1 MULTI-PHYSICS AND MULTI-SCALE MOD ELING AND SIMULATION OF THERMALFLUID AND ELECTROCHEMICAL TRANSPORT IN A SOLID OXIDE FUEL CELL By YAN JI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA AUGUST 2007
2 Copyright 2007 Yan Ji
3 ACKNOWLEDGMENTS First, I would like to express my greatest a ppreciation to my advisor, Dr. Jacob N. Chung, for his continuous support, encouragement, motivation and guidance throughout my work. Without his direction and support, this work would not have been possible. My sincere thanks are extended to my co mmittee members, Drs. James F. Klausner, Renwei Mei, Malisa Sarntinoranont, and Eric D. Wachsman. Their encouragement and valuable advice are greatly acknowledged. I would also like to recognize my fellow graduate associ ates Renqiang Xiong, Jun Liao, Yun Whan Na, Shalabh Maroo, Al phna Agarwal, Tae-seok Lee, Mo Bai and Peter Tai for their kindly assistance. I am deeply indebted to my parents and my husband for their never ending love, dedication and support through my long journey of study.
4 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................3 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 LIST OF TERMS... ABSTRACT....................................................................................................................... ............17 CHAPTER 1 INTRODUCTION.................................................................................................................19 1.1 Energy Crisis and Hydrogen Economy............................................................................19 1.2 History of Fuel Cell Development....................................................................................23 1.3 Solid Oxide Fuel Cell...................................................................................................... .28 1.4 Scope...................................................................................................................... ...........31 2 LITERATURE REVIEW AND OBJECTIVES....................................................................33 2.1 Literature Review for Fuel Cell Modeling.......................................................................33 2.1.1 System and Thermodynamics Modeling................................................................33 2.1.2 Coupled Electrochemical, Flow and Thermal Modeling for Stack........................34 2.1.3 Chemical Reaction Rate Modeling.........................................................................36 2.1.4 Hydrogen/oxygen Reduction Mechan ism and Electrode Modeling......................38 220.127.116.11 One-dimensional models based on mass transfer analysis...........................38 18.104.22.168 Oxygen reduction reaction and chemi cal kinetics modeling in cathode......39 22.214.171.124 Monte-Carlo simulations based on random packing of spherical particles.................................................................................................................40 126.96.36.199 Continuum micro-scale models base d on the statistical properties..............41 2.1.5 Molecular-level Modeling......................................................................................43 2.1.6 Unresolved Issues in SOFC Modeling...................................................................43 2.2 Objectives................................................................................................................. ........44 3 DEVELOPMENT OF A MACROSCALE MODEL AND PERFORMANCE OPTIMIZATION FOR A SOFC UNIT CELL......................................................................49 3.1 Electrochemistry Fundamentals........................................................................................49 3.2 Electrical Potential Loss.................................................................................................. .52 3.3 Three-dimensional Thermo-fluid Electro chemical Model for a SOFC Unit Cell............56 3.3.1 Development of Modeling......................................................................................56 188.8.131.52 Model features and assumptions..................................................................56 184.108.40.206 Thermal-fluid model....................................................................................58
5 220.127.116.11 Electrochemical model.................................................................................61 3.3.2 Solution Procedure and Boundary Conditions.......................................................64 3.4 Model Application for an Anode-supported SOFC..........................................................68 3.4.1 Choice of Parameters..............................................................................................68 3.4.2 Results and Analysis...............................................................................................68 18.104.22.168 Velocity field................................................................................................68 22.214.171.124 Temperature field.........................................................................................69 126.96.36.199 Concentration field and current density.......................................................70 188.8.131.52 Conclusion....................................................................................................72 3.5 Effect of Transport Scale fo r an Electrolyte-supported SOFC.........................................72 3.5.1 Problem of Statement.............................................................................................72 3.5.2 Parameters and Additional Equations....................................................................74 3.5.3 Simulation Results and Discussion........................................................................76 184.108.40.206 Baseline case................................................................................................76 220.127.116.11 Effects of gas flow channel dimension........................................................78 18.104.22.168 Evaluation of thin-film electrolyte...............................................................82 22.214.171.124 Optimization of rib width.............................................................................84 3.5.4 Conclusions............................................................................................................86 4 MICRO-SCALE MODELING FOR ELECTRODES...........................................................88 4.1 Random Medium Models and Objectives........................................................................88 4.2 Key Issues in Random Electrode Modeling.....................................................................94 4.2.1 Pore Structure.........................................................................................................95 4.2.2 Three Phase Boundary and Reacti on in Porous Electro-ceramics.........................96 4.2.3 Diffusion Mechanisms............................................................................................97 4.3 Random Network Modeling for Optim ization of Composite Cathode............................99 4.3.1 Physical Description and Assumptions..................................................................99 4.3.2 Derivation of Numerical Model...........................................................................101 126.96.36.199 Electrical network......................................................................................102 188.8.131.52 Diffusion and reaction rate.........................................................................106 4.3.3 Reconstruction of Microstruc ture and Boundary Conditions...............................108 4.3.4 Model Input Parameters.......................................................................................112 4.3.5 Numerical Results and Discussions......................................................................112 184.108.40.206 Contours of potential a nd oxygen concentration........................................112 220.127.116.11 Activation and concentr ation overpotentials..............................................116 18.104.22.168 Optimization of cathode design..................................................................117 4.3.6 Conclusions..........................................................................................................125 5 MULTI-SCALE MODELING OF A LOWTEMPERATURE SOLID OXIDE FUEL CELL WITH CONSIDERATION OF MI CROSTRUCTURE AND INTERFACIAL EFFECTS........................................................................................................................ ......127 5.1 Low Temperature SOFC and Modeling Objectives.......................................................127 5.2 General Descriptions and Model Assumptions..............................................................129 5.3 Macro-submodel.............................................................................................................132 5.3.1 Governing Equations............................................................................................132
6 5.3.2 Gas transport and Concentration Overpotential...................................................133 5.4 Micro-submodel............................................................................................................. .134 5.5 Model Parameters...........................................................................................................135 5.5.1 Effective Conductivity and TPB Length..............................................................136 5.5.2 Electrode Kinetics and Transfer Current Density................................................139 5.5.3 Mass and Heat Sources.........................................................................................140 5.6 Solution Algorithm and Parameters................................................................................140 5.7 Simulation Results and Discussion.................................................................................142 5.7.1 Baseline Case........................................................................................................142 5.7.2 Temperature Effects.............................................................................................145 5.7.3 Thickness Optimization........................................................................................147 5.7.4 Porosity.................................................................................................................151 5.7.5 Electrode Composite............................................................................................152 5.7.6 Particle Diameter..................................................................................................153 5.8 Conclusion................................................................................................................. .....155 6 CONCLUSION AND FUTURE WORK............................................................................157 LIST OF REFERENCES.............................................................................................................160 BIOGRAPHICAL SKETCH.......................................................................................................169
7 LIST OF TABLES Table page 3-1 Physical properties in porous materials and geometry parameters....................................68 3-2 Model input parameters for the baseline case....................................................................75 4-1 Model input parameters for the baseline case..................................................................112 5-1 Model input parameters for the baseline case..................................................................141
8 LIST OF FIGURES Figure page 1-1 Dramatic increase in transportation relate d petroleum use over the last three decades for the United States. Projections show that this trend will continue................................20 1-2 The increasing tendency for U.S. oil de mand and shortage of domestic production........20 1-3 Hydrogen economy........................................................................................................... .21 1-4 Comparison of total cumulative units from 1990-2005.....................................................23 1-5 The components of a typical PEM fuel cell.......................................................................25 1-6 Summary of fuel cell types................................................................................................27 1-7 Solid oxide fuel cell. A) Planar. B) Tubular....................................................................30 2-1 Multiscale computational framework................................................................................45 2-2 The random network model. A) Irregular microstructure of a composite electrode. Black circle = electronic conductor, blue circle = ionic conductor, empty space = pore. In an actual electrode, the shape of pa rticle may be spherical or cylindrical etc. B) Network discretizaiton of microstructure. Red rectangle denotes the resistance. Black node = electronic conduc tor, blue node = ionic conduc tor, white node = pore......47 3-1 The voltage of a fuel cell operating at about 800 degrees. (Larminie and Dicks, 2003)...53 3-2 A co-flow planar SOFC and the computational domain....................................................58 3-3 Equivalent electric circuit................................................................................................ ..64 3-4 Solution procedure......................................................................................................... ....67 3-5 Cross-section velocity contour s at selected planes. A) xvelocity at z=0.001m. B) yvelocity at x=0.005m, x=0.025m and x=0.045m (a rrow indicates the flow direction).....69 3-6 Temperature contours at selected pl anes (at x=0.005m, x=0.025m and x=0.049m, arrow denotes flow direction)............................................................................................70 3-7 Molar fraction and current dens ity contours at interfaces. A) O2 at cathode/electrolyte. B) CO at anode/electrolyte. C) H2 at anode/electrolyte. D) H2 at z=0.001m. E) CO at z=0.001m. F) Current density (A/m2).........................................71 3-8 Concentration contours. A) Oxygen at ca thode/electrolyte interf ace. B) Hydrogen at anode/electrolyte interface. C) Methane at anode/electrolyte interface. D) Temperature contour at z=0.001m (a rrow denotes flow direction)...................................77
9 3-9 Performance of SOFC. A) Current density distribution. B) Average polarizations........78 3-10 Effects of channel on mass transfer rate (at z=0.001m plane). A) H2 mass transfer coefficient on fuel channel. B) O2 mass transfer coefficient on air channel. C) Heat transfer coefficient on air channel......................................................................................79 3-11 Effects of channel on cell performance. A) Average temperatures. B) Average current. C) Output voltages comparisons..........................................................................80 3-12 Effects of temperature. A) Temperature effect on convent ional SOFC (baseline case). B) Scaling effect on the miniaturized-thi n-electrolyte SOFC unde r inlet temperature 850K. C) Efficiency comparison......................................................................................83 3-13 Effects of rib width on cell performa nce. A) Average top innterconnector temperature. B) Comparison of performa nces. C) Oxygen at cathode/ electrolyte interface with thinnest rib (ratio 1:7). D) Oxygen at cathode/ electrolyte interface with thickest rib (ratio 1:1)................................................................................................85 4-1 Images for anode and cathode of solid oxi de fuel cell. A) Low-resolution SEM topdown view of Ni-YSZ electrode on YSZ el ectrolyte disk. B)High-resolution SEM image of cross section of Ni-YSZ elect rode on YSZ. C) SEM image of cathode (LSM:YSZ=5:5). D) High-reso lution SEM of (LSM:YSZ=9:1)......................................89 4-2 A SOFC composite electrode. Three different clusters are formed. Black circle=electronic conductor, bl ue circle=ionic conductor. In actual electrode, the shape of particle may be spheri cal, cubic or arbitrary etc..................................................96 4-3 Three phase boundary regi on in cathodi c reaction............................................................97 4-4 Electrical analogue circuit showing the flux of the diffusion species within a porous medium......................................................................................................................... .....98 4-5 Random network model. Red rectan gle denotes the active TPB; Black circle=electronic conductor, blue circle=ionic conductor, white empty=pore (the circles have been removed for clarify). Three paths along the three phases for gas (black), O(red) and e(white) are represented by curves...............................................100 4-6 Electrical network......................................................................................................... ...103 4-7 Contact neck formed by two different kinds of particles.................................................103 4-8 Reconstruct of 3-D random mi crostructure. Model parameters: 0.4 elio:1:1 242418 Blue (0)=isolated particle, cy an (10)= transport pore, green (20)=transport electronic particle, ye llow (30)= transport ionic particle.........................109 4-9 Statistics of random micr ostructure (model parameters: 0.4 elio:1:1 A) Local variation in three kind of sites al ong the thickness. B) Effects of cross-
10 sectional mesh resolution on the transport pore volume fraction and transport solid volume fraction................................................................................................................110 4-10 Potential distribution (unit: V). A) For ioni c particles. B) For electronic particles.......113 4-11 Profiles of local relative exchange current density and relative polarization conductance.................................................................................................................... ..114 4-12 Oxygen concentration. A) Lower poten tial difference. B) Higher potential difference..................................................................................................................... ....116 4-13 Overpotential distribution. A) Ohmic and activation loss. B) Concentration loss........117 4-14 Effect of porosity on performance. A) Average local oxygen concentration. B) Average reaction rate.......................................................................................................118 4-15 Number of TPB and specifi c total resistance with respect to the porosity. A) Total number of active TPB. B) Specific total resistance........................................................119 4-16 Effect of thickness on reaction rate a nd concentration overpotential. A) Local average reaction rate. B) Local av erage concentration overpotential.............................121 4-17 Comparison among the model prediction, ex perimental data a nd continuum model......123 4-18 Effect of electrode fraction volume and pa rticle radius on total resistance. A) For different el B) el0.4 and different radius. C) el0.2 and different radius.........124 5-1 A planar solid oxide fuel cell. A) A un it cell. B) Computational domain, structural modeling of a solid oxide fuel cell, an d close-up of cermet TPB. The composite cathode (LSCF/SDC) and anode (Ni/SDC) are modeled by a random packing of binary particles (not on scaled)........................................................................................130 5-2 Temperature and molar contours. A) Te mperature. B) Oxygen at air channel and cathode layer. C) Hydrogen at anode layer and fuel channel. D) Water at anode layer and fuel channel (arrow denotes flow direction; TI/BI=top/bottom interconnect; An=anode; Ca=cathode; Air=air ch annel; Fuel=fuel channel)........................................143 5-3 Performance of SOFC. A) Ionic poten tial distribution. B) Electronic potential distribution. C) Activation ove rpotential. D) Concentra tion overpotential (unit: V)....144 5-4 Performance comparison with 500oC and 600oC. A) I-V performance. B) Exchange current density distribution over the cathode...................................................................146 5-5 Effect of thickness on cathode performa nce. A) I-V performance cathode. B) Concentration overpotential in the cathode. C) Current density distribution across the cathode.................................................................................................................... ...149
11 5-6 Effect of thickness on anode performance. A) Concentration overpotential in the anode. B) Current density di stribution across the anode................................................150 5-7 Effect of electrode poro sity on performance and overpot entials. A) Dependence of IV performance on porosity. B) Overpotential s (cathode). C) Overpotentials (anode)....151 5-8 Effect of anode composition on performance..................................................................153 5-9 Effect of particle sizes on performance. A) I-V performance. B) Averaged exchange current density distribution across the anode C) Averaged exchange current density distribution across the cathode.........................................................................................154
12 LIST OF TERMS areaA Active area [m2] iC Molar concentration of component i [-3mol m] or mass concentration [-3kg m] c Specific heat capacity [-1-1J kgK] ijD Binary mass diffusivity [2-1ms] k, iD Effective Knudsen diffusion coefficient for component i [2-1ms] hd Hydraulic diameter [m] E Nernst potential [V] F Faradays constant 96486.7[-1Cmol] F1 Forchheimer coefficient G Chemical potential [-1J mol] reformshiftelec,,HHH Enthalpy change of reactions [-1J mol] H Channel height [m] h Heat transfer coefficient [-2-1W mK] m, ih Mass transfer coefficient of component i [-1ms] i Local current denstiy [-2Am] I Current [A] i Current vector oai,oci, ca/an tri Exchange current density [-2Amor -1Am] L Channel length [m] or TPB length [-3m m]
13 xL,yL,zL Dimension of electrode [m] J Diffusion mass flux vector K Permeability of the porous medium [m2] k Thermal conductivity [-11WmK ] +-+rrshsh,,,kkkk Velocity constants for reforming or shift reaciton i M Molar weight of component i [-1kg mol] im Molar consumption or produc tion rate of component i at the electrode/electro lyte interface [-2-1mol ms] eln, ion Number fraction of electr onic or ionic particles en Number of electron P Pressure [Pa] P Probability of -type conductors in th e percolating cluster i p Partial pressure of component i [atm] 2r Op Partial pressure of oxygen at reaction site[atm] "q Heat flux at interface [-2Wm] elecq Heat flux from electrochemical reaction [-2Wm] ohre,QQ Volumetric heat generation [-3Wm] r Radius [m] Rg Universal gas constant 8.3143 [-1-1J molK] R Volumetric reaction rate [-3-1kgms]
14 total R Total resistance [ ] ,,,, Re,,int R axRazRcxRcz RinbR Resistance of electrodes, electrolyte and interconnectors r,sh,,ii R R Volumetric reaction rates of component i in reforming or shift reaction [-3-1molms] iS Net rate of production for component i T Temperature [K] V Velocity vector [ms-1] Xi Molar fraction of component i Yi Mass fraction of component i Z Coordination number Greek letters Thickness of electrode or neck of diameter [m] Transfer coefficient Contact angle [o] Electrical potential [V] Viscosity [2-1ms] Porosity Ratio of porosity to tortuosity Electrical conductivity [-1-1m ] or conductance [S] act,anact,ca, Activation overpotential fo r anode and cathode [V]
15 con Concentration overpotential [V] k Kozeny constant Conductivity [-1-1m ] Neck circumference [m] Size ratio int Interface velocity [-1ms] Density [-3kg m] Coordinate of the pore iI Distance for any two neighbor particles [m] Subscripts a Air an Anode b Bulk ca Cathode eff Effective el Electronic f Flow i Components in gas mixture (fuel channel mixture: 4CH,CO, i 2CO,22H,HO; air channel mixture: 22O,N i ) int Interface io Ionic s Solid
16 TPB Three phase boundary
17 Abstract of Dissertation Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTI-PHYSICS AND MULTI-SCALE MOD ELING AND SIMULATI ON OF THERMALFLUID AND ELECTROCHEMICAL TRANSPORT IN A SOLID OXIDE FUEL CELL By Yan Ji August 2007 Chair: Jacob N. Chung Major: Mechanical and Aerospace Engineering Solid oxide fuel cells are exp ected to be widely applicable for both small and large-scale power generation systems. The reason is that th e solid oxide fuel cell (SOFC) is simple, highly efficient, tolerant to impuritie s, and can at least partially in ternally reform hydrocarbon fuels compared to the conventional energy conversion systems. However, the SOFCs have not been commercialized widely due to some technical barriers. Most of limitations are primarily due to high operating temperatures (800-o1200C), expensive materials, di fficulties in fabrication and large thermal stresses within the cells, etc. Considering the complexity of the actual energy transport process in SOFCs, a mathematical mode l is usually a desirable tool for analyzing the phenomena inside a SOFC and finding optimum operating conditions to achieve a sufficiently high power density. I propose a multi-physics, multi-scale model stru cture by integrating three submodels, i.e., a macro-continuum model, a micro-scale model (random walk model) and an atomistic-level model. This multi-scale model has the capability of handling transport mechanisms on different length scales at the same time. The coarsest macr o-continuum model is first proposed to simulate all energy transport processes in an electrolyte-/anode-supported solid oxide fuel cell. It is based on continuum conservation laws. The main objectiv es are to examine the transport channel size
18 effects and to assess the potential of a thin-film-SOFC. Results demonstrate that decreasing the height of flow channels can lower the average solid temperature and improve cell efficiency. But this improvement is rather limited for the smallest channels. Compared with the conventional size SOFC, the miniaturized SOFC with a thin-film electrolyte has the advantages of a lower operating temperature and an excellent performance. The microscopic material behavior cant be explained effectively, because in the macrocontinuum model random, microstructure of elec trodes is partially simp lified to be homogenous. In order to overcome this limitation, I devel op a novel micro-level model (random walk model) to investigate the electrochemical performance in a composite electrode. My model takes into account the details of the specific electrode microstructure, such as random pore structure, active TPB (three phase boundary) site distribution, particle size and composition and their interrelationship to the charge transfer and ma ss transport processes. The pore structure and mass diffusion are also incorporated into this model. Finally I propose a multi-scale model by combining the developed macro-level model and micro-level model for a lower temperature SO FC. The outcome of numerical simulations provides a basic understanding of the infrastructure of transport mechanisms of momentum, heat and mass from microto macro-s cale dimension, together with cat alytic chemical reactions and electric charge and ion transpor t through membranes and electrodes. In this model, the macrosubmodel is developed on the basis of Chapter 3. Solutions from the macro-submodel are supplied to the micro-scale submodel as the glob al parameters. The new micro-scale submodel is still at the microscopic scale, but based on the pe rcolation theory and coordination theory. Based on this multi-scale model, the dependence of electrochemical performance on the global parameters and microstructures is assessed for the entire fuel cell stack.
19 CHAPTER 1 INTRODUCTION 1.1 Energy Crisis and Hydrogen Economy Like most of the world, the United States is dependent on petroleum for transportation, heating, and manufacturing. Auto mobiles, trains, and planes ar e fueled almost exclusively by gasoline and diesel fuel. However, the United States cant produce enough oil to meet the demand, over half of the oil we use is imported (58%). We produce just 9% of the global supply and account for only 3% of the worlds petroleum reserves; About 65% of those resources reserves are concentrated in the Middle East and over two-thirds ar e controlled by the OPEC members. Figure 1-1 and Figure 1-2 show the dram atic increase in U.S. oil demand during the last three decades. The rocketing oil price shocks and price mani pulation by the OPEC have cost American economy dearlyabout $7 trillion from 1979 to 2000, and each major price shock was followed by a recession. DOE (Department of Energy) estimates that by 2015, the demand for energy will grow by 54% worldwide and by 129% in developing Asia. Unless the United States takes dramatic steps to reduce fuel cons umption, this growing dependence on imported oil will put U.S.A at serious risk. A worldwide energy crisis due to the shortage of oil resources would disrupt this countrys economics. Therefor e, we must find new solutions to reduce our dependence on oil. In addition, burning fossil fuels such as gasoli ne or diesel contribu tes to a number of environmental problems. Spills from refining and transporting oil and petroleum products damage ecosystems and pollute groundwater and streams. According to U.S. Environmental Protection Agency, the fossil fuels burned to run cars and trucks, heat homes and businesses, and power factories are responsibl e for about 98% of U.S. car bon dioxide emissions, 24% of methane emissions, and 18% of nitrous oxide em issions. It has been estimated that 60% of
20 Americans live in areas where levels of one or more air pollutants ar e high enough to affect public health and/or the environment. Theref ore, these environmental and health problems require us to create new energy sources that can replace petroleum cleanly and inexpensively. Figure 1-1. Dramatic increase in transportation related petroleum use over the last three decades for the United States. Projections show that this trend will continue. 195019601970198019902000 0 1 2 3 4 5 6 7 8 Billion barrels per yearYear import production consume Figure 1-2. The increasing tendency for U.S. o il demand and shortage of domestic production. One promising alternative to fossil fuels is hydrogen. Th rough its reaction with oxygen, hydrogen releases energy e xplosively in heat engine s or quietly in fuel cells to produce water as its only byproduct. Hydrogen is abundant and di stributed throughout the wo rld without regarding to national boundaries; using it to create a hydrogen economy a future energy system based on
21 hydrogen and electricity. Figure 1-3 depicts the hydrogen economy as a network composed of three functional steps: production, storage, and use. Figure 1-3. Hydrogen economy. Most of hydrogen energy does not exist in nature as fuel H2, but in chemical compounds like water or hydrocarbons that mu st be transformed to yield H2 chemically. Hydrogen is a carrier of energy, like electrici ty, which should be produced from natural resources (water, fossil, renewable solar or thermal energy). The produced hydrogen can be stored in pressurized gas containers or as liquid in cryogeni c containers, but not in densities that would allow for practical applications. To the purpose of application, hyd rogen can be converted in to electricity by means of fuel cells or other devices. Unlike an in ternal combustion engine fuel cells rely on electrochemical reactions rather than combusti on of fuels to generate power. Compared with traditional power sources, fuel cells hold the fo llowing competitive advantages (Larminie and Dicks, 2003): H2O Solar/Wind Thermo-chemical cycles Fuel cells Biological or bio-inspired Customer electronics Electricity/Heat generation Fossil fuel reforming H2 H2 Gas/Hydride storage Production Use Storage
22 High efficiency: fuel cells are generally more efficient than combustion engines whether piston or turbine based. The efficiency can reach from 30% to 90% depending on the fuel cell system and if the surplus heat is util ized. Combustion-based energy generation first converts the fuel into heat, limited by the Car not efficiency, and then into mechanical energy, which provides motion or drives a tu rbine to produce energy. The additional steps involved in combustion generation allow energy to escape as heat, friction and conversion losses, resulting in lower overall efficiencies. Low emissions: the by-product of the main fuel cell reaction, when hydr ogen is the fuel, is pure water, which means a fuel cell can produce essentially zero emission. This is their main advantage when used in vehicles, as ther e is a requirement to reduce vehicle emission. If 10% of vehicles powered by fuel cells, carbon dioxide emissions would be reduced by 60 million tons a year. Silence: fuel cells can be made quiet. Because fuel cell systems have few moving parts, they could potentially be more reliable and require less maintenance than internal combustion engines, the most common e ngines in todays market. A fuel cell never runs down, it continues to pr oduce electricity as long as fuel is present. When a battery runs down, it has to under go a lengthy, inconvenient recharge time to replace the spen t electricity. However, fuel cells at the present time are still used on a limited basis. Cost and durability are the two biggest hurdles to commercialize fuel cells. The most widely de ployed fuel cells cost about $4500 per kilowatt. By contrast, a diesel generator costs rough $800 to $1500 per kilowatt, and a natural gas turbine can be even less. Alt hough much progress has been made, the cost of todays fuel cell technology must be reduced by a factor of ten to meet cost targets. For example, the target of the Department of Energy's Fossil Energy fuel cell program is to develop $400 per kilowatt (or less) SOFC power-generation system s for stationary, mobile, and military markets by the end of this decade. It is expected that lower cost fuel cells wi ll lead to widespread utilization. To compete with ot her power generation systems, stat ionary fuel cells must operate reliably for more than 40,000 hours. Fuel cells for transportation use must operate more than 5,000 hours. However, todays fuel cell sy stems can not achieve those targets. According to the Fuel Cell Todays worldwide survey, the fuel cell market has experienced an impressive 32% growth rate from 2004 to 2005 (Figure 1-4). With the government or private
23 funding support, Japan, China, Korean and Europ ean countries are making much more efforts in fuel cells and hydrogen industry. In summary, fuel cells and hydroge n offer the best path towards meeting our goals in energy util ization, environment conservation and economy development. Many scientists and researcher ar e taking technologies from res earch lab to demonstration to commercialization and widespread use, realizing a cleaner, sustainable energy future. 19901992199419961998200020022004 0 2000 4000 6000 8000 1 0000 1 2000 1 4000 16000 Years Cumulative Units 1990-2005 Figure 1-4. Comparison of tota l cumulative units from 1990-2005. 1.2 History of Fuel Cell Development Actually, the fuel cell is not a new technology, but has been around for 150 years. As early as 1839, Sir William Robert Grove (o ften referred to as the "Father of the Fuel Cell") discovered that it may be possible to gene rate current flow by taking tw o platinum (Pt) electrodes and immersing one end of each in a container of sulfuric acid with the other ends placed separately in sealed containers of oxygen a nd hydrogen. It was not until 1889 that two researchers, Charles Langer and Ludwig Mond, coined the term "fuel cell" as they were trying to engineer the first practical fuel cell using air and coal gas. While further attempts were made in the early 1900s to develop fuel cells that could conve rt coal or carbon into electric ity, the advent of the internal
24 combustion engine temporarily quashed any hope s of further development of the fledgling technology. Francis Bacon developed what was perhaps the first successful fuel cell device in 1932, with a hydrogen-oxygen cell using alkaline electr olytes and nickel electrodesinexpensive alternatives to the cata lysts used by Mond and Langer. Due to a substantial number of technical hurdles, it was not until 1959 that Bacons company first demonstrated a practical five-kilowatt fuel cell system. Also in the late 1950s, NASA began to build a compact electricity generator for use on space missions. NASA soon came to fund hundreds of research contracts involving fuel cell technology. Fuel cells now have a proven role in the space program after supplying electricity to several space missions. In more recent decades a number of manufacturers, including major automakers and various federal agencies, have supported ongoing research into the development of fuel cell technology for use in fuel cell vehicles (FCV) and other applica tions (fuel cell, 21st century power, 2005) A fuel cell power system has many components, but its heart part is th e stack. This stack is actually made of many thin, flat cells layered together like sli ces in a loaf of bread. Most individual fuel cells designed for use in vehicles produce less than 1.16 volts of electricity far from enough to power a vehicle. Therefore, multip le cells must be assembled into a fuel cell stack. (The term fuel cell is often used to refe r to the entire stack, but, strictly speaking, it refers only to the individual cells.) Each cell pro duces electricity and the out put of all the cells is combined to power the vehicle or for other app lications. The potential power generated by a fuel cell stack depends on the number a nd size of the individual fuel ce lls that comprise the stack. A single fuel cell consists of an anode, a cathode and an electrolyt e as shown in Figure 1-5. The
25 fuel is introduced at the anode the oxidant at the cathode, and the electrolyte electrically and physically separates the two electrodes. During a typical PEM fuel cell operation, the fuel is oxidized at the anode catalyst to produce ions and electrons The ions diffuse through the electrolyte to the cathode, where they react w ith the oxidant and the electrons to produce a molecular product, often water. If a wire is at tached between the anode and cathode, current will flow in that wire in proportion to th e amount of fuel and oxidant reacting. Figure 1-5. The components of a typical PEM fuel cell. The following reactions take place on the electrodes: Anode: +2H2H+2e (1.1) Cathode: +221/2O+2H+2eHO (1.2)
26 Overall cell reaction: 2221/2O+HHO (1.3) If assuming there are no constraints on s upplying fuel and oxidant to the anode and cathode, respectively, the electrical voltage of fuel cell re action is determined by the equation below: 0VVIR (1.4) where 0V is defined as open circuit voltage (OCV), which is equa l to the potential difference between the fuel and the oxida nt (oxygen). For hydrogen and oxygen, the upper limit of OCV on individual fuel cell is 1.229V at 100oC. The cell overpotential, is due to the non-ideal behavior of the fuel and oxidant at a given current, which depe nds on the material properties, operation temperature, pressure, cell geometry and so on. This term determinately influences the electrochemical performance of fu el cell and will be specifically discussed later. The third term is the voltage loss due to resistances in all of the solid components of a fuel cell. For different types of fuel cell, the above basic working principle is similar, but has a little different. The energy conversion efficiency of a fuel cell can be defined as: 0Efficiency/ VV (1.5) where Vis terminal output and 0V is the OCV from equation (1.4). There are several types of fuel cells that have been made, the most common are: Alkaline fuel cell (AFC) Polymer electrolyte membrane fuel cell (PEMFC) Phosphoric acid fuel cell (PAFC) Molten carbonate fuel cell (MCFC) Solid oxide fuel cell (SOFC)
27 Figure 1-6 (Larminie and Dicks, 2003) summarie s these five types of fuel cells. The low temperature systems, AFC, PEMFC and PAFC, re quire relatively pure hydrogen to be supplied to anode. Correspondingly, an exte rnal fuel processor needs to be included into the system, which not only increases the complexity and cost, but also decreases the overall efficiency. In contrast, the higher temperature systems, MCFC and SOFC, can electrochemically oxidize both CO and H2. Accordingly, the fuel-processing reaction can be integrated into the stack, which enables novel thermal management design featur es to provide improved efficiency. There are some detailed discussions about th e advantages and disadvantages for all types of fuel cell in Fuel Cell Systems Explained (Larminie a nd Dicks, 2003). Summaries of the current technological and commercial stat us of fuel cells are provide d by the Fuel Cell Handbook (2003) and Singhal et al. (2003). In this proposal, the fo cus will be on the SOFC and specific discuss is given below. Figure 1-6. Summary of fuel cell types.
28 1.3 Solid Oxide Fuel Cell Fuel cells were first developed through the U.S. space programs. During the 1960s and 1970s, Westinghouse Electric Corporation continued to develop the SOFC in the United States. In the 1980s, a satisfactory design appeared that yielded a commercially vi able stack design in the 1990s (tubular SOFC design). The first demons tration of a SOFC-powered electrical plant was conducted in 1997 by Westinghouse in Europe w ith an electrical conv ersion efficiency of more than 45%. Today, SOFCs are reachi ng precommercialization with some hundred residential stationary power units (about 1KW) being tested in Europe and larger unit (250KW) being evaluated by various utility company worldwid e. In general, solid oxide fuel cell can offer combined heat and power (CHP) at an electrical efficiency between 40-50%, and an overall LHV efficiency between 80 and 90%. Compared with ot her types of fuel cells, SOFC systems have much higher efficiency and few components. Actually, about 2030% of the volume of a solid oxide fuel cell system is made up of the cell stack. The rest of th e system is the balance of plant (BOP) which includes thermal insulation, heat exchangers, fuel processors, piping pumps, control system and so on. Thus, the BOP is the do minant part of the SOFC system and should be treated with some concerns (Singhal, 2003). Howeve r, in the present propos al, the fuel cell stack is only our interest. In the followings, the solid oxid e fuel cell refers to the entire stack, or the individual cell. A SOFC is a solid-state fuel cell composed of ceramic materials and metals. Only four components are needed: electrolyte, anode, cat hode and inter-connector. In general, the electrolyte is made from yttria -stabilised zirconia (YSZ) which acts as a conductor of oxide ions at temperature fromo5001000C. At porous cathode layer which is a typical perovskite material such as 0.80.23LaSrMnO (LSM), oxygen atoms are reduced to oxide ions by accepting the
29 electrons, which then are passed through electrolyte to the porous anode (Ni) region where the oxide ions react with hydrogen, re leasing electrons to an external circuit as shown in Figure 1-7. The inter-connector supports the structure and also serves as the current collector. SOFC systems are being improved by a number of companies with two major stack designs: planar and tubular (Figure 1-7). The tubular de signs are closer to commerc ialization and pioneered by Westinghouse Electric Corporati on. This design constitutes th e stack as a bundle of tubular electrode-electrolyte assemblies (Figure 1-7B). On e great advantage of tubular design is that high-temperature gas-tight seal is eliminated. Compared with t ubular designs, planar SOFCs are simpler to manufacture, consisting of flat plates bonded t ogether to form the electrode-electrolyte assemblies. This flat-plate design results in lower ohmic losses than that in the tubular arrangement. It leads to a superior stack perf ormance and a much high power density. Also, lowcost fabrication methods such as screen printing a nd tape casting can be us ed in this design. The major disadvantage for this design is the need fo r gas-tight sealing around the edge of the cell components. Although the planar design has attracti ons in terms of power density and efficiency compared with the tubular design, there are still much more improvements needs, or even some companies have abandoned such design because of inherent techni cal problems. Recently, several novel designs, such as flat tube HPD-SOFC and MOLB (mono-block-layer-built-type) are reported to improve the power density greatly and have higher fuel/oxidant utilization ratio. A
30 B Figure 1-7. Solid oxide fuel ce ll. A) Planar. B) Tubular. Although in the past 20 years, there have b een substantial progresses in materials, fabrication and system-technology of different SOFC systems, some limitations still bound the application of SOFC, especia lly on the large-scale producti on capacities (>300KW or MW). These limitations are primarily related to th e high operation temper atures and expensive materials (Larminie et al., 2003; Weber et al ., 2004). The high operating temperature has numerous advantages: there is no need for a metal catalyst, units enjoy fuel flexibility, and combined SOFC/gas-turbine power plant is possi ble for a higher efficiency. On the other hand, high temperature can lead to extremely stringe nt requirement for materials. The thermal expansion coefficients for components must be cl osely matched to reduce thermal stress due to larger temperature differences. For example, the thermal stresses within the cells during start up and across cells during operation lead to mechan ical failure of components at their interfaces. Therefore, in recent years, development of intermediate temperature SOFC (IT-SOFC, 500800oC) has become a new tendency. An intermedia te operating temperature results in a drastic increase in the internal cell re sistance, so the ohmic loss, especially at the electrolyte and polarization loss at the cathode, is lowered greatly. The thin-film or double-layer electrolyte and alternative electrodes are being developed and tested (Chen and Wu et al., 2004; Gorman and Anderson, 2005; Tang et al., 2005).
31 These existing technical barriers are stimulating researchers to apply different methods to overcome them and make solid oxide fuel cells be come commercially competitive. Mathematical modeling has been proven as a powerful tool to as sist the design, test a nd optimization of fuel cells. Especially, it is extremely difficult to meas ure the physical parameters within a fuel cell stack due to compactness and complexity. Thus, a well-validate, stable mathematical model can be an alternative to expensive, labor-intensi ve experimental testi ng and provides detailed information on the flow field, pressure variation, heat transfer and chem ical reactions, thus find best approaches to improve the electrochemical performance of a solid oxide fuel cell. 1.4 Scope This proposed research is focused on invest igating fluid dynamics and energy transport process in the SOFC stack and electrodes. Proper mathematical models will be developed to predict the flow field, thermal field, con centration, current and potential fields, and electrochemical behavior in the three phase boun dary (TPB). Based on these models, parametric studies will be employed to optimize the pe rformance of a solid oxide fuel cell. In Chapter 2, the background of solid oxide fuel cell modeling is briefly introduced. Previous works for different level models, from the system level, via the stack level, to the component-level, and finally to the molecular le vel, are reviewed and qualitatively assessed. Some unresolved problems in the existing mode ls are summarized. Then a multi-physics, multiscale model for the current research is proposed This complete model includes three submodels: macro-, microand atomistic modules. In Chapter 3, electrochemistry fundamentals fo r solid oxide fuel cell are reviewed first. Then, a macro-level 3D thermo-fluid/electro-chemical model is developed to simulate the heat and mass transfer phenomena in a thin-film co-f low solid oxide fuel cell. A network circuit model is applied to simulate the electrical potential, ohmic losses and activation polarization.
32 Finally, this model is applied to investigate and analyze the heat/mass tran sfer, chemical reaction and electrochemistry in an anode-supported fuel cell and the effect of transport scale on an electrolyte-supported fuel cell. In Chapter 4, a new 3D random walk model is first developed to simulate the performance in an porous composite cathode giving a complete description of the elec trode structure as well as the processes occurring therein. The pore distribution, compositi on, mass diffusion and current density distribution etc will be totally in cluded into this new micro-model. Results for a baseline case are demonstrated and analyzed. Th en, detailed investigatio ns are carried out to optimize the geometry parameters. A comparison be tween this model and experimental data is conducted. In Chapter 5, by integrating the macro-model developed in Chapter 3 with the micro-level model that statistically considers the electr ode microstructure, a multi-scale approach is specially developed. It accounts for transport me chanisms arising from different processes and length scales for a SOFC. Simulated results fo r a lower temperature SOFC are discussed. Chapter 6 concludes the research with a su mmary of overall work and a discussion of future work.
33 CHAPTER 2 LITERATURE REVIEW AND OBJECTIVES 2.1 Literature Review for Fuel Cell Modeling Various mathematical models provided by prev ious investigators ar e concerned with the theoretical fundamentals and practical opera tions of a SOFC. The modeling may focus on individual flow, chemical and electrochemical subsystem or on coupled integrated systems. Because the subsystems have different characteris tic lengths, so the modeling can be established on the different levels, from the system level, vi a the stack level, to the component-level, and finally to the molecular level. 2.1.1 System and Thermodynamics Modeling Many research groups have conduc ted theoretical studies for th e performance analysis of the SOFC. However, studies for the power gene ration system composed by the SOFC and other auxiliary components including refo rmer, compressor, turbines, and inverters, etc. were not active until concerns about the gas turbine-fuel ce ll hybrid power system were raised. In the early of 1980s, Appleby and Foulkes (1989) considered the possibility of integr ating fuel cells with gas turbines. But this kind of hybrid cycle had not been successfully applied into the power systems until Siemens-Westinghouse proposed the SureCell concept (Veyo and Forbes, 1998) and then built the first commercia l SOFC-GT system for Southern Californian Edison in 2000. Almost at the same time, modeling for distribute d power generation began to develop rapidly. The main objective of system modeling is to determine the energy efficiency and heat/power ratio of the whole system. Earl y work was reported by Harvey et al. (1993), Massardo and Lubelli (1998), Stephenson et al. ( 1997), and Veyo et al. (1999). The parametric study of Stephenson et al. (1997) focused specifi cally on the recuperate d SOFC/GT cycle. It concluded that high cycle effici encies are achieved by directly expanding the SOFC exhaust gas
34 at the gas turbine without firing supplemental fu el at the gas turbine combustor, emphasizing the synergy of SOFC and gas turbine integration. Ma ssardo and Lubelli (1998) demonstrated that the SOFC-GT cycles are higher (65-70% atmosphe ric cell; 74-76% pressurized cells) than the efficiency that achieved by the most advanced combined cycle plant. Similarly, Mitsos et al. (2004, 2005) compared different technology altern atives for mico-power generation based on a superstructure process, including hundreds of di fferent designs and identified conditions. However, these work oversimplified the ener gy transport in components and geometrical effect due to one-dimension c onsideration. In order to overcome these limitations, recent researchers have combined the 2-D SOFC stack m odel with system level models, such as Stiller et al. (2005), Palsson et al. (2000) and Calise et al. (2005). Pal sson et al. (2000) integrated a 2D SOFC model into a process simulation tool, Aspen Plus, as a user-defined model, whereas other components constituting the system were modeled as standard operation unit models. Parametric study was conducted to obtain the stack and overall sy stem behavior such as the effect of fuel flow on the stack and the effects of voltage and compressor pressure on the system. These system modeling analysis shows how st rongly system characteristics, such as efficiency, depend on accurate input data for the electrochemical model used in simulating stack performance. Also, such studies indicate that the ultimate design of the stack and the required accuracy of stack modeling are best determined after preliminary system design studies have been performed using preliminary st ack, reformer, and turbine models. 2.1.2 Coupled Electrochemical, Flow a nd Thermal Modeling for Stack As a matter of fact, the heat transfer, gas species diffusion, kinetics of chemical reactions and charge transfer within a solid oxide fuel cell are all interdependent and very complicated. Many studies have clarified that any change in fl ow, thermal field or potential field may result in substantial variation in electrochemical performance. Therefor e, if only one or several physical
35 mechanisms are considered in the models, it will lead to inaccuracy or errors in predicting the performance of fuel cells. Early papers used ba sic planar or tubular geometry and solved the transport equations incorporating the electrochemi cal reactions. Flow field was only solved by an one-dimensional model or neglected and only th e overall reaction was considered. One of the first significant publications on SO FC modeling is Lu and Maloneys published in 1988. In their simulation, they used a constant current density across the cell, and calc ulated the cell voltage from an input linear curve of current density vs voltage. The following work of Lu and Maloney (1988), Ahmed et al. (1991), Hirano et al.(1992) and Besse tte et al. (1995) de veloped the coupled thermal and electrochemical models, in which th ey evaluated the performance of a SOFC under different operating conditions. However, laminar convection heat transfer coefficients based on fully-develop flows were assumed and mass diffus ion in the radial direction was ignored. The mechanisms for different polarizations were not totally considered (Ahmed et al., 1991; Hirano et al., 1992) or oversimplifie d (Bessette et al., 1995). Although a one or two dimensional analysis can be useful for the general design, much more fuel cell modeling activities are related to 3D simulation in order to obtain more accurate information. The most recent and advanced models for a tubular SOFC have been presented by Li et al. (2003, 2004). Their models accounted for the actual geometry of the cell and all major phenomena, including momentum, heat and mass transport coupled with electrochemical and chemical reactions. However, these models negl ected the diffusion of oxygen through the thick porous cathode. Since the O2 diffusivity is much less than the H2 diffusivity (2-3 times less), the concentration loss in the cathode is significant, especially for the high current density case. Furthermore, the mechanisms of reactions were simplified. For example, the rates of the reforming and shift reactions were assumed to be in equilibrium and el ectrochemical reactions
36 were expressed entirely in term s of constant exchange current densities as adopted from Chan and Khor et al. (2001). Lu et al (2005, 2006) followed the appro ach by Li et al. (2003, 2004) to compare the performance of flat tube HPDSOFC with traditional tubular SOFC. In more advanced studies, much of the phys ical phenomenon or reaction mechanisms have been accounted for including species diffusion in electrodes, variation of partial pressure in electrodes, radiation effects, mechanical stress, electrons/ions transport, and polarizations etc. For example, Yakabe et al. (2001) took account the effect of the radiant heat exchange inside fuel/air channels into their model and estimated the internal thermal stress in cell components. In their another work (2004), the detail current path was simulated. It is found that the diagonal electric current inside the electr olyte and the inter-connector exists and the geometry of the interconnector affects the cell performance markedly. Listed below are related research publications emphasizing different physical mechanisms: Studies on intermediate temperature anode supported-SOFC and species diffusion in porous layer: Haberman et al. (2004, 2005), Aguiar et al. (2004, 2005). Simulation for the effect of flow configurati on and new channel design: Yuan et al. (2001), Ferguson et al.(1996), Iwata et al.(2000), Rechnagle et al (2003), Hwang et al.(2005). The influence of radiation heat transfer a nd thermal stress on fuel cells performance: Yakabe et al. (2001, 2004) Murthy et al. (2003). Dynamic behavior during start-up and shutdown: Aguiar et al. (2005), Gemmen et al. (2005), Iora et al. (2005) Damm et al. (2006). 2.1.3 Chemical Reaction Rate Modeling In order to obtain the temperature distributi on within cells, the local heat generation rate, which is dependent on chemical reaction rate, mu st be determined exactly. However, a common weakness of the above coupled ther mal-fluid models is that the chemical reactions are assumed to be in equilibrium and independent of the speci es concentration and pressure. In other words, the chemical reactions taking plac e in fuel cell are not always simple; methane fuel, for example,
37 in the presence of steam may undergo steam refo rming (or water gas shifting) upstream of the electrode reactions sites, so the overall heat generation may be due to a multitude of anode reactions such as: Electrochemical reaction: 2-22 2-2H+OHO+2e CO+OCO+2e (2.1) Methane reforming: 422CH+HOCO+3H (2.2) Water gas shifting: 222CO+HOCO+H (2.3) Moreover, actual shifting or reforming reacti ons are recognized to involve a series of intermediate steps (Bockris et al., 1993; Belyavev et al., 1995; Clarke et al., 1997; Ahmed et al., 2000). Unfortunately, which step governs the chemical process depends on the detailed experimental condition and electrodes microstructu re, etc. Effective chemical rate equation and reaction kinetic parameters can be determined by fitting a simplified kinetic model to the experimental data. Although kinetics of the reforming and shifting reactions have been studied widely, the rate equations presented in literatures still va ry greatly due to different catalyst amount and material structure. Achenbach et al. (1994) and Belyavev et al. (1995) found that the reforming reaction rate is first order in methane and zero order in water, for a 80wt.% ZrO2 & 20wt.% Ni cermet and for a mixed Ni-(5wt.%) ZrO2(2wt.%) CeO2 anode. Ahmed et al (2000) studied the two different Ni-YSZ anodes and found that the r eaction order in methane increases from 0.85 to 1.4 and the order in water changes from -0.35 to -0.8. Other kinetic expressions for reforming
38 reactions can be found in the following referenc es: Dicks et al. (2000) and Meusinger et al. (1998). A discussion concerning the comparison of rate equations a nd reaction orders in methane reforming reaction has been presented by Aguiar et al. (2004). For the shift reaction which takes place at high temperatures, previous studies have demonstrated that it reacts very quickly and is almost in equilibrium. The most frequently cite d date is Drescher and Lehnert et al.s (1998, 2000). They measured the forward velocity constants (+ shk ) and backward velocity constants (shk ) for the shift reaction at different temperatures. Just as mentioned above, the reaction equations strongly depend on material properties and opera tion conditions, so for a specific mathematic model, one should be careful on choos ing appropriate rate equations. 2.1.4 Hydrogen/oxygen Reduction Mechanism and Electrode Modeling As model have been developed, it became cl ear that electrochemical behavior was not accurately accounted for. A number of very impo rtant models were developed that described the electrochemically active region of the catalyst layer in detail. Some researchers extended the modeling effort to include a description of porous electrode microstructure. These models were aimed to improve the understanding of the electr ochemical reactions associated losses, and then assist in the design of optimized electrode stru ctures. In general, th ere are five different approaches to model the electrochemi cal behaviors in porous electrodes. 22.214.171.124 One-dimensional models based on mass transfer analysis At this level, the kinetics of electrode (relate d to activation polarization) is assumed to be rapid, and ohmic and mass transfer resistances ar e the dominant resistances in the electrodes. This major advantage of this approach is that analytical expressions can be obtained for the polarization behavior. A simple Bulter-Volmer equati on is always used to describe the activation overpotential. Chan et al. (2001) presented a complete polarization model and analyzed the
39 effects of cell component thickness on the potential loss. They found that the sensitivity of cell voltage due to the change of electrolyte thickness is the highest, then sensitivity of cell voltage due to the change of cathode thickness and fina lly the sensitivity of cell voltage due to the change of anode thickness. Resu lts showed that the activation polarization in the cathode is relatively larger for anode-supporte d SOFC, but this polarization is estimated based on empirical equation and constant exchange current density. Fu rthermore, due to the limitation of this model, some important effects, such as the porosity, to rtousity and pore size, were not evaluated. Using similar method, Kim et al. (1999, 2001) evaluate d the magnitude of concentration and activation overpotentials by introducing an important para meter called limiting current in an anodesupported SOFC. Unfortunately, be cause the insufficient information on the chemical kinetics (i.e., activation overpotential), the theoretical resu lts can not explain the e xperimental data very well. The simplicity of the mass-transfer approach makes it attractive and therefore it was used in some coupled models for stack, such as, Virkar et al. (2001), Li et al. (2003), and Haberman et al. (2004, 2005). However, for the optimization of microstructure or anal ysis of kinetics of electrodes, this approach is inaccura te and causes too much uncertainty. 126.96.36.199 Oxygen reduction reaction and chemic al kinetics modeling in cathode With the growing interests in IT anode-suppor ted SOFC, kinetic resistance in the cathode is believed to be one of the major bottleneck s in improving the electrochemical performance. Therefore, numerical modeling of oxygen re duction mechanism (or cat hodic reaction) is receiving much attention. Experimental studies have indicated that three possible paths for oxygen reduction: surface path, bulk path and el ectrolyte surface path (Sasaki et al.,1996; Odgaard et al., 1996; Fleig 2003.). Based on thes e experimental findings, this kind of model approach describes reasonable localized kinetics of cathodic reaction and establishes the relations between performance and relevant para meters. Using rate equation, several relations
40 such as current and voltage characteristics or pa rtial pressure dependencie s of polarization can be calculated for different kinetic si tuations, i.e., different rate-det ermining steps. When different elemental reaction steps play ra te-determining role, different relation between overpotential and current can be analytically or numerically obtained. This model is very helpful to analyze experimentally determined relations among polar ization resistance, external materials and geometrical parameters. In the su rface path, detailed simulations have been presented in Van Heuveln et al. (1997), Svensson et al. (1998), and Mizusa ki et al. (1987). For example, Svensson et al. (1998) proposed a two-dime nsional model to simulate transport of oxygen in the cathode with negligible oxygen ion conductivity in the bulk. Adsorption and surface diffusion of oxygen on the pore walls, interfacial diffusion of adsorbat es, and electrochemical reaction kinetics at the electrode/electrolyte in terface were considered. They found th at the extension of the reaction zone into the two-phase boundary, by diffusion of adsorbed intermediate oxygen along the electrode/electrolyte inte rface, is beneficial for the cathode performance. In modeling bulk path, both electro/ion current and electrochemical reacti on at interfaces and stro ichiometry variations in the bulk have to be consid ered. Related studies are reported by Fleig (2003) and Jamnik et al. (1999). 188.8.131.52 Monte-Carlo simulations based on ra ndom packing of spherical particles Composite electrodes are commonly used in the solid oxide fuel cell due to an increase in the three phase contact area (TPB) compared wi th pure electronic conduction electrode materials and as a result composite electr odes usually enhance the perfor mance. Monte-Carlo simulation was developed in response to the need for modeling such composite electrodes. The basic assumptions behind this method (Sunde, 2000; Abel et al., 1997) are: (1) composite electrode can be represented by discrete particles of electr olyte (ionic conductor) a nd electrode (electronic conductor) materials packed together complete ly at random; (2) current is conducted from
41 particle to particle through necks formed betw een them; (3) representative aggregates can be created on a computer with serial, random depos ition of spheres, leading to highly amorphous structures, or by regular lattices of a chosen st ructure, e.g., cubic. A network of resistors is formed throughout the entire computational dom ain. The electrical and electrochemical properties of the particles are then descri bed by such network. Two of the significant publications on Monte Carlo simulation are Sund es published in 1996 and 1998. He applied the Monte Carlo method to calculate the conductivity of composite el ectrodes. It is found that a sharp transition from low to high conductivity o ccurs at approximately 30% volume fraction for cubic lattices and for the random packing with un iform particle radius. Following Sundes work (1996a, b), Jeon et al. (2005) applied this method to simulate the perfor mance of a composite cathode with penetrating electrolyt e structure. To our best knowledge, this method represents the type of approach that best predicts the most important expe rimental features of composite electrodes. However, it is too ex pensive and only limited to problem with small sample size due to high computational cost. Furthermore, this method has not been performed on other types of electrodes, like mixed ionic and el ectronic conducting cathode (MIEC). 184.108.40.206 Continuum micro-scale models ba sed on the statistical properties This method is also suitable for composite el ectrodes. The basic assumptions in continuum micro-scale models are identical to assumptions (1) & (2) in the Monte-Carlo simulation. However, instead of using computer-generated packing where all particle positions and types are known, this method disregards the actual geomet ric details of the comp osite electrode. The electrode is described in term s of continuous, average quantitie s, in analogy with corresponding theory for singlephase porous electrodes used in aqueous electrochemistry (Sunde, 2000; Singhal and Kendall 2003; Fleig, 2003). The most important parameters, effective ionic and electronic conductivities, can be modeled by the following most co mmon statistical approaches:
42 (1) Percolation theory and coordination numbers th eory in random packing of bimodal spheres. Percolation theory gives a phenom enological equation for the conductiv ity of a system near to a metal-insulator (or cond uctor-perfect conductor) transition (K irkpatrick, 1973). Although in the binary particle system, only clus ters of electrolyte particles (suc h as YSZ) and electrode particles (such as Ni/LSM) extending through the entire co mposite are taken into account in this method, previous studies have successfully proven its va lidity in predicting conductivity for composite electrode (Bouvard et al., 1991; Kuo et al., 1995; Jiang et al., 1995; Sunde 2000; Costamagna et al., 1998; Chan et al., 2001, 2004). However, almost all reported results so far have been for onedimensional formulation and the direction normal to the electrode plane was considered; (2) Effective medium theory: an effective media th eory of a binary mixture is one where each ellipsoidal grain is, on average, surrounded by a mi xture which has the eff ective conductivity of the composite medium. It originates from Clausisu-Mossotti approximation and Bruggeman symmetric/asymmetric effective media theories (Landauer,1978; Meredith and Tobias, 1962). Mclanchlan and Newnham (1986, 1990) derived a general eff ective media (GEM) equation, which combines most aspects of both percola tion and effective media theories. Electrical conductivity obtained from GEM equa tion can quantitatively fit the experimental data. However, this model has sometimes been criticized because by their nature, they cant be expected to accurately describe the behavior of th e conductivity near critical points. The major advantage of continuous micro-model is the simplicity and its applicability in optimizing the design of electrodes microstructures. Percolation theory will be developed in a 2D domain to calculate effec tive electrical conductivity in th e proposed modeling work, and so will be specially described in Chapter 5.
43 2.1.5 Molecular-level Modeling The last type of modeling approach is the mo st elaborated one. Over decades, enormous progresses have been achieved in calculating the structures, the propertie s and the reactivity of solids starting from atomistic approaches. The mo lecular-level models have been advanced far into several categories (Singhal et al., 2003): (1) Empiri cal inter-atomic potential models. These methods originate from a given effective potentia l that describes the inter-atomic forces in a system of atoms using classical techniques. One of popular approach is the molecular dynamics method. (2) Quantum mechanical elect ronic structure cal culations, or the ab initio methods. This method is based on the self-consistent solution of the Schrodinger equation for a cluster of atoms and a set of boundary conditions. This level modeling needs overwhelming computational resources, so it is still difficult to integr ate with a macro-model or micro-model. 2.1.6 Unresolved Issues in SOFC Modeling Many progresses have been made in SOFC modeling as summarized above. The most suitable modeling depends on the problem under consideration. However, much work is still needed. For continued progress in this field, mode ls must be based on an accurate description of the fundamental principles underlying the various processes occurring at th e micro-scale level or molecular level. Constant improvement in SOFC modeling will allow models to be developed for making design decisions and performance predic tions and help commercialize SOFCs. Future models must properly address or solve some important issues or problems which still exist in the present models (Faghri and Guo, 2005; Bove and Ubertini, 2006): The physical properties, such as exchange curren t density or charge transfer coefficient for the oxygen reduction process, are critically im portant to model accuracy. In the simulation to study the activation polarization, the exchange current density is a parameter that relates to the activity of electrochemical reaction. Ho wever, the values used in the literature are quite different, or even cont radictory. Most models rely on estimated values for many required physical properties, mainly because pr operty data are evaluated under a limited set of conditions that may be different from the conditions of a given model. Actually, it is
44 impossible to simulate the performance of an actual fuel cell accurately based on such uncertainty in properties. Therefore, obt aining these property dates will improve the accuracy of existing models greatly. SOFC models can be improved by developing submodels and ai ming at a specific transport process which is too complex to be explicitly included in a general comprehensive model. By modeling a specific transport process, it is possible to gain a greater and more fundamental understanding of fuel cell operati on than would be possible through the use of macroscopic models alone. At the nano-scle of atoms and molecules, pred iction of material beha vior and interfacial interaction are becoming possibl e. Many significant advances are now being obtained in the understanding of composite stru ctures of electrodes and th ree phase boundaries (TPB). Therefore, in the further, it is expected th at models which combine the macroscopic and atomistic approaches will be developed. 2.2 Objectives The main objective of our rese arch is to develop a direct numerical simulation capability that is based on first theoretical principles and will provide a fundamental understanding of the infrastructure of transport mechanisms of momentum, heat, and mass from nano/microto macro-scale dimension, together with catalytic chemical reactions and membrane transport at atomistic level. We will focus on identifying the key underlying transport physics and improving our knowledge of their inherent c onsequences to facilitate breakt hroughs that are needed for the realization of hydrogen economy. Here, a macro, nano/micro and particle-scale modeling and simulation framework is proposed to achieve th e above objective. This multiscale computational framework as shown in Figure 2-1 includes th e following three numeri cal models that are intercoupled to cover the entire fuel cell stack. Macroscale Model. Certain transport phenomena that takes place within the SOFC stack, such as the air and fuel fluid flow, and heat tran sfer inside transport channels, can be simulated and analyzed by the continuum-scale computationa l fluid dynamics (CFD) technique. In this model, the electron and ion transport in the electrodes and membrane, respectively, are only
45 modeled based on a global (or lumped capaci tance) method and ther efore the specific microstructure and atomic/molecula r effects have been neglected. Figure 2-1. Multiscale co mputational framework. Nano/Micro Scale Model (Random Network Model). The microstructures of electrodes are not regular and thus all the mass and energy transports within electrodes take place in a disordered medium. The morphology of porous electrodes, i.e., its connectedness (the way pores are connected to one another) and its geometry (the shapes and sizes of the pores), plays a fundamental role in the transport and reaction in side the system which is often more important than the role of other influenc ing factors. However, in most of previous modeling work for electrodes, the non-homogeneous microstructure s are substantially si mplified to become approximately homogeneous. Some critical properti es, such as the electrical conductivity and diffusivity, are obtained by approxima te or statistical approaches, such as the effective medium theory and classical percolation theory. More over, the individual pore effect and surface Macro-model Atomistic-model Micro-model Computational domain : Interconnector & pure fluid region (air and fuel channels) Solve N-S equations, energy equation, and species equation to get: temperature, velocity, concentration, pressure & potential distributions (note: only for pure fluid region and interconnectors) Computation domain : electrodes & electrolyte By identifying microstructure to get two kinds of physical parameters: (i) Electrochemical parameters : electrical/ioni c conductivities, effective diffusivity, reaction rate, potential, activation overpotential, ohm ic and concentration overpotentials; (ii) Microstructure thermo-fluid parameters : temperature, concentration and pressure distributions Computation domain : Sampled Microscopic Local Region By using atomistic method to get: diffusion mechanisms for cathodic and anodic reactions Mass flux, temperature, pressure, potentials at anode/fuel and cathode/air interfaces More accurate properties and electrochemical reaction Expected subject
46 diffusion within microspores are not considered Due to inherent uncer tainties, errors and limitations in these methods, the predicted resu lts often can not explain the experimental phenomena satisfactorily. Therefore, a micro-scale model, which focuses on the electrodes and electrolyte, will allow us to investigate the effects of microstructure variations on the mixture gas diffusion from the bulk flow to the reaction site, the surface diffusion within micropores and the effective electrical conductivities. All of the thre e diffusion types, i.e., bulk di ffusion, surface diffusion and Knudsen diffusion, will be taken into acc ount. In this method, a SEM photograph of microstructure for an electrode can be mapped onto a random network of bonds connected to one another at nodes of the network. The network model incorporates pore-scale descriptions of the medium and the physics of pore-scal e activities. For example, a 3D random resistor network can be established to directly describe a composite cathode structure as show n in Figure 2-2 (only 2D view is given). By identifying the positions and shapes of grain particles and pores, the most accurate information about the microstructure can be described. Nodes denote the grain particle or pore, which include information about the spa tial dimension distribution. Resistances connect each pair of nodes. Here, the different types of resistances represent different transport mechanisms. For example, it can be mass resist ance, electrical resistance or polarization resistance. The local resi stance is evaluated on the basis of shap es of the grain/po re and position. The electrochemical reaction, ohimc resistance and polarization resistance, etc. can be calculated by this model. It will undoubtedly ga in a greater and more fundamental understanding of fuel cell operation than would be possibl e through the use of macroscopic models alone.
47 A B Figure 2-2. The random network model. A) Irre gular microstructure of a composite electrode. Black circle = electronic conductor, blue circle = ionic conductor, empty space = pore. In an actual electrode, the sh ape of particle may be spheri cal or cylindrical etc. B) Network discretizaiton of micr ostructure. Red rectangle denotes the resistance. Black node = electronic conductor, blue node = ionic conductor, white node = pore. Atomistic Model. The atomistic model is an integral part of the complete multi-scale model but it will not be a part of the current work. In view of keeping the completeness of the model framework (Figure 2-1), we give some ba sic explanation for this atomistic model. For oxygen reduction, not a single mechanism discove red so far can explain all the oxygen reduction processes in the cathode. Although th e O or H atomic diffusion is not the rate limiting step at macroscopic scales, it is anticipated that it c ould become significant in components that have been miniaturized to the micrometer or nanometer scales and/or at lower operating temperatures. To model the above mentioned atomic diffusion in metals requires the calculation of bond energies, cohesive energies, and atomic-scale forces that can be accomplished with computational methods derived directly from fi rst quantum mechanical principles (hence the name ab initio). For Ab initio quantum molecular methods, they are direct and most accurate, but the current issue is that they are computationally demanding and th erefore restricted to relatively microscopic local regions. We propose to use th e quantum molecular dynamics methods that will employ the density functional theory (DFT) by Hohenberg an d Kohn (1964). DFT methods are readily applied to periodic systems such as lattices, surfaces and interfaces. Inter-connector Electrolyte I II III
48 Specifically, the generalized gradient appr oximation (GGA) approach will be used to calculate the electron density, which has been show n to be the best for calculating bond lengths and energies. The DFT can be used to calculate atom ic energies and subsequent forces to be used in molecular dynamics (MD) simulations, wher e Newtons equations of motion are solved numerically for the transport of oxygen and hydrogen atoms. The goal of the atomistic effort is to study the diffusion of O and H atoms through the metal oxide lattices that make up the transport membranes. These lattice models can in clude various point defects (such as vacancies) and structural defects (such as grain boundaries) that could play a role in atomic and molecular diffusion. The multi-scale framework proposed here will be employed to integrate the atomistic, nano/micro, and macroscopc models. The macroscopic model will be computed based on the continuum conservation laws. The solution will be supplied to atomistic model as the global parameters. The micro-scale model establishe s the complex relationships among the various transport phenomena, which includes the transpor t of electron, ion and gas molecules through the electrodes, electrolyte and particularly at the three-phase boundary region. The atomistic model, in turn, will be executed strategi cally at sampled microscopic local regions in the electrode and electrolyte to supply the flux profiles to the microscopic model as boundary conditions. Such a process can iterate among the three modeling levels until the atomic, micro and macroscopic simulations offer a consistent result based on the input from one another. Finally, by integrating the three models, the dependence of electrochemical performance on the geometrical parameters (channel height, th ickness of electrodes, etc.), microstructures (porosity, volume fraction, composite ratio, etc. ), and electrochemical r eaction (surface diffusion, atomic diffusion) will be assessed for the entire system in a meaningful way.
49 CHAPTER 3 DEVELOPMENT OF A MACROSCALE MOD EL AND PERFORMANCE OPTIMIZATION FOR A SOFC UNIT CELL In this chapter, the electrochemistry fundament als for solid oxide fuel cell are reviewed in the first section. These fundamentals are rela ted to the ideal performance and thermodynamic operation of fuel cells, i.e., how operating conditio ns (such as pressure and temperature, etc) affect the performance of fuel ce lls. The electrical potential loss in the practical fuel cell process is discussed in detail in th e second section, which includes the ohmic, activation and concentration overpotentials caused by mass transf er. In the third section, a 3-D thermo-fluid electrochemical model is developed to simulate the heat/mass transport and electrochemical process in a SOFC unit cell. Finally, this mode l is applied to investigate and analyze the heat/mass transfer, chemical r eaction and electrochemistry in an anode-supported fuel cell and the effect of transport scale on an electrolyte-supported fuel cell. 3.1 Electrochemistry Fundamentals Before starting to model the energy transport process, one should well understand the basic electrochemical phenomenon in fuel cells. In general, a fuel ce ll possesses two types of energy: (1) free energy, G and (2) unavailable energy. Free energy earns its name because it is available for conversion into usable work. The unavailable en ergy is lost due to the increased disorder, or entropy, of the system. The maximum electrical work obtainable in a fuel cell operating at constant temperature and pressure is gi ven by the change in Gibbs free energy (G) of the electrochemical reaction: GnFE (3.1) where n is the number of electrons participating in the reaction. F is Faradays constant and E
50 is the ideal potential of the cell. For reactants a nd products in their standa rd states, i.e., 298K and 1atm: 00GnFE (3.2) For the general cell reaction: aAbBcCD (3.3) The Gibbs free energy which pertains to one mole of a chemical species is called chemical potential (g). For ideal gas with temperature T and pressure p the chemical potential can be expressed by: 0 0lng p ggRT p (3.4) where g R is the gas constant and 0 p is the standard pressure of 1a tm. It is noted that the unit of pressure p is atm. The overall Gibbs free energy cha nge in equation (3.3) can be expressed in terms of the standard state Gibbs free energy and partial pressure of individual components: ()()cDABGcggagbg (3.5) Substituting equation (3.4) into (3.5), then we can get: 00 0000 00 00 0 00(/)(/) ()()ln (/)(/) (/)(/) ln (/)(/)c CD cDABg ab AB c CD g ab ABpppp GcggagbgRT p ppp pppp GRT pppp (3.6) When equation (3.1) is substituted into equation (3 .6), the Nernst potential, which is also called theoretical electromotive force (EMF), is:
51 00 g 0 AB 00 CD(/)(/) ln (/)(/)ab cRT p ppp G EE nFnFpppp (3.7) where the ideal standard potential 0E is equal to 0G nF If the oxidation of hydrogen is the only electrochemical reaction in the solid oxide fuel cell, and it can be expressed by the following chemical equation: 222H+1/2OHO(gas) (3.8) The Nernst potential will be: 22 2000.5 HanodeOcathode g 0 0 HOanode(/)(/) ln 2(/) pppp RT EE Fpp (3.9) the ideal standard potential 0E depends on temperature, which d ecreases with the increasing of temperature. For example, for a solid oxide fuel cell (operating at o1100C) in which H2 and O2 react, this value is about 0.91 volts with gaseous water product. In the anode side of a solid oxide fuel cell, the electrochemical equilibrium exists in the gas species: 22 2000.5 Hanodeanode 0 g 0 HOanode(/)(/) 0ln (/)Opppp GGRT pp (3.10) then substituting Eq. (3.10) into Eq. (3.9), the Nernst potential can be obtained in another form: 22g 00 OcathodeOanode[ln(/)ln(/)] 2RT Epppp F (3.11) Compared with equation (3.11), equation (3.9 ) is more common to apply. Once the oxygen partial pressure on the cathode side, water and hydrogen partial pressure on the anode side are known, the EMF will be determined.
52 Some secondary parameters are very useful in assessing the overall pe rformance of a fuel cell and will be used in the fo llowing model, are given here: (1)Fuel utilization factor: 22 2(MassofHin MassofHout) 100(%) MassofHin (3.12) (2) Air ratio: Total oxygen mass Consumed oxygen mass (3.13) (3) Power density: Total power Cell volume (3.14) (4)Fuel efficiency: 2Total power MassofHin( H) (3.15) 3.2 Electrical Potential Loss However, in reality, the irreve rsibility related to chemical kinetics becomes a significant factor and the actual potential is lower than ideal potential. Figure 3-1 shows the characteristic shape of the voltage/current density graph resulting from four major irreversibilities. These will be outlined briefly here and then considered in more detail in the modeling development (section 3.3). Ohmic Polarization. The ohimic loss is caused by the electrical and contact resistances of the electrodes and the collecting components, as well as the ioni c conduction resistance of the electrolyte. If the electric al resistance in current collector is negligible (Iwata, et al. 2000), which leads to the consideration that the current is almost perpendicularly collected, the voltage loss can be described by Ohms law: ohmicaaccee()/iA (3.16) wherea c and e are resitivities for anode, cathode and electrolyte, respectively. a c and e are respective thickness for three layers. But th is is the simplest case. If the resistances in
53 current collectors, contact resist ance and internal crossover current are considered, an equivalent circuit comprising many resistors can be applied to describe the ohmic behavior. However, in actual situation (especially in the start-up or shut-down period), the current response is not instantaneous. Therefore, capacitance effect should be consider ed. For classical materials used in SOFC, such as YSZ electrolyte, Ni anode, a nd LSM+YSZ cathode, the major ohmic contribution is from electrolyte, because the ionic resistivity of the electrolyte is much larger than the electronic resistivitie s of the anode and cathode. The re lative contributi ons of various polarizations vary widely among different cell design, such as cathode-supported, and electrolyte-supported and anode-sup ported. Ohmic contribution is smaller in electrode-supported cells due to relatively thin elec trolyte thickness. Thus the anod e-supported fuel cell usually has better performances. 02004006008001000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 fall faster at higher currents fairly linear very small initial fall in voltageCell voltage (V)Curent density (mA cm-2) no loss voltage of 1.0V Figure 3-1. The voltage of a fuel cell operati ng at about 800 degrees. (Larminie and Dicks, 2003) Concentration Polarization. This type of polarization is due to the mass transport resistance between the channel fl ow region and the reaction site, which results in lower partial pressures for the reactants and higher partial pr essures for the products at the electrode surfaces.
54 Concentration polarization is gene rally largest at the cathode due to low diffusivity of oxygen. Moreover, at a high cell current de nsity, the increased requirements for the feed of the reactants and removal of the products can significantly ra ise the concentration po larization, and the cell output potential will drop accordingly. The concentration polarizati on at the cathode and anode ar e functions of the species effective diffusivity, current density and microstructure, etc.. In pa rticular, the microstructure of the electrode strongly influences the effective diffusivity and this pol arization, because the Knudsen diffusion, surface diffusion or even effect s of adsorption may occur in porous media. In terms of measurable parameters, the 1-D analytical expressions have been derived by Kim et al. (1999) and Chan et al. (2001). Th e two expressions based on the binary mass transfer principle are very explicit. However, in practice, the fuel is not pure hy drogen, but reformed hydrocarbon. This requires a numerical and tw o or three-dimensional solution including reforming and shifting reaction, as well as multi-component transport. This method will be addressed in the following model. Activation Polarization (or activation overpotential). Compared with concentration and ohmic polarization, this one is much more co mplicated. Activation pola rization is caused by the slowness of the electrode reaction taking place in the electrodes. As mentioned in Chapter 2, the electrode reaction, which is either the oxygen reduction in th e cathode or the hydrogen reduction in the anode, is a multi-step process. The pa rameters influencing the corresponding reaction can be pressure, temperature, material properties or geometry of electrodes, etc. Therefore, obtaining a unified reaction mechanism is unrealistic. Fo r example, the oxygen reduction in the cathode may involve oxygen diffusion, adsorption of oxygen on the electrode surface, diffusion of oxygen species along the surface towards the th ree-phase boundary, followed by complete
55 ionization and ionic transf er into the electrolyte or other possi ble diffusion paths. A wide variety of reaction mechanisms have been reported in the litera tures, like Sasaki et al.(1996), Odgaard et al.(1996) and Fleig (2003), etc. The lack of consistency is apparently due to the fact that electrode processes and mor phology are closely interrelated and also the fundamental mechanisms are not fully understood at the present time. However, it is possible to describe the ove rall process in a phenomenological approach. The most common approach is known as the Bu lter-Volmer equation that gives a quantitative relation between current de nsity and activation loss: eacteact 0 ggexp()exp[(1))] nFnF ii RTRT (3.17) where is the transfer coefficient; act is activation overpotential and 0i is the exchange current density. It should be noted th at the relationship between act and current density i is nonlinear and implicit. However, for limiting cases, i.e., current density is very high or small, act can be approximated as the Tafel equation or linear expression (Li et al., 2003; Singhal and Kendall, 2003; Brad and Faulkner, 2001). Experimental work combined with modeling has been conducted in order to analyze and determine the exchange current density under different conditions. Electrochemical impedance spectrosc opy (EIS) is the most commonly used method. It has been proved that exchange current density depends on the activity of the electrocatalyst (TPB), microstructure, shap e of electrodes, partial pressure and so on. Electrode polarizations cause si gnificant voltage losses in SO FCs and need to be reduced to improve efficiency. Previous studies have in dicated that the largest polarization contribution comes from the oxygen reduction process in the cathode. So, much more composites or MIEC cathodes, which can extend the TPB length, are b een using in fuel cells to reduce activation
56 losses. In addition, fine microstructure al so increases the TPB length and activity of electrochemical reaction, so the ac tivation losses can be reduced furt her. But, fine microstructure means smaller porosity and larger mass transfer resi stance. This may lead to larger concentration polarization. Therefore, an idea l electrode structure should be optimized. In Chapter 4, a detailed discussion on composite electrode kinetics and al so on calculating different overpotentials based on a micro-level model will be presented. 3.3 Three-dimensional Thermo-fluid Electrochemical Model for a SOFC Unit Cell In this section, a 3-D thermo-fluid electrochemical model is developed to simulate the heat/mass transport and electrochemical process in a SOFC unit. In future work, it is applied into optimizing design of an anode-supported a nd an electrolyte-supported SOFCs. 3.3.1 Development of Modeling 220.127.116.11 Model features and assumptions The model is developed to simulate a planar SOFC by a finite-volume approach (Patanka, 1980). Three-dimensional temperature, chemical c oncentration profiles, r eactant utilization and power outputs are yielded. Fuel composition may include any combination of hydrogen, methane, carbon dioxide, carbon monoxide, water, nitrogen or others. Air is c onsidered to be composed of oxygen and nitrogen. Although, the model is base d on co-flow and a planar SOFC, it is also applicable for cross or count er-flow configurations and t ubular SOFC by changing some geometrical parameters and boundary conditions. There are several assumptions and definitions that should be pointed out before describing the present model: Transport and gradients in all th ree-directions are considered. Energy transfer due to species diffusion in cha nnel flows and porous layers are neglected.
57 Because the single cell model represents a repea ting cell unit in the cente r of a larger stack, cyclic boundary conditions for current are imposed at the boundaries of model domain. Walls at the periphery of the single cell are assumed to be adiabatic. The kinetics of carbon monoxide oxidation at the r eaction site is generally slow if compared with hydrogen oxidation .Therefore only hydr ogen is assumed to take part in the electrochemical reaction, while carbon monoxide is oxidized through the shift reaction. The heat exchange by thermal radiation is cons idered negligible with respect to the heat balance inside the cell channels due to the dom inant effect of convective heat transfer and small cross-section of the channels w ith respect to the channel length. The stack of a SOFC is made up of many identi cal unit cells connected in series, most of which function under similar operation conditions. Therefore, to reduce computational work, only one unit cell is taken as the computational doma in. This typical single cell is assumed to be an interior and non-boundary unit. As a result, the heat loss to the surroundings is not considered and the only heat removal mechanism from the cell is through the gas streams. The schematic of the computational model is shown in Figure 3-2. Furthermore, only one half of the gas channels are modeled by applying the symmetry condition. A single cell consists of a porous layer of anode, a porous layer of cathode, a dense layer of electrolyte, and interc onnect (or rib). The interconnect supports the structure and also serves as the current collector. When air and fuel streams flow along the X -direction (co-flow arrangement), O2 gas from the air stream diffuses through the cathode and accepts electr ons from the external circui t at the cathode/electrolyte interface. Also, at the anode/el ectrolyte interface the oxygen i ons pass through the electrolyte and react with the H2, and the reaction product H2O diffuses back into the fu el stream. If the fuel is natural gas (CH4, CO2, CO, H2, and H2O), the reforming and shift reactions take place through the anode layer as follows: Reforming: 422CH+HO3H+CO +206kJ/mol (3.18) Water-gas shift:
58 222CO+HOH+CO41kJ/mol (3.19) Figure 3-2. A co-flow planar SO FC and the computational domain. 18.104.22.168 Thermal-fluid model In this model, the entire computational dom ain consists of three sub-domains: porous (electrodes), solid (interconnects and electrolyte) and pure fluid regions (channel flows). It has been demonstrated by previous experiments that the fluid flow in the porous media is a low speed mixture gas flow with a Reynolds number mu ch less than unity. Therefore, this flow is treated as laminar and incompressible. It is also assumed that the porous medium is homogeneous and in local thermodynamic equilibr ium with the fluid. In the porous region, the flow is governed by the Brinkman-Forchheimer ex tended Darcy model, which takes into account the effects of flow inertia as well as friction induced by macroscopic shear stresses. While in the fluid region, the Navier-Stokes e quations describe the flow beha vior. The conservation equations of mass, momentum, energy and species for flui ds in both the pure fluid region and porous domain are: f()0 V (3.20) 2 fff1()()fF p VVVVVV K K (3.21) ffeffreoh()() cVTkTQQ (3.22) L Computational domain Air Fuel To p interconnwall. Cathode Anode Electrolyte Symmetric boundary X Z Y Wr 0.5W H
59 f()iiiYVJS (3.23) 11n i iY (3. 24) f g()i i iP Y RT M (3.25) where parameter is set to unity for the flow in a porous medium (01 ) and to zero in regions without porous materials (1 ). Vand iY represent the velocity vector and the local mass fraction of each component, respectively. is the porosity and K is the permeability of the porous medium.1F the Forchheimer coefficient is calculated by50.5 11.8/(180) F The effective thermal conductivity effk is equal tofs(1) kk The heat source ohQ is due to ohmic loss and activation overpotenti als. Another source term,reQ, denotes the heat produced by reforming and shift reactions which only take place in the anode layer. There is no heat generation term in pure fluid region Thus, once the information of the electrical field and chemical reactions is obtained, the two h eat source terms can be exactly determined. iS is the net rate of production of component i by chemical reaction and is left for specific discussion later.iJ is the diffusion mass flux vector derived from the Stefan-Maxwell relations (Mills, 2001): 1 k,,eff,effg()n ijji i i i iijij jiXJXJ J P X DDDRT (3.26) wherei X is the mole fraction of component i and ,eff ijD is the effective binary diffusion coefficient of a mixture of components i and j and equal toijD is the ratio of porosity to
60 tortuosity.k, iD is the effective Knudsen diff usion coefficient of component i and calculated according to the kinetic theory of gas: g k,8 2 3i i R T Dr M (3.27) where r is the average pore radius. The density, viscosity, specific heat, and thermal conductivity are functions of both temperature and mass fraction of gas mixture. In a SOFC, the shift reaction, which takes place at high temperat ure, can be assumed to react very quickly and almost in equilibrium (Lehmert et al., 2000). The reforming kinetics is assumed to be quasihomogeneous. Lehmert et al. reported experimental results for the forward velocity constants (+ rk ,+ shk ) and backward velocity constants (rk ,shk ) for the two reactions at three different temperatures. Haberman et al. (2004, 2005) have developed empiri cal correlations for velocity constants with Arrhenius curve fit ting technique using experimental data of Lehmert et al. in the temperature range,900K1300K T The volumetric reaction rates of hydrogen in the shift reaction and methane in the reform ing reaction are calculated by: 2222+22 sh,HsgCOHOsgCOH()() R kRTCCkRTCC (3.28) 44222-43 r,CHrgCHHOrgCOH()() R kRTCCkRTCC (3.29) where iC (4222CH,CO,CO,H,HO i ) is the molar concentration of component i. The net rate of production for each component,iS can be stated as the followings: 444CHCHr,CHSMR (3.30) 42COCOr,CHCOsh,HSMRMR (3.31) 22422HOHOr,CHHOsh,HSMRMR (3.32) 22422HHr,CHHsh,H3R SMRM (3.33)
61 222COCOsh,HSMR (3.34) When the molar numbers of methane and hydrogen involved in the reactions are determined, the volumetric heat generation rate due to chemi cal reaction in Eq. (3.22) is expressed by: 42rereformr,CHshiftsh,HQHRHR (3.35) It should be noted that Eq. (3.35) is only appli cable to the anode layer. In Eq. (3.22), the second source term includes the heat generations from ohmi c loss in all solid or porous regions and the activation overpotentials in electrodes: for solid: 2 ohQIR (3.36a) for electrodes 2 ohactQIRI (3.36b) where I is the local current and R is the resistance. act is the activation overpotential. It is assumed that the electrochemical reaction occurs in the vicinity of the an ode/electrolyte interface, and so the thermodynamic heat ge neration from this reaction is: 2elecelecH()qHGm (3.37) here G is the chemical potential and 2Hm is the molar consumption rate of hydrogen. In the actual calculation, Eq. (3.37) is treated as a heat flux condition at the a node/ electrolyte interface. 22.214.171.124 Electrochemical model The electrochemical model predicts the local electrical potential and current density, which is subject to changes in geometry, local temp erature and gas compositi ons. Numerical simulation for a three-dimensional electrical field is directly analogous to the calculat ion of heat transfer using a finite volume method. A separate anode/electrolyte/ cat hode unit is modeled and the
62 discretization of equivalent electric circuit is shown in Figure 33. Current enters into electrical network from the bottom curren t collector (i.e., bottom interc onnect), then flows through the anode, electrolyte and cathode, finally to the external circuit by top current collector. In this model, it is assumed that the cathode and the anode are purely electronic conductor and electrolyte is purely ionic conducto r. However, this electrical netw ork is inappropriate for micromodel in Chapter 4, in which the composite electrodes will replace such one-phase electrode, because the electrochemical reaction is not limi ted at the interface betw een an anode and an electrolyte, but everywhere on the whole anode. Within the el ectrolyte layer, the current primarily flows from the anode to the cathode. In the anode layer, the ohmic resistance in normal direction (Y-direction) is neglec ted for thin anode. But if the anode is thick, such as anodesupported SOFC, this directional re sistance will be incorporated. Electrostatic potential within all computational elements must satisfy the Laplace equation: ()0 (3.38) where is the electrical or ionic conductivity and is the potential. In Figure 3-3, ,Eij denotes the ideal potential in each element. R axRazRcxRcz and Re denote the ohmic resistances for anode, cathode and electrolyte, respectively. Rint and Rinb are resistances from current collectors. Calculation of this electric field potential combines the following contributions. The first is ohmic losses in all the conducting materials, including the electro lyte, the electr odes, and the current collectors. The second is contact resistan ce at interfaces and interconnect resistance, which have been included into Re. The third is activation overpotentials act which is described by the well-known Butler-Volmer expression, in the anode and cathode layers (Brad and Faulkner, 2001):
63 eacteact 0 ggexp()exp[(1))] nFnF ii RTRT (3.39) where F is the Faradays constant. en is the number of electrons. is the transfer coefficient. 0i is the exchange current density. These parameters are given in Table 3-1. If is chosen as 0.5, activation overpotential for the anode a nd the cathode can be wr itten as follows: g 1 act,an 0asinh() 2 RT i Fi (3.40) g 1 act,ca 0csinh() 2 RT i Fi (3.41) The concentration overpotentials at electrodes are calculated as: 2 2O g con r Oln() 4 p RT Fp (3.42) where 2O p and 2r O p are the partial pressures of oxygen at electrode/channel interface and on the reaction site, respectively. The distribution of local current density must be calculated repeatedly and the calculation must be converged to satisfy the same output voltage ( Vout) between the upper and lower surfaces of the interconnects. Once the el ectrical potential at each node is known, the local current density can be calculated as: actcaan area() (,) Re Ei ixz A (3.43) with Nernst potential: 22 20.5 HO g 0 HOln() 2 p p RT EE Fp (3.44)
64 Figure 3-3. Equivale nt electric circuit. where Reis the electrolyte resistance and areaA is the active area. ca and an are electrical potentials at cathode/air and anode /fuel interfaces, respectively.2H p and 2HO p are partial pressures of hydrogen and water at interfaces betw een electrodes and electrolyte, respectively. 4 01.27232.764510 ET is a function of temper ature at reaction location. 3.3.2 Solution Procedure and Boundary Conditions For this model, there are two kinds of boundary conditions: boundary conditions for the overall computational domain and the interfacial condition at the interf ace between electrode and electrolyte. The first type of boundary condi tions is summarized as the followings: 1,2ca Rax1,3 1,3 an X Z Y Rax1,2 Rcx1,2 Rcx1,3 Rcx1,1 Rcz1,1 Rcz1,2 Re1,2 Re1,3 Raz2,1 2,1 ca 2,2 ca 1,3 ca 2,1 an 2,2 an 1,1 an 1,2 an Rax1,1 Raz1,1 Rcz2,1 Raz1,2 Rcx2,2 Rcz2,2 Raz2,2 Re1,1 Rint1,1 Rint2,1 Rinb1,1 Rinb2,1 VOUT VOUT E1,1 E2,1 E2,2 1,1 ca
65 Air inlet: 2in,airOin,0,0, uuvwYY ,in,airTT ,1atm p (3.45a) Fuel inlet: in,fuelin,242,0,0,(i=H,CH,CO,CO)iiuuvwYY in,fuelTT ,1atm p (3.45b) Air outlet: 20,0OY u xx (3.45c) Fuel outlet: 0,0iY u xx (3.45d) In addition, the non-slip for ve locity and impermeable conditi on for concentration field are applied on the walls. During calcul ating the electrical potential and temperature, the cyclic boundary conditions are imposed at the top and bo ttom of the model domains, while the lateral walls are assumed adiabatic and insulation. In actua l application, the fuel utilization is required to in the range of 60-80%, so the inlet fuel velo city in Eq. (3.45b) is always determined as: 42in,fuel in,CHin,Hin,COfuel(822)ILW u FYFYFYA (3.46) Here I is the average current density which is a given initial condition. L and W are the length and width of electrode. fuel is the fuel utilization and A is the cross section of the fuel channel. The boundary conditions at the interfaces: 2 222O OO,airOint, airC mDC y (3.47a)
66 2 222HOHO,fuelHOint, fuel HOC mDC y (3.47b) 22HOHmm (3.47c) where 2Om and 2HOm are the molar production or consump tion rate of oxygen and water at the cathode/electrolyte and anode/elect rolyte interfaces, respectively. int,air and int, fuel are the interfacial velocities induced by local mass flux and defined as: 22O,N int,air air i im v (3.47d) 22H,HO,etc int,fuel fuel i im v (3.47e) For each finite volume, the governing equations (3.20)-(3.25), (3.38) and electrochemical equations with appropriate boundary conditions are progressively solved by employing the SIMPLEC algorithm (Patankar, 1980). To evaluate the performance of the numerical method and the computational code, we first test it for in compressible gas flow in a smooth microchannel with no slip boundary condition. The numerical result s agree well with the an alytical results with a maximum difference of 2.71%. Additional comput ations are performed with coarse grids and fine grids to check the grid size effect on the numerical solution. The grids are refined near the wall region to obtain highly accurate numerical so lutions. For example, the grid numbers are set as 506432, 1009648and 2009664 The maximal error in temperature between 1009648 and 2009664 grid is 3.13%. By balancing between the computation time and accuracy, the grid size 1009648 is applied sufficient to achieve a reasonable accuracy. The detailed solution procedure is shown in Fi gure 3-4 and can be summarized as follows:
67 Mesh the computational domain and input the boundary condition, geometrical parameters and properties for materials. Determine the ohmic resistances, activati on overpotenital and interfacial boundary conditions by using the latest temperature, velocity, concentration and pressure values. Solve the discretized equations for electrical potential (Eq. (3.38)) to obtain local potential distribution. Figure 3-4. Solution procedure. The updated potential field is passed to th e energy equation to get heat source terms and interfacial mass flux. Solve the momentum and continuity equatio ns to update the local distributions of pressure temperature and concentrations. Set initial conditions Input geometrical parameters and properties Latest velocity and concentrationfields Latest electrical potential and local current density Evaluate heat sources Output voltage convergent? Yes No Convergent ? No Yes Input temperature, velocity, concentration and local current density Tem p erature field Coupled with ICs
68 The calculation is repeated until convergence is reached according to a user-defined residual error. Here we set the residual for velocity410 temperature610, concentration610, and potential610 3.4 Model Application for an Anode-supported SOFC In this section, the above m odel is applied to an anode-suppor ted SOFC. Then a parametric study examines the temperature field, concentrati on field, distribution of local current density and power density. 3.4.1 Choice of Parameters Air gas (79%N2 and 21%O2) is delivered to air cha nnel at 1000K with 2.5ms-1. Fuel (15.9%H2O, 64.8%H2, 4.8%CO2, 14.9%CO) is delivered to fuel channel at 1000K with 0.6ms-1. It is assumed that the H2 consumption rate is about double of the CO consumption rate at the anode/electrolyte interface. Some physical parameters are extracted from literatures (Bessette et al., 1995; Iwata et al., 2000) and given in Table 3.1. The poros ity, permeability and tortuosity for electrodes are 0.5, 101.710 m2 and 1.5, respectively. The cr oss sections of channels are 322(1.210)m and the length of unit cell is taken as 0. 05m .The average current density is set as 4000Am-2. Table 3-1. Physical prop erties in porous materials and geometry parameters Description Anode Cathode Electrolyte Interconnect Thickness (m) 31.210 65010 62010 30.410 Electric resistively ( m ) 52.9810 exp(1392/) T 58.11410 exp(500/) T 52.9410 exp(10350/) T 32.010 3.4.2 Results and Analysis 126.96.36.199 Velocity field Figure 3-5 shows the individual component of the gas velocity vector. From Figure 3-5A, it can be seen that the x-velocity in channel increases along the flow direction. This indicates that the development of the boundary la yer near the interface. In addi tion, mass injection due to the electrochemical oxidation reaction increases th e x-momentum flux in the fuel channel. In
69 contrast, the oxygen reduction in the cathode causes the x-momentum flux decrease in the air channel. The velocity magnitude in porous layer is still much smaller than channel flow, which means the diffusion is dominant in this region. In the most of anode region, the velocity is dominated by a stronger y-velocity, which is in the order of 0.001ms-1 as Figure 3-5B shown. A B Figure 3-5. Cross-section velocity contours at selected planes. A) x-velocity at z=0.001m. B) yvelocity at x=0.005m, x=0.025m and x=0.045m (arrow indicates the flow direction). Along the flow direction, the y-velocity at the fuel/anode interf ace reduces and less mass due to convection is brought into the fuel bulk flow. At the regi on near the lower corner, the yvelocity reaches maximum value. This may be caused by the bigger pressure gradient along the wall. However, the velocity magnitude is almo st close to zero at th e region underneath the interconnect. This weak convecti on effect will further worsen the mass transport demonstrated as below. 188.8.131.52 Temperature field The temperature contours for different planes are shown in Figure 3-6. Along the flow direction, the temperatur e increases monotonically from inlet to outlet due to the contribution of Joule heat and chemical reaction. The maximu m temperature is around 10 36K near gas outlet. The temperature increase is not as significant as expected, beca use the thick anode layer (1.2mm) and high mass rate in the air channel allevi ate the temperature increase. Such uniform X0 0.01 0.02 0.03 0.04 0.05 Z0 0.0005 0.001 0.0015 X Y Z y-velocity 0.012 0.008 0.006 0.004 0.002 0.0012 0.0005 0 -0.0008 -0.001 -0.002 -0.003 -0.0036 -0.004 -0.006 -0.0074 -0.008 -0.01 -0.012 -0.014 -0.019 Fuel channel Air channel Z X Y x-velocity 3.91441 3.00968 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.993455 0.95 0.9 0.55 0.4 0.2 Anode layer Fuel channel Air channel
70 temperature distribution will alleviate the risk of se vere thermal stress and thus help to ensure the structural integrity of the cell components. But low temperature will in crease the electrical resistance of electrolyte, which is a function of temperature. T hus, it is important to obtain an appropriate temperature distribut ion and thus improve the perfor mance of SOFC and efficiency. X0 0.01 0.02 0.03 0.04 0.05 Z0 0.0005 0.001 0.0015 X Y Z temperature 1034 1032.44 1032 1030 1028 1026.89 1026 1024 1022 1020 1018 1017.29 1016 1014 1012 1010 1008 1006 1004 1002 Figure 3-6. Temperature contour s at selected planes (at x= 0.005m, x=0.025m and x=0.049m, arrow denotes flow direction). 184.108.40.206 Concentration field and current density Figure3.7 gives the molar fraction distributions for different gas components. There is a non-uniform distribution for oxygen concentration di stribution at cathode/el ectrolyte interface as shown in Figure 3-7A. The molar fraction of oxyge n under the rib is lower than that in other areas. This is because oxygen can not diffuse well underneath the inter-connector. But if the width of rib is decreased, the c ontact resistance between rib and anode or cathode will increase and thus increase ohmic losses. Previous studies have demonstrated (Ferguson et al., 1996) the need of optimization of width of rib. At anode /electrolyte interface, th ere is a net decrease in molar fraction of CO from fuel inlet to outlet due to electrochemical reaction. The change in molar fraction of H2 is similar to that in CO. It should be noted that the rib does not cause the
71 remarkable concentration gradients for H2/CO/CO2 in the z-direction due to a thicker anode layer. However, in the y-direction, the concentration gr adients at anode are relatively significant. For example, the molar fraction of CO from the bulk fuel stream to the react ion location decreases by 7.9% at x = 0.0237m and z = 0.001m as shown in Figure 3-7E. Y Z Xmolar%-O2 0.186 0.184 0.182 0.18 0.178 0.176 0.174 0.172 0.17 0.168 0.166 0.164 0.162 0.16 0.158 AB Y Z Xmolar%-H2 0.63 0.62 0.61 0.6 0.59 0.58 0.57 0.56 0.55 0.54 CD EF Figure 3-7. Molar fraction and current density contours at interfaces. A) O2 at cathode/electrolyte. B) CO at anode/electrolyte. C) H2 at anode/electrolyte. D) H2 at z=0.001m. E) CO at z=0.001m. F) Current density (A/m2). X0 0.01 0.02 0.03 0.04 0.05 Z0 0.0005 0.001 0.0015 X Y Z 0.14 0.136541 0.128366 0.12 0.115666 0.112053 0.109266 0.1 0.095 0 Anode Fuel channel Air channel X0 0.01 0.02 0.03 0.04 0.05 Z0 0.0005 0.001 0.0015 X Y Z 0.64 0.63137 0.62 0.6 0.58159 2 0.58 0.56 0.54 0 Anode Fuel channel Air channel Y Z Xmolar%-CO 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0.1 0.095 X0 0.01 0.02 0.03 0.04 0.05 z0 0.0005 0.001 0.0015 3500 4000 4500 X Y Z 'CURRENT_DEN' 4600 4500 4400 4300 4200 4100 4000 3900 3800 3700 3600
72 It should be pointed that th ere is a nonlinear relationship between the concentration distribution and the EMF, so when concentrations of CO and H2 experience small increases, the EMF will increase much more. In addition, for an anode-supported SOFC, the main concentration polarization is in the anode layer, not in the fuel channel, because the effective diffusivity in porous region is much less than that in the fluid regi on. The diffusion of H2 is about 3 times faster than that of CO. Thus, lowering the concentration polarization of CO is very critical. The output voltage is 0.6335V in this ca se. The local current de nsity profile given in Figure 37F has a maximum value near the inle t, then becomes smaller and smaller along flow stream. The local current density near the symm etric line is relatively lower because of the longest current path. At the regi on under the rib, the current dens ity is higher and very smooth. Further calculations demonstrat e that only when the channel di mensions are reduced, the output voltage will be improved, but not significantly. This is because that the improvement in concentration at the reaction location is rather limited. 220.127.116.11 Conclusion Above results indicate that there are trade-o ffs among factors influencing performance of SOFC. Concentration, temperatur e, velocity and electrochemical reactions are highly coupled. Therefore, our further work is to find better appr oaches or other critical parameters to optimize the design of micro-so lid oxide fuel cells. 3.5 Effect of Transport Scale fo r an Electrolyte-supported SOFC 3.5.1 Problem of Statement In order to minimize the overall losses, alle viate the thermal stress and improve the electrolyte-supported SOFC ef ficiency, key parameters gove rning the polarizations and heat/mass transfer efficiency s hould be investigated. Basically, for a given operating condition, there are three kinds of mechanisms that cause the overall losses: (1) oh mic polarization, which
73 is from the ohmic resistances of all com ponents and contact inte rfaces; (2) Activation polarizations at the anode and cathode, which are related to charge transf er processes and depend on the nature of electrode-elect rolyte interfaces; (3 ) Concentration polarization, which is associated with the transport of gas species and thus depends on the channel dimensions, electrode geometry and the nature of electrode microstructure, i.e., porosity and tortuosity. For an electrolyte-supported SOFC, a la rger ohmic polarization in the electrolyte is a serious problem, so developing a thin-film electrolyte to substantially reduce the ohmic loss is a worthwhile approach (Chen et al., 2004; Tang et al., 2005; Gorman et al., 2005). In general, with the dimension of flow channel decreasing from conventional to microscale dimensions, the temperature /concentration gradient decreases an d the heat/mass transfer coefficient increases. Moreover, minimized sizes are important to develop low cost and portable SOFCs. One, therefore, expects that concentration loss and ope rating temperature in smaller fuel cells would be reduced effectively. However, only a few investigations address the issues of thermal stress, heat/mass transfer efficiency and cell performance for flow channels whose characteristic dimension is less than 1.0 mm (Tang et al., 2005; Li, 2005; Cha et al., 2004;). On the other ha nd, small scales may lead to thermal management difficulties and poor thermal cyclability, etc. Recently, Ramakrishna et al. (2005) developed a new type of SOFC which ha s a thin-walled geometry Because of a better longitudinal reactant distribution along the thin-walled geometry, they were able to improve the SOFC performance and achieve a ma ximum power density of 1.18Wcm 2. Li et al. (2005) proposed a new design for the fuel cell gas dist ributor by maximizing the access area of reactants and enhancing mass transfer, and were able to increase the maximum output power by 40% for a single cell compared to that of the old cell. Thes e potential advantages motivate us to investigate
74 the scale effects, especially miniaturized flow channels, on the performance of a SOFC. Other geometrical parameters, such as the rib width and electrolyte thickness, also have strong effects on the performance of a fuel cell and should be optimized. In the present study, a 3-D thermo-fluid electr ochemical model is deve loped to investigate the effect of transport scale on the fuel cell el ectrochemical performance. The geometry of the interconnect (or rib) and microstr uctures of electrodes are considered. Based on this model, the effects of channel height, thin-film elect rolyte, and rib thic kness on the performance characteristics of a SOFC are quan titatively investigated and the fo llowing issues are addressed: As the height of a flow channel is decreased whether or not the pow er density is improved and the solid temperature rise is suppressed? To what extent does the channel dime nsion affect the heat/mass transfer? Whether or not the thin-film electrolyte improves the performance of a SOFC? What is the optimal thickness of the rib? How do the above variations in geometry affect the thermal stress? 3.5.2 Parameters and Additional Equations Air (79% N2 and 21% O2) is delivered to the air channel. Fuel (15.22%CO, 4.52%CO2, 36.8%H2, 27.3%H2O, and 16.16%CH4) is delivered to the fuel channel. The average current density is set to 4000Am-2. To study the geometry effects, in each computational case one geometric parameter, such as the height of cha nnels, rib width or cell volume, is varied while keeping the other input parameters fixed. Some input parameters for the baseline case are extracted from literatures (Iwata et al. 2000; Habeman et al., 2004; Chan et al., 2001) and lis ted in Table 3-2. The physical properties of common materials, which are used for electrodes and electrolyte, strongly depend on temperature. Consequently, the performance of a SOFC is significantly affected by the temperature field. For instance, the resistivity of elect rolyte at 1200K could be one or two orders of magnitude smaller
75 than that at 900K. This temperature effect is included in the present m odel. Since electrolytesupported cells are always designe d to operate at over 1100K to a void a larger ohmic resistance in the electrolyte, the inlet fuel and air temper atures are chosen as 1100K in the present model. The local mass transfer coefficient m, ih for component i, heat transfer coefficient h, and average temperature T are defined as followings: m, ,int, i i biim h CC (3.48a) b intq h TT (3.48b) ATTdA (3.48c) where "q is the heat flux at the interface. int, iC and biC are the molar concentration at the interface and for the bul k flow of component i, respectively. A is the surface area. Table 3-2. Model input para meters for the baseline case Parameters and conditions Value Parameters and conditions Value Fuel inlet temperature (K) 1100 Ionic resistivity for cathode ( m) 58.11410 exp(500/) T Air inlet temperature (K) 1100 Interconnect resistivity ( m) 3110 Inlet pressure (Pa) 51.0110 Contact resistance (2m ) 6110 Fuel inlet velocity (-1ms) 0.6 Density of electrodes/electrolyte (-3kgm) 4400 Air inlet velocity (-1ms) 2.5 Density of interconnect (-3kgm) 5700 Cell length (m) 0.05 Thermal conductivity for anode/cathode (-1-1WmK) 12 Cell width (m) 33.210 Thermal conductivity for interconnect (-1-1WmK) 11 Rib width (m) 30.410 Thermal conductivity for electrolyte (-1-1WmK) 2.7 Channel width (m) 32.410 Porosity (%) 50 Anode thickness (m) 5510 Tortuosity 3
76 Cathode thickness (m) 5510 Anode exchange current density (Am-2) 6300 Electrolyte thickness (m) 615010 Cathode exchange current density (Am-2) 3000 Electric resistivity of electrolyte ( m) 52.9410 exp(10350/)TPermeability of the anode and cathode (m2) 101.710 Ionic resistivity for anode ( m) 52.9810 exp(1392/) T Average pore radius (m) 6110 3.5.3 Simulation Results and Discussion 18.104.22.168 Baseline case Figure 3-8 provides the molar fr action profiles for different ga s components at an average current density of -24000Am. As shown in Figure 3-8A, th e oxygen concentration distribution in the z-direction is highly non-uni form. The molar fraction of oxyge n under the rib is lower than those in other areas. This is because oxygen can not diffuse well undernea th the interconnect. On the other hand, the rib does not cause si gnificant concentration gradients for H2 in the z-direction since the diffusion of H2 is about two to three times faster than those of other components as Figure 3-8B indicates. Simila rly, the molar fraction of CH4 decreases from the fuel inlet to outlet due to the reforming reaction (Figure 3-8C). The concentration gradients in the y-direction which is normal to the electrolyte/anode interface, pr oduce some concentration loss, which will be discussed later. Figure 3-8D shows the te mperature contour on the z=0.001m plane. Along the flow direction, the temperatures in crease monotonically from the in let to the outlet due to the contribution of Joule heating a nd chemical reaction. For the re gion near the gas inlet, the temperature increases quickly and therefore causes a larger thermal stress than in other areas. On the air side, the temperature rise is not as large as that in the fuel side due to high mass fluxes. Figure 3-9A exhibits the current density contours. It is found that the local current density profile
77 has a peak value located roughly near the lower le ft corner. The local current density near the symmetric line is relatively lower because of the l ongest current path. In th e region under the rib, the current density level is higher and their profiles are very smooth. AB CD Figure 3-8. Concentration contour s. A) Oxygen at cathode/elect rolyte interface. B) Hydrogen at anode/electrolyte interf ace. C) Methane at anode/electrolyte interface. D) Temperature contour at z=0.001m (a rrow denotes flow direction). Figure 3-9B shows the calculated overpotential distri butions. It is found that the ohmic loss across the electrolyte and electrodes (marked as Ele.Ohmic) represents the major source of voltage loss, followed by activa tion overpotentials, interconnect ohmic loss and concentration loss. The activation overpotential at the cathode and the concentra tion loss are relatively small. Especially, the concentration pol arization is on the order of 10-4(mV) and almost negligible in this case. In addition, further ca lculation indicates that this pola rization is much higher near the rib due to a weak diffusion proce ss. Clearly, the ohmic heating in an electrolyte contributes most to the increase in temperature. Therefore, the key elements to be focused on for improvement should be the minimization of ohmic polarizat ion and suppressing temp erature rise without X Y 1160 1180 1200 1200 1220 1220 1230 1240 1250 1256 1258 1259.35 X Z 0.182 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2 0.204 X Z 0.115 0.12 0.125 0.13 0.135 0.14 0.15 0.155 0.157 0.16 X Z 0.29 0.3 0.31 0.32 0.376122
78 significantly degrading the cell pe rformance. In the following sections, for the purpose of improving power density, the effects of geometrical parameters on the SOFC performance will be analyzed. A 0.000.010.020.030.040.05 0 25 50 75 100 125 150 175 200 225 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Con. cat Act. cat Con. an Act. an Inter. Ohmic X (m)Polarization (mV) Ele. Ohmic Polarization (mV)B Figure 3-9. Performance of SOFC A) Current density distributi on. B) Average polarizations. 22.214.171.124 Effects of gas flow channel dimension 0.000.010.020.030.04 0 2 4 6 8 10 12 14 16 H=0.2mm H=0.5mm H=1mm H=2.5mm H=5mm H2 mass transfer coefficient (ms-1)X (m)A0.000.010.020.030.04 0 1 2 3 4 5 6 H=0.2mm H=0.5mm H=1mm H=2.5mm H=5mmO2 mass transfer coefficient (ms-1)x (m)B 3900 3920 3940 3960 3980 4000 4020 4040 4060 4060 4080 X Z
79 0.000.010.020.030.04 0 50 100 150 200 250 300 350 400 450 H=0.2mm H=0.5mm H=1mm H=2.5mm H=5mm Air heat transfer coefficient (WK-1m-2)x (m)C Figure 3-10. Effects of channel on mass tran sfer rate (at z=0.001m plane). A) H2 mass transfer coefficient on fuel channel. B) O2 mass transfer coefficient on air channel. C) Heat transfer coefficient on air channel. Reducing the channel sizes results in a shorter current path and also facil itates higher heat/mass transport rates. This will lead to changes in temperature and concentration distributions, and the power e fficiency of a SOFC. Figure 3. 10A-3.10C give the calculated mass/heat transfer coefficients on z=0.001m plan e for all cases. As shown in the figures, the channels with smaller heights experience greate r enhancement in the mass and heat transfer coefficients. The values of coefficients decrease sharply due to the entrance effect in the region near the channel inlet. Then, th e downstream values almost st ay relatively constant along the flow direction. Mass accumulation in flow channe ls results in a small increase in mass transfer coefficients. Moreover, the increase in heat transfer rate on the air side will help remove more heat and improve the cooling effect. We ma y conclude that higher mass tran sfer rates in smaller channels will significantly reduce the concentration pol arization in channel flows and improve cell performance.
80 0.000.010.020.030.040.05 1100 1125 1150 1175 1200 1225 1250 1275 R=0.1(cm, Tin=1100K H=5mm H=2.5mm H=1mm H=0.5mm H=0.2mm Average temperature (K)X (m)dimension decrease A0.000.010.020.030.040.05 3600 3700 3800 3900 4000 4100 4200 R=0.1(cm, Tin=1100K H=5mm H=2.5mm H=1mm H=0.5mm H=0.2mm Average current density (Am-2)X (m) dimension decrease B 0.0000.0010.0020.0030.0040.005 0.30 0.35 0.40 0.45 0.50 0.55 0.60 R=0.2(cm, Tin=1100K R=0.1(cm,Tin=1100K R=0.02(cm,Tin=1100K Output voltage (V)Channel hei g ht ( m ) C Figure 3-11. Effects of channe l on cell performance. A) Average temperatures. B) Average current. C) Output voltages comparisons. The average temperature profiles for interconne cts of various height s are shown in Figure 3-11A. When the channel dimension is decreased the average solid temp erature level steadily decreases. However, the temperature slopes of sm aller channels are much steeper than those of larger channels. This means that although the larg er channels have higher temperature levels, the more gradual temperature increases will alleviate the risk of severe thermal stress and thus help ensure the structural integrity of the cell components. However, higher temperature levels put a
81 more stringing requirement for materials and th us making fuel cells susceptible to thermomechanical failure. The average current density distributions for different channel sizes are shown in Figure 311B. It turns out that the location of the peak valu e for the local current density shifts towards the downstream direction of the flow as the channel dimension is d ecreased. Moreover, the current density profiles are flattened for larger channels. In some of previous research works (Li et al., 2003), the interconnect (or rib) was assumed to have very high electrical conduc tivities such that th e solid resistances ar e neglected. However, the values of ohmic resistances in interconnects could vary up to one order of magnitude for different materials. To determine the effect of the rib resistance on th e cell performance, two additional cases, in which rib resistances are modified to 0.2cm R and0.02cm R are investigated. Calculations show that the dist ributions of current de nsity and average solid temperature are not sensitive to the change of rib resistance. However, the terminal outputs strongly depend on the rib resistances as illust rated in Figure 3-11C. For the cell with the smallest rib resistance,0.02cm R the output increases slightly with decreasing channel dimension. As a contrast, outputs are improved remarkably with the reduction in channel dimensions for the case with higher rib resistances. It is interesti ng to note that the cell with the smallest channel dimension does not have the ma ximal terminal output. A plausible reason is that on one hand, a lower temperat ure increases the ohmic loss of solid part, which is a function of the operating temperature. While on the othe r hand, a shorter current path and a lower concentration loss may partially or totally counteract this ohmic pol arization increase. In addition, an extremely small structure will lead to severe thermal stresses from a large temperature
82 gradient. Comprehensively, the cell with the hei ght of 0.5mm shows the best performance in the present study. 126.96.36.199 Evaluation of thin-film electrolyte Temperature effects of the baseline case. As mentioned before, the operation temperature of a SOFC needs to be maintained at a high level, which is essential to ion conductance in the electrolyte and high electroche mical reaction kinetics. However, on the other hand, higher temperatures may cause larger th ermal stresses and electrode sintering. As a compromise, the temperature of a SOFC is usually controlled within a reasonable range. Figure 3-12A compares the temperature profiles and ohimc polarizations for two different gas inlet temperature for the baseline case. Compared with a 96K rib temperature rise in the flow direction for the case of a higher inlet gas temperature (1100K), a larger temperature rise of 130K was found for the case of a lower inlet gas temperature (1000K). Therefore, it is noted that the temper ature gradient in the ribs for the lower inlet gas temperature case is 30% higher, which induces more thermal stresses. The output for cell with lower inlet temperature ( 1000K) decreases to 0.495V from 0.5478V (1100K). In principle, lower flow temperatures retard reforming reaction speed. In addition, it is observed that the ohmic polarizati on is increased by over 50% due to reducing cells temperature. From the present simulation, it is concluded that higher inlet temperature improves the efficiency of SOFC, suppresses the temperatur e rise and thus avoids larger thermal stress, however, with the expenses such as stringent requirements fo r materials and making fuel cell susceptible to thermo-mechanical failure.
83 0.000.010.020.030.040.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 T=130k Ohmic polarization(V)X (m) Tin=1100K Tin=1000K T=96k Average temperature (K)A 0.000.010.020.030.040.05 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 800 850 900 950 1000 1050 1100 volume=0.169cm3 volume=0.0448cm3 volume=0.00832cm3 Ohmic polarization (V)X (m) Average temperature (K) B0.000.050.100.15 0.55 0.60 0.65 0.70 1 2 3 4 5 6 OutputOutput voltage (V) Cell volume (cm3 ) Power densityPower density (W/cm 3 ) C Figure 3-12. Effects of temperat ure. A) Temperature effect on conventional SOFC (baseline case). B) Scaling effect on the miniat urized-thin-electrolyte SOFC under inlet temperature 850K. C) E fficiency comparison. Temperature effects of a thin-film electrolyte. One way to regain a higher fuel cell performance at a lower operating temperature is to reduce the thickness of the electrolyte. Also, the above results indicate that a smaller fuel cell can reduce the concentration loss and improve the fuel cell performance greatly Therefore, a miniaturized SOFC with a thin-film type electrolyte is expected to offe r a better performance. In the following simulations, the thickness of electrolyte is set as10 m. Three different cell volumes were investigated, i.e.,30.169cm with
84 a cross section of 2(1.03.2)mm 30.0448cmwith a cross section of 2(0.51.6)mm and 30.00832cm with a cross section of 2(0.20.64)mm Figure 3-12B displays the average solid temperatures and the ohmic polarizations w ith a reduced inlet te mperature of 850K. The interconnect temperatures decrease with decreasin g cell volumes. Compared with the results for the baseline case (Figure 3-12A), the total temperatur e rises for the three cases are still at similar levels, about 105-135K. It seems that the thermal stress levels due to temperature gradients would not be reduced for a miniat urized SOFC. For the smallest cell, the ohmic polarization is higher than those for larger cells due to a lower operati ng temperature. Figure 3-12C demonstrates that a miniaturized SOFC im proves the output voltage and power density effectively. Similar results were reported in a recent work by Cha et al. (2004). They found that micochannels enhance fuel cell performance in terms of both power density and efficiency. Experimental results further revealed th at the cell performance peaks for the100 m channels and gradually decreases with furthe r reductions in the cell size. 188.8.131.52 Optimization of rib width When the width of the rib is decreased for a fixed cell width i.e., 3.2mm, the resistances in the rib and the active reaction area will increase. At t he same time, the concentration overpotential underneath the rib and the contact resistances w ill decrease. To achieve the maximum cell performance, the rib width should be optimized. Figure 3-13A displays the effect of rib width on the temperature field. Clearly, a wi der rib reduces the rate of temperature increase. For example, total temperature rise is suppre ssed by about 30K for the widest rib (0.8mm) as compared with a thinnest rib (0.2mm). Thus it ca n be concluded that a wider rib can alleviate thermal stresses. Figure 3-13B co mpares the outlet molar fractions of various gas components and the terminal output for different rib widths Having the highest average solid temperature,
85 the reforming and shift reactions are found to be most active in the cell with thinnest rib, consequently less H2 and more CH4 are consumed inside it. 0.000.010.020.030.040.05 1150 1175 1200 1225 1250 1275 width 0.2mm (ratio 1:7) width 0.4mm (ratio 1:3) width 0.6mm (ratio 1:67) width 0.8mm (ratio 1:1) Average temperature (K)X (m)A0.20.40.60.8 0.40 0.45 0.50 0.55 0.60 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 OutputMolar fraction Output voltage (V)Rib width (mm) H2 CO CH4B C D Figure 3-13. Effects of rib width on cell perf ormance. A) Average top innterconnector temperature. B) Comparison of performa nces. C) Oxygen at cathode/ electrolyte interface with thinnest rib (ratio 1:7). D) Oxygen at cathode/ electrolyte interface with thickest rib (ratio 1:1). Moreover, Figure 3-13C and Figur e 3-13D indicate that for cells with thinner ribs, the concentrations of oxygen are more uniformly distributed across the whole porous layer (zdirection), which eliminates inefficiencies th at would normally be caused by obstructed dead zones under the ribs. Comparing with a thicker rib, the molar fr action of oxygen near the diffusion layer surface is significantly higher for the cell with a thinner rib. Similar tendencies are found for other reactants. However, the decrease in concentrat ion loss for a thinner ri b is not strong enough to counteract the increase in ohmic loss and thus the terminal out put still decreases. So it is X Z 0.16 0.165 0.17 0.175 0.175 0.18 0.18 0.185 0.185 0.19 0.19 0.195 0.2 X Z 0.1877 0.19 0.1933 0.1965 0.2 0.204
86 expected that a smaller ratio of the channel wi dth to the rib width would serve to reduce the ohmic losses at the interface a nd improve cell performance. 3.5.4 Conclusions A 3D thermo-fluid/electrochemical model is developed to evaluate the heat/mass transfer and electrochemical performance for an elec trolyte-supported SOFC. Based on this model, detailed investigations are carried out to study the effects of operating te mperature and geometry on the distributions of temperatur e, local concentration, local cu rrent density and output voltage. The main conclusions reached are listed below: When the height of flow channels is decrea sed, the average solid temperature is reduced accordingly and the cell efficiency is impr oved due to both higher heat/mass transfer coefficient between the channel wall and flow stream and a shorter current path. However, a larger fuel cell has a more gradual temperatur e rise, which alleviates the risk of severe thermal stress and helps ensure the structural integrity of the cell components. If the channel height is shorter than 0.5mm, then the cell efficiency will not be improved any further. For a conventional SOFC, if the gas inlet temperature is higher, then higher cell efficiency will result, which causes an in creased cell terminal output, a lower solid temperature rise across the channel and sma ller thermal stress accordingly. However, highly elevated solid temper atures usually make stringi ng demands on materials. In comparison, the miniaturized SOFC with a thin-film electrolyte has advantages of a lower operation temperature and an excellent performance. Conservatively, we expect that the power density of a miniaturized SOFC with a thin-film electrolyte will have the potential to exceed 10Wcm-3. Of course, the average solid temperature may be relatively higher and therefore, thermal management will face some challenge. However, it is noted that if the radiation loss to environment is considered, the energy lo ss may be serious and performance will be lowered because of a larg er surface to volume ra tio. In addition, the thermal-mechanical reliability of thin films should be consider ed in actual applications. The smaller ratio of the channel width to the rib width contributes to the reduction of the ohmic loss at the interface and thus improves the cell performance. Overall speaking, the present model has dem onstrated that there is a trade-off among factors influencing the performance of a SOFC, which was not fully addressed by previous work. It should also be noted that re ducing the cell volume and the el ectrolyte thickness and optimizing the rib width are only some of the factors infl uencing the performance of a SOFC. Therefore,
87 finding better approaches or other critical parameters to optimize the design of a SOFC will be a right track for future work.
88 CHAPTER 4 MICRO-SCALE MODELING FOR ELECTRODES In this chapter, the previous micro-models for porous electrode are reviewed first and assessed. However, their models suffered from some common weaknesses. These models overemphasize the effect of selected parameters on the performance and do not comprehensively evaluate the overall performance. Further, the in fluence of pore distribution, thermal factor and current distribution are rarely considered. Therefor e, reliable models capable of handling realistic microstructural features and mass/heat trans port are essential for es tablishing a quantitative relationship between the microstr ucture and the electrical proper ties in composite electrodes. Accordingly, a micro-scale model consideri ng more detailed mass and energy transport processes will be developed in this chapter. Some simulated results are presented and analyzed. 4.1 Random Medium Models and Objectives Progress in solid oxide fuel cell essentiall y depends on the improvements of its four elements: the anode, the solid electrolyte membra ne, the cathode and the interconnect materials. Many investigations on the loss mechanisms by impedance spectroscopy as well as theoretical analysis have revealed that th e cathode governs the main part of the losses. In the case of pure electronic conducting electrode materi als like metals or some per ovskite type oxides (LSM), the electrochemical reactions are almost restri cted to the triple phase boundaries (TPB, gas/electrolyte/electrode) (Fleig, 2003). The tr ansport of oxide ions within the electrode materials is advantageous concerning the numbe r of possible reaction pathways. Therefore, good electrodes should be compos ed of either a composite consisti ng of an electronic and an ionic conducting phase or a mixed conducting metal oxide to expand the active ar ea into the electrode volume. Figure 4-1 shows an experimental observation for composite anode and composite cathode (Kim et al., 2000).
89 A B C D Figure 4-1. Images for anode and cathode of soli d oxide fuel cell. A) Low-resolution SEM topdown view of Ni-YSZ electrode on YSZ el ectrolyte disk. B)High-resolution SEM image of cross section of Ni-YSZ elect rode on YSZ. C) SEM image of cathode (LSM:YSZ=5:5). D) High-resolution SEM of (LSM:YSZ=9:1). The most critical parameter characterizing the performance of a composite electrode is its conductivity, which is determined by: (1) conductivities of pure ionic and electronic conductors, (2) the surface area availabl e for electrochemical reaction (or the size of TPB), (3) the composition and microstructure of electrode, i. e., the compositional volume ratio, porosity and particle size distribution, and (4 ) path of gas diffusion. For ex ample, Kenjo et al. (1991) found that there is a strong relation between the conductivity of YSZ a dded and polarization losses and suggested that th e internal LaMnO3-YSZ contacts are active for O2 reduction. Kim et al. (2000) indicated that polarization resist ance of the LSMYSZ composite electrodes is closely related to the connectivity of LSMYSZ particles and depends on their size ratio. The composite electrode showed low electrochemical act ivity without YSZ connectivity. However, the polarization resistance decreased abruptly when YSZ connec tivity was present. These papers emphasize the effect of volume fraction of elect rode particles and good electrical contact between the electrode
90 particles on electrical conductivity. Therefore, optimizati on of conductivity and polarization resistance of composite electrodes for SOFC in terms of the microstructure (i.e., composition, particle size, etc) will un doubtedly gain substantial understanding and predictions. However, it is observed from Figure 4-1 that the microstructure of an electrode is not regular and thus all mass and energy transport within electrodes take place in a disordered, not an ordered medium. This random characteristic ma kes modeling work become extremely difficult, because the accuracy of a model is dependent on how to describe this random geometry. Obviously, the most natural modeling methodology is directly based on an actual microstructure of such electrodes. When the random microstr ucture is obtained from experiments, the information about the reaction cu rrent distribution, ohimc resist ance, polarizati on resistance can be determined by a mathematical model. This method is called random network model, which denotes the most accuracy method. In principle, any disordered system can be mapped onto a random network of bonds connected to each other at sites or nodes of the network. The network models incorporate pore-scale de scriptions of the medium and the physics of pore-scale events (Berkowitz and Ewing, 1998). The second method is percolation theory, which is a geometrical and statistical approach. This theo ry has shed some light on the larger scale behaviors, accounting for random ness in porous medium geometry fluid properties, and their interplay. The behaviors are seen at the larger scales (e.g., assemblage of pores) that are not necessarily predictable at the scale of the individual pore like the random network model method. The essence of percolation theory is to determine how a given set of sites, regularly or randomly positioned in some space, is interconnected. Perc olation theory is review ed briefly by Lux (1993) and an introduction to this subject is given by Zallen ( 1983) and Stauffer (1985). The two methods are complementary and always combined to predict the percolat ion properties in porous
91 media. The third is continuum medium approach, which is the most simple and coarsest method. A continuum medium is made up of regular ly or arbitrarily shaped parts filling up all the space. In this method, the electrical conductivity can be expressed by an an alytical equation. Several different theories have been presented: Brick-laye r Model (BLM,), Maxwell-Wagner Theory (MWT), Bruggermans Effective Medium Theory (BEMT), Mclachlans Generalized Effective Medium Theory (GEMT, Mclachlan et al.,1990). Han and Choi have reviewed this subject (1998) in thei r recent publication. Although the models of electri cal conductivity of composite ha ve been studied for almost one hundred years, the modeling st udy for SOFC composite electro des is still a new topic and underdeveloped due to the complexity of sp ecies diffusion, electro chemical reaction mechanisms and the microstructure from sample to sample. The predicted results sometimes cant agree with experimental da ta very well under specific condi tions. One of the representative publications is Costamagnas analytical mode l (Costamagna et al., 1998) based on percolation theory. In his model, the composite electrodes are formed by a mixture of electronic conductor and ionic conductor particle s of small sizes. The electronic and ionic transports together with the electrochemical reaction are taken into considerations. Their results indicated that the reciprocal electrode resistance (i.e., c onductivity) reaches a maximum va lue in correspondence to a composition near to the percolation threshold of the electronic conducting phase for a thick electrode. This means that there exists an optimum value for electrodes composition and thickness. These results agree with the experiment al data of Kenjo et al. (1991, 1992). However, there are some limitations or oversimplifications in their model: (1) the mass transfer in porous medium is completely neglected due to the assumptions of faster diffusion and smaller particle size. However when the thicknesses of electrode or current density are greater, the overpotential
92 due to mass resistance is not negligible. In pa rticular, for IT-SOFC, the concentration loss is significant due to a thick anode layer. (2) Sin ce only sample-spanning electrolyte and electrode cluster are included in this model, the theory ca n not be valid outside the composition regime in which isolated clusters are present, i.e., the volum e fraction of electrolyte particles is below their percolation threshold fraction; (3) The model is only valid for high ratios of thickness to particle radius; (4) This model is only 1-D and the effect s of temperature and fluid flow on resistance are not included. (5) In actual SOFC composite electrodes, one-third of the volume is occupied by pores. This pore network guides th e fuel to the electrochemical reaction sites, i.e., the metalelectrolyte grain pairs. However, the species conc entration is considered to be uniform and the mass resistance due to toutoris ity and pore distribution is ne glected. (6) The mechanism of electrochemical reaction is not in vestigated. Therefore, it is diffi cult to make an exact prediction of percolation thresholds and consequently of the composition in correspondence to which a minimum electrode resistance is obtained. However, this model still gives a good starting point, i.e., percolation theory, for electr ode simulation due to simplicity and explicit. In order to further apply it to actual SOFC electr odes, some modificati ons are proposed by Chan et al. (2001, 2004) and Jeon et al. (2006). The primary improvement is the electrochemical reaction in which the exchange current density is not constant like the model by Costam agna et al. (1998). The effect of grain boundary on conductivity is also taken into account. Jeon et al. (2006) extended the original model to an actual multila yer electrode structure. Recently, Deseure et al. (2005) used a one-dimension flooded homogeneous model to optim ize the cathode performance. However, the results obtained from this model maybe only valid for some specific cases due to the introduction of many assumptions and empirical parameters.
93 In the aspects of random network modeling, Sunde et al. (1996a, 1996b, 2000) applied the Monte-Carlo method to calculate the conductivit y of composite electrodes. To my best knowledge, this method represents the type of approach that best predic ts most important experimental features of composite electrodes. Ho wever, in their work, the important pore effect is still negligible and only polarization and ohmic resistances are simulated. Similarly, Abel and Loselevich (1997) applied a resi stor network model to optimi ze performance of anode. They found that key determining parameter for the electr ical properties of the anode is the ratio of elementary reaction resistance of a metal-electrolyte grain junction to the ion transport resistance of the electrolyte grains. However, their mode l suffers from some common weaknesses as the follows. These models overemphasize the effect of selected parameters on the electrodes performance, not comprehensively evaluate ove rall performance. For ac tual electrodes, many parameters may affect substantially the perfor mance, such as pore distribution, composition, particle size, grain boundary and local current distri bution etc. However, several important factors, such as the effects of pore distribution and current distribution are rarely considered in most of published papers. In addition, the electroc hemical reaction is simplified to become linear. Although the performance of el ectrodes are closely linked to material properties and fabrication process, the mechanisms of heat transfer and fluid flow in porous medium still influence significantly the performance of porous electrodes. For exampl e, on the cathode, the concentration loss due to lower oxygen diffusivity may be significant. In the anode, the heat generation from chemical/electrochemical reacti ons may change substantially the ohmic loss or activation loss. In pa rticular, for an anode-supported SOFC chemical reactions, such as reforming or shift reactions must be incorpor ated. However, these thermal factors and mass
94 diffusion have been always neglected. Therefore, reliable models capable of handling realistic microstructural features and mass/heat trans port are essential for es tablishing a quantitative relationship between the microstr ucture and the electrical properties in electroceramics. These models have not been applied for th e simulation for a SOFC unit consisting of electrode/electroly te/interconnections. By combining a m acro-model (such as the model in chapter 3) with a micro-level model incorporatin g the electrode microstruc ture, it is possible to gain a greater and more fundamental understandi ng of fuel cell operation than through the use of macroscopic models alone. In order to overcome the above mentioned li mitations in the previous models, a novel 3D random walk model will be developed and also applied to optimize the performance in a porous composite cathode resulting in a complete descrip tion of the electrode structure as well as the processes occurring therein. Th e pore distribution, compositi on, mass diffusion and current density distribution, etc will be completely integrated into the new micro-model. 4.2 Key Issues in Random Electrode Modeling The phenomenon that takes plac e in the electrodes, for ex ample in an anode, can be summarized as follows: (i) diffusion of reactive ga s from the boundary of fuel channel to the reaction site through the pores and transport of the reaction products (water) b ack to the gas channel; (ii) Transport of el ectron from the interconnetors to the reaction site through an electronic conductor; (iii) Electr ochemical reaction at the three phase boundary where the gas, the electronic conductor and ionic conductor are present at the same time; (4) Transport of ions from the reaction site to the el ectrolyte through the i onic conductor. The similar framework holds for the cathodes. Therefore, a good performa nce of cathode/anode should have well contact between ionic and electronic partic les, faster diffusion rate and hi gh levels of ionic and electronic
95 conductivities. Before developing a random electrode model, the following several key issues should be taken into account. 4.2.1 Pore Structure Surface area and pore structure are the two important propertie s of composite electrodes. Gas reaction catalyzed by solid materials occurs on the exterior surface of the porous catalyst support. The specific rate of electrochemical or chemical reaction is a f unction of the accessible surface area. The greater the amount of accessible surface area, the larger the amount of reactant converted per unit time and per uni t mass of electrode. Fo r the second factor, the porous structure comprises properties like pore size distribution, connectivity and shap e of pores. The porous structure can be obtained by e xperimental methods, like nitroge n sorption measurements or mercury porosimetry (Berkowitz et al., 1998). For SOFC porous electrodes, on the one hand, the larger pore can guide the fuel/air into the react ion site and reduce the diffusion resistance very well. On the other hand, the larger pore, i.e ., greater porosity, occupies much more space and thus the contact area between el ectronic and ionic particles is reduced. Too lower porosity or smaller particle size may lead to strong transpor t limitations for the fuel/air supply. Therefore, the size of the grains is usua lly not much smaller than 1 m and the porosity is as high as 3040% in actual electrodes. In addi tion, there are three different type s of solid clusters formed in SOFC composite electrodes as s hown in Figure 4-2: (i) the firs t one is that the same type particles (ionic or electronic pa rticles) touch one another to ex tend through the entire electrode. This kind of cluster is called percolating clus ter. Under such condition, a good conductivity is reached. (ii) The second is that a cluster is only connected to its corresponding bulk phase type (inter-connector or elec trolyte). (iii) The thir d cluster is completely isolated from its corresponding bulk phase. The exis tence of this kind of cluste r increases the polarization
96 resistance in the electrode. The di stribution of such clusters in electrodes is strongly influenced by the particle ratio, the volume ratio and the si ntering process etc. As mentioned above in section 4.1, these microstructures will affect the electrochemical behavior of electrodes substantially. Therefore, in the following micromodel, a 3-D random network will be used for simulate the pore structure and solid clusters. Figure 4-2. A SOFC composite electrode. Th ree different clusters are formed. Black circle=electronic conductor, bl ue circle=ionic conductor. In actual electrode, the shape of particle may be spherical, cubic or arbitrary etc. 4.2.2 Three Phase Boundary and Reaction in Porous Electro-ceramics The edge of electronic/ionic in terface that makes contact with the gas phase (as shown in Figure 4-3) is often described as the triple-phase boundary (TPB). Much work has been devoted in recent years to develope the concept of triple phase boundary (TPB) as a means of understanding the complex interplay among the particle size, pore si ze distribution and electrochemical reactions. In the cathode, oxygen mo lecules are generally thought to be adsorbed somewhere onto one or more solid surfaces, where they undergo catalytic and/or electrocatalytic reduction steps to form partially reduced ionic/at omic species. Before, after, or between partial
97 reduction steps, these species must translate alon g surfaces, interfaces, or inside the bulk of the electrode material to the elec trolyte, where they are fully incorporated as electrolytic O2. However, how and where any of these processes take place and what steps are rate determining for a particular electrode is only partially unde rstood. Up to now, not any single mechanism has been discovered which can explain all oxygen redu ction processes in the cathode. Many factors may influence the oxygen reduction: composition of sample, microstructure, and experimental conditions etc. Therefore, augments in the liter ature concerning which theory is correct for a given cathode system usually are based on lim ited data and some sp ecific conditions. Two excellent reviews on the oxygen reduction mechan ism have been presented by Adler (2004) and Fleig (2003). Similar level of understanding holds for hydrogen reduction at anodes. In view of the complexity and uncertainty in the electroc hemical reactions, only the most reasonable or appropriate oxygen/hydrogen reduction is selected and simulated in the following micro-model for electrodes. Figure 4-3. Three phase boundary region in cath odic reaction. 4.2.3 Diffusion Mechanisms Diffusion of fluid mixtures insi de a porous matrix also aff ects the performance of SOFC electrodes. Within the pores, there are three di stinct diffusion mechanisms for mass transfer
98 (Mills, 2001; Krishna et al., 1997): (1) molecular diffusion: this type becomes significant for large pore sizes and high system pressures. Here molecule-molecule collisions dominate over molecular-wall collisions; (2) Knudsen diffusion becomes predominant when the mean-free path of the molecular species is much larger than the pore diameter and hence molecule-wall collisions become important; (3) Surface diffusion of adsorbed molecula r species along the pore wall surface. This mechanism of transport b ecomes dominant for micropores and for strongly adsorbed species. In general, micropores have diameters smaller than 2nm; macropores have sizes greater than 50nm. The bulk and Knudsen diffusion mechanisms occur simultaneously. The surface diffusion occurs in parallel to the other two mechanisms and its contribution to the total species flux may be quite significant in many cases. The surface diffusion contribution can be important for components with high adsorption st rength. Within the micropores, the dominant mechanism is surface diffusion. It is for this reas on that surface diffusion is also referred to as micropore diffusion in the literature. The pressure gradient inside the pa rticle is not always negligible and this pressure gradient gives rise to viscous or Darcy flow. Figure 4-4. Electrical analogue circuit showing the flux of the diffusion species within a porous medium. Figure 4-4 shows the various contributions to the flux of the species inside the particle. The combination can be visualized by an electric al analogue circuit. For the diffusion process in a SOFC composite electrode, it can affect the electrodes at anothe r level (Costamagna et al.,1998) Bulk+Knudsen diffusion Surface diffusion Viscous diffusion Total flux
99 through the micropores present in the particles th at form the electrode, limiting the supply of reactants to the electron conductor /ionic conductor interface. Some of the studies assert that one of two limiting situations arises: either the activ e area for the electrochemi cal reaction is a small area around the three phase boundary (T PB) or it is coincident with the whole two phase contact region between electrode and electrolyte. Whether the former or the latter or an intermediate situation arises, it depends on the ratio between th e rate of the electrochemical reaction and the rate of the process of diffusion of the gaseous reactants at the interf ace between electrode and electrolyte. Therefore, the different diffusion mechanisms s hould be incorporated in model. 4.3 Random Network Modeling for Opti mization of Composite Cathode In this section, a 3D random network model will be presented to simulate the transport process in a composite electrode (cathode). We then apply this model to optimize the performance of a cathode. 4.3.1 Physical Description and Assumptions The networks of porous cathode are complex, and in practice it is necessary to adopt simplified models which reproduce th e essential features, but ignore many of the details, of a real network. In the present model, an idealized 3-D microstructure is constructed as shown in Figure 4-5 (Only 2-D view is given). The microstructure takes the form of f ace-centered cubic lattice (FCC). This kind of structure represents a dense p acking of particles of the same size (electrolyte, electrode, pore). The practical cathode has a rand om topology, i.e., its local coordination number, which is the number of bonds c onnected to a site, is a chaotic and almost random variable. For example, Sunde (1996a, b), Zalc et al. (2003), Constantinides et al. (1989) and Ciobanu et al. (1999) employed the arbitrary resistor networ k with non-uniform grain size and shape to represent the porous structures. However, some studies (Winterf eld et al.1981; Sahimi et al., 1985) have demonstrated that as long as the average coordination number of a topologically
100 disordered network is equal to the coordination number of a topologically regular network, transport and main other properties are identical for the two network syst ems. Therefore, a 3D regular lattice is used here. However, the spatia l distributions for grain particles and pores are still random. In Figure 4-5, blue, black, and while sites represent the elec trolyte, electrode and pore particles, respectivel y. The electrons from the interconnect or through the electrolyte clusters and the oxygen through contiguous pore passage ar e supplied to the act ive TPB (explained later), where the oxygen ions are generated by el ectrochemical reaction. Then the oxygen ions are transferred to the electrolyte me mbrane by the electrode clusters. Figure 4-5. Random network model. Red rectangle denotes the active TPB; Black circle=electronic conductor, blue circle=ionic conductor, white empty=pore (the circles have been removed for clarify). Three paths along the three phases for gas (black), O(red) and e(white) are represented by curves. In developing this model, several assumptions and definitions are made and summarized as follows: Not all of pores are open fo r reaction. Some of pores ma y be inaccessible to one or more of the species so that the effective TP B length and effective diffusivity are reduced. When the porosity is small, the pore forms sma ll and isolated clusters. As the porosity is increased enough, there would be some cluste rs spanning the network from one end to the other. This kind of pore clusters is termed as transport pore cluster, since it allows the gas to transport across the entire me dium. A pore belonging to a transport pore cluster is called a transport pore. The sec ond kind of pores, which is called dead-end interconnector electrolyte
101 pore, is interconnected only fr om one side (current collec tor). Although these pores can often be penetrated, their contributions to tran sport is negligible to transport. The third kind of pores is isolated pore, which is tota lly isolated from its neighbors of transport pore and can not contribute to transport of mass across the porous medium. Here, this common realistic situation is considered and pores are distributed randomly and identified as transport, dead-end or isolated pores. Three types of clusters in a so lid phase, i.e., percolating cl uster (I), singly connected cluster (II) and isolated clus ter (III) are identified. The first kind of clusters, i.e., a percolating cluster, is that th e same type of particles (ionic or electronic particles) touches one another to extend through the entire electrode. Under such a condition, a good conductivity is reached. The second is that a cluster is only connected to its corresponding bulk phase type (i. e., inter-conn ector or electrolyte) The last kind of clusters, isolated cluster, is completely isolated from its corre sponding bulk phase. The existence of this kind of clusters increases the polarization resistan ce in the electrode. The distribution of these clusters in electrodes is influenced strongly by the particle ratio, volume ratio and sintering process etc. Gas is considered to be an id eal two-component mixture of O2 and a dilute gas (N2). Within the pores, there are three distinct diffusion mechanisms for the mass transfer (Mills, 2001): molecular diffusion, Knudsen diffusion and surface diffusion of adsorbed molecular species along the pore wall surfac e. The molecular and the Knudsen diffusion are dominant mechanisms at the cathode and w ill be taken into account in this model. Only the DC part of the effective conductivity is considered. The AC part of the conductivity due to double layer effect (i .e., capacitance effect) is neglected. The cathodic reaction site is limited to the vicinity of an active bond. This concept is first introduced by Alder et al. (1997). The bond between two neighboring sites, one of which is an electrolyte and the other an electrode site, is ac tive if both of the sites belong to a percolating cluster or a singly c onnected cluster and also the bond has a neighboring empty site (pore), which belongs to the trans port pore cluster or dead-end pore cluster. Isothermal and steady state operation. 4.3.2 Derivation of Numerical Model According to the conservation principle, th e current flux entering a control volume should be equal to those leaving this volume plus the accumulative current within it. Therefore, the governing equations for charge transport in th e solid phase (two phases) and gas species transport in the void space, respectively, are: eff()0 (4.1)
102 tot0JR (4.2) Here and totJ are the potential within solid particles and the total mass flux vector, respectively. R is the rate of consumed mass due to the electrochemical reaction. eff is the effective conductance. The conductance can be genera lly expressed, at each site (i, j, k), in the discretized sense, as: eff(,,)fijk (4.3) where (,,) f ijk is a phase function which is defined as unity in the solid phase and zero in the gas phase, respectively. Actually, equation (4.1) is applied to both of the two solid phases and (4.2) is valid for pore space. The two equations are coupled by the el ectrochemical reaction on the active three phase boundary. He nce, the idealized random micros tructure (Figure 4-5) can be considered to split into two subnetworks: an el ectrical network for solid phase and a pore network on the void space. The species concentrations and chemical reaction rate (Eq. 4.2) will be calculated within the pore network, whereas th e charge transfer, resist ances and overpotentials (Eq. 4.1) will be analyzed w ithin the electrical network. 184.108.40.206 Electrical network Figure 4-6 illustrates a two-dime nsional electrical network. Node s (i, j ,k) represent solid particle sites. Between each pair of solid pa rticle, the current is conducted through resistance with the effective conductanceeff There are three kinds of conductances: the conductance between the electr olyte particles io-io the conductance between the electrode particles el-el and the conductance between tw o different particlesio-el Here, the subscript io refers to electrolyte and el to electrode particles. These conductances depend on the neck formed by
103 contact of two particles. A neck of diameter 2 formed between two cont acting spheres with the same radius r and thus the neck circumference, is equal to 2 as shown in Figure 4-7. Figure 4-6. Electrical network. Figure 4-7. Contact neck formed by two different kinds of particles. The conductance between the same types of particles can be approximated as: ioio-io io-io4 elel-el el-el4 (4.4) io and el are ionic conductivity and el ectronic conductivity, respectiv ely, which are dependent on the temperature. The conductance io-el between the different kind particles is given by: 1 io-el io-ioel-elp111  22 (4.5) where p (unit: S) is the polarization co nductance. For an inactive TPB, p is set as infinite. While for an active TPB, p depends on the geometry of TPB and the activity of cathodic 2 electrode electrolyte ,,ijk Y ( j ) X ( i ) 1,,ijk 1,,ijk ,1,ijk 1,1,ijk 1,1,ijk,1,ijk
104 reaction. In the previous works (Sunde, 1996a, b; Jeon et al., 2005), p is assumed to be a constant and proportional to the polarizati on conductivity p and three phase boundary lengthTPB However, Sunde has implied that his mo del has significant erro r due to neglecting the effect of gas diffusion on electro-chemical reaction and p determined by a simplified geometry, not the electrolyte-electrode contacts in the composite electrode. Thereafter, a new approach is used to evaluate this value consid ering the effect of gas diffusion, temperature, chemical reaction and TPB geometry. act is the activation overpoten tial and defined as the potential difference between two sites: actelio (4.6) As a matter of fact, the activation overpotential in equation (4.6) includes the voltage loss due to mass diffusion, thus act should be corrected as: actelioconc (4.7) So the polarization conductance corrected for ohmic drop can be expressed by: 11 actelioconc p trTPBtrTPBii (4.8) here TPB is still equal to2 and also can be given by: TPB22sin 2r (4.9) where r is the radius of the particle; is the contact angle between the ionic and electronic conductor. Chen et al. (2004) found th at this angle can be estimated aso60. The concentration overpotential is therefore calculated as:
105 2 2int O g conc r Oln() 4 p RT Fp (4.10) where 2int O p and 2r O p are the partial pressures of oxygen at cathode/ channel inte rface and reaction site (TPB), respectively. In equation (4.8), there is still one pa rameter, the exchange transfer current,tri, left to be determined. As a matter of fact, oxygen reduction reaction has been recognized to involve a series of consecutive elemental steps including, i.e., dissociation, ch arge transfer, surface diffusion of oxygen intermediate species, and incorporation of oxygen ions into the YSZ electrolyte. The current dens ity depends on the partial oxygen pressure, rate constants in elementary reaction steps, detailed experimental condition and microstructure of electrodes, etc. It is difficult to evaluate this value theoreti cally; instead, fitting from experimental data is generally used (Chen et al., 2004): 4 1/4 trOactOactact 222.0510 (,,)7874exp()exp()exp() ifTppff T (4.11) here p O 2is the partial oxygen pressure, and T is the absolute temperature. g/ f FRT where g R is the gas constant and F is the Faradays constant. Fi nally, the current conservation equation (equation 4.1) can be replac ed by the following discretized form: ijijij jj()0 I (4.12) where i and j are the potentials for particle i and its neighbors j, respectively. ij I denotes the current flowing from site i to its neighboring site j. ij is the conductance between site i and j. By iteratively solve equations (4.4)(4.12), the distribution of potenti al in the composite cathode will
106 be obtained. Once the potential di stribution is determined, the tota l resistance for the cathode is estimated as: topbot tot totalR I (4.13) It should be noted that the to tal resistance includes all three overpotentials, i.e., ohmic loss, activation loss and concentration loss. If the polarization conductiv ity is set to be infinite (p ), the total ohmic resistance ohm R will be easily evaluated. Then, the total polarization resistance p R is obtained by subtracting ohm R from tot R i.e., p totohm R RR 220.127.116.11 Diffusion and reaction rate The multi-compnent diffusion in the pore network is described by the dusty-gas model. Just as discussed in section 4.2.3, the combin ation of the three diffusion mechanisms exists within the voids. The total flux of any sp ecies is obtained by combining the separate contributions as presented in Figure 4-4. Becau se the bulk and Knudsen diffusion are dominant mechanisms in the composite cathode which has larger particle sizes (0.15 m ), the surface diffusion is neglected in this model. The total mass transfer in the single pore is a sum of diffusive and viscous flux (21kg ms) totdiffvisJJJ (4.14) The first term on the right hand side (diffJ ) is the mass flux due to the bulk and the Knudsen diffusion, which can be expressed by an N-dimensional matrix notation: e1 diffp[(,)] dC JBrC d (4.15)
107 where C and are the mass concentration vector and coordinate of the por e, respectively. The N-1 dimensional square matrix e B has the elements: e i ij(ij) b ij, X B D N1 e k ii Kb k1,ki iiik1 X B DD (4.16) here i X represents the mole fraction of component i; N is the total number of components. b ijD is the binary diffusivity of components i and j, andK iiD is the effective Knudsen diffusivity of component i. Both of the two diffusivities can be cal culated based on the same method in the Chapter 3. The only difference is that the tortuos ity is equal to unity he re. The viscous flux is given by the following expression: 2 p tot vis8r dC JCRT d (4.17) here is the viscosity and T is the temperature. totC is the total molar concentration and given by: i tot 1 iN iC C M (4.18) The volumetric source term R -3-1(kg ms)due to electrochemical reaction is related to the transfer current density tri, TPB length and pore size, which can be expressed by: trTPB 3 p4 4 3i R rF (4.19) It is apparent that only th e gas phase site having activ e TPB has the source term R The pore structure can be treated as a th ree-dimensional pore network, and the single pores are connected at the nodes of the network. The flux of each sp ecies entering a node must be equal to the flux
108 leaving the node plus the accumulative mass. Ther efore, at the inner nodes of the network, an equation similar to the Kirchhoffs law holds: iI I iMJR (4.20) where M denotes all the pores that are connected at node I, and iIJ are fluxes leaving or entering that node I. Actually, Eq. (4.20) is the discritized form of Eq. (4.2). For a two binary component system (O2 and N2), equation (4.20) can be rewritten in the following expression by combining Eq. (4.15) and (4.17): 222 2 222 22 O,IO,iO,I p tot,itot,I CC O,I I K 33 iMiM iIOO2,NiI pp1 1 1616 8 33XCC r CC DD CRTR DD rr (4.21a) 1/2 2 21O NM M (4.21 b ) iI is the distance between any si te I and any neighboring pore i. 2O,I X is the molar fraction of oxygen at site I. In this model, each pair of any tw o neighboring sites is assumed to be connected by a cylindrical channel with length iI and diametercD. Finally, equations (4.12) and (4.20) have to be solved simultaneously by a fi nite difference scheme for the entire network. 4.3.3 Reconstruction of Microstructure and Boundary Conditions Fig. 4-8 shows a typical constr ucted microstructure for the 242418 cubic FCC lattice. The size of the lattic e is represented byxy,NN and zN layers in the x, y, and z directions, respectively. The z-direction is taken as the primary direction of current flow and the dimension in this direction is by definition e qual to the thickness of electrode, zL.
109 X0 1E-05 2E-05 3E-05 4E-05 Y0 1E-05 2E-05 3E-05 4E-05 5E-05 Z0 5E-06 1E-05 1.5E-05 2E-05 2.5E-05 X Y Z 30 20 10 0 Figure 4-8. Reconstruct of 3-D rando m microstructure. Model parameters: 0.4 elio:1:1 242418. Blue (0)=isolated particle, cy an (10)= transport pore, green (20)=transport electronic particle, ye llow (30)= transport ionic particle. The FCC lattice is generated by assigning each site an electrode, electrolyte or pore particle, depending on the value of a random numb er generated within the interval [0, 1] uniformly. If the number is smaller than a given porosity, the corresponding site is set to be occupied by pore phase; If this number is between and el x the corresponding site is electrolyte. Otherwise, it is occ upied by electrode. The radius of a particle is in the range from 0.1 to 5 m. Once the microstructure is constructed, the connectivity and the different clusters need to be identified. For this purpose, the Hoshen-Kopelman algor ithm (1976) is used to identify the percolating cluster, isolated cl uster, dead-end cluster and active TPB. The detailed process can be found in references (Hoshen et al.1976, Keil, 1999). All of the
110 quantitative results will be obtai ned as an average over six differe nt random micros tructures with different seed numbers. 0.00.20.40.60.81.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Local average site fractionZ/LZ Transport pore sites Transport electronic and ionic sites Isolated sitesA 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00.20.40.60.8 Transport pore volume fraction Porosity 16*16*18 24*24*18 36*36*18 reference lineTransport solid volume fraction B Figure 4-9. Statistics of random microstructure (model parameters: 0.4 elio:1:1 A) Local variation in three kind of sites al ong the thickness. B) Effects of crosssectional mesh resolution on the transport pore volume fraction and transport solid volume fraction. Fig. 4-9A gives the local percen tage variation for three differe nt sites. These data suggest that the pore, electronic and i onic sites are distributed across the thickness randomly and the algorithm used to describe the microstructure is unbiased. All of the elec trochemical variables in
111 the z-direction are of interest, so the edge effect of xand y-dimension on the electrochemical behavior should be minimized. Fig. 4-9B shows the dependence of the effective porosity on the x and y dimensions. It shows that when the total number of cells in the x and y directions is increased from 24 to 36, the transport pore volume fraction varies only slightly. Therefore, the number of lattices is chosen at 24 that is considered to be sufficient in the x and y directions, and it is employed for the following calculations. Also, it is noted that when the natural porosity is larger than 0.3, the effective porosity is almost identical to the natural porosity. However, when the natural porosity is larger than 0.5, the volume fraction of t ransport solid particles (electronic and ionic) dramatically decreases since the isolated a nd dead-end solid cluste rs dominate the whole electrodes. Therefore, it is c oncluded that 0.3 is the lower bo und of the porosity. In practical applications, the porosity is alwa ys in the range of 0.35-0.45. The boundary before the first z layer represen ts the equipotential surface of the current collector (metal), which means that all bonds connected with it ar e linked to the metal and gas transport channels and the potential is set as 0 .The boundary after the la st z layer contacts the solid bulk electrolyte membrane, which stands for another equipotenal surface and is set to be at zero voltage. In the x and y directions, sy mmetric boundary conditions are employed. The boundary conditions are summarized as below: At ,,0,0, x y x yxLyL: 20,0OC nn (4.22) At the electrode/metal interface 0:z 22OO,0el0tot,,CCII (4.23)
112 At the electrolyte/cathode interface :zzL 2O io0,0C n (4.24) Note that in equation (4.23), only the potentials for electrode particle s are set as finite values, and the potentials for electrolyte pa rticles should be considered as inner nodes and obtained by solving the governing equations. Similar cases hold for boundary condition (4.24). 4.3.4 Model Input Parameters The input parameters, including the transport and kinetic parameters are extracted from literatures (Costamagna et al., 1998; Chen et al., 2004; Ji et al ., 2006) and listed in Table 4-1. The conductivities for LSM and YSZ are consider ed to be dependent on the temperature. Air (79% N2 and 21% O2) is delivered to cathode. Table 4-1. Model input para meters for the baseline case Parameters Value Parameters Value Air inlet temperature (K) 1100 Contact angle o60 LSM conductivity (-1S m) 78.85510 exp(1092.5/) T T YSZ conductivity (-1S m) 53.3410 exp(10300/) T 0 (voltage at electrode/metal interface) (V) 0.2 Bulk oxygen concentration (-3kg m) 0.0825 xyzLLL 242418 Particle diameter ( m) 6210 Porosity 0.4 4.3.5 Numerical Results and Discussions 18.104.22.168 Contours of potential and oxygen concentration To study the potential dist ribution, the mass diffusi on and the significance of overpotentials around the ove rall cathode region, we first select a microstructure with the seed number and as the baseline case. Although results obtained from this random
113 microstructure is different from that based on an average of several microstructures with different seed numbers, the errors are so small that will not change the values of potential and oxygen concentration significantly. X Y Z0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 A X Y Z 0.19 0.15 0.1 Figure 4-10. Potential distribution (unit: V). A) Fo r ionic particles. B) For electronic particles.
114 In this case, the external voltage difference is set as 0.2V, the porosity is 0.4 and the composition ratio of electrode phase is 0.5. The average current density is calculated as 0.806Acm-2. The 3-D contours of potential ar e plotted in Fig. 4-10. It turns out that the potential in the ionic phase decreases grad ually from about 0.11V to zero ac ross the electrode thickness (zdirection). For electronic phase (Fig. 4-10B), however, the vari ation in potential is almost negligible. These distributions attribute to the higher conductivity fo r LSM and very lower conductivity for YSZ. Such a larger potential di fference between LSM and YSZ is just caused by the electrochemical reaction in the vicinity of TPB. Distributions of corresponding average cross-se ctional exchange density and polarization conductance through the cathode, normalized to the current and the polari zation conductance at the first layer, are displayed in Fig. 4-11. In th e present study, the exchange current is modeled as a function of activation overpotentia l and other operation parameters which is different from the assumption of constant values the study by Sun (1996a,b). The exchange current density increases monotonically and reaches the p eak value at the bottom of electrode. 0.00.20.40.60.81.0 0 1 2 3 4 5 6 7 1 2 3 4 Relative polarization conductanceRelative exchange current densityZ/LZ Figure 4-11. Profiles of local relative exchange current dens ity and relative polarization conductance.
115 This tendency indicates that the strongest el ectrochemical reaction takes place in the region near the bottom end of the cathode where the larger pot ential difference and maximal number of active TPB exist. For a pure phase cathode, the el ectrochemical reaction is only confined to a very narrow zone, i.e., the electrolyte /cathode interface; while for a composite cathode, the ionic phase extends to the bulk LSM phase and hence in creases the reaction site s. Also, we found that the higher active TPB number, the la rger the polarization conductance is Fig. 4-12 compares the oxygen concentra tion under a lower and a higher potential differences. Under the lower poten tial difference, the oxygen concentration is fairly uniform, decreasing from 0.08259 -3kgm to roughly 0.075-3kgm across the electrode thickness. While for the case of a higher pote ntial difference, i.e., 0.4V, the oxyge n gradually decreases to a lower level of oxygen concentration due to a larg er oxygen consumption rate. The uniform oxygen distribution on a x-y plane validates that the percolating cluster dominates the whole region and makes the oxygen supplied to the active TPB very well. X Y Z0.0826 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 A
116 X Y Z0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 B Figure 4-12. Oxygen concentration. A) Lowe r potential difference. B) Higher potential difference. 22.214.171.124 Activation and concentration overpotentials Fig. 4-13A illustrates the pr ofiles of average ohmic and act ivation overpotentials with respect to different external voltages. An incr ease of voltage difference (i.e., average current density), the sum of ohmic and activation loss also increases. Although both of the numbers of active TPB on the gas/cathode and electrolyte/c athode interfaces are close to the maximum values, the overpotenial next to the electrolyte/cathode surface, i. e.,/1ZZL, is still much higher because of higher polarizatio n potential difference in this region. Actually, the activation loss contributes to the major portion of th e overpotential. The corresponding average concentration loss is given in Fig. 4-13B. It indica tes that this loss is negligible compared with the activation loss under the lower current density. Howe ver, when the current density is enhanced due to a larger potential difference, the concentration loss can not be neglected any more. For example, when the potential is 0.4V the maximal concentration loss is about 0.0325V that is responsible for 9% of the total loss.
117 0.00.20.40.60.81.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.2V 0.25V 0.35V 0.4V Activation and ohmic overpotential on TPB (V)Z/LZA 0.00.20.40.60.81.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.2V 0.25V 0.35V 0.4VAverage local concentration overpotential (V)Z/LZB Figure 4-13. Overpotential distri bution. A) Ohmic and activation loss. B) Concentration loss. 126.96.36.199 Optimization of cathode design I) Porosity effect Fig. 4-9 implies that the cathode should ha ve the best performance with the porosity exceeding 0.3 in terms of the effective porosity. But how the porosity exactly influences the mass transfer and electrochemical behavior still needs to be further investigated. The average
118 oxygen concentration distribution an d reaction rate as a function of porosity are shown in Fig 414A and Fig. 4-14B, respectively. 0.00.20.40.60.81.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6Average oxygen concentration (kg m-3)Z/LZA 0.00.20.40.60.81.0 0 100 200 300 400 500 600 700 800 Average reaction rate (Kg m-3s-1)Z/LZ 0.1 0.2 0.3 0.4 0.5 0.6B Figure 4-14. Effect of porosity on performance. A) Average local oxygen concentration. B) Average reaction rate. The porosity is varied from 0.1 to 0.6 while keeping other parameters the same as that listed in Table 4-1. Clearly, when the porosity is at a relatively lower level, i.e., 0.1 or 0.2, the oxygen penetration depth is considerably sm all, only 25% of the cathode thickness for 0.1 which will lead to a larger concentration overpot enital. The peak value of reaction rate occurs
119 near the gas/electrode interface. This is due to the fact that an isolated pore dominates the whole electrode and blocks the gas supply to the TPB, and then substantially reduces the gas penetration depth. Therefore, th e availability of oxygen supply and reaction sites is confined to a very narrow zone next to the gas/electrode inte rface. Nevertheless, when the porosity is equal to 0.3, the reaction rate begins to have a conti nuous tendency to increase along the thickness and also possesses the highest reaction rate. When the porosity is enhanced to 0.4 or more, the cathode performance deteriorates du e to insufficient reaction sites. 0.10.20.30.40.50.6 0 2,500 5,000 7,500 10,000 12,500 15,000 0 20 40 60 80 100 Percentage of active TPB number (%) Active TPB number (#contact)Porosity A 0.00.10.20.188.8.131.52 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 Specific total resistance (mcm2)PorosityB Figure 4-15. Number of TPB and specific total resistance with respect to the porosity. A) Total number of active TPB. B) Specific total resistance.
120 Fig. 4-15A shows the statistical characters of the active TPB with respect to the porosity. It is observed that when the porosity ranges fr om 0.25 to 0.45, the percentage of active TPB number lies between 77% and 91%, which means the most of transport ionic or electronic particles are available to the elect rochemical reaction. Therefore, in light of the above results, it can be concluded that the con centration loss can be neglecte d and the cathode performance reaches the best when the porosity is in the ra nge of 0.25-0.45. In the results reported by Sunde (1996a, b), it is assumed that all of solid part icles are always availa ble for reaction and the oxygen supply is not limited in the vicinity of a TP B. Therefore, their results are only valid for the porosity range in the middle le vel and also in general the resi stance is underestimated in their study. The corresponding total resi stance excluding the mass transfer resistance vs. porosity is plotted in Fig. 4-15B. The maximum performance is achieved when 0.3 When the porosity is equal to zero, the total resistance drastically increases by almost ten times, since there are still some active TPBs on the gas/cathode interface. On the other hand, the reaction sites completely disappear and the resistance goes to infinity when not any solid phase exists in the cathode. The above results have sufficiently demonstrated th e porosity 0.5 is the upper limit and 0.3 is the lower limit of porosity in a composite cathode. II) Layer thickness effect To discover the dependence of the reaction rate and overpot entials on the cathode thickness, several cases with diffe rent thicknesses are employed. Fig. 4-16 gives the comparison results. When the thickness is less than 30 m, the curve for the reaction rate is almost flat, i.e., almost all active bonds of the cathode are used; no matter how far they are from the gas/cathode interface. However, when the thickness is increased beyond30 m, only the zone which is close to the electrolyte/cathode is well utilized for reaction as shown in Fig. 4-16A.
121 0.00.20.40.60.81.0 0.1 1 10 100 5.26m 10.16m 29.76m 59.15m 88.55m 117.94m Average reaction rate (Kg m-3s-1)Z/LZA 0.00.20.40.60.81.0 0.0000 0.0015 0.0030 0.0045 0.0060 0.0075 0.0090 0.0105 0.0120 5.26m 10.16m 29.76m 59.15m 88.55m 117.94m Concentration overpotential (V)Z/L Z B Figure 4-16. Effect of thickne ss on reaction rate and concentrat ion overpotential. A) Local average reaction rate. B) Local av erage concentration overpotential. The same tendency is also demonstrated in Co stamagna et al.s study (1998). A parameter, i.e, effectiveness factor is defined as a criterion to indi cate what extent the electrode is utilized. When is greater than 0.3 and one of two pha ses has much higher resistivity than the other phase, the major part of the electrode be comes useless and the activation overpotential (or
122 reaction) is concentrated on the electrolyte/cathode interf ace. Moreover, with increasing thickness, the concentration loss will become more and more significant as shown in Fig. 4-16B due to a longer diffusion path and poor gas supply. Comprehensively considering the above results, it is suggested that the cat hode thickness should be less than 89 mhere to achieve minimized concentration loss and uniform reaction rates. III) Combined effect of compositi on, particle size and thickness One of the important criteria indicating the electrode performance is the total specific resistance. Some experimental results have demons trated the possibility of optimized parameters to minimize the resistance. In this section, we wi ll consider the combined effects of composition, particle size and thickness on this resistance. Th e validation of the model with the experimental data and other theoretical study is presented in Fig. 4-17. We have inco rporated experimental parameters from the literature and some reasonable approximated values (Costamagna et al., 1998; Kenjo et al., 1991, 1992) for an ED B/Pt cathode into our model, e.g., 0.4 o900CT,o60,0.4 el0.1 mr, io0.1 mr 6-1 0,el2.510Sm, -1 0,io20Sm. Kenjo et al. (1991, 1992) does not give a relationship between overpot ential and exchange current density such as equation ((4-11), but an appr oximation for exchange current density, i.e., -2 0400Ami was reported by the same authors and this value has been adopt ed in our model. The predicted results are in good ag reement with the experimental data reported by Kenjo et al. (1991, 1992). When the thickness exceeds a certain value *L, the specific polarization levels off, keeping an almost constant trend. This can be explained by that the elec tronic conductor is close to a pure conductor elio the charge flows through an electronic conductor without any ohmic losses and the most of electrochemical reac tions occurs in a narrow region next to the electrolyte/cathode with the thic kness is increased as shown in Fig. 4-16A. Therefore, the
123 electrode resistance does not increase as the ohm ic losses are negligible. Compared with the results from a continuum theory, there is a signi ficant derivation as the el ectrode is very thin (10 m). This derivation becomes more remarkable for the case of a lower electrode volume fraction0.32 The resistance calculated from our model is higher than that from Costamagna et al. (1998). The continuum theory implies an assumption that only percolation cluster dominates the performance and the isolated and dead-end clusters are negligible. Therefore, for a very thin electrode in which that the percola ting cluster almost vanishes, this theory tends to underestimate the resistance. For a thicker electrode, the consiste ncy between the two models is satisfactory. For the case with0.32el our model reveals a smaller value of *L, while Costamagna et. al indicates a larger *L value and gradual decrease in resistance before 75 mZ. Simulated results sufficiently indicate that our model can well predict the experimental data. 010203040506070 200 400 600 800 1000 experimental data,el=0.32 experimental data,el=0.4 continuum theory, el=0.32 continuum theory, el=0.4 simulated result, el=0.32 simulated result, el=0.4Specific total resistance (mcm2)Thickness (m) Figure 4-17. Comparison among the model prediction, experimental data and continuum model. Fig. 4-18 presents the combined effects of electrode volume fraction, thickness and radius of particle on the total resistance. The continuum theory and the theory of coordination
124 number demonstrate that the percolation threshol d varies from 0.294 to 0.709 as the radius ratio eliorrequals to unity. Within these thresholds, a reasonable performance of the cathode can be achieved. 020406080100120 0 250 500 750 1000 1250 1500 1750 2000 el=0.20 el=0.25 el=0.30 el=0.35 el=0.40 el=0.50Specific total resistance (mcm2)Thickness (m)A 020406080100120140160 100 200 300 400 500 600 700 800 900 r=0.5m r=1m r=2m Specific total resistance (mcm2)Thickness (m) B020406080100120 600 900 1200 1500 1800 2100 2400 2700 3000 r=0.5m r=1m r=2mSpecific total resistance (mcm2)Thickness (m) C Figure 4-18. Effect of electrode fraction volume a nd particle radius on tota l resistance. A) For different el B) el0.4 and different radius. C) el0.2 and different radius. The resistance distribution with in the cathode as a function of the electrode fraction is displayed in Fig. 4-18A. The porosity, radius and electrode volume fraction are fixed at0.4 elio1 mrrand el0.5 respectively. When the electrode fraction is equal to 0.35, 0.4 and
125 0.5, respectively, the resistance maintains a consta nt value after a certain thickness value and the cathode thickness corresponding to the optimum performance is in the range from 17.5 mto40 m.The maximal performance is achieved at el0.4, and 29.14 mZ. However, for cases with el0.3 the resistances become very large due to insufficient reaction sites. If considering a cathode only composed of the LSM phase with parameters: 29.76 m, ZL elio1 m, =0.4,rr the calculated total resistance is about 21695m cm. Compared this value with the data shown in Fig. 4-18A, the advantage of a composite cathode is clear. From Fig. 4-18B and Fig. 4-18C, it can be observed that with the decreasing of the radius, the optimal thickness corresponding to mi nimal total resistance also decreases when the electrode fraction is within the thresholds. This implies that the larger particle size, the thicker the cathode is needed to reach an optimum performa nce. While beyond the suggested percolation, i.e.,el0.2 the optimum thickness does not exist fo r all three cases. Therefore, the performance of a cathode can be improved by decr easing the particle size and thus resulting in the enhancement in the TPB length per unit volume. However, very small LSM or YSZ particles correspondingly increase the gas diffusion path a nd make the pore size become smaller, so the concentration overpotential will increase, especially fo r a much thicker cathode. 4.3.6 Conclusions A 3D random micro-scale model ta king into account of th e gas diffusion and pore structure has been developed to evaluate the ma ss transfer and electroche mical performance for a composite cathode in a solid oxide fuel cell. Then detailed investigatio ns are carried out to optimize the geometry parameters. A comparison w ith literature experime ntal data shows a good agreement.
126 Model results demonstrate that the concentration loss is in significant only when the porosity is in the middle range and the cathode is thin under a lowe r current density. The electrochemical reaction behavior is strongly influenced by the porosity value. Only when the porosity is in the range from 0.25 to 0.45, the ga s penetration depth is la rger, reaction rate is relatively uniform and almost all of the TPB sites become active. The larger thickness is favorable for the performance by increasing the number of the effec tive reaction sites and decreasing the specific resistance. Smaller part icle sizes reduce activation overpotential by increasing the TPB length per unit volume of th e cathode. The minimum valu e in total specific resistance decreases as the particle radius is d ecreased. But at the same time, too small particles will increase the gas diffusion resistance. In th e present study, the best performance is achieved at el0.4, and 29.14 mZwhen 0.4 There is a good agreement between our predicted results with the continuum theory and experimental data. It shows that the pred icted results on total specific resistance are reasonably larger than those obtai ned from the continuum theory for a very thin cathode. The presented model is still valid for the cathode with a lower ratio between the thickness and particle diameter compared w ith the continuum theory. Also, the current model removes some weaknesses in the model by Sunde (1996a,b), su ch as ignoring the diffusion resistance and specific pore structure. Therefore, this model can be applied to much more general situations.
127 CHAPTER 5 MULTI-SCALE MODELING OF A LOW-TE MPERATURE SOLID OXIDE FUEL CELL WITH CONSIDERATION OF MICROSTRUCTURE AND INTERFACIAL EFFECTS We have proposed a multi-physics, multi-scale macro, nano/micro and particle-scale model structure in the Chapter 2. This model is a long term plan. In this chapter, our works focus on how to combine the macro-model in the Chap ter 3 with the new micro-level model developed in this Chapter, and then use this model to optimize the SOFC performance at micro-scale level. 5.1 Low Temperature SOFC and Modeling Objectives The superior efficiency of a solid oxide fuel cell in comparison with other energyconversion systems is due to its higher operation temperature, which may exceed o1000C in some designs (Larminie et al., 2003; Singhal et al., 2003). As the operation temperature of a SOFC is reduced, the activation resistance in the electrodes and the ohm ic resistance in the electrolyte will contribute to the majority of the total cell resistances (Cai et al., 2002). For example, if reducing the opera ting temperature of a SOFC tooo550C-800C, the cathode resistance was 70-85% of the total cell resist ance for anode-supported SOFCs with standard LSM-YSZ cathode on YSZ electrolyte (Tsai et al., 1997). However, lowering the operation temperature is an inevitable path for the commercialization of th is technology, which allows for a broader choice of materials, reduces the fabrication cost, minimizes the interfacial diffusion between electrode and electroly te, and improves the response time for start-up and shut-down period, etc.. For this purpose, research has been focused on de veloping thin film electrolyte (Hertz et al., 2004; Chen et al ., 2004), adopting alternative electr olyte or electrodes with higher conductivity (Shao et al., 2005, Wang et al., 2004; Zhang et al., 2006) and finding new fabrication processes for el ectrodes (Gorman et al., 2005). 0.50.53-SmSrCoO (SSC), 1-0.81-3-LaSrCoFeOxxy (LSCF) and 0.50.50.80.23-BaSrCoFeO (BSCF) have been experimentally
128 proven to provide a superior performance as a cathode material for SOFC or its combination with Gd0.2Ce0.8O1.9 (GDC) and 0.850.152CeSmO (SDC) to form a composite cathode (Watanabe et al., 1997; Liu et al., 2006; Hua ng et al., 1997). For example, the measured total electrode area specific resistance is only 0.1022cm at o600C for a cell with a SDC electrolyte thickness of 20 m and a composite cathode (75wt.% S CC +25wt.%SDC, Zhang et al., 2006). Adding SDC particles to Ni is effective for improving the activity of Ni anode through increasing the TPB area and maintaining the appropria te porosity (Wang et al., 2004). However, this kind of composite el ectrode exhibits a complex geometry/morphology, it is frequently a demanding task to predict how th e interplay of materials properties and geometry (e.g., porosity, particle size, a nd electrode thickness) affects the polarization resistance. A micromodel, which focuses on the electrode and electrol yte region, is therefor e crucial to understand the relations between performance and relevant microstructure parameters in much more details (Costamagna et al., 1998; Sunde, 2000; Jeon et al., 2006). For a lower temperature SOFC, the temperature becomes a crucial factor and ev en an insignificant variation in non-homogenous temperature distribution or flow field within a SOFC might change the performance substantially. Unfortunately, previous micro-models can not account for this temperature or geometry effect at macro-level (e.g. stack dimension) because of inherent model weaknesses and assumptions. Especially, when considering the goals of our model, i.e., how to optimize the entire cell design including global macroand microstructure para meters, only few influencing factors in a model will lead to significant inaccuracies or errors in predicting overal l SOFC performance. Therefore, it is desirable to develop a multi-scale model whic h has the capability of handling the transport mechanisms on different length scales at the same time.
129 The current study aims at developing a new multi-scale model, which will be employed to integrate a micro-scale submodel with a macr o-scale submodel and account for transport mechanisms arising from different processes an d length scales for a lower temperature SOFC. Recently, the authors have developed a multi-physic s macro-model to inves tigate the effect of transport scale on the performance of a SOFC un it (Ji et al., 2006, Chapter 3).The macrosubmodel is developed on the basis of the previous work. Then solution will be supplied to the micro-scale submodel as the global parameters The micro-scale submodel establishes the complex relationships among the various transpor t phenomena in the pore level, which includes the transport of electron, i on and gas molecules through the electrodes, electrolyte and particularly at the three-phase boundary region. Finally, by inte grating the two submodels to form a mutiscale model, the dependence of electrochemical performance on the global parameters (temperature, thickness of electrodes) and microstructures ( porosity, volume fraction, composite ratio, etc.) will be assessed for the en tire fuel cell stack in a meaningful way. 5.2 General Descriptions and Model Assumptions A typical single anode-supported SOFC unit cell consists of a thin layer of anode, a thick anode substrate layer, a porous layer of cathode, a dense layer of electrolyte, and interconnects (or ribs) as illustrated in Fig. 51A. Previous simulation results from the macromodel (Chapter 3, Ji. et al., 2006) have sufficiently demonstrat ed that the variations for temperature, velocity and concentrat ion in the z-direction (normal to xy plane, refer to Fig. 5-1A) are insignificant if the rib thickness is small. So here, the rib thickness is assumed to be small and a two-dimensional model is considered for savi ng much computational time. The schematic of the 2-D computational model is shown in Fig. 51B. To study the different physical processes at various length scales, the curr ent model consists of two lengt h scales: a macro-scale submodel and a micro-scale submodel. The macro-scale submodel is associated with the entire
130 computation domain. While the micro-scale su bmodel is used only in the electrode and electrolyte regions. A Ni/SDC composite anode, a SDC electroly te and a LSCF/SDC composite cathode are considered in the present simulation for simulating a lower temperature solid oxide fuel cell. A B Figure 5-1. A planar solid oxide fuel cell. A) A unit cell. B) Computational domain, structural modeling of a solid oxide fuel cell, an d close-up of cermet TPB. The composite cathode (LSCF/SDC) and anode (Ni/SDC) are modeled by a random packing of binary particles (not on scaled). TPB (reaction site) ca an RLSCF RSDC rNi RSDC Interconnect Interconnect Electrolyte Electronic conductor (LSCF particle) Ionic conductor (SDC particle) Electronic conductor (Ni particle) e-e-H2 H2O O2 O2 -Pore Pore X Y Air Fuel Air Fuel Top interconnect Cathode Electrolyte Anode Bottom interconnect Symmetric Boundary X Z Y
131 In Fig. 5-1B, the composite electrodes are m odeled as a random packing system made of electronic particles (LSCF or Ni ), ionic particles (SDC) and interstitial pores. Current is conducted from particle to particle through the interface. Although a real porous microstructure is different from this random system, the previo us studies have successfu lly proven its validity in predicting electrochemical performance for a composite electrode (C ostamagna et al., 1998; Sunde 2000; Jeon et al., 2006; Bouvard et al., 1991). In the cathode region, oxygen which diffuse s through the air channel and interconnected pores, is reduced somewhere in the vicinity of three phase boundary (TPB) region to oxygen ions via the overall half -cell reaction: -x 2ooO(g)+4e+2V2O (5.1) where oVand x oOare vacancies and oxygen ions in the SDC phase. Likewise, the hydrogen reduction reaction in the TPB of the anode occurs as: 2-22H(g)+OHO+2e (5.2) The system terminal output is critically depende nt on the activation, c oncentration, and ohmic polarizations and can be expressed as: an+caan+ca+chan+ca+ele+inn cellactconohmcontact()VE (5.3) with the Nernst potential E is given below: 22 20.5 HO g 0 HOln() 2 p p RT EE Fp (5.4) where an+ca actis the activation overpotential due to electrochemical reaction at TPB. an+ca+ch con is the concentration overpotential due to mass transport in the el ectrodes and channel flows. an+ca+ele+inn ohm is the ohmic overpotential across electr odes, electrolyte and inter-connectors. contact is the
132 contact resistance. 2H p and 2HO p are the partial pressures of hydrogen and water in bulk channel flows, respectively. 4 01.27232.764510 E T depends on local temperature. However, it is rather difficult to clearly identi fy one kind of overpotential from others because they are highly inter-related in an actual energy transport process Therefore, completely understanding and modeling those losses are critical to predict the cell s electrochemical performance accurately, which is one of the main objectives in the present work. Modeling is performed with the following assumptions: Steady state condition. Mass/heat transport and gradients in tw o-directions (2D) are considered. Energy transfer due to species diffusion in ch annel flows and porous layers are neglected. Because the single cell model repr esents a repeating cell structur e in the center of a larger stack, cyclic boundary conditions for curren t are imposed at the boundaries of model domain. Walls at the periphery of the single cell are assumed to be adiabatic. For electrodes, each of the two conducting phases is considered as continuous and homogenous having an effective conductiv ity dependent on the micro-parameters. Identical and uniform catalyst activity thr oughout electrodes. Releva nt electrode (anode SDC/Ni and cathode SDC/LSCF) parameters are extracted from the literatures. In view of that molecular diffusion and Knudsen diffusion are comparable in the electrodes, both of them are cons idered together. 5.3 Macro-submodel 5.3.1 Governing Equations In the Chapter 3, the macro-s cale submodel has been developed to simulate the thermal, flow and species fields in the entire computational domain. In the pure fluid region (channel flows), the Navier-Stokes equations desc ribe the flow behavior In the porous region (electrodes), flow is treated as laminar, incompressible and governed by the Brinkman-
133 Forchheimer extended Darcy model. The conser vation equations of mass, momentum, energy and species for fluids in both pure fl uid region and porous domain are: f()0 V (5.5a) 2 ffff1()() F p VVVVVV K K (5.5b) ffeffreoh()() cVTkTQQ (5.5c) f()iiiYVJS (5.5d) 11n i iY (5. 5e) f g()i i ip Y RT M (5.5f) Here the meanings of all symbols are the same as those in equations (3. 20)-(3.25) of the Chapter 3. 5.3.2 Gas transport and Concentration Overpotential The gas transport through porous media is main ly controlled by the molecular diffusion, Knudsen diffusion and viscous flow. For compos ite electrodes in fuel cells, the Knudsen diffusion is as important as bulk diffusion due to th at the mean free path of molecular species is comparable to the pore size. Thus, the two diffe rent diffusion mechanisms should be considered in the present model. Assuming that the porous medium is in local thermodynamic equilibrium with the fluid, the extended Stefan-Maxwell rela tion is still used to determine the diffusion mass flux vectoriJ as in Chapter 3:
134 1 ,k,eff,effg()n ijji i i i iijij jiXJXJ J P X DDDRT (5.6) The calculation on the effective binary coefficient,eff ijDand the Knudsen diffusion coefficient ,k iD has been presented in the Chapter 3. The concentration overpotentials, which are induced by the gaseous transport at electr odes and channel flows, are calculated as: 2 2int O g ca con r O(,)ln() 4 p RT xy Fp, 22 22intr HHO g an con rint HHO(,)ln() 2pp RT xy Fpp (5.7) 2 2b O g fuel con int Oln() 4 p RT Fp, 22 22bint HHO g air con intb HHOln() 2pp RT Fpp (5.8) where b i p int i p and r i p (222O,HO,Hi ) are the partial pressures of component i at the air/fuel channel flow, the electrode/cha nnel interface and the reaction site (TPB), respectively. 5.4 Micro-submodel There are three different groups of cl usters formed in electrodes: (i) the first is composed of the same type particles (ionic or el ectronic particles) that in contact with one another to extend through the enti re electrode. This kind of cluste r is called percolating cluster. Under such condition, a good conductivity is reac hed.The second is that a cluster is only connected to its correspo nding bulk phase type (inter connect or electrolyte) .The third cluster is completely isolated from its corresponding bulk phase. The existence of this kind of cluster increases the polarization resistance in the electrode. In the present micro-porous electrode mode l, the electronic phase (Ni/LSCF) or ionic phase (SDC) are considered as formed by the fi rst kind of clusters, which means percolating clusters dominate the electrode performance. So this assumption requires the electrodes have a high ratio of electrode th ickness to particle diameter that is always greater than 100. When the
135 ratio is roughly lower than 100, the model accuracy may be affected by some degree of uncertainty and Monto-Carlo simula tion is suitable for thin electr odes (Sunde, 1996a, b; Ji et al., 2007). The second type is neglecte d and the third type is assume d to be taken into account by parameter choices in the following derivation. The charge transfers along the electronic or ionic phase and the transfer current is intrinsically between the two phases. According to Ohms law, the current balance in electrodes can be established as: el,effelVeli (5.9a) io,effioV ioi (5.9b) ca/an TPBtract(,,)iLiYTioelii (5.9c) Where V is the potential and i(Am-2) is the current vector per unit area of the electrode. io,eff and el,effare the effective con ductivity of the ionic and electroni c conductor phase, respectively. TPBL(-3mm) is the active TPB length per unit volume of the electrode. ca/an tri (-1Am) is the transfer current density per uni t TPB length in the cathode or a node, which is a function of the local activation overpotential an/ca act, temperature and concentra tion. The local activation overpotential act is equal to: actioelcon()VV (5.10) Local concentration loss con is estimated by equation (5.7). 5.5 Model Parameters There are still some unknown parameters or terms to be determined in equation (5.5) and equation (5.9), such as the heat genera tion sources, the effective conductivities, the TPB length and the transfer current density, etc. The classical percolation theory, coordinate number
136 theory and experimental work on electrode kine tics will be applied to obtain these important parameters. 5.5.1 Effective Conductivity and TPB Length According to the results of the percolation theory (Kir kpatrick, 1973), and effective medium theory (Mclachlan et al., 1990), the eff ective conductivity of elec trode particle in a random packing of bimodal particles above the percolation threshold is: m el el,effel,0 el,c1 (1) 1 (5.11) Similarly, the ionic conductivity for the electrolyte phase is: m io io,effio,0 io,c1 (1) 1 (5.12) where el is the volume fraction of the electrode part icles with respect to all solid volume (void excluded). el,c is the volume fraction for electrode pa rticles corresponding to the percolation threshold. io,c is the volume fraction for ionic part icles corresponding to the percolation threshold. m is the adjustable parameter. ,0 is the conductivity for -phase material. However, the neck formed between same types of particles re duces this value. It is assumed that the bulk conductivity corresponds to the maximal TPB length, so it can be estimated as: min TPB ,0,bulk,bulk,bulk TPB,maxmin2sin 2 sin 22r l lr (5.13) where is the contact angle and is set as 60o, and minr is the radius of the smaller particle. However, when the porosity is decreased or incr eased, how is the conductivity in SOFC affected? Here, an approximate approach is introduced to take into account the contribution of pore.
137 According to Archies law (Archile, 1942; Wong et al., 1984), the eff ective conductivity is strongly dependent on the poros ity of the pure materials: ,eff(1)t (5.14) here t is an empirical parameter. For a spherical po re, this value is set as 1.5. In actual case, equation (5. 11) is obtained with an inherent assumption, i.e., the por osity is estimated as 0.4, so the conductivity of an electrode particle can be modified as: el,0 m el el,eff el,c(1) 1 (1) (10.4)1t t (5. 15) According to effective medium theory (EMT) (M clachlan et al., 1990), the adjustable parameter m also can be chosen as 1.5. Thus the final forms for conductivities are: 1.5 el el,effel,bulk el,c1 (1) sin1 20.61 (5.16) 1.5 io io,effio,bulk io,c1 (1) sin1 20.61 (5.17) In the above equations, the percolation threshold el,c and io,c can be determined by the coordination number theory and has been given by Kuo et al. (1995) and Suzuki et al. (1983). The approach to estimate TPB length per unit volume TPB,VL (-3mm ) was developed by Sunde (1996a,b), Costamagna et al. (1998) an d Bouvard et al. (1991) using the relationship between percolation and part icle coordination number: ioel TPB,Vminioelioel 02sin 2ZZ LrNnnPP Z (5.18) where ioP and elP are the probability of el ectronicor ionic-type co nductors in the percolating
138 cluster. Suzuki et al. (1983) proposed a more accurate expression toP in the range 0.1546.464 (ioel/dd is the diameter ratio between ionic and electronic particles): 0.4 2.54.236 1 2.472 Z P (5.19) 0 2(1)Zn Z nn (5.20) where Z is the coordination number betw een same type particles, and0 Z is the average total coordination number in a random packi ng of binomial spheres and equals 6.ion and ion are number fractions of electronic and ionic partic les, which are related with volume fractions el and io N is the total number of particles per unit volume.io Z and el Z are the coordination numbers of electronic conduction and i onic conductor. They are given by: 33 elelel1 4/3[(1)]N rnn (5.21) 2 0 io 2 elel(3) 3 (1)Z Z nn (5.22) 0 el 2 elel(3) 3 (1)Z Z nn (5.23) It should be pointed out that e quations (5.18)-(5.20) only take in to account the contribution of percolating cluster to conductivity and TPB length. As long as is far away from threshold and P is also really high, this kind of clusters dominates the response of the electrodes performance and thus the equations are reasonable.
139 5.5.2 Electrode Kinetics a nd Transfer Current Density In principle, oxygen reduction and hydr ogen oxidation reactions have been recognized to involve a series of consecutive elemental step s including, i.e., dissoci ation, charge transfer, surface diffusion of oxygen/hydrogen intermediate species, etc.. There is still no common consensus in the mechanisms and kinetics of the two reactions and sometimes contradictory because the elemental steps strongly depend on the specific experime ntal conditions and microstructure, ect. (Bieberl e et al., 2001; Fleig, 2003; Alder, 2004; Ji ang et al., 1998). Especially, the experimental data on electroche mical kinetics of Ni/SDC anode and LSCF/SDC cathode at lower temperatures are very limited and incomplete at present. The relationship between the ionic current densities in the composite electrodes and the charge-transfer overpotentials can be described by Bulter-Volmer equation: eacteact 0 ggexp()exp[(1))]nFnF ii RTRT (5. 24) where F is the Faradays constant. en is the number of electrons. is the transfer coefficient. 0i is the exchange current density, which ge nerally depends on the local temperature and composition at the electrodes. For oxygen reduc tion in LSCF/SDC cathode, Liu et al. (2006) found the exchange current density ca 0i is between -210Am and -2650Amat the temperatures ranging from o500C too700C, but they did not furt her report the effect of the partial oxygen pressure on this value. Here we adopt this range. For SDC/Ni anode under lower operation temperature, the anodic exchange current density is set as -2100Am on the basis of literature data reported by Watanabe et al. (1997) This paper focuses on presenting a theoretical framework instead of making quantitative co mparison with experiments, so the constant exchange current
140 densities are still helpful in qualitatively studying the mechan istic issues that govern the performance of SOFCs. 5.5.3 Mass and Heat Sources In mass transport equation (5.5d), the net rate of production for each component,iS, can be stated as the following: 22ca HOHOel1 2SM Fi (5.25a) 22ca HHel1 2SM Fi (5.25b) 22an OOel1 4SM Fi (5.25c) The volumetric heat generated from elect rochemical reactions is expressed by: anan reelecel1 () 2QHG Fi (5.26) Here equation (5.26) is only for anode layer. G is the chemical potential. The heat generated from ohmic loss exists in all of solid parts (elect rodes, interconnector a nd electrolyte), so it can be expressed by: 2an/ca ohelioeffact0 QR iii (5.27) Here eff R is ohmic resistance, which is related to effective conductivity and geometry. 5.6 Solution Algorithm and Parameters The SIMPLEC (Patankar, 1980) method is applied to solve th e discretized equations of momentum, energy, concentration and electrical pot ential. Air (79%N2 and 21%O2) is delivered to the air channel. Fuel (85%H2 and 15%H2O) is delivered to the fuel channel. Some input parameters for the base line case are ex tracted from th e literatures (Watanabe et al., 1997; Liu et al., 2006; Zh an et al., 2001; Huang et al., 1997; Anselmi-
141 Tamburini et al., 1998) and listed in Table 5-1. The physical properties of common materials, which are used for electrodes and electrolyte, st rongly depend on temperat ure. Consequently, the performance of a SOFC is affected by the temperat ure field significantly. This temperature effect is included in the present model. Since low-te mperature anode-supported fu el cells are always designed to operate below 1000K, the inlet fuel and air temperatures are chosen as 773K in the baseline simulation. The single unit cell is used in the numerical evaluation and the unit is assumed to be placed in an elec tric furnace exchanging radiative heat transfer between the cell outer surface and the furnace inner surface. Table 5-1. Model input para meters for the baseline case Parameters Value Parameters Value Fuel inlet temperature (K) 773 Particle/pore size ( m) 1 Air inlet temperature (K) 773 Thermal conductivity for anode (-1-1WmK) 12 Inlet pressure (Pa) 51.0110 Thermal conductivity for interconnector (-1-1WmK) 11 Fuel inlet velocity (-1ms) 0.5 Thermal conductivity for SDC (-1-1WmK) 2 Air inlet velocity (-1ms) 2.5 Thermal conductivity for cathode (-1-1WmK) 2.7 Cell length (m) 0.1 Porosity (%) 40 Interconnector height (m) 30.510 Tortuosity 3.5 Channel height (m) 31.010 Anode exchange current density (Am-2) 100 Anode thickness (m) 650010 Cathode exchange current density (Am-2) 500 Cathode thickness (m) 65010 Electric conductivity for Ni (-1S m) 63.27101065T Electrolyte thickness (m) 62010 Ionic conductivity for SDC (-1S m) 7 56.410/ 0.86 exp 8.6173410 T T
142 5.7 Simulation Results and Discussion 5.7.1 Baseline Case To provide a basis for comparison, the thermo-fluid transport and electrochemical performance of an anode-supported SOFC is fi rst investigated usi ng the parameters and operating conditions listed on Tabl e 5-1. Fig. 5-2A shows the temperature contour along the channel. The temperature at the inlet is se t at 773K and it increases monotonically to around 940K due to the contribution of Joule heating and chemical reac tion. The overall temperature level is considered to be in the low temperatur e range of a SOFC operatio n. On the air side, the temperature rise is not as large as that in the fuel side due to higher mass flow rates. Inside the interconnects, the temperature distribution in the ydirection is fairly uniform due to high thermal conductivity. The molar fraction contours for the major reactants and products at an average current density of 4000Am-2 are plotted in Fig. 5-2B--5-2D. It reflects a fact that the diffusion of oxygen from the air channel to th e cathode creates a significant c oncentration gradient (Fig. 52B). This concentration loss at the cathode is detrimental to th e cell performance and it can be minimized by reducing the channel dimension as shown in the Chapter 3. On the other hand, the hydrogen and water vapor in the fuel channel do not cause significant concentration gradients in th e y-direction since the diffusion of H2 is about two to three times faster than that of oxygen as Fig. 5-2C and Fig. 5-2D indicate. The molar fraction of H2 decreases while that of H2O increases, respectively, from the fuel channel inlet to the outlet due to the electrochemical reaction (Fig. 5-2C). However, in the anode layer, the concentration gradients become significant because of combin ation effect of the Knudsen diffusion and the binary diffusion.
143 A B C D Figure 5-2. Temperature and mola r contours. A) Temperature. B) Oxygen at air channel and cathode layer. C) Hydrogen at anode layer and fuel channel. D) Water at anode layer and fuel channel (arrow denotes flow di rection; TI/BI=top/bottom interconnect; An=anode; Ca=cathode; Air=air ch annel; Fuel=fuel channel). The spatial distributions of electronic and ionic potentials are shown in Fig. 5-3A and Fig. 5-3B. In this case, the porosity and the compositi on ratio of the electrode phase are 0.4 and 0.5, respectively. The terminal output is0.465V. It is found that the electr onic potential is higher than the ionic potential in the anode and lower than the ionic po tential in the cathode as shown in Fig. 5-3A. Moreover, the electronic potential distribution across th e cathode and anode is really uniform. While in the ionic phase, it turns out th at the potential gradually decreases across the entire fuel cell thickness. These distributions are attributed to th e higher conductivity for LSCF/Ni and much lower conductiv ity for SDC. In the electrol yte, the potential drop is significant, e.g. in the range of 0.07V-0.18V, due to the lowe r operation temperature compared with a higher-temperature SOFC. Th is result implies that the thickness of electrolyte must be minimized for the reduced-temperature fuel cell in order to lower the ohmic overpotential. An+Fuel Y 790 800 800 810 820 850 860 860 860 870 880 890 900 920 935 939X TI BI Ca+Air Y 0.185 0.188002 0.192697 0.197696 0.2 0.202986 0.205X Ca+Air 0.58 0.6 0.62 0.66 0.68 0.72 0.82 Y X An Fuel Y 0.2 0.22 0.28 0.3 0.32 0.36X An Fuel
144 1.07246 1.06 1.02554 1 0.987 0.9 0.8 0.7 0.6 0.5 0.46827 0.45 0.4 0.3 0.2 0.1 0 1.05 1.020 1 0.95 0.9 0.85 0.8 0.75 0.65 0.6 0.55 0.5 0.45 0.4 0.3 0.25 0.15 0.1 0 A B 0.27 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.01 0.008 0.002 0.0002 2E-06 0.0085 0.008 0.0075 0.007 0.0065 0.006 0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 C D Figure 5-3. Performance of SO FC. A) Ionic potential distribu tion. B) Electronic potential distribution. C) Activation overpotential. D) Concentr ation overpotential (unit: V). Distributions of correspondi ng activation and co ncentration polarizations through the cathode and anode are displayed in Fig. 5-3C and Fig. 5-3D, resp ectively. In the cathode, the activation overpotential keeps increasing and reache s the peak value at the interface between the electrolyte and the cathode. This tendency indi cates that the strongest electrochemical reaction takes place in the region next to the electrolyte where the largest poten tial difference and the maximal number of active TPB exist. This result agrees with those from the previous studies Anode Cathode El Anode Cathode El Anode Cathode El Anode Cathode El
145 (Jeon and Nam et al., 2006; Ji et al., 2006) using the Monte Carlo simulation. While in the anode, the activation overptential is almost negligible, i.e. on the order of610 in most of the region and jumps to a remarkable level in the zone next to the electrolyte/cathode interface. This result means a large part of the anode does not actively participate in th e electrochemical reaction, i.e. it is useless. Nevertheless, the cathode seems to be much more efficient because the activation overpotential distribu tion is more even through th e entire cathode layer. The corresponding concentration loss is give n in Fig. 5-3D. It indi cates that this loss is unnoticeable compared with the activation lo ss in the cathode. But in the anode, the concentration loss becomes substan tial and is about ten times larger than that in the cathode due to the larger thickness of anode. Clearly, the activation loss in the cathode contributes most to the overpotential for the curr ent anode-supported low-temperature SOFC unlike in an electrolytesupported fuel cell, where the ohmic loss is domin ant. Therefore, for the current case, the key elements for improving the performance should be focused on optimizing the electrode microstructure. 5.7.2 Temperature Effects In general, if the operating temper ature for a fuel cell is reduced, the performance will deteriorate due to the increase in ohmic loss. In th is section, investigation will be carried out to study the dependence of fuel cell performance on the gas inlet temper ature. Two gas inlet temperatures of o500C and o600Care investigated. The other i nput parameters and properties are still kept the same as those in the baseline case. The simulated performance is shown in Fig. 5-4A and Fig. 5-4B. It indicates that th e maximum power density is about 0.256 Wcm-2 for o500C and 0.34Wcm-2 foro600Cand the terminal output is about 0.749V for o500C and 0.866V for o600C. Compared with the higher temper ature SOFC such as an electrolyte-
146 supported fuel cell, this kind of reduced-temperature fuel cell has a considerably lower performance due to a higher ohmic loss and a weak electrochemical reaction activity. The corresponding current density distributi on is displayed in Fig. 5-4B. 0.00.20.40.60.81.01.21.41.61.82.02.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 Power density (W/cm2) Output (V)Current density (A/cm2) 500oC 600OC A 0.00.20.40.60.81.0 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Exchange current density (A/m)Y/LC I0=1000A/m2, 500oC I=02000A/m2, 500oC I0=4000A/m2, 500oC I0=8000A/m2, 500oC I0=15000A/m2, 500oC I0=1000A/m2, 600oC I0=2000A/m2,600oC I0=4000A/m2, 600oC I0=8000A/m2, 600oC I0=15000A/m2, 600oC electrolyte/cathode interfaceB Figure 5-4. Performance comparison with 500oC and 600oC. A) I-V performance. B) Exchange current density distribution over the cathode.
147 Clearly, the electrochemical reaction str ongly depends on the temperature. As the temperature is increased too600C, the profiles of exchange current density in the cathode become flattened, which means more regions are involved in the r eaction and the cathode becomes more efficient. Another point is that as the current density in creases, the relative importance of TPB near the electrolyte becomes more prominent, and the region far away from the electrolyte is almost use less having the weakest reaction rate. Similar case holds for the anode, but the useless region in the anode is much larger. 5.7.3 Thickness Optimization In order to study the dependence of SOFC performance, ove rpotential and exchange current density (or reaction activ ity) on the electrode thickness, several cases with different cathode and anode thicknesses are carried out. Fi g. 5-5 presents the comparison results with different cathode thicknesses. The I -V performance curves in Fig. 5-5A exhibit insignificant dependence on the cathode th ickness. When current density is lower, i.e., 1000Am-2 or 2000Am-2, the performance is enhanced with a larger cathode thickness, but it is not remarkable. When the current density exceeds 8000Am-2, such differences are so unno ticeable that the curves are almost identical. A thicker cathode possesses increased TPB number that results in the improvement of the activity of chemical r eaction; on the other hand, the concentration overpotential also is increased at the same time as shown in Fig. 5-5B. However even for a thicker cathode, the co ncentration loss in the cathode is still insignifican t, e.g., in the order of 103V~10-4V, so the decrease in activation loss overw eighs the increase in concentration loss and the overall cathode performance is better when the curr ent density is not very high. Although for a thicker cathode, the concentra tion overpotential starts to beco me significant due to a longer diffusion path as the current density is very high, such increase still is not so substantial that it
148 can be completely counteracted by the reduction in activation overpoten tial. Additionally, the temperature rise in a thic ker cathode is slightly larger, which offers an extra improvement in the performance to some extents. 0.00.20.40.60.81.01.21.184.108.40.206 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Output (V)Current density (A/cm2) LC=10m LC=25m LC=50m LC=100m A 0.00.20.40.60.81.0 0.0000 0.0005 0.0010 0.0015 electrolyte/cathode interfaceAveraged concentration overpotential (V)Y/LC LC=10m LC=25m LC=50m LC=100m B
149 0.00.20.40.60.81.0 0.0000 0.0002 0.0004 0.0006 0.0008 Current density at cathode (A/m)Y/LC LC=10m LC=25m LC=50m LC=100melectrolyte/cathode interface C Figure 5-5. Effect of thickness on cathode perf ormance. A) I-V performance cathode. B) Concentration overpotential in the cathode. C) Current density di stribution across the cathode. Fig. 5-5C shows that a thinne r cathode provides a much more uniform density distribution and thus results in higher cathode utilization. However, after we review the results from Fig. 55A that an extremely thin cathode (e.g.,10 m) has an overall lower performance due to insufficient reaction sites, it is suggested that th e cathode thickness shoul d be between 25 m and 100 m to achieve an optimal combination of a minimized concentration loss and a uniform reaction rate. These results are in good agreem ent with the previous theoretical studies (Costamagna et al., 1998; Jeon et al., 2006; Ji et al., 2006) and experimental observations by Kenjo et al. (1991). For the anode, similar results can also be obt ained as illustrated in Fig. 5-6. But the difference is that the concentration overpotentia l becomes more significant than that in the cathode, especially for the case of a higher current density. For th e thicker anodes, valley regions occur and the central part of a node layer can not be utilized ve ry well due to the limitation of
150 mass diffusion. However, the electrochemical reaction seems to o ccur throughout the whole anode at thickness less than100 m. 0.00.20.40.60.81.0 0.000 0.002 0.004 0.006 0.008 0.010 Concentration overpotential (V)Y/L A LA=50m LA=100m LA=250m LA=500mA 0.00.20.40.60.81.0 -10 -8 -6 -4 -2 Log(Average current density) (A/m)Y/LA LA=50m LA=100m LA=250m LA=500mB Figure 5-6. Effect of thickne ss on anode performance. A) C oncentration overpotential in the anode. B) Current density distribution across the anode.
151 5.7.4 Porosity In this section, th e effect of electrode porosity on the fuel cell performance is investigated. A more realistic electrode geometry is consider ed by setting the anode and the cathode thickness as 500 m and 50 m, respectivel y. The porosity is va ried between 0.25 and 0.6. Fig. 5-7A gives the dependence of I-V performance on the porosit y. It turns out that reducing the electrode porosity l eads to a slight increase in te rminal output for both the cathode and the anode. On the one hand, a smaller porosity leads to an increase in solid volume fraction and thus TPB length, so the activ ation overpotential is effectivel y decreased as shown in Fig. 57B and Fig. 5-7C. Figure 5-7. Effect of electrode porosity on performance and ov erpotentials. A) Dependence of I-V performance on porosity. B) Overpotentials ( cathode). C) Overpot entials (anode). 0.200.250.300.350.400.450.500.550.600.65 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 Output (V)Output (V)Porosity cathode with average I0=1000A/m2 anode with average I0=1000A/m2 cathode with average I =11000A/m2 anode with average I =11000A/m20.00.20.40.60.81.0 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 con =0.3 =0.4 =0.5 =0.6 I0=1100A/m2,act Overpotential (V)Y/LA I0=11000A/m2,act 0.00.20.40.60.81.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 =0.3 =0.4 =0.5 =0.6conI0=1000A/m2, act Overpotential (V)Y/LC I0=11000A/m2, act
152 In particular, the reduction of activation overpotential in the cathode is much more remarkable, which leads to a higher performance improvement in the cathode than that in the anode. However, on the other hand, an extremely sm all porosity limits the mass transfer from the channels to the reaction sites and increases th e concentration loss and pressure loss in the electrodes by reducing the binary diffusivity Knudsen diffusivity and permeability. This negative effect becomes severer for the a node with a larger current density, 11000Am-2, so the anode with the smallest porosity 0.25 does not ha ve too much improvement compared with that with porosity 0.3. Therefore, it is not desirabl e to improve the fuel cell efficiency by reducing porosity, because it causes a larger mass transport loss and pressure loss in the porous anode layer. This is the main reason why the porosity is always in the range of 0.3~0.4 in the practical application. In the previo us study (Chapter 4) based on the Mona-Carlo simulation gas is assumed to penetrate into the anode or cathode a nd also include the dead effect or inactive TPB. However, in the current study, it is assume d that all pores are open to gas reaction and not any dead effects, so the decreasing of output is just caused by the decr easing of TPB length and the concentration effect is not very significant. 5.7.5 Electrode Composite First of all, it should be noted that the c ontinuum theory and coordination number theory have demonstrated that the percolation threshol d varies from 0.294 to 0.709 as the radius ratio equals to unity. Therefore, the current model based on the two theories is invalid beyond this percolation range. In the study of the compositi on effect, the volume fraction number of the electronic phase ranges between 0. 299 and 0.7. Fig. 5-8 presents the effect of electronic phase volume fraction in the anode on the terminal output. The maximal performance is achieved atel0.4~0.45 For cases with el0.2970.35 and el0.50.7 the output is relatively
153 lower; it means the charge resistance is greatl y increased due to insuffi cient reaction sites and smaller TPB length. For the cathode, further calc ulation also shows the similar optimal value. These results match well with the previous experimental and theoretical results (Costamagna et al., 1998; Ji et al., 2006).However, when the co mposition is beyond the percolation threshold, our previous study has indicated th at the specific tota l resistance is very large and the overall performance deteriorates. 0.30.40.50.60.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Output (V)Volume fraction of electronic particle el I0=4000A/m2 I0=1000A/m2 I0=11000A/m2Anode Figure 5-8. Effect of an ode composition on performance. 5.7.6 Particle Diameter Fig. 5-9 shows the influence of particle size on the terminal output and exchange current density for the ionic and electroni c phases. The thicknesses of cathode and anode still are kept as 50 m and 500 m, respectively. From Fig. 5-9A, it is observed that a smaller particle diameter results in better performance, primar ily due to the increase of the active three phase boundary length per unit volume. However, the im provement is less significant at the lower
154 average current density, e.g.,-21000Am, which is due to the rela tively smaller contribution of activiation overopotential to th e overall potential loss. 0.00.20.40.60.81.01.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Output (V)Current density (A/cm2) del=dio=0.1m del=dio=0.5m del=dio=1m del=dio=2m del=dio=4mA B C Figure 5-9. Effect of particle sizes on performance. A) I-V performance. B) Averaged exchange current density distribution across the anode C) Averaged exchange current density distribution across the cathode. 0.00.20.40.60.81.0 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 Log( Itpb,anode) (A/m)Y/LA d=0.1m d=0.5m d=1m d=2m d=4m Anode, I0=4000A/m20.00.20.40.60.81.0 -9 -8 -7 -6 -5 -4 -3 -2 -1 Log( Itpb,cathode) (A/m)Y/LC del=dio=0.1m del=dio=0.5m del=dio=1m del=dio=m del=dio=4mCathode, I0=4000A/m2
155 Apparently, for the larger particle size a nd the smaller density of TPBs, the exchange current density per unit length is higher which is due to the cons ervation of charge. For the anode with the smallest particle size, i.e., 0.1 m, only the end regions of th e anode play important roles in generating the current and so the electrode reac tion mainly occurs in the reaction zone close to the anode/electrolyte and anode/fue l interfaces. In particular, the a node/electrolyte zone is much more remarkable. If comparing Fig. 5-9B with Fig. 5-6B, the valley region shrinks with increasing particle sizes. It indicates that a larger particle size impr oves the activity of the electrochemical reaction at the TPBs and the r eaction is much more uniform across the whole anode. Similarly, if the particle size in the cathode is extremely small, the zone close to the electrolyte/cathode interface plays a significant role With the increase of particle radius, the reaction zone extends into the bulk of the com posite cathode due to the decrease of the active length of TPB. In addition, extremely small par ticle sizes will have very small pore sizes in electrodes and result in high con centration overpotentials. Therefor e, Fig. 5-9B and Fig. 5-9C can be considered as useful criteria to determin e the electrode thickness with different particle sizes. 5.8 Conclusion A multiscale model that consists of two sub-models, i. e., a micro-scale sub-model and a macro-scale sub-model, has been developed for the investigation of a lowtemperature SOFC. The macro-scale sub-model is based on the con tinuum conservation laws The micro-scale submodel addresses the complex rela tionships among the transport phenomena in the electrodes and electrolyte, which include the transport of electrons, ions and gas molecules through the composite electrodes, electrolyte and three-phase boundary region. The mode l is used to assess the dependence of electrochemical performan ce on the temperature, global geometrical
156 parameter and material microstruc tures (porosity, volume fraction, composite ratio, etc.). The important findings are listed below: For the current low-temperature, anode-s upported fuel cell, a major portion of the overpotential comes from the ac tivation loss in the cathode. In general, when the system operating temperat ure is lowered, the performance is reduced mainly due to the increased ohmic loss in the electrolyte layer and re duced electro-chemical reactivity which renders portions of the electrodes inac tive. Either new material or reducing the layer thickness is needed. In order to balance the competing effects, the cathode thickness is s uggested to be in the range of 25 m to 100 m. For the anode, the thickness should be limited to some certain value in order to avoid excessive con centration loss and lower utilization. When the porosity is reduced, the activation ove rpotential is lowered but the concentration loss and pressure drop are increased. The optimal range for the porosity is between 0.3 and 0.4. The optimal electronic phase volume fraction is determined in the range of 0.35 to 0.5. In general, the fuel cell performance increases as the particle size is reduced. While larger particles produce wider r eaction zones and more uniform reaction activities.
157 CHAPTER 6 CONCLUSION AND FUTURE WORK 6.1 Conclusion In order to identify the key underlying transport mechanisms and improve our understanding of their inherent c onsequences to facilitate breakt hroughs that are needed for the realization of high power density and highly efficiency solid oxide fuel cells, a macro, nano/micro and particle-scale modeling framework is proposed to achieve the above objective. The energy transport and electrochemical process in a SOFC are analyzed and modeled from the macro-scale stack level to the microstructure le vel. Major conclusions are reached as follows. In the stack level, a 3D thermo-fluid/electro chemical model is deve loped to evaluate the heat/mass transfer and electroch emical performance for an a node-supported and an electrolytesupported SOFC, respectively. Detailed numerical si mulations are carried out to study the effects of operating temperature and geomet ry on the distributions of temp erature, local concentration, local current density and output voltage. When th e height of flow channels is decreased, the average solid temperature is reduced accordingly and the cell efficiency is improved due to both a higher heat/mass transfer coefficient between the channel wall and flow stream and a shorter current path. However, the thermal stress levels ar e still high. Compared with traditional SOFCs, the miniaturized SOFC with a thin-film electrolyte has advantages of a lower operation temperature and a much higher power density. In order to improve the scientific unders tanding of the electr ochemical reactions associated losses, and then assi st in the design of optimized el ectrode structures, A 3D random micro-scale model taking into account of the gas diffusion and pore structure is developed to evaluate the mass transfer a nd electrochemical performance for a composite cathode. A FCC random microstructure is numerically generated. The pore structure and active TPB concepts
158 are incorporated into this model. Preliminary re sults demonstrated that the concentration loss is insignificant only when the porosity is in the middl e range and the cathode is thin under a lower current density. Also, th e electrochemical reaction behavior is strongly influenced by the porosity value. Only when the porosity is in th e range from 0.25 to 0.45, the gas penetration depth is larger, reaction rate is relatively uniform and almost all of the TPB sites become the active TPB sites. Larger thickness is favorable for the pe rformance by increasing the number of the effective reaction sites and decreasing the spec ific resistance. Smalle r particle sizes reduce activation overpotential by increa sing the TPB length per unit volume of the cathode. But too small particles increase the gas diffusion resi stance. There is a good agreement between our predicted results with the continuum theory and experimental data. A multi-scale model is employed to integrate the microscopic and macroscopic models and accounts for transport mechanism arising from different processes a nd length scales in a lower-temperature SOFC. This approach can pred ict the performance of the whole SOFC stack in the macro-scale while allowing preservation of important microstructu ral properties of porous electrodes. Similar results are al so obtained from this model method. However, it has capability of predicting some important electrode characte ristics which cant be obtained from the macrocontinuum model and the micro-scale model sepa rately. For example, for a thick anode in a reduced temperature SOFC, thickness should be limited to some certain value in order to avoid excessive concentration loss and lower utilizat ion. Therefore, this multi-scale model is complementary and synergetic w ith the previous two models.
159 6.2 Suggested Further Study Further research efforts will focus on improvi ng the accuracy and efficiency of the multiphysics, multi-scale model. There are several importa nt aspects needed to be addressed in order to improve the predictive capability of propos ed models for solid oxide fuel cells: Although the atomistic model ha s not been included in the proposed work, some basic approaches will be expected to be establishe d in further work. The atomistic model is an integral part of the long-term multi-scal e model (Fig. 2-1). For oxygen reduction or hydrogen reduction reaction, not a single mechanism discovered so far can explain all the reduction processes in the elec trodes. To model the above me ntioned atomic diffusion in metals, it is suggested that ab initio molecular dynamics method that employs the density functional theory (DFT) by H ohenberg and Kohn (1964). Specifically, the generalized gradient approximation (GGA) a pproach should be used to ca lculate the electron density, which has been shown to be the best for cal culating bond lengths and energies. This model can also be applied into other material behavior simulations. Establish the connection between the micro-sc ale and the atomistic model. Like the multiscale model, efforts should focus on creating a similar matching condi tion to link the two models. The future multi-scale model needs to be completed and validated by means of experimental results. Because the proposed multi-scale model is established on the percolation theory and statistic al method, the fine microstruc ture is always neglected. Theoretical results must be compared with experimental work and then find the model limitation. The future modeling efforts s hould also be put on the react ion mechanisms and materials optimization. To our best knowledge, there sti ll lacks an effective modeling approach to simulate the reaction on three phase boundary (TPB). These modeling works will be extended to other types of fuel cell systems, such as polymer electrolyte membrane fuel cell (PEMFC) or direct methanol fuel cell (D MFC). Additionally, it w ould be worthwhile considering the current fuel cell modeling work for microgravity applications. Under microgravity condition, the flow and heat transfer characteristics will be changed significantly; so the current model will be pa rtially invalid and we have to face new challenges and arising issues in models applied to future space shuttle.
160 LIST OF REFERENCES Abel, J., Kornyshev, A. A., Lehnert, W., 1997. Co rrelated resistor networ k study of porous solid oxide fuel cell anodes. J. Electrochem. Soc. 144, 4253-4259. Alder, S. B., 2004. Factors governing oxygen reduc tion in solid oxide fuel cell cathodes. Chem. Rev. 104, 4791-4843. Achenbach, E., Riensche, E., 1994. Methane/steam re forming kinetics for solid oxide fuel cells. J. Power Sources 71, 283-288. Aguiar, P., Adjiman, C. S., Brandon, N. P., 2004. Anode-supported intermediate temperature direct internal reforming solid oxide fuel cell. I: model-based steadystate performance. J. Power Sources 138, 120-136. Aguiar, P., Adjiman, C. S., Brandon, N. P., 2005. Anode-supported intermediate temperature direct internal reforming so lid oxide fuel cell. II: mode l-based dynamic performance and control. J. Power Sources 138, 136-147. Ahmed, K., Foger, K., 2000. Kine tics of internal steam reformi ng of methane on Ni/YSZ-based anodes for solid oxide fuel cells. Catalysis Today 63, 479-487. Ahmed, S., Mcpheeters, C., Kumar, R., 1991. Thermal-hydraulic model of a monolithic solid oxide fuel cell. J. Electrochem. Soc. 138, 2712-2719. Anselmi-Tamburini, U., Chiodelli, G., Arimondi, M., Maglia, F., Sp inolo, G., Munir, Z. A., 1998. Electrical properties of Ni/YSZ cermets obtained thr ough combustion synt hesis. Solid State Ionics 110, 35-43. Appleby, A. J., Foulkes, F. R., 1989. Fuel Cell Handbook. Van Nostrand Reinhold, New York. Archie, G. E., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. Am. Inst. Min. Metall. Eng 146, 54-62. Belyavev, V. D., Politova, T. I., Marina, O. A., Sobyanin, V. A., 1995. Internal steam reforming of methane over Ni-based elect ode in solid oxide fuel cells. Applied Catalysis133, 47-57. Berkowitz, B., Ewing, R. P., 1998. Percolation theory and networ k modeling applications in soil physics. Survey in Geophysics 19, 23-72. Bessette, N. F., Wefer, W. J., Winnick, J., 1995. A mathematical model of a solid oxide fuel cell. J. Electrochem. Soc.142, 3792-3780. Bieberle, A., Meier, L. P., Gaucker, L. J., 2001. The electrochemistry of Ni pattern anodes used as solid oxide fuel cell m odel electrodes. J. Electroc hem. Soc. 148, A646-A656. Bockris, J. OM., Khan, S. U. M., 1993. Surface electrochemistry a molecular level approach. Plemum Press, New York.
161 Bouvard, D., Lange, F. F., 1991. Relation betwee n percolation and particle coordination in binary powder mixtures. Acta Metall. Mater. 39, 3083-3090. Bove, R., Ubertini, S., 2006. Modeling solid oxide fu el cell: approaches, techniques and results. J. Power Sources 159, 543-559. Brad, A. J., Faulkner, L. R., 2001. Electrochem ical methods: fundamentals and applications. Wiley, Phoenix. Cai, Z., Lan, T. N., Wang, S ., Dokiya, M. 2002. Supported Zr(Sc)O2 SOFCs for reduced temperature prepared by slurry coating and co-firing. Solid State Ionics 152/153, 583-590 Calise, F., Palombo, A., Vanoli, L., 2006. Desi gn and partial load energy analysis of hybrid SOFC-GT power plant. J. Power Sources 158, 225-244. Cha, S. W., OHayre, R., Lee, S. J., Saito, Y., Pr inza, F. B., 2004. Geometric scale effect of flow channels on performance of fuel cells J. Electrochem. Soc. 151, A1856-A1864 Chan, S. H., Khor, K. A., Xia, Z. T., 2001. A co mplete polarization model of a solid oxide fuel cell and its sensitivity to the change of ce ll component thickness. J. Power Sources 93, 130-140. Chan, S. H., Xia, Z. T., 2001. Anode micro model of solid oxide fuel cell. J. Electrochem. Soc. 148, A388-394. Chan, S. H., Xia, Z. T., 2004. Cathode micromodel of solid oxide fuel cell. J. Electrochem. Soc. 151, A164-172. Chen, X., Wu, N. J., Smith L., I gnatiev, A., 2004. Thin-film heterost ructure solid oxide fuel cells. Appl. Phys. Letters 84, 27002702. Chen, X. J., Chan, S. H, Khor, K. A., 2004. Simula tion of a composite cathode in solid oxide fuel cells. Electrochimica Acta 49, 1851-1861. Ciobanu, C., Liu, Y., Wang, Y., Patton, B. R., 1999. Numerical cal culation of electrical conductivity of porous electro ceramics. Journal of Electroceramics 3, 17-23. Clarke, S. H., Dicks, A. L., Pointon, K., Smith, T. A., Swann, A., 1997. Ca talystic aspects of the steam reforming of hydrocarbons in internal reforming fuel cells. Catalysis Today 38, 411-423. Constatinides, G. N., Payatakes, A. C., 1989. A three dimensional network model for consolidated porous media: basic stud ies. Chem. Eng. Comm. 81, 55-81. Costamagna, P., Costa, P., Antonucci, V., 1998. Micro-modelling of solid oxide fuel cell electrodes. Electrocheimica Acta 43, 375-394.
162 Damm, D. L., Fedorov, A. G., 2006. Reduced-o rder transient thermal modeling for SOFC heating and cooling. J. Power Sources 159, 956-967. Deseure, J., Bultel, Y., Dessemond, L., Siebert, E., 2005. Theoretical opt imization of a SOFC composite cathode. Electrochimica Acta 50, 2037-2046. Drescher, I., Lehnert, W., Meusinger, J., 1998. Structrural properties of SOFC anodes and reactivity. Electrochimica Acta 43, 3059-3068. Dicks, A. L., Pointon, K. D., Siddle, A., 2000. Instrinsic reaction ki netics of methane steam reforming on a nickel/zirconia a node. J. Power Sour. 86, 523-530. Dullen, F. A. L., 1973. Porous media fluid tran sport and pore structure. Academic Press, New York. Faghri, A., Guo, Z., 2005. Challenges and opportuniti es of thermal management issues related to fuel cell technology and modeling. In t. J. Heat and Mass 48, 3891-3920. Ferguson, J. R., Fiard, J. M., Herbin, R., 1996. Three-dimensional num erical simulation for various geometries of solid oxide fuel cells. J. Power Sources 58, 109-122. Fleig, J., 2003. Solid oxide fu el cell cathodes: Polarization m echanisms and modeling of the electrochemical performance. A nnual Rev. Mater. Res. 33, 361-382. Fukunaga, H., Ihara, M., Sakaki, K., Yamada, K., 1996. The relationshi p between overpotenial and the three phase bou ndary length. Solid St ate Ionics, 104, 1179-1185. Gemmen, R. S., Johnson, C. J., 2005. Effect of load transients on SO FC operation current reversal on loss of load. J. Power Sources 144, 152-164. Gorman, B. P., Anderson, H. U., 2005. Processing of composite thin film solid oxide fuel cell structures. J. Am. Ceram. Soc. 88, 1747-1753. Haberman, B. A., Young, J. B., 2004. Three-dime nsional simulation of chemically reacting gas flows in the porous support struct ure of an integrated-planar solid oxide fuel cell. Int. J. Heat and Mass Transfer 47, 3617-3629. Haberman, B. A., Young, J. B., 2005. Numerical investigation for the ai r flow through a bundle of IP-SOFC modules. Int. J. H eat and Mass Transfer 48, 5475-5487. Han, D. G., Choi, G. M., 1998. Co mputer simulation of the electri cal conductivity of composites: the effect of geometrical arrangem ent. Solid State Ionics 106, 71-87. Harvey, S. P., Richter, H. J., 1993. Improved Gas Turbine Power Plant efficiency by use of recycled exhaust gases and fuel cell tec hnology. ASME Heat Tran sfer Division 266, 199 207.
163 Hertz, J. L., Tuller, H. L., 2004. Electrochemical Characterization of thin films for a micro-solid oxide fuel cell. Electroceramics 13, 663-668. Hirano, A., Suzuki, M., Ippommatsu, M.,1992. Eval uation of a new solid oxide fuel cell system by non-isothermal modeling. J. El ectrochem. Soc. 139, 2744-2751. Hohenberg, P., Kohn, W., 1964. Inhomogeneous Electron Gas. Phys. Rev. 136, 864B. Hoshen, J., Kopelman, R., 1976. Percolation and cl uster distribution. I: cluster multiple labeling technique and critical con centration algorithm. Phys ical Review B 14, 3438-3445. Huang W., Shuk P., Greenblatt, M., 1997. Properties of sol-gel prepared 1-xx2-x/2CeSmO solid electrolytes. Solid St ate Ionics 100, 23-27. Hwang, J. J., Chen, C. K., Lai, D. Y., 2005. De tailed characteristic comparison between planar and MOLB-type SOFCs. J. Power Sources143, 75-83. 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 propert ies. Steady-state and dynamic analysis. Chemical E ngineering Science 60, 2963-2975. 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 distributions Solid State Ionics 132, 297-308. Jamnik, J., Maier, J., Pejovnik, S., 1999. A power ful electrical network model for the impedance of mixed conductors. Electrochim. Acta. 44, 4139-4145. Jeon, D. H., Nam, J. H., Kim, C. J., 2005. A random resistor networ k analysis on anodic performance enhancement of solid oxide fuel cells by penetrating electrolyte structures. J. Power Sources 139, 21-29. Jeon, D. H., Nam, J. H., Kim, C. J., 2006. Mi crostructural optimization of anode-supported solid oxide fuel cells by a comprehensive microscal e model. J. Electro. Society 153, A406-A417. Ji, Y., Yuan, K., Chung J. N., Ch en Y. C., 2006. Effects of trans port scale on heat/mass transfer and performance optimization for solid oxide fuel cells. J. Power Sources 161, 380391. Ji, Y., Yuan, K., Chung J. N., 2007. Monte-carl o simulation and performance optimization for the cathode microstructure in a solid oxide fuel cell. J. Power Sources, in press Jiang, S., Wanger, J. B., 1995. A theoretical m odel for composite electr olytes-II: percolation model for ionic conductivity enhancemen t. J. Phys. Chem. Solids 56, 1113-1124. Jiang, Y., Wang, S., Zhang, Y., Yan, J., Li, W., 1998. Electrochemical reduction of oxygen on a strontium doped lanthanum manganite elect rode. Solid State I onics110, 111-119. Kaviany, M., 1999. Principles of heat transf er in porous media. Springer, New York.
164 Keil, F. J., 1999. Diffusion and relation in porous network. Catalysis Today 53, 245-258. Kenjo, T., Osawa, S., Fujikawa, K., 1991. High temperature air ca thodes containing ion conductive oxides. J. Electrochem. Soc. 138, 349-356. Kenjo, T., Nishiya, M., 1992. LaMnO3 air cathodes containing ZrO2 electrolyte for high temperature solid oxide fuel cells Solid State Ionics 57, 295-302. Kikuchi, R., Yano, T., Takeguchi, T., Eguchi, K ., 2004. Characteristics of anodic polarization of solid oxide fuel cells under pressurized conditions. So lid State Ionics 174, 111-117. Kirkpatrick, S., 1973. Percolation and conduction. Rev. Mod. Phys. 45, 574-588. Kim, D. K., Kim, G. D., Moon, J. W., Lee, H. W., Li, K. T., Kim, C. E., 2000. The effect of percolation on electrochemical performa nce. Solid State Ionics 133, 67-77. Kim, J. W., Virkar, A. V., Fung, K. Z., Mehta, K., Singhal, S. C., 1999. Polarization effects in intermediate temperature, anode -supported solid oxide fuel cells J. Electrochem. Soc. 146, 69-78. Krishna, R., Wesselingh, J. A., 1997. The MaxwellStefan approach to mass transfer. Chemical Engineering Sciences 52, 861-911. Kuo, C., Gupta, P. K., 1995. Rigidity and conduc tivity percolation thresh olds in particulate composites. Acta. Metall. Mater. 43, 397-403. Landauer, R., 1978. Electrical conductivity in inhomogeneous media. Am erican Institute of Physics Conference Proceedings, No. 40, Elect rical Transport and Optical Properties of inhomogeneous Media. Edited by J. C. Garland and D B. Tann er. American Institute of Physics, New York, 2-45. Larminie, J., Dicks, A., 2003. Fuel cell systems explained. 3rd ed., JohnWiley & Sons Inc., Chichester, West Sussex. Lehnert, W., Meusinger, J., Thom, F., 2000. M odeling of gas transport phenomena in SOFC anodes. J. Power Sources 87, 57-63. Li, P. W., Chyu, M. K., 2003. Simulation of the chemical/electroch emical reactions and heat/mass transfer for a tubular SOFC in a stack. J. Power Sources 124, 487-498. Li, P. W., Schaefer, L., Chyu, M. K., 2004. A numerical model coupling the heat and gas species transport pro cesses in a tubular SO FC. ASME J. of Heat Transfer 126, 219-229. Li, P. W., Chen, S. P., Chyu, M. K., 2005. N ovel gas distributors and optimization for high power density in fuel cells. J. Power Sources 140, 311-318. Liu, J., Co, A. C., Paulson, S., Birss, V. I., 2006. Oxygen reduction at sol-gel derived 0.80.20.80.23LaSrCoFeO cathodes. Solid State Ionics 177, 377-387.
165 Loselevich, A., Kornyshev, A. A., Lehnert, W., 1997. Degradation of solid oxide fuel cell anodes due to sintering of metal particles: correlated percolation model. J. Electrochem. Soc.144, 3010-3019. Lu, C. Y., Maloney, T. M., 1988. Abstract: Mathem atical modeling of solid oxide fuel cells. Fuel Cell Seminar, 78. Lu, Y., Schaefer, L., Li, P. W., 2005. Numerical study of a flat-tube high power density solid oxide fuel cell: Part I. heat/mass transfer a nd fluid flow. J. Power Sources140, 331-339. Lu, Y., Schaefer, L., Li, P. W., 2006. Numerical study of a flat-tube high power density solid oxide fuel cell: Part II. cell performance and stack optimization. J. Power Sources 153, 6875. Lux, F., 1993. Models proposed to explain the el ectrical conductivity of mixtures made of conductive and insula ting materials. J. Materi al Science 28, 285-301. Mclachlan, D. S., 1986. Equations for conductivity of macroscopi c mixtures. J. Phys. C: Solid State Phy.19, 1339-1354. Mclachlan, D. S., Blaszkiewicz, M, Newhanm, R. E., 1990. Electrica l resistivity of composites. J. Am. Ceram. Soc. 73, 2187-2203. Massardo, A. F., Lubelli, F., 1998. Internal reform ing solid oxide fuel cell-gas turbine combined cycles (IRSOFC-GT), Part A: cell model and cycle thermodynamic analysis. ASME paper 98-GT-577. Meredith, R. E., Tobias, C. W., 1962. Conductio n in heterogeneous systems. Advances in Electrochemistry and Electroc hemical Engineering, Vol 2. Edited by C. W. Tobias. Interscience, New York, 15-47. Meusinger, J., Riensche, E., Stimming, U., 1998. Reforming of natural gas in solid oxide fuel cell systems. J. Power Sources 71, 315-320. Mizusaki, J., Amanno, K., Yamauchi, S., Fu eki, K., 1987. Electr ode reaction at Pt, O2(g)/stabilized zirconia interf aces. Part I: theoretical cons ideration of reaction model. Chemical Engineering Science 22, 313-322. Mills, A.F., 2001, Mass tr ansfer, Prentice Hall. Mitsos, A., Hencke, M. M., Barton, P. I., 2005. Product engineering for man-portable power generation based on fuel cells AIChE Journal 51, 2199-2219. Mitsos, A., Palou, R. I.., Barton, P. I., 2004. Alternatives for micropower generation processes.Industrial and Engineerin g Chemistry Research. 43, 74-84. Murthy, S., Fedorov, A. G., 2003.Ra diation heat transfer analysis of the monolith type solid oxide fuel cell. J. Power Sources 86, 453-458.
166 Noren, D. A., Hoffman, M. A., 2005. Clarif ying the Butler-Volmer equation and related approximation for calculating activation losses in solid oxide fuel cell models. J. Power Sources 152, 175-181. Odgaard, M., Skou, E., 1996. SO FC cathode kinetics investigated by the use of cone shaped electrodes: the effect of polar ization and mechanical load. Solid State Ionics 86-88, 12171222. Palsson, J., Selimovic, A., Sjunnesson, L., 2000. Combined solid oxide fuel cell and gas turbine systems for efficient power and heat genera tion. J. Power Sources 86, 442-448. Patankar, S. V., 1980. Numerical he at transfer and fluid flow. Hemisphere Press, New York. Ramakrishna, P. A., Yang, S., Sohn, C. H ., 2006. Innovative design to improve the power density of a solid oxide fuel cell. J. Power Sources 158, 378-384. Recknagle, K. P., Williford, R. E., Chick, L. A., Khaleel, M. A., 2003. Three-dimensional thermo-fluid electrochemical modeling of planar SOFC stack s. J. Power Sources 113, 109-114. Sahimi, M., Tsotsis, T. T., 1985. A percolation model of catalyst deactivation by site coverage and pore blockage. Journal of catalysis 96, 552-562. Sasaki, K., Wurth, J. P., Gschwend, R., Godick emeier, M.,Gaucker, L. J., 1996. Microstructure property relations of solid oxide fuel cell cathodes a nd current collectors. J. Electrochem. Soc. 143, 530-543. Shao, Z., Haile, S. M., Ahn, J., Ronney, P. D., Zhan, Z., Barnett, S. A., 2005. A thermally selfsustained micro solid oxide fuel-cell stack with high power density. Nature 435, 795-798. Singhal, S. C., Kendall, K., 2003. High temperature solid oxide fu el cells. Elsevier, New York. Stauffer, D., 1985. Introduction to percolati on theory. Taylor and Francis, London, U.K. Stephenson, D., Ritchey, I., 1997. Parametric study of fuel cell and gas turbine combined cycle performance. ASME Paper 97-GT-340. Stiller, C., Thorud, B., Seljebo, S., Mathisen, Q., 2005. Finite-v olume modeling and hybrid-cycle performance of planar and tubular solid oxide fuel cells. J. Power Sources 1041, 227-240. Sunde, S., 2000. Simulation of composite electrodes in fuel cells. J. Electrocrramics 5, 153-182. Sunde, S., 1996a. Monte Carlo simulations of co nductivity of composite electrodes for solid oxide fuel cells. J. Electrochem. Soc. 143, 1123-1132. Sunde, S., 1996b. Monte Carlo simulations of polar ization resistance of co mposite electrodes for solid oxide fuel cells. J. Electrochem. Soc.143, 1930-1938.
167 Suzuki, M., Oshima, T., 1983. Estimation of th e coordination number in a multicomponent mixture of spheres. Power Technology 35, 159-166. Svensson, A. M., Nisancioglu, K., 1998. Current distribution at porous electrode-solid oxide electrolyte interface. J. Electrochem. Soc.145, 3130-3135. Tanner, C. V., Fung, K. Z., Virkar, A. V., 1997. The effect of porous composite electrode structure on solid oxide fuel cell performance, I: theoretical an alysis. J. Electrochem. Soc. 144, 21-40. Tang, Y., Stanley, K., Wu, J., Ghosh, D., Zha ng, J., 2005. Design consid eration of micro thin film solid-oxide fuel cells. J. Micromech. Microeng. 15, S185-S192. Tsai, T., Barnett, S. A., 1997. Increase solid-oxide fuel cell power density using interfacial ceria layers. Solid State Ionics 98, 191-196. Van Heuveln, F. H., Bouwmeester, H. J. M., Van Berkel, F. P. F., 1997. Electrode properties of Sr-doped LaMnO3 on Yttria-stabilized zirconia, Part I: three-phase boundary area. J. Electrochem. Soc. 144, 126-133. Van Heuveln, F. H., Bouwmeester, H. J. M ., 1997. Electrode properties of Sr-doped LaMnO3 on Yttria-stabilized zirconi a, Part II: electrode kinetics. J. Electrochem. Soc. 144, 134-140. Veyo, S. E., Forbes, C. A., 1998. Demonstr ations based on Westinghouses prototype commercial AES design. Proceedings of the Th ird European Solid Oxide Fuel Cell Forum, 79-86. Veyo, S. E., Lundberg, W. L., 1999. Solid Oxide Fuel Cell Power system cycle. ASME Paper 99-GT-356. Virkar, A. V. Chen, J., Tanner, C. W., Kim, J. W., 2000. The role of electrode microstructure on activation and concentration pol arizations in solid oxide fuel cells. Solid State Ionics 131, 189-198. Wang, S., Ando, M., Ishihara, T., Takita, Y., 2004. High performance 0.150.852-Ni-SmCeO cermet anodes for intermediate temperatur e solid oxide fuel cells using LaGaO3 based oxide electrolytes. Solid State Ionics 174, 49-55. Watanabe, M., H. Uchida and M. Yoshida, 1997 Effect of ionic conductivity of zirconia electrolytes on the polari zation behavior of ceria-based anode s in solid oxide fuel cells. J. Electrochem. Soc.144, 1739-1743. Weber, A., Tiffee, E. I., 2004. Materials and co ncepts for solid oxide fuel cells (SOFCs) in stationary and mobile applications J. Power Sources 127, pp. 273-283. Winterfeld, P. H., Scriven, L. E., Davis, H. T., 1981. Percolation a nd conductivity of random two-dimensional composites. J. Phys C: Solid state Phys.14, 2361-2376.
168 Wolfgang, G. B., 2005. A new co mputational approach for SOFC impedance from detailed electrochemical reactiondiffusion models. Solid State Ionics176, 997-1011. Wong, P., Koplik, J., Tomanic, J. P., 1984. C onductivity and permeability of rocks. Physical Review B 30, 6606-6614. Yakabe, H., Ogiwara, T., Hishinuma, M., Yas uda, I., 2001. 3-D model calculation for planar SOFC. J. Power Sources 102, 144-154. Yakabe, H., Sakurai, K., 2004. 3D simulation on the current path in planar SOFCs. Solid State Ionics 174, 295-302. Yuan, J. L., Rokni, M., Sunden, B., 2001. Simula tion of fully developed laminar heat and mass transfer in fuel cell ducts with different cros s-sections. Int. J. Heat and Mass Transfer 44, 4047-4058. Zalc, J. M., Reyes, S. C., Iglesia, E., 2003. Monte-Carlo simulations of surface and gas phase diffusion in complex porous structur es. Chem. Eng. Sci. 58, 4605-4617. Zallen, R., 1983. The physics of amorphous solids. Wiley, New York. Zhan, Z., Wen, T., Tu, H., Lu, Z., 2001. AC impe dance investigation of samarium-doped ceria. J. Electrochem. Soc. 148, A427-A432. Zhang, X., Robertson, M., Yick, S ., Deces-Petit, C.,Styles, E., Qu, W., Xie, Y., Hui, R., Roller, J., Kesler, O., Maric, R., Ghosh, D., 2006.0.50.530.20.81.9SmSrCoOSmCeO composite cathode for cermet-supported thin 0.20.81.9SmCeO electrolyte SOFC operating below o600C. J. Power Sources 160, 1211-1216.
169 BIOGRAPHICAL SKETCH Yan Ji was born in Shannxi, China. After rece iving her Bachelor of Engineering in Airconditioning and Refrigerat ion from Xian Jiaotong University of China, she received Master of Science in Cryogenic Engineering and Refrigeration from Chinese Academy of Sciences in 2002. In pursuit of a Ph.D. degree in mechanical engine ering, Yan Ji began her study at the University of Florida in 2004.