Reduced Modelling of Planar Fuel Cells (eBook)

Dr. He received his B. Eng. Degree in Mechanical Engineering (major in Industrial Design) and Ph.D. degree in Engineering Mechanics from Nanyang Technological University in Singapore in 2009 and 2014, respectively. Currently, Dr. He is a research fellow in Energy Research Institute at Nanyang Technological University (ERIAN). He is strong at multiphysics modelling and simulation, especially at developing fast and efficient models for fuel cells whose performance typically involves multiphysical behaviours such as transport phenomena, thermal dynamics, and electrochemical performance. His research interests also include electrostatic precipitation, steam soaking, and acoustic agglomeration to remove particulate matters from air. Dr. Hua Li received his B.Sc and M.Eng degrees in Engineering Mechanics from Wuhan University of Technology, P.R.C., in 1982 and 1987, respectively. He obtained his Ph.D degree in Mechanical Engineering from the National University of Singapore in 1999. From 2000 to 2001, Dr. Li was a Postdoctoral Associate at the Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign. At the end of 2005, he was a Visiting Scientist (on invitation) at the Department of Chemical and Biomolecular Engineering of Johns Hopkins University. From 2001 to 2006, he was a Research Scientist in the A*STAR Institute of High Performance Computing. Dr Li joined Nanyang Technological University (NTU) as an Assistant Professor in June 2006 and he was promoted to Associate Professor in March 2013. He is currently in the School of Mechanical & Aerospace Engineering at NTU. His research interests include the multiphysics modelling of soft matters (smart hydrogel in bioMEMS and biological cell in microscale fields); development of highly efficient numerical computational methodology (meshless & multiscale algorithms); simulation of sustainable energy (building energy efficiency and fuel cell system); and dynamics (high-speed rotating shell and composite materials structure). He has sole-authored a monograph book entitled “Smart Hydrogel Modelling” published by Springer, co-authored two monograph books entitled “Meshless Methods and Their Numerical Properties” by CRC Press and “Rotating Shell Dynamics” by Elsevier, and 2 book chapters, one on MEMS simulation and the other on hydrogel drug delivery system modelling, and authored/co-authored over 140 articles published in peer-reviewed international journals. He received the Silver Award in HPC Quest 2003 - The Blue Challenge presented by IBM & IHPC in 2003. He is also extensively funded by agencies and industry, for example, the principal investigator of a computational BioMEMS project awarded under A*STAR’s strategic research programme in MEMS. Dr. Karl Erik BIRGERSSON is an associate professor at the Chemical and Biomolecular Department and an affiliate of the Engineering Science Programme at the National University of Singapore (NUS) and the Solar Energy Research Institute of Singapore. He was awarded his Ph.D. in Fluid Mechanics from the Royal Institute of Technology (KTH), Stockholm, in 2004. He also holds an MS degree in Chemical Engineering from KTH (1998) and a Licentiate degree (2003). He was a postdoctoral fellow (2004-2005) and research engineer (2005-2006) at the Institute of High Performance Computing, A*Star, Singapore. He specializes in mathematical modelling and transport phenomena; his current research focuses on electrochemical energy systems and organic solar cells. He has published around 150 papers (journal, conference and book chapters). Besides research, he enjoys teaching and experimenting with teaching strategies; he has been awarded twelve teaching awards since 2006 in NUS.

Foreword 6
Preface 8
Acknowledgment 10
Contents 11
About the Authors 15
Symbols 17
1 Introduction 23
1.1 Background 23
1.1.1 Proton Exchange Membrane Fuel Cell (PEMFC) 27
1.1.2 Planar Solid Oxide Fuel Cell (P-SOFC) 30
1.2 Motivation of Model Reduction 32
1.3 Book Outline 35
References 37
2 Literature Review 43
2.1 Introduction 43
2.2 Single-Cell Modelling 43
2.2.1 Polarizations 44
2.2.1.1 Reversible Potential 45
2.2.1.2 Activation Polarization 47
2.2.1.3 Ohmic Polarization 49
2.2.1.4 Concentration Polarization 50
2.2.1.5 Leak Polarization 51
2.2.2 Transport Phenomena in Electrodes 51
2.2.2.1 Volume Averaging for Porous Media 52
2.2.2.2 Mass Transfer 53
2.2.2.3 Momentum Transfer 56
2.2.2.4 Energy Transfer 57
2.2.2.5 Reaction Zone 58
2.2.3 Spatial Dimension 59
2.2.3.1 Zero-Dimensional Model 59
2.2.3.2 One-Dimensional Model 60
2.2.3.3 Two-Dimensional Model 60
2.2.3.4 Three-Dimensional Model 61
2.3 Stack Modelling 62
2.4 Model Simplification 63
2.5 Numerical Methods 64
2.6 Sensitivity Analysis 65
2.7 Remarks 66
References 68
3 Full Three-Dimensional Modelling of PEMFC and Planar SOFC 77
3.1 Introduction 77
3.2 Three-Dimensional Two-Phase PEMFC Model 77
3.2.1 Assumptions 78
3.2.2 Mathematical Model 79
3.2.2.1 Governing Equations 79
Conservation of Mass 79
Conservation of Momentum 80
Conservation of Species 81
Conservation of Charge 82
Conservation of Energy 83
3.2.2.2 Boundary Conditions 84
3.2.2.3 Constitutive Relations 85
Expressions for Water Content and Activity 85
Agglomerate Model for the Cathode 88
3.2.3 Remarks 88
3.3 Three-Dimensional P-SOFC Model 90
3.3.1 Assumptions 90
3.3.2 Modelling Domains 91
3.3.3 Mathematical Model 91
3.3.3.1 Governing Equations 91
Conservation of Mass 92
Conservation of Momentum 93
Conservation of Species 94
Conservation of Charge 94
Conservation of Energy 95
3.3.3.2 Boundary Conditions 96
3.3.3.3 Constitutive Relations 99
3.3.4 Numerical Implementation 99
3.3.5 Model Validation 101
3.3.6 Numerical Convergence Test 104
3.3.7 Remarks 106
References 107
4 Development of Reduced PEMFC Models 110
4.1 Introduction 110
4.2 Spatially-Smoothed Isothermal Two-Phase PEMFC Model 111
4.2.1 Spatial Smoothing 111
4.2.1.1 Momentum Transport in Flow Field 113
4.2.1.2 Diffusive Transport in the Flow Field 115
4.2.1.3 Diffusive Transport in the Backing Layer 116
Derivation of Correlation Factors Based on a Laplace Equation 116
Application of the Derived Correlation Factors 118
4.2.1.4 Charge Transport in the Solid Part of the Flow Field 119
4.2.1.5 Reduction in Dimensionality 119
4.2.2 Numerical Implementation 121
4.2.3 Model Verification 122
4.2.4 Remarks 127
4.3 Asymptotic Non-isothermal Two-Phase PEMFC Model 130
4.3.1 Mathematical Formulation 130
4.3.1.1 Full Non-isothermal PEMFC Model 130
4.3.1.2 Asymptotic Reduction 133
4.3.1.3 Stream Function of the Flow Field 136
4.3.2 Numerical Implementation 137
4.3.3 Calibration, Verification, and Validation 139
4.3.4 Thermal Decoupling 141
4.3.5 Computational Cost and Efficiency 145
4.3.6 Remarks 146
4.4 Reduced Non-isothermal PEMFC Stack Model 148
4.4.1 Mathematical Formulation 148
4.4.1.1 Full Cell Model 149
4.4.1.2 Asymptotic Cell Model 151
4.4.2 Numerical Implementation 155
4.4.2.1 Implementation of Full and Reduced Models 155
4.4.2.2 Automated Model Generation 156
4.4.3 Model Verification 157
4.4.3.1 Verification Without Perturbations Between Cells 158
4.4.3.2 Verification with Perturbations Between Cells 160
4.4.4 Computational Cost Analysis 163
4.4.5 Remarks 165
4.5 Aggregate Measure for Local Current Density Coupling in Fuel Cell Stacks 166
4.5.1 Mathematical Formulation 166
4.5.2 Analysis 168
4.5.3 Model Verification 171
4.5.4 Remarks 174
4.6 Computationally-Efficient Hybrid Strategy for Mechanistic Modelling of PEMFC Stacks 174
4.6.1 Mathematical Formulation 175
4.6.2 Hybrid Coupling Methodology 176
4.6.3 Numerical Implementation 177
4.6.4 Model Verification 178
4.6.5 Computational Cost and Efficiency 180
4.6.6 Remarks 182
References 183
5 Development of Reduced P-SOFC Models 187
5.1 Introduction 187
5.2 Asymptotic Spatially-Smoothed Isothermal (ASSI) P-SOFC Cell Model 187
5.2.1 Spatial Smoothing with Correlation Factors Derived Based on a Full Cell Model 188
5.2.1.1 Momentum and Mass Transport in Flow Field 190
5.2.1.2 Mass and Charge Transport in Backing Layer 192
5.2.2 Asymptotic Reduction 195
5.2.2.1 Nondimensionalization 196
5.2.2.2 Reference and Scale Factors 199
Reference Factors 199
Scale Factors for Velocity and Pressure Drop 200
Scale Factors for Diffusivity and Correlation 202
Scale Factors for Potential and Current Density 203
5.2.2.3 Asymptotic Spatially-Smoothed Formulation 204
5.2.3 Numerical Implementation 204
5.2.4 Model Verification 206
5.2.5 Computational Cost Analysis 211
5.2.6 Remarks 212
5.3 Advanced Spatially-Smoothed Model 213
5.3.1 Novel Variation Factor to Capture the Variability of Dependent Variables Along Cell Width 213
5.3.2 Full and Reduced Cell Models 218
5.3.3 Numerical Implementation 220
5.3.4 Model Verification 220
5.3.5 Remarks 223
5.4 Asymptotic Spatially-Smoothed Non-isothermal (ASST) P-SOFC Cell and Stack Models 224
5.4.1 Cell and Stack Modelling 225
5.4.2 Spatially-Smoothed Energy Equation 226
5.4.2.1 Local Thermal Equilibrium 227
5.4.2.2 Local Thermal Non-equilibrium 228
5.4.3 Asymptotic Reduction 231
5.4.3.1 Local Thermal Equilibrium 231
5.4.3.2 Local Thermal Non-equilibrium 234
5.4.4 Numerical Implementation 235
5.4.5 Model Verification 237
5.4.5.1 Cell Performance 237
5.4.5.2 Stack Performance 238
Stack Without Perturbation Between Cells 239
Stack with Perturbation Between Cells 241
5.4.6 Remarks 244
References 245
6 Integrated Stochastic and Deterministic Sensitivity Analysis: Correlating Variability of Design Parameters with Cell and Stack Performance 246
6.1 Introduction 246
6.2 Monte Carlo Simulation of a P-SOFC Single Cell 246
6.2.1 Quasi-3D Asymptotic Spatially-Smoothed Isothermal (ASSI) Single-Cell Model 247
6.2.2 Monte Carlo Simulation 250
6.2.3 Numerical Implementation 253
6.2.4 Statistical Results and Sensitivity Analysis 254
6.2.4.1 Determination of Sample Size 254
6.2.4.2 Varying Modelling Parameters Individually 255
6.2.4.3 Varying Modelling Parameters Simultaneously 258
6.2.5 Remarks 264
6.3 Monte Carlo Simulation of a P-SOFC Stack 266
6.3.1 Quasi-3D Spatially-Smoothed Non-Isothermal (SST) Stack Model 266
6.3.2 Monte Carlo Simulation 271
6.3.3 Numerical Implementation 272
6.3.4 Statistical Results and Sensitivity Analysis 273
6.3.4.1 Varying Modelling Parameters Individually 274
6.3.4.2 Varying Modelling Parameters Simultaneously 275
Global Stack Performance 276
Local Stack Performance 278
6.3.5 Remarks 283
References 286
7 Conclusions 289
7.1 Conclusions from the Present Work 289
7.2 Recommendations for Future Work 293
References 294
Appendix A: Scaling Analysis for Current Collector 295
Appendix B: Scaling Analysis for Flow Field 296
Appendix C: Scaling Analysis for Backing Layer 299
Appendix D: Scaling Analysis for Reaction Zone Layer 302
Appendix E: Scaling Analysis for Electrolyte 306