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(1)AGH University of Science and Technology D e p ar t m e n t o f Fu n d a m e n t a l R e s e ar c h i n E n e r g y E n g i n e e r i n g PhD Thesis. Three-dimensional numerical analysis of transport phenomena in a Positive-Electrolyte-Negative assembly of a Solid Oxide Fuel Cell Tomasz Prokop, MSc.. Supervisors: prof. Janusz S. Szmyd. prof. Shinji Kimijima. Auxiliary Supervisor: dr. Grzegorz Brus, associate professor. Kraków, 2020.

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(3) Acknowledgements. I’m most grateful for the guidance of my Supervisors, Professor Shinji Kimijima from Shibaura Institute of Technology in Tokyo, and Professor Janusz Szmyd from AGH University of Science and Technology in Krakow. Their immeasurable support, the accurate suggestions, and wide expertise made my research possible. I would like to further thank Professor Kimijima for the hospitality and care I received during my stay at Power and Energy Systems Laboratory of Shibaura Institute of Technology. I cannot downplay the input of my auxiliary supervisor at AGH, Doctor Grzegorz Brus (Associate Professor at AGH), who has provided me with guidance and instructions since the time I was undertaking a master’s degree, and who managed most of the research projects, in which I was involved. I would like to acknowledge Professor Naoki Shikazono, who kindly accepted me in his research group at the University of Tokyo’s Institute of Industrial Science. My stay at Professor Shikazono lab, as well as his highly professional and benign guidance, allowed me to obtain irreplaceable, and defining experience in this crucial stage of my scientific career. I would also like to thank the members of Professor Shikazono’s lab who helped me during my stay: Dr. Anna ´ a˛z˙ ko, Dr. Yosuke Komatsu, Dr. An He, Professor Junya Onishi, and others. Sci I am grateful for the input of Dr. Marcin Mozdzierz who provided data such as stack performance experiment, and reconstructions of microstructures. I would also like to express my gratitude for Dr. Mozdzierz’s enlightening advice and reliable support. Furthermore, I acknowledge the contribution of Dr. Katarzyna Berent, who performed expert nanotomographic studies, without which none of the publications we co-authored would be possible. The research presented in this work was conducted under the Joint Doctoral Diploma Program between Shibaura Institute of Technology in Japan and AGH University of Science and Technology in Poland. Therefore I would like to acknowledge Professor Akito Takasaki, former Director of the Center for International Programs and current Dean of Graduate School of Engineering and Science at SIT for laying the foundations for fruitful partnership between the two institutions. i.

(4) ii. I would like to express sincere gratitude for the financial support, which made this work possible. The research and the dissemination of results was aided by the National Science Centre of Poland as part of the project "A three dimensional analysis of local microstructure evolution in a Solid Oxide Fuel Cell stack" (SONATA-10, Grant No. UMO-2015/19/D/ST8/00839). Elements of the presented research were performed during ’Easy-to-assemble Stack Type (EAST): Development of solid oxide fuel cell stack for innovation in polish energy sector’ project, carried out within the FIRST TEAM programme (project number First TEAM/2016-1/3) of the Foundation for Polish Science, co-financed by the European Union under the European Regional Development Fund. The samples discussed in this thesis were investigated at AGH University of Science and Technology’s Academic Centre for Materials and Nanotechnology. The necessary computing resources were provided by PL-Grid Infrastructure.. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(5) Contents. Abstract.................................................................................................................................. vii Streszczenie............................................................................................................................ ix Nomenclature ........................................................................................................................ xi List of abbreviations ............................................................................................................. xv List of figures ......................................................................................................................... xvii List of Tables..........................................................................................................................xxxi 1. Introduction...................................................................................................................... 1. 1.1. Microstructure-scale models..................................................................................... 5. 1.1.1. Continuous electrode models....................................................................... 8. 1.1.2. Non-continuous phase distribution models.................................................. 9. 1.1.3. Mass conservation in SOFC electrodes ...................................................... 13 1.2. Positive-Electrolyte-Negative Assembly models..................................................... 14 1.3. Chapter conclusions................................................................................................. 15 2. Aim and scope ................................................................................................................. 17 3. Mathematical model ....................................................................................................... 19 3.1. Conservation equations............................................................................................ 19 3.1.1. Conservation of charge ............................................................................... 21 3.1.2. Conservation of mass .................................................................................. 22 3.1.3. Reaction current .......................................................................................... 25 3.2. Overpotential ........................................................................................................... 27 3.3. Boundary conditions and case assumptions ............................................................ 28 3.4. Performance parameters .......................................................................................... 34 3.5. Chapter conclusions................................................................................................. 35 4. Numerical Model............................................................................................................. 39 4.1. Mesh construction.................................................................................................... 39 iii.

(6) iv. CONTENTS. 4.2. Microstructure quantification................................................................................... 39 4.2.1. Local pore diameter .................................................................................... 39 4.2.2. Triple Phase Boundary density quantification ............................................ 45 4.2.3. Double Phase Boundary density quantification .......................................... 46 4.2.4. Tortuosity quantification ............................................................................. 47 4.3. Discretization of model equations ........................................................................... 48 4.3.1. Discretization of charge conservation equations ........................................ 49 4.3.2. Discretization of mass conservation equations ........................................... 49 4.3.3. Discretization of the reaction current term ................................................. 49 4.3.4. Final form of the coefficient matrix ............................................................ 52 4.4. Solution scheme....................................................................................................... 54 4.5. Error analysis ........................................................................................................... 54 4.6. Falsification attempts .............................................................................................. 57 4.6.1. Approximate analytical solution ................................................................. 57 4.6.2. Comparison to the analytical solution......................................................... 60 4.6.3. Comparison to a continuous model............................................................. 62 4.6.4. Mesh resolution sensitivity study................................................................ 62 4.7. Chapter conclusions................................................................................................. 65 5. Model validation.............................................................................................................. 67 5.1. Input data acquisition............................................................................................... 67 5.2. Chapter conclusions................................................................................................. 68 6. Microstructure-scale parametric study ........................................................................ 71 6.1. Anode....................................................................................................................... 71 6.1.1. Overpotential decomposition ...................................................................... 73 6.1.2. Desired active layer thickness..................................................................... 73 6.2. Cathode .................................................................................................................... 91 6.2.1. Overpotential decomposition ...................................................................... 91 6.3. Positive-Electrolyte-Negative assembly .................................................................. 105 6.4. Chapter conclusions................................................................................................. 116 7. Heterogeneity impact study ........................................................................................... 117 7.1. Chapter conclusions................................................................................................. 122 8. Cross-electrolyte phenomena ......................................................................................... 129 T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(7) CONTENTS. v. 8.1. Conclusions.............................................................................................................. 143 9. Interpretation of an Aging Experiment ........................................................................ 145 9.1. Input data acquisition............................................................................................... 145 9.2. Simulation results .................................................................................................... 147 9.3. Chapter conclusions................................................................................................. 148 10. Summary and conclusions.............................................................................................. 155 10.1. General remarks....................................................................................................... 155 10.2. Suggestions for future work..................................................................................... 159 Bibliography .......................................................................................................................... 161 Appendices............................................................................................................................. 175 A. Material and gas properties ........................................................................................... 177 A.1. Nickel-Yttrium Stabilized Zirconia Anode ............................................................. 177 A.2. Lanthanium Strontium Cobalt Ferrite-Gallium Doped Ceria Cathode.................... 177 A.3. Gas properties .......................................................................................................... 178. T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(8) vi. T. Prokop. CONTENTS. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(9) Abstract. In this dissertation, a three-dimensional, microstructure-scale model of a Solid Oxide Fuel Cell Positive-Electrolyte-Negative assembly is constructed, validated and applied to a series of research problems. Empirical data from open literature are used to compute conductivities and exchange current densities for anodic and cathodic materials. Butler-Volmer model is applied to compute local reaction rates. the free-molecular, and the continuum-flow diffusion are accounted for using Cylindrical Pore Interpolation Model. The model equations are discretized using the Finite Volume Method, and solved using the Successive Over-Relaxation method with local linearization. The computational domain is based on digital microstructure reconstructions obtained using the Focused Ion Beam Scanning Electron Microscopy. The methods are implemented in an in-house code written in C++. The model is validated against empirical data from a commercial Solid Oxide Fuel Cell stack. Simulations at different operation parameters are performed, the impact of microstructure inhomogeneities is assessed, and transport in thin-electrolyte cells is studied. The model is applied to explain unusual results of a long term operation experiment, in which stack performance enhancement, rather than deterioration was observed. The decrease of pore tortuosity due to microstructure evolution was identified as the cause.. vii.

(10) viii. T. Prokop. CONTENTS. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(11) Streszczenie. W niniejszej rozprawie opisana jest konstrukcja, walidacja i szereg aplikacji trójwymiarowego, mikroskalowego modelu układu anoda-elektrolit-katoda ogniwa paliwowego z elektrolitem stałotlenkowym (SOFC). Wła´sciwo´sci elektrochemiczne modelowanych materiałów anodowych i katodowych oparto o dane eksperymentalne z otwartej literatury. Do obliczenia lokalnych szybko´sci reakcji stosowany jest model Butlera-Volmera. Transport dyfuzyjny w porach o s´rednicy zbliz˙ onej do s´redniej drogi swobodnej czastek ˛ gazu opisany jest za pomoca˛ Modelu Interpolacyjnego Cylindrycznych Porów (CPIM). Dyskretyzacja modelu odbywa si˛e za pomoca˛ Metody Obj˛eto´sci Sko´nczonych, uzyskane równania rozwiazywane ˛ sa˛ za pomoca˛ metody Sukcesywnej Nadrelaksacji z lokalna˛ linearyzacja.˛ Domena obliczeniowa oparta jest na cyfrowych rekonstrukcjach z nanotomografii FIB-SEM. Metody numeryczne zaimplementowano w autorskim kodzie C++. Model został poddany walidacji poprzez porównanie wyników z danymi eksperymentalnymi z komercyjnego stosu ogniw SOFC. Model posłuz˙ ył do przeprowadzenia studium parametrycznego, oceny wpływu niejednorodno´sci mikrostruktury i analizy transportu w ogniwach o cienkim elektrolicie. Zastosowany został równiez˙ do wyja´sniania nietypowych wyników eksperymentu starzeniowego, w którym zaobserwowano polepszenie pracy ogniwa. Ustalono, z˙ e przyczyna˛ był spadek kr˛eto´sci porów na skutek ewolucji mikrostruktury.. ix.

(12) x. T. Prokop. CONTENTS. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(13) Nomenclature. A. area density, in (m2 m−3 ). A. t x t matrix of coefficients. a. Finite Volume Method coefficient. b. column vector of constant terms. C. any constant. c. molar concentration, in (mol m−3 ). D. mass diffusivity, in (m2 s−1 ). F. Faraday constant, F = NA e, in (C mol−1 ), F = 96 485.332 89 C mol−1. I. source term. i. volumetric current density, in (A m−3 ). J. flux vector, in (mol s−1 ). j L. current density (mean charge transfer rate), in (A2 m−1 ) p Momentum transfer model parameter, in ( kg/mol/s). l. length density, in (m m−3 ). M. molar mass, in (kg mol−1 ). N. total number. n. quantity. p. partial pressure, in (Pa). R. universal gas constant, in (J mol−1 K−1 ), R = 8.314 459 8 J mol−1 K−1. r. vector of residuals. T. temperature, in (K). V. volume, in (m3 ). v. component diffusion volume. x. column vector of variables. K. length, in (m). xi.

(14) xii. Nomenclature. x, y, z. spatial dimensions, in (m). X. Molar fraction. Greek symbols. α. charge transfer coefficient. β. linear equation coefficient. Γ. linear equation coefficient. δ. oxygen non-stoichiometry, in (m). ∆. between two adjacent control volumes’ faces, in (m). ε. relative active layer thickness. η. overpotential, in (V). κ. Wilke model expression. µ. dynamic viscosity, in (kg m−1 s−1 ). µˆ. molar chemical potential, in (J mol−1 ). π. Archimedes’ constant, π = 3.141 592 653 5. ρ. density, in (kg m−3 ). σ. conductivity, in (S m−1 ). τ. tortuosity factor. φ. electric potential, in (V). Φ. electric potential of a phase, in (V). Ψ. molar volume. ψ. volume fraction. ω. relaxation coefficient. Subscripts act. activation. ano. anode. avg. average. b. boundary. bcw. backward reaction. cath. cathode. conc. concentration. CV. control volume. D. center of the lower control volume. dpb. double phase boundary. E. center of the east control volume. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(15) xiii. Nomenclature. e. interface between control volumes P and E. eff. effective. el. electron conducting. frw. forward reaction. H2. hydrogen. H2O. water. i, j, k. general indices. ion. ions conducting phase. K. Knudsen diffusion. mesh. mesh. N. center of the north control volume. n. interface between control volumes P and N. num. numerical. O2. oxygen. ohm. ohmic. P. center of the central control volume. W, E, N, S, U, D center of the control volume located: "West", "East", "North, "South", "Over", "Below" the central control volume pore. pore phase. S. center of the south control volume. s. interface between control volumes P and S. str. straight path. tpb. triple phase boundary. U. center of the upper control volume. W. center of the west control volume. w. interface between control volumes P and W. Superscripts ano. anode. bulk. bulk property. cath. cathode. diff. diffusion. dpb. adjacent to double phase boundary. eff. effective. el. electron conducting phase. eq. equilibrium T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(16) xiv. Nomenclature. ion. ions conducting phase. iter. iteration. long. extention. mesh. mesh. previous. previous value. tpb. adjacent to triple phase boundary. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(17) List of abbreviations. CCD. charge-coupled device. CFD. computational fluid dynamics. CHP. combined heat and power. CPIM. cyllindrical pore interpolation model. CT. computer tomography. CV. control volume. DGM. dusty gas model. DPB. double phase boundary. FDM. finite difference method. FEG-SEM. field emision gun scanning electron microscopy. FIB. focused ion beam. FIB-SEM. focused ion beam scanning electron microscope. FM. Fick model. FSG. Fuller-Schettler-Giddings correlation. FVM. finite volume method. GDC. gadolinia-doped ceria. HHV. higher heating value. HOR. hydrogen oxidation reaction. LHV. lower heating value. LSCF. lanthanum strontium cobalt ferrite. LSM. lanthanum strontium manganite. LBM. lattice-Boltzmann method. MIEC. mixed ionic-electronic conductor xv.

(18) xvi. List of abbreviations. OCV. open circuit voltage. ORR. oxygen reduction reaction. PEN. positive-electrolyte-negative assembly. PFIB-SEM. xenon plasma focused ion beam. RAM. random access memory. RN, ResNet. resistor network. SEM. scanning electron microscope. SMM. Stefan-Maxwell model. SOEC. solid oxide electrolyzer cell. SOFC. solid oxide fuel cell. SOFC-GT. solid oxide fuel cell - gas turbine. TPB. triple phase boundary. X-Ray-CT. X-ray computer tomography. YSZ. yttria-stabilized zirconia. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(19) List of figures 1.1. Principle of a hydrogen fuel cell. . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. A thermal system equivalent to a fuel cell. . . . . . . . . . . . . . . . . . . . .. 2. 1.3. Hydrogen-oxygen redox reaction in a Solid Oxide Fuel Cell. . . . . . . . . . .. 6. 1.4. The continuous electrode theory approach (a) and non-continuous phase distribution approach (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 3.1. The computational domain in context of a typical SOFC device . . . . . . . . .. 20. 3.2. Applicability of the charge conservation equation - ion and electron conducting phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 3.3. Applicability charge conservation equation - gas diffusion . . . . . . . . . . .. 22. 3.4. Applicability of the reaction current term . . . . . . . . . . . . . . . . . . . .. 26. 3.5. Restrictions on interphase charge transfer . . . . . . . . . . . . . . . . . . . .. 30. 3.6. The integration domain for calculating total overpotential from local values at Triple Phase Boundaries (TPB), and Double Phase Boundaries (DPB) . . . . .. 34. 3.7. Applicability of model equations in context of the computational domain . . . .. 37. 4.1. The process of microstructure reconstruction based on Focused Ion Beam Scanning Electron Microscopy (FIB-SEM) tomography . . . . . . . . . . . . . . .. 4.2. 40. Example numerical mesh corresponding to the YSZ phase in an anodic microstructure (10 µm × 1.6 µm × 1.6 µm) . . . . . . . . . . . . . . . . . . . . . .. 41. 4.3. Algorithm for computing the distance between a mesh node and phase interface. 42. 4.4. Algorithm for computing the local value of micro-channel (e.g. pore) radius . .. 43. 4.5. Method of computing the local value of micro-channel (e.g. pore) - fragment of a three-dimensional cross section of a mesh based on irregular geometry . . . .. 44. 4.6. Mesh resolution sensitivity in the case of the cathodic TPB . . . . . . . . . . .. 45. 4.7. Volume Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 4.8. Marching Cube Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 4.9. Random Walk Simulation used to compute tortuosity . . . . . . . . . . . . . .. 47. xvii.

(20) xviii. LIST OF FIGURES. 4.10 Set of equations with the seven-diagonal matrix of coefficients . . . . . . . . .. 50. 4.11 Local node notation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 4.12 Averaged, absolute, relative values of residuals in the cross sections of an SOFC anode for an example set of results. Neumann boundary condition. Simulation parameters: Overpotential: j = 3040 A m−2 , p = 100 000 Pa, pH2 O = 97 000 Pa, pH2 O = 3000 Pa T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. 4.13 Averaged, absolute, relative values of residuals in the cross sections of a PositiveElectrolyte-Negative assembly for an example set of results. Simulation parameters: Overpotential: 0.125 V, p = 100 000 Pa, pH2 O = 20 000 Pa, pO2 = 23 000 Pa T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. 4.14 Numerical model verification. Ionic potential and exchange current density fields. Reference (Ref) parameters and boundary conditions: φel,b = 0.05 V, p = 100000 Pa, pH2 O = 3000 Pa, pH2 = p − pH2 O , thickness: 62.6 µm, T = 1073 K, applied in all the cases unless specified otherwise. . . . . . . . . . . . . . . . .. 61. 4.15 Comparison to a one-dimensional, continuous model. Ionic potential and exchange current density fields. Simulation parameters: j = 2000 A m−2 , p = 100 000 Pa, pH2 O = 60 000 Pa, thickness: 250 µm, T = 1023 K. . . . . . . . . .. 62. 4.16 Current density distributions in the microstructure of an anode computed for meshes of varying resolutions. Simulation parameters: j = 2000 A m−2 , p = 100 000 Pa, pH2 O = 60 000 Pa, T = 1023 K. . . . . . . . . . . . . . . . . . . .. 64. 4.17 Current density distributions in the microstructure of a positive-electrolytenegative assembly computed for meshes of varying resolutions. Simulation parameters: Overpotential: 0.05 V, p = 100 000 Pa, pH2 O = 60 000 Pa, pO2 = 23 000 Pa T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. 64. A comparison of experimental and simulated Current-Voltage (IV) curves for different temperatures. Operation parameters: p = 100000 Pa, pH2 = 60000Pa, pN2 = p − pH2. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. A comparison of experimental and simulated Current-Voltage (IV) curves for different partial pressures. Operation parameters: T = 1023 K . . . . . . . . .. 6.1. 69. An example 3D reconstruction of a Ni-YSZ-GDC-LSCF positive-electrolytenegative assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. T. Prokop. 69. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell. 72.

(21) LIST OF FIGURES. 6.2. xix. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=100 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3. 74. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=300 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4. 74. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.5. 75. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.6. 75. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.7. 76. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=3000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.8. 76. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=100 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.9. 77. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=300 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . .. 77. T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(22) xx. LIST OF FIGURES. 6.10 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . .. 78. 6.11 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78. 6.12 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 6.13 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=3000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 6.14 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=100 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80. 6.15 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=300 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80. 6.16 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 6.17 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell. 81.

(23) LIST OF FIGURES. xxi. 6.18 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 6.19 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=3000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 6.20 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=100 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83. 6.21 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=300 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83. 6.22 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84. 6.23 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84. 6.24 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85. 6.25 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=3000 A m−2 p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85. 6.26 Overpotential decomposition. Reference cell. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 923 K. . . . . . . . . . .. 86. T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(24) xxii. LIST OF FIGURES. 6.27 Overpotential decomposition. Reference cell anode. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 973 K. . . . .. 86. 6.28 Overpotential decomposition. Reference cell anode. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1023 K. . . . 6.29 Overpotential. decomposition.. Reference. cell. anode.. 87. Simulation. parameters:p=100000 Pa, pH2 = 60000 Pa, electrode thickness: 240 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87. 6.30 Potential of the ion-conducting phase simulated for electrodes of different thicknesses. Simulation parameters: T = 1023 K Total pressure: p=100000 Pa, pH2 = 60000 Pa.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88. 6.31 Anodic voltage losses at different electrode thicknesses. Simulation parameters: T = 973 K Total pressure: p=100000 Pa, pH2 = 60000 Pa. . . . . . . . . . . .. 89. 6.32 Anodic voltage losses at different electrode thicknesses. Simulation parameters: T = 1023 K Total pressure: p=100000 Pa, pH2 = 60000 Pa.. . . . . . . . . . .. 89. 6.33 Anodic voltage losses at different electrode thicknesses. Simulation parameters: T = 1073 K Total pressure: p=100000 Pa, pH2 = 60000 Pa.. . . . . . . . . . .. 90. 6.34 Best performing simulated anode thickness at various operating temperatures. Simulation parameters: Total pressure: p=100000 Pa, j = 100 A m−2 , pH2 = 60000 Pa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90. 6.35 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=100 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92. 6.36 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=300 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92. 6.37 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=500 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell. 93.

(25) LIST OF FIGURES. xxiii. 6.38 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=1000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93. 6.39 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=2000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94. 6.40 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=3000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94. 6.41 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=100 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 6.42 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=300 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 6.43 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=500 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96. 6.44 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=1000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96. 6.45 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=2000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(26) xxiv. LIST OF FIGURES. 6.46 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=3000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 6.47 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=100 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98. 6.48 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=300 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98. 6.49 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=500 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99. 6.50 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=1000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99. 6.51 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=2000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.52 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=3000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.53 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=100 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(27) xxv. LIST OF FIGURES. 6.54 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=300 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.55 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=500 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.56 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=1000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.57 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=2000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.58 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the cathode. Reference cell cathode. Simulation parameters: j=3000 A m−2 , p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.59 Overpotential decomposition. Reference cell cathode. Simulation parameters: p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 923 K. . . . . 104 6.60 Overpotential decomposition. Reference cell cathode. Simulation parameters: p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 973 K. . . . . 104 6.61 Overpotential decomposition. Reference cell cathode. Simulation parameters: p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1023 K. . . . . 104 6.62 Overpotential. decomposition.. Reference. cell. cathode.. Simulation. parameters:p=100000 Pa, pO2 = 21000 Pa, electrode thickness: 55 µm, T = 1073 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.63 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa T = 923 K. . . . . . . . . . . 106 6.64 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pO2 = 21000 Pa, T = 923 K. . . . . . . . . . . . . . . . . . . . 107 T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(28) xxvi. LIST OF FIGURES. 6.65 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 973 K. . . . . . . . . . . 108 6.66 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 973 K. . . . . . . . . . . 109 6.67 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 1023 K. . . . . . . . . . 110 6.68 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 1023 K. . . . . . . . . . 111 6.69 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=500 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 1073 K. . . . . . . . . . 112 6.70 Current density and ion-conducting phases’ potential in a Positive-ElectrolyteNegative assembly. Reference cell. Simulation parameters: j=1000 A m−2 p=100000 Pa, pO2 = 21000 Pa, pH2 = 60000 Pa, T = 1073 K. . . . . . . . . . 113 6.71 Ion-conducting phase potential and ionic current density in a Ni-YSZ-GDCLSCF positive-electrolyte-negative assembly of an SOFC - a three dimensional visualization. Case parameters: j=1000 A m−2 T = 1073 K, pO2 = 23000 Pa, pN2 = 77000Pa, Anode thickness: 40 µm, Cathode thickness: 13 µm (including a 3 µm GDC separative layer), Electrolyte thickness: 10 µm. . . . . . . . . . . . 114 6.72 Ion-conducting phase potential and ionic current density in a Ni-YSZ-GDCLSCF positive-electrolyte-negative assembly of an SOFC - a three dimensional visualization. Case parameters: j=2000 A m−2 T = 1073 K, pO2 = 23000 Pa, pN2 = 77000Pa, Anode thickness: 50 µm, Cathode thickness: 40 µm (including a 3 µm GDC separative layer), Electrolyte thickness: 10 µm. . . . . . . . . . . . 115 7.1. Parametric study cases - type, size and placement of the virtual disturbance within the microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118. 7.2. Ionic potential and current densities in a homogeneous anode. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: a) 30 µm b) 60 µm. T = 1073 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(29) LIST OF FIGURES. 7.3. xxvii. Ionic potential and exchange current density fields in a homogeneous anode. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: a) 30 µm b) 60 µm. T = 1073 K . . . . . . . . . . . . . . . . . . . . . . . 122. 7.4. "Center-type" disturbance parametric study - Ionic potential and exchange current density fields. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: 30.0 µm. T = 1073 K . . . . . . . . . . . . . . . . 123. 7.5. "Corner-type" disturbance parametric study - Ionic potential and exchange current density fields. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: 30.0 µm. T = 1073 K . . . . . . . . . . . . . . . . 124. 7.6. "Corner-type", ion-conducting (YSZ) disturbance parametric study. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: 30.0 µm. T = 1073 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. 7.7. "Corner-type", ion-conducting (YSZ) disturbance parametric study - charge transfer rate. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: 30.0 µm. T = 1073 K . . . . . . . . . . . . . . . . . . . . 126. 7.8. "Corner-type" non-conducting disturbance parametric study - 3D visualization of ionic potential and exchange current density fields for cases for r = 7.5 µm cases: a) h = 0 µm b) h = 3.75 µm c) h = 7.5 µm. Boundary conditions: φel,b = 0.05 V, pH2 = 97000 Pa, pH2 O = 3000 Pa. Thickness: 30.0 µm. T = 1073 K . . 127. 8.1. Total cell overpotential. Simulation parameters: p=100000 Pa, pH2 = 20000 Pa, pO2 = 21000 Pa, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . 130. 8.2. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 10 µm. . . . . . . . . . . 131. 8.3. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 5 µm. . . . . . . . . . . . 131. 8.4. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 2.5 µm. . . . . . . . . . . 132. 8.5. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 1 µm. . . . . . . . . . . . 132 T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(30) xxviii. 8.6. LIST OF FIGURES. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 0.5 µm. . . . . . . . . . . 133. 8.7. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 0.25 µm. . . . . . . . . . 133. 8.8. Potential and current distributions in several cases of thin electrolyte analysis. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, Overpotential: 0.125 V, electrolyte thickness 0.1 µm. . . . . . . . . . . 134. 8.9. The distribution of ion-conducting phase potential in cells with different electrolyte thicknesses. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, j = 2000 A m−2 . . . . . . . . . . . . . . . . . . 135. 8.10 The distribution of ion-conducting phase potential in cells with different electrolyte thicknesses. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, j = 1000 A m−2 . . . . . . . . . . . . . . . . . . 136 8.11 The distribution of ion-conducting phase potential in cells with different electrolyte thicknesses. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, j = 500 A m−2 . . . . . . . . . . . . . . . . . . . 137 8.12 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 10 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.13 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.14 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 2.5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.15 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 1 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.16 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 0.5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(31) LIST OF FIGURES. xxix. 8.17 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 0.25 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.18 Polarization curves and overpotential composition. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K, electrolyte thickness 0.1 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.19 The voltage losses on the electrolyte. Simulation parameters: p=100000 Pa, pH2 = 20000 Pa, pO2 = 21000 Pa, T = 1023 K. . . . . . . . . . . . . . . . . . 141 8.20 The cell overpotential excluding the voltage losses on the electrolyte. Simulation parameters: p=100000 Pa, pH2 = 60000 Pa, pH2 = 23000 Pa, T = 1023 K . . . 142 8.21 Overpotential components at different electrolyte thicknesses as percentages of the total losses, and as percentages of the losses on electrodes. Simulation parameters: p=100000 Pa, pH2 = 20000 Pa, pO2 = 21000 Pa, T = 1023 K., η = 0.125 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.1. Locations of measurement sites within the stack. . . . . . . . . . . . . . . . . 147. 9.2. Anodic polarization curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149. 9.3. Overpotential composition - the reference cell microstructure. Simulation parameters: p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , anode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150. 9.4. Composition of the overpotential - Cell 1 microstructure. Simulation parameters: p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , anode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150. 9.5. Composition of the overpotential - Cell 5 microstructure. Simulation parameters: p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , anode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151. 9.6. Composition of the overpotential - Cell 9 microstructure. Simulation parameters: p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , anode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151. 9.7. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Reference cell. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . 152 T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(32) xxx. LIST OF FIGURES. 9.8. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Cell 1. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152. 9.9. The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Cell 5. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153. 9.10 The distribution of ion-conducting phase potential and volumetric exchange current density in the active layer of the anode. Cell 9. Simulation parameters: j=2000 A m−2 p=100000 Pa, pH2 O = 1000 Pa, pH2 = p − pH2 O , electrode thickness: 240 µm, T = 1023 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.11 A 3D visualization of the active layer ion-conducting phase potential before (a), and after (b–d) the experiment. Simulation parameters: p = 100,000 Pa, pH2 = 60,000 Pa, anode thickness: 240 µm, T = 1023 K. . . . . . . . . . . . . 154. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(33) List of Tables 3.1. Scheme of the Dirichlet boundary conditions - Nickel (Ni) - Yttrium Stabilized Zirconia (YSZ) anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2. Scheme of the Neumann boundary conditions - Nickel (Ni) - Yttrium Stabilized Zirconia (YSZ) anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3. 31. Scheme of the Neumann boundary conditions - Lanthanium Strontium Cobaltite Ferrite (LSCF) - Gadolinum Doped Ceria (GDC) cathode . . . . . . . . . . . .. 3.4. 30. 31. Scheme of the Neumann boundary conditions - Positive-Electrolyte-Negative assembly of a cell composed of a Nickel (Ni) - Yttrium Stabilized Zirconia (YSZ) anode, YSZ electrolyte, and Lanthanum Strontium Cobaltite Ferrite (LSCF) Gadolinum Doped Ceria (GDC) cathode . . . . . . . . . . . . . . . . . . . . .. 3.5. 32. Scheme of Fourier Boundary Conditions - Positive-Electrolyte-Negative assembly of a cell composed of a Nickel (Ni) - Yttrium Stabilized Zirconia (YSZ) anode, YSZ electrolyte, and Lanthanium Strontium Cobaltite Ferrite (LSCF) Gadolinum Doped Ceria (GDC) cathode . . . . . . . . . . . . . . . . . . . . .. 33. 4.1. Parameters of the microstructure used in the model falsification attempts . . . .. 60. 4.2. Parameters of the microstructure used in the comparison to the continuous model 63. 4.3. The electrode parameters used in the mesh resolution sensitivity study . . . . .. 63. 5.1. Anodic microstructure sample parameters . . . . . . . . . . . . . . . . . . . .. 68. 5.2. Cathodic microstructure sample parameters . . . . . . . . . . . . . . . . . . .. 68. 7.1. Microstructure and disturbance parameters in the cases studied . . . . . . . . . 119. 8.1. The electrode parameters used in the cross-electrode phenomena study . . . . . 129. 9.1. Anodic microstructure sample parameters . . . . . . . . . . . . . . . . . . . . 146. A.3.1Basic gas properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 A.3.2Gas viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 xxxi.

(34) xxxii. T. Prokop. LIST OF TABLES. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(35) Chapter 1. Introduction. Fuel Cells are energy conversion devices, which convert the chemical energy of fuels into electrical energy. At present, fuel cells tend to have higher energy efficiency than equivalent energy conversion systems which include a thermal cycle. This is accomplished by separation of of oxidation side and reduction side of the combustion reaction. An example of a typical fuel cell reaction is provided in Equations (1.1 - 1.3): 2H2 + O2 → 2H2 O. (1.1). O2 + 2e− → 2O2−. (1.2). 2H2 + 2O2− → 2H2 O + 2e−. (1.3). A comparison between a fuel cell (see Fig. 1.1) and a conventional, equivalent combustionbased energy conversion system (see Fig. 1.2) reveals a number of differences between the two energy conversion processes. Unlike the latter, a fuel cell does not require a heat exchanger, a thermal engine, or an electrical generator to produce electrical energy. Furthermore, since the half-reactions are separated, combustion-related irreversibilities are reduced or avoided. While the maximum theoretical efficiency is the same for a given chemical reaction, the thermodynamic losses occurring in fuel cells tend to be lower than in devices which use combustion process [1]. The half-reaction separation, as well as lower operational temperature (in comparison to combustion) also leads to another advantage of fuel cells - elimination of toxic combustion products [1]. Solid Oxide Fuel Cell (SOFC) is the type of a fuel cell, in which the electrolyte is composed of ceramic material, and electrodes are made of ceramic, or ceramic-metal composite (cermet). Due to the nature of ionic conductivity in these materials, SOFCs reach 700–1000 degree Celsius during operation. Thus, they are classified as high temperature fuel cells. Another feature of Solid Oxide Fuel Cells is that they can operate using hydrogen, carbon monoxide or a mixture of these gases - syngas. High temperature of SOFC devices means that they are compatible with carbohydrate reformers. Therefore, systems using Solid Oxide Fuel Cells can also be powered by methane gas [2, 3] or other alkanes. This eliminates the need for hydrogen storage and transport infrastructure. Interestingly, researchers have also proposed SOFC systems 1.

(36) 2. Anodic Channel. Anode Current Collector. Anode. Electrolyte. Cathode. Cathode Cathodic Current Channel Collector. H2. O2 H2O. Figure 1.1. Principle of a hydrogen fuel cell.. Gas reagent channel. Combustion Chamber. Heat Exchanger. Boiler. Thermal cycle. Mechanical Transmission. H2 O2. H2O. Figure 1.2. A thermal system equivalent to a fuel cell.. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell. Generator.

(37) 3. using carbohydrates which are liquid in standard conditions [4, 5], and even powdered biomass [6]. Solid Oxide Fuel Cells are characterized by high efficiency. Even in the case of relatively ˜ small (3kW) systems, researchers have been able to demonstrate chemical-to-electrical energy conversion efficiency of 50-70% %(DC,LHV) [7–9]. Subscript ’LHV’ signifies that the efficiency was measured in relation to lower heating value (LHV) of the fuel. Subscript DC means that the power output had the form of Direct Current (DC). Somekawa et al. have reported on a high fuel utilization system, in which electrical efficiency of 77.8%(DC,LHV) was achieved [10]. This quality is mirrored in commercially available SOFC technologies, such as Kyocera’s 3kW co-generation system, whose electrical efficiency is claimed to be at 52%(AC,LHV) . The system shares technology with a 700W unit by Kyocera ENE-FARM Type ”S” [11]. Large scale SOFC auxiliary co-generation systems tend to be more efficient. For example Bloom Energy Server ES5 by BloomEnergy available at 200-300kW scale is characterized by efficiency of 53-65%(AC,LHV) [12]. Subscript ’AC’ signifies alternating current generation efficiency. By comparison, the most efficient combined cycle power plant in the world - Nishi-Nagoya Thermal Power Station No. 7-1, designed by Toshiba Energy Systems & Solutions Corporation, and operated Chubu Electric Power Co., Inc. - has achieved 63.08 % efficiency [13]. The plant consisted of three 7HA.02 Combined Cycle Gas Turbines designed by General Electric Company (384MW, 63.5%(AC,LHV) in combined cycle, 42.5%(AC,LHV) in single-cycle). [14]. However, SOFCs are also able to operate in a combined cycle. While the high operating temperature is challenging in terms of engineering smaller SOFC systems, it also provides an opportunity to utilize the high quality heat in a Gas Turbine (GT) - Combined Cycle Installation ("Triple" Combined Cycle). Additionally, the waste heat from a combined SOFC-GT may be employed in a Combined Heat and Power (CHP) scheme [15]. Kobayashi et al. [16] proposed an 800 MW, gas firing triple combined-cycle plant with an efficiency of 70%(DC,LHV) , as well as a power plant employing coal gasification, while generating 700MW of electric power at 60%(DC,LHV) efficiency. Integrated Gasification Combined Cycle involving an SOFC device may also be used efficiently in a plant with inherent carbon capture. A hybrid system of this kind has been demonstrated at Kyushu University, where a 250kW SOFC-MGT unit operates at Net efficiency of 55%(AC,LHV) , and total heat efficiency of up to 73%(AC,LHV) , as reported by Kobayashi et al. [17]. The interest in high efficiency energy conversion systems is likely going to increase, as obtaining the fossil fuels has been becoming more difficult. According to a review article by Hall et al. [18] energy return on energy invested of oil, gas, and coal (with some exceptions) has been steadily declining for the past several decades. For example, in 1970s, for every 1 oil barrel equivalent of energy expended, the Canadian petroleum industry would extract 60 barrels of oil. In 2010s, the value has fallen to around 15. While these numbers are affected by the T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(38) 4. state of technology, government policies and market phenomena, the discussed tendency holds in almost all cases. This suggests that, even if the primary energy supply remains steady, a strong motivation exists to use the natural resources to the full extent. Increasing efficiency also means less greenhouse gas and pollutant emissions per unit of energy generated. In the case of SOFCs this effect is reinforced since, due to the separation of combustion half-reactions in SOFCs, the formation of some pollutants, such as NOx compounds, is largely avoided in SOFC systems.. A natural use for smaller SOFC devices is auxiliary and backup power generation [12, 19]. Such power systems have also been suggested for distributed generation programs [20]. This is facilitated by the fact, that, SOFC devices generate little-to-no toxic air pollutants due to the half-reaction separation. Another advantage of SOFC systems is that they can be fitted to operate as electrolysis devices, thereby making them viable for energy storage installations [21]. Jensen et al. [22] have proposed a system for large-scale plant, which would consist of Solid Oxide Electrolyzer/Cell installation and underground storage of CH4 and CO2 . According to the authors [22], the technology could, in some cases match the cost and efficiency of pumped hydro energy storage plants. The efficiency is an important advantage of SOFC devices, as the interest in renewables creates demand for energy storage and backup power devices [23]. While it is a common opinion that fuel cell devices of this type make poor power sources for vehicles due to relatively low power density [24] and susceptibility to mechanical stresses, SOFCs have been suggested for use in aeronautics and space exploration [25], as well as land vehicles, such as trains, and cars [26, 27].. Despite the aforementioned advantages, we are yet to see large-scale implementation of SOFC in the mainstream energy sector. The reluctance of businesses and customers to adapt SOFC technologies comes from relatively low investment return rate, compared to conventional energy conversion devices. Since degradation of stack properties during its prolonged performance is still not documented in full extent, more research is needed in that regard. The availability of statistical information pertaining to unit operation is generally lacking. The material and component optimization loop is slow and costly, which makes the Research and Development more demanding in terms of time, manpower and material. Scaling up the SOFC unit production to the levels allowing for substantial decrease in prices is also going to require development of techniques for reliable, automated (or semi-automated) stack production. The aforementioned difficulties, such as low predictability and high research cost, are all, at least in part related to one defining feature of a modern Solid Oxide Fuel Cell — the complex, ceramic-metal microstructure of its electrodes. T. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(39) 1.1. Microstructure-scale models. 5. 1.1. Microstructure-scale models The complexity of microstructure is one of the major challenges in the mathematical modeling of SOFC devices. SOFC microstructure consists of distinct phases, characterized by specific transport function. The ceramic-metal anode is composed of three phases: electron-conducting metal (such as Nickel), oxygen ion conducting ceramic material (such as Yttrium-Stabilized Zirconia - YSZ, or Gadolinium-Doped Ceria - GDC), and open porosity, where the diffusion of gaseous reagents (such as hydrogen, and water vapor) takes place. As a consequence, the anodic reaction may only take place in proximity of the boundary shared by these three phases. Such a microstructure feature is known as the Triple Phase Boundary. An analogous problem is present at the cathodic side of the assembly. However, the cathodes are commonly made of a Mixed Ion-Electron Conductor (MIEC) type materials, such as Lanthanium-Strontium Manganite (LSM) or Lanthanium-Strontium-Cobalt-Ferrite (LSCF). In such a case the reaction occurs in proximity of the boundary between the MIEC, and the pores, which provide the influx of molecular oxygen. This boundary is known as the Double Phase Boundary (DPB). A symbolic representation of SOFC transport phenomena, in the context of the microstructure is presented in Figure 1.3. During long term operation of a Solid Oxide Fuel Cell, its microstructure changes. The impact of microstructure evolution during long term operation on the device’s performance has been investigated by several research groups. Cayan et. al. has constructed and validated a model of SOFC degradation due to an accumulation of impurities from syngas [28]. Sezer et. al. later used a similar technique to research the impact of phosphine induced nickel migration [29]. Nevertheless, significant changes in phase distribution may also occur due to the impact of high temperature over a prolonged operating period [30]. Cold standby and shut down cycling also cause accelerated degradation of SOFC anode [31]. The degradation depends on the fuel type used. Stoeckl et al. [32] investigated the degradation in a carbon-oxide powered SOFC device. However, more research is required to understand the deterioration phenomena fully. Brus et al. [33–35] measured morphological parameters of an SOFC anode microstructure before and after a prolonged period of operation, during which the SOFC stack performance was registered. The experiment demonstrated an increase of cell terminal voltage after 3700 hours of continuous operation, as well as radical changes of TPB length density, average grain size, phase tortuosity factor and volume fraction [33]. Their studies indicated that despite the significant decay of TPB length density the reported stack performance increased [33]. This phenomenon cannot be explained using only microstructural studies and requires employing an advanced simulation of transport phenomena inside the anode’s microstructure. To date, there have been very few studies, in which microstructure-scale transport phenomena were analyzed by means of microstructure-scale simulation - notably, by Jiao et al. [36] for a Ni-YSZ anode, T. Prokop 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.

(40) 6. 1.1. Microstructure-scale models. H 2O H2 Hydrogen Oxidation. e-. Reaction at. Anode. Triple Phase Boundary. ElectroMotive Force. Electrolyte. O2-. Oxygen Reduction. O2-. Reaction at Double Phase. e-. Cathode. Boundary. O2. Figure 1.3. Hydrogen-oxygen redox reaction in a Solid Oxide Fuel Cell.. and by Shimura et al. [37] for an LSCF cathode. Miyoshi et al. [38] used a similar technique to analyze a Lanthanum Strontium Manganite (LSM) cathode aging under the influence of chromium poisoning. So far, the microscale transport phenomena were not simulated in a case where an aging experiment showed performance enhancement. The virtual reconstruction of an SOFC microstructure may be modified or artificially generated. This can be advantageous in simulations aimed at investigating an influence of a specific microstructure feature, such as a mesostructure or an anisotropically placed grain. Chen et al. [39] have simulated an influence of meso- and macro-pores on reagent concentration and catalyst utilization by editing an artificially generated pore network. Hsu et al. [40] have compared real and synthetic microstructures to investigate the origin of inhomogeneity in heterogeneous electrodes. Microstructure-scale models may be applied to optimize the shape of mesoscale structures introduced in the electrode to improve performance. This has been the subject of studies conducted by several research teams. [41–44]. Iwai et al. [41] investigated the impact of grooves introduced on an Cathode-Electrolyte interface. The shape of grooves was optimized numerically using a topology optimization method by Yamada et al. [45]. Then, Iwai et al. [41] manufacT. Prokop. 3D numerical analysis of transport phenomena in a Solid Oxide Fuel Cell.