Analytical Modelling of Fuel Cells

ContentsPreface Introduction 1 Fuel cell basics1.1 Fuel cell thermodynamics1.1.1 The physics of the fuel cell effect1.1.2 Open-circuit voltage1.1.3 Nernst equation1.1.4 Temperature dependence of OCV1.2 Potentials in a fuel cell1.3 Rate of electrochemical reactions1.3.1 Butler-Volmer equation1.3.2 Butler-Volmer and Nernst equations1.3.3 Tafel equation1.4 Mass transport in fuel cells1.4.1 Overview of mass transport processes1.4.2 Stoichiometry and utilization1.4.3 Quasi-2D approximation1.4.4 Mass conservation equation in the channel1.4.5 Flow velocity in the channel1.4.6 Mass transport in gas diffusion/backing layers1.4.7 Mass transport in catalyst layers1.4.8 Proton and water transport in membrane1.5 Sources of heat in a fuel cell1.6 Types of cells considered in this book1.6.1 Polymer electrolyte fuel cells (PEFCs)1.6.2 Direct methanol fuel cells (DMFCs)1.6.3 Solid oxide fuel cells (SOFCs)2 Catalyst layer performance2.1 Basic equations2.1.1 The general case2.1.2 First integral2.2 Ideal oxygen and proton transport2.3 Ideal oxygen transport2.3.1 Basic equations2.3.2 Integral of motion2.3.3 Equation for proton current2.3.4 Low cell current2.3.5 High cell current2.3.6 The general polarization curve2.3.7 Condition of negligible oxygen transport loss2.4 Ideal proton transport2.4.1 Basic equations2.4.2 The x-shapes and polarization curve2.4.3 Large zeta (zeta greater than 1)2.4.4 Small zeta (zeta less than 1)2.5 Optimal oxygen diffusion coe±cient2.5.1 Reduction of the full system2.5.2 Optimal oxygen diffusivity2.6 Gradient of catalyst loading2.6.1 Model2.6.2 Polarization curve 2.7 DMFC anode2.7.1 The rate of methanol oxidation2.7.2 Basic equations and the conservation law2.7.3 The general form of the polarization curve2.7.4 Small variation of overpotential in the active layer2.7.5 Active layer of variable thickness2.8 Heat balance in the catalyst layer2.8.1 Heat transport equation in the CL2.8.2 Reduction to boundary condition2.8.3 Solution to the heat transport equation2.9 Remarks on Chapter 23 One-dimensional model of a fuel cell3.1 Voltage loss due to oxygen transport in the GDL3.2 One-dimensional polarization curve of a cell3.3 One-dimensional model of DMFC3.3.1 Feed molecule concentration in the active layers3.3.2 One-dimensional polarization curve of DMFC3.4 Heat transport in the MEA of a PEFC3.4.1 General assumptions3.4.2 Equations3.4.3 Exact solutions3.4.4 Temperature profiles3.4.5 How to measure thermal conductivities of MEA layers3.4.6 One-sided fluxes from the MEA3.4.7 Heat crossover through the membrane3.5 Heat transport in the MEA of a DMFC3.5.1 Assumptions3.5.2 Equations3.5.3 Heat transport under open-circuit conditions3.5.4 How to measure thermal conductivities of MEA layers3.5.5 Temperature profiles and discussion3.5.6 Exact solutions4 Quasi-2D model of a fuel cell4.1 Gas dynamics of channel flow4.1.1 Momentum balance in the cathode flow4.1.2 The limit of low flow velocity4.2 A model of PEFC4.2.1 Oxygen concentration and local current along the channel4.2.2 Cell polarization curve4.2.3 Water crossover and the polarization curve4.2.4 Local polarization curves4.3 A model of PEFC with water management4.3.1 Model and governing equations4.3.2 Solution at constant flow velocity4.3.3 Close to the limiting current density4.3.4 The general case4.3.5 Model validation 4.3.6 Limiting current, optimal feed composition4.3.7 Constant oxygen stoichiometry4.4 Degradation wave4.4.1 Model4.4.2 Wave propagation4.4.3 Cell potential4.4.4 Two scenarios of cell performance degradation4.4.5 Accelerated testing of aging phenomena4.5 Gradient of catalyst loading along the oxygen channel4.5.1 Low cell current4.5.2 High cell current 4.5.3 The effect of transport loss in the GDL4.6 A model of SOFC anode4.6.1 Basic equations and the local polarization curve4.6.2 Hydrogen concentration in the channel4.6.3 Cell voltage4.6.4 Low current: z-shapes4.6.5 Low current: Polarization curve4.6.6 High current: z-shapes and polarization curve4.6.7 Remarks4.7 A model of DMFC4.7.1 Continuity equations in the feed channels4.7.2 Solution for the case of lambda^a = lambda^c4.7.3 Cell depolarization at zero current: mixed potential4.7.4 Cross-linked feeding4.7.5 Oxygen and methanol utilization, mean crossover current density4.7.6 Remarks4.8 DMFC: The general case of arbitrary lambda a and lambda c 4.8.1 Equation for local current4.8.2 Numerical solution4.9 DMFC: Large methanol stoichiometry, small current4.9.1 The shape of the jumper4.9.2 Plateau4.9.3 Critical air flow rate4.9.4 Experimental verification5 Modelling of fuel cell stacks5.1 Temperature field in planar SOFC stack5.1.1 General assumptions5.1.2 The general equation for bipolar plate temperature5.1.3 Heat balance in the air channel5.1.4 Heat balance in the BP5.1.5 Cell polarization curve5.1.6 Boundary conditions5.1.7 Method of asymptotic expansion5.1.8 Asymptotic solution5.1.9 Local current5.1.10 Example: Oxide-dominated stack resistivity5.1.11 Remarks5.2 Temperature gradient in SOFC stack5.2.1 Stack and air temperatures5.2.2 Temperature gradient5.3 Thermal waves in SOFC stack5.3.1 Basic equations5.3.2 Stability analysis5.3.3 Flow temperature is constant5.3.4 Solution: The general case5.3.5 Role of boundary conditions5.3.6 Remarks5.4 Heat effects in DMFC stack5.4.1 General assumptions5.4.2 Equations for stack and flow temperature5.4.3 Asymptotic solution: The general case5.4.4 Optimal stack temperature5.5 Mirroring of current-free spots in a stack5.5.1 Equation for bipolar plate potential5.5.2 Cell polarization curve5.5.3 Spot shape and numerical details5.5.4 Numerical results5.5.5 Analysis of equations: The length of mirroring5.5.6 Remarks5.6 Hybrid 3D model of SOFC stack5.6.1 Thermal model5.6.2 Electric problem5.6.3 Numerical details5.6.4 Numerical results5.6.5 Analysis of governing equations5.6.6 Temperature stratification5.6.7 The mechanism of anomalous heat transport5.7 Power generated and lost in a stack5.7.1 The nature of voltage loss in bipolar plates5.7.2 Power dissipated in a bipolar plate5.7.3 Power dissipated in a thin bipolar plate5.7.4 Useful power generated by the individual cell5.7.5 Illustration: A 1D caseBibliographyAbbreviationsNomenclature *