubes and the environment; • Plug-flow form for mass and energy balances. Each tube is a sequence of cells electrically connected in series and this justifies the hypothesis that the current is the same for all the cells. Since the fuel flows in series through the various cells of the tube, fuel compositions vary from one cell to the next. As a consequence, cell voltage also varies from cell to cell, and this can be evaluated through a detailed evaluation of the local electrochemical kinetics of the reactor, as described in Section 6.2 of this work. When the model of the electrochemical kinetics is included into the overall model of the tube, an additional type of resistance is taken into consideration, i.e. the contact resistance (an approach
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used in other literature papers about fuel cell simulation, see for example [9, 15]). This type of resistance in principle takes into account the fact that a full size stack is usually affected by small defects, i.e. pinholes, delaminations, etc., which negatively affect the performance. As a consequence the performance of a full size stack is not simply proportional to the performance of small scale experimental single cells. The contact resistance is assumed to be constant (0.37 ohm cm2 ), i.e. completely independent of the operating conditions of temperature, concentration, etc. In addition to the local electrochemical kinetics, the model of the tube includes the following mass and energy balances (for more details about mass and energy balances in SOFCs see Chapter 3): ∂ni = ri , ∂x i
ni · cp,i ·
∂Tg ∂ni hB + · (Tg − Ts ) + · (Tg − Ts ) = 0, cp,i · ∂x ∂x a i
1 1 hB(Tg − Ts ) − s g s
(6.22) (6.23)
∂ 2 Ts H + Vcell J + λx · = 0, nF ∂x 2 k λk · lk λx = , k lk h=
Nuλ . dh
(6.24)
(6.25) (6.26)
Mass balances (Equation (6.22)) allow the evaluation of the gaseous flow compositions over the fuel cell tube taking into account the consumption and production rates (ri ) due to the electrochemical (Equations (6.1–6.3)) and chemical (Equations (6.4) and (6.5)) reactions. In all the equations above ni represents the flux (in mol cm−2 s−1 ) of the gaseous species i. The energy balances of the gaseous flows (Equation (6.23)) allow the evaluation of the temperature of the reactant gases (Tg ), which varies along the flow direction because of gas-solid heat transfer due to mass transfer (second term in Equation (6.23)), and convective heat exchange between solid walls (at temperature Ts ) and bulk of gases (third term in Equation (6.23)); B is the ratio between the gas-solid effective heat-exchange area and the cell area, a is the thickness of the channel to which Equation (6.23) is applied (anode or cathode side), and h is the heat transfer coefficient calculated from the Nusselt number for laminar flow (Equation (6.26)). The solid energy balance (Equation (6.24)) takes into account, in order, the forced convective gas-solid heat exchange, the heat sources due to the electrochemical and chemical reactions (the H appearing in Equation (6.24) accounts for both the chemical and the electrochemical reactions) and the heat conduction within the solid structure. In the first term of Equation (6.24) s represents the solid thickness, i.e. the thickness of the tube. The solid thermal conductivity λx is evaluated considering the layered structure (Equation (6.25)), where lk is the thickness and λk is the thermal conductivity of each single layer. All the
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S. Grosso et al. Table 6.3 Operating conditions for the simulation of the IP-SOFC tube. Fuel flow rate Air flow rate Molar fuel composition Fuel intel temperature Air intel temperature Current density Pressure
4.3 [Nl min −1 ] 25 [Nl min−1 ] 50% H2 , 18%H2 O, 2% CH4 , 10% CO2 , 20% CO 800 [◦ C] 800 [◦ C] 280 [mA cm−2 ] 1 [atm]
material properties (λk , cp , etc.) have been evaluated on the basis of literature data [3, 11]. The system of equations is integrated numerically through a Fortran program, using a finite difference method (Equation (6.22) and (6.23)) to a relaxation method for Equation (6.24). Thus, the following procedure is applied to solve the equations: (i) set an initial solid-temperature distribution; (ii) calculate the current density and solve the mass balances; (iii) solve the energy balance of the gaseous phase to determine the temperature distribution of the anodic and cathodic gases; (iv) solve the temperature balance of the solid to find a new temperature field. Steps (ii) to (iv) are repeated until the solid temperature distribution is calculated within the given threshold of accuracy (i.e. the variation of solid temperature from one iteration to the next one is smaller then the desired threshold). During each step of the relaxation procedure, the integration of the balance equations is done through a finite difference method. The tube simulation can be run under either isothermal conditions (by excluding Equation (6.24) from the numerical integration and using a pre-defined field of temperatures) or adiabatic conditions (in this case, Equation (6.24) is used to evaluate the temperature profile). In the isothermal case, it is possible to simulate the electrochemical performance (in terms of I-V curves) at different imposed operational temperatures; the option of running the simulation with isothermal solid temperature distribution is often applied, since this is the condition under which some experimental data are obtained at RRFCS (see Figure 6.11). All the other results reported in this section have been obtained under adiabatic conditions (i.e. perfect insulation of the vessel where the tube is contained), since this is a realistic practical operating condition for the tube when included into the plant.
6.3.2 Model Results and Comparison with Experimental Data Figures below report some simulations results obtained for a tube, having 20 cells per side and operated under the conditions given in Table 6.3.
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Fig. 6.9 Distribution of temperature (a) and voltage (b) in a 20-cell tube.
Figure 6.9a shows the temperature distribution within the solid, along the x coordinate of the tube. The abscissa of the plot indicates the cell numbers along the tube (cells are numbered from bottom to top, so that cell no. 1 is the cell located at the bottom of the tube, where fresh air is introduced). Figure 6.9a shows that the temperature of the solid increases from bottom to top due to the heat dissipated by the electrochemical reaction. The voltage distribution along the tube follows quite faithfully the solid temperature distribution, as reported in Figure 6.9b, and this is mainly due to the fact that voltage losses (reported in Figure 6.9b) decrease with increasing temperature. Furthermore, Figure 6.10a reports the behaviour of all the different types of overpotentials along the tube; apart from contact resistance which, as already discussed, assumes a constant value, all the other overpotentials decrease along the tube due to the increase in the temperature of the solid [10]. Finally, in Figure 6.10b, the distribution of the concentrations within the anodic gases is reported as a function of the x coordinate. Indeed, at the bottom of the tube, H2 concentration increases and H2 O decreases due to the reforming reaction which is faster than the electrochemical one. After only a couple of cells, CH4 is almost totally consumed so that from this point the only effects are due to H2 consumption and H2 O production due to the electrochemical reaction (since the equilibrium of the shifting reaction
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Fig. 6.10 Distribution of voltage losses (a) and fuel compositions (b) in a 20-cell tube.
Fig. 6.11 Comparison between modelling results and experimental data (operating pressure: 1 atm; operating temperature 950◦ C).
is included in the simulation as well, it has some effect, in particular on the CO concentration). The modelling results have also been compared with experimental data obtained from a tube operated into a muffle furnace kept at 950◦C, modified to enable fuel and air to be fed to the test tube, and to allow electrical measurements to be made.
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The experimental apparatus has been designed, built and operated by RRFCS; due to the peculiarities of the experimental tubes, the model presented in Section 6.3.1 needed some adaptation to be suitable to properly simulate the configuration of the fuel channels internal to the support [18]. Characteristic V-I curves were measured at atmospheric pressure with different fuel flows of H2 /H2 O/CO/CO2 mixtures and with air used as the oxidant. Under these operating conditions, the tube is considered to be isothermal in all the simulations, since, due to the effective radiation between the tube and the muffle, the tube is expected to have a temperature very close to that of the muffle itself. A comparison between experimental and simulation results is reported in Figure 6.11, which compares three V-I curves obtained with a fuel flow of 10 Nl min−1 and different fuel compositions (Case 1: 19% CH4 , 54% H2 , 27% CO; Case 2: 1.4% CH4 , 65.6% H2 , 33% CO; Case 3: 66.7% H2 , 18.7% CO; 14.6% CO2 ; compositions reported here are dry compositions; actually the fuel is fully humidified at ambient temperature before being fed into the experimental tube). Figure 6.11 reports a very good agreement between simulated and experimental results, and in particular a peculiar trend displayed by the experimental data is very well followed by the simulation, i.e. the fact that the lower the hydrogen content in the fuel, the higher the performance. This behaviour may be explained if one considers the water content of the fuel mixtures rather than the hydrogen content. Mixture 2 contains mainly hydrogen and carbon monoxide, with small percentages of methane and water. This mixture practically will not react further on either the anodes or the support. Mixture 3 however contains some 14.6% carbon dioxide. At such high temperatures this mixture is fully expected to undergo the water gas shift reaction on the anodes and the porous ceramic support. Since this reaction is exothermal, the equilibrium will be shifted toward the production of carbon monoxide and water. Thermodynamic calculations indicate that at 950◦ C mixture 3 will shift to produce an equilibrium mixture containing 55.5% H2 , 12.2% H2 O, 5.4% CO2 , 26.8% CO and 0.1% CH4 . Compared with mixture 2 it is now obvious why mixture 3 gives poorer performance despite having a slightly higher initial hydrogen content. If mixture 1 is now considered, it can be seen that it contains 19% methane but only approximately 3% water. The methane and water are expected to react at the fuel cell anodes via the steam reforming reaction. The large excess of methane will result in almost complete depletion of the water content of the fuel gas. It is this depletion of the water content and not the hydrogen concentration that leads to an increase in the Nernst voltage and hence to improved performance of the fuel cells. Thus, a correct simulation of the reforming and shifting reactions is essential in order to be able to predict the experimental performance under these operating conditions; the good agreement between the experimental and simulation results confirms that equilibrium is a good assumption for the reforming and shifting reactions.
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6.4 Bundle Simulation For the purpose of bundle simulation, a two-dimensional (2-D) model including radiation is developed [19]. In the 2-D coordinate system, the x direction remains the fuel-flow direction, and the y direction is the air-flow direction. The 2-D model is derived from the 1-D model, whose equations are reported in Section 6.3.1. As well as the 1-D model, the 2-D model embeds the electrochemical simulation of the single cells as described in Section 6.2 of the present chapter. The equations of mass and energy balance of the gaseous flows are re-written in two dimensions, and are integrated through a method very similar to that used for the 1-D model. For the purpose of integrating the 2-D equations, the single tube has been divided into several elements along the fuel flow direction (x direction) and air flow direction (y direction), so that all the quantities calculated by the model, i.e. voltage, current density, voltage losses, gas and solid temperatures are evaluated in a 2-D matrix of points in the tube plane. On the contrary, any variations of the above-mentioned quantities, and in particular of the solid temperature, along the tube thickness are neglected. As a consequence of this, the two faces of each tube are characterised by having identical temperature distributions. This assumption is justified by the fact that the tubes are extremely thin (about 3.5 mm), despite the low thermal conductivities characterizing ceramic materials. An additional change made to the equations is that they include heat transfer by radiation between one tube and the adjacent ones in the bundle, following a treatment which is discussed in detail in the subsequent Section 6.4.1, the purpose of which is to show the approach followed to simulate the radiative heat exchange between pairs of facing tubes within the stack. The thermal fluxes due to radiation are then included in the energy balance of the solid structure and these affect the steady state temperature distributions over the tubes. Temperature distributions, in turn, have a strong influence both on the chemical and electrochemical phenomena and on the mechanical stresses in the ceramic materials.
6.4.1 Radiation Effects The first assumption made is that each element of a tube exchanges heat only with the two elements located in front of it respectively on the previous and on the subsequent tubes (Figure 6.12). The equation used to calculate the radiative heat exchange (qi,j ) between two facing elements is the following one: qij =
σ (Ti4 − Tj4 ) 1−εi εi Ai
+
1 Ai Fij
+
1−εj εj Aj
,
(6.27)
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Fig. 6.12 Simulation of radiative heat exchange between tubes.
Fig. 6.13 Procedure for the calculation of the view factors.
where εi and εj are the emissivities of the elements i and j , Ai and Aj are the areas of the elements i and j , σ is the Stefan–Boltzman constant, and Fij is the view factor. The denominator in Equation (6.27) consists of a sum of three thermal resistances, the first and the last of which are surface resistances involving the emissivities, while the second one is a geometric resistance depending on the view factor between the two elements. A value of 0.3 has been chosen for the emissivity of the cell and a value of 0.85 for the interconnection and for the inactive edges [11, 20]. In calculating the view factor between two facing rectangular elements, the approach proposed by [11] has been followed, so that it can be considered as the geometric mean between the two following view factors: • View factor between two squares with the side equal to the shortest side of the rectangle. • View factor between two squares with the side equal to the longest side of the rectangle. A schematic explanation of this procedure is given in Figure 6.13. In the light of this, it is necessary to calculate the view factor between two facing squares. Values for such view factors are tabulated in the literature [20] as a function of L/d, where L is the side of the square and d is the distance between the two sur-
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Fig. 6.14 View factors for facing squares as a function of L/d. Table 6.4 Operating conditions for the simulations of the IP-SOFC stack. Fuel flow rate Air flow rate Molar fuel composition Fuel inlet temperature Air inlet temperature Current density Pressure
4.3 [Nl min−1 ] 75 [Nl min−1 ] 70% H2 , 18% H2 O, 2% CH4 , 10 % CO2 800 [◦ C] 900 [◦ C] 280 [mA cm−2 ] 1 [atm]
faces. For the sake of convenience it has been decided to approximate the tabulated view factors with a continuous function by means of a least squares method. The function used is the following one: F
L L/d − 0.2335 1.6614 = . d L/d + 0.9535
(6.28)
In Figure 6.14 we show a good match between a plot of Equation (6.28) (solid line) and the tabulated values (crosses). Equations (6.27) and (6.28) have then been included in our Fortran code in order to investigate the influence of radiation on the temperature distributions.
6.5 Results In this section, some results obtained from the simulation of one single bundle are reported. The bundle is composed by 10 tubes, each carrying 15 cells per side, operating under the operating conditions reported in Table 6.4. Figure 6.15 shows a comparison of the temperature distributions over the first, the second and the tenth tube in the cases with and without radiation. The main effect that is possible to observe in the maps in Figure 6.15 is the stronger similarity among
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Fig. 6.15 Temperature distributions over the first, second and tenth tube (respectively first, second and third column) in the following cases: no radiation taken into account (1st row), radiation taken into account (2nd row). Table 6.5 Effects of radiation on efficiency, mean voltage and mean temperature.
Efficiency Mean voltage Mean temperature in the bundle
No radiation
With radiation
54.22% 0.677 [V] 984 [◦ C]
54.43% 0.684 [V] 983 [◦ C]
temperature distributions in the different tubes of the same bundle, when radiation is taken into account. Nevertheless, this rearrangement of temperature over the tubes does not significantly affect the mean temperature of the bundle, as displayed in the last column of Table 6.5. Table 6.5 also shows the effect of radiation on some meaningful quantities such as the efficiency of the bundle and the mean voltage over a single cell. Even if some slight variations may occur, we can say, from a practical point of view, that the effect of radiation is negligible if we are considering the overall electrochemical performance.
6.6 Conclusive Remarks Modelling of the IP-SOFC tube involves different levels of detail, such as the electrode, the single cell, the stack, and the system level. At the electrode level, the main
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phenomena taken into account in the models are the electrochemical reaction, the charge conduction and the mass transfer. At the tube and bundle level, the models include the local mass and energy balances. The scope of IP-SOFC modelling is multi-fold: it leads to a better understanding of the physical-chemical phenomena occurring in the electrodes, in the fuel cell and in the stack; it makes it possible to predict the local behaviour of the cell; it makes it possible to identify dangerous effects (for example, hot spots) which may lead to damage; it can serve as the basis for planning experimental campaigns; it provides a useful tool for optimisation of fuel cell systems, in terms of both operating conditions and design parameters. The model of the IP-SOFC bundle has been the basis of the model of the IPSOFC stack (Figure 6.2) [18]. Further, in the framework of a collaborative project between RRFCS and the Thermochemical Power Group of the University of Genoa, the model of the IP-SOFC bundle has been incorporated into a tool for the simulation of pressurised IP-SOFC/ gas turbine hybrid systems designed for operation in the range 250 kW-20 MW [21].
Acknowledgements The authors thank Rolls-Royce Fuel Cell Systems for financial support during this work. Many thanks also to Gerry Agnew, Robert Collins, Colin Berns and Olivier Tarnowski of Rolls-Royce Fuel Cell Systems for their valuable contribution to this work. This work has been partially developed within the framework of the European Contracts FP-V ENK5-CT-2000-00302 “IM-SOFC-GT” and NNE52001-791 “PIP-SOFC”, and the authors would like to thank all the partners of these projects. Also, the support received by the British Council-CRUI Partnership Programme for Young Researchers (2005) is gratefully acknowledged.
References 1. Gardner F.J., Day M.J., Brandon N.P., Pashley M.N., Cassidy M. (2000) SOFC technology development at Rolls-Royce. Journal of Power Sources 86, 122–129. 2. Garche J. (Ed.) (2008) Encyclopedia of Electrochemical Power Sources. Elsevier Science, Oxford, UK. 3. Bossel U.G. (1992) Facts & figures. An international energy agency SOFC task report. Berne. 4. Costamagna P., Arato E. (1997) Modelling of the ohmic resistance of tubular solid oxide fuel cells. In: Solid Oxide Fuel Cells V, Stimming U., Singhal S.C., Tagawa H., Lehnert W. (Eds.). The Electrochemical Society, Pennington, NJ, USA, pp. 1339–1348. 5. Pepper D.W., Heinrich J.C. (2006) The Finite Element Method. Taylor & Francis, New York. 6. Comsol (2005) Comsol Multiphysics User’s Guide. Comsol AB, Stockholm. 7. Comsol (2006) Comsol New Features Guide. Comsol AB, Stockholm. 8. Repetto L., Agnew G., Del Borghi A., Di Benedetto F., Costamagna P. (2007) Detailed simulation of the ohmic resistance of SOFCs. Journal of Fuel Cell Science and Technology, in press.
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9. Costamagna P., Honegger K. (1998) Modeling of a solid oxide heat exchanger integrated stack and simulation at high fuel utilization. Journal of the Electrochemical Society 145, 3995–4007. 10. Costamagna P., Selimovic A., Del Borghi M., Agnew G. (2004) Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC). Chemical Engineering Journal 102, 61–69. 11. Perry R., Green D.W. (1998) Chemical Engineers’ Handbook. McGraw-Hill, New York. 12. Bird R.B., Stewart W.E., Lightfoot E.N. (1960) Transport Phenomena. John Wiley & Sons, New York. 13. Debenedetti P.G., Vayenas C.G. (1983) Steady state analysis of high temperature fuel cells. Chemical Engineering Science 38, 1817–1829. 14. Vayenas C.G., Debenedetti P.G., Yentekakis Y., Hegedus L.L. (1985) Cross-flow solid-state electrochemical reactors: A steady-state analysis. Industrial & Engineering Chemistry Fundamentals 24, 316–324. 15. Achenbach E. (1994) 3-Dimensional and time-dependent simulation of a planar solid oxide fuel cell stack. Journal of Power Sources 49, 333–348. 16. Bessette N.F., Wepfer W.J., Winnick J. (1995) A mathematical-model of a solid oxide fuelcell. Journal of the Electrochemical Society 142, 3792–3800. 17. Larrain D., Van Herle J., Favrat D. (2006) Simulation of SOFC stack and repeat elements including interconnect degradation and anode reoxidation risk. Journal of Power Sources 161, 392–403. 18. Costamagna P., Cerutti F., Di Felice R., Collins R. (2004) Utilization of gases from biomass gasification in a reforming reactor coupled to an integrated planar solid oxide fuel cell: Simulation analysis. Thermal Science 8, 127–142. 19. Grosso S., Collins R., Costamagna P. (in preparation) Steady-state modelling of the integratedplanar solid oxide fuel cell. 20. Faggiani D. (1946) Trasmissione del calore. Tamburini, Milano. 21. Magistri L., Traverso A., Cerutti F., Bozzolo M., Costamagna P., Massardo A.F. (2005) Modelling of pressurised hybrid systems based on integrated planar solid oxide fuel cell (IP-SOFC) technology. Fuel Cells – From Fundamentals to Systems 5, 80–96.
Chapter 7
Tubular Cells and Stacks Vittorio Verda and Chiara Ciano Department of Energy Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
7.1 Introduction Solid Oxide Fuel Cells are manufactured in different geometries. Industrial research is focused on planar and tubular cells. Planar SOFCs reach larger power densities than tubular SOFCs, but present some criticalities due to the seals and their behaviour under thermal gradients. To solve this problem, some manufacturers have proposed tubular geometries. The tubular SOFC developed by Siemens Westinghouse is shown in Figure 7.1. This design has no seals and its shape is that of a pipe with a closed end whose wall is constituted by the Positive electrode, Electrolyte, Negative electrode (PEN) layers. Inside the cell there is the air injection channel (also indicated as injection tube). Air enters the injection channel and reaches the closed end of the cell where it inverts its direction starting to flow upwards in the annular section between the cathode layer and the outer side of the injection tube. The fuel flows on the outer side of the cell. Tubular SOFCs vary in length which generally ranges from 30 to 150 cm. There are also micro-tubular cells with a length of approximately 10 cm (Sammes et al., 2005). Data reported in the paper by Li and Chyu (2003) show that the thickness of the PEN area does not depend on the cell length. On the contrary, the diameter varies linearly with the length. An advantage of this practice is a volume reduction in the case of small cells. Moreover, it is possible to keep about the same operating conditions, independently of the cell length. Figure 7.2 shows the Reynolds number inside the cell with and without the injection tube. The active area is: A = π · D · L ∝ L2 .
(7.1)
If the current density and the gas utilization are assumed as constant, the mass flow rate per unit active area is constant. Without injection tube, the gas velocity does not depend on the cell length: R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 207–238. © Springer Science+Business Media B.V. 2008
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Fig. 7.1 Bundle of tubular SOFCs.
Fig. 7.2 Reynolds number in the cell: comparison between two cases.
Fig. 7.3 Current flow path and voltage distribution at 0.656 V across a tubular SOFC section (left) and a detail (right).
u=
m ρ ·π ·
D2 4
∝
A ρ·π ·
D2 4
∝
π · D2 ρ ·π ·
D2 4
= const.
(7.2)
When the injection tube is used, it is possible to choose its diameter so that the Reynolds number is about constant in the air channels. A similar consideration can
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Fig. 7.4 HPD5 bundle.
Fig. 7.5 Current flow path and voltage distribution at 0.656 V across an HPD5 SOFC section (up) and a detail (down).
be made for the outer gas channels. A drawback of this practice is that the manufacturing process becomes more complex. Figure 7.3 shows the current flow path and the potential distribution in a cross section of a tubular cell. The American Department of Energy, through the Solid State Energy Conversion Alliance (SECA) programme, has funded the research necessary to reduce the production cost of tubular cells. Siemens Westinghouse, as a milestone in the SECA programme, developed the High Power Density SOFCs (HPD, Figure 7.4), an evolution of tubular cells. While keeping the important feature of being sealless, HPD cells provide larger current density. Their shape is that of a flattened tube; inside there are several vanes separated by ribs of cathodic material. The ribs allow for current flow paths shorter than in the tubular SOFC. This lowers the ohmic losses. The vanes serve as air channels; the fuel flows on the outer side of the cell. Among HPD cells, the one chosen for the first prototype phase is the HPD5 (five air channels) cell. As of September 2006, a fuel cell system composed of six bundles
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Fig. 7.6 Delta9 bundle.
Fig. 7.7 Current flow path and voltage distribution at 0.656 V across a Delta9 SOFC section (up) and a detail (down).
of six cells each was under testing at Siemens laboratories. The gross power output of such system is 3.4 kW and the average operating temperature is slightly below 1000◦C (Vora, 2006). Tested HPD5 cells have an active area of 900 cm2 and 75 cm of active length. Even compared to the longest tubular SOFC (active area of 850 cm2 ), a single HPD5 cell presents a higher power density in the same conditions of temperature and cell voltage. The potential distribution and the current path flow of an HPD5 cross section are shown in Figure 7.5. The evolution of HPD is the Delta shaped SOFC. This configuration takes its name from the section of the air channels, which is triangular, thus recalling the Greek letter (Delta). Each cell is composed of several Deltas, as shown in Figure 7.6. Compared to the previously described geometries, this one is characterized by larger current density and a larger power per volume ratio. Cells may be bundled very tightly taking advantage of their shape. The air flows in the internal vanes whereas the fuel flows in the spaces left after bundling together two cells.
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Fig. 7.8 Polarization curves: comparison between Tubular, HPD ad Delta9 SOFCs.
This design is still under preliminary modeling and testing. Research is focused on the further improvement of the power density, avoiding the use of high temperature seals. The first geometry proposed by Siemens, shown in Figure 7.6, is the Delta9 cell: a width of 10 cm and an active surface of 1200 cm2 . Figure 7.7 shows current path flow and potential field across a section of such cell. Currently, other variations of the Delta design are under study and development. Figure 7.8 shows the diagram of voltage versus current. As can be seen, the new designs, HPD5 and Delta9, both present an increase in current density with respect to the traditional tubular configuration. This allows lower production costs. As for the power density of SOFC systems, a comparison between the different kinds of cells has been made (Vora, 2006). The measured specific power (at 1000 K, fuel utilization ratio of 80% and 0.65 V) of a tubular SOFC bundle (24 cells) is around 0.13 in W/cm3 , whereas that of an HPD5 bundle (six cells) is of 0.17 in W/cm3 . The Delta9 configuration, with bundles of nine cells, reaches over 0.4 W/cm3 . A comparison is shown in Figure 7.9. As can be seen, although the power available from a single HPD5 cell is much higher than that from a tubular cell, on a power density basis the two designs do not differ very much. The reasons for this stand in the different numerousness of the bundles (24 for the tubular SOFC and six for the HPD5), the very similar active areas and the greater volume of the HPD5 bundle. A quick comparison of the different bundles and their volumes can be seen in Figure 7.10. Other manufacturers have proposed different solutions. The cell manufactured by Acumentrics is similar to the Siemens Westinghouse tubular SOFC; the differences mainly reside in the fuel flowing inside the cell without inlet pipe and in the tube having open ends. Reforming is internal to the tube. The Acumentrics cell operates in the same temperature range than the previously mentioned SOFCs, but
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Fig. 7.9 Power density per bundle of different kinds of SOFCs.
Fig. 7.10 Comparison of different bundles.
its start up time is limited to 45 minutes because it is dimensionally smaller. For further information on this cell, the reader may refer to the Acumentrics web site (www.acumentric.com). Mitsubishi Heavy Industries (MHI) produces a segmented-in-series tubular SOFC. In this design, the active area of the cell is segmented along the support porous tube, generally aluminate. The cells are connected in electrical series. The interconnection acts as a sealing and electrical contact between the anode of a cell and the cathode of the subsequent one. Inside the tube, the fuel flows from one cell to the next one. The oxidant flows outside (Singhal and Kendall, 2003). The segmentation in series allows a higher voltage per tube than the tubular design. The current density is therefore lower. MHI constructed both atmospheric and pressurized 10 kW stacks, achieving power densities of around 140 mW/cm2 (Anonymous, 2004). Another design for this kind of cell is the Integrated Planar SOFC (IP-SOFC) developed by Rolls Royce. It is a segmented-in-series cell with flattened tubes. For details on the modeling of such cell, the reader may refer to Gardner et al. (2000), Costamagna et al (2004), Haberman and Young (2005), Bujalski et al. (2006), Young (2007) or to Chapter 6.
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Table 7.1 Dimensions of the layers in different kinds of cells (Campanari and Iora, 2004; Bharadwaj et al., 2005). Layer
Tubular SOFC
HPD SOFC
Cathode Electrolyte Anode Length
2.2 [mm] 0.04 [mm] 0.1 [mm] 300; 500; 1500 [mm]
1.6 [mm] 0.04 [mm] 0.12 [mm] 750 [mm]
Toto industries, in Japan, manufacture micro-tubular SOFCs that are 15 cm long. The operating temperature is around 500◦C. Start up time at room temperature is possible in 5 minutes. These systems are well suited to provide power in the small range size. The electrical efficiency is around 55% (Saito et al., 2005).
7.2 Model Approaches Tubular cells are difficult to model mainly because of the large ratio between their length and the thickness of the electrodes and electrolyte. Some typical dimensions are reported in Table 7.1. A suitable mesh for modeling fluid dynamics, heat transfer, mass transfer and current conservation using a 3D geometry would require very large computational resources. Some simplifications are usually necessary. Most of the models available in the literature are axial symmetric. A second simplification refers to the discretization adopted for the electrodes and electrolyte. Some of the models consider the cathode, electrolyte and anode as a whole and adopt an axial discretization. Electronic/ionic resistivity is computed as the average value of the single resistivites, calculated at the local temperature (Campanari and Iora, 2004). Using this approach means to simplify the solution of mass transfer in the porous media and the conservation of current. Authors have shown that about 200 elements are sufficient to describe the behaviour of a cell 1.5 m long using a finite volume approach (Campanari and Iora, 2004). In other models, the porous media is meshed to solve the equations for mass transfer, while current conservation is modelled by means of a resistive network. In this case authors have used about 40000 elements to build the model using a finite volume approach (Li and Chyu, 2003). Ciano et al. (2006) have used a finite element approach to model a tubular cell 0.3 m long. The equations are available in Ciano et al. (2006). Table 7.2 shows the partial differential equations and the mesh characteristics. This model is computationally demanding and the equations have been solved by adopting an iterative procedure. Initial guess values for temperature and current density are assumed (current density is calculated by means of a lumped model, as the function of the average temperature and the cell voltage). Momentum equation and continuity equation are
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Partial Differential Equations
Air channels
Navier–Stokes (2 eq.), Continuity, Energy, Conservation of species (1 eq.) Energy Brinkmann (2 eq.), Continuity, Energy, Conservation of species (1 eq.), Current conservation Energy, Current conservation Brinkmann (2 eq.), Continuity, Energy, Conservation of species (4 eq.), Current conservation Navier–Stokes (2 eq.), Continuity, Energy, Conservation of species (4 eq.)
Tube Cathode Electrolyte Anode Fuel channel
Number of elements 40000 5000 305000 260000 370000 27000
solved first. Using these results together with the initial guess values, conservation of species is solved. Finally, the current conservation is solved. The initial guess values are then replaced with the ones just found and the iterative procedure runs until the errors are small enough. To model a complete stack, which may be constituted of more than 1000 cells, it is necessary to adopt a different approach. In this chapter a finite difference model is presented. Only energy equation and current conservation are solved. This allows one to examine possible improvements in the stack configuration design that can be achieved by taking advantage of the relation between temperature and electronic/ionic resistivity, heat transfer and chemical reactions, etc. In addition, this model can be used for analyzing the effects of possible anomalies and performance degradation.
7.3 Single Cell Modeling In this section the characteristic phenomena occurring in the tubular cells are shown and the differences with respect to the planar cells are stressed. The model equations are recalled from Chapter 3.
7.3.1 Axial Symmetric CFD Model Axial symmetric representation is typically considered for single tubular fuel cell modeling because it allows one to design, mesh and compute only a portion of the cell taking advantage of the symmetry condition. The partial differential equations are written in cylindrical coordinates and solved in two dimensions (radial and axial). This assumption is possible when there are no significant circumferential variations in the boundary conditions. Such a hypothesis is reasonable when the
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Table 7.3 Boundary conditions used in the fluid-dynamic model of the air channel of Figure 7.11. Boundary
Condition
Value
Air inlet Air outlet Cell axis Air supply pipe walls Cathodic walls
inlet velocity profile outlet pressure axial symmetry no slip velocity
9.81 m/s 1 bar v=0 v = velocity in the porous electrode
cell is not installed in a stack or when the operating conditions of the surrounding cells are similar (i.e. a cell in the middle of the stack). When the cell is close to the external boundary of the stack a 3D model should generally be used.
7.3.1.1 Effects of the Geometry on Fluid-Dynamics Fluid dynamics in a tubular fuel cell has significant effects on different other phenomena such as the electrochemical reactions, which need species to be transported to the reaction site, and mass and heat transfer. In the porous structures, an equation that accounts for the fluid flowing in the pores – such as Brinkman’s or Darcy’s equations (Equation (3.4)) – must be used; usually the velocities are low. Fluid-dynamic is modelled through Navier–Stokes equation in the channels (Equation (3.3)). Generally, the flow field is assumed as a laminar flow, due to the relatively low velocities that the air reaches inside the cell. Nevertheless, Campanari and Iora (2004) performed a fluid dynamic calculation of the flow in the air injection tube and in the annular section of the cell; the results indicated a transition from laminar to turbulent flow: the values of the Reynolds number found were in some cases above 1000, whereas the transition between laminar and turbulent flow is stated to be in the range between Re = 750 and Re = 2700. The regime of the flow affects the heat exchange between the gas and the solid material and the diffusion of chemical species. Li and Suzuki (2004) too performed similar calculations and found values of the Reynolds number that were consistent with a regime transition in the air injection tube, but not for the annular section (Re = 385 with a velocity lower than 7.82 m/s). Li and Chyu (2003) state that the assumption of laminar flow is to be rejected. Other researchers, such as Haynes and Wepfer (2001) previously and Stiller et al. (2005) later, assume laminar flow. Concerning the flow in the porous media, the use of the Brinkman equation allows to set the continuity in the velocity profiles in the channel and in the porous medium (Nield and Bejan, 1999). Figure 7.11 shows the air velocity field at the closed end of a 30 cm long tubular cell is presented. The boundary conditions set to obtain this result are presented in Table 7.3. The same field for a HPD cell may be found in the paper by Lu et al. (2005).
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Fig. 7.11 Velocity field at the closed end of a tubular SOFC.
7.3.1.2 Effects of Geometry on the Diffusion of Species The diffusion of species is modelled through the partial differential equation expressing the conservation of species (Equation (3.25)). The diffusion coefficient may be computed in different ways which are shown in Chapter 3. The difficulty with diffusion coefficients is that available experimental values are measured at temperatures lower than those of an operating cell. Therefore, semi-empirical, theoretical methods have been used to extend available data to practical operating conditions. Some authors (Fischer and Seume, 2006) use the Chapman–Enskog formulation; some others (such as Campanari and Iora, 2004; Verda et al., 2005; Sánchez et al., 2006) use Fuller’s correlation to calculate the ordinary diffusion of species in mixtures, Di,j , as proposed by Fuller et al. (1966). All of these formulations, however, relate Di,j with temperature, pressure, molar weights of the gases and some other parameters variable from method to method. In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself; therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). The consumption of the species is accounted for by means of Faraday’s law, which relates the local current density to the consumption of the reactants and the
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Fig. 7.12 Mass fraction of the fuel constituents.
generation of the products in terms of molar flows (Equation (3.46)). What is differently modelled by the various authors it is where the reaction sites are and which species generates the electric current. In Figure 7.12 the mass fractions of the fuel constituents along a 1.5 m cell are shown in a typical operating condition. The diffusion coefficients of the species in the mixtures are computed using the method proposed by Fullers et al. (1966). For the mixtures in the porous electrodes, Equation (3.58b) has been used in which porosity was set to 0.5 and tortuosity to 3 (Campanari and Iora, 2004). The fuel inlet is constituted by about 0.7% which is reformed in the first part of the cell. Hydrogen mass fraction is almost constant on the first 20 cm, in fact hydrogen consumed by the electrochemical reaction is almost equal to the hydrogen generated by chemical reactions. Water mass fraction increases. CO2 increases after an initial slight decrease, while an opposite behaviour is registered for CO.
7.3.1.3 Effects of Geometry on Temperature Heat generated inside the cell, due to Joule effect and the electrochemical reactions, is released into the inlet air and fuel. In particular, the inlet air, while flowing in the injection pipe, receives heat by the up flowing air in the annular section. Considering a single cell, the temperature profile of the gases and solid parts is presented in the next section. The differences in the various models reside in the choices regarding the mechanisms through which heat is transferred and generated. Since heat transfer has great influence on the overall performance of the cell, as already mentioned, the correct modeling of the temperature field is crucial.
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Heat exchange in a SOFC occurs through all the three possible mechanisms: conduction, convection and radiation. Convection is never neglected as it is considered to be the principal mechanism in heat transfer inside a SOFC, but the importance of radiation has been underestimated for a long time. VanderSteen and Pharoah (2004) describe the effect that radiation has on the temperature gradient and distribution. In their paper they show that not considering the radiative effects between solid-solid surfaces and solid-gas interfaces leads to predicted values of the temperature field much higher than considering the radiative effects does. Li et al. (2004), as well as Fischer and Seume (2006), accounted for radiation, but only between the airinducing tube and the cathode porous material, neglecting the radiative exchange that takes place on the fuel channel between the anodic wall and the gases in the fuel flow. Sánchez et al. (2006) consider the analysis of radiative heat transfer as mandatory for an accurate fuel cell model; the reason for their statement is the high view factor between injector tube and cathode and their high temperatures. Furthermore, they model not only the radiative heat exchange between air feed tube and inner cathode walls, but also the one between anode walls and the fuel. This step is justified since carbon dioxide and water steam are present in the fuel mixture (up to 75% in volume), thus the gas mixture cannot be regarded as a transparent body. Their results show that anodic gas radiation is to be considered, particularly when current density is high. In this case there are large mass flows able to act as heat sinks with high heat capacity. When this happens, temperature on the anode side is lower than in the case of low current densities. As the temperature decreases, the electric resistivities of the anode and the electrolyte rise and this negatively affects the performance of the cell. Albeit the authors present a 2D axisymmetric model of a tubular SOFC, they recognize that a fully 3D study must be done if the influence of peripheral cells on the global performance of the stack is to be evaluated (Sánchez et al., 2006). However, on whether considering radiative heat exchange or not, the conclusions of many authors greatly differ: some authors (Campanari, 2001; Li and Chyu, 2003; Campanari and Iora, 2004) completely neglect heat radiation, others state their interest in modeling it (Li and Suzuki, 2004). Ota et al. (2003), quantifying and comparing the contributions of the different heat transfer mechanisms, positively conclude that the rate of heat radiation from the cell to the air feed tube is high. The paper by Haynes and Wepfer (2001) highlights the importance of heat exchange in SOFCs. To achieve an accurate temperature field, they propose a model that takes into account conduction, convection and radiation. The authors show how important the radiative heat exchange is. In particular, on heat transfer from the inner surface of the cell, radiation plays a much more important role than convection, which contributes for about 10%. Currently, the tendency in SOFC modeling seems to be to take into account all three heat transfer mechanisms. Figure 7.13 shows the different contributions on the anodic side. The heat fluxes per unit area for a tubular cell 1.5 m long are shown. The total contribution due to convection is about 35%. The convective heat flux decreases along the cell mainly
7 Tubular Cells and Stacks
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Fig. 7.13 Heat transfer on the anodic side of a tubular cell.
because of the increasing fuel temperature. The radiation heat fluxes have been split in two parts. One is the heat transfer between the surface and the surrounding surfaces; the other one is the heat transfer between the surface and the gas. In the first half of the cell, the surface-surface contribution to radiation decreases since the temperature difference between adjacent surfaces tends to reduce. In the second half of the cell the difference between adjacent surfaces tends to increase and so does the radiative heat flux. The gas-surface exchange term decreases in the first part of the cell because of the increase in the gas temperature; then, this trend is partially contrasted by the increase in the gas emissivity. The latter is related to the increase in the partial pressure of H2 O and CO2 as shown in Figure 7.12. On the same track and particularly significant is the previously mentioned work by Fischer and Seume (2006) that aims at locating and quantifying heat sources and sinks in the cell. The different phenomena are modelled in order to give the most precise information about the temperature field. The mechanisms considered as heat sources and sinks are the ohmic, activation and concentration losses; the endothermic methane reforming reaction and the exothermic water gas shift reaction; the electrochemical reactions. On this last topic especially the model by Fischer and Seume differs from those of the other authors since it does not consider the overall reaction taking place at the electrolyte-anode interface; instead it regards the two semi-reactions at both electrode-electrolyte interfaces. To estimate the single-electrode entropy changes, the Seebeck coefficient is used. In their study, the authors report that Seebeck coefficients of YSZ samples have been experimentally measured. A constant mean value of −0.463 mV/K could be found using a YSZ electrolyte. As for the electrodes, their influence on the Seebeck coefficient is considered to be negligible since the transported entropy in the SOFC electrode materials is small compared to that in the oxide ion conducting electrolyte. The half
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reactions turn out to be the greatest heat source (on the cathode side) and sink (on the anode side) of the cell. Fischer and Seume also performed a calculation of the temperature profile assigning all heat generated by the electrochemical oxidation of hydrogen to the anodic triple phase boundary, which is a simplification widely used in the literature. The comparison between the two profiles showed that the difference in absolute temperature is irrelevant, but relevant is the difference between the two resulting temperature gradients. In particular, the one obtained by considering two differently located reactions is higher than the one obtained with the simplified model. Since temperature gradients, more than temperature itself, are responsible for the thermal stress to which the cell is subjected and which could compromise its mechanical resistance, the exact quantification of their magnitude could be essential to the performance and maintenance of the cell. Lockett et al. (2004) present one of the first models on micro-tubular SOFCs that are 10 cm long. Although the cell is very short, there is a temperature gradient along it; this means that in some areas of the cell the temperature is not optimal. The advantage of these cells versus the longer ones is that the start up time of the micro-tubular cells is very short (minutes).
7.3.2 Three-Dimensional Model The use of 3D models for tubular cells is particularly interesting, especially for studying the new geometries. In the literature the applications are very few and the reason is the large computational load. Here some of the results of current research conducted by the research group at Politecnico di Torino are shown. A first application is the 3D model of the tubular cell by Siemens. The calculated temperature field is shown in Figure 7.14. This result has been obtained by solving fluid dynamics and heat transfer only. A constant current density of 1700 A·m−2 has been considered. A fuel mass flow rate of 3.11 · 10−5 kg·s−1 enters the cell at 520◦C. The fuel composition (mass fractions) is: 0.7% CH4 , 35.9% CO2 , 29.9% CO, 27.9% H2 O, 4.4% H2 , 1.2% N2 . An air mass flow rate of 1.5 · 10−4 kg·s−1 enters the cell at 830◦C. To solve the model, inlet velocities and temperatures are imposed as boundary conditions. Outlet pressures are also imposed. External surfaces are assumed as adiabatic. No turbulence model has been considered. A second application is being conducted with the aim of developing new geometries. With this goal the entropy generation minimization technique (Bejan, 1996) has been applied to a tubular cell whose shape is made to change in order to reduce as much as possible the sources of irreversibility. Figures 7.15 and 7.16 show the velocity profiles and the voltage drop in a tubular cell characterized by trapezoidal shape channels, which is the geometry considered as the base case for entropy generation minimization (Sciacovelli, 2006).
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Fig. 7.14 Temperature profile in a 1.5 m tubular cell.
Fig. 7.15 3D velocity field in the cell (m/s).
7.4 Stack Modeling Stacks of tubular SOFCs may consist of numerous cells plus additional devices such as the reformers. In the case of Siemens CHP100 unit, currently installed in Torino,
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Fig. 7.16 3D potential drop in the cell (V).
the stack consists of 1152 tubular cells and 52 in-stack reformers. Non negligible thermal gradients have been registered inside the stack, as shown in Figure 7.17. In this figure, the deviation with respect to the average temperature is shown for a horizontal cross section, at about 0.75 m from the cell bottom. There are several causes for such temperature gradients: (1) the stack is not adiabatic towards the ambient (external surface temperature is about 50◦C); (2) there is a significant heat flux towards the reformers; (3) fuel enters the stack through two ejectors positioned on the same side (right side of Figure 7.17), thus a larger fuel mass flow rate is expected to flow through the reformers close to the injection; (4) the cells have different pedigrees: some of them have run more than 35000 hours, while others have been substituted after about 20000 hours when the system was installed in Torino. A detailed model is required to evaluate distributions of temperature and current in the stack. Temperature gradients are particularly important in stack operation in order to avoid hot spots that can cause failures. In addition, temperature gradients must be checked in stack design because of their effects on the cell performance. To build such model some simplifications must be introduced. In the following section the main characteristics of a stack model are introduced. In this model the energy equation and the conservation of current are solved, while information about mass transfer is introduced.
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Fig. 7.17 Deviation with respect to the average temperature in a stack cross section.
In the case of stacks, radiation plays an even more role important than in the case of a single cell. In fact, in addition to differences in the axial temperature gradients there are differences between the operating temperatures of adjacent cells. A detailed explanation of a radiation model is proposed in the last section.
7.4.1 Network Model The SOFC model introduced in this section only solves the energy equation and the current conservation. The necessary information concerning fluid dynamics and diffusion of species is set through specific assumptions. This model is used for stack modeling purposes. A possible different use consists in determining the boundary conditions to be set in a CFD model in order to account for the position of the cell in the stack. Each cell is divided into vertical portions. Each portion is constituted of the following elements: cold air, tube (t), hot air (a), cell (i, j ), fuel (f ), as shown in Figure 7.18. The cell elements are considered as constituted of four slices. This choice allows one to consider that each cell is in contact with four fuel channels. In a stack, fuel in these channels may be in different conditions. The energy equation is written for each portion of the elements previously introduced. The energy equation for the hot air is: −Qcv,a = ma,a+1 · ha − ma−1,a · ha−1 + ma,i · ha −
4 k=1
m(i,a)k · hi,k .
(7.3)
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Fig. 7.18 Representation of a cell portion.
Fig. 7.19 Representation of the terms of Equation (7.1).
Terms in Equation (7.3) are shown in Figure 7.19. Subscript a refers to the air portion, subscripts a + 1 and a − 1 refer to the upper and lower air portion, index k refers to the four cell slices. Qcv,a is the convective heat flow exchanged with the tube and with each cell slice, which can be calculated: Qcv,a = αint · Aint · (Ta − Tt ) +
4
αext,k · (Ta − Ti,k ),
(7.4)
k=1
where α is the convective heat transfer coefficient and A the heat transfer surface; the subscripts ‘int’ and ‘ext’ refer to the internal and external surfaces respectively. The mass flow rates appearing in Equation (7.3) have the following meaning: ma,a+1 is the fluid flow exiting the ath element and entering the upper element (a + 1); ma−1,a is the fluid flow entering the ath element from the lower element (a − 1); ma,i is the fluid flow entering in the porous cathode; mk(i,a) is the fluid flow exiting each of the four slices of the porous cathode. The mass flow rate exchanged with the cathode can be evaluated on the basis of results obtained through a CFD simulation.
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Fig. 7.20 Net mass flows of nitrogen under different current conditions.
To evaluate the net mass flow rate entering the cathode it is possible to apply the Faraday law (Equations (3.46)). The mass flow rate entering the cathode is larger, in fact oxygen tends to diffuse towards the electrolyte. On the other hand nitrogen tends to diffuse towards the air channel. Figure 7.20 shows the ratio between the mass flow rate exiting the cathode, due to nitrogen diffusion, and the total mass flow rate entering the cathode. The blue curve refers to constant current density, while the red curve refers to the current density calculated by using the axial symmetric model shown in the previous section. The curves have been obtained for a 20 cm long cell portion; the air mass flow is 3.42 · 10−5 kg·s−1 and its temperature is 1000◦C. Air is considered as a transparent gas, thus there is no radiative term in the energy equation. The energy equation for the general cell slice is written: −Qcv,i −Qr,i −Cc,i +Qg,i −Wel = −ma,i ·ha +mi,a ·hi(a) −mf,i ·hf +mi,f ·hi(f ) . (7.5) The first three terms on the left hand side are the net convective, radiative and conductive heat transfers, whose expressions are reported in Equations (7.6–7.8). The fourth term is the heat generated by the chemical/electrochemical reactions (m · T · s) and by Joule effect, while the last term is the electrical power generated in the slice.
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Fig. 7.21 Total net flow exiting the ith slice of a cell.
The convection heat transfer takes place between the internal surface and the hot air and between the external surface and the fuel flow: Qcv,i = αint,i · Aint,i · (Ti − Ta ) + αext,i · Aext,i · (Ti − Tf ).
(7.6)
The radiation heat fluxes are exchanged between the internal and external surfaces of the cell slice and the surrounding surfaces. Moreover, fuel should not be considered as a transparent gas because of the significant partial pressures of CO2 and H2 O. This means that radiation heat fluxes are exchanged between the external surface of the cell slice and the fuel volumes. The total net flux exiting the ith cell slice can be expressed as (see Figure 7.21): Qr,i = Qr(i,t −1) + Qr(i,t ) + Qr(i,t +1) + · · · + Qr(,l−1) + Qr(i,l+1) + · · · + Qr(i,m−1) + Qr(i,m) + Qr(i,m+1) + · · · + Qr(i,i−1) + Qr(i,i+1) + · · · + Qr(i,n−1) + Qr(i,n) + Qr(i,n+1) + · · · + Qr(i,f −1) + Qr(i,f ) + Qr(i,f +1) + · · · + Qr(i,λ−1) + Qr(i,λ) + Qr(i,λ+1) + · · · + Qr(i,µ−1) + Qr(i,µ) + Qr(i,µ+1) + · · · + Qr(i,ν−1) + Qr(i,ν) + Qr(i,ν+1) + · · · .
(7.7)
For sake of simplicity, only the terms associated with the heat exchange between surfaces whose barycentres are at the same height and the surfaces immediately above and below are explicitly shown. The net exchange with the outer elements is not negligible if a fine discretization is assumed, as discussed in Section 7.5. The internal surface exchanges heat with the tube (t − 1, t, t + 1, . . .), with the adjacent portions of the internal surface (l − 1, l, l + 1, . . . , m − 1, m, m + 1, . . .), the upper and lower surfaces (i − 1, i + 1, . . .) and the opposite surfaces (n − 1, n,
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Fig. 7.22 Internal surfaces involved in radiation heat transfer.
Fig. 7.23 External surfaces involved in radiation heat transfer.
n + 1, . . .). A schematic of the internal surfaces involved in radiation heat transfer is shown in Figure 7.22. The external surface is convex. It exchanges heat with the external surfaces of the other cells facing the fuel channel (in this case indicated as λ−1, λ, λ+1, . . . , µ−1, µ, µ + 1, . . . , ν − 1, ν, ν + 1, . . .) and with the gas volumes (f − 1, f, f + 1, . . .), as shown in Figure 7.23. The conductive heat flux can be written as
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Sl,i 2Sn,i Sn,i Sl,i (Ti − Ti−1 ) + (Ti − Ti+1 ) + (Ti − Tm ) + (Ti − Ti ) , li li Ri Ri (7.8) where Ri is the average radius of the cell, Sl and Sn are the heat transfer surfaces as indicated in Figure 7.21, which also shows the energy fluxes and mass flows to be considered in the energy equation. The slice also exchanges mass with the adjacent slices, but these terms have been neglected in the model. As a last example the energy equation for the general fuel element is considered:
Qc,i = λ
Qcv,f + Qr,f = −mf −1,f · hf −1 + mf,f +1 · hf + mf,i · hf − mi,f · hi +
ν
mf,j · hf − mj,f · hj .
(7.9)
j =λ
The convective heat flux can be written Qcv,f = α · A · (Ti − Tf ) +
ν
[α · Aj · (Tj − Tf )].
(7.10)
j =λ
Equation (7.10) is constituted by as many terms as the number of cells facing the fuel channel. Radiation heat transfer accounts for the heat fluxes exchanged between the gas volume and the surrounding surfaces and the surrounding gas volumes: Qr,f = Qr(i−1,f ) + Qr(i,f ) + Qr(i+1,f ) + · · ·
ν
Qr(j −1,f ) + Qr(i,f )
j =λ
+ Qr(i+1,f ) + · · · + Qr(f −1,f ) + Qr(f +1,f ) + · · · .
(7.11)
The first three terms on the right hand side represent the radiation heat transfer between the gas volume and the fuel cell surfaces. Summation in j appearing in Equations (7.10) and (7.11) accounts for the contributions of the other surfaces constituting the fuel cavity. In the case of a cavity constituted of four fuel cells, the summation is constituted of the contributions due to surfaces . . . l − 1, l, l + 1, . . . , m − 1, m, m + 1, . . . , n − 1, n, n + 1. The boundary conditions for the thermal model are the inlet air and fuel temperatures as well as the heat fluxes exchanged with the extern (ambient or reformers). The electrical model is solved for the cell slices in order to determine the current distribution. The equation of current conservation is written for a cell slice in the general form: ∇ 2 = 0, (7.12) where is the electronic/ionic overpotential. Equation (7.12) can be rewritten by considering cylindrical coordinates:
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σr ∂ ∂ 2 ∂ 2 σc ∂ 2 + σr 2 + σz 2 + 2 2 = 0, r ∂r ∂r ∂z r ∂c
(7.13)
where σr , σz and σc are the radial, axial and circumferential electronic/ionic conductivities. This equation can be solved by using a finite difference scheme (see Incropera and De Witt, 2002). Cells are divided into elements as shown in Figure 7.24. Derivatives are expressed as: ∂ c,r−1,z − c,r+1,z = , ∂r 2 · r
(7.14)
∂ 2 c,r−1,z + c,r+1,z − 2 · c,r,z = , 2 ∂r r 2
(7.15)
c,r−1,z + c,r+1,z − 2 · c,r,z ∂ 2 = , ∂z2 z2
(7.16)
c,r−1,z + c,r+1,z − 2 · c,r,z ∂ 2 = . (7.17) 2 ∂c c2 r, z and c are the radial, axial and circumferential dimensions of the elements. Equation (7.13) becomes: σr
c,r−1,z − c,r+1,z c,r−1,z + c,r+1,z − 2 · c,r,z + σr 2 · r · r r 2 c,r,z−1 + c,r,z+1 − 2 · c,r,z + σz z2 + σc
c−1,r,z + c+1,r,z − 2 · c,r,z = 0. r 2 c2
(7.18)
When there are multiple layers between the nodes, the conductivities are calculated by considering proper averaged values, i.e. by considering if the layers are connected in series or in parallel. The isolated surfaces represented in Figure 7.24 avoid current shortcuts. The boundary conditions for the electrical model are the overpotentials on the cathodic side and on the anodic side. In both cases these conditions are set at the interconnections. The first overpotential is set to zero, while the latter is calculated as the difference between the open reversible voltage and the cell voltage, namely an = Vrev − Vcell .
(7.19)
When the stack is modeled, boundary conditions are set only for the boundary cells; in the case of the other cells, the nodes at the interconnections are linked together when two cells are connected in series. Figure 7.24 also shows the connections between the cells in a stack. The results obtained by applying this model to the case of a single cell are shown in Figure 7.25.
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Fig. 7.24 Representation of a cell slice (a) and connections between the cells (b).
Fig. 7.25 Temperature of air, fuel and solid structure along the cell.
7.4.2 Radiative Heat Transfer The zone method is an effective way to model radiation heat transfer when geometries are sufficiently simple as in this case (Hottel and Sarofim, 1967). The system is divided into subsystems, the zones, which can be either surfaces (in the case of solids) or volumes (in the case of non-transparent gases). In the case of the tubular SOFC, the zones are the internal and external surfaces of the cell slices, the external surface of the tube elements, the fuel elements. Each zone is considered as characterized by a unique temperature. A zone model is particularly suitable for to use in a model like the finite difference model introduced in Section 7.4.1. Using zone models, surfaces are treated as gray surfaces. Emissivity is usually in the range of 0.93–0.96, depending on the operating temperature and the material. Air is considered as a clear gas thus it does not affect the radiation between the surfaces. In contrast fuel should be considered as a participating medium. In zone models, participating gases are treated as a weighted mixture of one clear gas and one or more gray gases. The weights can be expressed as a function of the temperature.
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Fig. 7.26 View factors for a cavity constituted of four cells as the function of the length of the cell slice.
7.4.2.1 Element Discretization Radiation heat transfer is assumed to be significant between each portion of the cells and those portions of other cells that are in the same plane, in the upper plane and in the lower plane. The error associated with this assumption increases when small vertical portions are considered. Figure 7.26 shows the view factors, for a cavity constituted of four cells, as the function of the length of the cell slice (vertical discretization). The four view factors have the following meaning: (1) Fap – view factor between a cell and the two adjacent cells in the same horizontal plane; (2) Fop – view factor between opposite cells in the same horizontal plane; (3) Fad – view factor between a cell and the two adjacent cells in the upper and lower plane; (4) Fod – view factor between opposite cells in the same horizontal plane. When the element length is larger than 0.04 m, the summation of these coefficients is about 0.97, which means that the error is less than 3%. The calculation of these coefficients is shown in Section 7.4.2.2.
7.4.2.2 Radiative Heat Transfer between Non-Participating Media When the gas emissivity is sufficiently small, the calculation is quite simple. The procedure can be presented by introducing an equivalent electric network, as shown in many textbooks (see for example Incropera and DeWitt, 2002). To calculate the heat transfer fluxes it is necessary to calculate the equivalent resistances due to the non-black surfaces (Rε ) and those associated with the view factors (RF ):
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Fig. 7.27 View factor cylinder-plane.
Rε = RF =
1−ε , ε·A
(7.20)
1 . F12 · A
(7.21)
The latter can be calculated as the function of the position of each couple of surfaces (zones), depending on the surfaces forming the cavity. Four expressions can be considered to obtain the view factors for modeling tubular cells and stacks: (1) view factor of two tangent cylinders of length l and diameter D (Wiebelt and Ruo, 1963; Sparrow and Cess, 1978): π −2 , 2π Y Y 1 cos−1 Z (2X2 ) cos−1 XZ − 2L + Y sin−1 X1 − π2 Z Fc,c = C ·
C =1−
1− X = 1+ π 6
π 2
π
;
Y = L2 −X2 +1;
Z = L2 +X2 −1;
L=
(7.22)
, (7.23)
2l . (7.24) D
Expression (7.22) can be used when 0.5 < 2l/D < 10. (2) View factor of a cylinder of length l and diameter D (surface 1) and a tangent plane (surface 2). This view factor is used for calculating the radiative heat transfer between the cell and the in-stack reformer or the stack walls (see Figure 7.27). (3) View factor of the internal surface of a cylinder of length l to itself, with a coaxial cylinder of the same length (Sparrow et al., 1962).
7 Tubular Cells and Stacks
F1−1
233
1 −1 R1 π(R = − R ) + cos 2 1 πR 2 R2 [(1 + 4R22 ) × (R22 − R12 )]1/2 R1
+ 2R1 tan−1 [2(R22 − R12 )1/2 ] , − (1 + 4R22 )1/2 tan−1
(7.25)
where R1 is the ratio between the external cylinder and its length and R2 is the ratio between the internal cylinder (coaxial) and its length. (4) View factor of the internal surface of a cylinder of length l to the external surface of a coaxial cylinder of the same length internal to the first one (Brockman, 1994). 1 π R2 1 2 (R1 − R22 − 1) cos−1 + πR2 − AB F1−2 = 1 πR 2 R1 2 − 2R2 tan−1 (R12 − R22 )1/2 + ((1 + A )(1 + B )) 2
2
1/2
tan
−1
(1 + A2 )B (1 + B 2 )A
1/2 ,
(7.26)
where R1 is the ratio between the external cylinder and its length, R2 is the ratio between the internal cylinder and its length, A is the summation of R1 and R2 , and B is R1 minus R2 . In the case of the Siemens CHP 100 stack, it is worth to develop the expressions for two possible cavity configurations: (1) cavity constituted of four cylinder slices; (2) cavity constituted of two cylinder slices and one plane. View factors are obtained from the expressions previously introduced. In the first case the following expressions are used: Fapc = 2 · Fc,c (l),
(7.27)
Fopc = 2.3135 · Fc,c (l),
(7.28)
Fadc = 2 · Fc,c (2l) − Fapc ,
(7.29)
Fodc = 2.3135 · Fc,c (2l)Fopc .
(7.30)
In the second case the following expressions are used: l Fapp = 0.405 · tg−1 6 · , D 2l l − tg−1 6 · . Fadp = 0.405 · tg−1 6 · D D
(7.31) (7.32)
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Fig. 7.28 Schematic of the view factors.
Figure 7.28 shows the meaning of the view factors expressed by Equations (7.26– 7.32).
7.4.2.3 Radiative Heat Transfer between Participating Media When the gas emissivity is not negligible, two options are proposed: (1) gas is considered as a gray gas (or a mixture of gray gases); (2) gas is considered as a weighed sum of gray gases, whose weights depends on the temperature. In the first case the procedure consists of the following steps: 1. construct the gas model; 2. calculate the direct exchange areas of each pair surface-surface (ss), surface-gas (sg), gas-gas (gg); 3. calculate the total exchange areas (separately for each gas);
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4. set the energy balance of each zone and the boundary conditions. Gas emissivity can be calculated as the function of the gas temperature, the partial pressure of CO2 (pc ) and H2 O (pw ) and the characteristic length of the cavity (L). ε = 0.9 · 1.05 · (1 − e−K·L·0.3048),
(7.33)
where 1.05 is a correction factor here introduced in order to reduce the difference between the results obtained by Equation (7.3) and the charts, available for example in Hottel and Sarofim (1967), in the temperature range between 800◦ C and 1100◦C and K is the gas absorption coefficient, calculated as Ganapathy (1991): T pw + pc K = (0.8 + 1.6 · pw ) · 1 − 0.38 · · . (7.34) 1000 L In the case of a pair surface-surface, the direct exchange area is the ratio between the heat flux and the difference between the emissions of the two surfaces, i.e. Qr,ij cos βi · cos βj · t (r) · dAi · dAj si sj = = , (7.35) Ei − Ej π · r2 Ai Aj where E is the surface emission, r is the distance between the barycentre of the two surfaces, β is the angle between the normal to the surface and the segment joining the barycentre of the two surfaces, t is the transmission coefficient, which is calculated as t (r) = exp − K · dr . (7.36) In the case of transparent gas, the direct exchange area is equal to the product of the view factor and the surface area, namely si sj = Fij · Ai . In the case of pairs surface-gas and gas-gas the direct exchange areas are: Qr,ij Ki cos βj · t (r) · dVi · dAj gi sj = = , Ei − Ej π · r2 Vi Aj gi gj =
Qr,ij = Ei − Ej
Vi
Aj
Ki · Kj · t (r) · dVi · dVj . π · r2
(7.37)
(7.38)
(7.39)
The total exchange areas allow one to account for the surface emissivity. The calculation of these quantities is shown in Hottel and Sarofim (1967). In the second case, an iterative procedure is required. It consists of the following steps: 1. construct the gas model as a mixture of gray gases and one transparent gas; 2. calculate the direct exchange areas of each pair surface-surface, surface-gas, gasgas (separately for each gas);
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calculate the total exchange areas (separately for each gas); assume guess values for the temperature of zones; calculate the direct flux areas; set the energy balance of each zone and the boundary conditions; calculate a new set of temperatures.
The direct flux areas account for the dependence of the weights considered in the gas model on the temperature. This is the reason for using an iterative procedure whose accuracy depends on the size of the zones and the number of gases considered for the gas model. The interested reader can find all the details in Hottel and Sarofim (1967).
Acknowledgements We wish to thank Professors Michele Calì and Romano Borchiellini of Politecnico di Torino for their support and advice and Ing. Gianmichele Orsello of Turbocare S.p.A. for providing data and technical information. We would like to thank Ing. Francesco Colella, Ing. Adriano Sciacovelli and Ing. Fabio Zanghirella for their precious contribution. Special thanks to Maria Giovanna Domenichini and George Oduor Oloo for their help and the time they dedicated to us.
References Anonymous (2004) Fuel Cell Handbook, seventh edition, US Department of Energy, National Energy Technology Laboratory, Morgantown. Bejan A. (1996) Entropy Generation Minimization. The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes, CRC Press, Boca Raton, FL. Bharadwaj A., Archer D.H., Rubin E.S. (2005) Modeling the performance of flattened tubular solid oxide fuel cell. Journal of Fuel Cell Science and Technology 2, 52–59. Brockmann H. (1994) Analytic angle factors for the radiant interchange among the surface elements of two concentric cylinders. International Journal of Heat Mass Transfer 37(7), 1095– 1100. Bujalski W., Paragreen J., Reade G., Pyke S., Kendall K. (2006) Cycling studies of solid oxide fuel cells. Journal of Power Sources 157, 745–749. Campanari S. (2001) Thermodynamic model and parametric analysis of a tubular SOFC module. Journal of Power Sources 92, 26–34. Campanari S., Iora P. (2004) Definition and sensitivity analysis of a finite volume SOFC model for a tubular cell geometry. Journal of Power Sources 132, 113–126. Ciano C., Calì M., Melhus O., Verda V. (2006) A model for the configuration of a tubular solid oxide fuel cell stack. In Proceedings of 2006 ASME International Mechanical Engineering Congress and Exposition IMECE2006, ASME, American Society of Mechanical Engineers, Paper IMECE2006-16141:1–9. Costamagna P., Selimovic A., Del Borghi M., Agnew G. (20004) Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC). Chemical Engineering Journal 102, 61–69.
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Fischer K., Seume J. (2006) Location and magnitude of heat sources in solid oxide fuel cells. In Proceedings of the 4th International ASME Conference on Fuel Cell Science, Engineering and Technology, FUELCELL2006, ASME, American Society of Mechanical Engineers, Paper FUELCELL2006-97167:1–12. Fuller E.N., Schettler P.D., Giddings J.C. (1966) A new method for prediction of binary gas-phase diffusion coefficients. Industrial and Engineering Chemistry 58, 19–27. Ganapathy V. (1991) Waste Heat Boiler Deskbook, The Fairmont Press, Atlanta. Gardner F.J., Day M.J., Brandon N.P., Pashley M.N., Cassidy M. (2000) SOFC technology development at Rolls-Royce. Journal of Power Sources 86, 122–129. Haberman B.A., Young J.B. (2005) Numerical investigation of the air flow through a bundle of IP-SOFC modules. International Journal of Heat and Mass Transfer 48, 5475–5487. Haynes C., Wepfer W.J. (2001) Characterizing heat transfer within a commercial-grade tubular solid oxide fuel cell for enhanced thermal management. International Journal of Hydrogen Energy 26, 369–379. Hottel H.C., Sarofim A.F. (1967) Radiative Transfer, McGraw-Hill, New York. Incropera F.P., DeWitt D.P. (2002) Fundamentals of Heat and Mass Transfer, Wiley, Hoboken. Li P.W., Chyu M.K. (2003) Simulation of the chemical/electrochemical reactions and heat/mass transfer for a tubular SOFC in a stack. Journal of Power Sources 124, 487–498. Li P.W., Schaefer L., Chyu M.K. (2004) A numerical model coupling the heat and gas species’ transport processes in a tubular SOFC. Journal of Heat Transfer 126, 219–229. Li P.W., Suzuki K. (2004) Numerical modeling and performance study of a tubular SOFC. Journal of the Electrochemical Society 151, A548–A557. Lockett M., Simmons M.J.H., Kendall K. (2004) CFD to predict temperature profile for scale up of micro-tubular SOFC stacks. Journal of Power Sources 131, 243–246. Lu Y., Schaefer L., Li P. (2005) Numerical study of a flat-tube high power density solid oxide fuel cell. Part I: Heat/mass transfer and fluid flow. Journal of Power Sources 140, 331–339. Nield A.D., Bejan A. (1999) Convection in Porous Media, second edition, Springer-Verlag, New York. Ota T., Koyama M., Wen C., Yamada K., Takahashi H. (2003) Object-based modeling of SOFC system: Dynamic behavior of micro-tube SOFC. Journal of Power Sources 118, 430–439. Saito T., Abe T., Fujinaga K., Miyao M., Kuroishi M., Hiwatashi K., Ueno A. (2005) Development of tubular SOFC at TOTO. In Proceedings – Electrochemical Society, v PV 2005-07, Solid Oxide Fuel Cells IX: Cells, Stacks, and Systems, Proceedings of the International Symposium, ECS, Electrochemical Society Inc., pp. 1039–1040. Sammes N.M., Du Y., Bove R. (2005) Design and fabrication of a 100 W anode supported microtubular SOFC stack. Journal of Power Sources 145, 428–434. Sánchez D., Chacartegui R., Muñoz A., Sánchez T. (2006) Thermal and electrochemical model of internal reforming solid oxide fuel cells with tubular geometry. Journal of Power Sources 160, 1074–1087. Sciacovelli A. (2006) Thermo-fluid dynamic model of a planar solid oxide fuel cell, Graduation Thesis, Politecnico di Torino [in Italian]. Singhal S.C., Kendall K. (2003) High Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications, Elsevier, Oxford. Sparrow E.M., Cess R.C. (1978) Radiation Heat Transfer, Hemisphere, Washington DC. Sparrow E.M., Miller G.B., Jonsson V.K. (1962) Radiative effectiveness of annular-finned space radiators, including mutual irradiation between radiator elements. Journal of Aerospace Science 29(11), 1291–1299. Stiller C., Thorud B., Seljebø S., Mathisen Ø., Karoliussen H., Bolland O. (2005) Finite-volume modeling and hybrid-cycle performance of planar and tubular solid oxide fuel cells. Journal of Power Sources 141, 227–240. VanderSteen J.D.J., Pharoah J.G. (2004) The effect of radiation heat transfer in solid oxide fuel cells modeling. Combustion Institute/Canadian Section, Spring Technical Meeting, Queen’s University, May 9–12, 2004.
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Verda V., von Spakovsky M.R., Siegel N. (2005) Development of a detailed planar SOFC CFD model for analyzing cell performance degradation. In: Proceedings of the 18th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Trondheim, Norway, Kjelstrup S., Hustad JE, Gundersen T., Røsjorde A., Tsatsaronis G., Academic Press, pp. 1193–1202. Vora S.D. (2006) SECA program review. Paper presented at Seventh Annual SECA Workshop, Philadelphia, PA. Wiebelt J.A., Ruo S.Y. (1963) Radiant-interchange configuration factors for finite right circular cylinder to rectangular plane. International Journal of Heat Mass Transfer 6, 143–146. Young J.B. (2007) Thermofluid modeling of fuel cells. Annual Review of Fluid Mechanics 39, 193–215.
Chapter 8
Modeling of Combined SOFC and Turbine Power Systems Eric A. Liese1 , Mario Luigi Ferrari2 , John VanOsdol1, David Tucker1 and Randall S. Gemmen1 1 National Energy Technology Laboratory, U.S. Department of Energy, P.O. Box 880, Morgantown, WV 26507, U.S.A. 2 DiMSET Thermochemical Power Group, Università di Genova (University of Genoa), via Montallegro 1, 16145 Genova, Italy
8.1 Introduction An SOFC outputs a significant amount of high quality heat which has drawn much attention as to how to use this heat in the most efficient manner possible. Since a gas turbine (GT) produces work from the expansion of a heated gas, it is one choice to consider for combination with an SOFC. A hybrid system in this field of research generally refers to a combined fuel cell and some form of heat engine. Gas turbines (GT) have been the most commonly investigated heat engine, although the use of a Sterling engine and steam turbines have also been considered. The goal of this chapter is to discuss modeling considerations specific to gas turbine SOFC hybrid systems. Two broad categories of hybrid systems are the pressurized and the unpressurized fuel cell hybrid. In the pressurized configuration, the fuel cell is located between the compressor and expander of the turbine system and thus at some elevated pressure. In the atmospheric configuration, the waste heat from the fuel cell goes to a heat exchanger located between the compressor and expander. Also in this system, the air from the turbine exhaust (the expander) is used at the cathode inlet. While keeping the fuel cell at atmospheric pressure poses much less risk, the pressurized configurations offer more potential for increased efficiency, and perhaps reduced cost, and therefore have been more often investigated. Thus, this chapter will discuss the pressurized option, beginning with an overview of how pressure affects SOFC and GT performance. Then steady-state configuration models will be presented, and finally non-linear dynamic modeling of gas turbine and hybrid systems will be discussed.
R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 239–268. © Springer Science+Business Media B.V. 2008
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Fig. 8.1 The effect of pressure on SOFC Nernst voltage.
8.2 Fuel Cells and Pressure Effects Pressure can affect the operation of a fuel cell in several ways, such as in its thermodynamic performance, transport of reactants, kinetics of reactants, etc. Here, only the more prominent pressure benefit to the fuel cell resulting from the Nernst potential will be considered. The Nernst potential is given by: ⎡ ⎤ 1/2 p p RT O H E = E0 + ln ⎣ 2 2 ⎦ , (8.1) 2F pH O 2
where E 0 is the standard state potential of the hydrogen and oxygen electrochemical reaction, R is the universal ideal gas constant, T is the fuel cell temperature, F is Faraday’s constant (96,485 coulombs per mole-e− ), and pi is the constituent partial pressure. As an example, the effect of pressure on the Nernst potential is shown in Figure 8.1, assigning a fuel mole fraction of 0.5 for hydrogen and 0.1 for water (balance inert), an oxygen mole fraction of 0.21 (balance inerts) to Equation (8.1), and assuming equal total pressure on both the cathode and anode. The pressure effect is logarithmic according to the square root of pressure. For a given system, the benefit to fuel cell operation with increasing pressure can be viewed in several ways. The current can be assumed fixed and thus the power and efficiency of the fuel cell will increase due to the increasing voltage following Equation (8.1). Or, if the voltage is assumed fixed, the current and power will increase. For this latter scenario, the change in efficiency of the system will depend on the relative magnitudes of the cell overpotentials, and the possible pressure effects on those losses. There are drawbacks to fuel cell pressurization such as the potential for faster cell degradation and the need for a larger pressure vessel wall thickness which leads to increased capital costs. It is quite possible that pressure vessel code requirements
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Fig. 8.2 Schematic of a recuperated, intercooled turbine system.
could be the primary deciding factor in determining the operating design pressure for a commercially viable system.
8.3 Pressure Ratio Effects of Simple, Recuperated and Intercooled Gas Turbines Figure 8.2 shows a gas turbine system that has intercooling between a two-stage compressor and recuperation of the turbine exhaust. Inserting the isentropic relationship, Equation (8.2), into the definitions for compressor and turbine power gives Equations (8.3) and (8.4) respectively: Tout = Tin
Pout Pin
k/ k−1 ,
m ˙ (cp Tc_in − cp Tcisen _out ), ηc mc ˙ p_ave Tc_in Pc_out kave −1/ kave −1 , Pc = − ηc Pc_in
(8.2)
Pc = m(c ˙ p Tc_in − cp Tc_out ) =
˙ p Tt _in − cp Tt _out ) = mη ˙ t (cp Tt _in − cp Ttisen Pt = m(c _out), Pt _out kave −1/ kave ˙ p_ave Tt _in −1 , Pt = −ηt mc Pt _in
(8.3)
(8.4)
where k is the specific heat ratio, ηc and ηt are the compressor and turbine efficiencies respectively. Typically, an average specific heat and specific heat ratio across the turbomachinery is assumed, which requires an iterative solution (assuming constant values
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based on the inlet becomes inaccurate at higher pressure ratios). To calculate the system efficiency, the excess shaft power (Pt , minus Pc ) is divided by the heat input to the combustor, which is the mass flow and enthalpy downstream of the combustor, station 5, minus the mass flow and enthalpy upstream of the combustor – station 4 for the recuperated case (with or without intercooling), and station 1 for the simple cycle. The enthalpies are obtained knowing the temperatures at the outlet of the expander, compressor and/or the heat exchanger. The temperatures at the outlet of the turbomachinery are obtained from the relationships given in Equations (8.3) and (8.4). For the recuperated case, the definition of heat exchanger effectiveness, Equation (8.5) where hot-side and cold-side mass flows and specific heats are the same, is used to determine the temperature at station 4. Effectiveness =
T4 − T 3 . T6 − T3
(8.5)
First, it is instructive to examine the performance of a recuperated system that has only one compressor (i.e., remove the IC and C2 from Figure 8.2) and compare this to a simple cycle GT (i.e., also remove the recuperator from the diagram). Consider an isentropic compressor efficiency of 85%, isentropic turbine expander efficiency of 90%, recuperator effectiveness of 88% and no pressure losses. A fixed turbine inlet temperature of 1200 K will be assumed for various pressure ratios. This value is based on an assumed 1000 K SOFC inlet temperature, and a 200 K temperature rise from the SOFC inlet to the turbine inlet. The 200 K temperature increase from the cathode inlet to the turbine inlet is reasonable to assume given a cathode temperature difference across the cell of 150 K, and another 50 K temperature increase from anode exhaust combustion. Thus, 1200 K will be used as a base case for the turbine inlet temperature, and for sensitivity, values of 1100 and 1300 K will also be analyzed. Figure 8.3 plots the efficiency versus pressure ratio at three turbine inlet temperatures (1100, 1200 and 1300 K) for the recuperated case, without intercooling, and the simple cycle. Also plotted are the power curves that apply to both cycle configurations. The circles highlight the crossover of the efficiency curves. To the left, the recuperated case is more efficient and to the right the simple cycle case. The recuperated cases have a substantially higher efficiency at their maximum, and this maximum occurs at substantially lower pressure ratios. As expected the turbine inlet temperature has a significant impact on peak efficiency and in the simple cycle cases, the rate at which the efficiency decreases after the maximum. In pressurized configurations, a higher fuel cell cathode outlet temperature will result in a higher turbine inlet temperature. Due to cost considerations, fuel cell manufacturers want to reduce the fuel cell temperature so that lower temperature materials can be used. According to Figure 8.3, this will have a negative impact on GT efficiency and power, and perhaps the overall hybrid system depending on how the fuel cell is operated. More observations can be made from the 1200 K recuperated case calculations (without intercooling) tabulated in Table 8.1. First, in order to obtain a temperature of 1000 K for the cathode inlet from the compressor adiabatic heating alone, a PR
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Fig. 8.3 Comparison of a simple and recuperated GT cycle for a turbine inlet temperature of 1100, 1200 and 1300 K. Power based on 1 kg/s air flow rate. Table 8.1 Simple cycle results for 1200 K turbine inlet temperature. Power based on 1 kg/s air flow rate. PR
Pt − Pc (kW)
Tc,exit (K)
Trecup,exit (K)
1.5 2 6 12 14 20 40 60
76 120 211 208 201 176 94 25
343 377 533 655 685 757 913 1012
1007 952 775 690 673 – – –
of 60 bars is necessary. As seen in Figure 8.3, at this pressure the GT power and efficiency is significantly lower than optimum, since the TIT is so low. Therefore, we would expect that for this particular cathode inlet temperature requirement, the system efficiency will be quite low. Second, Table 8.1 shows that if a recuperator were used, heat exchange would only be possible up to about a PR of 12:1. Beyond this, the turbine exhaust gas temperature, EGT, is too close to the compressor exit temperature. The lower the pressure ratio, the greater the temperature difference and thus the more heat that can be recuperated (but requiring an increasing amount of heat exchange area as well). In this case, the pressure ratio would have to be quite low, between 1.5 and 2, in order for the EGT to be high enough to heat the compressor air. Figure 8.4 compares the recuperated cycle with and without intercooling. Intercooling will reduce the overall compressor power since the average density is reduced. Since there is recuperation, the lower compressor outlet temperature should not hurt system efficiency (intercooling is not shown for the simple cycle since the
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Fig. 8.4 Comparison of a recuperated GT cycle with and without intercooling for a turbine inlet temperature of 1200 K.
result of intercooling without recuperation is always a reduced efficiency). In this example, it is assumed that the inlet temperature of the second compressor, C2, is cooled to the inlet temperature of C1, 300 K. Each compressor is operating at the square root of the final desired pressure ratio at the outlet of C2 (for three compressors, this would be the cube root and so on). The figure shows that intercooling improves both the power and efficiency. In conclusion, it would seem that either recuperated case would always be preferred. However, the cost of a recuperator can be significant. Also, the analyses performed fixed the turbine inlet temperatures at fairly low values. Typically with larger heavy-duty turbines the temperature is pushed to material limits and heat recuperation from the turbine exhaust is done in a combined cycle (steam bottoming cycle). Such a system can achieve close to 60% LHV efficiency on natural gas. Section 8.4 will consider whether a steam bottoming cycle is potentially appropriate for a hybrid system. A steam cycle is generally not economical for smaller systems.
8.4 Hybrid System Steady-State Configurations The analysis in Section 8.3 provided a general understanding of GT performance with a turbine inlet temperature representative of current SOFC exhaust temperatures. An analysis of an entire fuel cell and gas turbine system will now be examined, where the fuel cell essentially replaces the combustor in Figure 8.2, and the GT will produce some fraction of the total power. What remains to be defined is the method by which the air to the fuel cell cathode is brought to an acceptably high temperature for operation. Three basic pressurized SOFC/GT configurations are chosen
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Fig. 8.5 Combined diagram for steady-state modeling study.
here for a comparative analysis based on the work by Tucker et al. [1]. The three configurations are labeled by their means of inlet cathode air preheating: 1. Heat of Compression. The turbomachinery is operated at a sufficiently high pressure ratio in order to provide the required temperature to the cathode inlet. Compressor airflow is used to maintain the temperature difference between the cathode air inlet and cathode air outlet (fuel cell air T ). 2. Cathode Recycle. Air from the cathode exhaust is recycled to the cathode inlet and increases the air temperature from the compressor by mixing. As with the previous case, this method requires certain compressor airflow to achieve the required T across the fuel cell. Two methods for achieving cathode recycle are commonly considered. A high temperature blower powered by an electric motor can be used to recycle the flow. Also, an ejector uses the compressor as the primary flow to recycle the cathode air. These comparisons will assume a blower is used. 3. Recuperation. A recuperator is used to exchange heat between the turbine exhaust and compressor exit air. As in the case for the heat of compression, fuel cell T is managed via the compressor air mass flow. In the simulations, the recuperator was not able to provide all of the heat necessary to preheat the cathode flow to the desired temperature. Methods for adding additional heat to the system are supplementary fuel added to a combustor before the fuel cell or after, or cathode recycle can also be used to make up the difference. Here, supplementary fuel was added after the fuel cell, to the post-combustion system. A combined diagram is shown in Figure 8.5. In this examination, the cathode recycle case will have no recuperator and the recuperator case will have no cathode recycle. However, it is quite possible that a hybrid system could use both. The anode exhaust is combusted at the turbine inlet in all cases. This could be done in other locations, such as the recycle loop, which would reduce the recycle required. It is apparent that a number of other configurations could be imagined (such as intercooling); however, each will likely be a modification of one of the base configurations given here.
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Molar Fraction
H2 CO2 H2 O N2
0.836 0.115 0.013 0.036
8.4.1 Boundary and Initial Conditions This analysis is aimed at a syngas fuel source coming from a coal gasifier, therefore no reformer section is needed. If the fuel source had significant amounts of methane, a methane-steam reformer would be necessary to convert the methane to hydrogen in order to avoid possible coking within the SOFC. Note that a fuel cell with internal reforming will need less cooling air because the overall reforming reaction is endothermic. A rough estimate is 40% of the fuel cell’s waste heat would go towards reforming, assuming natural gas. Assuming that the temperature difference across the fuel cell would have the same requirement, all cases with reforming will have a lower gas turbine flow, and thus power, and thus total system efficiency (which is apparent since less waste heat is available to produce work). In Liese and Gemmen [2], internal and external reforming systems are compared. A system with an external reformer can be as efficient as one with internal reforming, but with increased system complexity. The fuel gas composition molar fraction is shown in Table 8.2. All minor constituents and contaminants have been neglected for this study. The gas composition chosen for this study was based on a representation of Pittsburgh #8 coal (LHV = 30.75 MJ/kg) gasified in an dry, entrained flow gasifier in order to produce a syngas composed primarily of carbon monoxide and hydrogen [3]. It is further assumed that CO shift and CO2 capture takes place upstream of the fuel cell anode, and that the efficiency of both processes is approximately 90%. In calculating the overall system efficiency, the lower heating value of the fuel gas composition shown in Table 8.2 was added to any other fuel energy input to the system, and the total fuel energy used was divided into the net electric output of the system. The parameter values used for the turbine and balance of plant are shown in Table 8.3. Note, today’s compressor efficiencies are typically higher (86–89%) than what has been used in this analysis. Each hybrid system configuration uses a lumped parameter SOFC model. The SOFC model used in these studies is taken after Berry et al. [4]. In this model the activation loss (voltage) is described by: RT J ηact = ln , (8.6) 2F J0
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Table 8.3 Values of turbomachinery parameters used in model. Parameter
Value
Compressor isentropic efficiency Turbine isentropic efficiency Gas turbine mechanical efficiency Generator efficiency Inverter efficiency Fuel cell pressure drop Combustor and manifold pressure drop Recuperator pressure drop Blower efficiency
80.0% 90.0% 98.0% 97.0% 98.0% 2.5% 5.0% 2.5% 70.0%
Table 8.4 Constants used for resistance calculation. j
Layer
Thickness δj (m)
αj (Ohm-m)
βj (K)
1 2 3
anode cathode electrolyte
0.00150 0.00010 0.00001
0.00002980 0.00008144 0.00002940
–1392 600 10350
where J is the current density, and the exchange current density is assumed to have the following linear temperature-dependent relationship: J0 = 13595.0 + 15 · T (K) A/m2 . The concentration loss uses the following relationship: J RT ηconc = − ln 1 − . 2F 4000
(8.7)
(8.8)
The ohmic loss is defined as: ηohm = J ·
3
γj ,
(8.9)
j =1
where the temperature-dependent resistance is assumed to have the following characteristic: γj = δj αj e(βj /T ) Ohm·m2 , (8.10) with the constants used in Equation (8.10) given in Table 8.4. In this case, the operating temperature of the fuel cell (used in the Nernst potential, and overpotential calculations) is assumed to be the outlet cathode gas temperature. Initial fuel cell parameters used in the various simulations are summarized in Table 8.5. The process for analyzing and comparing configuration options on hybrid systems can be simplified by fixing certain parameters. In these simulations the fuel
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Value
Unit
Initial cell voltage at 101.325 kPa Initial current density at 101.325 kPa Stack power Stack fuel utilization Cell temperature Fuel cell inlet temperature Fuel cell T
0.7 500 240 80 1123 973 150
V mA/cm2 MW % K K K
utilization is fixed and the current adjusted until a stack power of 240 MW is obtained (the cell voltage changes in response to the operating conditions). The recuperator cold-side outlet temperature has been constrained to 20 K below the hot-side inlet temperature. The inlet air flow rate is adjusted to keep the temperature difference across the fuel cell at 150 K. For the recycle model, the recycle rate is adjusted to maintain the cathode inlet temperature of 973 K and for the recuperated model the fuel to the post-combustor is adjusted to maintain this temperature. A part-load analysis will not be done. As a note, it would be a good assumption to fix SOFC temperature constraints for a part-load analysis as well, since operating the fuel cell at a fixed temperature is better for the durability of the fuel cell. However, it would not be a good assumption to constrain other parameters, such as the turbomachinery efficiencies, for a part-load or dynamic analysis.
8.4.2 Steady-State Model Results Results are shown for a range of pressure ratios. For the recuperated case, the efficiency increases as the pressure ratio is reduced as shown in Figure 8.6. The efficiencies are high because most of the input fuel is hydrogen, and the energy needed to produce it from some source is not included. While the system efficiency increases for the recuperated case, note that the GT power declines sharply as shown in Figure 8.7. Therefore, not only would a recuperator need increasing amounts of surface area in order to transfer the heat, and volume to maintain constant frictional losses, but the system would have an increasingly lower power rating. Thus the system capital cost per kilowatt would increase. To understand the peak recuperated case efficiency, note that as the pressure ratio decreases, the turbine is taking less energy from the air, increasing the temperature going to the recuperator. Thus, ultimately less auxiliary fuel is needed at the postcombustor to keep the temperature at the cathode inlet fixed at 973 K. At a pressure ratio between 3 and 4, fuel to the auxiliary fuel is no longer necessary to preheat the air to the cathode inlet. This is the point of maximum efficiency for this system. The decrease in fuel cell efficiency with pressure is not enough to offset the improved recuperated GT efficiency, which is incidentally near its peak efficiency
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Fig. 8.6 Efficiency comparison.
(Section 8.3, Figure 8.3). This is comparable to the result by Nishida et al. [5] for a similar recuperated system (but with an internal reforming fuel cell and plots that use constant temperature curves; whereas, the results here are without a reformer and at constant fuel cell power). Nishida et al. [5] also show results for internal preheating (i.e. a lower allowable fuel cell inlet temperature and higher T across the fuel cell), with the effect being to move the maximum efficiency to a higher pressure ratio (from 5 to 7) and a broader peak. This is logical since a lower temperature would be needed from the turbine exhaust and thus less expansion. Also, the system efficiency with internal preheating would be higher. The simplest way to understand this is to know that the GT power will be higher at these pressure ratios (assuming all other parameters are equal) as shown in Section 8.3, Figure 8.3. Another advantage to operating at a slightly higher pressure ratio is that there is more potential for added power (but with decreases system efficiency) by auxiliary firing the turbine system. This can be seen again in Figure 8.3 by comparing the power curves between the two constant temperature lines at a pressure ratio of 3 with a pressure ratio of 7. There is a larger difference at the pressure ratio of 7. As seen in Figure 8.6, the cathode recycle case demonstrates the potential for reasonably high efficiency without the expense or complexity of an exhaust gas recuperator, but at the expense of GT power, as shown in Figure 8.7 comparison (the SOFC power is the same for both cases, 240 MW). A high temperature blower; however, could also be expensive. The blower power required is included in the efficiency calculations (70% blower efficiency). The back pressure that must be overcome is considered to be the fuel cell pressure drop only, which is likely optimistic. As shown in Figure 8.8, the amount of recycle necessary to maintain a cathode inlet temperature of 973 K is high and decreases as PR increases until the point where the heat of compression is reached where no recycle is necessary.
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Fig. 8.7 GT power comparison.
Fig. 8.8 Recycle air flow rate.
Figure 8.9 shows the turbine exhaust gas temperatures for the cathode recycle and recuperated cases. The cathode recycle has a comparatively higher quality of heat. Systems with temperatures above 800 K are a good candidate for heat recovery steam generation. Thus, up to a PR of about 20 there is some potential for the addition of a steam cycle to the cathode recycle analysis, which would increase both system power and efficiency. More detailed studies, including economic, are warranted. A combination of cathode recycling and recuperation, or steam generation, could be the best way to optimize efficiency, power and cost for a particular power plant size.
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Fig. 8.9 Turbine inlet and exhaust gas temperature comparison.
8.5 Hybrid Systems Dynamic Modeling To best achieve the benefits of hybrid systems, improved dynamic system models are needed. Much of the opportunity for innovation and ultimate commercial success for this technology lies in the area of system dynamics and control. To achieve commercial success, it is critical that the technical issues surrounding system dynamics are identified. Dynamic models can play a helpful role in that regard. Chapter 9 describes dynamic modeling of the primary device, the SOFC itself. This chapter’s section will expand on this to discuss a full non-linear hybrid system dynamic model.
8.5.1 Connecting Individual Models and Timescale Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model.
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The more common method could best be described as the Equation Oriented Method (EOM). A good example of this method’s use is by Schobeiri et al. [6], who applied it to detailed gas turbine modeling. Individual models are divided into two main types of elements, flow elements and pressure elements. An example of a flow element would be a pipe with a particular pressure drop. The change in mass flow rate with respect to time will depend on a given upstream and downstream pressure. For incompressible flow (Mach numbers below 0.3), the lumped momentum differential equation used to calculate mass flow rate for a pipe is simplified to ∂m ˙ m ˙ 2R Across Tout m ˙2 Tin = (Pin − Pout ) − Cf + − , (8.11) ∂t Across L Pin Pout L 2DρAcross where Across is the cross-sectional area, L the length, D the diameter, and Cf the friction coefficient. The temperature at the exit of any model would use an appropriate energy equation. An example of a pressure element would be a tank or pressure vessel with a specified volume. This volume would include any volume from the neighboring flow elements. The change in pressure with respect to time will depend on a given upstream and downstream pressure from the flow element neighbors. In the case of a tank (or volume), the energy and mass conservation equations are generally combined as shown: ⎤ ⎡ n m ∂T 1 k Cpin,i ˙ ⎣ = kTin,i − T − (1 − k) m ˙ in,i m ˙ out,j T ⎦ − Q, ∂t ρV Cp ρV Cp j =1
i=1
⎡ kR ⎣ ∂P = ∂T V
n i=1
m ˙ in,i
Cpin,i − Cp
m j =1
(8.12)
⎤ m ˙ out,j T ⎦ −
(k − 1) ˙ Q, V
(8.13)
where k is the specific heat ratio, V the volume, Q˙ the heat flow out of the element, n the number of inlets, m the number of outlets, and the non-subscript symbols are the condition in the plenum. This structure assures correct pressure-flow solutions and, thus keeps the model consistent. Software packages, such as ASPEN Dynamics, will ensure this correct coupling. In general, two flow calculating devices cannot be connected directly, but must have a pressure (typically a volume) element in between. Two flow devices can be connected if a single equation can be written that describes the pressure drop over the connected section. For instance, some programs allow two pipe models to be connected. Another method, less common, will be called here the Pressure Node approach. A company called, TRAX, uses this method in their dynamic modeling software package called ProTRAX. Their literature discusses this method’s mathematical derivation. With this approach, the flow elements described previously are connected by either pressure elements (such as a volume) or by what are called, pressure
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nodes. The strength of this approach is that flow elements can be connected directly without specifying a volume. Each flow element must not only calculate flow, but the partial of flow with respect to pressure at each boundary and a ‘compressibility’ term. But for quasi-steady flow, this will reduce to an algebraic equation. The partial derivative acts as a predictor for a matrix pressure solution. The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some ‘compressibility’ to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. The decision process of putting together models based the above methods is helped by some observations about timescale and solution stiffness. The modeler should decide on a timescale of interest. The characteristic times, or first order time constants, for a typical flow and a pressure element based on lumped momentum and mass conservation respectively are shown in the following equations [7, 8]:
ρ τflow = L 1.49 · , (8.14) 2(Pin − Pout ) ρV , (8.15) m ˙ where the coefficient, 1.49, in Equation (8.14) is a corrective factor that is justified by Traverso [9]. Note that the derivation of Equation 8.14 ignores the friction coefficient (i.e. a large pipe diameter is assumed which is why τflow is independent of the cross-sectional area). For representative hybrid flow conditions one finds that τflow is on the order of 0.01 s. Hence, if one is only interested in thermal timescales of order 1 s, the momentum equation could be simplified to its quasi-steady form. The filling time constant is indicative of a time constant for mass exchange inside a volume. For τfilling, any change in concentration would have this time constant. Equation (8.15) does not express a time constant of the pressurization/depressurization phenomenon. However, it has been shown that a pressurization/depressurization is about 4–5 times greater than the filling time [8]. Thermal characteristic time constants are also given by: ρV , (8.16) τthermal filling = k·m ˙ ρV cv τwall transfer = , (8.17) h · As τfilling =
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τsolid thermal =
(Mass · cp )solid , h · As
(8.18)
where V is the gas volume, k, the specific heat ratio, h, the heat transfer coefficient, As , the surface area of the component, cv , is the constant volume specific heat capacity of the gas. Equation (8.16) is a fraction of Equation (8.15) and represents a time constant for the temperature change at the outlet of a volume. Equation (8.17) is the gas thermal wall transfer time constant and depends on the amount of gas in the component and the heat transfer coefficient and Equation (8.18) is the solid thermal time constant and depends on the component’s mass. As an example, if only quasi-steady flow elements are used with volume pressure elements, a model’s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork’s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)
8.5.2 Turbomachinery Modeling The work of Schobeiri et al. [6] can be used to create a fairly detailed dynamic model of axial turbomachinery stage by stage. At the opposite end of the detailed approach is the previous steady-state example in Section 8.3 that described the behavior of the compressor and expander components of the gas turbine by fixing the efficiencies and using thermodynamic equations to calculate the power supplied to (expander) or taken from (compressor) the shaft. Left-over power was considered to be more or less the generator power, minus estimated losses. However, the following approach is often used. A turbomachinery map can be either generated from experimental data, scaled from maps of other related turbomachinery, or obtained from the manufacturer. For proprietary reasons, such maps can be difficult to obtain, but many modelers use whatever maps are available in the literature and scale such maps to desired conditions, although scaling can lead to inaccuracies, especially for smaller turbomachinery (because of Reynolds effects). Section 8.5.3.3 will help demonstrate this point. A software program called GasTurb by Joachim Kurzke provides some maps and can also scale and digitize maps. If a sufficient map is not available, a one-dimensional design and analysis code could be used to generate a map. Generally, the compressor and expander maps plot referred or corrected mass flow rate, speed and torque. For instance, the mass flow rate and rotational speed are
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Fig. 8.10 Generic compressor performance map.
plotted in the quasi-dimensionless forms shown in Equations (8.19) and (8.20). √ θ , (8.19) ˙ m ˙ referred = m δ N Nreferred = √ , θ
(8.20)
where δ is the inlet pressure/101.3 kPa and θ is the inlet temperature/288 K. In order to get the real mass flow from the map, multiply by delta and divide by the square root of theta. The pressure and temperature condition immediately before the turbomachinery is used, not the ambient measurement. For instance, at the turbine expander, the inlet temperature and pressure will be quite elevated and will change dramatically with a change in some process condition such as combustor firing rate; therefore, the look-up value on the map will change. The book Gas Turbine Performance by Walsh and Fletcher [10] has excellent treatment on turbomachinery maps. An example of a compressor map is shown in Figure 8.10. This map fully defines the pressure-flow-efficiency-rotational speed relationship of the compressor. Employing the Beta-line or R-line method, maps can be digitized into tabular form as described by Walsh and Fletcher [10]. The Betaline method is helpful to ensure numeric stability with such maps that would be otherwise problematic because of the near zero and infinite slope at the ends of the constant speed lines. Note that the compressor model can either be a flow element or a pressure element. That is, from speed and pressure it is possible to obtain flow and efficiency from the map, or from speed and flow it is possible to obtain pressure and efficiency from the map. The choice depends on what is more convenient, that is, what type of elements are modeled upstream and downstream.
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Fig. 8.11 Generic turbine performance map.
Typically, the expander has a separate flow and efficiency map as shown in Figure 8.11; although, a number of other options are available such as a torque map instead of efficiency (which may be better for use at low pressure ratios and speeds). For the expander, since the corrected flow is constant for most of the operating area (i.e. a zero slope of the flow map as shown in Figure 8.11), the best element representation is a flow element; that is, obtain the flow based on pressure ratio and speed. For gas driven turbomachinery, it is sufficient to assume that quasi-steady behavior exemplifies the transient characteristic. That is, the maps are sufficient to describe the dynamic pressure-flow characteristic. Such a model, however, would not model surge characteristics [11, 12]. Stall occurs left of the indicated stall line in Figure 8.10, a condition where flow detachment is likely (the likelihood depending on the type of turbomachinery and the equipment around the compressor). Such surge modeling would involve a special characterization of the compressor map, as well as modified and additional differential equations. Because a surge event occurs on a small timescale, along with reverse flow, it is best to have a separate model to primarily look at the surge characteristics of a system. The shaft speed, N, will change when there is a power difference on the shaft: ∂N Pt − Pc + PGenerator + Losses = , ∂t Inertia · N
(8.21)
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where, if the power terms are in watts, the inertia in kg-m2, and N in rad/s, the resulting integration of Equation (8.21) will give the speed in rad/s. Typically, the generator is the primary source of the inertia. For a heavy-duty gas turbines, Rowen [13] lists some typical inertia values.
8.5.3 Recuperated Gas Turbine with Volume Example 8.5.3.1 Experiment and Model Setup This example will demonstrate the implementation of two previously discussed topics; that is, the use of turbomachinery maps and the use of the EOM modeling approach. This section shows work from a collaboration between the National Energy Technology Laboratory (NETL, U.S., D.O.E.) and the University of Genoa (DiMSET-TPG) that compares two transient models, programmed using MATLAB-Simulink, with the data of an experimental facility at NETL, called Hyper. While the first model has been implemented at the NETL obtaining realtime performance [14], the second one has been implemented using the TRANSEO tool [15], developed at TPG in previous works showing good performance in the calculation reliability [16]. The experimental facility, shown in Figure 8.12, is a physical simulator of a 250 kW Direct Fired Fuel Cell Gas Turbine Hybrid System with turbomachinery, two recuperators in parallel (primary surface, counter-flow heat exchangers), and two vessels used to generate the volume capacity effect of the stack and the post-combustor. The thermal effect of the SOFC is simulated with a natural gas-fed combustor, controlled by a real-time fuel cell model [17, 18]. The fuel cell model is not used in this example. Table 8.6 shows a list of components that were modeled, and the calculations performed including the state variables that must be numerically integrated. The experimental test and model comparison have been carried out without bleed and hot bypass, but with a cold bypass valve opening of 40%. The use of the cold-air bypass valve (FV-170) allows compressor discharge to be directed into the turbine inlet, bypassing the heat exchangers, air plenum and fuel cell simulator. FV-170 is a nominal 15.4 cm ID Fisher-Rosemont V-150 Vee-Ball control valve with a full range slew rate of 1.5 seconds. Table 8.7 shows some data characteristics of the experimental system and parameters used for the transient calculations. The system was operated as a conventional constant speed machine, with extensive piping between compressor and turbine. In this particular example turbine speed control was taking place. The closed loop feedback is turbine speed in rpm as measured by an optical pickup on the generator box (nearly instantaneous). In this test, the experimental speed controller and the simulated speed controller (in both models), used a proportional gain of 0.001 and an integral gain of 0.001 × 0.75 with an input of speed error in rpm and output in fuel valve %. For the experiment presented, the fuel flow rate in grams per second
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Fig. 8.12 Flow diagram of the Hyper facility at NETL. Table 8.6 List of modeled components for recuperated GT. Module
Calculation
1. Compressor
outlet P , outlet T , shaft load ∂Tmetal ∂m ˙ dt, dt ∂t ∂t ∂Tgas ∂P ∂Tmetal dt, dt, dt ∂t ∂t ∂t ∂m ˙ dt, Tadiabatic ∂t ∂Tgas ∂P ∂Tmetal dt, dt, dt ∂t ∂t ∂t
2. Heat exchangers and associated piping 3. Plenum: fuel cell 4. Combustor and associated piping 5. Plenum: post-combustor 6. Turbine 7. Shaft, generator
mass flow rate, outlet T , shaft power ∂N dt ∂t
8. Bleed, cold bypass, hot bypass and fuel valve
m, ˙ x˙
9. Cooling flow leakage
m ˙
was determined from the fuel valve position in percent by a third order polynomial with coefficients from highest to lowest order: 1.35E-4, –9.09E-3, 6.04E-1, –3.69. For example, a 39.5% valve position gives a fuel flow rate of 14.3 g/s.
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Table 8.7 Data used in the recuperated GT simulations. Parameter
Value
Unit
Nominal rotational speed Nominal compressor pressure ratio Shaft components mechanical inertia Plenum volume: fuel cell (V 301) Plenum volume: post-combustor (V 304) Total recuperator mass: (E 305, E 300) Plenum mass: fuel cell Plenum mass: post-combustor Recuperator nominal mass flow rate Cold bypass nominal mass flow rate Compressor/turbine leakage
40500 3.5 0.027 2 0.6 36.2 1310 1800 1.0 0.54 10.5%
rpm – kg·m2 m3 m3 kg kg kg kg/s kg/s –
Table 8.8 Time constants of the transient phenomena. Component
Type
Characteristic Time [sec]
Fuel cell plenum
Filling Pressurization/depressurization Thermal filling Thermal wall transfer Solid thermal
3.27 4–5*(3.27) 2.41 7.06 1208
Post-combustor plenum
Filling Pressurization/depressurization Thermal filling Thermal wall transfer Solid thermal
0.49 4–5*(0.49) 0.37 3.47 3472
Recuperator (matrix)
Solid thermal
45.33
Shaft
Mechanical
3.27
There is a broad range of time constants (calculated from Equations (8.14) to (8.18) in Section 8.5.1) for the various component of the system, as shown in Table 8.8. Because of the large mass of the metal vessels, the solid thermal time constants are long. While a change in gas temperature in these vessels will take a long time to affect the solid temperatures, the same is not true for the outlet gas temperature, of which both the gas thermal wall transfer and thermal filling time constant is more indicative. In this particular model since the larger temperature changes are occurring in the post-combustor, the influence of the heat transfer coefficient on the results were important to consider. The post-combustor volume has the smallest time constant and we can guess that an appropriate numerical integration time step will be at least an order of magnitude less. The shaft mechanical time constant is an order of magnitude less than the fuel cell plenum pressurization/depressurization time constant. Thus it is expected that the shaft inertia will not have an effect on the shaft speed dynamics. The recuperator solid thermal time con-
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Fig. 8.13 Rotational speed comparison.
Fig. 8.14 Fuel cell plenum mass flow rate comparison.
stant is interesting. While it will not likely affect any shorter response during turbine speed control, there should be a fairly short term effect on the thermal response.
8.5.3.2 Results for a 15 kW Load Decrease The results show the transient effects due to a 15 kW step decrease to the generator (from 60 to 45 kW), which occurred at time zero. Figure 8.13 compares the dynamics of the shaft speed and shows that both models are in good agreement with the experimental data with regards to frequency. This is not surprising since the system volume will have the primary influence on the frequency response, and the mod-
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Fig. 8.15 Fuel cell inlet temperature comparison.
eling of this is fairly straightforward. As for the amplitude, the model developed with the TRANSEO tool generates a closer result to the experiment. Looking at the air flow rate, Figure 8.14, the TPG model predicts the magnitude more closely. The NETL model air flow rate is high by approximately 7%, but the initial fuel flow is exact (Figure 8.16); therefore, it is likely that something has produced an error in the overall energy balance in the NETL model, such as the efficiency of compressor or turbine, or heat loss to the cathode pressure vessel V 301, or postcombustor vessel, V 304. The NETL model was initially calibrated at zero load conditions. One difficulty in reconciling the NETL model comes from attempting to match numerous conditions while also using only a simple scaling procedure of the original turbomachinery maps to one design point; whereas, the TRANSEO model uses experimental data exclusively to generate the map. Map error and its effect on steady-state dynamic behavior will be discussed in Section 8.5.3.3. Another important comparison is in regards to the temperatures. The fuel cell plenum inlet temperature is shown in Figure 8.15. It is a significant parameter for the hybrid system emulation, because it is used as an input for the real-time fuel cell model (see Figure 8.12). The temperature is decreasing because the perturbation is a decrease in generator load. The response is characteristic of the thermal heat capacity of the recuperators. The initial temperature increase shown by the TPG model (within the first 10 seconds) is due to a slight rise in the temperature of the plenum due to compression heating. The experimental data model does not show this since the thermocouple is in the piping prior to the plenum. The NETL model gives the result prior to the plenum and is therefore more comparable to the experimental result. The blip at the 20 second mark is due to a feedback from the recuperator. The fuel flow rate is adjusting in an opposite manner to the speed as seen in Figure 8.16. The first increase in fuel flow (from 10 to 25 seconds) is increasing the recuperator hot-side inlet temperature and thus has the fairly immediate impact on the cold-side
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Fig. 8.16 Fuel mass flow rate comparison.
Fig. 8.17 Combustor inlet temperature comparison.
outlet temperature (since it is a counterflow heat exchanger) as shown by the blip. Note that any change in compressor temperature is not having an impact (or the temperature in Figure 8.15 would rise at the start of the transient) since it is being washed out by the heat capacitance of the heat exchanger. It is not as evident in the experimental data probably due to limitations of the thermocouple response. Figure 8.17 shows TE 307, the combustor inlet temperature. The only difference between TE 307 and TE 326 (Figure 8.16) should be the heat loss of the pressure vessel used to simulate the cathode volumes, V 301 and V 304. In the TPG more attention was focused on the simulation of the heat losses in vessels and in pipes, using average heat coefficient values. Only by an extensive model calibration effort was it possible to calculate the combustor inlet temperature (TE 307) with a high
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Fig. 8.18 Fuel cell inlet temperature: compressor efficiency variation.
Fig. 8.19 Fuel cell inlet temperature: turbine efficiency variation.
accuracy. In this case, there is a brief rise in temperature at the start of the transient shown in both models and the experimental data. This is the volume compression effect mentioned previously. Again, the second response is due to the recuperator feedback effect.
8.5.3.3 Importance of Performance Map Precision It will now be shown that the performance map precision for both compressor and turbine can be important, and explains some of the discrepancy between TPG and NETL models. For this reason, some additional simulations have been carried out
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Fig. 8.20 Rotational speed comparison: case with the turbine efficiency increased.
with the TPG model in order to study the effects of compressor or a turbine output results based on an efficiency variation typical of manufacturer map tolerance. Figures 8.18 and 8.19 show the values of fuel cell inlet temperature, obtained at steadystate conditions, versus the normalized values of compressor and turbine efficiency, respectively. For these results the electrical power demand has been maintained at 60 kW and the valves have been set as given previously. These figures show that the temperature result can be significantly different for a slight difference in turbomachinery efficiency. In both cases, the machine efficiency increase changes the power balance between compressor and turbine and, in order to maintain a constant rotational speed, generates a fuel mass flow rate reduction. Consequently, the turbine inlet temperature decrease generates a turbine exhaust temperature reduction and, because the cycle is recuperated, there is a compounding fuel cell inlet temperature decrease. This influence of performance map variation justifies the importance of obtaining, if possible, accurate map data for use in the simulation models as done with the TPG model. In order to show the effect, TPG model has been used to re-simulate the 15 kW load step decrease with a 1% turbine efficiency increase. Figure 8.20 shows a slight increase in the amplitude of the rotational speed transient behavior. However, as with the NETL model, the frequency is not much affected. Therefore, it is likely that some of the amplitude error from the NETL model comes from performance map. It is also possible that some of the difference comes from a different heat transfer model for the NETL post-combustor, V 304.
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Fig. 8.21 Hybrid system dynamic model.
8.5.4 Pressurized Hybrid Example The following example is not as rigorous as the previous model, which had the benefit of some critical validation steps; however, it will still show the benefits of a dynamic model, even without such advantage. Here, two events are studied; a load loss from the fuel cell, and a load loss from the turbine in the hybrid system. These two events are considered to have the potential for significant damage to the system.
8.5.4.1 Assumptions The following assumptions are made in this model: 1. Quasi-steady, pressure-driven flow. 2. The alternator windage losses and viscous shaft losses are not modeled. 3. The fuel source is methane and a reformer is necessary. For this study, the reformer is externally heated such that it is not a heat load on the system. This allows one less physical feedback mechanism on the dynamic system, which will simplify the result and interpretation of the results. (For real application studies, the coupling is needed for peak efficiency reasons.)
8.5.4.2 Description of the Model One-dimensional models of a solid oxide fuel cell (see Chapter 9) and a methanesteam reformer [19, 20] were incorporated into the ProTRAX programming environment for transient studies. Lumped parameter ProTRAX sub-models were used for the remaining system components (heat exchangers, turbomachinery, valves, etc.). A schematic of the model is provided for reference in Figure 8.21.
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Fuel Cell
Parameter
Value
Unit
Parameter
Value
Unit
FC power Alternator power Gas turbine Shaft speed Pressure ratio Compressor flow TIT Compressor eff. Turbine eff. Alternator eff. Inertia of shaft components
186 130 – 80 3.5:1 0.664 1280 80 86 96 0.03
kW kW – krpm – kg/s K % % % kg·m2
Fuel utilization O2 utilization Unit cell area Number of cells Inlet steam flow Steam temp. Inlet methane flow Methane temp. Anode inlet temp. Cathode inlet temp. Current ave. cell temperature
85.4 14.4 1.04 140 0.045 823 0.008 823 1004 967 1103
% % m2 – kg/s K kg/s K K K K
Table 8.10 Reformer exit fuel composition. Compound
Molar Fraction
H2 O CO H2 CH4 CO2
0.443 0.053 0.432 0.004 0.068
The base case conditions for the model are shown in Table 8.9, with the reformer outlet fuel composition shown in Table 8.10. The conditions are similar to those given for a 260 kW thermodynamic model by Campanari [21], with a 207 kW output for the fuel cell and 53 kW for the turbine alternator. There is a considerable difference between the alternator power in this model (130 kW) and Campanari’s (53 kW). The model in this study uses external heating for the reformer so that its heat load is not included in the analysis, which accounts for the discrepancy.
8.5.4.3 Results for a Fuel Cell Load Loss A sudden loss of electrical load to the fuel cell in a hybrid system will immediately terminate the electrochemical reaction in the fuel cell, dropping the fuel utilization to zero. This will result in all of the fuel supplied to the fuel cell being consumed in the post-combustor, and the resulting thermal energy being absorbed by the turbine. In a SOFC, this fuel will consist primarily of hydrogen and CO even for a fuel cell system with internal reforming since the reforming will not be immediately affected by the loss of electrical load. Even if the fuel is shut off immediately, the fuel plenum and anode have a sufficient volume of pressurized fuel to provide a sustained increase in thermal output to the turbine. The model results for a sudden loss of load
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Fig. 8.22 Loss of load to the fuel cell with fuel shut off.
to the fuel cell in a hybrid system are shown in Figure 8.22. In this scenario, the fuel supply to the reformer was shut off, while a purge flow was immediately added to purge the system. Increased electrical load is used in the model to maintain a constant turbine shaft speed. The model predicts a sudden sharp increase in turbine inlet temperature persisting for over ten seconds. In a hybrid system where the turbine is sized at only 10 to 20% of the fuel cell capacity, with half the efficiency of the fuel cell, this represents a burst of thermal energy exceeding the capacity of the turbine by two to five times. In a hybrid system lacking the ability to overload the turbine generator or alternator, excessive turbine inlet temperatures would cause increased turbine speed and ultimately result in damage to the turbine. The increased turbine speed would result in increased air flow and pressure on the cathode side of the fuel cell. While these effects provide some mitigation of the temperature increase to the turbine by both lowering the mixture temperature and slowing the fuel flow across the anode, without the reaction in the fuel cell to generate heat, this increased flow could cause excessive cooling rates and result in damage to the ceramic materials of the fuel cell.
References 1. Tucker D., VanOsdol J., Liese E., Lawson L., Ford J.C., Haynes C. (2006) Evaluation of methods for thermal management in a coal-based SOFC turbine hybrid through numerical simulation. In Proceedings of the Seventh International Colloquium on Environmentally Preferred Advanced Power Generation, Irvine, CA, ICEPAG2006-24022. 2. Liese E.A., Gemmen R.S. (2005) Performance comparison of internal reforming against external reforming in a solid oxide fuel cell, gas turbine hybrid system. ASME Journal of Engineering for Gas Turbines and Power 127, 86–90.
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3. Lokurlu A., Pruscheck R. (1999) Simulation of advanced SOFC power plants with gas and steam turbine cycles on the basis of integrated coal gasification (IGFC). In Proceeedings of the 6th International Symposium on Solid Oxide Fuel Cells (SOFC IV), Honolulu, HI, pp. 439– 446. 4. Berry D.A., James R.E., Gardner T.H., Shekhawat D. (2003) Systems analysis of diesel-based fuel cells. In Proceedings of the ASME First International Conference on Fuel Cell Science, Engineering and Technology, Rochester, NY. 5. Nishida K., Takagi T., Kinoshita S., Tsuji T. (2002) Performance evaluation of multi-stage SOFC and gas turbine combined systems. In Proceedings of the ASME Turbo Expo, Amsterdam, the Netherlands, GT-2002-30109. 6. Schobeiri M.T., Attia M., Lippke C. (1994) GETRAN: A generic, modularly structured computer code for simulation of dynamic behavior of aero- and power generation gas turbine engines. ASME Journal of Engineering for Gas Turbines and Power 116, 483–494. 7. Magistri L., Ferrari M.L., Traverso A., Costamagna P., Massardo A.F. (2004) Transient analysis of solid oxide fuel cell hybrids part C, whole cycle model. In Proceedings of the ASME Turbo Expo, Vienna, Austria, GT2004-53845. 8. Traverso A., Trasino F., Magistri L. (2006), Dynamic simulation of the anodic side of an integrated solid oxide fuel cell system. In 8th Biennial ASME Conference on Engineering Systems Design and Analysis, Turin, Italy, ESDA2006-95770. 9. Traverso A. (2004) TRANSEO: A new simulation tool for transient analysis of innovative energy systems. Ph.D. Thesis, TPG-DiMSET, University of Genoa. 10. Walsh P.P., Fletcher P. (1998) Gas Turbine Performance, Blackwell Science Ltd. 11. Gravdahl J.T., Egeland O. (1997) A Moore–Greitzer axial compresor model with spool dynamics. In Proceedings of the 36th Conference on Decision and Control, San Diego, CA. 12. Hahn A. (2002) Modeling and control of solid oxide fuel cell, gas turbine power plant systems. Thesis, University of Pittsburgh School of Engineering. 13. Rowen W.I. (1983) Simplified mathematical representations of heavy-duty gas turbines. ASME Journal of Engineering for Gas Turbines and Power 105, 865–869. 14. Shelton M., Celik I., Liese E., Tucker D., Lawson L. (2005) A transient model of a hybrid fuel cell, gas turbine test facility using simulink. In Proceedings of the ASME Turbo Expo, Reno-Tahoe, NV, 2005-GT-68467. 15. Traverso A. (2005) TRANSEO Code for the dynamic simulation of micro gas turbine cycles. In Proceedings of the ASME Turbo Expo, Reno-Tahoe, NV, 2005-GT-68101. 16. Traverso A., Calzolari F., Massardo A.F. (2003) Transient behavior of and control system for micro gas turbine advanced cycles. In Proceedings of the ASME Turbo Expo, Atlanta, GA, 2003-GT-38269. 17. Liese E.A., Smith T.P., Haynes C., Gemmen R.S. (2006) A dynamic bulk SOFC model used in a hybrid turbine controls test facility. In Proceedings of the ASME Turbo Expo, Barcelona, Spain, GT2006-90383. 18. Smith T.P., Haynes C.L., Wepfer W.J., Liese E.A., Tucker D. (2006) Hardware based simulation of a fuel cell turbine hybrid response to imposed fuel cell load transients. In Proceedings of the ASME International Mechanical Engineering Conference and Exhibition, Chicago, IL, IMECE2006-13978. 19. Gemmen R.S., Liese E., Rivera J.G., Brouwer J. (2000) Development of dynamic modeling tools for solid oxide and molten carbonate hybrid fuel cell gas turbine systems. In Proceedings of the ASME Turbo Expo, Munich, Germany, 2000-GT-0554. 20. Liese E., Gemmen R.S., Jabbari F., Brouwer J. (1999) Technical development issues and dynamic modeling of gas turbine and fuel cell hybrid systems. In Proceedings of the ASME Turbo Expo, Indianapolis, IN, 99-GT-360. 21. Campanari S. (1999) Full load and part load performance prediction for integrated SOFC and microturbine systems. In Proceedings of the ASME Turbo Expo, Indianapolis, IN, 99-GT-65.
Chapter 9
Dynamic Modeling of Fuel Cells Randall S. Gemmen National Energy Technology Laboratory, U.S. Department of Energy, P.O. Box 880, Morgantown, WV 26507, U.S.A.
9.1 Introduction The analysis of fuel cells can be divided into two areas, steady state modeling and dynamic modeling. Prior chapters largely focused on the fundamental principles and present practices for modeling the steady state performance of fuel cells and fuel cell systems. For these situations, stable conditions exist at all points in the system, and process performance parameters, such as efficiency, are precisely described. This chapter opens the door to the study of the dynamic behavior of fuel cells and fuel cell stacks. (Full system dynamic analysis, where both stacks and other balance of plant dynamics are modeled, is considered separately in Chapter 8. In this chapter, important capacitive elements in the fuel cell process are identified and modeled. While some inductive-type behavior has been noted within the literature (see Section 9.2.1), little advancement has been made to develop physical models to support such transient behavior for the cell per se; therefore, our attention is focused primarily on capacitive behavior only. These capacitive elements control the rate at which process parameters change due to changes in other coupled process parameters. For example, thermal capacitance controls the rate of cell temperature change due to an imbalance between internal energy sources and boundary energy transfers. There is significant motivation for the dynamic modeling of SOFC components and systems (Bistolfi et al., 1996; Benhaddad and Protkova, 2005). The author of this chapter has participated in the experimental testing of both individual cells and complete fuel cell systems, and it is usual that the greatest concern for failure is during a change in state from one steady operating condition to another. Often the greatest concern exists during startup or shutdown where the unit passes through one of its largest transitions. However, it is not just the magnitude of transition that is important. The speed at which the transition occurs is also of concern. Transitions from one steady state to another that occur slowly (relative to criteria to be given later), can be assumed to be quasi-steady in which the cell/stack simply passes through a series of steady states. Transitions that occur more quickly relative to the R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 269–322. © Springer Science+Business Media B.V. 2008
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criteria result in significantly different cell/stack operation. It is the work of dynamic modeling engineers to clearly resolve these differences so that accurate and useful information can be provided to fuel cell designers. Dynamic models, when properly developed, can provide important information to the fuel cell designer regarding the internal conditions of the cell during real-world transient operation. If properly developed, the dynamic model can provide detailed spatial information at any point in the cell and at any point of the transition. Such information will allow the fuel cell designer to engineer systems having greater reliability. As is evident from prior chapters, several levels of models are useful for dynamic studies, from zero-dimensional (lumped) models to multi-dimensional models. It happens that for purposes of fuel cell performance prediction, one-dimensional modeling can be very effective, and in some cases improves performance prediction only slightly over lumped models (Iora et al., 2005). This results from the simple fact that both convective and diffusive transports, each occurring in their own distinct regions within fuel cells (channels and electrodes, respectively), behave nearly one-dimensional. Further, the electrochemical reactions are tightly controlled by the control of an external load. This removes an otherwise tight coupling between local temperature and chemical kinetics that exists in such studies as combustion science. Finally, while computational speeds continue to increase, and the use of parallel processing is widespread, it remains true that to achieve timely results, transient modeling of fuel cells can most effectively be performed with one-dimensional modeling. It is for these reasons, and due to relative simplicity, that this chapter will focus on lumped and one-dimensional tool development for the transient analysis of cells and stacks. The chapter begins with a presentation of available evidence of cell and system transient behavior. Based on this evidence, parameters that have been proposed to account for various capacitive behaviors within the cell and fuel cell system are presented. A brief discussion for how such transient behavior affects the safe operation of a fuel cell and system is then given. Models are then developed for predicting the behavior of the cell and system, and, in the final section of this chapter, examples of their application are given.
9.2 Transient Characteristics of Fuel Cell Operation This section examines some experimental evidence for the transient behavior of individual cells evaluated on single-cell test stands and stacks contained within full systems. Results from models developed to help understand some of the detailed physical effects that influence cell performance are also examined. The goal is to introduce, albeit in brief, some of the principle dynamic characteristics of single cell and full stack performances. Single cell studies are important since they isolate the cell in a well understood and controlled environment thereby removing the effects of other external processes (e.g., reforming) which may have their own transient behavior that affects cell operation (e.g., controlling cell input fuel composition). Such
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Fig. 9.1 Transient behavior of a YSZ-Pt point electrode undergoing anodic (0.1 V versus air) and cathodic (−0.15 V versus air) step changes in voltage (Bay and Jacobsen, 1997).
isolated transient cell studies allow the detailed analysis of internal cell processes (e.g., diffusion, electrochemistry, and ohmic losses). In the end, however, full system studies must be performed since they provide the data that characterize the full performance of the unit under development for commercialization. As discussed in more detail below, the analysis of stack transient data is complicated by the inherent coupling with the numerous other system components.
9.2.1 Individual Electrode and Cell Performance Figure 9.1 shows the long-term transient behavior of a yttrium stabilized zirconia – platinum (YSZ-Pt) ‘point electrode’ in air at 1000◦C when subject to step potential changes (Bay and Jacobsen, 1997). Results for both anodic (i > 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
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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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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.
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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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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,
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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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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
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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
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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., <850◦C) are sufficiently slow so as to be negligible. Table 9.1 shows that the convective transport has a timescale on the order of 10−1 s. This time is the time it takes for gas to pass through the cell. Hence, for calculation times T much shorter than this duration, convective transport can be assumed stationary (constant), and Equations (9.10) and (9.11) can be ignored. On the other hand, for time steps t longer than this duration, the quasi-steady formula (where the left hand sides to (9.10) and (9.11) are both zero) can be used. For most cell performance problems, these model equations for the gas phase are normally integrated with the model equation for the solid components (e.g., cell and interconnect) which is presented next.
9.3.3.4 Solid Body Thermal Energy Transport The only dynamic phenomena of interest within the solid materials of an SOFC (cell, interconnects, end plates, etc.) is thermal energy transport. Here, the same formulation used for the gas-phase, Equation (9.10), can be used, except that here the velocity, V , is zero. While the solid body components do not show chemical reactions, Ri , there is internal heat generation through electrical ohmic losses. Hence, the model equation is: qi dAi dρCT = + G . (9.12) dt Af dx Table 9.1 shows that the solid body thermal transport has a timescale on the order of 103 s. This time is the time it takes for the temperature of a cell to reach equilibrium following a thermal perturbation (as can be induced, for example, by a load change). Hence, for calculation times T much shorter than this duration, the change in temperature of the SOFC can be assumed zero, and Equation (9.10) can be ignored. On the other hand, for time steps t longer than this duration, the quasi-steady formula (where the left hand side to (9.10) is zero) can be used. The various subsections given above present the basic individual transient physical processes occurring in fuel cell operation. Any of these may be included with the others to develop a complete dynamic model for a given study. In the next section we show several examples of how this can be done, starting form the simplest problem to more detailed problems. Applications of these models are given in Section 9.5.
9.4 Example Models As already mentioned in this chapter and elsewhere, there are different levels (details) of modeling commonly used (Bistolfi et al., 1996). The selection of any one particular level depends on the scale of the problem being studied (e.g., full plant
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Fig. 9.7 Thermodynamic analysis of an SOFC. ‘CV’ represents the control volume for the analysis.
(Padulles et al., 2000; Petruzzi et al., 2003; Ferrari et al., 2005) versus single cell (Achenbach, 1994, 1995; Haynes, 2002; Ivers-Tiffeé et al., 2004; Xue et al., 2005; Aguiar et al., 2005). In this section we begin with a simple lumped model for the thermal dynamics of an SOFC. Such models are used in the study of full plant dynamics where simplicity and speed for each submodel is key to providing timely results. Further, when coming to dynamic modeling for the first time, it is wise to start with the simplest of models, and only later add more complexity. Such an approach offers lower probability of generating programming errors, and enables the development of a useful model for application and study more quickly.
9.4.1 Lumped Thermodynamic SOFC Model One of the most important SOFC behaviors considered today is thermal dynamics. In this section we analyze the fuel cell via one of the most simplified methods? thermodynamic analysis. While we sacrifice some details, one key strength to this simplified approach is how, as we will see below, it brings clarity to how the various process parameters affect the thermal behavior of a fuel cell. We will arrive at a single analytic equation that relates key problem variables to the thermal behavior of the cell. Another strength of the model is the speed of solution, which may be required for some applications that are focused on the thermal response of a system. To begin, we select a control volume around the entire cell and consider the transient thermal behavior via a conservation of mass and energy. The calculated outputs from the model will be the SOFC temperature versus time. Figure 9.7 shows a schematic of the problem to be analyzed. The cell voltage, Ec , is assumed given as:
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Ec = EN − R(I )I,
(9.13)
where EN is the ideal Nernst potential (based on the quasi-steady average composition (Bove et al., 2005)), R(I ) is the known (say via experimental measurement) current dependent resistance of the cell, and I is the load current. The energy equation for this control volume is obtained from Equation (9.10) after being integrated in the x-direction over the length of the cell: d(MC T¯c ) = [mh] ˙ in − [mh] ˙ out − Pgen = m ˙ in (g + T s)in − m ˙ out (g + T s)out − Pgen , dt (9.14) where MC is the heat capacity of the internal cell materials within the control ˙ air hair is the total enthalpy flow due to both fuel and volume, mh ˙ = m ˙ f hf + m air entering/leaving the control volume, and Pgen is the generated electrical power removed from the control volume. While the above equations can together be used to provide a temporal solution to the thermal problem, a slightly simplified solution will be derived as follows. From conservation of mass and an identity for the exit Gibbs free energy, we have: m ˙ in = m ˙ out = m; ˙ gout = g(Tout , Xout ) = g(Tin , Xout ) + (g(Tout , Xout ) − g(Tin , Xout )).
(9.15)
The functional notation for the Gibbs free energy (g(T , X)) shows that it is only a function of temperature and mole fractions at the given location. No pressure term is shown in the function for the Gibbs free energy, since pressure is assumed constant for this analysis. The energy equation now becomes: d(MC T¯c ) = −m[g(T ˙ in , X] dt − m[g(T ˙ ˙ s)] − Pgen . (9.16) out , Xout ) − g(Tin , Xout )] − m[(T Here, the operator describes the change in the associated parameter from outlet to inlet. The first term on the right hand side (which is the change in Gibbs free energy due to isothermal electrochemical reaction) describes the maximum theoretical work that is possible from the process. If we subtract from this the actual work, Pgen , we have the internal heat generation due to all internal loss mechanisms: 2 ˙ Qgen = −m[g(T in , X)] − Pgen = I R(I ).
(9.17)
Using this equation for Qgen and the identity g = h − T s, we finally have: I d(MC T¯c ) = I 2 R(I ) − (Tin s) − m[h(T ˙ out , Xout ) − h(Tin , Xout )]. dt nF
(9.18)
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The second term on the right is a heat generation/sink term given by the electrochemical reaction (so called fundamental entropy heating which is evaluated at the inlet temperature). It is positive or negative depending on the sign on the entropy change, s, of the overall fundamental electrochemical reaction (e.g., H2 + 12 O2 = H2 O). By lumping all heat generation terms into parameter, Q(I ), we have: d(MC T¯c ) = Q(I ) − m[h(T ˙ (9.19) out , Xout ) − h(Tin , Xout )]. dt Equation (9.19) is the most basic equation for the dynamic thermal analysis of an SOFC cell/stack. As one would expect, the equation shows that under steady state, the change in enthalpy from inlet to exit is equal to the internal heat generation within the system. Aside from the original assumption of a lumped analysis, thus far there have been no other assumptions or approximations to the model. The model relies completely on basic thermodynamic principles, a known cell performance R(I ), and rigorous mathematical operations. To solve the model, we need to know the bulk mass and heat capacity of the cell, M and C, respectively; the reactant supply flow rate (m ˙ = fuel flow + air flow); the inlet temperature and pressure and the change in stream composition due to the electrochemical reaction, X, so that the change in enthalpy can be calculated; the electrical load current, I ; and the inlet and exit temperatures, Tin and Tout. We apply our model to an SOFC operating with fixed flow inlet conditions at 750◦C, 1 ATM, anode fuel of pure H2 , and pure air at the cathode inlet. The assumed initial conditions are the steady state conditions for zero electric load current. At time t = 0 the load current on the SOFC is increased from 0A to 100A, after which the steady state fuel and air utilization are Uf = 80% and Ua = 15%, respectively. The chemical reaction is given by H2 + 12 O2 → H2 O. Therefore, the exit composition (in mole fraction) of each stream at steady state is given by, respectively: XH2 out = 1 − XO2
out
=
I /(2F ) ; m ˙ H2 in /MWH2
XH2 O out = 1 − XH2
−0.21m ˙ AIR in /MWAIR − I /(4F ) ; m ˙ AIR in /MWAIR − I /(4F )
(9.20)
out ,
XN2 out = 1 − XO2
out ,
(9.21)
where MW is the molecular weight. The total supply flow rate is given by: ˙ AIR = m ˙ =m ˙ H2 + m
MWAIR I MWH2 I + . 2F (Uf ) 4F (Ua )0.21
(9.22)
The change in entropy due to electrochemical reaction is: 1 s = sH2 O (Tin , P ) − sH2 (Tin , P ) + sO2 (Tin , P ) 2 = −55490.4 [J(kg molH2 )−1 K−1 ].
(9.23)
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R.S. Gemmen
The remaining unknowns are cell temperature, Tc , and exit temperature, Tout . Therefore one additional equation is needed to solve the thermal problem. Because this is a lumped model, we could decide to relate these two temperatures in several ways. One way is to assume that the gas exit temperature is the same as the cell temperature. This would imply that most of the heating occurs toward the front (inlet) of the cell and that by the time the gas exits the cell the outlet gas temperature and the cell temperature have both reached the same temperature. (Such an approach is not unreasonable given that for high fuel utilization cases most of the current (and therefore heat generation) occurs at the inlet to the cell.) Another way, which we will take here, is to assume that the cell temperature is an average between inlet and outlet temperatures: Tc = (Tin + Tout )/2. Later, we will compare this approach to a more detailed one-dimensional model approach and show the significance of taking on this assumption. To arrive at an analytical solution to Equation (9.19), we assume an ideal calorically perfect gas, h = Cp T , and now have (assuming steady inlet temperature conditions): d(MC(Tout + Tin )/2) d(MCTout/2) = = Q(I ) − mC ˙ p [Tout − Tin ], dt dt
(9.24)
where Cp is the exit mixture specific heat (Cp = mass-weighted specific heat of anode and cathode flows). The steady state solution to Equation (9.24) is: Tout_ss =
Q(I + ) + mC ˙ p Tin , mC ˙ p
(9.25)
where I + is the load current just after t = 0. The general time dependent solution is: t Cp m Tout (t) = Tout_ss + T exp −2 ˙ , (9.26) MC where T = Tout(0) − Tout_ss =
Q(I − ) − Q(I + ) , mC ˙ p
(9.27)
where I − is the load current just before t = 0. Therefore the cell temperature is: 1 Q(I + ) Q(I − ) − Q(I + ) t Cp m + exp −2 ˙ . (9.28) T¯c (t) = Tin + 2 mC ˙ p mC ˙ p MC Equation (9.28) shows that the time constant for this transient is proportional to MC/(2Cp m). ˙ Hence, larger cells (having more heat capacity) will have longer time constants, while cells with greater flow rates will have shorter time constants. Further, cells having larger increases in current (greater heat addition) will have larger temperature changes, while cells with greater flow rates will have smaller magnitudes of temperature change. While there are many independent design requirements for avoiding cell failure, one important objective is to have cells experience
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the smallest temperature changes and at the slowest rate possible. The basic analysis presented here would suggest that to achieve slow temperature changes fuel cell designers should consider having as large as possible heat capacity cells (e.g., thick interconnects). Going to higher flow rates also helps to reduce the magnitude of temperature change. Note, however, that higher flow rates also speed up the changes, and overall does not control the magnitude of the rate of change of cell temperature (|dTc /dt|). That is, the same peak temporal gradient occurs regardless of the flow rate, but the peak will occur for a shorter period of time at higher flow rates. A sample case for this model is analyzed and compared to the coupled lumped model results later in Section 9.5.1. In closing, we see that certain simplified models are beneficial, especially when they permit analytical solutions to be derived showing the influence of all key governing parameters. As seen in the above analysis, however, certain information is unavailable such as information on the transient cell voltage. In addition, effort is needed to independently assess the overall cell resistance, R(I ), which, in general, is also a function of other parameters (reactant concentration, temperature, and pressure). This can be remedied by adding additional equations as the following section shows.
9.4.2 Coupled Lumped SOFC Thermal Model We consider here Equations (9.10), (9.11) and (9.12) in the analysis of a coupled lumped SOFC. Because we now resolve the anode and cathode regions in this analysis (thereby temporally resolving the reactant concentrations, pressures and temperatures), the cell voltage versus. time can also be determined (using the Nernst equation). The anode gas phase lumped model is provided by integrating Equations (9.10) and (9.11) in the x-direction: dρCp T¯ L = [ρCp T V ]in − [ρCp T V ]out + dt d N¯ i L = ni·in − ni·out + dt
qi Ai , Af
(9.29)
si Ai . Af
(9.30)
Here, for the time being, we explicitly assume that no internal energy is being generated through chemical reactions (Gg = 0). Similar model equations are used for the lumped cathode gas model. The solid body thermal equation is: dρC T¯ L= dt
qi Ai ¯ L. +G Af
(9.31)
Two such solid body equations are needed, one for the cell and one for the interconnect. In the above equations, a bar above a given variable, e.g., T¯ , indicates the
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spatial average (lumped) temperature of the given component (solid, anode gas, or cathode gas) over the associated length (for T ) or area (for q and s ), and L represents the length of integration (i.e., the cell length) in the x-direction. The heat flux at any given surface is comprised of both convective and radiative components. The convective heat flux between the gas and solid is obtained through the use of a convective coefficient: qg = −qs = h(T¯s − T¯g ),
(9.32)
where subscript g can represent anode (a) or cathode gas (c), and subscript s can represent cell (c) or interconnect (i). The convective coefficient, h, can be found from engineering textbooks (Mills, 1995) for the type of channel geometry used. As shown for laminar, fully developed flow, the square duct Nusselt number is 3.66, and it increases with increasing channel aspect ratio. The heat flux due to radiation transfer between the solid cell and interconnect bodies is given by: qc = −qi = σ Fv (T¯i4 − T¯c4 ), (9.33) where σ is the Stefan–Boltzmann constant, Fv is the radiation view factor, assumed to be ∼ 1 given the aspect ratios (channel thickness)/(channel length) for this problem. The mole flux of species at the bounding surfaces of the gas channel is governed by Faraday’s law: (9.34) sH 2 = −sH 2 O = −I /(A2F ), sO 2 = −I /(A4F ).
(9.35)
The above equations allow us to solve for Tc , Ti , Tan , Tca , NH2 , NH2 O , NO2 (bars have been dropped) as a function of time. As in the model of the prior section, an additional assumption is needed; namely, that the ‘lumped internal’ conditions for the anode and cathode gases will be the average of the inlet and outlet values. The given parameters for this analysis are all the cell design parameters (geometry, materials, properties, etc.), the input temperatures, pressures, mass flow, and compositions of the anode and cathode gases, and the load current on the cell. Such a simple set of coupled ordinary differential equations is readily solved via Matlab-Simulink, and a sample case is presented in Section 9.5.
9.4.3 Flow-Wise One-Dimensional Electrochemical Energy Conversion The next level of transient modeling improves the spatial accuracy beyond the lumped analysis given in prior sections by breaking up the original control volume that previously spanned the entire cell into numerous additional control volumes, or nodes (see Figure 9.8). However, for purposes of computational efficiency, it is noted that once the number of computational nodes becomes greater than ∼10, additional
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Fig. 9.8 Finite control volume layout for the one-dimensional model.
nodes provide only minor changes to the predicted fuel cell performance (Gemmen et al., 2000). One of the main benefits gained by resolving the streamwise conditions is that we will no longer have to estimate the averaged (lumped) properties as we did for the lumped models. For example, estimates for the average gas conductivities were required to calculate the average anode and cathode heat transfer coefficients. Gas conductivity depends on both temperature and gas composition, and both temperature and composition change significantly in the streamwise direction making it difficult to accurately assess the average conductivity. Further, since the overall transport & electrochemistry process is non-linear, finding proper model-average properties becomes very difficult – using simple spatial averaging for all properties will not likely be generally viable. To achieve the one-dimensional model, the exact same model equations presented in the coupled lumped model are employed for each of the individual control volumes shown in Figure 9.8. The only additional model features needed are to relate the individual axial nodal currents, ii , to the total current (which is simply given by I = ii ), and to derive a model for the individual nodal currents. For the latter, the same techniques shown in prior chapters can also be used; specifically, we assume constant voltage over the entire cell (this assumes good electric conductivity of the electrodes), and then iterate on the nodal current, ii , to balance the total voltage, Vcell +Vloss(ii ), with the local Nernst voltage, EN,i (Vcell +Vloss(ii ) = EN,i ). This is the most computationally efficient approach, although total current can also be specified by the user in which case the afore-mentioned approach is used for iteration until the total current is achieved. While general solution packages such as Matlab-Simulink are good for solving simple problems having few coupled parameters, it is often more efficient to generate custom solvers for more complicated problems. For the examples shown in Sec-
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tion 9.5, the coupled differential equations were represented via finite differences and solved numerically through an explicit first-order forward Euler method. The model equations are solved at every time step based on the given instantaneous boundary conditions. The solution starts by solving all nodal electrochemical reaction rates (current densities) given the specified cell voltage (or total current). If a quasi-steady activation loss is assumed (i.e., no double-layer dynamics to be solved), an iterative approach is used to determine the cell current densities at each node – node current at each time step is iterated so as to ensure a uniform cell voltage (e.g., to within 4 microvolt). The balance equation to be iterated at each node is: Vcell = EN (I ) − ηj (I ). The Nernst parameter, EN (I ), is a function of node current, I , through the consumption of reactants with I . The loss terms, ηj , are the ohmic, concentration, and electrochemical over-potential, all of which are functions of node current. A combination Newton and simple bisection method is used to converge to the desired solution. Once the current is known at each node, the dynamic equations are stepped forward one time step for all nodes. This approach allows the transient solution for the one-dimensional gas and solid bodies that comprise a fuel cell. Any number of control volumes are possible along the streamwise direction for each of the four major components: fuel electrode gas, air electrode gas, cell, and separator plate. Because of the separation between cathode and anode, this one-dimensional model approach is not limited to the co-flow configuration shown in Figure 9.8. Provided conserved transport of oxygen ions from the cathode to the anode is achieved, as well as heat transfer across components, this approach can solve for any geometry: co-flow, counter-flow, and crossflow. For the cross-flow geometry, an array of fuel electrode channels and air electrode channels are generated and assembled to form such a cell along with associated two-dimensional arrays for the cell and interconnect nodes (Ahmed et al., 1991). For the co-flow and counter flow geometries, only one channel of both fuel electrode and air electrode gas is required.
9.4.4 Transverse One-Dimensional Diffusion Transport though the Electrode While not of primary concern for cell performance prediction, dynamic modeling of the transport through the electrode has been used to assess the impact of such things as inverter ripple on cell conditions (Gemmen, 2003). Other needs for such modeling may also arise as fuel cells become more widely applied. The problem under consideration is shown schematically in Figure 9.9. The governing equation for this problem was given previously in Equation (9.8), where the species fluxes, ni , are determined from Equation (9.7). Equation (9.7), for i = 1 to J − 1, provides a linear set of independent ordinary differential equations. To solve for all J components, an additional equation is needed. For this we use the conservation of species:
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Fig. 9.9 Electrode transport model setup.
nT =
J
ni .
(9.36)
j =1
Equations (9.7) and (9.36) can be used to solve for the spatial distribution of mole fluxes and mole fractions for all J components. For the solution, we note that Equation (9.7) can be recast into matrix notation as (Solcova et al., 1999): H · n + Ng = 0, (9.37) where n = {n1 , n2 , n3 , . . . , nJ }T is the vector of gas phase mole fluxes (molecular + permeation transport). The components for the submatrix J − 1 × J of H are given by: ⎫ J ⎪ 1 yi Bi ym ⎪ Hii = K + K + ; (m = i) ⎪ ⎪ ⎪ ⎪ Dim Di Di ⎪ ⎪ m=1 ⎪ ⎪ ⎪ yi Bi yi ⎪ ⎪ Hij = L − ; (i = j ) ⎪ ⎬ Dim Dj , (9.38) ⎪ B0 /DiK ⎪ ⎪ ⎪
Bi = − ⎪ ⎪ ⎪ 1 + B0 Jm=1 DymK ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ε p 2 ⎪ ⎭ B0 = r τ 8µ
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where p [Pa] is the pressure and µ [Pa·s] is the mixture dynamic viscosity. The above submatrix to H provides an independent set of transport equations derived from Equation (9.7). To provide the final J th independent equation, the last row of H is given by: HJj = 1. (9.39) Finally, the vector Ng is given by: Ng = {dN1 /dx, dN2 /dx, dN3 /dx, . . . , dNJ −1 /dx, −nT }T ,
(9.40)
which is the vector of gas phase mole density gradients, but with the last element being the negative of the total mole flux. Following the same procedures outlined in the prior section, the conservation equation, Equation (9.8), was represented in finite difference form: ε
k−1 k (Ni,m − Ni,m )
t
=−
k−1 (nk−1 i,m+1 − ni,m−1 )
2x
+ Rik−1 ,
(9.41)
where t is the time step, x is the spacing between nodes. The superscript k indicates the time index. To avoid oscillations in the numerical solution, the locations for node properties (e.g., specie concentration, temperature and pressure) are offset by 1 2 x from the flux node locations following (Patankar, 1980). The gradient term in Equation (9.8) is given by central differencing (as shown), except at the end nodes where forward or backward differencing is required. The specified boundary conditions are specie concentration values at the freestream and reactant consumption at the electrolyte (see Equations (9.34) and (9.35)). Initial conditions are given for all parameters at t = 0. To integrate in time, at any time t an L-U decomposition matrix solver is used to solve for specie fluxes, ni, in Equation (8) given known distributions of specie concentrations, Ni . The specie concentrations at t + dt are then calculated via Equation (9.41) by explicit time integration.
9.5 Model Applications & Results As already mentioned in this chapter, there are a variety of tools available to aid in the solution of dynamic models. Depending on the scale of the problem (e.g., full plant versus single cell) different tools can be selected. In this section we apply the above models to several different problems, and demonstrate the use of different solution methods.
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Fig. 9.10 Temperature history of cell components due to 55 amp step change in load. (a) Actual cell component temperatures during the transient; (b) the difference between results when radiation is included and when it is ignored for the same transient.
9.5.1 Cell Heat-up on Load Change – Coupled Lumped Model Analysis Matlab-Simulink was used to develop a solver for the ‘Coupled Lumped SOFC Thermal Model’ problem presented in Section 9.4.2. This solver was then applied to the problem of predicting the cell thermal transient due to a load change. The Simulink subsystems developed for this model are shown in Appendix A9.1, along with the list of model input parameters. The results for a 55 amp load change are shown in Figures 9.10 and 9.11. Figure 9.10 shows the temperature histories for the cell, interconnect, anode exit gas and cathode exit gas. As can be seen the thermal conditions reach their new equilibrium by approximately t = 800 s, resulting in an exponential time constant of τ = 266 s. The temperature change for the gas stream is about 150◦C. A calculation of the thermal time constant based on the fully lumped model (Equation (9.28) shows
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Fig. 9.11 Comparing ‘base’ cell concept with a cell having half a heat capacity and a cell having twice the flow.
τ = MC/(2Cp m) ˙ = 239 s, which compares reasonably well with the coupled lumped model results. Likewise, the temperature change determined by the fully ˙ p = 169◦ C, and this too agrees lumped model shows T = [Q(I − ) − Q(I + )]/mC reasonably well with the coupled lumped model results. Because radiation has been discussed in the literature as a significant contributor to heat transfer (Yakabe et al., 2001; Damm and Fedorov, 2005; Daun et al., 2006), the effect of radiation is briefly considered here. Yakabe et al., which performed a three-dimensional model analysis for an SOFC, found approximately 17 K difference in the average cell temperatures between results where radiation was included and when it was ignored. They also note that the addition of radiation flattens the temperature profile of the cell (a flattened profile is implicit for lumped models such as presented here). For the present lumped model results, Figure 9.10b shows the difference between results with radiation and those without radiation. As can be seen, the lumped model results show cell and interconnect temperatures having very minor differences whether radiation is included or not – cell temperature is slightly cooler and the interconnect temperature is slightly warmer. The differences between the results of Yakabe et al. and the present lumped model indicate that streamwise radiation is important for conditions where streamwise temperature gradients exist, and that for improved predictions (by circa 10%), gas phase radiation within the flow channels should be included in the model. For problems with small temperature gradients (as implicit within the lumped model), radiation has little affect. Finally, Figure 9.11 shows the effect of decreasing the heat capacity of the cell (say, by selecting a different material or reducing the size of the interconnect/cell), and the effect of increasing the flow rate. All other parameters of the model were held constant. The results show that by decreasing the heat capacity by 1/2, the time constant has likewise decreased by n´ as expected by the fully lumped model, Equation (9.28). The results also show that increasing the flow rate will both decrease the temperature change by 1/2 as well as the time constant, again in agreement with the fully lumped model. Such model-to-model comparisons are called ‘model verifica-
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Fig. 9.12 Node definitions for the cell, anode gas, cathode gas and interconnect calculation domains.
tion’, and such an exercise is helpful to confirm the viability of a newly programmed model (Iora et al., 2005) and is vital when experimental data is lacking.
9.5.2 Load Perturbation Study – One-Dimensional Model Analysis Here we solve the same problem as in Section 9.5.1, but now using the ‘OneDimensional Model’ developed in the C++ computer language. The node numbers for the calculation domain are shown in Figure 9.12. The setup parameters used for this model are shown in Appendix A9.2 (the parameter names are descriptive so as to define their usage). Unlike the lumped model, the one-dimensional model used a controlled input cell voltage. The cell voltage was specified to be the same as that of the lumped model results in Section 9.5.1 at steady state (circa 0.8 V). The initial conditions assumed zero load, and at time t = 0, the load was applied. The results of the model are shown in Figures 9.13 to 9.16. Figure 9.13 shows the cell temperatures. (Note, because of good heat conduction to the interconnect, and convection to the anode gas, these two parameters closely follow the cell temperature.) Immediately upon addition of the load, all cell nodal temperatures begin to rise. The temperatures of the nodes closer to the inlet rise more quickly due to the relatively high heat generation in the leading portion of the cell as shown in Figure 9.14. These same nodes also reach their steady state values more quickly. The cathode gas temperature nodal histories are shown in Figure 9.15 and follow the same behavior. The steady state temperature profiles of the cell and cathode gas are shown in Figure 9.16. Also shown on the figure are the profiles at 250 s and 500 s. As can be seen at early portions of the transient (e.g., 250 s) the cathode gas passing through the cell is first heated by the relatively warm cell temperatures, and is then cooled as it passes toward the cell exit by the relatively cool cell temperatures. By
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Fig. 9.13 Nodal cell (anode electrode+electrolyte+cathode electrode) temperature histories for the one-dimensional model.
Fig. 9.14 Nodal heat generation histories for the one-dimensional model.
the time the cell reaches steady state, the cathode gas has a monotonic increase in temperature as it passes through the cell. Because of the greater dimensionality of the problem being solved by the one-dimensional model, the behavior exhibited here is far more complex than that what can be predicted by the lumped model. Figure 9.17 compares the lumped model results to the one-dimensional model results. Recall that in the derivation of the lumped model, a relation between the inlet and exit gas temperature was required – see the discussion leading into Equation (9.24). For the lumped model, we assumed a simple averaging of Tc = (Tin + Tout )/2.
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Fig. 9.15 Nodal cathode gas temperature histories for the one-dimensional model.
Fig. 9.16 Temperature profiles at 250 s, 500 s, and steady state for the one-dimensional model.
The curve labeled 1-Dim–‘A’ in Figure 9.17 plots the instantaneous average of all nodes for the one-dimensional model results. The curve labeled 1-Dim–‘B’ plots the instantaneous average of the inlet and exit nodes for the one-dimensional model results. From Figure 9.17 we see that the lumped model results closely match curve ‘A’ only at the early portion of the transient, and they closely match curve ‘B’ at the steady state portion of the transient. These results indicate that the lumped model accurately predicts the average energy being absorbed over the cell at the early portion of the transient. At later times, the lumped model under predicts the average energy absorbed over the cell (this is consistent with the curvature in the steady state temperature profile shown in Figure 9.16.) Since the cell temperature follows
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Fig. 9.17 Comparison of the lumped model with the one-dimensional model results (‘A’ = average over all nodes; ‘B’ = average of inlet and exit nodes).
fairly closely the cathode gas temperature, the lumped model approaches the average of the inlet and exit temperatures (curve ‘B’) at steady state – conservation of energy requires that the difference in inlet and exit gas temperatures (∼energy) be proportional to the thermal energy generated within the cell. Since both models assume the same type of cell performance, both models result in similar temperature rise, and their average inlet+exit temperature values are closely the same. This model comparison highlights why it is important to understand the limitations of different models. While lumped models have their benefits for certain applications, the information they provide need to be carefully considered to avoid making improper conclusions.
9.5.3 Load Loss Studies It was mentioned previously that thermal transients are one of the more important considerations of solid oxide fuel cell developers. While this is true, reactant transients coupled with thermal transients can also be important to understand. The previous model applications examined the effect of load increase on cell thermal performance. In this section we examine the effects of severe load loss on cell operation (Gemmen and Johnson, 2005). The cell geometry under consideration is shown in Figure 9.18. Here, an array of one-dimensional models are used for both the cathode and anode. The initial conditions for the model are a steady state load applied to achieve an operating cell voltage of 0.70 V. Under these conditions a temperature rise across the cell exists due to the heat generated within the cell (see Section 9.5.2). At time t = 0,
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Fig. 9.18 10 cm × 10 cm cell geometry for transient load loss study (Gemmen and Johnson, 2005).
Fig. 9.19 Contour showing region of current reversal following load decrease (0.70 to 0.96 V) (Gemmen and Johnson, 2005). (Cross-flow)
the load was changed to provide a cell voltage of 0.957 V (close to open circuit). At this point negligible external load exist on the cell, and it is often assumed that no reactants are then being consumed. Figure 9.19 shows a contour plot of the cell current density immediately following this load transient. As the figure shows, a significant portion of the cell exhibits reverse current. This current is driven by the thermal gradient that exists on a cell when under load. This temperature difference supports a 1/2 Nernst potential gradient over the cell (EN ∼ E0 (T )+RT /nF ln(PH2 PO2 /PH2 O )). While the reactant concentrations are nominally the same over the cell when little consumption occurs, the temperature gradient can continue to support a Nernst potential gradient which circulates the current within the cell. In actuality, these very
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Fig. 9.20 Hydrogen concentration after unload event (Gemmen and Johnson, 2005). (Cross-flow)
Fig. 9.21 Simulink electrochemical model with R-C model capacitance behavior. The ‘BV Fcn’ block is the quasi-steady Butler–Volmer overpotential equation giving current through the Rct charge transfer resistor as a function of charge transfer overpotential, η.
currents result in a variation in local reactant concentrations (Figure 9.20) until a balance between the driving potential (EN ) and internal voltage losses is achieved. This analysis is an example of how modeling can help to enlighten fuel cell design-
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Fig. 9.22 Coupled lumped model with electrode capacitance.
ers as they consider the impact of different events on the cell operation. Prior to this modeling work, such behavior was unrecognized given the difficulty of obtaining accurate in situ measurements at SOFC conditions. Later to these modeling studies, experimental work demonstrated and confirmed that temperature differences can indeed drive current flow (Gemmen and Johnson, 2006).
9.5.4 Electrochemical Transient The Coupled Lumped Simulink model presented in Section 9.5.1 was extended to analyze for the double layer capacitance behavior using the model given in Section 9.3.3. The Simulink block for calculating the capacitance behavior is shown in Figure 9.21. Results for a load decrease and increase are shown in Figure 9.22. Figure 9.23 shows the same data for the load decrease case but over an expanded timebase to more clearly show the transient behavior due to the electrochemical capacitance. Similar results are found for the load increase case.
9.5.5 Electrode Species Transport Studies The final example problem to be considered is the electrode diffusion model shown in Section 9.4.4. This model was used to assess the effect of an oscillating load on the diffusion of reactants in an anode electrode. Depending on the power elec-
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Fig. 9.23 Same results shown in Figure 9.22 with expanded timebase for load decrease transient. This figure more clearly shows the short term capacitive ‘charging’ for the double layer.
tronics and electrical filtering used on a fuel cell system (e.g., inverters and dc/dc converters), such oscillating loads may be imposed on a fuel cell stack. The parameters used in the model are shown in Table 9.3. As shown, a square wave current load was applied for this problem having a frequency of 120 Hz. The freestream gas composition used is based on an entrained coal gasifier (Hoffman, 1981). Fortyone nodes were used through the electrode; where the 0th node was located at the electrolyte interface, and the 41st node was located at the freestream interface (see Figure 9.9). Results from the model analysis are shown in Figure 9.24. Because of a water gas shift reaction for CO, the mean concentration of hydrogen within the electrode increases from node 40 to node 30 (in the direction from freestream into the anode) as CO is converted to H2 . The concentration of hydrogen at the electrolyte interface (node 0) is found to have an 8% variation due to the oscillating current load. This variation decreases as the distance from the electrolyte increases, and the oscillations away from the electrolyte are phase-shifted slightly (lag) relative to the conditions at the electrolyte. Because of these significant levels of oscillation in reactant concentration at these conditions, fuel cell designers might apply filtering as part of the power conditioner design, and/or operate the system at other conditions (e.g., higher frequency helps to reduce these effects.)
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Table 9.3 Dynamic electrode model parameters. Parameter CO (vol. %) H2 (vol. %) CO2 (vol. %) H2 O (vol. %) N2 (vol. %) CH4 (vol. %)
29.1 28.5 11.8 27.6 2.1 0.009
Electrode thickness, L Tortuosity, τ Porosity, ε Permittivity, = ε/τ Mean pore diameter, r Operating temperature, T Operating pressure, P Average current load, i Square wave amplitude Square wave frequency
0.002 m 3.6 0.56 0.156 1.07 µm 800◦ C 101000 Pa 300mA/cm2 300mA/cm2 120 Hz
Fig. 9.24 Effect of square wave current load on reactant diffusion through the anode electrode. Node ‘0’ is at the electrolyte interface, and node 40 is at the freestream interface.
9.6 Closing The goal of this chapter was to open the door to dynamic model analysis of fuel cells. Many of the key known capacitance elements in fuel cell operation were identified and model equations derived. Through the use of these models the following key issues for dynamic modeling have been identified:
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1. A wide range of timescales exist given the wide range of physical phenomena and length scales that exist in fuel cells. 2. To generate the most effective model for a given transient problem, it is wise to work to simplify a dynamic model by identifying which timescales can be ignored. 3. For simple lumped problems, as often used in control studies, packaged solvers such as Simulink can be beneficial. 4. For larger more complex problems, a dedicated custom solver can be a better approach in order to control the solution methods and speed of solution. 5. Comparing the results from simple to relatively more complex models is wise to ensure that proper modeling and analysis is achieved. 6. Finally, dynamic modeling provides unique insights to cell and system behavior that cannot be provided by other (steady state) modeling. Such modeling can be used to both support/confirm/explain already existing experimental results, as well as lead in identifying complex fuel cell dynamic behavior previously unrecognized.
Appendix A9.1 – Coupled Lumped Simulink Model
Fig. A9.1 Main Simulink SOFC model page.
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Fig. A9.2 Simulink ‘CoupledLumpedCell’ submodel page.
Fig. A9.3 Simulink ‘ThermalModel’ submodel page.
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Fig. A9.4 Simulink ‘Anode H2/H2O Conservation’ submodel page.
Fig. A9.5 Simulink ‘Cathode O2 Conservation’ submodel page.
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Fig. A9.6 Simulink ‘CellVoltage’ submodel page.
Fig. A9.7 Simulink ‘CellVoltage’ submodel page.
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Fig. A9.8 Simulink ‘Diffusion Losses’ submodel page.
Fig. A9.9 Simulink ‘ElecChemOverPot’ submodel page.
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Fig. A9.10 Simulink ‘LoadBox’ submodel page.
Fig. A9.11 Simulink ‘RadiationFromCell’ submodel page.
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Fig. A9.12 Simulink ‘Tan’ submodel page (anode gas energy conservation; a similar form exists for the cathode).
Fig. A9.13 Simulink ‘Tcell’ submodel page (cell energy equation conservation; a similar form exists for the interconnect).
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Appendix A9.2 – Parameters Used for One-Dimensional Model Analysis
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References Achenbach E. (1994) Three-dimensional and time dependent simulation of a planar solid oxide fuel cell stack. Journal of Power Sources 49, 333–348. Achenbach E. (1995) Response of a solid oxide fuel cell to load change. Journal of Power Sources 57, 105–109. Adler S.B., Wilson J.R., Schwartz D.T. (2003) Nonlinear harmonic response of mixed-conducting SOFC cathodes. In Solid Oxide Fuel Cells VIII (SOFC-VIII), Electrochemical Society Proceedings, 2003-07, S.C. Singhal and M. Dokiya (Eds.), The Electrochemical Society, Pennington, NJ, pp. 516–524. Aguiar P., Adjiman C.S., Brandon N.P. (2005) Anode-supported intermediate-temperature direct internal reforming solid oxide fuel cell II Model-based dynamic performance and control. Journal of Power Sources 147, 136–147. Ahmed S., McPheeters C., Kumar R. (1991) Thermal hydraulic model of a monolithic solid oxide fuel cell. Journal of the Electrochemical Society 138(9), 2712–2718. Bay L., Jacobsen T. (1997) Dynamics of the YSZ-Pt interface. Solid State Ionics 93, 201–206. Benhaddad S., Protkova I. (2005) Failure analysis as an essential component of an SOFC development program. In Solid Oxide Fuel Cells IX (SOFC-IX), Electrochemical Society Proceedings, Quebec PQ, 15–20 May, Vol. 1, Cells, Stacks, and Systems, S.C. Singhal and J. Mizusaki (Eds.), The Electrochemical Society, Pennington, NJ, pp. 923–930. Bistolfi M., Malandrino A., Mancini N. (1996) The use of different modeling approaches and tools to support research activities: An industrial example. Computers & Chemical Engineering 20, S1487–S1491. Bove R., Lunghi P., Sammes N.M. (2005) SOFC mathematicmodel for systems simulations Part one: from a micro-detailed to macro-black-box model. International Journal of Hydrogen Energy 30, 181–187.
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Damm D.L., Fedorov A.G. (2005) Radiation heat transfer in SOFC materials and components. Journal of Power Sources 143, 158–165. Daun K.J., Beale S.B., Liu F., Smallwood G.J. (2006) Radiation heat transfer in planar SOFC electrolytes. Journal of Power Sources 157, 302–310. Ferrari M.L., Traverso A., Magistri L., Massardo A.F. (2005) Influence of the anodic recirculation transient behavior on the SOFC hybrid system performance. Journal of Power Sources 149, 22–32. Gemmen, R.S., Johnson C.D. (2005) Effect of load transients on SOFC operation – Current reversal on loss of load. Journal of Power Sources 114, 152–164. Gemmen R.S., Trembly J. (2006) On the mechanisms and behavior of coal syngas transport and reaction within the anode of a solid oxide fuel cell. Journal of Power Sources 161, 1084–1095. Gemmen R.S., Johnson C.D. (2006) Thermal gradient induced current recirculation on load change in solid oxide fuel cells. In Proceedings of the ASME Conference on Fuel Cell Science, Engineering and Technology, June 19–21, 2006, Irvine, CA, USA, Paper No. FUELCELL200697187. Gemmen R.S., Liese E.A., Rivera J.G., Jabbari F., Brouwer J. (2000) Development of dynamic modeling tools for solid oxide and molten carbonate hybrid fuel cell gas turbine systems. In Proceedings of the 2000 IGTI Turbo Expo Conference and Exposition, May 8–11, Munich, Germany. Gemmen R.S. (2003) Analysis for the effect of inverter ripple current on fuel cell operating condition. Journal of Fluids Engineering 125, 576–585. Gupta G.K., Dean A.M., Hecht E.S., Zhu H., Kee R.J. (2005) The influence of heterogeneous chemistry and electrochemistry on gas-phase moleculear-weight growth and deposit formation. In Solid Oxide Fuel Cells IX (SOFC-IX), Electrochemical Society Proceedings, Quebec PQ, 15–20 May, Vol. 1, Cells, Stacks, and Systems, S.C. Singhal and J. Mizusaki (Eds.), The Electrochemical Society, Pennington, NJ, pp. 679-688. Haynes C (2002) Simulating process settings for unslaved SOFC response to increased load demand. Journal of Power Sources 109, 365–376. Hendriks M.G.H.M., ten Elshof J.E., Bouwmeester H.J.M., Verweij H. (2002) The electrochemcial double-layer capacitance of yttria-stablized zirconia. Solid State Ionics 146, 211–217. Hoffman E.J. (1981) Coal Gasifiers, The Energon Company, Laramie, Wyoming. Iora P., Aguiar P., Adjiman C.S., Brandon N.P. (2005) Comparison of two IT DIR-SOFC models: Impact of variable thermodynamic, physical, and flow properties steady-state and dynamic analysis. Chemical Engineering Science 60, 2963–2975. Ivers-Tiffeé E., Weber A., Schmid K., Krebs V. (2004) Macroscale modeling of cathode formation in SOFC. Solid State Ionics 174, 223–232. Jørgensen M.J., Mogensen M. (2001) Impedance of solid oxide fuel cell LSM/YSZ composite cathodes. Journal of the Electrochemical Society 148(5), A433–A442. Mason E.A., Malinauskas A.P. (1983) Gas Transport in Porous Media: The Dusty Gas Model, Chemical Engineering Monographs, Vol. 17, Elsevier, New York. Mills A.F. (1995) Basic Heat and Mass Transfer, Richard D. Irwin, Boston, MA. Padulles J., Ault G.W., McDonald J.R. (2000) An integrated SOFC plant dynamic model for power systems simulation. Journal of Power Sources 86, 495–500. Patankar S.V. (1980) Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences, W.J. Winkowycz and E.M. Sparrow (Eds.), McGraw-Hill Book Company, New York. Petruzzi L., Cocchi S., Fineschi F. (2003) A global thermo-electrochemical model for SOFC systems design and engineering. Journal of Power Sources 118, 96–107. Qi Y., Huang B., Chuang K.T. (2005) Dynamic modeling of solid oxide fuel cell: The effect of diffusion and inherent impedance. Journal of Power Sources 150, 32–47. Solcova O., Snajdaufova H., Hejtmanek V., Schneider P. (1999) Textural properties of porous solids in relation to gas transport. Chemistry Papers 53(6), 396–402. Warren L.W. (1969) Verification and application of the dusty gas model. PhD Thesis, University Microfilms Inc, Ann Arbor, MI.
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Xue X., Tang J., Sammes N., Du Y. (2005) Dynamic modeling of single tubular SOFC combining heat/mass transfer and electrochemical reaction effects. Journal of Power Sources 142, 211– 222. Yakabe H., Ogiwara T., Hishinuma M., Yasuda I. (2001) 3-D model calculation for planar SOFC. Journal of Power Sources 102, 144–154.
Chapter 10
Integrated Numerical Modeling of SOFCs Mechanical Properties and Stress Analyses H. Yakabe Product Development Department, Tokyo Gas Co. Ltd., Tokyo 116-0003, Japan
Nomenclature d h ht mm n p q r si sh t xi ui u˜ j √ g D E E0 Ei Fh,j G HA HF Hm HS J
distance between diffraction plane ≡ ht + mm Hm thermal enthalpy mass fraction of mixture consistuent m unit vector normal to the boundary piezometric pressure heat loss term reaction rate momentum source components energy source time Cartesian coordinate (i = 1, 2, 3) absolute fluid velocity component in direction xi relative velocity between fluid and local coordinate frame that moves with velocity ucj determinant of metric tensor diffusion constant Young’s modulus cell operating voltage Nernst potential at the ith cell diffusional energy flux in direction xj Gibbs energy total enthalpy of air in the channel total enthalpy of fuel in the channel heat of formation of constituent m enthalpy in the electrolyte current density
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K L MA MF R S Si Q T ρ τij σ ε ν λ κ η δ ψ
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equilibrium constant length of a sample total mass of air in the channel total mass of fuel in the channel resistance entropy rate of production or consumption of species i heat source term temperature density stress tensor component stress strain or emissivity Poisson’s ratio thermal conductivity heat transfer coefficient overpotential oxygen vacancy tilting angle
10.1 Overview of Stresses in SOFCs 10.1.1 Introduction to Stresses in Single Cell and Stack Solid oxide fuel cells (SOFCs) are composed of ceramic components and the mechanical reliability of these components is a significant issue in the development of SOFCs. In particular, the durability of the components against stresses during operation is a serious problem. To evaluate the mechanical reliability of SOFCs, the magnitude of the stresses in cell components and the strength of the components against the stresses must be considered precisely. Stresses in SOFCs are categorized into four types. 1. A thermally induced residual stress. The origin of the residual stresses is a mismatch of thermal expansion behaviors among the components. Rigid connection of each component with different thermal expansion coefficients causes residual stresses. For example, the electrolyte and electrodes are fabricated and connected at a high temperature. If the thermal expansion behaviors are not identical among the components, residual stresses will occur in the cell at room temperature. For stacks, similar residual stresses will occur by a mismatch of thermal expansion behavior among cells and other stack components. 2. A thermally induced stress.
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Fig. 10.1 Pictures for the different designs of SOFCs and the related cross-sectional diagrams; (a) electrolyte-supporting planar cell, (b) anode-supported planar cell, (c) tubular cell, (d) segmented cell in-series design.
In procedures for switching the SOFCs on or off, or during their operation, a temperature distribution occurs in cells or cell stacks; this temperature distribution causes thermal stresses inside the cell stacks. 3. A Physically or chemically induced stress. Due to changes in the temperature or ambient conditions, occasionally, there is a change in the physical properties of the cell components. For example, it is known that the structure of 3YSZ, which is used as the electrolyte, undergoes changes in its structure between monoclinic and orthorhombic at several hundred ◦ C [1]. Ni in the anode is in a metallic state under a reducing gas and in an oxidized state under an oxidizing gas. It is known that the structure of doped-LaCrO3, which is an interconnector, changes between orthorhombic and rhombohedral at a certain temperature [2]. In addition, its lattice constant changes with regard to the ambient partial pressure of oxygen [3]. These changes in the physical state due to the ambient conditions induce stresses inside the cell components. 4. A stress from mechanical loads on cell components. In the case of cell stacks of planar cells in particular, many cells are piled vertically to obtain a high voltage and a mechanical load is sometimes placed on the cells on top of the stack. This load induces a stress inside the cell stacks. These stresses affect the mechanical reliability of SOFCs and sometimes result in the destruction of cells or stacks. Therefore, reducing stresses and improving the durability of the stack components against these stresses are significant objectives in the development of SOFCs.
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10.1.2 Cell and Stack An SOFC consists of a solid electrolyte and electrodes. Typical materials for the electrolyte, cathode, and anode are yttria stabilized zirconia (YSZ), LaMnO3, and a cermet (a composite of Ni and YSZ), respectively. A single cell generates about 1 V under the open circuit conditions, and hence, a large number of single cells have to be connected in series to obtain a large output voltage. The connected cells are called a “module” or a “stack”. Several different designs (such as electrolytesupporting, anode-supported, Tubular design, and segmented cell in-series design) depending on the cell geometry are under development. Figure 10.1 shows crosssectional diagrams of the different designs of SOFCs and related pictures. (a) Planar design (electrolyte-supporting) The planar cell has a simple basic design. A characteristic feature of the electrolytesupporting cell is that a thick electrolyte supports both the electrodes. The thick electrolyte leads to a large ohmic resistance at the electrolyte. The ionic conductivity of the electrolyte increases with temperature and hence, it must be operated at a high temperature, for example, around 1000◦C. Ceramic interconnectors and auxiliaries are typically employed to stack the cells. Since all the components are ceramics, the reliability of the cell stacks against thermal stresses is a serious concern and an important design goal. Recently, Mitsubishi Material Co. has developed several kW systems using doped LaGaO3 as the electrolyte [4]. By using the doped LaGaO3 , which shows a high ionic conductivity of oxygen ions at a lower temperature, the cell operating temperature is reduced below 800◦C and an alloy can be used as the interconnector. (b) Planar design (anode-supported) To solve the problem in the electrolyte-supporting cell, anode-supported cells have been developed. Since a thin electrolyte is fabricated on a thick anode, the ohmic resistance of the electrolyte can be reduced. The low resistance of the electrolyte implies that the anode-supported cell can be operated at a low temperature below 800◦C. As a result, commercial alloys can be used as the interconnectors and auxiliaries. Using the alloys improves the mechanical reliability of the cell stack. For several years, Versa Power Systems have developed several kW systems with anode-supported cells and have demonstrated the performance of such systems [5]. (c) Tubular design Thus far, tubular design cells are the most reliable cells against thermal stresses and chemical degradation of the performance. Siemens Power Generation has demonstrated the performance and reliability of SOFC systems with tubular cells [6]. The 100 kW CHP system has been operated for more than 20000 h, and it was reported that no degradation of the system was observed. To prove the feasibility of a GT/SOFC hybrid system, a 220-kW hybrid system with an SOFC generator integrated with a micro gas turbine has been demonstrated at the University of California; this system has shown a conversion efficiency of around 49% (HHV).
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(d) Segmented cell in-series design The segmented cell in-series design is a unique concept for achieving a high output voltage of SOFCs. Many cells are fabricated on an electrically insulating substrate, and each cell is connected in series. Although the cell stack must be fabricated using a special technique, a high voltage is obtained easily and many disadvantages in assembling the cells are solved [7].
10.1.3 Mechanical Properties of Cell and Stack Components Typical materials for the electrolyte are YSZ, samaria doped ceria (SDC), and LaGaO3 . The intrinsic property of the thermal expansion behavior of an electrolyte depends only on the material species. However, the other mechanical properties (Young’s modulus, Poisson’s ratio, and strength) depend on the morphology through the manufacturing processes. Accordingly, the reported mechanical properties are not unique. The reported thermal expansion coefficient (TEC) and other mechanical properties for the electrolyte materials are listed in Table 10.1. Presently, only a few materials are used as the anode, and metal Ni is conventionally used as the anode material. In order to ensure that the TEC of the electrolyte and anode agree, a cermet (a composite of Ni and YSZ) is used for the anode. With increasing YSZ ratio, the TEC of the anode becomes close to that of YSZ but the electric conductivity of the anode decreases. Usually, 40∼60 molar% of YSZ is mixed with Ni. Certain oxides with a perovskite structure are generally applied to the cathode. For high temperature type SOFCs, doped-LaMnO3 is used as the typical cathode. For low temperature SOFCs, LaSr(CoFe)O3 or La(NiFe)O3 are used as the cathode. Doped LaCoO3 has a high electric conductivity and shows an excellent catalytic performance. However, the TEC of LaSrCoO3 is larger than that of the electrolyte, and so Fe is substituted to reduce the TEC of the cathode.
Table 10.1 Mechanical properties of YSZ.
3YSZ 8YSZ 8ScSZ GCO
TEC
E (Elastic constant) (GPa)
ν (Poisson’s ratio)
Mechanical strength (MPa)
11.5a (from R.T. to 1000◦ C) 10.5a (from R.T. to 1000◦ C) 10.7c (from R.T. to 1000◦ C) 12.96d (from R.T. to 1200◦ C)
207b 180b
0.33b 0.24b
200d
0.328d
960b 236b 275c 120∼169d
a Measured
with a dilatometer. from Ref. [48]. c Taken from Ref. [49]. d Taken from Ref. [50]. b Taken
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For high temperature SOFCs, a conventional commercial alloy cannot be used as the interconnector because they are easily oxidized under a high temperature oxidizing atmosphere. Therefore presently, special rare metals or doped-LaCrO3 are used as the interconnector. For low temperature SOFCs, a commercial alloy can be used as the interconnector and thus, various alloys are considered as the interconnector. Austenite stainless steels have a high TEC in comparison to YSZ, and only ferritic stainless steels are considered as a candidate for the interconnector.
10.2 Procedure for Integrated Modeling 10.2.1 Introduction to Stress Simulations The fundamental equations treated in structural analyses are the mechanical equilibrium, strain-displacement relation, and stress-strain relation. The equilibrium equations in an elementary volume can be expressed: ∂τxy ∂τzx ∂σx + + + X = 0, ∂x ∂y ∂z ∂σy ∂τyz ∂τxy + + + Y = 0, ∂x ∂y ∂z ∂τyz ∂σz ∂τzx + + + Z = 0, ∂x ∂y ∂z
(10.1)
where σi and τj i are the direct and share stresses along i-direction, and X, Y and Z are the body force components. In general, the strain vector {ε} is written using a differential operator matrix [A] and the displacement vector {U }: {ε} = [A]{U }.
(10.2)
Equation (10.2) can be rewritten as: ⎤ ⎧ ⎫ ⎡ ∂ εxx ⎪ ⎪ ∂x 0 0 ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎢ ∂ ⎢ 0 ∂y ⎪ 0 ⎥ εyy ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎥⎧U ⎫ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨ x⎪ ⎨ εzz ⎪ ⎬ ⎢0 0 ∂ ⎥ ⎬ ∂z ⎥ ⎢ ⎥ U =⎢ ∂ ∂ . y ⎪ γxy ⎪ ⎪ ⎢ ⎪ 0 ⎥ ⎩ ⎭ ⎪ ⎥⎪ ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎥ Uz ⎪ ⎪ ⎢ ⎢0 ∂ ∂ ⎥ ⎪ γyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ∂z ∂y ⎪ ⎪ ⎩ ⎭ ∂ ∂ γzx ∂z 0 ∂x
(10.3)
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The relationship between the stress and strain in ceramics is treated as an elastic condition. The total stress is defined from the total elastic strain as {σ } = [D el ]{εel },
(10.4)
where σ is the total stress, D el is the fourth-order elasticity tensor and εel is the total elastic strain. These three equations are solved in the structural analyses. Depending on the number of symmetry planes for the elastic properties, a material classified as either isotropic or anisotropic. The simplest form of linear elasticity is the isotropic case, and the stress-strain relationship is given as ⎤⎧ ⎡ ⎧ ⎫ ⎫ ν ν 1 0 0 0 1−ν 1−nu σ εxx ⎪ ⎪ ⎪ ⎪ xx ⎪ ⎪ ⎪ ⎪ ν ⎪ ⎪ ⎪ ⎪ ⎢ 0 0 0 ⎥ 1 ⎪ σyy ⎪ εyy ⎪ 1−ν ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎥ ⎢ 1 0 0 0 ⎥ εzz ⎬ E(1 − ν) σzz ⎢ = , 1−2ν ⎢ 0 0 ⎥ σxy ⎪ γxy ⎪ ⎪ ⎥⎪ (1 + ν)(1 − 2ν) ⎢ 2(1−ν) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ 1−2ν ⎪ σyz ⎪ ⎪ γyz ⎪ ⎪ ⎪ 0 ⎦⎪ ⎪ ⎣ symmetric ⎪ ⎪ ⎪ ⎪ 2(1−ν) ⎩ ⎩ ⎭ ⎭ 1−2ν σzx γzx 2(1−ν)
(10.5) where E is the Young’s modulus, and ν is the Poisson’s ratio. Thus the Young’s modulus and Poisson’s ratio of materials are necessary to solve the equations. Here, stress analyses are conducted using the finite element method. The principle of the stress analyses using the finite element method have explained in detail elsewhere [8]. The model is divided into finite elements. From Equation (10.1) the following equation can be derived: T T ¯ δ{ε} {σ }dV − δ{U } {F }dV − δ{U }T {T¯ }dS = 0. (10.6) V
V
S
{F¯ } is the body force vector per unit volume, {T¯ } is the surface force vector per unit area, V is the volume and S is the area. With a nodal displacement vector {d} the displacement vector {U } is written as {U } = [M]{d},
(10.7)
where [M] is the shape function matrix. Using Equations (10.2) and (10. 6), {ε} is expressed by: {ε} = [B]{d}, (10.8) where [B] is the strain-displacement matrix. From Equations (10.4) and (10.6–10.8), the following formula is achieved: T T el T T ¯ δ{d} [B] [D ][B]{d}dV − δ{d} [M] {F }dV − δ{d}T [M]T {T¯ }dS = 0. V
V
To satisfy Equation (10.9) for an arbitral δ{d},
S
(10.9)
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Fig. 10.2 The flow of the structural analyses for SOFCs. CFD and structural analyses are coupled and the results of the simulation using CFD are transferred to the structural analyses as the boundary conditions.
[B]T [D el ][B]dV {d} = V
S
[M]T {T¯ }dS +
[M]T {F¯ }dV
(10.10)
V
is required. Thus by applying Equation (10.10) to all the divided elements and summarizing them, the equilibrium equation is obtained. The flow of the structural analyses is illustrated in Figure 10.2. The stress simulation is coupled with the other simulations such as fluid dynamics, and all the simulations are integrated. Before performing the stress simulations for SOFCs, the origin of the stress must be made clear and the stress conditions of the cells or stacks must be clarified. The following four stress conditions are considered here: (a) Thermally induced residual stresses The initial and final conditions of a cell or a cell stack must be defined. A bi-layer of the electrolyte and electrode is fabricated at a high temperature and it is then cooled to room temperature. Thus, to calculate the residual stress in a cell at room temperature, the fabrication temperature and room temperature must be defined as the initial and final conditions. (b) Thermally induced stresses With respect to the thermally induced stresses in cells, a temperature distribution in the cell is calculated first. The thermal stress in the cell is then simulated using the
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data of the temperature distribution in the cell. The temperature distribution in the cell is calculated for steady-state or non-steady-state conditions. (c) Chemically induced stresses The origin of chemically induced stresses is a change of the lattice distance of materials due to changes in the physical state. For example, it is known that the lattice constant of LaCrO3 increases under a reducing atmosphere [3]. On the basis of the analogy between the thermal expansion and chemical expansion, the chemical expansion behavior can be simulated along with the thermal expansion behavior. (d) Stresses caused by mechanical loads Stresses are induced by piling the cells or binding each bundle. Mechanical loads or pressure are also considered in the stress simulations. During cell operation, thermally induced stresses are superposed on the stresses induced by the mechanical loads. This state can be treated by the stress calculations.
10.2.2 CFD Modeling for Cell Operation To simulate the thermally induced stresses in a cell, temperature distribution data in the cell must first be known. The cell performance and related temperature distribution in a cell during these operations are calculated using a computational fluid dynamics code.
10.2.2.1 Thermo-Fluid Model Here, the thermo-fluid analyses are performed using the computational fluid dynamics code “STAR-CD” (Computational Dynamics Ltd.) [9]. In STAR-CD, the algebraic finite-volume equations are solved. The solid and fluid parts are divided into small discrete meshes, and in each mesh, the following differential equations governing the conservation of mass, momentum, and energy are solved. • The mass and momentum conservation (the “Navier–Stokes” equations): 1 ∂ √ ∂ ∂p g ρuj + (ρ u˜ j uj − τij ) = − + si . √ g ∂t ∂xj ∂xi
(10.11)
• The enthalpy conservation equation: ∂ 1 ∂ √ g ρh + (ρ u˜ j h − Fhj ) √ g ∂t ∂xj 1 ∂ √ ∂p ∂ui = √ gp + τij + sh . g ∂t ∂xi ∂xj
(10.12)
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The specific heat of the fluid is treated as the polynomial function of temperature, while the thermal conductivity of the fluid and solid and the specific heat of the solid part are assumed to be independent of the temperature. For the whole model, the following mass and energy balances should be considered: • The mass balance of the total gas: dMA = MaIN − MAOUT + SiA , dt i
dMF = MFIN − MFOUT + SiF . dt
(10.13)
i
• The enthalpy balance: dHF = HFIN − HFOUT + QF − qF − qFEXT, dt dHA = HAIN − HAOUT + QA − qA − qAEXT, dt dHS = QS + qA + qF − qSEXT. dt
(10.14)
Cell stacks are usually operated at 700∼1000◦C, and at such high temperatures, a radiant heat exchange would have a significant effect on the heat exchange inside the channels. In STAR-CD, the radiant heat exchange can be treated as follows. To calculate the radiant heat exchange inside the channels, the boundaries of the calculation domain are all subdivided into contiguous non-overlapping areas called “patches”. A number of beams are emitted from the nominal center of each patch, and each beam is traced through the fluid until it intercepts an opposing patch. Gray body surfaces are assumed for small areas dAi and dAj , as shown in Figure 10.3, and the amount of radiant energy transfer to or from each patch is then calculated from the radiant transport equation: dqi−j = cos φi cos φj
dAi dAj εi εj (Ei − Ej ), πr 2
(10.15)
where dqi−j is the net energy exchange between patches i and j ; r is the distance between the patches; Ei and Ej are the radiation energy of the patches i and j , respectively; φi and φj are the angles between a normal to the surface and a line drawn between the patches; and εi and εj , the surface emissivity for the patches i and j . Using the Stefan–Boltzmann law E = σ T 4, where σ is the Stefan–Boltzmann constant, Equation (10.15) is rewritten as
(10.16)
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Fig. 10.3 Radiative heat exchange dqi−j between two domains dAi and dAj with different temperatures at the interconnector.
dqi−j = cos φi cos φj
dAi dAj εi εj σ (Ti4 − Tj4 ), πr 2
(10.17)
where Ti and Tj are the surface temperatures at the patches i and j , respectively. For all the patches, the net energy exchange given by Equation (10.17) should be calculated. When dealing with radiant heat exchange patches, the temperatures vary between high and low. When the heat exchange angles φi and φj in Equation (10.17) are close to π/2, dqi−j determined by Equation (10.17) would be a small value. If the hot and cold areas are far from each other in the cell, the contribution of the radiant heat exchange to the entire heat exchange becomes small, and hence, it is advantageous to neglect the radiant heat exchange in the calculation. This is because when the radiant heat exchange is considered, the amount of time needed to calculate the accurate result would be equal to ten times that for the calculation without considering the radiant heat exchange. In this case, the entire energy balance equation in the steady-state is simplified as follows: HAOUT + HFOUT = HAIN + HFIN .
(10.18)
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10.2.2.2 Electrochemical Model Since the STAR-CD does not include an electrochemical module at present, the electrochemical calculations on SOFCs are treated using subroutines produced inhouse. The reactions taken into account are given as follows: (at the anode) steam reforming reaction: CH4 + H2 O ↔ CO + 3H2 .
(10.19)
H2 O + CO ↔ H2 + CO2 .
(10.20)
H2 + O2− → H2 O + 2e− ,
(10.21)
CO + O2− → CO2 + 2e− .
(10.22)
shift reaction: electrochemical oxidation:
(at the cathode) 1 O2 + 2e− → O2− . (10.23) 2 Electrochemical reactions are assumed to be instantaneous at both the electrodes and occur at the interfaces between the electrodes and electrolyte. It has been reported that the shift reaction is fast enough [10] and in the present model, it is assumed to be chemical equilibrium at any point in the anode. For the steam reforming reaction (10.20) at the anode, some formulae describing the reaction rate have been proposed [11, 12]. To model the steam reforming reaction, it is better to measure the reaction rate for the methane reforming reaction using a practical Ni/YSZ cermet. In the present calculations, the following empirical formula is used as the reaction rate: −E 1.3 r(1) = k0 × exp × PCH × PH1.2 × ρan , (10.24) 4 2O RT where E is the activation energy; k0 , a constant; P , the partial pressure; and ρan , the density of the anode. The equilibrium constant for the shift reaction (10.20) is expressed as α − S0 α ln T β γ 2 −H0 Kshift = exp − + + T + T , (10.25) RT R R 2R 6R where H0 is the heat of reaction; S is the entropy change; and α, β, and γ are the coefficients of the temperature-dependent terms in the function for the specific heat .
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Fig. 10.4 Modeling of the electric circuit for the cell. A ladder circuit model is employed. E0 is the cell operating voltage; Ei , Nernst potential; Ri , the ohmic resistance; ηI , the overpotential; and Ii , the electric current.
The reaction rates of the shift reaction (10.20) are determined for the forward and backward processes: r(2)f = k1 × PCO × PH2 O , (10.26) r(2)b = k1 × Kshift × PCO2 × PH2 ,
(10.27)
where k1 is a constant. A large number is used for k1 so that r(2)f would be equal to r(2)b . The electric current density J is expressed by Faraday’s law as follows: J = 4F NO2 = 2F (Nf ),
(10.28)
where NO2 and Nf are the molar flux rates of oxygen and fuel consumed at the cathode and anode, respectively. It has been reported that the electrochemical oxidation rate of H2 is about two times higher than that of CO [13]. As a consequence, in the model, it is assumed that the H2 consumption rate is double that of the CO consumption rate. If the shift reaction is sufficiently fast, the shift reaction will be in an equilibrium state in the anode, and as a result, the assumption for the ratio of the H2 consumption to the CO consumption will not affect the simulation. The diffusion of the gaseous species in the porous electrodes is considered by specifying the molecular diffusivity for each specimen and specifying the momentum source term in a linearized form. The calculation for the diffusion of the gaseous species in porous electrodes is reported in details in Chapter 3. For simple calculations of the electric current, in the simplest case, a ladder circuit model is sufficient for the simulation. A discrete anode/electrolyte/cathode unit shown in Figure 10.4 is employed for the modeling. The direction of the electric current path in the electrolyte is vertical to the interfaces between the electrolyte and electrodes, and the in-plane path is treated as negligible. The electric current Ii in the ith discrete cell is determined by the operating cell voltage Vcell as follows: Vcell = Ei − Ri Ii − ηi ,
(10.29)
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where Ei is the Nernst potential; Ri , the ohmic resistance;, and ηi , the overpotential. The calculated Nernst potentials include the concentration overpotential itself and thus, ηi mainly consists of the activation overpotential. If the resistance of the electrolyte is sufficiently larger than those of the other components, an overall resistance model can be used. In the overall resistance model, the experimentally measured I-V data for a single cell is employed and the overall resistance is modeled using a unique function as follows: ri =
T (ohm cm−2 ). (−434 + 1.25 × 106 × exp(−7185/T )) + 0.04
(10.30)
It is assumed that there is no temperature gradient in the discrete anode/ electrolyte/cathode unit, and so the Nernst potential is a function of the average temperature Ti in the ith anode/electrolyte/cathode unit and the oxygen partial pressure PO2 at both the electrodes. Accordingly, the Nernst potential is given as Ei =
PO (c) RT ln 2 , 4F PO2 (a)
(10.31)
where R is the gas constant; F , the Faraday constant; and PO2 (c) and PO2 (a), the partial pressure of oxygen at the electrolyte/cathode and electrolyte/anode interfaces, respectively. In more precise calculations for the cell performance, the resistance of the cell has been estimated by considering subdividing the cell into cell components (i.e., electrolyte, anode, cathode, and interconnector) [14, 15]. If there is a large distribution of the electromotive force in a single cell, the electric current may flow diagonally in the electrolyte, and the ladder circuit model cannot simulate the diagonal electric current in the electrolyte precisely. In addition, in the calculation of the cell performance, it is difficult to take the geometry of the cell components (e.g., the pitch of the interconnector rib) into consideration. In order to treat the diagonal electric current in the electrolyte precisely and consider the effects of the cell geometry on the cell performance, it is indispensable to simulate the electric current path in the cell components. Thus, a new calculation technique on the cell performance has been presented [16]. In the new model, the three-dimensional electric current path in the cell components is simulated precisely. The calculation of the electric current path inside the single cell and the calculations of chemical and thermo-fluid phenomena are connected to each other, i.e., it is a fully connected model. Using this model, it is possible to simulate the diagonal electric current in the electrolyte, and in addition, the geometry of the cell components can be considered in the calculation of the cell performance. Contact resistances among each cell component are also considered in the calculation. Here, a detailed calculation using the new model will not be mentioned.
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10.2.2.3 Calculation Methods for Stresses in SOFCs Stress calculations are carried out by the finite element method. Here, the commercial finite method code “ABAQUS” (Hibbit, Karlsson, and Sorensen, Inc.) is used. Other codes such as MARC, ANSYS are also available. To calculate the stresses precisely, appropriate meshes and elements have to be used. 2D and shell meshes are not enough to figure out stress states of SOFC cells precisely, and thus 3D meshes is suitable for the stress calculation. Since the division of a model into individual tetrahedral sometimes faces difficulties of visualization and could easily lead to errors in numbering, eight-cornered brick elements are convenient for the use. The element type used for the stress simulation here is three-dimensional solid elements of an 8-node linear brick. In the coupled calculation between the thermo-fluid calculation and the stress calculation a same mesh model have to be used. Consequently same discrete 3D meshes used for the thermo-fluid analysis are employed for the stress calculation. Using ABAQUS, the deformations and stresses in a material under a load are calculated. Besides this treatment, the initial and final conditions of models can be set as the boundary conditions and the structural change can thus be treated. For the calculation of the thermally induced stress, thermally induced strain is additionally considered. The total strain vector {ε} is written using the elastic strain vector {ε el } and thermally induced strain vector {ε th } as {ε} = {ε el } + {ε th }.
(10.32)
Accordingly, Equation (10.10) is rewritten as: [B]T [D el ][B]dV {d} V
[M]T {T¯ }dS +
= S
V
[M]T {T¯ }dV +
[B]T [D el ]{εth}dV .
(10.33)
V
Thus Equation (10.33) is solved as the new equilibrium equation. To calculate the thermal expansion behavior of the model, the thermal expansion coefficient is necessary as the calculating parameter. If the temperature of the model is even, the initial temperature and the final temperature are used as just a calculating parameter. If a temperature distribution exists in the model, the temperature distribution data is dispensable for the stress calculation. The temperature is firstly calculated by a CFD and the calculated data is used as the boundary condition in the stress calculation. If the thermal expansion coefficient is temperature dependent, the temperature dependence must be considered in the calculation. Here the temperature data at the nodes is transferred from STAR-CD to the ABAQUS.
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Fig. 10.5 Model geometry of a cell stack. There are internal manifolds in the separator, and fuel and air are introduced to each cell through the manifolds.
10.3 Calculation of Temperature Distribution in SOFCs 10.3.1 Temperature Distribution in a Simple Model 10.3.1.1 Model Geometry and Calculating Conditions Here, as an example of CFD, an electrolyte-supporting cell is modeled. In general, a single-cell stack is constructed from a cell (anode/electrolyte/cathode) and an interconnector. An interconnector with channel ribs is employed as a guide for air and fuel to the cell. The cell stack is assembled using separators as an auxiliary, and airand fuel-manifolds are formed to introduce the air and fuel. Figure 10.5 shows the schematic diagram of a cell stack with internal manifolds formed in the separators. Since a considerable amount of time is required to simulate the cell performance for the entire stack, a simplified single-unit cell model with bipolar channels is useful for a simple simulation. For the model, only the counter-flow and the co-flow models are applicable. For the cross-flow pattern, the whole model has to be used because the gas flow is asymmetric. Considering the symmetry conditions in the calculation, half of the single repeating unit located in the middle of one center stack is used. The single cell stack and single unit model are illustrated in Figure 10.6. In this model, the thickness of the electrolyte, cathode, and anode are 0.1 mm, 0.15 mm, and 0.1 mm, respectively. As mentioned above, in this simulation, one repeating unit located in the middle of the center stack is selected and modeled. In the middle of the center stack, the boundaries between the single unit and adjacent units are regarded to be thermally adiabatic. Hence, for the thermal boundary conditions at the surfaces of the solid part connecting the next units, adiabatic boundary conditions are employed. In a practical cell stack, the fuel and air are introduced into the channel through man-
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Fig. 10.6 Model geometries of a single stack cell and a single unit. A repeating unit with anode/electrolyte/cathode and the interconnectors is modeled.
ifolds. The cell stack illustrated in Figure 10.5 has horizontal guide parts formed by the separators from the manifolds to the channels. The edges of the single unit face the guide parts, and the heat exchange between the edges and the environment around the stack is indirect. To simplify the calculation an adiabatic boundary condition is employed at the edges. For the boundary between the solid and fluid, the following continuity condition is imposed: λ(x)∇Ts (x) · n = κ(Tf (x) − Ts (x)),
(10.34)
where λ is the thermal conductivity; κ, the heat transfer coefficient; n, the unit vector normal to the boundary; and Ts and Tf , the temperature of the solid and fluid at x on the boundary, respectively. The simulations of the cell performance are carried out under the six different calculating conditions listed in Table 10.2. Calculating condition #1 is set as the standard case and all the calculated results are compared with to this. The parameters used for the calculations are also listed in Table 10.3.
340
Table 10.2 Calculation conditions for cell performance using the single-unit model. Fuel
Channel Thermal conductivity length (Wm−1 K−1 ) Sample Velocity Temperature Composition Velocity Temperature (mm) Separator Electrolyte (m/s) (K) (m/s) (K) #1 #2 #3 #4 #5 #6
0.4 0.57 0.4 0.4 1.2 0.4
1273 1273 1273 1273 1273 1273
Air
H2 O/CH4 = 2 pre-reformed H2 O/CH4 = 2 H2 O/CH4 = 2 H2 O/CH4 = 2 H2 O/CH4 = 2
3 3 3 3 9 3
1123 1123 1123 11 1123 1123
100 100 100 100 300 100
1.8 1.8 10 1.8 1.8 1.8
2.25 2.25 2.25 2.25 2.25 2.25
Flow pattern (V)
Operating voltage
counter-flow counter-flow counter-flow co-flow counter-flow counter-flow
0.7 0.7 0.7 0.7 0.7 0.65
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Table 10.3 Parameters used for cell performance calculation. Dan
k0
E
H0
S
9.3 × 102 (kg/m3 )
1.09 × 1019 (mol/kg)
1.91 × 105 (J/Kmol)
−4.179 × 104 (J)
−4.337 × 10 (J)
α
β
γ
k1
× 10−1
−9.306 (J)
× 10−2
2.383 (J)
× 10−5
−1.220 (J)
1.2 × 104 (mol/m3 s)
Fig. 10.7 The calculated profiles of (a) CH4 , (b) H2 , (c) CO, and (d) H2 O concentrations in the anode for the standard case (#1) in Table 10.2.
10.3.1.2 Results and Discussion Firstly, the standard cell operating conditions and parameters for the counter-flow case are specified so that the fuel utilization would be around 80% (the flow rate of air is set so that the oxygen utilization would be around 30%). With the selected parameters for the operating conditions, the concentrations of the gaseous species, the Nernst potential, the distribution of the current density, the average current density, the overall fuel utilization, and the temperature distribution are calculated. Figures 10.7a–10.7d show the calculated concentration profile for CH4 , H2 , CO, and H2 O at the anode, respectively. Here, it should be noted that the radiant heat exchange inside the channels is neglected in this calculation. It can be seen that most
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Fig. 10.8 The simulated temperature profiles in the cell and at the electrolyte for standard case (#1) in Table 10.2 without considering the radiant heat exchange.
of the supplied CH4 is reformed within the first 10 mm from the fuel inlet, suggesting that the steam reforming reaction is much faster than the velocity of the fuel flow. Consequently, the H2 and CO concentrations rapidly increase within this area, and then gradually decrease as a result of consumption by electrochemical oxidation while flowing toward the fuel outlet. In the area below the interconnector rib, the gaseous species are transported only by diffusion in the porous electrodes. This leads to lower H2 and CO and higher H2 O concentrations than in the channel area. Although in the present analyses, the effect of the rib width on the cell performance has not been studied, increase in the overall potential drop with an increase in the rib width has been reported [17]. 80% of the supplied fuel is consumed at the fuel outlet. At only a short distance from the fuel inlet, the cell temperature drops rapidly because of the endothermic steam reforming reaction, as shown in Figure 10.8. In the middle area where the steam reforming reaction is complete, the cell temperature increases because of the exothermic electrochemical reaction. At the downstream of the fuel, the cell temperature decreases along the fuel stream, since the heat generation is small; in addition, the hot fuel is cooled by the incoming air. Here, the effect of radiant heat exchange inside the channels on the simulated temperature distribution is discussed. Figure 10.9 shows the temperature distribution at the electrolyte calculated by considering a radiant heat exchange inside the channels. When Figure 10.9 is compared with Figure 10.8, two characteristic features can be observed. Firstly, when the radiant heat exchange is taken into consideration, the temperature distribution in the cell becomes flat. This is because radiant heat exchange occurs when the temperatures in the channels differ between high and low temperature volumes. The difference in the maximum temperature in the cell for ε = 0 and ε = 0.8 is 30 K (ε represents here the emissivity). The value of 30 K is equal to about 10% of the difference in the temperature along the fuel flow at the electrolyte. A further difference is that when the radiant heat exchange is considered, the
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Fig. 10.9 The simulated temperature profiles in the cell and at the electrolyte for standard case (#1) in Table 10.2 considering the radiant heat exchange of (a) ε = 0.2 and (b) ε = 0.8.
area where the temperature is the maximum moves to the fuel downstream. When the radiant heat exchange is not considered, the areas with the maximum and minimum temperatures are very close, as can be observed in Figure 10.8. When the radiant heat exchange is considered, the radiant heat loss at the hottest area is larger because the temperature is higher and the radiation angle φ is close to zero, as shown in Figure 10.10. As the distance from the fuel inlet where the temperature is the minimum to the area with the maximum temperature increases, the value of the radiant heat exchange becomes small. Thus, the area with the maximum temperature may shift to the downstream of the fuel flow when the radiant heat exchange is considered. Figures 10.11a and 10.11b indicate the Nernst potential and current density profile at the electrolyte/anode interface, respectively. The Nernst potential decreases along the fuel stream, reflecting the decrease in the H2 and CO concentrations. The lack of H2 and CO is reflected by the Nernst potential being lower in the area below the interconnector rib than in the channel. At the fuel inlet, instead of the high potential, the current density is not as high because the cell temperature is low; thus, the cell resistance is high. Along the fuel flow, the current density increases once and reaches a maximum value of about 0.9 A/cm2 . At the fuel downstream, both the cell temperature and Nernst potential is lower, and the current density has an accordingly lower value. Although the maximum current density is as high as 0.9 A/cm2 , the minimum current density value is as low as 0.073 A/cm2 in the area below the interconnector rib and thus, the average current density reaches 0.39 A/cm2 . Next, we focus on the effects of gas flow pattern. When the air inlet temperature in the co-flow pattern is the same as that in the standard case (counter-flow pattern),
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Fig. 10.10 An illustration of the radiant heat exchange between areas with high and low temperatures near the fuel inlet.
Fig. 10.11 The calculated (a) Nernst potential profile and (b) current density profile at the electrolyte/anode interface for the standard case (#1) in Table 10.2.
the drop in the electrolyte temperature due to the reforming of endothermic steam is larger, resulting in an increase in the sheet resistance. Accordingly, the fuel utilization becomes smaller than that in the standard case. In order to increase the fuel utilization to that in the standard case, the air inlet temperature has been modified to 1173 K. All the other operating conditions are the same as in the standard case. Figure 10.12 exhibits the temperature profile in the cell and at the electrolyte for the co-flow case. In contrast to the standard counter-flow case, the electrolyte temperature increases monotonically from the fuel inlet to the fuel outlet. As a result, the potential maintains a high value from the fuel inlet to the middle area of the
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Fig. 10.12 The simulated temperature profile in the cell and at the electrolyte for the co-flow case (#4) in Table 10.2.
Fig. 10.13 The calculated (a) Nernst potential profile and (b) current density profile at the electrolyte/anode interface for the co-flow case (#4) in Table 10.2.
fuel flow, and decreases rapidly near the fuel outlet, as shown in Figure 10.13 a. Although the current density profile shows a maximum value in the middle area, similar to the standard case, the difference between the maximum and minimum current densities, 0.46 A/cm2 , is smaller than 0.88 A/cm2 in the standard case, as can be observed in Figure 10.13b.
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Table 10.4 Calculation conditions for cell performance using the single-unit model. Sample
#1 #2 #3 #4 #5 #6
Fuel utilization (%)
Average current density (Acm−2 )
79 77 82 77 77 92
0.398 0.308 0.41 0.386 0.386 0.463
Cell temperature (K) at fuel outlet at air outlet maximum 1213 1200 1291 1344 1162 1232
1326 1511 1320 1338 1331 1443
1488 1536 1398 1352 1571 1617
Maximum tensile stress (MPa) in separator in electrolyte 1.96 0.57 0.45 0.79 2.64 2.41
67 7.7 79 26 95 77
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All the simulated results for the selected cases in Table 10.2 are summarized in Table 10.4. To validate the numerical simulation on the cell performance, a benchmark test on the numerical model for the cell performance has been carried out by IEA (International Energy Agency) participants [17]. Since the calculating conditions (e.g., the temperatures of the introduced fuel and air, fuel composition, singlecell properties, cell operating voltage, and thus fuel utilization) in the present simulation are different from those in the IEA benchmark test, the present results cannot be compared quantitatively with the results obtained by IEA. However, the present results such as the concentration profile for the gaseous species, and the reduction in the temperature difference in the cell by employing the co-flow pattern are qualitatively in agreement with the previously reported results [18].
10.3.2 Temperature Distribution in a Stack A cell stack is comprised of cells (anode/electrolyte/cathode) and interconnectors. Interconnectors with channel ribs are adopted as guides for air and fuel to the cell. The cell stack is assembled using an auxiliary, and air and fuel manifolds are formed to introduce air and fuel into the cell. In the previous section, a simplified single unit model has been used and the cell performance is calculated. In this section, stack models are employed to simulate practical tests on mechanical strength against thermal stresses induced by internal heat evolution. In the practical tests, one-, three-, or ten-cell stacks are set in an electric furnace and they are operated at 1000◦C with H2 . By increasing the heat generation inside the stack via an increase in the electric current, the durability of the stack against thermal stresses is evaluated. Here, one-, three-, and ten-cell stack models are employed and practical heat evolution experiments are simulated. The stack structure of the one-cell stack is illustrated in Figure 10.14. In the cell-stack model calculation, thermo-fluid analyses are carried out similar to the previous single-unit model calculations, and the electrochemical reactions and diffusion of the gaseous species in the porous electrodes are also considered. The only difference between the single-unit model calculation and present cell-stack model calculation is that the geometry for the cell-stack model is coarsely divided into a finite number of elements. In the previous calculation using a single-unit model, one repeating unit located at the center of a stack is modeled and the boundaries between a single unit and the adjacent units are regarded to be thermally adiabatic. For the present cell stack model, on the other hand, a radiant heat exchange condition is assumed for the entire stack surface. The stack surface exchanges heat energy with an inner wall of an electrical furnace, where the temperature is evenly 1000◦C. For the cell resistance, the resistance of each cell component is not considered separately, and the sheet resistance given in Equation (10.30) is also used for the calculation. The current density dependence of the polarization resistance is neglected in the calculation. The fuel is H2 :H2 O = 0.8:0.2, and the flow rates of fuel and air are set so that the fuel and air utilization would be around 50% and 30%, respectively.
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Fig. 10.14 Geometry of a single-stack.
Fig. 10.15 Simulated temperature distribution at the electrolyte during an operation of the singlecell stack model. The stack is assumed to be operated at 1000◦ C in an electric furnace.
10.3.2.1 Results and Discussion Figure 10.15 shows the simulated temperature distribution in the electrolyte for the single-cell stack model. In this calculation, the cell operating voltage is set at 0.16 V at which the electrolyte sheet cracked in the internal heat evolution test. In Figure 10.15, it can be observed that the temperature is almost 1273 K outside the anode/electrolyte/cathode area where heat is generated. Near the fuel inlet at the channel, the temperature increases steeply and the maximum temperature spreads over a wide area to the downstream of the fuel flow; the maximum temperature difference in the electrolyte is around 60 K.
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Fig. 10.16 Simulated temperature distribution at the electrolyte during an operation of the threecell stack model. The stack is assumed to be operated at 1000◦ C in an electric furnace.
Figures 10.16 and 10.17 show the simulated temperature distribution in the electrolyte for the three-cell and ten-cell stack models, respectively. For the three-cell and ten-cell stack models, the flow rates of air and fuel are three times and ten times those for the single-cell stack model, respectively. The other calculation conditions are the same as those employed in the single-cell stack model. In Figure 10.16, the upper and lower diagrams correspond to the upper and the middle cells in the threecell stack, respectively. It can be observed that the maximum temperature is higher in the middle cell than in the upper cell. This is because the generated heat in the middle cell dissipates indirectly to outside the cell stack through the upper and lower cells. This indirect dissipation of the generated heat from the middle cell to the outside cell is smaller in the ten-cell stack, and as a result, the generated heat remains in the middle cell. Accordingly, the temperature in the middle cell in the ten-cell stack
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Fig. 10.17 Simulated temperature distribution at the electrolyte during an operation of the ten-cell stack model. The stack is assumed to be operated at 1000◦ C in an electric furnace
becomes higher than that in the three-cell stack, as shown in Figure 10.17. The maximum temperature difference in the 5th cell from the top is as large as 150 K for the ten-cell stack model. Here, it should be noted that the difference in the maximum temperature of the cell stack between the three-cell and ten-cell stack models is smaller than that between the single-cell and three-cell stack models. This suggests that the boundaries between the middle and adjacent cells are almost thermally adiabatic and the heat dissipation from the middle cell through the adjacent upper and lower cells is small even in the three-cell stack model. Thus, it can be supposed that the maximum temperature in a multi-cell stack model with more than ten cells may be almost the same as that in a ten-cell stack model.
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Fig. 10.18 Photograph of the anode-supported cell (left figure) and its structure (right figure).
10.4 Evaluations of Stresses in Single Cell & Stack 10.4.1 Residual Stresses in Single Cell & Stack As an example of residual stresses in cells, a stress in an anode-supported cell is presented here. Anode-supported cells are suitable for operations at lower temperatures because of the substantially lower ohmic resistance of the electrolyte. By lowering the operating temperature of SOFCs below 800◦C, conventional metal alloys can be used for interconnectors or other components; this can result in high mechanical reliability of a cell stack and lower manufacturing costs. Thus, anodesupported SOFCs have been receiving considerable interest [19–23]. The anodesupported cells are generally fabricated by co-firing the thin electrolyte on the anode substrate at 1400∼1500◦C [18–22]. A photo of the fabricated anode-supported cell and its structure is shown in Figure 10.18. Ni/YSZ and YSZ are used as the anode substrate and electrolyte, respectively. The TEC of Ni/YSZ and YSZ do not match well and, as a result, the mismatch induces a large residual stress in the cell at room temperature. The thermally induced residual stress may induce micro-cracks or a delamination of the electrolyte at the interface under a thermal cycle. These damages of the cell decrease the cell performance, and sometimes cause destruction of the cell. Thus, it is significant to evaluate the residual stress in the anode-supported cell and reduce the thermal stress under the thermal cycles. As will be mentioned in the next section, the residual stresses in the electrolyte of the anode-supported cell at room temperature have been measured by X-ray stress measurements. Here, only the calculations of the residual stresses are presented. In addition to the occurrence of the residual stress, cells warp toward the cathode side because of the mismatch of the thermal expansion behavior between the electrolyte and anode. Thus, the deformations of cells at room temperature are also
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Fig. 10.19 Simulated residual stress at the electrolyte of the anode-supported cell at room temperature. Negative stress means that the stress is compressive. Table 10.5 Mechanical properties and cell sizes used for the stress simulation.
Electrolyte Anode
Young’s modulus (GPa)
Poisson’s ratio
TEC (K−1 )
Thickness (mm)
206a 96b
0.33a 0.3c
10.56 × 10−6d 12.22 × 10−6d
40 × 40 × 0.03 40 × 40 × 2
a From
Ref. [48]. 4-point bending tests at room temperature. c Assumed value. d Assumed from the temperature dependence of TEC below 1000◦ C [52]. b From
simulated and the degrees of the deformations are calculated. In a practical cell stack, to achieve good contact among each cell, a mechanical load is placed on top of the stack. To investigate the change in the warp under a mechanical load, a load test was performed [24]. Using the numerical method, the load test is also simulated. Numerical calculations for the residual stresses in the anode-supported cells are carried out using ABAQUS. After modeling the geometry of the cell of the electrolyte/anode bi-layer, the residual thermal stresses at room temperature are calculated. The cell model is divided into 10 by 10 meshes in the in-plane direction and 20 submeshes in the out-plane direction. In the calculation, it is assumed that both the electrolyte and anode are constrained each other below 1400◦C and that the origin of the residual stresses in the cell is only due to the mismatch of TEC between the electrolyte and anode. The model geometry is 50 mm × 50 mm × 2 mm. The mechanical properties and cell size used for the stress calculation are listed in Table 10.5. Figure 10.19 shows the simulated stress distribution in the electrolyte for the anode-supported cell at room temperature. Except for the edge, the stresses are al-
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Fig. 10.20 Effect of the temperature at which the electrolyte and the anode are constrained on the calculated residual stress in the electrolyte. The stress is compressive stress.
most homogeneous values of 640 to 700 MPa over the electrolyte plane. It is not clear in the figure but it is found that the stress is slightly larger near the corner than that at the center; this is qualitatively consistent with the aforementioned result of the X-ray stress measurements. In this calculation, the electrolyte and anode are assumed to be constrained each other at 1400◦C. However, the exact temperature at which the electrolyte and anode are constrained is unclear, and the assumption of this temperature affects the results of the simulation, as shown in Figure 10.20. With a change in the temperature at which the electrolyte and the anode are constrained, the calculated stress at the center of the electrolyte varies from 550 to 670 MPa. From these observations, the temperature at which the electrolyte and anode are constrained appears to be between 1250◦C and 1400◦C. The as-grown cells are usually extremely warped toward the cathode side at room temperature. To reduce the warp, the cells are flattened at 1400◦C with a distributed load on the surface. The flattening treatment could reduce the warp at room temperature indicating that the sample is in a plastic state at 1400◦C. In contrast, it was reported that the sintering of YSZ does not proceed well below 1250◦C [25]. Thus, it is considered that the temperature at which both the electrolyte and anode are constrained is between 1250◦C to 1400◦C. When the temperature at which the electrolyte and anode are constrained is assumed to 1400◦C, the calculated residual stress is close to the measured value. Next, the effect of cell size on residual stress is studied. In the present calculation, the thicknesses of the electrolyte and anode are fixed as 30 µm and 2 mm, respectively, and the residual stresses are calculated with various plane sizes. The result of the simulation is displayed in Figure 10.21. When the cell size is more than 5 mm, the calculated stress is almost independent of the cell size. In practical stress
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Fig. 10.21 Effect of the plane sample size on the calculated residual stress in the electrolyte. The thicknesses of the electrolyte and the anode are fixed as 30 µm and 2 mm, respectively. The stress is compression.
Fig. 10.22 Effect of the thicknesses of the electrolyte and the anode on the calculated residual stress in the electrolyte at room temperature. The plane sample size is fixed at 40 mm × 40 mm. The stress is compression.
measurements, the cells are cut into small pieces but the sizes of the samples are larger than 5 mm. Accordingly, the effect of the sample cut on the residual stress may be negligible. The effect of the thickness of the electrolyte and anode on the residual stresses is also studied. The residual stresses in the electrolyte are calculated for different combinations of the thickness of the anode and electrolyte, and the results are displayed in Figure 10.22. The combination of the thin anode and thick electrolyte merely decreases the residual stress in the electrolyte. This is desirable from the point of view of the mechanical reliability of cells. On the other hand, for a practical cell,
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Fig. 10.23 Simulated stress distribution under standard condition #1 in Table 10.2; (a) at the electrolyte and (b) at the interconnector.
the thick electrolyte causes an increase in the electric resistance in the electrolyte. Thus, an optimal thickness of the electrolyte has to be selected after considering both the electric resistance and residual stress.
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Fig. 10.24 The tensile stress profile in the electrolyte for the external-reforming case (#2) in Table 10.2.
10.4.2 Thermally Induced Stresses in Single Cell & Stack under Cell Operations 10.4.2.1 Stresses in Single Cells Thermally induced stresses in the electrolyte and interconnector under the six different operating conditions shown in Table 10.2 are simulated. The distributions of the principal stresses in the electrolyte and interconnector are calculated from the simulated temperature distributions. The results for the standard counter-flow case #1 are shown as an example in Figure 10.23. A stress concentration can be observed near the fuel inlet due to the steep temperature drop caused by the endothermic steam reforming reaction. The maximum principal stress in the interconnector is about 2 MPa, which is sufficiently low to avoid damage of the interconnector. On the other hand, the principal stress in the electrolyte is almost 70 MPa. When 8YSZ is used as the electrolyte, the internal stress might cause cracks or destruction of the electrolytes. From the measurements of the tensile strength on the tape-cast 8YSZ, the compensated tensile strength of the electrolyte sheets at room temperature was estimated to be about 80 MPa [27], and it has been reported that the mechanical strength of 8YSZ becomes low at high temperatures [28, 29]. Thus, the calculated stress may be a critical value of the failure of the electrolyte. All the calculated maximum tensile stresses in the electrolyte and interconnector for the 6 cases listed in Table 10.2 are also summarized in Table 10.4. For all the cases, the maximum tensile
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Fig. 10.25 The tensile stress profile in the electrolyte for the co-flow case (#4) in Table 10.2.
stresses in the electrolyte are several tens of MPa while those in the interconnector are less than 10 MPa. The stress concentration might occur near the minimum temperature point, and thus, the magnitude of the maximum tensile stress can be related to the magnitude of the steep temperature drop caused by the prevailing steam reforming reaction. For the external reforming case, the maximum tensile stress is one order of magnitude smaller than those for the internal methane reforming cases because there is no steep temperature drop from the largely endothermic reaction. In addition, the stress concentration occurs at a different region from that for the internal reforming case, as shown in Figure 10.24. For the co-flow case, it can be observed in Figure 10.25 that the maximum tensile stress in the electrolyte is significantly reduced as compared to the counter-flow case. This is due to the fact that the steam reforming reaction is less influential than in the counter-flow case, and the electrolyte temperature gradually increases along the fuel flow, as shown in Figure 10.12. In the case of an interconnector with a high thermal conductivity, although the calculated temperature difference in the cell is smaller, the maximum tensile stress is higher than that for the standard case, as shown in Figure 10.26. This is because near the fuel inlet, the temperature gradient along the direction perpendicular to the gas flow is steeper. The steeper temperature gradient is because the highly thermally conducting interconnector carries the heat from a high temperature region to a minimum temperature region. Thus, these results indicate that the steam reforming reaction is the main cause of the origin of the thermal stress, and reducing the induced temperature drop near the fuel inlet is an advantageous way of avoiding failure of the
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Fig. 10.26 The tensile stress profile in the electrolyte for the case of using an interconnector with a high thermal conductivity (#3) in Table 10.2.
electrolyte sheet. If a mechanical load exists or the cell components constrain each other, the maximum stress will increase further. However, it is possible to reduce the tensile stress in the electrolyte by employing the co-flow pattern.
10.4.2.2 Stresses in Cell Stacks Here, the effects of cell size on thermal stresses are first considered. The calculated temperature distribution data using a single-unit model is expanded to a multi-unit model. By using the simulated temperature distribution in the electrolyte for the conditions in Table 10.2, the thermal stresses in the electrolyte of a multi-channel cell are calculated. Figure 10.27 shows the comparison of the thermal stresses in the electrolytes of the 4-channel and 16-channel models under standard condition #1 listed in Table 10.4. Considering the symmetry of the problem, symmetric boundaries are set at the left of both electrolytes and only half the models are drawn in this figure. The maximum stress in the electrolyte increases with the number of channels, and in the 16-channel model, the maximum stress of 250 MPa concentrates at the center channel and near the fuel inlet. This stress concentration is attributed to the steep temperature drop near the fuel inlet due to the endothermic steam reforming reaction, as previously mentioned. The freedom of deformation is constrained more at the center channel (i.e., the left side of the electrolytes in the figure) than
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Fig. 10.27 Effect of the stack size on the thermal stress at the electrolyte. The calculation is carried out for the standard condition #1 in Table 10.2.
at the edge channels, and hence, the stress concentrates at the center channel. Further, it can be observed that a stress concentration occurs at the edge shown with an arrow. The reason for this stress calculation can be explained as follows. Along the fuel flow, the fuel temperature first drops and then increases to reach a maximum. This is because the fuel at the fuel outlet is cooled by the incoming air. In the area where the temperature is maximum, the electrolyte expands more, and hence, the displacement from the center of the electrolyte in the direction perpendicular to the gas flow is largest at the edge. As a result, the edge expands outward to result in large tensile stresses. Since the magnitude of the expansion at the edge increases with the number of channels, the stress concentration at the edge increases with the number of channels. Figures 10.28a–10.28c display the simulated stress distribution in the electrolyte using the 16-channel model for the standard condition (#1), co-flow case (#4), and pre-reforming case (#2), respectively. In comparison with the counter-flow case, the co-flow case is characterized by a maximum stress occurring near the fuel outlet. This is because the temperature increases monotonically from the fuel inlet to the fuel outlet, and the maximum temperature occurs close to the fuel outlet. Furthermore, the temperature drop near the fuel inlet caused by the steam reforming reaction is not very steep, and thus, the stress concentration near the fuel inlet is not significant. The maximum stress can be reduced to about 1/5 of that for the standard condition. For the pre-reforming case (Figure 10.28c), the temperature does not drop near the fuel inlet. As a result, the stress concentration at the fuel inlet is not large. In addition, the fuel temperature reaches a maximum value at a short distance from the fuel inlet, and thus, the stress concentration at the edge, which appears in Figure 10.27, is not remarkable. This is because a constraint against strain is weaker at
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Fig. 10.28 Calculated temperature and stress distributions at the electrolyte for a 16-channel model under three different conditions; (a) for standard condition (#1), (b) for co-flow case (#4), (c) prereforming case (#2) in Table 10.2.
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Fig. 10.29 Calculated stress distribution at the electrolyte for the single-cell stack model. The stress is calculated from the data shown in Figure 10.15. To magnify the deformation of the electrolyte at the edge, the deformed shape of the electrolyte is illustrated at ×10 magnification.
Fig. 10.30 Photograph of the ten-cell stack used for the heat evolution test. The cell was destructed after the test.
the corner than at the edge. All the calculated results are summarized in Table 10.2. From these stress simulations, the following points have been clarified: (i) The increase in the cell size in the perpendicular direction to the gas flow merely increases the tensile stress in the electrolyte. (ii) The steam reforming reaction induces a large tensile stress in the center channel and near the fuel inlet. (iii) A temperature maximum induce a relative more expansion of the cell and causes a large tensile stress at the edge of the cell.
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Fig. 10.31 Calculated stress distribution at the electrolyte for the three-cell stack model. The stresses are calculated from the data shown in Figure 10.16.
Stack Model Figure 10.29 shows the thermally induced stress in the electrolyte under the stack operation calculated from the temperature distribution shown in Figure 10.15. A tensile stress of about 100 MPa is concentrated at the edge and shown by an arrow. The origin of this stress concentration is the same as that explained in the multi-channel model calculation; the electrolyte expands more in the area where the temperature is the maximum and the edges are expanded. In Figure 10.29, to magnify the deformation of the electrolyte at the edge, the deformed shape of the electrolyte is illustrated at ×10 magnification. Figure 10.30 shows a photo of the cell stack used for the heat evolution test. The thermally induced stress causes the cell stack to break. The damage at the edge near the fuel inlet is serious. This is
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Fig. 10.32 Calculated stress distribution at the electrolyte for the three-cell stack model. The stresses are calculated from the data shown in Figure 10.17.
quantitatively consistent with the result of the stress calculation indicating that the stress concentrates at the edge near the fuel inlet. Figures 10.31 and 10.32 show the stress distribution in the electrolyte for the three-cell and ten-cell stack models, respectively, as calculated from the temperature distributions shown in Figures 10.16 and 10.17. It is observed that the maximum tensile stress in the electrolyte increases with the temperature difference in the electrolyte. For the 5th cell from the top in the ten-cell stack model, where the temperature difference in the electrolyte is the largest, the maximum tensile stress is as large as 220 MPa; this is much larger than that for the single-cell stack. This suggests that if the number of cells increases, the cell stack will be damaged even under smaller heat generation. To obtain an even temperature distribution, a separator or an interconnector which has a good thermal conductance must be used for the stack. Metallic interconnectors are good candidate because the thermal conductance
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is higher than that of LaCrO3 . However for the electrolyte-supporting cells operated at a high temperature of 1000◦C, metallic interconnectors are not applicable because of the poor durability under a high temperature oxidizing atmosphere. Since for the anode-supported cells operated around 750◦ C metallic interconnectors can be used and thus they are prospective for mechanically reliable SOFCs.
10.4.3 Chemically Induced Stresses in Interconnector 10.4.3.1 Introduction In SOFCs, interconnecting materials must be electrically conducting and chemically stable both in reducing and oxidizing atmospheres at high temperatures. Doped lanthanum chromites have been used as a typical interconnector material because they meet the requirements such as high chemical stability and high electrical conductivity in both reducing and oxidizing atmospheres at a high temperature. However, it has been reported that in doped lanthanum chromites, a number of oxygen vacancies are generated under high-temperature reducing atmospheres, which expands the crystal lattice depending on the concentration of the oxygen vacancy [30, 31]. For the lanthanum chromites doped with alkaline-earth metals, the specimen length measured in a strongly reducing atmosphere such as the fuel environment in the SOFC was larger than that measured in air by 0.1% [32]. Under a practical operation of SOFCs, the doped-lanthanum chromites are exposed to an oxidizing atmosphere on one side and a reducing atmosphere on the other side. As a result, an oxygen potential gradient occurs in the interconnector, and the interconnector expands non-uniformly depending on the oxygen potential gradient inside. This non-uniform expansion induces a deformation of the interconnector, resulting in poor electrical contact between the interconnector and electrodes, failure of the gas seal, and generation of internal stresses. In the worst case, the deformation generates cracks or self-destruction of the interconnectors. Consequently, elucidating the expansion behavior and estimating the magnitude of the internal stresses of the interconnector under practical cell operating conditions are of high significance for the development of SOFCs with high mechanical reliability. A numerical model to simulate the lattice expansion behavior of the doped lanthanum chromites under a cell operating condition has been proposed, and the deformation of the lanthanum chromite interconnectors has been calculated [33]. In the model, the sample deformation is calculated from the profile of the oxygen vacancy concentration in the interconnector. Under a practical cell operation, the oxygen vacancy concentration in the interconnector distributes unevenly from the air side to the fuel side. The distribution of the oxygen vacancy concentration in the interconnector depends on both the temperature distribution in the interconnector and the profile of the oxygen partial pressure at the interconnector surface. Here, a numerical model calculation for the expansion behavior of the LaCrO3 interconnector under a practical cell operation is carried out, and the uneven distribution of
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the oxygen vacancy concentration in the interconnector is considered. In the model, firstly, a practical cell operation is simulated, and the temperature distribution in the interconnector and the profile of the oxygen partial pressure at the surface of the interconnector are calculated. As the next step, the oxygen vacancy profile in the interconnector is simulated by solving Fick’s first law for the oxygen vacancy. Then, simulations for the expansion behavior of the interconnector are carried out using the calculated profiles of the temperature and the oxygen vacancy in the interconnector. A single-unit model with bipolar channels is used to simulate the cell performance, and a model geometry with multiple channels is used to calculate the expansion behavior of the interconnector.
10.4.3.2 Calculation Model In Section 10.3.1, the cell performance of a planar type SOFC using a single-unit model with bipolar channels for a counter-flow or co-flow pattern is simulated. In the present calculation, to calculate the deformation or internal stress of the interconnector, the same model geometry is employed. The oxygen partial pressure at the surface of the interconnector and the temperature distribution in the interconnector are first calculated using the single-unit model. The expansion behavior of the interconnector is simulated according to the following procedure. The procedure is for Sr-doped lanthanum chromites. The flow chart of the procedure is shown in Figure 10.33. (1) Calculation of the oxygen vacancy concentration at the interconnector surface On the basis of the point defect theory, the oxygen vacancy concentration (mole fraction) δ on the fuel and air side surfaces of the interconnector are calculated [34]. In an equilibrium state, the formation of the oxygen vacancy can be described as follows using Kröger–Vink notation [35]: 1 × ˙ O2 + V¨O + 2Cr× Cr ↔ OO + 2CrCr , 2 Kox =
[Cr˙Cr ]2 [O× O] 2 ¨ [Cr× Cr ] [VO ]PO2
1/2
,
(10.35)
(10.36)
where Kox is the equilibrium constant and PO2 is the oxygen partial pressure. Imposing the following electroneutrality condition and the conservation law for the Cr sites, [Sr˙La ] = [Cr˙Cr ] + 2[V¨O ], [Cr˙Cr ] + [Cr× Cr ] = 1. Equation (10.36) is written as
(10.37)
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Fig. 10.33 The flow chart of the calculating procedure for the deformation of LaCrO3 interconnector.
Kox =
(2 − 2δ)2 (3 − δ) 1/2
(1 − x + 2δ)2 δPO2
,
(10.38)
where x is the mole fraction of doped Sr and δ is the concentration of the oxygen vacancy. Thus, the δ profile at the sample surface can be obtained by solving Equation (10.38). The equilibrium constant Kox is given by H 0 G0 S 0 = exp exp − , (10.39) Kox = exp − RT R RT where G0 , S 0 , and H 0 are the standard Gibbs energy, entropy, and enthalpy change for the oxygen vacancy formation reaction, respectively. Consequently, δ depends on both the temperature and PO2 at the surface and in the bulk volume of the interconnector. The results of the previous calculations of the distributions of PO2 and temperature at the interconnector surface are employed to calculate the distribution of δ at the surface of the interconnector.
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(2) Calculation of the distribution of the oxygen vacancy concentration in the interconnector Under an oxygen potential gradient, the oxygen vacancies diffuse from the fuel side to the air side of the interconnector plate, and thus, the oxygen vacancies distribute in the specimen according to the oxygen potential. Under a static state, the oxygen vacancies and electron holes counter-diffuse so that the electroneutrality condition is satisfied, and the diffusion of the oxygen vacancy can be described using the chemical diffusion coefficient Dchem . By using the analogy between the thermal diffusion and oxygen vacancy diffusion, the δ distribution in the specimen is simulated by the finite volume method. The simulation is carried out using STAR-CD. Fast surface kinetics are assumed, and thus, the δ profile at the surface calculated in the previous step is used as the boundary conditions. In this calculation, the dependency of the chemical diffusion coefficient Dchem on the oxygen vacancy concentration δ is taken into account. On the basis of the point defect theory and the ambipolar diffusion theory [36], Dchem can be derived as follows [37]: 4δ , (10.40) Dchem = Dv 1 + x − 2δ where Dv is the diffusion coefficient of the oxygen vacancy. In the derivation, the hopping conduction of the oxygen vacancies inside the interconnector [33], the conservation of the charge neutrality condition in the mixed conduction of the oxygen vacancies and the holes, and the unity of the electronic transference number are assumed. It has been reported [34] that Dv is experimentally given by H , (10.41) Dv = D0 exp − RT where H is the activation energy and the pre-exponential constant D0 is the frequency factor. (3) Determination of the local strain distribution in the interconnector The local strain distributions in the interconnector are determined from the calculated δ distributions. L/L (L is the length of the divided specimen and L is the expansion of the divide specimen) is almost the same meanings as the strain. Assuming the following linear relationship between the relative expansion L/L and δ, L = kδ, (10.42) L the δ distributions are converted to L/L distributions in the interconnector. The proportional factor k employed in this simulation is determined from the reported relationship between L/L and δ for the doped LaCrO3 under uniform atmospheres with various oxygen partial pressures [32]. (4) Conversion of the strain distribution to the equivalent temperature distribution The strain distributions calculated in step (3) are converted to the equivalent
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Fig. 10.34 The illustration of the technique to change the strain distribution data to the equivalent temperature distribution data.
temperature distributions in the interconnector. The overall specimen deformation is given by the superimposition of the local strains in the specimen. In order to simulate the overall specimen deformation, the finite element method is employed. Because it is difficult to calculate the specimen deformation directly from the strain profile, it is assumed that the strain induced by the chemical lattice expansion can be treated in the same way as the thermally induced expansion. In this way, the equivalent temperature distribution in the specimen, which causes the same local strain distribution as that obtained in the previous step, is calculated in the present step. In order to convert the strain distributions to the equivalent temperature distributions, the strain values at arbitrary node points of the finite elements are divided by thermal expansion coefficient λ. The configuration of this technique is illustrated in Figure 10.34. In the figure only the strain in the x direction is considered. (5) Calculation of the deformation of the interconnector and internal stresses From the calculated temperature distributions, the deformation and related internal stresses of the interconnector are calculated using ABAQUS. Two specimen compositions have been selected in order to calculate the chemical expansion behavior of the lanthanum chromite interconnector. They are La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ and La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnectors with a
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large and small chemical expansion, respectively. The physical and mechanical properties of these materials are listed in Table 10.3. For both specimens, a small amount of Ni or Co is doped to promote sintering and obtain highly dense samples. When the 3D-transition metal is heavily doped to LaCrO3 , the dopant affects the electronic state of the Cr ion. Accordingly, the effects of the dopant should be considered in the calculation of the δ profile at the sample surface, using Equation (10.40). For the present specimens, however, the amount of the doped Ni or Co is small; hence, the effect of the dopant in the calculation can be neglected [38, 39]. All the parameters used for the stress calculation on the doped-LaCrO3 are listed in Table 10.6.
10.4.3.3 Results and Discussion δ distribution in the interconnector Firstly, the simulation of the δ distribution in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector under the standard counter-flow condition (#1) in Table 10.2 is reported. Using the previously calculated temperature and PO2 distribution data (shown in Figure 10.35), the δ profile at the interconnector surface is calculated as the first step, and then the δ distribution inside the interconnector is simulated as the second step. Figure 10.36 shows the contour plot of the calculated δ distribution in the interconnector for the standard condition (#1). In Figure 10.35, it is to be noted that the upper surface of the interconnector drawn at the upper position and the lower surface of the lower interconnector are connected continuously, and only half the single-unit model is drawn. As shown in Table 10.6, G0 in Equation (10.39) is a negative value, and hence, the equilibrium state for La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ expressed by Equation (10.35) shifts to the right side with an increase in the temperature and PO2 . PO2 in the fuel channel increases along the fuel flow, and the maximum temperature appears near the fuel inlet in the temperature profile. Hence, δ is maximum near the fuel inlet and decreases steadily along the fuel flow. At the interface between the interconnector and anode, PO2 is higher than that in the gas channel. This is because in the porous anode, the gaseous species can move only by diffusion, and thus, the steam and CO2 generated at the electrolyte/anode interface cannot move to the gas channel smoothly. Consequently, δ is lower at the interconnector/anode interface than at the channel area. In this simulation, it should be noted that the chemical diffusion coefficient of the oxygen vacancy is a function of δ itself, as expressed by Equation (10.40), which indicates that the diffusion rate of the oxygen vacancy is faster near the fuel side than the air side. Thus, the calculated δ distribution in the interconnector is different from that calculated using δ-independent Dchem , and δ increases non-linearly from the air side to the fuel side. Figure 10.37 displays the simulated δ distribution in the interconnector assuming that Dchem is equal to Dv . It can be observed that the width between the contour lines is narrow near the fuel side and wide near the air side as compared with that in Figure 10.36.
370
Table 10.6 Physical and mechanical properties used for the stress simulations. Composition
H 0 (kJ/mol)
S 0 D0 H k (J/mol/K) (×10−8 m2 /s) (kJ/mol)
La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ –289a –96a 12b c c La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ –293 ± 37 –92 ± 9 3.5 ± 1.0c
Young’s Poisson’s Thermal expansion modulusd ratio coefficient (GPa) (×10−6 K−1 )
17b 1.5 ± 0.5c 46 ± 4c c 65 ± 6 2.2 ± 0.4c 45 ± 4c
0.3c 0.3e
9.65 ± 0.1c 9.78 ± 0.1c
a Taken
from Ref. [37]. from Ref. [34]. c This study. d From 4-point bending tests at room temperature. e Assumed value. b Taken
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Fig. 10.35 The calculated (a) temperature and (b) PO2 distributions at the interconnector for the standard counter-flow condition #1 in Table 10.2 using the single-unit model.
Next, as other examples of the simulations for the calculating conditions in Table 10.2, the calculated δ distribution in the interconnector for the pre-reforming and co-flow pattern cases are shown in Figures 10.38a and 10.39a, respectively. In
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Fig. 10.36 The contour plot of the calculated δ distribution in the interconnector for the standard counter-flow condition #1 in Table 10.2.
Fig. 10.37 The contour plot of the calculated δ distribution in the interconnector for the standard counter-flow condition #1 in Table 10.2 assuming that Dchem is equal to Dv .
Figures 10.38b and 10.39b, the previously calculated temperature profiles in the interconnector are also shown. In Figure 10.38a, it is characteristic that δ is maximum at the fuel inlet and decreases steadily along the fuel flow. This is because both the
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Fig. 10.38 The simulated (a) δ distribution and the (b) previously calculated temperature distribution in the interconnector for the pre-reforming case #2 in Table 10.2.
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Fig. 10.39 The simulated (a) δ distribution and the (b) previously calculated temperature distribution in the interconnector for the co-flow case #4 in Table 10.2.
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fuel temperature and PO2 decrease monotonously from the fuel inlet to the fuel outlet. As compared with standard condition #1, the maximum δ in the interconnector is large for the pre-reforming case. One reason is that the maximum temperature in the interconnector is higher than that for the standard case, and another reason is that PO2 is the largest at the fuel inlet where both the temperature and the H2 and CO concentrations are maximum. For the co-flow case, the fuel temperature decreases first and then increases gradually with the fuel flow as shown in Figure 10.39b, while PO2 decreases monotonously from the fuel inlet to the fuel outlet. Accordingly, δ reaches its maximum value in the middle area along the fuel flow in the δ profile. It can be observed that the gradient of δ is gradual, and in addition, the magnitude of the maximum δ is small as compared with those for the standard and pre-reforming cases. Figures 10.40a–10.40c show the simulated δ distribution in the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector for the standard, pre-reforming, and co-flow cases, respectively. As compared with the results for the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector, the magnitudes of the maximum δ are small under all the conditions whereas the contour plots of δ are similar to those for the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector. Comparing the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ and La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnectors, Kox is almost the same, and the difference in the maximum δ in the interconnector is due to the difference in the amount of the doped Sr. For the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector, x is twice that of the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ , and δ depends on x, as seen from Equation (10.38). With an increase in x, δ also increases so that the charge neutrality condition will be met; hence in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector, δ is greater than that in the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector. In this study, the magnitude of the chemical expansion under reducing atmospheres at a high temperature is assumed to be linearly proportional to δ, and thus, the magnitude of the chemical expansion of the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector will be smaller than for La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ .
10.4.3.4 Expansion and Stress From the above-mentioned δ profile, the chemical expansion of the interconnector and related internal stresses are calculated. In this calculation, it is assumed that no mechanical loads are placed on the interconnector and the interconnector can deform freely (i.e., without constraint condition). The simulated distribution of the internal stress in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector is shown in Figure 10.41a under the standard condition. In the figure, only half the model (i.e., model geometry with 8 channels) is drawn, and the left side of the drawn interconnector corresponds to the symmetrical center. It can be observed that a stress concentration occurs near the fuel inlet or at the right edge. In order to clarify the contribution of the internal stress caused by the temperature distribution to the total internal stress, the thermal stress profile in the interconnector is calculated and shown in Figure 10.41b. By comparing Figure 10.41a with Figure 10.41b, it is evid-
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Fig. 10.40 The simulated δ distribution in the La0.9 Sr0.1 Cr0.98 Co0.0 2O3 interconnector for (a) the standard, (b) pre-reforming, and (c) co-flow cases.
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Fig. 10.41 The calculated distribution of the principal stress in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector for the standard counter-flow case #1 in Table 10.2; (a) the stress is calculated considering both the temperature and δ distributions in the interconnector, (b) the stress is only the thermal stress. The model geometry with 16-channels is used, and half the model is drawn in the figure.
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ent that the stress concentration at the edge is mainly attributed to the temperature distribution since the magnitude of the stress at the edge is almost the same for both Figures 10.41a and 10.41b. On the other hand, near the fuel inlet, the internal stress is larger when both the temperature and the δ distribution are taken into account. Hence, the stress concentration near the fuel inlet might be due to both the temperature and δ distribution. Figures 10.41 and 10.43 display the calculated profile of: (a) the total stress, and (b) the thermal stress in the interconnector for the prereforming and the co-flow cases, respectively. For the pre-reforming case, although the maximum temperature in the interconnector is higher than that for the standard case, the maximum tensile stress caused by both the temperature distribution and chemical expansion is smaller than that for the standard case. This is because the contribution of the thermal stress to the total stress near the fuel inlet is smaller than that for the standard case, as shown in Figure 10.42b. For the co-flow case, as compared with the result for the standard condition, it can be observed that the stress concentrated region spreads widely from the fuel inlet to the middle channel length, and in addition, the stress concentration near the edge is not obvious. This is because the temperature drop caused by the endothermic methane reforming reaction is not steep as compared with the standard case, and the maximum temperature region is located at the middle channel length. The maximum tensile stress caused by both the temperature distribution and chemical expansion is smaller than that for the standard case. The reason is that the maximum temperature in the interconnector is lower than that for the standard case, and thus, the oxygen vacancy in the interconnector is also smaller. Figures 10.44a–10.44c show the calculated profiles of the total stress in the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector for the standard, pre-reforming, and coflow cases, respectively. For all the cases, the calculated maximum stresses in the interconnector are smaller than those for the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector. This is due to the small contribution of the chemical expansion to the total stress. For the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector, as compared with the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector, the oxygen vacancy concentration inside is small, as shown in Figure 10.40. Hence, the chemical expansion due to a small stress is smaller than that for the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnectors. All the calculated stresses in the interconnector are listed in Table 10.7. From this simulation, the following findings are obtained. First, for both La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ and La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnectors, the total stresses are greater than the thermal stresses. In particular, when the thermal stresses are relatively small (e.g., the co-flow case or pre-reforming case), the total stresses are two to four times greater than the thermal stresses. Hence, the chemical expansion of the interconnector will induce large tensile stresses in the interconnector itself. However, by employing the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector, it is possible to reduce the chemical expansion, and thus, the total stress. In the case of co-flow in particular, the total stress can be lowered to around 20 MPa by employing a co-flow pattern. Secondly, although the internal methane reforming causes a large thermal stress in the interconnector, it dose not directly induce it in the interconnector. Since the magnitude of the chemical expansion is largest near
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Fig. 10.42 The calculated distribution of the principal stress in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector for the pre-reforming case #2 in Table 10.2; (a) the stress is calculated considering both the temperature and δ distributions in the interconnector, (b) the stress is only the thermal stress. The model geometry with 16-channels is used, and half the model is drawn in the figure.
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Fig. 10.43 The calculated distribution of the principal stress in the La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ interconnector for the co-flow case #4 in Table 10.2; (a) the stress is calculated considering both the temperature and δ distributions in the interconnector, (b) the stress is only the thermal stress. The model geometry with 16-channels is used, and half the model is drawn in the figure.
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Fig. 10.44 The calculated distribution of the principal stress in the La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ interconnector for (a) standard, (b) pre-reforming, and (c) co-flow cases. The model geometry with 16-channels is used, and half the model is drawn in the figure.
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Table 10.7 Calculated thermal and chemical stresses in doped LaCrO3 under cell operation conditions. Calculated stress (MPa) Condition
Thermal stress
ID #1 #2 #3 #4 #5 #6
60 22 14 12 27 66
Total stress La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ
La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ
91 66 64 48 68 103
65 40 36 23 40 81
the maximum temperatures, isolating the area where the temperature is the maximum from the area at which the methane reforming occurs (e.g., the co-flow case) is effective to reduce the total stress in the interconnector.
10.5 Measurements for Stresses and Deformation of the Cells 10.5.1 Experimental Techniques for Stress Measurements Here, the evaluation of the residual stresses in the electrolyte of the anode-supported cell at room temperature is reported. The X-ray diffraction method is used to measure the residual stresses in the electrolyte of the anode-supported cell, and a synchrotron radiation is used as an excellent X-ray source; this enables the estimation of the residual stresses in the electrolyte with a high accuracy. It is clarified that the measured stresses are close to the calculated stresses. The X-ray diffraction pattern for the electrolyte of the anode-supported cell is measured and the residual stresses were estimated by the sin2 ψ method [40–44]. When a sample is sustaining stresses, the inter-planar spacing d between the specified diffracting planes of a crystallite in a microscopic grain can be expressed as d=
1+ν ν d0 σ sin2 ψ + 1 − (σ1 + σ2 ) d0 , E E
(10.43)
where d0 is the inter-planar spacing under a stress-free condition; ψ, tilting angle of the specified diffracting plane in the crystallite from the normal to the sample surface; ν, Poisson’s ratio; E, Young’s modulus; σ , the stress in the grain; and σ1 and σ2 , the principal stresses in the direction 1 and 2. Modifying Equation (10.43) σ is expressed by 1 ∂d E × × . (10.44) σ = 1+ν d0 ∂ sin2 ψ
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Fig. 10.45 Schematic diagram of the X-ray stress measurement, iso-inclination method, and fixed ψ0 method. The tilt of the sample is (a) +ψ and (b) −ψ.
To derive Equation (10.44), it is assumed that the sample is isotropic and elastic. In addition, there is no stress perpendicular to the sample surface, because only the stresses near the sample surface are measured using the X-ray diffraction method. In the stress measurement, the iso-inclination method and the fixed ψ 0 method are employed. The configuration for the diffraction measurements and stress analyses is indicated in Figure 10.45. In Figure 10.45, ψ 0 is the angle between the normal to the sample surface and the incident beam and 2ψ is the scattering angle of the specified diffracting plane in the crystallite. In the fixed ψ 0 method, 2θ scan is carried out at a fixed ψ 0 . +ψ and −ψ are determined as follows (+ψ: the sample normal is tilting to the diffraction beam, −ψ the sample normal is tilting to the incident beam). To eliminate the effect of an orientation of the crystallite or an irregularity of the grain on the stress measurements, the sample is fluctuated by ±1 degree around the fixed ψ 0 with a sufficiently high speed as compared with the scanning speed of 2θ . Synchrotron radiation is used as an X-ray source. The synchrotron radiation is an ultra-bright, highly directional, and collimated light. In addition, a wide spectrum of wavelengths ranging from infrared to X-rays can be used. Using the synchrotron radiation as the beam source in the residual stress measurements has the following advantages [43]: 1. A beam with an arbitrary wavelength is available, and thus, stress measurement for an arbitrary diffracting plane with a high accuracy is possible. 2. The beam is a monochromatic light, and thus, it is not necessary to consider the effect of Kα1,α2 doublet. The experiment is performed at the BL09XU line at SPring-8, which is a standard Xray undulator beam line. For the stress measurements, an X-ray energy of 8.05 keV (λ = 0.1541 nm), which corresponds to the Cu Kα line, is selected to compare the measured results with those measured using a commercial X-ray diffraction apparatus. The attenuation length of the X-ray of λ = 0.1541 nm in ZrO2 at a fixed angle of 90 degrees is calculated to be around 15 µm. The typical thickness of the
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Fig. 10.46 Side view of the warped anode-supported cell. The electrolyte is located on the upper side.
electrolyte is around 30 µm, which is almost triple that of the attenuation length of the X-ray in ZrO2 , and thus, the contribution of the anode to the diffraction beam is negligible. The beam size at the sample point is adjusted to 1 mm (H) × 1 mm (V) using W slits. The diffracting plane used for the stress measurements (531) of 8YSZ is selected. The anode-supported cells used for the stress measurements were fabricated by co-firing both the electrolyte and anode at 1500◦C [45]. The 8YSZ electrolyte is first screen printed or dip-coated on the green NiO/YSZ substrate. To avoid generations of cracks and micro-pores in the electrolyte, the particle sizes of the raw powders for the NiO/YSZ substrate and YSZ electrolyte are carefully selected. After co-firing, the anode-supported cell is cooled down to room temperature. A typical cell size is 40 mm × 40 mm × 2 mm, and the thickness of the electrolyte is about 20–40 µm (the image is shown in Figure 10.18). As compared to the anode substrate, the electrolyte is sufficiently thin for the assumption that the direction of the main stress in the electrolyte is parallel to the film surface. To investigate the stress distribution in the anode-supported cell, two 10 mm × 10 mm pieces are cut from the center and corner of the sample. This is due to the restriction on the size of the sample stage. Here, the effect of cutting the sample on the residual stress is discussed again. From the stress calculations in Section 10.4.1, the effect of the cut sample on the residual stress may be negligible. In this measurement, the residual stresses for seven different samples are measured. The characteristics of the samples are listed in Table 10.8. Samples #1 and #2 are cut from an as-grown cell. Usually, the as-grown cell warped toward the electrolyte side, as shown in Figure 10.46. If the cells warp, it is difficult to properly contact each cell well, and the electric connection between the cells
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Table 10.8 Properties of samples used for stress measurement. ID
Electrolyte material
Manufacturing process
Post treatment
Anode state
Position
#1 #2 #3 #4 #5 #6 #7
8YSZ 8YSZ 8YSZ 8YSZ 8YSZ 8YSZ 8YSZ
screen-print screen-print screen-print screen-print screen-print screen-print dip-coat
as-sintered as-sintered flattened flattened flattened flattened flattened
oxidized oxidized oxidized oxidized reduced reduced oxidized
center corner center corner center corner corner
Fig. 10.47 Comparison of the lattice structure and relevant lattice constant between NiO and Ni.
becomes poor. Hence, the warp is reformed and the cell is flattened using a distributed load on the electrolyte at 1400◦C. Samples #3 and #4 are cut from the flattened cell. Before a cell operation, the anode of the cell has to be exposed to a reducing atmosphere to reduce NiO to Ni in the anode. Samples #5 and #6 are the reduced samples. The lattice constant of Ni is smaller than that of NiO by 20%, as illustrated in Figure 10.47. As a result, the reduction of the anode causes a shrinkage of the anode and may change the residual stress in the electrolyte. For samples #1–6, the electrolyte is screen printed on the surface of the anode substrate first and cofired. Only for sample #7, the electrolyte is dip-coated on the surface of the anode substrate.
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Fig. 10.48 Configuration of the specimen surface shape measurement.
Fig. 10.49 Configuration of the load-deformation test. The cell is sandwiched between two rigid plates, and a load is placed on the top of the plate.
10.5.2 Experimental Techniques for Deformation Measurements The deformation of the cell (i.e. the warp of the cell) is measured using a laser beam. The picture of the configuration of the measurement is shown in Figure 10.48. A sample is set on an X-Y stage and a spot of laser beam is irradiated to the surface of the sample. A detector detects the reflection of the beam and the distance between the detector and the spot using the Doppler effect. The sample stage can move in a horizontal direction and the laser spot trace the surface of the sample. The relative vertical distance from the reference point to the sample surface is measured and the sample surface shape can be imaged.
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Fig. 10.50 X-ray diffraction pattern for the electrolyte of the anode-supported cell. The diffraction peaks by arrows are used for the stress measurements.
10.5.3 Load-Deformation Tests A change in the warp of the cell with a load is measured. The configuration of the load test is shown in Figure 10.49. The cell is located on a flat rigid plate with the cathode side up, and the cathode side of the cell is pushed with another flat rigid plate. The load is placed on the cell through a plate in a perpendicular direction to the cell surface, and the displacement of the upper plate from the initial position at which the plate contacts the cell is measured with increasing load. In addition to the load test, the contacting pressure between the cell and plate is measured using a sensor sheet that can detect the area pressure. The sensor sheet is inserted between the cathode and plate, and the contacting pressure between the cell and plate is measured with increasing load.
10.5.4 Numerical Model Validation by Experiments Results 10.5.4.1 Stress Measurements for Single Cell & Stack Figure 10.50 shows the result of θ –2θ scan for the electrolyte of the sample #3 in Table 10.8. First, the sample is set so that the sample surface is parallel to the incident beam, and then the incident beam is focused on the center of the sample. All the peaks in the diffraction pattern can be assigned to the cubic YSZ structure. There are no diffraction peaks from the anode substrate.
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Fig. 10.51 Shift of the diffraction peak of (531) plane with ψ.
Fig. 10.52 d −sin2 ψ plot measured for the electrolyte of the anode-supported cell. The diffraction peak of (531) plane is used to calculate the plane distance d.
Figure 10.51 shows the change in the diffraction peak of the YSZ (531) plane with a change in ψ measured for sample #3. With increasing ψ, the peak position shifts to a higher angle, indicating that the inter-planar spacing d decreases and the stress type is compressive. To determine the center of the peak, the peak profiles are fitted with the Gaussian function, and the d −sin2 ψ diagram thus obtained is shown in Figure 10.52. The open circles and open squares represent positive and negative ψ, respectively, and the solid line is the linear least-squares fit. Although the data are slightly scattered d decreases linearly with an increase in sin2 ψ. From the slope of
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Table 10.9 Stiffness of cubic YSZ used for calculation of elastic constant.
Electrolyte a Taken
C11 (GPa)
C12 (GPa)
C44 (GPa)
(1 + ν)/E (10−4 /GPa)
400a
110a
55a
182
from Ref. [47].
Table 10.10 List of the results of the stress measurements for the electrolyte. Sample ID
#1
#2
#3
#4
#5
#6
#7
Calculated stress [MPa]
774
672
633
677
658
678
676
the d − sin2 ψ diagram, the residual stress can be estimated using Equation (10.44). In the X-ray stress measurements, the stress in a sample can be estimated using the diffraction from the specified diffracting plane of crystallites, which orients to the specified direction. Therefore, the elastic constant E and ν in Equation (10.44) is different from those measured by a mechanical method, and these elastic constant should be measured using the X-ray diffraction method. Here, the X-ray elastic constant is calculated by the Voight model [46] using the reported elastic constant for cubic YSZ [47]. The reported elastic constant for cubic YSZ and the calculated E/(1 + ν) value are listed in Table 10.9. For sample #3, the estimated residual stress in the electrolyte is a compressive stress of 633 MPa. For all the other samples, the estimated residual stresses in the electrolyte are listed in Table 10.10. From this measurements and stress simulation for the electrolyte of the anode-supported cell, it has been clarified that the residual stresses in the electrolyte at room temperature are compressive stresses of around 650 MPa. As mentioned in Section 10.4.1, the simulated residual stresses are from 640 to 700 MPa and this measurement revealed the validity of the simulations on the residual stress. From the present results, the following information is obtained: (a) From a comparison of #1 and #2, the residual stress is larger at the center of the cell. The origin of the difference is the macro deformation of the cell (i.e., the warp of the cell). (b) By a flattening treatment, the residual stress at the center of the cell is lowered, and the distribution of the residual stress in the electrolyte becomes small. This means that the electrolyte is in a plastic state at 1400◦C. (c) The effect of the reducing treatment of the anode on the residual stress is small. This may be because the Young’s modulus of the anode became small due to the reduction. In other words, the effects of the softening of the anode may cancel the effect of the expansion on the residual stress. (d) The residual stress is independent of the fabrication method of the electrolyte. This is reasonable because the origin of the residual stress may be only the mismatch of TEC between the electrolyte and anode.
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Fig. 10.53 Measured surface shape of the anode-supported cell; (a) on the anode side and (b) on the cathode side.
Fig. 10.54 Calculated change in the magnitude of the warp for the anode-supported cell at room temperature. The warp is plotted with a relative thickness between the electrolyte and anode.
(e) The strength of YSZ against the compressive stress is more than 1 GPa [26], and thus, these residual stresses are not sufficiently large to induce a failure of the electrolyte.
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Fig. 10.55 Measured displacement of the top position of the cell from the initial position with increasing load.
10.5.4.2 Deformation Measurements Figure 10.53 shows the measured surface shape of the warped cell. The cathode side is a convex shape and the top of the cell is warped by more than 2 mm from the corner position in a perpendicular direction to the plane for the 10 mm × 10 mm sample. From the simulation, it is found that the magnitude of the warp depends on the relative thickness between the anode substrate and electrolyte, as shown in Figure 10.54. The warp increases with the increase in the ratio of the anode thickness to electrolyte thickness. The calculated value of the warp is very close to the measured one indicating that this simulation is reliable in predicting the warp with various combinations of the thickness between the anode and electrolyte.
10.5.4.3 Load-Deformation Tests The measured magnitude of the warp of the cell decreases under a load in the perpendicular direction to the cell surface, as shown in Figure 10.55. In the figure, at a small load, the top position of the cell changes linearly with the load indicating a linear decrease in the warp with the load. At a load more than 200 N, the plate position does not change with the load. This means that the center of the concave anode side touches the bottom plate and the cell becomes flat. The results of the measurements for the contacting pressure that is mentioned later revealed that the cell can be flattened with a load greater than 200 N. Figure 10.56 shows the results of the measurement for the contacting pressure between the cathode and flat plate under the load tests. It can be seen that at a small load, contacting pressure appears only at the center of the cell. As the load is increased, the contacting area spreads across the
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Fig. 10.56 Measured surface pressure at the cell surface with increasing load. The contacting area spreads with the increasing load.
Fig. 10.57 Calculated stress distribution at the anode under a load of 1000 N. The deformed cell shape is drawn.
cell. At a load more than 100 N, the edge of the cell contacts the flat plate, which suggests that the cell is almost flattened. At a load more than 500 N, the change in
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the contacting area is slight, which is consistent with the results of the load test. The load test is simulated using the finite element method. The model used for the simulation is the warped cell and the existence of the residual stress is assumed. The simulation technique is almost the same as that mentioned in Section 10.4.1. The only difference is that a mechanical load is assumed to be put on the center of the cathode (i.e. at the center of the warped cell). As the boundary condition the four corner point of the below surface of the cell is constrained in the perpendicular direction. The result of the simulation is shown in Figure 10.57. The deformed shape of the cell and the stress distribution in the anode at a load of 1000 N are plotted in the figure. It can be seen that the center of the cell is flat and the cross sectional shape looks like the “M” character. This means that the edge and center part of the cell contact well the bottom flat plate and thus the result of the simulation is consistent with the result of the load test. From the simulation it is found that a tensile stress of around 60 MPa occurs in the anode under a load, which is below the mechanical strength of the anode material.
References 1. Patil, R.N. and Subbarao, E.C., Acta Cryst A26, 1970, 535. 2. Oikawa, K., Kamiyama, T., Hashimoto, T., Shimojyo, Y. and Morii, Y., Structural phase transition of orthorhombic LaCrO3 studied by neutron powder diffraction, Journal of Solid State Chemistry 154, 2000, 524. 3. Yasuda, I. and Hishinuma, M., Electrical conductivity and chemical diffusion coefficient of strontium-doped lanthanum manganites, Journal of the Electrochemical Society 143, 1996, 1583. 4. Ishihara, T., Matsuda, H. and Takita, Y., Doped LaGaO3 Perovskite type oxide as a new oxide ionic conductor, Journal of the American Chemical Society 116, 1994, 3801. 5. Tang, E., Prediger, D., Pastula, M. and Borglum, B., The status of SOFC development and Versa Power Systems, in Proceedings of Solid Oxide Fuel Cells IX, The Electrochemical Society Proceedings, Electrochemical Society, Pennington, NJ, 2005, p. 89. 89(2005). 6. George, R. and Casanova, A., Developments in Siemens Westinghouse SOFC program, in Proceedings of 2003 Fuel Cell Seminar, 2003, p. 895. 7. Yakabe, H., in Proceedings of INDIN06, Singapore, 2006, in press. 8. Zienkiewicz, O.C. (Ed.), The Finite Element Method, third edition, McGraw-Hill Book Company, 1976. 9. Yakabe, H., Ogiwara, T., Hishinuma, M. and Yasuda, I., 3-D nodal calculation for planar SOFC, Journal of Power Sources 4426, 2001, 1. 10. Stolten, D., Froning, D. and de Haart, L.G.J., in Proceedings of European Solid Oxide Fuel Cell Forum IV, 2000, p. 347. 11. Achenbach, E., Three-dimensional and time-dependent simulation of a planar solid oxide fuel cell stack, Journal of Power Sources 52, 1994, 282. 12. Lee, A.L., Zabranski, R.F. and Huber, W.J., Internal reforming development for solid oxide fuel cells, Ind. Eng. Chem. Res. 29, 1990, 766. 13. Matsuzaki, Y. and Yasuda, I., Electrochemical characteristics of a Ni-YSZ cermet electrode of YSZ in a H2 -H2 O-CO-CO2 system, in Proceedings of Solid Oxide Fuel Cells VI, S.C. Singhal and M. Dokiya (Eds.), 1999, p. 560. 14. Herbin, R. and Fiard, J.M., Three-dimensional numerical simulation of the temperature, potential and concentration for various geometries of SOFCs, in Proceedings of 1st European Solid Oxide Fuel Cell Forum, U. Bossel (Ed.), 1994, p. 317.
394
H. Yakabe
15. Selimovic, A, in Proceedings of European Solid Oxide Fuel Cell Forum IV, 2000, p. 375. 16. Yakabe, H. and Sakurai, T., 3D simulation on the current path in planar SOFCS, Solid State Ionics, 174, 2004, 295. 17. Bernier, M., Herbin, R. and Gehain, E., Stack and cell modeling with SOFC3D: A computer program for the 3D simulation of solid oxide fuel cells, in Proceedings of European Solid Oxide Fuel Cell Forum II, B. Thorstensen (Ed.), 1996, p. 203. 18. Herbin, R., Fiard, J.M. and Ferguson, J.R., Three dimensional numerical simulation of the temperature, potential and concentration for various geometries of SOFCs, in Proceedings of 1st European Solid Oxide Fuel Cell Forum, U. Bossel (Ed.), 1994, p. 317. 19. Primdahl, S., Jorgensen, M.J., Bagger, C. and Kindl, B., Thin anode supported SOFC, in Proceedings of Solid Oxide Fuel Cells VI, S.C. Singhal and M. Dokiya (Eds.), 1999, p. 793. 20. Ouweltjes, J.P., van Berkel, F.P.F., Nammensma, P. and Chrisite, G.M., Development of 2nd generation, supported electrolyte, flat plate SOFC components as ECN, in Proceedings of Solid Oxide Fuel Cells VI, S.C. Singhal and M. Dokiya (Eds.), 1999, p. 803. 21. Stover, D., Diekmann, U., Flesch, U., Kabs, H., Quadakkers, W.J., Tietz, F. and Vinke, I.C., Recent developments in anode supported thin film SOFC at Research Center Julich, in Proceedings of Solid Oxide Fuel Cells VI, S.C. Singhal and M. Dokiya (Eds.), 1999, p. 812. 22. Grobal Thermoelectric Inc., SOFC developments at global thermoelectric, in Abstracts of 2002 Fuel Cell Seminar, 2002, p. 974. 23. Stevenson, J.W. and Singh, P., Solid oxide fuel cell development at Pacific Northwest National Laboratory, in Abstracts of 2002 Fuel Cell Seminar, 2002, p. 890. 24. Yakabe, H. and Sakurai, T., Analyses on the performance of anode-supported SOFC stacks, in Proceedings of 2003 Fuel Cell Seminar, 2003, p. 295. 25. Badwal, S.P.S., Bannister, M.J. and Hannink, R.H.J. (Eds.), Science and Technology of Zirconia V, Technology Publishing Company, 1993. 26. Kato, T., Wang, N.S., Negishi, A., Momma, A., Kasuga, Y. and Nozaki, K., in Proceedings of the 3rd International Fuel Cell Conference, 1999, p. 461. 27. Ogiwara, T., Yakabe, H., Hishinuma, M. and Yasuda, I., in Proceedings of European Solid Oxide Fuel Cell Forum IV, 2000, p. 193. 28. Greiner, H., Keim, E., Kleinlein, W. and Weis, E., Mechanical testing of thin electrolyte materials for solid oxide fuel cells using the concentric ring test, in Proceedings of Solid Oxide Fuell Cells II, F. Grosz, P. Zegers, S.C. Singhal and O. Yamamoto (Eds.), 1991, p. 705. 29. Lowrie, F.L. and Rawlings, R.D., Room and high temperature failure mechanisms in solid oxide fuel cell electrolytes, Journal of European Ceramic Society 20, 2000, 751. 30. Flandermeyer, B.K., Nasrallah, M.M., Sparlin, D.M. and Anderson, H.U., High Temp. Sci. 20, 1985, 259. 31. Mizusaki, J., Yamauchi, S., Fueki, K. and Ishikawa, A., Nonstoichometry of the Perovskitetype La1−x Srx CrO3−δ , Solid State Ionics 12, 1984, 119. 32. Yasuda, I. and Hishinuma, M., Lattice expansion of acceptor-doped lanthanum chromites under high-temperature reducing atmospheres, Electrochemistry 68, 2000, 526. 33. Yakabe, H., Hishinuma, M. and Yasuda, I., Static and transient model analysis on expansion behavior of LaCrO under an oxygen potential gradient, Journal of The Electrochemical Society 147, 2000, 4071. 34. Yasuda, I. and Hishinuma, M., Electrochemical properties of doped lanthanum chromites as interconnectors for solid oxide fuel cells, Journal of The Electrochemical Society 143, 1996, 1583. 35. Kröger, F.A. and Vink, H., Relations between the concentrations of imperfections in crystalline solids, Solid State Phys. 3, 1956, 307. 36. van Roosbroeck, W., The transport of added current carriers in a homogeneous semiconductor, Phys. Rev. 91, 1953, 282. 37. Yasuda, I. and Hishinuma, M., Electrical conductivity and chemical stability of calcium chromate hydroxyl apatite, Ca5 (CrO4)3 OH, and problems caused by the apatite formation at the electrode/separator interface in solid oxide fuel cells, Solid State Ionics 80, 1995, 141.
10 Integrated Numerical Modeling of SOFCs
395
38. Yasuda et al. plotted the data of the electrical conductivity at 1273 K in pure oxygen as a function of the dopant concentration for 12 samples of non-doped, Sr-doped, and Ca-doped LaCrO3 , and revealed that the conductivity of the alkaline-earth metal doped LaCrO3 is proportional to the concentration of the doped alkaline-earth metal. The conductivities of La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ and La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ fall on the straight line, which means that Ni or Co doped on the Cr-site will be treated as behaving similarly to Cr3+ . 39. S 0 and H 0 of La0.8 Sr0.2 Cr0.95 Ni0.05 O3−δ and La0.9 Sr0.1 Cr0.98 Co0.02 O3−δ listed in Table 10.3 are close to the reported values of La0.8 Sr0.2 CrO3−δ and La0.9 Sr0.1 CrO3−δ [J. Mizusaki, S. Yamaguchi, K. Fueki and A. Ishikawa, Solid State Ionics 12, 1984, 119.], respectively. 40. Noyan, I.C. and Cohen, J.B., X-ray diffraction study of the residual stress-strain distributions in shot-peened two-phases brass, Mater. Sci. Eng. 75, 1983, 179. 41. Noyan, I.C. and Cohen, J.B., Determining stresses in the presence of nonlinearities in interplanar spacing VS. Sin∗∗ 2 psi, Adv. X-Ray Anal. 27, 1984, 129. 42. Tanaka, K., Yamamori, Y., Mine, N. and Suzuki, K., in Proceedings of the 32nd Japan Congress on Materials Research, 1989, p. 199. 43. Yoshioka, Y., Stress measurement in stainless steel by use of monochromatic Cr-Kβ X-rays and a position sensitive detector, Adv. X-ray Anal. 24, 1981, 167. 44. Barral, M., Sprauel, J.M., Lebrun, J., Maeder and Megtert, S., On the use of synchrotron radiation for the study of the mechanical behaviour of materials, Adv. X-ray Anal. 27, 1984, 149. 45. Yasuda, I., Uratani, M., Itoh, K., Yakabe, H. and Matsuzaki, Y., in Proceedings of Fuel Cell Seminar, 2000, p. 574. 46. Voight, W., Lehrbuch der Kristallphysik, Teubner, Berlin, 1928, p. 962. 47. Botha, P.J., Chiang, J.C.H., Mjawara, P.M. and Ngoepe, P.E., Behavior of elastic constants, refractive index, and lattice parameter of cubic zirconia at high temperatures, Journal of Applied Physics 73(11), 1993, 7268. 48. Greiner, H., Keim, E., Kleinlein, W. and Weis, E., Mechanical testing of thin electrolyte materials for solid oxide fuel cells using the concentric ring test, in Proceedings of Solid Oxide Fuel Cells II, F. Grosz, P. Zegers, S.C. Singhal and O. Yamamoto, The Electrochemical Society, Pennington, NJ, 1991, p. 705. 49. Mizutani, Y., Tamura, M., Kawai, M., Nomura, K. and Nakamura, Y., Characterization of the Sc2 O3 -ZrO2 system and its application as the electrolyte in planar SOFC, in Proceedings of Solid Oxide Fuel Cells IV, M. Dokiya, O. Yamamoto, H. Tagawa and S.C. Singhal (Eds.), The Electrochemical Society, Pennington, NJ, 1995, p. 301. 50. Atkinson, A. and Selquk, A., Mechanical properties of ceramic materials for solid oxide fuel cells, in Proceedings of Solid Oxide Fuel Cells V, U. Stimming, S.C. Singhal, H. Tagawa and W. Lehnet (Eds.), The Electrochemical Society, Pennington, NJ, 1997, p. 671. 51. Yakabe, H. and Yasuda, I., Model analysis of the expansion behavior of LaCrO3 interconnector under solid oxide fuel cell operation, Journal of The Electrochemical Society 150, 2003, A35. 52. Ogiwara, T., Yakabe, H., Hishinuma, M. and Yasuda, I., Assessment of mechanical reliability of planar SOFCs, in Proceedings of European Solid Oxide Fuel Cell Forum IV, 2000, p. 193.
Fuel Cells and Hydrogen Energy 1.
R. Bove and S. Ubertini: Modeling Solid Oxide Fuel Cells. Methods, Procedures and Techniques. 2008 ISBN 978-1-4020-6994-9
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