Computational Fluid Dynamics Modelling of. Solid Oxide Fuel Cell Stacks

Transcription

1 Computational Fluid Dynamics Modelling of Solid Oxide Fuel Cell Stacks by Robert Takeo Nishida A thesis submitted to the Department of Mechanical and Materials Engineering in conformity with the requirements for the degree of Master of Applied Science Queen s University Kingston, Ontario, Canada September 2013 Copyright c Robert Takeo Nishida, 2013

2 Abstract Two computational fluid dynamics models are developed to predict the performance of a solid oxide fuel cell stack, a detailed and a simplified model. In the detailed model, the three dimensional momentum, heat, and species transport equations are coupled with electrochemistry. In the simplified model, the diffusion terms in the transport equations are selectively replaced by rate terms within the core region of the stack. This allows much coarser meshes to be employed at a fraction of the computational cost. Following the mathematical description of the problem, results for single-cell and multi-cell stacks are presented. Comparisons of local current density, temperature, and cell voltage indicate that good agreement is obtained between the detailed and simplified models, verifying the latter as a practical option in stack design. Then, the simplified model is used to determine the effects of utilization on the electrochemical performance and temperature distributions of a 10 cell stack. The results are presented in terms of fluid flow, pressure, species mass fraction, temperature, voltage and current density distributions. The effects of species and flow distributions on electrochemical performance and temperature are then analyzed for a 100 cell stack. The discussion highlights the importance of manifold design on performance and thermal management of large stacks. i

3 Acknowledgments I would like to express my sincere gratitude to Dr. Jon Pharoah and Dr. Steven Beale for their ideas and guidance. I would like to thank Dr. Hae-Won Choi, Dr. Mobin Baboli, Duncan Gawel, and Jake Garvey for many helpful conversations during the research process. I would also like to acknowledge Helmut Roth for sharing his technical knowledge. Finally, I would like to express a very special thank you to Rachel Lenaghan for her endless patience, support and encouragement. ii

4 Contents Abstract Acknowledgments Contents List of Tables List of Figures Nomenclature i ii iii vi vii ix Chapter 1: Introduction Motivation Background Information Background of Solid Oxide Fuel Cells SOFC Fundamentals Electrochemical Performance Conservation Equations Problem Statement Scope Contributions Organization of Thesis Chapter 2: Literature Review Modelling of Fuel Cells Electrochemistry and Heat Transfer Coupled Flow and Electrochemistry Stack Scale Modelling Research Opportunity Reactant Distribution Temperature Distribution iii

5 2.3.3 Other Considerations Chapter 3: Modelling Techniques Detailed Model Conservation Equations Electrochemistry Domain Decomposition Solution Procedure Simplified model Conservation Equations Transport Properties Domain Decomposition Closure Chapter 4: Comparison of Detailed and Simplified Models Geometry and Mesh Operating Conditions Co-Flow Counter-Flow Stack Geometry Closure Chapter 5: Performance of Large Stacks Geometry and Mesh Ten Cell Stack Operating Conditions Results and Discussion Effects of Utilization on Temperature and Cell Voltage Large Stack Performance Effects of Flow on Temperature and Performance Effects of Variable Properties Closure Chapter 6: Conclusions and Future Work Conclusions Comparison of Detailed and Simplified Models Performance of Large Stacks Future Work Bibliography 85 iv

6 Appendix A: Conservation Equations 96 A.1 Detailed model conservation equations A.2 Simplified model conservation equations Appendix B: Geometry and Transport Properties 98 B.1 Geometry and transport properties B.2 Channel properties and inter-phase transfer terms v

7 List of Tables 3.1 Boundary conditions for the conservation equations Boundary conditions for the conservation equations at the electrode/electrolyte interfaces Single-cell operating conditions Comparison of electrolyte values for single cell in co-flow Comparison of electrolyte values for single cell in counter-flow Comparison of voltage values for a three cell stack in co-flow Large stack operating conditions Variable properties vi

8 List of Figures 1.1 Planar fuel cell components (a) Characteristic voltage vs current graph and (b) Power curve Front view of detailed model mesh for single cell and three cell stack Current density (A/m 2 ) at electrolyte for (a) detailed and (b) simplified models in co-flow Temperature (K) at electrolyte for (a) detailed and (b) simplified models in co-flow Current density (A/m 2 ) at electrolyte for (a) detailed and (b) simplified models in counter-flow Temperature (K) at electrolyte for (a) detailed and (b) simplified models in co-flow Ten cell stack, elevation view for air (left) and fuel (right) Ten cell stack, elevation view Voltage vs utilization Average cell temperature vs utilization cell stack, elevation view for air (top) and fuel (bottom) cell stack, elevation view vii

9 5.7 Convective mass flux, voltage and average cell temperature by cell number of a 100 cell stack (a) Hydrogen species fraction and (b) density by cell number of a 100 cell stack Utilization of (a) O 2 and (b) H 2 by cell number of a 100 cell stack Convective mass flux of H 2 by cell number of a 100 cell stack Convective mass flux, voltage and average cell temperature by cell number for constant properties viii

10 Nomenclature A area, m 2 B driving force c p specific heat, J/kg K D diffusion coefficient, m 2 /s D h hydraulic diameter, m E Nernst potential, V f friction factor F Faraday s constant, , Coulomb/mol F j distributed resistance to momentum, kg/m 3 s g mass transfer conductance, kg/m 2 s G Gibb s free energy of formation, J/mol h heat transfer coefficient, W/m 2 K H enthalpy of formation, J/mol, conduction length, m i current, A k thermal conductivity, W/mK m mass, kg M molecular weight, kg/mol Nu Nusselt number p pressure, Pa q heat source, W r correction factor, Ω R gas constant, , J/mol, resistance, Ω Re h Reynolds number S source term, conduction shape factor, entropy, J/mol Sh Sherwood number t time, s T temperature, K u velocity, m/s U T overall heat transfer coefficient, W/m 2 K V cell voltage, V V volume, m 3 ix

11 v x y electron valence molar fraction mass fraction Greek Letters α transfer coefficient, heat transfer coefficient, W/m 3 K ε volume fraction η voltage losses, V Γ diffusive exchange coefficient κ permeability, m 2 µ dynamic viscosity, kg/ms φ general scalar ρ gaseous mixture density, kg/m 3 τ wall shear stress, Pa utilization ζ i Subscripts 0 reference value a air act activation ASR area specific resistance con concentration e electrolyte f fuel, fluid i gas species, interconnect in inlet n neighboring region, general direction ohm ohmic out outlet rxn reaction s solid t transferred substance state T reaction temperature w wall Superscripts per unit length per unit area per unit volume per unit time x

12 1 Chapter 1 Introduction 1.1 Motivation Energy is critical to most economic, environmental and developmental issues facing the world today. Sustainable energy systems are fundamental to global prosperity. As energy conversion devices, fuel cells offer the opportunity to create clean energy systems which are scalable and distributable to meet our energy needs. Fuel cells provide a bridge between chemical and electrical energy currencies and offer distinct advantages over conventional energy conversion technologies. They promise better efficiencies than the conventional internal combustion engine as fuel can be converted directly to electricity by a chemical reaction. Fuel cells are scalable and, unlike batteries, they will continue to produce electrical energy as long as the fuel is supplied. To develop fuel cell technologies, models are created and validated with experiments. Computational fluid dynamics (CFD) modelling is used as a tool to understand the detailed interactions among flow geometry, fluid dynamics, heat transfer and electrochemical reactions. Improved and validated modelling techniques offer

13 1.2. BACKGROUND INFORMATION 2 better predictions of performance, a deeper understanding of internal processes and faster optimization of fuel cell designs. An operating fuel cell stack produces and consumes both heat and chemical species to generate electrical power. Overall performance is affected by the distributions of heat and species which in turn are affected by manifold design and stack size. By modelling coupled equations of fluid flow, heat transfer and electrochemistry at the stack scale, a designer can gain a deeper understanding of critical concepts in fuel cell stack design as well as predict fuel cell performance. 1.2 Background Information In a fuel cell, chemical energy from fuel is converted directly to electrical energy by an electrochemical reaction. All major fuel cell types operate on the same basic principles, but operate at different temperature ranges, use different materials and differ in fuel tolerance and performance characteristics. The principle of a fuel cell is that two electrochemical half-reactions occur on opposite sides of an electrolyte. The reaction is completed by ions which carry charge across the electrolyte. Based on the reaction, electrons are freed, conducted in an external circuit and harnessed as electrical work. The following background information is largely gathered from Fuel Cell Fundamentals [1] Background of Solid Oxide Fuel Cells The high operating temperature of solid oxide fuel cells (SOFCs) offers advantages and disadvantages compared with low temperature fuel cells such as polymer electrolyte

14 1.2. BACKGROUND INFORMATION 3 membrane fuel cells (PEMFCs). SOFCs are operated at temperatures of around 800 C 1000 C. The high temperature allows internal reforming of fuels other than pure H 2, such as methane, propane, butane and other light hydrocarbons. Also, the kinetic activity and charge transport are both improved at high temperatures. As a result, SOFCs have a relatively high power density and commercially available systems reach upwards of 60% electrical efficiency. Furthermore, SOFCs produce high quality waste heat in the exhaust stream which can be used in combined heat and power applications to reach very high system efficiencies of greater than 80% [2, 3]. The high operating temperature also causes high thermal gradients, thermal cycling, and the need for matching thermal expansion coefficients of the component materials causing mechanical and reliability concerns. Tubular fuel cells show improved durability at high temperatures, however planar fuel cells offer a higher power density when used in stacks. Only the planar configuration is considered in this work. Solid oxide fuel cells are used in applications in which the advantages are maximized. SOFC systems are used for distributed power generation to replace conventional diesel generators based on their quiet operation and volumetric efficiency. In new community developments where it is not cost-effective to run high power transmission lines or in remote communities, combined heat and power SOFC systems provide a good alternative. Another application of SOFCs is alongside intermittent generation technologies. Wind and solar energy is harnessed and used to electrolyze water into hydrogen gas and oxygen. The hydrogen is then used in SOFCs in offpeak hours as clean energy. Other applications include use as auxiliary power units on refrigerated trucks and elsewhere where the characteristic fuel flexibility is desirable.

15 1.2. BACKGROUND INFORMATION 4 Figure 1.1: Planar fuel cell components SOFC Fundamentals A planar fuel cell repeat unit is made up of four essential parts as seen in Figure 1.1 [4]: electrolyte, anode, cathode, and interconnect. A stack consists of multiple repeat units. Air is distributed to the cathode reaction sites and fuel to the anode reaction sites via gas distribution in the interconnect manifolds. Gas molecules are transported from the surface of the porous electrodes to the reaction sites where the ions, electrons and gas molecules each exist. The reaction sites are where the ionic conducting phase intersects the electron conducting phase and the gaseous pore phase called triple phase boundaries (TPB). At the cathode TPB, oxygen gas combines with electrons to create oxygen ions (O 2 ) in an oxygen reduction reaction. The oxygen ions carry charge across the ion conducting electrolyte to the anode TPB where they produce water and electrons in a hydrogen reduction reaction. The two electrochemical half reactions in a SOFC

16 1.2. BACKGROUND INFORMATION 5 operated on hydrogen and oxygen are 1 2 O 2 + 2e O 2 (1.1) H 2 + O 2 H 2 O + 2e (1.2) The electrons are transported through the electron conducting phase to the interconnect and into an external circuit to harness the electrical power. The unreacted hydrogen and oxygen, and gaseous water by-product are transported out through the manifolds. Solid oxide fuel cells use a thin ceramic membrane for an electrolyte which allows the transport of oxygen ions. The most common SOFC electrolyte is yttria-stabilized zirconia (YSZ) which allows sufficient ionic conductivity when operated at high temperatures. The electrolyte also provides a spatial separation of fuel (at the anode) and oxidant (at the cathode) to prevent fuel crossover. The electrolyte is kept thin to minimize conduction losses while retaining mechanical integrity. The anode is where the fuel oxidation half reaction takes place. A common anode material is a (Nickel) Ni-YSZ cermet. The nickel provides the required electronic conductivity and catalytic activity while the YSZ provides ionic conductivity and a porous structural framework which matches the thermal expansion coefficient (TEC) of the electrolyte. The cathode is where the oxygen reduction half reaction takes place. In YSZ-based SOFCs, a common cathode material is strontium-doped LaMnO 3 as it provides the required physical and chemical stability, electrical conductivity and catalytic activity. The interconnect region distributes gases to each electrode as well as conducts

17 1.2. BACKGROUND INFORMATION 6 electrons. Interconnects are commonly made up of chromium metal alloys with good electrical conductivity, mechanical stability and thermal conductivity. A high thermal conductivity is important for reducing thermal gradients. However, metallic interconnects have a significant TEC mismatch with the ceramic components Electrochemical Performance The electrical power extracted from a fuel cell is measured by the current and voltage. The maximum voltage, E, available for work is based on the chemical potential of the reactant species, in this case H 2, O 2 and H 2 O, and is given by the Nernst equation [1] E = E T + RT a v i nf ln reactants a v i products (1.3) for gas constant R, temperature T, number of electrons transferred n, Faraday s constant F, species activity a, and stoichiometric coefficient v i. The reversible potential E T is based on Gibb s free energy at a given temperature. There are three main causes for the actual voltage of a fuel cell to be reduced from the ideal voltage and these losses increase with the amount of current: activation losses η act, ohmic losses η ohm, and concentration losses η con. The actual output voltage V of the cell is shown by the following equation V = E η act η ohm η con (1.4) The activation loss refers to the energy required to put the reactant species in the activated state so the reaction can occur. The energy required to overcome the activation barrier can not be used for useful work and represents a loss. Ohmic loss

18 1.2. BACKGROUND INFORMATION 7 refers to the intrinsic resistance to ionic and electronic charge flow. This resistance to energy transport is dissipated as heat. The concentration loss refers to the losses in mass transport between the reactant and product concentrations at the reaction sites rather than in the bulk flow. Also, reaction kinetics are affected by the concentrations of reactants and products at the reaction sites further reducing the ideal (Nernst) potential. Each of three losses increase with the amount of current drawn, but affect the voltage differently as shown by the polarization curve in Figure 1.2a. Equations which represent the effects on current can be found in Fuel Cell Fundamentals [1]. The effect of activation losses on voltage is exponential with increasing current. The ohmic loss has a linear relationship with current. At higher currents, the depletion of reactant species at the reaction sites is more severe, causing a larger concentration voltage loss. Figure 1.2b shows a power density curve which is the voltage of the polarization curve multiplied by the current density at each operating point. The power output reaches a peak operating state for voltage and current output Conservation Equations Computational fluid dynamics modelling offers a powerful predictive tool for the performance of fuel cells when the conservation equations are coupled with equations representing electrochemical performance. The governing equations of CFD models start with the conservation laws. The general conservation equation has the form (ρφ) t + (ρ uφ) = n α n (φ n φ) + (Γφ) + Ṡ (1.5)

19 1.3. PROBLEM STATEMENT 8 Figure 1.2: (a) Characteristic voltage vs current graph and (b) Power curve It is convenient to refer to the terms in Eqn. 1.5 in order as transient, convection, inter-phase transfer, diffusion, and source. The transient term is not employed in this work as it is considered steady-state. The inter-phase transfer term is a rate term which allows the assumption that the transport is known between two phases. This term is employed to supplant the diffusive term in any given direction to reduce calculation time and is the principle behind the distributed resistance analogy (DRA) model [5]. 1.3 Problem Statement The benefits of using SOFC technology are clear; however, the high operating temperatures and the inherently complicated processes within a solid oxide fuel cell present design challenges. Experimental advances are limited because experimental tests are expensive and time consuming, and it is difficult to look at a range of designs and operating conditions [6]. Therefore it is necessary to employ models to contribute to

20 1.3. PROBLEM STATEMENT 9 the development of fuel cells [7]. There have been significant advances in fuel cell modelling at the component and cell scales in recent years [8, 9, 10] for a multitude of design considerations. However, fuel cells at the stack scale have not been well explored to date despite the unique design requirements. The uniform distribution of reactant species and the removal of excess heat and reaction products are a matter of primary concern for electrochemical performance and durability in fuel cell stacks [11]. Flow and pressure maldistributions can occur, due to poor design, leading to variations in local current density, and reductions in stack voltage [12, 13]. Thermal stresses due to temperature gradients contribute to cell degradation [14, 5, 15]. The temperature distribution also has a strong effect on the electrochemical performance and electronic and ionic conduction [5] illustrating how the flow, temperature distribution, and electrochemical performance are inherently coupled. Two solid oxide fuel cell (SOFC) stack models were developed to solve the momentum, species and heat transport equations fully coupled with electrochemistry. The models capture the inherently coupled interactions allowing insight into the two major design considerations at that scale: removal of heat and reaction products, as well as the the distribution of reactant species at each cell in the stack. Unlike most other studies, the models in this work include the effects of internal and external manifolding, as well as stack size on electrochemical performance and temperature distributions.

21 1.4. SCOPE Scope The goal of this work is to investigate how flow geometry and thermo-fluid interactions impact electrochemical performance in solid oxide fuel cells at the stack scale. To accomplish this, two computational fluid dynamics models are created with the capability to investigate multiple cells in a stack with internal and external manifolding: the detailed model, and the simplified model. A detailed, stack scale model is created by altering an existing detailed, cell scale model in which the full conservation equations are coupled with electrochemistry. For larger stacks, it is essential to trade-off detail for improved calculation time. Integrating a high resolution mesh at the electrolyte scale for multiple cells in a stack can very quickly lead to a mesh too large for practical use. In the simplified model, the diffusion terms in the momentum and heat equations are selectively replaced by rate terms within the core region of the stack. A mass transfer conductance term is applied to calculate the wall values required for the Nernst equation. This method allows a substantial reduction of mesh size and computation time so that large stacks may be investigated. The detailed cell scale model was verified by comparison with the International Energy Agency benchmark of simulations [16] which provides standard model parameters and results for code verification. The simplified model is verified by displaying that good agreement is obtained between the detailed and simplified models for a variety of parameters. Results are presented in terms of distributions of local current density, species mass fraction, temperature and cell voltage for a single cell and a three cell stack. The simplified model is used in a small stack of 10 cells with external manifolds to

22 1.4. SCOPE 11 determine the effects of utilization of air and fuel on electrochemical performance and temperature distribution. The results are presented in terms of fluid flow, pressure, species mass fraction, temperature, voltage and current density distributions. The simplified model is then demonstrated for large stacks of 100 cells. The effects of flow distribution and species transport on electrochemical performance and temperature distributions are analyzed. The assumptions are analyzed for three transport properties which vary with species concentration and temperature. A comparison is made for how well a cold-flow model can be used to predict performance of an operating fuel cell. The discussion highlights the importance of manifold design on performance and thermal management of large stacks Contributions There are three main contributions of this work: the adaptation of a detailed cell model to a stack model, the creation of the simplified model in OpenFOAM with a substantial increase in detail over the original model and the demonstration of the ability to investigate key design questions for large solid oxide fuel cell stacks. The adaptation of a detailed cell model to the detailed stack model requires accounting for different voltages and current distributions at each cell and therefore different heat and species sources at each cell. This is among the first models which incorporate the full momentum, species and heat transport equations coupled with an electrochemical model for an SOFC stack with internal and external manifolds. The simplified model was originally implemented in the CFD program PHOENICS for a geometry without internal channels and assumed constant properties [5]. The current implementation is an advance from previous work in that significantly more

23 1.5. ORGANIZATION OF THESIS 12 physical realism is incorporated and is coded using the CFD program, OpenFOAM. More complex geometries are employed by using geometric shape factors to account for the ribbed interconnectors and averaging of multiple porous transport layers for the heat and mass transport coefficients. In this work, it is demonstrated that major design questions for solid oxide fuel cell stacks can be explored with manageable computation times, namely: What is the effect of utilization of air and fuel on electrochemical performance and temperature distributions in a small stack? What are the dominant processes for temperature distribution and electrochemical performance for a large stack? And, how well do assumptions hold up for each of the fluid properties leading to a fully cold-flow model when predicting performance of an operating stack? 1.5 Organization of Thesis After the introduction, Chapter 2 contains a review of the relevant literature on stack scale, SOFC modelling. The method behind each of the CFD models is presented in Chapter 3. A comparison of results verifying the simplified model is given in Chapter 4. Chapter 5 shows the effect of utilization on electrochemical performance and temperature distribution for a stack of 10 cells and investigates stacks on the order of 100 cells. Finally, Chapter 6 concludes the work with mention of future work in this area of research.

24 13 Chapter 2 Literature Review To develop solid oxide fuel cell technology, the effects of design and operation parameters at each length scale should be considered. Experiments are expensive and time consuming when characterizing the effect of each parameter on performance, therefore more practical prediction tools are needed [17, 18]. Mathematical models are created from the sub-component to system levels to capture the effects of parameters unique to each length scale. The weakness of modelling is that it is not guaranteed to exactly predict physical phenomena especially when key approximations and assumptions are used [9]. However, with careful consideration of results and comparison with experiment, models are indispensable design tools. The following describes the advancements leading up to the state of the art in comprehensive CFD modelling of solid oxide fuel cells. Relevant works are highlighted which show the importance of modelling specifically at the stack scale. Then, areas in need of further research are outlined.

25 2.1. MODELLING OF FUEL CELLS Modelling of Fuel Cells There have been significant advances in fuel cell modelling at the cell and component scales in recent years [8, 9]. Andersson et al. [10] provide a review of modelling at several different length scales. They highlight the importance of integrating models at multiple length scales. While there are an increasing number of researchers developing models at the stack scale, research has mainly been focused on the cell and component scales to this point. In 1994, Achenbach [19] provided a benchmark for modelling work in which he considered a 3D, time-dependent, stack model of a SOFC with internal reforming, gas recycling, and conductive, convective and radiative heat transfer. Distribution of gases and current density were predicted for cells in cross-flow, co-flow, and counterflow configurations with 10 internal channels and for a large stack in cross-flow. However, in this work, he did not consider the effects of fluid flow, relying on analytical equations for mass and heat transfer. Based on his own work and others, Achenbach authored the International Energy Agency (IEA) benchmark of simulations [16] which provides standard model parameters and results for code verification in cooperation with modellers in many European countries. In 1996, Ferguson et al. [20] demonstrated a single cell model of a SOFC using a finite volume approach to predict local distributions of temperature, electric potential, and concentration. The authors argue that the finite-volume method is more suitable than the finite element method (FEM) as the source terms are accounted for in a natural way in writing the balance of fluxes through a control volume. They studied the effect of rib width and electrode thickness on overall efficiency for cross-flow, co-flow, and counter-flow configurations and compared their results against the IEA

26 2.1. MODELLING OF FUEL CELLS 15 benchmark. Many works following Achenbach s original work focus on combinations of electrochemistry, heat transfer, species transport or fluid flow with varying degrees of approximation. The models include combinations of internal reforming, detailed analyses of kinetics, conservation of electrical charge, and stress calculations. With the improvement in computer power, analytical modelling has transitioned to CFD modelling covering larger length scales leading to comprehensive stack models Electrochemistry and Heat Transfer The following three research groups developed analytical models of species, charge, and heat transfer to study the effects of the activation, ohmic, and concentration overpotentials on cell performance. In 2000, Iwata et al. [21] developed a three dimensional analytical model considering species, charge, and heat along the flow directions to examine the effects of gas re-circulation ratio, pressure, and physical properties. Temperature and current density profiles for a planar SOFC in cross-flow, co-flow, and counter-flow were solved for an equivalent circuit of Nernst potential, and activation and concentration overpotentials. Chan et al. [22] created a general polarization model using Butler Volmer, and ordinary and Knudsen diffusion for concentration overpotential to test the sensitivity to thickness of fuel cell components. They showed that anode supported cells provide the lowest overpotential values. In 2003, Tanner and Virkar [23] examined the dependence of SOFC stack resistance as a function of interconnect contact spacing, contact areas, component thickness, and transport properties at interfaces. They derived an area specific resistance (ASR) for a repeat unit neglecting concentration polarization. They found that closely spaced

27 2.1. MODELLING OF FUEL CELLS 16 interconnect contacts and parallel channels were the best to minimize ASR. However, this is expected as they did not take into account the effect of those interconnect parameters on fuel, or temperature distribution. The following research groups developed models using the finite volume method (FVM) to focus on mass transport and temperature distributions for internal reforming reactions. In 2003, Ackmann et al. [24] used FLUENT and MATLAB to model mass transport by diffusion and the temperature distribution in the porous structure only. Electrochemical kinetics for an internal reforming and shift reaction were simulated for the anode and cathode substrate. They claimed that the model can be used to optimize mass transport in each electrode by changing structural parameters. In 2007, Janardhanan et al. [25] included direct internal reforming and assumed one dimensional plug flow through the channel and one dimensional species transport through the porous layers. They highlighted the importance of the temperature distribution and boundary conditions on the overall performance. In 2006, Liu et al. [26] examined temperature and current density distributions in a single cell molten carbonate fuel cell (MCFC) with 8 maldistributed inlet flows for a cross-flow configuration. They found that the maldistributions slightly affect mean temperature and mean current density, but strongly affect the distribution fields of each. They found that for a deviation of 25%, cell temperature variations were up to 12% greater than those in the uniform pattern. And, for a deviation of 25%, current density variations were up to 37% greater than the uniform pattern. From the same research group, Yuan and Liu [27] used the same methodology applied to an single cell SOFC. They found non-uniform air inlet flow creates hot spots with a cell plane temperature variation over of 10%. Non-uniform inlet flow creates cell plane current

28 2.1. MODELLING OF FUEL CELLS 17 density variations of between 10% to 50%. They also stated that non-uniform inlet flow always decreases average current density. While this is still a single cell model, it demonstrates the importance of understanding how non-uniform inlet flows can affect the current density distributions in a fuel cell Coupled Flow and Electrochemistry In the early 2000s, the following research groups modelled the effects of concentration, heat transfer, and fluid flow as well as electrochemistry in SOFCs in a half channel geometry. Yakabe et al. [15] considered a half channel, single cell, anode supported SOFC with hydrogen, carbon monoxide, and a water shift reaction to estimate the concentration polarization. They used a finite volume method to determine the fluid flow and distribution of reactant species assuming Darcy s law in the anode. They highlighted that, without great difficulty, experimental results can only show the average value over the anode, whereas modelling can show a distribution of concentration. Yakabe et al. [28] extended this work to simulate the concentration, temperature, potential, and current density distributions of a planar SOFC in co-flow and counter-flow. The conservation of mass, energy, and electrical charge were incorporated to determine the effect of cell size, operating voltage, and thermal conductivity on performance. Based on the the resulting temperature distributions, stress distributions were calculated using separate FEM software. They concluded that steam reforming can cause large internal stresses. Similarly, Yuan et al. [29] developed an in-house CFD model to solve gas flow and heat transfer on one half of a PEMFC duct. This analysis was extended to anode supported SOFCs [30], and intermediate temperature SOFCs [31] as only single phase flow was considered in both

29 2.1. MODELLING OF FUEL CELLS 18 cases. They employed a Fanning friction factor and Nusselt number to account for convective transport in the momentum and heat equations respectively. They investigated the effects of thickness of the porous layers, current density, permeability, and thermal conductivities on the fluid transport and heat transfer. They highlighted the importance of concentration polarization in anode supported cells, and the effect of duct geometry on gas distribution. The following research groups modelled a single cell comparing flow configurations for a simplified positive electrode - electrolyte - negative electrode (PEN) region. Recknagle et al. [32] created a 3D model which calculates the distribution of gases, temperature, current distributions, and utilization for cross-flow, co-flow, and counterflow configurations with internal manifolds. Cyclic boundary conditions were imposed at the top and bottom of a repeat unit cell to simulate a stack. They used the Navier stokes equations for flow, species and temperature, and coupled the information with solutions for electrochemistry. However, the PEN region was considered to be only one element in the stack direction, and an assumption was made for the effect of individual channels on the flow. Their results showed that the co-flow case had the most uniform temperature distribution and the smallest thermal gradients. Similarly, Campanari and Iora [33] performed an electrochemical and thermal analysis for cross-flow, coflow, and counter-flow geometries for a single cell, planar SOFC. They incorporated ohmic, activation, diffusion losses and a hydrocarbon kinetic model. They used a simplified heat transfer equation in a coarse mesh considering the PEN, air, fuel, and electroyte to be one finite volume thick. They were able to locate temperature hot spots and compared the performance of co-flow and counter-flow configurations. They highlight the possibility of optimizing air ratio for efficiency while keeping in mind

30 2.1. MODELLING OF FUEL CELLS 19 local hot spot temperatures. Xia et al. [34] coupled electrochemical kinetics with fluid dynamics to simulate heat and mass transfer using FLUENT and an in-house code. They modelled a one cell stack of a planar SOFC and performed a uniformity analysis for distributions of temperature, current density, and overpotential. They found that the counter-flow configuration had the most uniform current density and temperature distributions and a higher power output, fuel utilization, and fuel efficiency than the co-flow case. This is in contrast to the conclusion of Recknagle et al. [32] that the co-flow configuration provided the most uniform temperature distribution. In the following works, finite volume models were compared with experimental results. In 2004, Larrain et al. [35] created a generalized 2D model of a planar SOFC repeat unit which can calculate distributions of concentration, reaction rate, and temperatures using a finite difference method. They solved the species and energy equations without CFD and solved for velocity using an uncoupled Darcy equation. They claim this is beneficial for optimization as geometric parameters can be explored without regenerating a new mesh. However, the method used did not allow mesh refinement near the inlet causing errors and mesh dependence. In the following year, Van Herle et al. [36] showed a limited comparison of their SOFC model with experiment for a short stack. Autissier et al. [17] used FLUENT to solve for current density, flow, temperature, and concentration distributions in a small SOFC cell using the mass, momentum and energy equations including radiation coupled with chemical and kinetics equations. The results are compared with experiment for current and potential showing a good accuracy. They highlight the importance of geometry design on the supply of reactants to the active surface. Bedogni et

31 2.1. MODELLING OF FUEL CELLS 20 al. [37] considered a circular-planar intermediate temperature SOFC using an inhouse FVM code. They included mass and energy balances to provide predictions of current, voltage, and total resistance and compared with experimental results. Fluid flow was considered using a separate and uncoupled CFD code to help explain the experimental results. The model was calibrated by adding new resistance terms based on voltage, current and temperature values. They claim to be able to use the model to predict cell behaviour based on different operating conditions. Wang et al. [38] modelled a planar, electrode supported, single cell SOFC with the conservation of mass, momentum, energy, and electric charge in three dimensions. They employed equations for activation, ohmic, and concentration losses and compared the resulting power curve with experiment. They found that the mass transport resistance in the anode region was the dominant source of loss at high utilizations. Many of the above groups used limited experimental data to claim their models were representative of actual performance. The following two research groups presented models with electrochemistry coupled with the full transport equations. However, each of the descriptions were unclear, and showed very limited results. In 2003, Pasaogullari and Wang [39] used FLUENT to couple electrochemical kinetics with gas dynamics and species transport. They were able to calculate flow field, species concentration, potential, and current distributions. The conservation of mass, momentum, species, energy, and charge were employed for a 5 channel cell in cross-flow. However, this model and the results are not well documented. In 2005, Hwang et al. [40] coupled electrochemical kinetics with fluid dynamics in a study of mono-block-layer built (MOLB) SOFCs. In this design, trapezoidal flow channels with a corrugated PEN and interconnect region are

32 2.2. STACK SCALE MODELLING 21 employed to combine the sealing advantages of tubular and the power density of planar stacks. They employed the conservation of mass, momentum, energy, species, and charge to solve for local variations of overpotential and current density in the electrodes. This comprehensive application of the finite element method is not well documented. In 2010, Jeon et al. [41] compared two different approaches for obtaining diffusion coefficients on the performance of a single cell in cross-flow, co-flow, and counterflow. The methodology included coupling the momentum, species, and heat transport equations with electrochemistry using OpenFOAM. They found that the different diffusion coefficents have a strong effect on the entire solution and that the counterflow case has the most uniform temperature distribution. They found that the local fluctuations in mass fraction of oxygen based on the internal channel placement were large compared with the difference between the inlet and outlet. This effect was much less pronounced in hydrogen transport. However, they showed a higher utilization of oxygen than hydrogen which does not represent real operating conditions. 2.2 Stack Scale Modelling Stack scale modelling is focused on the two main design considerations at that scale: distributions of temperature and reactant species. Early cold flow models showed only fluid distributions in the fuel cell geometry without the effects of electrochemistry and mass transfer. Since then, researchers have incorporated heat and species transport, however the literature on this subject is not extensive. A small amount of researchers are examining the effects of flow paths on the temperature and reactant distributions using fully coupled CFD models.

33 2.2. STACK SCALE MODELLING 22 In 1994, Costamagna et al. [13] in association with Achenbach, studied the flow distribution in fuel cell stacks of up to 100 cells with external manifolds by comparing numerical, analytical, experimental results. Pressure and velocity distributions were estimated for various Reynolds numbers and geometric shapes. They highlighted the importance of manifold design in overall performance in a fuel cell stack. In , Boersma and Sammes [42, 43] developed an analytical model of fuel cell stacks based on hydraulic resistances. For a counter-flow case, resistances were assumed for flow through the manifold channels and for the splitting and combining of flows in the feed channels. They were able to point out the ratio of cells receiving less than the average mass flow to those getting more than average. Throughout the 2000s, several cold-flow models were developed to investigate the effects of geometry on flow. Beale et al. [44] used the distributed resistance analogy to investigate the flow fields for 10 cell and 50 cell SOFC stacks. The authors suggest that for a large stack it is necessary to maintain high viscous losses and pressure gradient within the core to ensure flow uniformity. Kee et al. [45] solved mass and momentum conservation equations in a 2D model with laminar flow in the cells of a generic fuel cell. Different friction factors were employed to capture the flow in the channels and the external manifold regions. They present flow uniformity as a function of several design parameters and highlight that channel geometry has a strong impact. Koh et al. [46] developed an algorithm in 2D to solve the pressure and flow distribution of an generic fuel cell stack. They employed Darcy s law in the channels and modeled a 100 cell MCFC stack. This is somewhat applicable to SOFCs since it is a strictly cold flow model. They found that the wall friction pressure loss is negligible compared with the pressure recovery in inlet manifolds or loss in outlet

34 2.2. STACK SCALE MODELLING 23 manifolds due to mass combining or dividing at flow junctions. They found that the non-uniform distribution is more significant at the cathode as a larger mass flow of air is used compared with fuel. Huang and Zhu [47] compared an analytical solution of a resistance network with a CFD model of a U-type generic fuel cell stack. They assumed uniform temperature, laminar flow, and no local losses such as dividing, combining, expanding, or contracting. They emphasize that flow uniformity between cells in a stack could be improved by reducing the pressure loss in the header compared with the channels. Chen et al. [48] developed a 2D model to analyze PEMFC stacks which is partially applicable to SOFC stacks as the working fluid was single-phase air and heat and electrochemical reactions were ignored. They found that larger channel resistance, larger manifold widths, and lower feeding rate could enhance flow uniformity. They state that the excessive pressure drop of larger channel resistance is not beneficial, therefore enhanced manifold width is a better solution. It is difficult to support this claim as the impact on utilization and electrochemical operation was ignored. The following research groups studied manifolding in fuel cell stacks by comparing models with experiment. Costamagna et al [13] experimentally measured pressure and velocity for various Reynolds numbers and geometries to validate their model. Chernyavsky et al [49, 50] studied the effects of flow non-uniformities in PEM fuel cell stack headers using cold-flow models complemented with Particle Image Velocimetry (PIV) measurements. The authors state that the flow, especially in the outlet header, has a substantially unsteady character. Lebaek et al. [51, 52] and the group of Grega [53, 54] compare simulations with experimental PIV measurements strictly for flow distribution in manifolds showing good overall agreements. The authors

35 2.2. STACK SCALE MODELLING 24 emphasize the importance of inlet manifold geometry and flow conditions on flow distribution and head loss. In 2003, Beale and Lin [7] compared three methodologies combining heat, species, and fluid flow for solutions of a planar SOFC stack: a detailed numerical model (DNM), distributed resistance analogy (DRA), and a presumed flow method (PFM). The main difference is that the DRA and PFM have prescribed momentum, heat and mass transfer coefficients (based on Fanning friction factor, Nusselt and Sherwood numbers) in the core region of the stack while solving the full Navier-Stokes equations in the manifolds. The comparison was made for a single rectangular duct without internal channels. Results were compared for a 10 cell stack showing good agreement. In 2005, The DRA method was expanded in detail in Beale and Zhubrin [5]. Another comparison between the DNM and DRA was made for planar air and fuel ducts with no internal manifolds. They emphasize the advantages of employing educated assumptions when designing an intricate system of channels or larger stacks. In 2004, Khaleel et al. [18] used the finite element method to couple electrochemistry with a flow and thermal analysis. They were able to generate current density, heat, species concentration, and flow distributions. However, the article focuses on the development of the model and does not show many results. Only distributions of temperature were shown in the article. They do show a voltage plot for a 30 cell stack stating that the outmost top and bottom 2-3 cells deviate substantially from the average cell voltage. In 2008, Yuan [12] studied the performance of a ten cell SOFC stack with nonuniform flow rate in the stack direction. It was found that since the inlet flow rate of fuel is controlled to reduce wastage, it dominates the voltage and current density

36 2.2. STACK SCALE MODELLING 25 distributions. A deviation of 50% from the mean flow rate n f is used, meaning 0.5 n f at the bottom cell, and 1.5 n f at the top cell. When this deviation is applied to the fuel inlet flow rate, the voltage varies by up to 15% for a ten cell stack with constant mean current density at each cell and an adiabatic boundary condition. In practice, higher flow rates of inexpensive air are used which result in larger cooling effects than fuel. The air flow therefore dominates the temperature distributions at each cell. When a deviation of 50% is applied to the air inlet flow rate, a variation of 16 is found from the top cell to the bottom cell for an adiabatic boundary condition. While this study considered the use of reformed methane and carbon monoxide as well as internal manifolds, the anode, cathode, and electrolyte regions were considered one unit, and the calculations were done in two dimensions with inter-phase transfer terms. Furthermore, a simple mass balance was used and the momentum equations are not considered. In 2009, Bi et al. [6] performed CFD calculations for flow distribution in SOFC stacks with different numbers of cells. They tested the effects variations in channel height and length, height of the repeating cell unit, and manifold width on flow uniformity. They found that the ratio of outlet manifold width to inlet manifold width is a key parameter for flow uniformity which depends on the number of cells in the stack. This article is beneficial as it focuses on a single parameter that can used in stack design to improve flow uniformity. However, this study used a simple mass balance and did not include species or heat transport which both have large effects in stack design. They state that almost all previous studies were based on simplified models helpful for overall characteristics and guidelines involving numerous approximations which may or may not be applicable to realistic 3D stack configurations.

37 2.2. STACK SCALE MODELLING 26 In 2011, Lai et al. [55] demonstrated a quasi-two dimensional model for current density, species concentration, and temperature distributions of large SOFC stacks which generates solutions in a relatively reduced computation time. The authors assume that the physical fields across the in-plane direction remain largely unchanged, therefore it can be assumed that a single channel can reasonably approximate the three dimensional solution in certain conditions. They also assume that the mass diffusion is small relative to the bulk flow throughout the SOFC and they do not model internal or external manifolds assuming uniform flow distributions. Results are compared with the IEA benchmark and another 6 cell stack model showing good agreement. They then display results for a hypothetical model of a 96 cell stack due the lack of detailed published data on tall SOFC stacks. Their results show voltage and temperature distributions for each cell in the stack. They investigate the effects of internal reforming, measurement plates, and flow maldistributions only at selected cells. While comprehensive models of large stacks are rare, the assumption of a uniform flow distribution and the exclusion manifold effects limits the applicability of the results. In 2011, Peksen [56] presented a three dimensional thermofluid and thermomechanical analysis of a 36 cell SOFC stack. In this analysis, the reaction is not reflected in the flow distribution and the heat sources are assumed. They claim that the thermofluid and thermomechanical models are coupled, however, only the resulting temperature distribution is transported to the thermomechanical analysis. The flow was found to be fairly uniform between cells resulting in a limited thermal stress effect for the stack direction. In early 2013, Peksen et al. [57] presented the transient thermomechanical response of a short SOFC stack including the effects of a wire

38 2.3. RESEARCH OPPORTUNITY 27 mesh, metal interconnect, and sealant materials. They indicate thermal stress within the start-up phase is most critical. While work from this group is focussed mainly on thermal stress analysis, there is promise for advanced SOFC stack models with complex multiphysics. A comprehensive model which includes species, heat, and momentum transport coupled with an electrochemical model for a SOFC stack with internal and external manifolds has not been found in the literature to date. 2.3 Research Opportunity After a review of the relevant literature on cell and stack scale modelling, it was found that researchers recognize the importance of modelling at the stack scale. As computational power improves, and modelling techniques become better validated with experimental work, more comprehensive models can be created to characterize some of the important processes at the stack scale. A recurring theme throughout the literature are two major concerns for the performance and durability of SOFC stacks: reactant and temperature distribution. The following summarizes the motivation for modelling at the stack scale and looks at upcoming research topics to illustrate where the work described in this thesis fits in Reactant Distribution Fuel cell stacks generally exhibit lower performance than the sum of individual cells treated in isolation in an idealized fashion [11]. Differences in the voltage output have been observed between the top and the bottom of the device partly caused by the non-uniform distribution of the feeding gas along the stack [13]. Flow and pressure

39 2.3. RESEARCH OPPORTUNITY 28 maldistributions occur, due to poor manifold design, leading to variations in temperature and local current density, and reductions in total stack voltage [12]. Cell areas covered by concentrated fuel become electrochemically active, leading to increased local temperatures, and increased reaction rate. Cell areas with depleted fuel become inactive, cooler and have slower localized reaction rates. Increased fuel flow increases uniformity of reaction rates across the active area and decreases utilization. Decreased fuel flow increases utilization, but can cause local depletion (cold-spots) creating large temperature non-uniformities [32]. A lack of reactants at reaction sites can cause confined but irreversible damage to a cell and strongly limit the performance of the fuel cell [13]. Geometry strongly affects maldistributions of reactants at reactant sites between cells and across channels [6, 12, 5, 17, 58, 32, 34, 26, 32, 46]. Nonuniform distributions occur when pressure gradients in the manifolds are not small in comparison with those in the fuel cell passages [6, 5]. Inertial effects alone will cause the pressure gradient across the stack at the top to be less than across the bottom resulting in variations in the flow field [5]. To minimize this tendency, designers should ensure the cell passages are small in comparison with the manifolds. However, overall net pressure drop should be as low as possible to reduce parasitic power needed to drive a blower [6, 45]. The effects of geometry on reactant distribution become more pronounced with larger stacks. Construction of 100 cell stacks is costly to undertake [46], therefore, modelling is essential. It is well understood that in large stacks, a transition to turbulent flow can occur. Bi et al. [6] assume a Reynolds of greater than 2000 is considered turbulent. They state that a 40 cell stack can give a Reynolds number of approximately 4000 for air and 500 for fuel. Koh et al. [46] assume the manifolds are

40 2.3. RESEARCH OPPORTUNITY 29 mostly turbulent in a 100 cell stack and that flow in the channels is laminar Temperature Distribution One major aspect of SOFCs is the fact that they work as an electrochemical generator as well as a heat exchanger [9]. At high temperatures, thermal management of SOFCs is of great importance. Thermal stresses due to temperature gradients, thermal shocks, and heat cycles contribute to cell degradation [14, 5, 15]. Mismatches in the thermal expansion coefficients of cell components can cause cracking and sealing problems, and repetitious thermal cycling may result in cracking of components and lead to gas leakage [15, 41]. Thermal management and response to fluctuating inlet compositions are necessary for material stablility [17]. Temperature gradients are generated by the distribution and utilization of fuel and air [32] and must be minimized. Air and fuel inlet temperatures affect reaction rates, temperature, and fuel utilizations. Therefore, management of air and fuel flow and temperature distribution is critical to stable operation. Parameters for managing heat are geometry, internal or external steam reforming, thermal conductivity, and operating voltage [15]. Even if fluid flow and chemical reaction rates are completely uniform, there will be temperature gradients and the metallic interconnect should be highly conductive for thermal management [7]. The temperature distribution also has a strong effect on the electrochemical performance [5], and electronic and ionic conduction. An uneven temperature distribution found in the stacking direction of an SOFC model results in an increased ohmic loss in the electrolyte and increased overpotential at the electrodes in the colder cells [19].

41 2.3. RESEARCH OPPORTUNITY Other Considerations The practical operation of fuel cells relies on real physical processes and can stray from the idealized cases which are modelled. The following design issues are areas outside the scope of this work. - A variation in current density causes longer current paths in some regions of the stack increasing the ohmic losses lowering cell voltages [19]. - Design parameters such as the spacing between interconnect contacts affect the current path [23]. - Thermal radiation has a significant effect and should be taken into account for effective thermal management [59]. - The reactions of internal reforming can result in temperature gradients, and other considerations [19, 28, 24, 60]. - Stress distributions due to thermal gradients and sealing pressure cause mechanical instability for practical installations [28]. - Time dependence for startup, thermal cycling, response to varying inlet conditions, and stack degradation should be studied [17, 61, 62]. - Current free spots cause a redistribution of current over a defective cell [63]. This can affect adjacent cells in a stack [64]. - Current free spots are caused by: - flow maldistributions. - resistive oxides forming on the interconnect [23, 64]. - delamination of the catalyst layer from the membrane due to thermal cycling/rapid heating [62] - chromium deposition. - A proper coupling of properties displayed at the microscale with those at a larger scale is not well investigated [10].

42 2.3. RESEARCH OPPORTUNITY 31 The models described in Chapter 3 provide tools for the analysis of the two major concerns in SOFC stacks: temperature and reactant distribution. They provide a foundation for the addition of further design considerations.

43 32 Chapter 3 Modelling Techniques Two computational fluid dynamics models are considered in this work: a detailed, and a simplified model. The detailed model incorporates the full conservation equations to provide a high level of detail in the manifolding and reaction regions of a fuel cell stack. In the simplified model, assumptions are made for the fluid flow, heat transfer, and mass transfer in the core region of the stack. For the momentum transport equation, a Fanning friction factor is applied in place of the viscous terms. A rate term is applied to replace the diffusive term in the energy equation. A resistance to mass transfer is assumed for the species transport between the bulk species in the channel and the active region. By making assumptions in the regions of the internal channels and the active porous regions in the computational mesh, overall solutions for very large stacks and cell sizes can found in a much shorter computation time. First, the detailed stack model is described fully. Then, the deviations from the detailed model found in the simplified model are described. The basis for the detailed stack model is derived from a detailed cell model which was previously developed by others in the 3D finite-volume CFD program OpenFOAM [41, 65]. In this work, the present author expanded the cell model to multiple layers of cells to create a

44 3.1. DETAILED MODEL 33 detailed stack scale model. The present author also created the simplified model in OpenFOAM. 3.1 Detailed Model The conservation equations and their interaction with electrochemistry are presented. Then, the geometry and solution procedure are described Conservation Equations The general conservation equation found in Eqn. 1.5 is applied for mass, momentum, energy, and species transport. The conservation equations, as applied in the detailed model, are summarized in Appendix A.1. Continuity and Momentum Mass conservation is given as the continuity equation for a unit volume in the form (ρ u) = 0 (3.1) where ρ is fluid density and u is the three dimensional local velocity vector. The momentum equations in three dimensions are applied in the form of the full Navier Stokes equations with an additional Darcy flow term [66]. (ρ u u) = p + (µ u) µ u (3.2) κ where p is pressure, and µ is viscosity. Viscosity is considered constant in the fluid regions. After and including section 5.3, viscosity is considered to be variable as

45 3.1. DETAILED MODEL 34 explained in section The Darcy term is applied within the porous electrodes and current collector layers for a constant permeability κ of the porous media. The mixture density is calculated using the ideal gas law for the distribution of molar fractions x i of the fluid species in air and fuel ρ = p RT xi M i (3.3) where R is the universal gas constant and M i is the molar mass of each species. The air region is a binary mixture of O 2 and N 2 and the fuel region is a binary mixture of H 2 and H 2 O. Heat Transfer The energy conservation equation is expressed in terms of temperature T and is applied in the form (ρc p ut ) = (k T ) + q (3.4) where c p is specific heat, k is thermal conductivity, and q is a heat source. Both the convective and diffusive terms are applied in the air and fuel regions. In the solid regions, only the diffusive term is applied. The volumetric heat source term q is calculated based on the heat due to the electrochemical reactions ( q = i HT H e 2F + V cell ) (3.5) where F is Faraday s constant. The generated heat is that of the reaction enthalpy H T minus the electrical work for a given area current density i and voltage difference V cell. The generated heat is applied throughout the electrolyte region by

46 3.1. DETAILED MODEL 35 converting from an area source to a volumetric source. For this planar case, the area divided by volume is the inverse of the height of the electrolyte, H e. The specific heat c p,i is calculated as a 7th order polynomial function of temperature throughout the fluid for each component species and is molar averaged to get the overall specific heat c p of each fluid [67]. The enthalpy of formation of each species is calculated by integrating the polynomial from a reference enthalpy of formation [68]. H T = H H H O 2 H H2 0 (3.6) H i = T K (c p,i ) dt + H 0 (3.7) The thermal conductivity k is given as a constant value for each of the air, fuel, porous media, and interconnect regions. Thermal conductivity is considered to be variable after and including section 5.3 as explained in section In the porous region, the thermal conductivity is calculated simply as an arithmetic volume average of the thermal conductivities of the solid and fluid phases in the porous media based on porosity [69]. Species Transport Species transport within the gas phase is in the form (ρ uy i ) = (ρd y i ) + ṁ i (3.8) where y i is the mass fraction of each species, D is diffusivity, and ṁ i is the species source. The species transport equation is solved for binary diffusion of O 2 in the

47 3.1. DETAILED MODEL 36 O 2 N 2 mixture. Similarly, H 2 transport is solved in the binary H 2 -H 2 O mixture. The sum of the species mass fractions for each binary mixture is one. Mass fractions, y i, can be converted to molar fractions x i by multiplying by the molar mass. The Fuller-Schettler-Giddings (FSG) correlation is used to obtain the diffusivity for the gases in the channels [70]. In the porous media, the effective diffusivity is obtained using Bosanquet s relationship to combine the effects of binary and Knudsen diffusion, as outlined in Jeon et al. [41]. The electrochemical reactions are assumed to occur at the electrode/electrolyte interface. The species source term, m i, is applied as a boundary condition based on Faraday s law for the local current density i as follows ṁ i = M ii vf (3.9) for molar mass M i, electron valence v, and Faraday s constant F. In the electrochemical half reactions, species source term is applied as a sink for H 2 and O 2 and a source for H 2 O. The mass flux through a fixed area boundary is defined for a system of species i in terms of mass flow rate and mass fraction from Fick s first law [70] ( ṁ i = y i ṁ i + ) ṁ j ρd y i (3.10) j i The first term on the right hand side of Eqn represents the convective mass flux of species i generated by the electrochemical reactions. The second term on the right hand side represents the mass flux due to the concentration gradient at the interface. The mass flux of the gas mixture through the cathode and anode interfaces can be

48 3.1. DETAILED MODEL 37 calculated by rearranging Eqn to the following form ρd y i = ṁ i (1 y i ) y i ṁ j (3.11) This equation is applied as a velocity and species boundary condition. j i Boundary Conditions The velocity, temperature, and mass fraction of species are prescribed at the inlet. Inlet velocities can be preliminarily calculated based on a target utilization ζ i of inlet flow rates of O 2 for air and H 2 for fuel. ζ i = ṁi,in ṁ i,out ṁ i,in (3.12) The flow is presumed uniform for the velocity field and boundary conditions of zero gradient are applied for temperature and species mass fraction at the outlet. The boundary conditions are shown in Table 3.1 for momentum, heat, and species transport in air and fuel. To account for the species sources and sinks due to the electrochemical reactions, the boundaries at the electrode/electrolyte interfaces are specified based on Eqn 3.11 and shown in Table Electrochemistry The reduction of oxygen and the oxidation of hydrogen are the electrochemical halfreactions considered in this work. Since they occur on either side of a very thin electrolyte, they are treated together as the Nernst potential, or the maximum amount

49 3.1. DETAILED MODEL 38 Table 3.1: Boundary conditions for the conservation equations Type Momentum Heat Species Air inlet u a = U a,in, pa n = 0 T a = T a,in y O2 = y O2,in, y N2 = y N2,in Air outlet u a n = 0, p a = 1atm Ta n = 0 y O2 n = 0, y N 2 n = 0 Fuel inlet u f = U f,in, p f n = 0 T f = T f,in y H2 = y H2,in, y H2 O = y H2 O,in Fuel outlet u f = 0, p n f = 1atm T f = 0 y H2 n n = 0, y H 2 O n = 0 External Boundaries u = 0, p n = 0 T = 0 n y i n = 0 of voltage that can be derived from the overall reaction. The Nernst potential, E, from Eqn. 1.3 is shown here neglecting the effect of pressure E = E T + RT ( 2F ln xh2 xo 0.5 ) 2 x H2 O (3.13) where the mole fractions, x i, and temperatures, T, are at the electrode/electrolyte interfaces. The cell potential at the reaction temperature E T is given by the following formula [68] E T = G rxn = H rxn T S rxn 2F (3.14) where the Gibb s free energy G rxn is given by the formation enthalpy H rxn less the entropy T S rxn generated by the H 2 - O 2 reaction at the reaction temperature, T.

50 3.1. DETAILED MODEL 39 Table 3.2: Boundary conditions for the conservation equations at the electrode/electrolyte interfaces Type Momentum Species Cathode/electrolyte interface u a = m O 2 ρ aa y O2 n = m O 2 (1 y O2 ) ρd a Anode/electrolyte interface u f = m H 2 +m H 2 O ρ f A y H2 n = m H 2 (1 y H2 ) m H 2 O y H 2 ρd f The formation enthalpy and entropy are calculated at a constant reference temperature of 925K. The operating cell voltage V for a given cell within a stack is calculated using the following formula V = E ir ohm η a η c = E ir ASR (3.15) where the anodic η a and cathodic η c overpotentials are combined with the local ohmic resistance R ohm to come up with a lumped area specific resistance R ASR of the cell. The area specific resistance is calculated using a polynomial fit to experimental data used for comparisons with the simplified model [71, 72]. The current density distribution and cell voltage are solved using a procedure in which the cell voltage is corrected at each iteration. The current density distribution is calculated by rearranging Eqn into the following formula [73] i = E V R ASR (3.16) Then, the mean current density is calculated by integrating the average the current

51 3.1. DETAILED MODEL 40 density distribution over the active area. This mean current density i is used to correct the voltage V by comparing with the target mean current density i 0 at each iteration using the following formula V = V old + r (i i 0) (3.17) where r is a correction factor and any reasonable value will produce convergence Domain Decomposition The geometry under consideration is an anode-supported SOFC made up of metallic interconnects, air and fuel channels, and a positive electrode - electrolyte - negative electrode (PEN). The PEN in the detailed model is composed of the cathode current collector layer (CCL), cathode functional layer (CFL), electrolyte, anode functional layer (AFL), and anode CCL. The geometry is specified in Appendix B.1 along with some physical properties. To properly capture the transport in the porous media as well as the channels, the detailed model requires a high mesh resolution in the fluid regions at and near the PEN. In the simplified model, the air, fuel, PEN, and interconnect regions are considered to be one computational cell thick for each cell in the stack. Volume fractions are employed to account for the volume difference between the computational cell and the volume of each region. As described in Jeon et al. [41], the detailed model is solved on two computational domains; there is one global domain for the entire SOFC, and there are subdomains for each of the air, fuel, electrolyte, and interconnect regions. The air and fuel regions each include porous media of their respective electrode and current collector layers.

52 3.2. SIMPLIFIED MODEL 41 The pressure, momentum, and species mass fractions are each solved on the fluid domains. The temperature is solved on the global domain and is mapped to each respective region. The equations are converted to steady state linear algebraic equations in Open- FOAM. In each case, the domain is tessellated with a rectilinear computational mesh. The choice of solvers and discretization schemes can be found in Jeon et al. [41] Solution Procedure First, the mesh, constants, solution fields, boundary conditions and other parameters are initialized. A target mean current density is prescribed and a cell voltage is guessed. The solution algorithm proceeds as follows: 1. Solve the velocity and pressure fields using the PISO algorithm [74]. 2. Calculate mass diffusivity and solve the species transport equation. 3. Map subdomain fields to global domain. 4. Solve energy equation for global temperature. 5. Solve Nernst equation, lumped internal resistance, current density, voltage and electrochemical heating. 6. Calculate species and velocity flux boundary conditions from electrochemical sources. 7. Map global temperature to subdomains. 8. Repeat until convergence is obtained. 3.2 Simplified model The simplified model is an application of the distributed resistance analogy [5]. The deviations from the detailed model in the conservation equations are shown. Then,

53 3.2. SIMPLIFIED MODEL 42 the differences in geometry and domain decomposition are described Conservation Equations The simplified model incorporates four distinct solution domains: air, fuel, PEN, and interconnect. The full conservation equations are applied in the fluid at the external manifold regions, however, in the core region (along the internal channels) the interphase transfer terms are applied in place of the diffusive terms. Volume fractions are also applied in the core region to account for the volume difference between the mesh and the fuel cell geometry. The conservation equations, as applied in the simplified model, are summarized in Appendix A.2. Continuity and Momentum In the simplified model, a volumetric mass source is applied to the conservation of mass equation in the bulk flow of air and fuel regions along the channel. Mass conservation is given as the continuity equation in the form (ερ u) = εṁ (3.18) where ε is the volume fraction of the fluid. The volumetric mass source in the continuity equation ṁ is the sum of the mass sources and sinks of each species in the binary mixture ṁ = m i (3.19) H e where H e is the height of the electrolyte and m i is computed from Faraday s law in Eqn The area source is converted to a volumetric source by dividing by the

54 3.2. SIMPLIFIED MODEL 43 height of the electrolyte. The momentum equation is shown here (ερ u u) = ε p εf ( u n u) + (εµ u) (3.20) where the term F represents a resistance to momentum, u is considered the bulk velocity of the fluid within the core region, and the velocity at the channel walls u n is zero. The resistance is applied as a source term based on solutions for one dimensional pressure drop due to wall shear stress for rectangular channels in laminar flow. The pressure drop is given by dp dx = 4 τ w = 4 f 1 D h D h 2 ρ u2 = 2 fre Dh 2 h µ u (3.21) where τ w is the wall shear stress, f is the Fanning friction factor, and Re h is the Reynolds number based on the hydraulic diameter, D h. Then, the term F, is given by F = 2 fre Dh 2 h µ (3.22) and fre h is approximated by solutions for rectangular channels [75] based on aspect ratio. Heat Transfer The inter-phase heat transfer terms are calculated between each two region combination of air, fuel, PEN, and interconnect except for air to fuel since they do not come in direct contact. The energy conservation equation is expressed in terms of

55 3.2. SIMPLIFIED MODEL 44 temperature and is applied in the form (ερc p ut ) = ε n α n (T n T ) + (εk T ) + ε q (3.23) where c p is heat capacity, α n is the inter-phase heat transfer coefficient, k is thermal conductivity, q is the source term, and T n is the temperature of the neighboring region. The inter-phase transfer terms are computed as αv = U T A (3.24) where A is the area for heat transfer and V is the volume of the region under consideration. U T is the overall heat transfer coefficient obtained by harmonic averaging the convection and diffusion terms, for example; 1 U T = 1 h + H k s S (3.25) where S is a conduction shape factor, and H is the conduction length for the solidphase thermal conductivity k s. In the case of the inter-phase transfer between the interconnect and PEN regions, the harmonic averaging occurs for the different diffusive k values for the two solid regions and the convective term is excluded. The convection heat transfer coefficient for a given fluid region h is calculated using Nusselt correlations for fully-developed laminar flow in a rectangular channel, namely h = Nuk f D h (3.26)

56 3.2. SIMPLIFIED MODEL 45 where Nu is the Nusselt number and k f is the diffusion heat transfer coefficient for the fluid phase [75]. Species Transport Species transport within the gas phase is in the form (ερ uy i ) = (ερd y i ) + εṁ i (3.27) where ṁ i is a volumetric mass source, D is diffusivity. Similar to the detailed model, source terms are computed from Faraday s law for each ion s chemical reaction at the assumed reaction site. The species mass fractions at the electrolyte wall y i,w needed for the Nernst equation are computed from [76] y i,w = y i + y i,t B 1 + B (3.28) where y i,t is the mass fraction of species i at the transferred substance state (T state) [77] and y i is the bulk species fraction in the channels. B is a mass transfer driving force used approximately as B = exp (ṁ g ) 1 (3.29) where g represents a mass transfer conductance. Similar to the calculation of the heat transfer coefficient, g is calculated by harmonic averaging the convection and diffusion terms 1 g = 1 g f + H ρd s S (3.30)

57 3.2. SIMPLIFIED MODEL 46 where S is a diffusion shape factor [78], and H is the conduction length for the porous diffusivity D s. The convection mass transfer coefficient g f for a given fluid region is calculated using Sherwood correlations for fully-developed laminar flow in a rectangular channel, namely g f = Sh ρd f D h (3.31) where Sh is the Sherwood number and D f is the diffusion mass transfer coefficient for the fluid phase [79]. The diffusivities for both the solid D s and fluid D f are considered constant and known in this calculation. Examples of channel properties and inter-phase transfer terms can be found in Appendix B Transport Properties The work after and including section 5.3 was performed using variable properties for viscosity µ and thermal conductivity k for air and fuel. The viscosity and thermal conductivity are both calculated as 7th order polynomial functions of temperature throughout the fluid for each component species [67]. The Wilke method is used as the mixing rule for viscosity [80]. For thermal conductivity, the similar Mason and Saxena modification of the Wassiljewa equation is used [81, 82] Domain Decomposition In the simplified model, four subdomains with the ability to overlap on a global domain are created for air, fuel, PEN, and interconnect. Internal mapping between subdomains is determined during grid generation. This strategy gives flexibility for the specification of internal and external manifold geometries while still allowing a solution within each of the overlapping PEN, fuel, and air regions.

58 3.3. CLOSURE 47 The mesh is one computational cell thick for each cell in the stack. This is so there are only bulk values in the air and fuel channels, and only wall values representing the electrolyte surface in the PEN region. The temperature distribution in the PEN region and the mole fractions of reactants and products at the fluid channel walls are used to generate Nernst potentials, and therefore voltages and current density distributions for each cell in the stack. 3.3 Closure Two distinct mathematical models were described in terms of the conservation equations coupled with electrochemistry and their implementation in OpenFOAM. The detailed model incorporates the full conservation equations whereas the simplified model includes assumptions for fluid flow, heat transfer, and mass transfer in the core region of the stack. These assumptions allow a reduction of the computational mesh in the core region resulting in a drastically reduced computation time. Results from the detailed and simplified models are compared against one another in Chapter 4 to show that the simplified model is a practical option for the investigation of SOFC stacks. Then, the simplified model is used to analyze the effects of flow on temperature distributions and electrochemical performance of large stacks.

59 48 Chapter 4 Comparison of Detailed and Simplified Models Comparisons are made between the detailed and simplified models for co-flow and counter-flow configurations for a 4mm 100mm cell. In each case, species fraction, current density, voltage, and temperature are compared. Then, cell voltages are compared from a three cell stack for varying inlet conditions between cells. 4.1 Geometry and Mesh Single cell and three cell stack configurations are used to compare the detailed and simplified models. The geometry of each cell is a single channel with internal lands and channels. This geometry is chosen to minimize redundancy when investigating parallel channels. The active area for each cell is m 2 based on a 4mm 100mm geometry. Comparisons are made in co-flow and counter-flow configurations. Figure 4.1 shows the mesh used for a single channel and a three cell stack using the detailed model. After a study of grid independence, a mesh size of 307,200 ( ) is used for the 4mm 100mm geometry and 921,600 ( ) for the three cell stack. For the simplified model, a mesh size of 200 ( ) is used for the 4mm 100mm geometry and 600 ( ) for the three cell stack.

60 4.1. GEOMETRY AND MESH 49 Figure 4.1: Front view of detailed model mesh for single cell and three cell stack It was found that while the simplified model allowed a smaller computational mesh size and reduced computational times, it required more iterations to converge based on the large amount of inter-phase transfer terms in both the species and heat transport equations. This, however, is related to the specifics of the solver algorithm implemented in OpenFOAM, and could be improved eg. with a partial elimination algorithm or a co-located scheme. The single cell detailed model took an average of 70 minutes on one core of an eight core, Intel Xeon processor with 2.66 GHz processor speed, and 16GB of RAM using Mac OSX. Based on the large difference in mesh size, the simplified model showed an average computation time of 1 minute which is 1.4% that of the detailed model.

61 4.2. OPERATING CONDITIONS 50 Table 4.1: Single-cell operating conditions Mean current density (A/m 2 ) 6000 Inlet mass fraction H Inlet mass fraction O Inlet temperature (K) 1000 Outlet pressure (Pa) Inlet flow rate of fuel (kg/s) x10 8 Inlet flow rate of air (kg/s) 7.911x10 6 Utilization H Utilization O Operating Conditions Uniform inlet and outlet flows are presumed and the effects of external manifolds are not considered in this comparison. The velocity, temperature, and mass fraction of species are prescribed at the inlet. Inlet velocities can be preliminarily calculated based on a target utilization of inlet flow rates of O 2 for air and H 2 for fuel. The operating conditions are shown in Table 4.1. The fuel region consists of H 2 and H 2 O, and the air region consists of O 2 and N 2 only. For the comparisons, a target mean current density of 6000A/m 2 is used. The flow rates and utilizations are based on inlet velocities of 7.5m/s and 0.75m/s for air and fuel, respectively, and an an inlet cross sectional area of 2mm 1.5mm. 4.3 Co-Flow Figure 4.2 shows the current density distribution at the reaction region for the (a) detailed and (b) simplified models at a mean current density of 6000A/m 2. The left side of the electrolyte is where the fuel and air inlets are located. Near the inlet,

62 4.3. CO-FLOW 51 Figure 4.2: Current density (A/m 2 ) at electrolyte for (a) detailed and (b) simplified models in co-flow where there is a higher concentration of reactant species, the current density is higher and is reduced towards the outlet. There are local maxima of current density along the centre of the cell in the detailed model as this region of the electrolyte is adjacent to the cell channels. The current density is reduced on the edges of the cell adjacent to the interconnect region where porous media mass transport limits the presence of reactant species. The range of local values across the channel at any point are resolved in the detailed model. In the simplified model, the range of local values across the cell are not resolved and instead the bulk value is shown. The bulk value is the average of the range of values at any given point along the channel. Figure 4.3 shows the temperature distribution at the electrolyte region for the detailed and simplified models. At the inlet on the left, where the inlet air and fuel temperatures are at 1000K, there is a lower temperature at the electrolyte. The temperatures of the gases along the channels increase towards the outlet as electrochemical reactions provide heat to the system reaching a maximum temperature at the outlets. The simplified model shows a slightly higher temperature on average at the electrolyte region possibly due to an underestimation of an inter-phase transfer

63 4.3. CO-FLOW 52 Figure 4.3: Temperature (K) at electrolyte for (a) detailed and (b) simplified models in co-flow coefficient. Table 4.2 shows a comparison of key values between the detailed and simplified models. The comparisons are made for the minimum and maximum bulk values from each model at the electrolyte surface. The bulk values from detailed model are calculated at any point in the channel as the integral average of the values along a 4mm line representing the width of the cell. The bulk values generated by the simplified model are used directly. The two models show good agreement at an average current of 6000A/m 2 with less than 3% percent difference for each of the extreme values. The overall cell voltage is accurate to 0.33%. Data was also gathered at constant utilization for current density values ranging from 3000A/m 2 to 6000A/m 2. The maximum and minimum values for species fraction, temperature and current density were more accurate at lower average current density.

64 4.4. COUNTER-FLOW 53 Table 4.2: Comparison of electrolyte values for single cell in co-flow Detailed Simplified % difference y O2,min y O2,max y H2,min y H2,max T min (K) T max (K) p air,in (Pa) p fuel,in (Pa) u air,out (m/s) u fuel,out (m/s) i min (A/m 2 ) i max (A/m 2 ) V (V) Counter-Flow The same geometry was tested in counter-flow and showed similar results. Figure 4.4 shows the current density distribution at the electrolyte for the detailed and simplified models. The left side of the electrolyte is where the fuel inlet and air outlet are located. The presence of hydrogen species dominates the current density distribution with a maximum current density at the fuel inlet and minimum current density at the fuel outlet. Figure 4.5 shows the temperature distribution at the electrolyte region for the detailed and simplified models. The air flow dominates the temperature distribution with a minimum temperature at the air inlet on the right and a maximum temperature at the air outlet on the left. Table 4.3 shows a comparison between the detailed and simplified models at a

65 4.4. COUNTER-FLOW 54 Figure 4.4: Current density (A/m 2 ) at electrolyte for (a) detailed and (b) simplified models in counter-flow Figure 4.5: Temperature (K) at electrolyte for (a) detailed and (b) simplified models in co-flow mean current density of 6000A/m 2 in counter-flow. The two models show good agreement at an average current of 6000A/m 2 with less than 2.5% percent difference for each of the extreme values. The overall cell voltage is accurate to 0.29%. Data was also gathered at constant utilization for current density values ranging from 3000A/m 2 to 6000A/m 2. The maximum and minimum values for species fraction, temperature and current density were once again more accurate at lower average current density.

66 4.5. STACK GEOMETRY 55 Table 4.3: Comparison of electrolyte values for single cell in counter-flow Detailed Simplified % difference y O2,min y O2,max y H2,min y H2,max T min (K) T max (K) p air,in (Pa) p fuel,in (Pa) u air,out (m/s) u fuel,out (m/s) i min (A/m 2 ) i max (A/m 2 ) V (V) Stack Geometry The detailed and simplified models are compared for a three cell stack in co-flow. A different flow rate is specified for each cell within the stack resulting in different voltages at each cell. For a flow rate ṁ at the bottom cell, a flow rate of 0.9ṁ is specified at the middle cell, and 0.8ṁ at the topmost cell. The range of voltages given in Table 4.4 are based on a mean current density of 6000A/m 2 and flow rates of 1.05e-05 5 kg/s and 9.15x10 8 kg/s corresponding to 10m/s and 1m/s for air and fuel, respectively, at the bottom cell. Comparisons of other parameters are similar to the single cell case. It can be seen that the voltage decreases with decreasing flow rate to each cell as expected. The current density range is larger at the cells with lower flow rates, and therefore higher utilizations. Varying the inlet flow rate respresents the pattern

67 4.6. CLOSURE 56 Table 4.4: Comparison of voltage values for a three cell stack in co-flow Detailed Simplified % difference Cell 1 (V) Cell 2 (V) Cell 3 (V) seen when external manifolds are present resulting in a different flow rate to each cell within the stack. The effects will become more pronounced at larger stack sizes. 4.6 Closure Results were compared between the detailed and simplified models for a variety of parameters. A single channel cell was considered in co-flow and counter-flow configurations. Then a three cell stack was compared for varying inlet conditions between cells. The simplified model shows a good overall comparison with the detailed model in each case based on solutions for species fraction, current density, temperature, and cell voltage. The simplified method allowed a smaller computational mesh size and computation times, but required more iterations based on the number of interphase transfer terms between regions. It was found that on average, the simplified model required 1.4% the computation time of the detailed model making it possible to investigate larger cell and stack sizes, and include external manifolds corresponding to practical fuel cell installations. The simplified model is used in Chapter 5 to investigate some of the key design questions for SOFC stacks up to 100 cells.

68 57 Chapter 5 Performance of Large Stacks For varying flow rates, the effects of air and fuel utilization on electrochemical performance and temperature distribution are determined for a 10 cell stack. The results are presented in terms of fluid flow, species mass fraction, pressure, temperature, voltage, and current density distributions for a 10 cell and a 100 cell stack. The effects of flow on the electrochemical performance and temperature distributions are analyzed in terms of voltage and utilizations at each cell in the stack. The discussion highlights the importance of manifold design on performance and thermal management in large stacks. 5.1 Geometry and Mesh A 10 cell stack geometry with external manifolds is used. Comparisons are also made for 40 and 100 cell stacks. Each cell within the stack is considered to be a single channel. The active area for each cell is m 2 based on a 4mm x 100mm geometry. The total height of each cell is 11.3mm for a total of 113mm for the 10 cell stack. Inlet and outlet manifold channels which are 100mm long distribute the gases to the base of the stack. The manifolds each have a cross-sectional area of 4mm

69 5.2. TEN CELL STACK 58 20mm. Comparisons are made in co-flow and counter-flow configurations. Four distinct subdomains are created for air, fuel, PEN, and interconnect. This allows the air and fuel to occupy the same space (something which is, of course, not physically possible in a real geometry). This strategy gives flexibility for the specification of internal and external manifold geometries while still allowing a solution within each of the overlapping PEN, fuel, and air regions. Performance calculations for the 10, 40, and 100 cell stacks took an average of 15, 30, and 60 minutes, respectively, on one core of an eight core, Intel Xeon processor with 2.66 GHz processor speed, and 16GB of RAM using Mac OSX. 5.2 Ten Cell Stack Calculations were performed on a 10 cell stack showing the effects of different flow rates to each cell on the temperature distribution and electrochemical performance. For a range of utilizations, the voltage and temperature distributions are analyzed Operating Conditions Uniform inlet and outlet flows are presumed. The velocity, temperature, and mass fraction of species are prescribed at the inlet. Inlet velocities are calculated, based on target utilizations of O 2 for air and H 2 for fuel. Operating conditions are shown in Table 5.1. The fuel consists of H 2 and H 2 O, and the air of O 2 and N 2 only. A mean current density of 6000A/m 2 is prescribed. The flow rates are based on inlet velocities of 3m/s and 0.7m/s for air and fuel, respectively.

70 5.2. TEN CELL STACK 59 Table 5.1: Large stack operating conditions Mean current density (A/m 2 ) 6000 Inlet mass fraction H Inlet mass fraction O Inlet temperature (K) 1000 Outlet pressure (Pa) Inlet flow rate of fuel (kg/s) 1.708x10 6 Inlet flow rate of air (kg/s) 8.438x10 5 Utilization H Utilization O Results and Discussion Results are presented in terms of velocity, pressure, species mass fraction, temperature, current and voltage distributions for co-flow of the working fluids. Figures 5.1a and 5.1b show air and fuel velocity magnitudes in the external manifolds, respectively. It can be seen that at the inlet manifold on the left, the gas velocities of both air and fuel decrease towards the top of the stack. Across the core region, the superficial velocity of the flow through the fuel stack is shown. To calculate the channel velocities, the superficial velocity must be divided by the volume fraction of the gas. At the outlet, the external manifold velocity increases towards the outlet at the bottom right. Higher flow rates of air and fuel are exhibited through the bottom cell vs the topmost cell. Figures 5.1c and 5.1d show the pressure distributions of air and fuel respectively. A higher pressure drop is seen across the core than in the manifolds. The pressure drop within the stack is mainly due to the viscous resistance in the smaller channels of the core region. This represents the pressure distribution of a good manifold geometry. When pressure drop along the stacking direction begins to match the

71 5.2. TEN CELL STACK 60 Figure 5.1: Ten cell stack, elevation view for air (left) and fuel (right)

72 5.2. TEN CELL STACK 61 Figure 5.2: Ten cell stack, elevation view

73 5.2. TEN CELL STACK 62 pressure drop across the core, a poor distribution of reactant species will be seen in the stack direction. Figures 5.1e and 5.1f show the mass fractions of O 2 within the air and H 2 within the fuel, respectively. The mass fractions of the reactant species decrease from the inlet on the left to the outlet on the right as the reactants are consumed across the core of the stack. The species mass fraction of hydrogen decreases more quickly than the fuel since during the fuel side reaction, water is produced which further reduces the mass fraction of hydrogen relative to the total composition. Figures 5.2a and 5.2b show the temperature distributions of air and fuel. As the gases flow through the core region of the stack, they are heated by the electrochemical reactions. It can be seen that for the fuel region, some heat is conducted back into the inlet manifold against the flow. Figures 5.2c and 5.2d show the local temperature distributions within the PEN and interconnect regions. Both distributions clearly show a lower temperature at the bottom left increasing towards the top right. This is partly due to the larger mass flow of air and fuel through the bottom cells relative to the top cells. Figure 5.2e shows the local current density distribution. The local current density is higher at the inlet where a higher concentration of reactant species is located and a maximum at the top left corner of the stack. As the reactants are consumed from left to right, the local current density is reduced. The overall voltage at each cell of the 10 cell stack is shown in Figure 5.2f where the voltage is lowest at the bottom cell (cell 1) and is clearly increasing towards the top cell (cell 10).

74 5.2. TEN CELL STACK Effects of Utilization on Temperature and Cell Voltage The utilization of H 2 is varied from the base case by changing the flow rate of fuel at the inlet while keeping the mean current density constant for both the co-flow and counter-flow configurations. It was found in each case that cell voltage increased from the bottom cell towards the top cell. Cell voltage at the bottom cell (cell 1) and the topmost cell (cell 10) are shown as a function of H 2 utilization in Figures 5.3a and 5.3b. It is seen that the difference between the cell 1 and cell 10 values increase with increasing utilization. The difference in voltage between cell 1 and cell 10 as a percentage of average cell voltage increases from 0.02% to 0.38% in co-flow and 0.02% to 0.41% in counter-flow corresponding to Figure 5.3. Figures 5.3c and 5.3d show voltage increasing with O 2 utilization, then decreasing. As O 2 utilization increases with decreasing air flow rate. The first part of the curve is the region where as the air flow rate is lowered, the overall stack temperature is increased, thereby increasing the voltage. This is where the electrochemical reaction is limited by the presence of hydrogen in the reaction. The second part of the curve is where the reaction becomes limited by the presence of oxygen at the reactant sites. This causes the voltage to decrease as there is less oxygen available for the reaction. This illustrates the inherently coupled nature of flow, heat, mass transfer, and electrochemistry for the problem at hand. Also, as oxygen utilization increases, the difference between cell 1 and cell 10 also increases, but not in a direct relation. The difference in voltage between cell 1 and cell 10 as a percentage of average cell voltage increases from 0.11% to 0.44% in co-flow and from 0.11% to 0.36% in counter-flow corresponding to Figure 5.3. Figures 5.4a and 5.4b show the average cell temperature of cell 1 and cell 10 as

75 5.2. TEN CELL STACK 64 Figure 5.3: Voltage vs utilization a function of H 2 utilization for co- and counter-flow configurations. As the fuel flow rate decreases, the average cell temperature increases. However, it is not a direct correlation. The difference in temperature between cell 1 and cell 10 also increases as a function of H 2 utilization. The difference in average temperature between cell 1 and cell 10 as a fraction of average stack temperature increases from 0.09% to 0.21% in co-flow and 0.12% to 0.24% in counter-flow corresponding to Figure 5.4. Similarly, Figures 5.4c and 5.4d show average cell temperature as a function of O 2 utilization. It can be seen that there is a very strong correlation between average cell temperature and O 2 utilization. The difference in average temperature between cell

76 5.3. LARGE STACK PERFORMANCE 65 Figure 5.4: Average cell temperature vs utilization 1 and cell 10 as a percentage of average stack temperature increases from 0.13% to 0.57% in co-flow and from 0.15% to 0.55% in counter-flow corresponding to Figure Large Stack Performance A large stack of 100 cells is investigated to determine the effect of maldistributions of air and fuel between cells on the temperature distribution and electrochemistry. The key effects of flow on average cell temperature, cell voltage, and utilization are discussed. Then, the effects of incorporating variable transport properties on flow, temperature, and voltage distributions are shown. A cold-flow model is analyzed to

77 5.3. LARGE STACK PERFORMANCE 66 determine how well it can be used to predict the performance of a fuel cell stack Effects of Flow on Temperature and Performance Calculations were performed on 40 cell and 100 cell stacks for the same utilization and mean current density boundary conditions as given in Table 5.1. To reach the same utilization, higher flow rates were used for the larger stacks resulting in higher Reynolds numbers, and higher pressure at the inlet. The 40 cell stack showed the similar effects as the 100 cell stack, but the ranges of quantitative results were not as extreme, therefore the 100 cell stack is displayed. Figure 5.5 shows the the velocity, pressure and temperature distributions for air and fuel in a 100 cell stack. Figures 5.5a and 5.5b show the the velocity and pressure drop for air. In the large stack, there is a higher pressure drop in the manifolds relative to the smaller stack of 10 cells, resulting in a larger variance in flow rate between cells. The temperature distribution for air can be seen in Figure 5.5c. The fuel had much lower flow rates with a much lower viscosity than the air, therefore the inertial effects were lower. Figures 5.5d and 5.5e show the velocity and pressure drop for fuel in a 100 cell stack which is lower than the pressure drop for air. The temperature distribution of the fuel in Figure 5.5f is dominated by the temperature distribution of the air. The air had much higher flow rates with a more uneven distribution between cells resulting in a larger effect on the temperature distribution throughout the fuel cell stack. The large difference in flow rate between cells in the 100 cell stack has a large effect on the temperature distribution. Figures 5.7a and 5.7b show the amount of convective mass flux through the inlet of each cell in the 100 cell stack for both air

78 5.3. LARGE STACK PERFORMANCE 67 and fuel. Figure 5.7a shows the higher convective mass flux of air at the inlet of the first cells being reduced towards the top of the stack. For the 100 cell stack, the bottom cell received 2.4x the convective mass flux of air relative to the top cell. The large difference in the flow rate of air resulted in a difference of 65 C between the bottom and top cell for the 100 cell stack as seen in Figure 5.7c. This is in comparison with a 2 C difference for the 10 cell stack. The fuel had a lower flow rate and was more evenly distributed between cells than the air. The bottom cell received 1.25x the convective mass flux relative to the top cell for fuel. Figure 5.7b shows the convective mass flux of fuel through the inlet of each cell decreasing towards the top of the stack. Figures 5.6a and 5.6d show the O 2 and H 2 mass fraction throughout the 100 cell stack. The change in air flow rate of 2.4x between the bottom cell and top cell is reflected in the O 2 distribution. Since the lower cells are receiving a higher flow rate, but are running at the same current density, the lower cells have a lower utilization relative to the higher cells. The higher cells have a lower flow rate, therefore have a higher utilization resulting in the lowest mass fractions of O 2 at the outlets of the top cells. A similar effect can be seen in the H 2 distribution, but it is much less pronounced. There is a more even balance of H 2 utilization between cells. Figures 5.6b and 5.6c show the temperature distributions in the interconnect and PEN regions. The temperature distributions display the effects brought on by the air flow distribution. Figure 5.6e shows the current density distribution within the stack. The top cell has a higher current density range from 5166A/m 2 to 7095A/m 2 compared with the bottom cell which ranges from 5453A/m 2 to 6618A/m 2. The lower flow rates and higher utilizations at the top cells result in a larger range of current. For the same

79 5.3. LARGE STACK PERFORMANCE 68 Figure 5.5: 100 cell stack, elevation view for air (top) and fuel (bottom)

80 5.3. LARGE STACK PERFORMANCE 69 Figure 5.6: 100 cell stack, elevation view