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Mathematical modeling of polymer exchange membrane fuel cells

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Title:
Mathematical modeling of polymer exchange membrane fuel cells
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Book
Language:
English
Creator:
Spiegel, Colleen
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University of South Florida
Place of Publication:
Tampa, Fla
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Subjects

Subjects / Keywords:
PEM fuel cell
Flow field
Thermal model
Microchannels
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: Fuel cells are predicted to be the power delivery devices of the future. They have many advantages such as the wide fuel selection, high energy density, high efficiency and an inherent safety which explains the immense interest in this power source. The need for advanced designs has been limited by the lack of understanding of the transport processes inside the fuel cell stack. The reactant gases undergo many processes in a fuel cell that cannot be observed. Some of these processes include convective and diffusional mass transport through various types of materials, phase change and chemical reaction. In order to optimize these variables, an accurate mathematical model can provide a valuable tool to gain insight into the processes that are occurring.The goal of this dissertation is to develop a mathematical model for polymer electrolyte-based fuel cells to help contribute to a better understanding of fuel cell mass, heat and charge transport phenomena, to ultimately design more efficient fuel cells. The model is a two-phase, transient mathematical model created with MATLAB. The model was created by using each fuel cell layer as a control volume. In addition, each fuel cell layer was further divided into the number of nodes that the user inputs into the model. Transient heat and mass transfer equations were created for each node. The catalyst layers were modeled using porous electrode equations and the Butler-Volmer equation. The membrane model used Fick's law of diffusion and a set of empirical relations for water uptake and conductivity.Additional work performed for this dissertation includes a mathematical model for predicting bolt torque, and the design and fabrication of four fuel cell stacks ranging in size from macro to micro scale for model validation. The work performed in this dissertation will help improve the designs of polymer electrolyte fuel cells, and other polymer membrane-based fuel cells (such as direct methanol fuel cells) in the future.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Colleen Spiegel.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 354 pages.
General Note:
Includes vita.

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aleph - 002047028
oclc - 497023043
usfldc doi - E14-SFE0002730
usfldc handle - e14.2730
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ABSTRACT: Fuel cells are predicted to be the power delivery devices of the future. They have many advantages such as the wide fuel selection, high energy density, high efficiency and an inherent safety which explains the immense interest in this power source. The need for advanced designs has been limited by the lack of understanding of the transport processes inside the fuel cell stack. The reactant gases undergo many processes in a fuel cell that cannot be observed. Some of these processes include convective and diffusional mass transport through various types of materials, phase change and chemical reaction. In order to optimize these variables, an accurate mathematical model can provide a valuable tool to gain insight into the processes that are occurring.The goal of this dissertation is to develop a mathematical model for polymer electrolyte-based fuel cells to help contribute to a better understanding of fuel cell mass, heat and charge transport phenomena, to ultimately design more efficient fuel cells. The model is a two-phase, transient mathematical model created with MATLAB. The model was created by using each fuel cell layer as a control volume. In addition, each fuel cell layer was further divided into the number of nodes that the user inputs into the model. Transient heat and mass transfer equations were created for each node. The catalyst layers were modeled using porous electrode equations and the Butler-Volmer equation. The membrane model used Fick's law of diffusion and a set of empirical relations for water uptake and conductivity.Additional work performed for this dissertation includes a mathematical model for predicting bolt torque, and the design and fabrication of four fuel cell stacks ranging in size from macro to micro scale for model validation. The work performed in this dissertation will help improve the designs of polymer electrolyte fuel cells, and other polymer membrane-based fuel cells (such as direct methanol fuel cells) in the future.
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Microchannels
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Mathematical Modeling of Polymer Exchange Membrane Fuel Cells by Colleen Spiegel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Shekhar Bhansali, Ph.D. Ken Buckle, Ph.D. Lee Stefanakos, Ph.D. Julie Harmon, Ph.D. Yogi Goswami, Ph.D. Date of Approval: November 4 2008 Keywords: PEM fuel cell, flow field, thermal model, microchannels Copyright 2008, Colleen Spiegel

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DEDICATION To my husband, Brian, who inspires me to be a better person every day, encourages me to pursue all of my dreams, and has the patience and endurance to stand by me while I work at them. To my son, Howard, who had to endure e ndless sleepless nights (in the womb and out) while I was completing this dissertation.

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ACKNOWLEDGMENTS To my parents, Chris and Shirley, and my in-laws, Mark and Susan (pseudo parents), for helping to watch Howard while I completed this dissertation. To Dr. Bhansali, who encouraged me to pursue my passion. Thank you for encouraging me to stick with the t opic that I am most passionate about.

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i TABLE OF CONTENTS LIST OF TABLES ............................................................................................................ vii LIST OF FIGURES ........................................................................................................... xi ABSTRACT ..................................................................................................................... xix 1 INTRODUCTION ............................................................................................................1 1.1 Background Information ....................................................................................4 1.1.1 Polymer Exchange Membrane ............................................................8 1.1.2 Gas Diffusion Layer ............................................................................8 1.1.3 Catalyst Layer .....................................................................................9 1.1.4 Bipolar Plates ....................................................................................10 1.1.4.1 Flow Field Designs ............................................................11 1.1.5 Stack Design and Configuration .......................................................14 1.1.6 Operating Conditions ........................................................................16 1.1.7 Polarization Curves ...........................................................................18 1.2 Previous Modeling Approaches .......................................................................20 1.2.1 MEA-Centered Approach .................................................................21 1.2.2 Channel-Centered Approach .............................................................26 1.3 Summary and Comparison of PEM Fu el Cell Mathematical Models .............30 1.4 Dissertation Objectives and Outline ................................................................33 2 GENERAL THEORY AND EQUATIONS ...................................................................36 2.1 Thermodynamics ..............................................................................................38

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ii 2.2 Voltage Loss Due to Activation Polarization ..................................................43 2.3 Voltage Loss Due to Charge Transport ...........................................................48 2.4 Voltage Loss Due to Mass Transport...............................................................53 3 HEAT TRANSFER MODEL .........................................................................................57 3.1 Model Development.........................................................................................58 3.1.1 Background and Modeling Approaches ............................................59 3.1.2 Methodology .....................................................................................60 3.2 Definitions of Segments and Nodes .................................................................61 3.2.1 Boundary Conditions ........................................................................63 3.2.2 Model Assumptions ..........................................................................64 3.3 Energy Balances and Thermal Resist ances for Each Fuel Cell Layer .............64 3.3.1 End Plates, Contacts, and Gasket Materials ......................................64 3.3.1.1 Thermal Resistances ..........................................................66 3.3.1.2 Heat Flow From Fluid/Gases in the Layer to the Solid .....68 3.3.2 Flow Field Plate ................................................................................69 3.3.3 Anode/Cathode Gas Diffusion Layer ................................................73 3.3.4 Anode/Cathode Catalyst Layer .........................................................75 3.3.5 Membrane .........................................................................................76 3.4 Heat Generated by Electrical Resistance .........................................................78 3.5 Heat Transfer to Gases .....................................................................................79 3.6 Convective Heat Transfer Coefficient .............................................................81 4 MASS, CHARGE, AND PRESSURE DROP MODEL .................................................86 4.1 Methodology ................................................................................................... 89

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iii 4.2 Definitions of Segments and Nodes ................................................................ 90 4.3 Boundary Conditions ...................................................................................... 92 4.4 Model Assumptions ........................................................................................ 93 4.5 General Mass Balance Equations.................................................................... 93 4.6 Pressure Drop ...................................................................................................97 4.7 Charge Transport ...........................................................................................100 4.8 Flow Field Plate Layers .................................................................................101 4.8.1 Diffusive Transport From th e Flow Field Channels to the Gas Diffusion Layer ...............................................................102 4.8.2 Calculation of Pressure Drop ..........................................................105 4.9 Anode/Cathode Diffusion Layer ....................................................................107 4.10 Anode/Cathode Catalyst Layer ....................................................................110 5 POLYMER ELECTROLYTE MEMBRANE MODEL ...............................................118 5.1 Model Development.......................................................................................122 5.1.1 Background and Modeling Approaches ..........................................123 5.1.2 Methodology ...................................................................................126 5.2 Definitions of Segments and Nodes ...............................................................127 5.3 Boundary Conditions .....................................................................................128 5.3.1 Model Assumptions ........................................................................129 5.4 Mass and Species Conservation .....................................................................129 5.5 Charge Transport ...........................................................................................135 5.6 Pressure in the Polymer Membrane ...............................................................137 5.7 Momentum Equation .....................................................................................139

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iv 5.8 Gas Permeation ..............................................................................................139 6 BOLT TORQUE MODEL ............................................................................................141 6.1 The Mechanics of Bolted Joints .....................................................................143 6.2 Calculating the Force Required on the Stack for Optimal Compression of the GDL ...............................................................................................147 6.3 The Stiffness of Bolted Joints ........................................................................151 6.4 Calculating the Tightening Torque ................................................................154 6.5 Relating Torque to the Total Clamping Pressure Applied to the Stack .........155 6.6 Torque Tightening Parameters .......................................................................156 6.7 Electrochemical Performance of PEM Fuel Cell Stacks ...............................160 7 DESIGN AND FABRICATION OF MI CRO FUEL CELL STACKS ........................169 7.1 Background and Approaches .........................................................................170 7.2 Design and Production of the Micro Fuel Cell Stack ....................................171 7.3 Microchannel Fabrication Process .................................................................176 7.3.1 The Two Stage DRIE Process .........................................................178 7.3.2 Single Cell Fuel Cell St ack Performance Tests ..............................179 8 FUEL CELL MODEL RESULTS ................................................................................185 8.1 Heat Transfer Portion of the Overall Fuel Cell Stack Model ........................190 8.1.1 Temperature Distribution of Various Stack Sizes ..........................194 8.1.2 Stack Temperature Dist ribution Over Time ...................................196 8.1.3 Temperature Distribution in a Single Cell ......................................198 8.1.4 Variation of Operating Current Density .........................................199 8.1.5 Effect of the Inlet Gas and Coolant Temperatures ..........................201

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v 8.2 Mass and Charge Transfer and Pressure Drop Portion of the Overall Fuel Cell Stack Model......................................................................................205 8.2.1 Total Mass Flow Rates ....................................................................206 8.2.2 Pressures Through Fuel Cell Stack .................................................209 8.2.3 Velocity Distribution Through the Fuel Cell Stack ........................215 8.2.4 Hydrogen Transport ........................................................................219 8.2.5 Oxygen Transport ...........................................................................223 8.2.6 Water Transport ..............................................................................224 8.3 Membrane Portion of the Overall Fuel Cell Stack Model .............................231 8.3.1 Effect of Current Density ................................................................233 8.3.2 Effect of Temperature .....................................................................234 8.3.3 Effect of Water Activity at the Catalyst/Membrane Interfaces ...... 235 8.4 Electron Transport .........................................................................................237 8.5 Overall Fuel Cell Model Validation ..............................................................238 9 SUMMARY AND FUTURE WORK ..........................................................................241 REFERENCES ...............................................................................................................246 APPENDICES .................................................................................................................260 Appendix A Fuel Cell Layer Parameters Used for Model ..................................261 Appendix B Diffusion Coefficients .....................................................................271 Appendix C Derivation of Overall Heat Transfer Coefficient .............................272 Appendix D Control Volume Energy Rate Balance ............................................275 Appendix E Energy Balances Around Each Node ..............................................279

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vi Appendix F Derivation of Mass Transport in the Flow Channels and Through the Porous Media [4] ................................................................................283 F.1. Convective Mass Transport From Flow Channels to Electrode .......284 F.2 Diffusive Mass Transport in Fuel Cell Electrodes ............................285 F.3 Convective Mass Transport in Flow Structures.................................288 F.3.1 Mass Transport in Flow Channels ....................................289 Appendix G Heat Transfer Model .......................................................................294 Appendix H Mass Transfer Analysis ...................................................................319 Appendix I Pressure Drop Analysis .....................................................................335 Appendix J Polymer Membrane Layer ................................................................341 Appendix K Parameters for 16 cm2 Fuel Cell Stack ...........................................347 Appendix L Typical Outputs fo r Each Fuel Cell Layer .......................................349 ABOUT THE AUTHOR ....................................................................................... End Page

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vii LIST OF TABLES Table 1.1 Basic PEM fuel cell components .....................................................................7 Table 1.2 Operating conditi ons of PEMFCs in literature ..............................................17 Table 1.3 Comparison of the characte ristics of recent mathematical models ................31 Table 3.1 Polynomial coefficien ts for calculating dynamic viscosity ...........................82 Table 3.2 Polynomial coefficients for calculating thermal conductivity .......................82 Table 3.3 Polynomial coefficients fo r calculating specific heat capacity and formation enthalpies .......................................................................................85 Table 6.1 Material properties used for material stiffn ess and compression calculations for stack #1 ...............................................................................157 Table 6.2 Material properties used for material stiffn ess and compression calculations for stack #2 ...............................................................................158 Table 6.3 Material properties used for material stiffn ess and compression calculations for stack #3 ...............................................................................159 Table 6.4 Bolt properties used for bolt stiffness and torque calculations ....................160 Table 6.5 Calculated for ce, tightening torque, and contact pressure for stack #1 ........163 Table 6.6 Calculated for ce, tightening torque, and contact pressure for stack #2 ........165 Table 6.7 Calculated for ce, tightening torque, and contact pressure for stack #3 ........166 Table 7.1 Prototype stack dimensions ..........................................................................172 Table 7.2 Flow field pl ate channel dimensions ............................................................173

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viii Table A.1 Parameters used for the end plate layers .......................................................261 Table A.2 Parameters used for the anode end plate .......................................................262 Table A.3 Parameters used for the cathode end plate ....................................................263 Table A.4 Parameters used for the current collector ......................................................264 Table A.5 Parameters used for the flow field layers ......................................................265 Table A.6 Parameters used for cooling channels ...........................................................266 Table A.7 Parameters used for surroundings .................................................................267 Table A.8 Parameters used for hydrogen, oxygen and water ........................................267 Table A.9 Parameters used for GDL layer .....................................................................268 Table A.10 Parameters used for the catalyst layers .........................................................269 Table A.11 Parameters used for the membrane layer ......................................................270 Table B.1 Values for the various gas phase coefficients ...............................................271 Table G.1 Heat transfer equations for th e end plate, manifold and gasket layers .........294 Table G.2 Gas temperature calculati ons for the end plate, manifold and gasket layers .................................................................................................296 Table G.3 Heat transfer coefficient for th e end plate, manifold and gasket layers ........298 Table G.4 Heat transfer calculations for the flow field plate layers ..............................299 Table G.5 Gas temperature calculati ons for the flow field plate layers .........................301 Table G.6 Heat transfer coefficien t for the flow field plate layers ................................304 Table G.7 Heat transfer equations for the gas diffusion layers ......................................305 Table G.8 Gas temperature heat transfer equations for the gas diffusion layers ...........307 Table G.9 Heat transfer equa tions for the catalyst layers ..............................................310 Table G.10 Gas temperature heat transfer equations for the catalyst layers ....................312

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ix Table G.11 Heat transfer equations for the membrane layer ..........................................314 Table H.1 Mass transfer equations for the e nd plate, manifold and gasket layers ........319 Table H.2 Mole fraction calculations for the end plate, manifold and gasket layers .....320 Table H.3 Mass transfer calculations for the flow field layers .......................................322 Table H.4 Mass transfer calculatio ns for the gas diffusion layers ..................................327 Table H.5 Mass transfer calculatio ns for the catalyst layers ..........................................331 Table I.1 Pressure drop calculations for the end plate, terminal and gasket layers ......335 Table I.2 Pressure drop calcula tions for the flow field layers ......................................337 Table I.3 Pressure drop calcula tions for the gas diffusion layers .................................339 Table I.4 Pressure drop calcu lations for the catalyst layers ..........................................340 Table J.1 Polymer electrolyte me mbrane layer mass balance equations ......................341 Table J.2 Calculation of mole fractions and molar flow rates for the PEM layer ........342 Table J.3 Diffusive flux and pot ential relations for the PEM layer ..............................344 Table J.4 Pressure, velocity and di ffusive flux equations for the PEM layer ...............345 Table J.5 Gas permeation equations for the PEM layer ................................................346 Table K.1 Material properties used for the anode layers of the 16 cm2 fuel cell stack ..347 Table K.2 Material properties used for the cathode layers of the 16 cm2 fuel cell stack.........................................................................................................348 Table L.1 Typical outputs of the anode end plate, terminal and c ooling channel layer after 30 sec .....................................................................................................349 Table L.2 Typical outputs of the anode fl ow field and GDL layers after 30 sec ............350 Table L.3 Typical outputs of the anode cat alyst and membrane layers after 30 sec ......351 Table L.4 Typical outputs of the cathode catalyst and GDL layers after 30 sec ............352

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x Table L.5 Typical outputs of the ca thode flow field layer after 30 sec ..........................353 Table L.6 Typical outputs of the cathode e nd plate, terminal and cooling layers after 30 sec .............................................................................................................354

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xi LIST OF FIGURES Figure 1.1 A single PEM fuel cell [2] ...............................................................................5 Figure 1.2 An exploded view of a polymer electrolyte membrane fuel cell stack [3] ......6 Figure 1.3 A serpentine flow field design [2] .................................................................12 Figure 1.4 A parallel flow field design [2] .....................................................................12 Figure 1.5 Multiple serpentine flow channel design [2] .................................................13 Figure 1.6 Interdigitated flow channel design [2] ...........................................................14 Figure 1.7 Typical fuel cell stack configuration (a two-cell stack) [2] ...........................15 Figure 1.8 A Z-type manifold [4] ...................................................................................16 Figure 1.9 Example of a PEMFC polarization curve [4] ................................................19 Figure 1.10 Parameters that need to be solved in a mathematical model [4] ...................30 Figure 2.1 Hydrogen–oxygen fuel cell pola rization curve at equilibrium [4] ................37 Figure 2.2 Nernst voltage as a function of temperature [4] ............................................39 Figure 2.3 Nernst voltage as a function of activity of hydrogen ....................................41 Figure 2.4 Nernst voltage as a function of activity of oxygen ........................................41 Figure 2.5 Effect of the exchange cu rrent density on the activation losses [4] ..............45 Figure 2.6 Effect of the transfer coefficient on the activation losses [4] ........................46 Figure 2.7 Butler-Volmer activation losses [4] ...............................................................48 Figure 2.8 Cell voltage and current density based upon land to channel [4] ..................51

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xii Figure 2.9 Cell voltage and current density due to electrolyte thickness (microns) [4] .................................................................................................52 Figure 2.10 Ohmic loss as a function of electrolyte thickness (cm) [4] ...........................53 Figure 3.1 Illustration of a polymer electrolyte membrane (PEM) fuel cell with heat generation from the catalyst layers [2] ..........................................................58 Figure 3.2 Schematic of the PEMFC stack and the nodes used for model development .................................................................................................62 Figure 3.3 End plate energy balance ...............................................................................65 Figure 3.4 Anode and cathode fl ow field plate energy balance ......................................70 Figure 3.5 GDL energy balance ......................................................................................74 Figure 3.6 Catalyst energy balance .................................................................................75 Figure 3.7 Membrane energy balance .............................................................................77 Figure 3.8 Energy balance for channels or void space in the fuel cell layers .................79 Figure 4.1 Fuel cell layers (flow field, gas diffu sion layer, and catalyst layer) that have convective and diffusive mass transport [4] .........................................87 Figure 4.2 Mass, energy and charge balance around a layer ..........................................89 Figure 4.3 Slices created for mass, char ge and pressure drop portion of the model ......91 Figure 4.4 Mass balance illustration fo r the channels or void space in the fuel cell layers ...............................................................................................94 Figure 4.5 Cathode flow fiel d plate mass/charge balance ............................................101 Figure 4.6 Entire channel as the contro l volume for reactant flow from the flow channel to the electrode layer [4] .......................................................103 Figure 4.7 GDL mass/charge balance ...........................................................................108

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xiii Figure 4.8 Catalyst laye r mass/charge balances ...........................................................111 Figure 4.9 Cell current vers us effectiveness factor .......................................................115 Figure 4.10 Superficial flux density of hydrogen [4] .....................................................116 Figure 5.1 Illustration of th e chemical structure of Nafion [4] .....................................119 Figure 5.2 Membrane transport phenomena [4] ...........................................................120 Figure 5.3 A pictorial illustration of the water uptake of Nafion [4] ............................122 Figure 5.4 Slices create d for 1-D membrane model .....................................................127 Figure 5.5 Lambda ( ) versus activity ........................................................................131 Figure 5.6 Cell voltage and current de nsity based upon electrolyte RH ......................131 Figure 5.7 Membrane thickness and water content ......................................................136 Figure 5.8 Membrane thic kness and local conductivity ...............................................137 Figure 5.9 Pressure profile for tr ansport through pol ymer membrane .........................138 Figure 6.1 Flow chart of bolt torque model ..................................................................142 Figure 6.2 The forces exerted by the clamped materials (fuel cell layers) on the bolt and nut............................................................................................143 Figure 6.3 The forces exerted by the clamped materials and bolt ................................145 Figure 6.4 Compressive sti ffness zones underneath a bolt h ead in a fuel cell stack ....146 Figure 6.5 Conductivity and permeabil ity as a function of GDL compressed thickness [93] ...............................................................................................150 Figure 6.6 Dimensions used in the bolt and layer stiffness calculations ......................152 Figure 6.7 Fuel cell stack si zes that were tested (a) 16 cm2, (b) 4 cm2, and (c) 1 cm2 active areas ..................................................................................161

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xiv Figure 6.8 Polarization curves with ti ghtening torques of 28 oz-in to 44 oz-in for stack #1 ..................................................................................................162 Figure 6.9 Polarization curves with ti ghtening torques of 6 oz-in to 14 oz-in for stack #2 ...................................................................................................164 Figure 6.10 Polarization curves with tight ening torques of 1 oz-in to 6 oz-in for stack #3 ...................................................................................................165 Figure 7.1 Single cell design and its components [98] ..................................................174 Figure 7.2 Flow chart of research methodology [98] ....................................................175 Figure 7.3 Flow chart of the RIE process used for the creation of the flow field plates [98] .....................................................................................................177 Figure 7.4 Micro flow field channe ls in silicon flow field plate ...................................178 Figure 7.5 Through-hole added to micro fl ow field channels in silicon flow field plate ......................................................................................................178 Figure 7.6 SEM images of micro flow fi eld channels and through holes, (a) 20 m, (b) 50 m, and (c) 200 m width channels ..................................................179 Figure 7.7 Prototypes of the si ngle cell fuel cell stacks [98] .........................................180 Figure 7.8 I–V curve of the cell performance tests [98] ................................................181 Figure 7.9 Fuel cell power density curv es for 20 1000 m channel widths and depths [98] .............................................................................................181 Figure 8.1 Schematic of the PEMFC stack and its components for model development .................................................................................................186 Figure 8.2 Overall diagram of MATLAB code created .................................................188 Figure 8.3 Illustration of fuel cell stack layer numbering ..............................................189

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xv Figure 8.4 Schematic of the numbering of layers and flows for the PEMFC model .....190 Figure 8.5 Temperature por tion of overall model ..........................................................191 Figure 8.6 Temperature dist ribution in a 20 cell fuel ce ll stack, a) surface plot of the temperature distributi on as a function of position and time, (b) temperature distribution at t = 300s .........................................192 Figure 8.7 Temperature distribut ion in a 250 cell fuel cell st ack, (a) surface plot of the temperature distribution as a functio n of position and time, (b) temperature distribution at t = 300 ....................................................................................193 Figure 8.8 Temperature distri bution at the end of 60 sec onds for (a) 5 (b) 10 (c) 20 (d) 50 and (e) 100 cell stacks ........................................................................195 Figure 8.9 Temperature dist ribution at different times (a) 10 (b) 30 (c) 60 (d) 300 and (e) 600 seconds .......................................................................................197 Figure 8.10 Temperature distribut ion through a single fuel ce ll, with using a (a) 1, (b) 10, (c) 32 and (d) 64 nodes per layer ......................................................199 Figure 8.11 Stack temperature profile for base conditions at various time for (a) i = 0.1 A/cm2 (b) i = 0.6 A/cm2 (c) i = 1.0 A/cm2 ...................................200 Figure 8.12 Stack gas temperature profile for base conditions at 1200 s for (a) i = 0.1 A/cm2 (b) i = 0.6 A/cm2, and (c) i = 1.0 A/cm2 ...........................201 Figure 8.13 Effect of heating the fuel cell stack layers on the inlet gas temperature .....202 Figure 8.14 Effect of heating the inlet gas temperature on the temperature of the fuel cell stack .........................................................................................203 Figure 8.15 Comparison of the effect of coolant on the stack temperature ....................204 Figure 8.16 Relative humidity of the gas st reams in the fuel cell stack .........................205

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xvi Figure 8.17 Mass transfer and pressure drop portion of the model ................................206 Figure 8.18 Mass flow rates through a 20 cell fuel cell stack, (a) surface plot of the mass flow rate distribution as a function of position and time, (b) mass flow distribu tion at t =300 s ................................................................208 Figure 8.19 Comparison of total mass flow ra tes with pressures of 1, 2 and 3 atm ........209 Figure 8.20 Pressure distribution through a 20 cel l fuel cell stack, (a) surface plot of the pressure distribution through a 20 cell stack as a function of position and time, (b) pressure distribution at t = 300 s ...............................211 Figure 8.21 Pressure distribution through a single ce ll fuel cell stack, (a) surface plot of the pressure distribution through a single cell stack as a function of position and time, (b) pressure distribution at t = 300 s ...............................212 Figure 8.22 Pressure distributi on for a 20 cell fuel cell stack with initial pressure of (a) 3 atm, (b) 2 atm, and (c) 1 atm ................................................................214 Figure 8.23 Pressure distribution thro ugh a single cell fuel cell stack ............................215 Figure 8.24 Velocity distribution through a 20 cell fuel cell stack, (a) surface plot of the velocity distribution through a 20 cell stack as a function of position and time, (b) velocity distribution at t = 300 s ...............................216 Figure 8.25 Velocity profile in the flow field, gas diffusion, catalyst and membrane layers of a single fuel cell, (a) surface plot as a functi on of position and time, (b) velocity distribution at t = 10 s ......................................................217 Figure 8.26 Velocity of a single cell ................................................................................218 Figure 8.27 Velocity of the MEA layers at different pressures .......................................219

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xvii Figure 8.28 Hydrogen mole fraction in the anode gas flow channel, electrode backing layer and catalyst layer ...................................................................220 Figure 8.29 Hydrogen mole fraction due to the va rying current density in the anode gas flow channel, GDL layer and catalyst layer ...........................................221 Figure 8.30 The concentration of hydrogen in th e anode gas flow channel, electrode backing layer and catalyst layer ...................................................................222 Figure 8.31 Hydrogen and oxygen concentration in the MEA fuel cell layers ...............222 Figure 8.32 The mole fraction of oxygen in the anode gas flow channel, gas diffusion layer and catalyst layer ..................................................................223 Figure 8.33 The mole fraction of oxygen in th e cathode gas flow channel, gas diffusion layer and catalyst layer .................................................................................224 Figure 8.34 Effect of current de nsity on water mole fraction ..........................................225 Figure 8.35 Effect of time on water mole fraction ...........................................................226 Figure 8.36 Water concentration as a function of time at 3 atm and i = 1 A/cm2, (a) 60 s and (b) 600 s ....................................................................................227 Figure 8.37 The concentration of water in the anode gas flow channel, electrode backing layer and catalyst layer ...................................................................228 Figure 8.38 Water concentration as a function of pressure .............................................229 Figure 8.39 Hydrogen, oxygen and water concen tration at 3 atm, i = 0.1 A/cm2 ...........230 Figure 8.40 Hydrogen, oxygen and water concen tration at 3 atm, i = 1 A/cm2 ..............231 Figure 8.41 Flow chart of membrane model ....................................................................232 Figure 8.42 Effect of current density on water concentration (a) 0.1 A/cm2 (b) 0.9 A/cm2 (c) comparison of 0.1 A/cm2, 0.5 A/cm2 and 0.9 cm2 .........234

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xviii Figure 8.43 Effect of temperature on water concentration (a) 353 K (b) 323 K (c) comparison of 343 K, 348 K, 353 K and 358 K .....................................235 Figure 8.44 Water concentration in the membra ne with varying wate r activity at the membrane/cathode catalyst layer interface ..................................................236 Figure 8.45 Water concentration in the membra ne with varying wate r activity at the membrane/cathode catalyst layer interface ..................................................236 Figure 8.46 The solid phase poten tial in the PEM fuel cell .............................................237 Figure 8.47 Comparison between fuel cell model and experiments at 298 K and 1 bar .......................................................................................................239 Figure 8.48 Comparison between fuel cell mode l and experiments at various temperatures .................................................................................................240 Figure C.1 Schematic for overall heat transfer coefficient derivation ............................272 Figure D.1 Illustration of the control volume conservation of energy principle ............276 Figure E.1 Schematic of the PEMFC stack and the nodes used for model development .................................................................................................279 Figure F.1 Fuel cell layers (flow field, ga s diffusion layer, cata lyst layer) that have convective and diffusive mass transport ..............................................284 Figure F.2 Control volume for reactant flow from the flow channel to the electrode layer ..............................................................................................................289 Figure F.3 Entire channel as the contro l volume for reactant flow from the flow channel to the electrode layer ...............................................................292

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xix MATHEMATICAL MODELING OF POLYMER EXCHANGE MEMB RANE FUEL CELLS Colleen Spiegel ABSTRACT Fuel cells are predicted to be the power delivery devices of the future. They have many advantages such as the wide fuel select ion, high energy density, high efficiency and an inherent safety which explai ns the immense interest in th is power source. The need for advanced designs has been limited by the lack of understanding of the transport processes inside the fuel cell stack. The reactant gase s undergo many processes in a fuel cell that cannot be observed. Some of these proce sses include convective and diffusional mass transport through various types of materials, phase change and chemical reaction. In order to optimize these variables, an accurate mathematical model can provide a valuable tool to gain insight into th e processes that are occurring. The goal of this dissertation is to develop a mathematical model for polymer electrolyte-based fuel cells to help contribu te to a better understandi ng of fuel cell mass, heat and charge transport phenomena, to ulti mately design more efficient fuel cells. The model is a two-phase, transient mathemati cal model created with MATLAB. The model was created by using each fuel cell layer as a control volume. In addition, each fuel cell layer was further divided into the number of nodes that the user inputs into the model. Transient heat and mass transfer equations were created for each node. The catalyst

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xx layers were modeled using porous electrode equations and the Butler-Volmer equation. The membrane model used Fick’s law of diff usion and a set of empirical relations for water uptake and conductivity. A dditional work performed for th is dissertation includes a mathematical model for predicting bolt torque, a nd the design and fabrication of four fuel cell stacks ranging in size from macro to mi cro scale for model validation. The work performed in this dissertation will help impr ove the designs of polymer electrolyte fuel cells, and other polymer membrane-based fuel cell s (such as direct methanol fuel cells) in the future.

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1 1 INTRODUCTION Energy is a vital part of modern society, enabling life after dar k, the movement of people and goods, and the continuous advancement of technology. Available conventional energy sources, such as crude oil and natural gas, have been used to serve the growth of the population for stationary and transportation purposes. However, the use of fossil fuels for power has resulted in many negative consequences; some of these include severe pollution, exte nsive mining of the world’s re sources, and political control and domination of countries that have exte nsive resources. All the while, the global demand for power will increase rapidly due to the large growth in global population. In addition, there is approximately 30 years le ft of fossil fuels to provide energy for transportation and stationary applications. A power source is needed that is energy efficient, has low pollutant emissions and has an unlimited supply of fuel. There are many types of renewable energy te chnologies that have been researched for several decades; some of these include hydro, wind, solar, tidal and biofuels. However, conventional energy sources like petroleum-based products have not been replaced because these alterna tives have lower reliability, low concentration and costly implementation. For example, wind energy may be only available in certain geographical locations, and may not be uniform or steady. So lar has enormous potential to be a major local energy source; neverthe less, the photovoltaic arrays can be costly due to the competing cost of polysilicon with electronic manufacturers.

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2 In spite of these challenges, there is a growing interest in renewable energy worldwide. Many of these sources can be re plenished continuously, which enhances the security of the energy supplies. There is also an increasing concern for the environment that makes many of these alternative ener gy options attractive. These factors have increased the research and development for seeking new power sources and energy technologies around the world. Hydrogen is a clean fuel, and in principa l, can be produced abundantly and safely. It can be created from many types of ener gy sources, unlike gasoline, which can only be refined from crude oil. Although hydrogen has less volumetric energy density than gasoline, the energy density can be increased by storing it in pre ssurized tanks, or in liquid or solid forms. Hydrogen can also be us ed like gasoline, directly in an internal combustion engine. In comparison, fuel cell technology can be used to directly create electrical energy. Fuel cells are now closer to commercializat ion than ever, and th ey have the ability to fulfill all of the global power needs while meeting the efficiency and environmental expectations thereof. Of the many types of fuel cells, the type most commonly used for transportation and portable applications is polymer electrolyte membrane (PEM) fuel cells. PEM-type fuel cells traditionally use hydr ogen as the fuel, but also have the ability to use many types of fuel – these range fr om hydrogen to ethanol to biomass-derived materials. These fuels can either be directly fed into the fuel cell, or sent to a reformer to extract pure hydrogen, which is then directly fe d to the fuel cell. PEM fuel cells operate at temperatures between 20 a nd 80 C, which enable a startup time comparable with the internal combustion engine. PEM fuel cells are able to obtain net pow er densities of over

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3 1 kW/liter, which makes them competitive wi th the internal combustion engine for transportation applications [1]. There are numerous advantages and challenges for PEM fuel cells. Some advantages include: 1. Fuel cells have the potential fo r a high operating efficiency. 2. There are many types of fuel sources and methods of supplying fuel to a fuel cell. 3. Fuel cells have a hi ghly scalable design. 4. Fuel cells produce no pollutants. 5. Fuel cells are low maintenance because they have no moving parts. 6. Fuel cells do not need to be rechar ged, and they provide power instantly when supplied with fuel. Some limitations common to all fuel cell systems include the following: 1. Fuel cells are costly due to the need for materials with very specific properties. There is an issue with finding low-cost replacements. 2. Fuel reformation technology can be e xpensive, heavy and requires power in order to run. 3. If another fuel besides hydrogen is fe d into the fuel cell, the performance gradually decreases over time due to catalyst degradation and electrolyte poisoning. Mathematical modeling studies can aid in overcoming these challenges. Since fuel cells are very small, and many of the layers have thicknesses in the micron range, local values of significant prope rties such as concentration, pressure and current density cannot be directly measured. The creation of mathematical models can help supply

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4 information about the processes that are occu rring inside of the fu el cell. In addition, mathematical models can help to eliminate unnecessary time-consuming experimental investigations due to a better understanding of the phenomena that oc cur inside the cell. This understanding leads to better designs and optimized operating conditions. In practice, it is essential to combine experiment al prototyping with si mulations to achieve the optimal design cycle. 1.1 Background Information Typical fuel cells operate at a volta ge ranging from 0.6 – 0.8 V, and produce a current per active area (curre nt density) of 0.2 to 1 A/cm2. A fuel cell consists of a negatively charged electrode (anode), a posit ively charged electrode (cathode), and an electrolyte membrane. Hydrogen is oxidized on the anode and oxygen is reduced on the cathode. Protons are transported from the a node to the cathode through the electrolyte membrane, and the electrons are carried to the cathode over the ex ternal circuit. The electrons are transported through conductive mate rials to travel to the load when needed. On the cathode-side, oxygen reacts with protons and electrons forming water and producing heat. Both, the anode and cathode, cont ain a catalyst to crea te electricity from the electrochemical process as shown in Figure 1.1.

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5 Figure 1.1 A single PEM fuel cell [2] The conversion of the chemical energy of the reactants to electrical energy, heat and liquid water occurs in the catalyst layers which have a thickness in the range of 5 to 30 microns ( m). A typical PEM fuel cell has the following reactions: Anode: H2 (g) 2H+ (aq) + 2e (1) Cathode: O2 (g) + 2H+ (aq) + 2e H2O (2) Overall: H2 (g) + O2 (g) H2O (l) + electric energy + waste heat (3) Reactants are transported by diffusion and convection to the catalyzed electrode surfaces where the electrochemical reactions take place. The water and waste heat generated by the fuel cell must be continuous ly removed, and may present critical issues for PEM fuel cells.

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6 Since most applications have voltage or power requirements that cannot be satisfied by a single cell, many cells are connected in series to make a fuel cell stack. These repeating cells are separa ted by flow field plates. Incr easing the number of cells in the stack increases the voltage, while increasi ng the surface area of the cells increases the current. A PEM fuel cell stack is made up of bipolar plates, membrane electrode assemblies (MEA), and end plates as shown in Figure 1.2. Figure 1.2. An exploded view of a polymer electro lyte membrane fuel cell stack [3] The bipolar plates are constr ucted of graphite or metal, and they simultaneously distribute gases through flow channels to th e MEA while transporting electrons to the load. The gas flow channels allow the anode and cathode reactants to enter the MEA, where the electrochemical reactions occur. Ther efore, the active area of the fuel cell is normal to the y-direction. The MEA t ypically has a thickness of 500 – 600 m, and consists of five layers: th e proton exchange membrane, the anode and cathode catalyst layers and the anode and cathode gas diffusion layers. The co mponents in the fuel cell

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7 stack are summarized in more detail in Tabl e 1.1. Sections 1.2 th rough 1.6 describe the PEM components, stack, operating conditions and basic testing in more detail. Table 1.1 Basic PEM fuel cell components ________________________________________________________________________ Component Descri ption Common Types ________________________________________________________________________ Proton Exchange Membrane Enables hydrogen protons Nafion membrane to travel from the anode to 112, 115, 117 the cathode. Catalyst Layers Breaks the fuel into protons Platinum/carbon and electrons. The protons catalyst. combine with the oxidant to form water at the fuel cell cathode. The electrons travel to the load. Gas Diffusion layers Allows fuel/oxida nt to travel Car bon cloth or Toray through the porous layer, paper. while collecting electrons. Flow Field Plates Distributes the fuel and Graphite, stainless oxidant to the gas diffusion steel. layer. Gaskets Prevent fuel leakage, and Silicon, Teflon helps to distribute pressure evenly. End plates Holds stack layers in place. Stainless steel, graphite, polyethylene, PVC ________________________________________________________________________

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8 1.1.1 Polymer Exchange Membrane The polymer electrolyte membrane is e ssential for a PEM fuel cell to work properly. When fuel enters the fuel cell stack, it travels to the catalyst layer where it gets broken into protons (H+) and electrons. The electrons trav el to the external circuit to power the load, and the hydrogen protons trav el through the electrolyte until they reach the cathode to combine with oxygen to form water. The PEMFC electrolyte must meet the following requirements in order fo r the fuel cell to work properly: 1. High ionic conductivity 2. Present an adequate barrier to the reactants 3. Be chemically and mechanically stable 4. Low electronic conductivity 5. Ease of manufacturab ility/availability 6. Preferably low-cost The membrane layer contains the solid polymer membrane, liquid water, water vapor and trace amounts of H2, O2, or CO2 depending upon the purity of the H2 coming into the system. 1.1.2 Gas Diffusion Layer The gas diffusion layers (GDL) are between the catalyst layer and the bipolar plates in the fuel cell stack. They provide electrical contact between electrodes and the bipolar plates, and distribute reactants to th e catalyst layers. The layers also allow reaction product water to exit the electrode surface and permit the passage of water

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9 between the electrodes and th e flow channels. The gas diffusion layers provide five functions for a PEM fuel cell: 1. Electronic conductivity 2. Mechanical support for the proton exchange membrane 3. Porous media for the catalyst to adhere to 4. Reactant access to the catalyst layers 5. Product removal. The diffusion layer is made of electrically conductive porous materials such as carbon or Toray paper. The thickness of th e diffusion layer is usually 0.25 – 0.40 mm. The conductivity of the paper can be impr oved by filling it with electrically conductive powder, such as carbon black. To help remove water from the pores of the carbon paper, the diffusion layer can be tr eated with PTFE. Some fuel cell developers forgo the diffusion layer altogether, and platinum is sputtered directly on the proton exchange structure. 1.1.3 Catalyst Layer The fuel cell catalyst layers are where th e electrochemical reactions occur. As mentioned previously, at the anode catalyst layer, the hydrogen is broken into protons and electrons. At the cathode catalyst layer, oxygen combines with the protons to form water. The catalyst layer should have a hi gh surface area, and preferably be low cost. These catalyst layers are often the th innest in the fuel cell (5 to 30 m), but are often the most complex due to multiple phases, porosity, and electrochemical reactions. It is a

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10 challenge to find a low-cost catalyst that is effective at creating electricity from the electrochemical reactions. The catalyst layers are usually made of a porous mixture of carbon supported platinum or platinum/ruthenium. In order to catalyze reactions, catalyst particles must have contact with the protoni c and electric conductors. There also must be passages for reactants to reach catalyst si tes and for reaction products to exit. The contacting point of the reactants, catalyst, and el ectrolyte is conventionally re ferred to as the three-phase interface. In order to achieve acceptable reaction rates, th e effective area of active catalyst sites must be several times higher than the geometrical area of the electrode. Therefore, the electrodes are made porous to form a three-dimensional network, in which the three-phase interfaces are located. The reactions in the catalyst layers are exothermic; therefore, heat must be transported out of the cell. The heat can be removed through the convection in the flow channels, and conduction in the solid portion of the catalyst layers, gas diffusion media and bipolar plates. Si nce liquid water is produced by the PEM fuel cell, the condensation and evaporation of water affects the heat transf er in a PEM fuel cell. Therefore, the water and heat management in the fuel cell are closely linked. 1.1.4 Bipolar Plates After the membrane electrode assembly (MEA) has been pulled together, the cell(s) must be placed in a fuel cell stack to evenly distribute fuel and oxidant to the cells, and collect the current to power the desired devi ces. In a fuel cell with a single cell, there

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11 are no bipolar plates (only singl e-sided flow field plates). Ye t, in fuel cells with more than one cell, there is usually at least one bipolar plate (flow fields exist on both sides of the plate). Bipolar plates perform many roles in fuel cells. They distribute fuel and oxidant within the cell, separate the individual cells in the st ack, collect the current, carry water away from each cell, humidify gases, a nd keep the cells cool. Bipolar plates also have reactant flow channels on both sides, forming the anode and cathode compartments of the unit cells on the opposi ng sides of the bipolar plate. In order to simultaneously perform these functions, specific plate mate rials and designs are used. Commonly used designs can include straight, se rpentine, parallel, interdigitat ed or pin-type flow fields. Materials are chosen based upon chemical co mpatibility, resistance to corrosion, cost, density, electronic conductiv ity, gas diffusivity/impermeab ility, manufacturability, stack volume/kW, material strength, and thermal c onductivity. The materials most often used are stainless steel, titanium, nonporous gra phite, and doped polymers. Several composite materials have been researched a nd are beginning to be mass produced. 1.1.4.1 Flow Field Designs In fuel cells, the flow field should be designed to minimize pressure drop, while providing adequate and evenly distributed mass transfer through the gas diffusion layer to the catalyst surface for reaction. The three mo st popular channel configurations for PEM fuel cells are serpentine, para llel, and interdigitated flow, which are shown in Figures 1.3 through 1.6. The serpentine flow path is continuo us from start to finish. An advantage of the serpentine flow path is that it reache s the entire active area of the electrode by eliminating areas of stagnant flow. A disadvantage of serpentin e flow is the fact that the

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12 reactant is depleted through the length of the channel, so th at an adequate amount of the gas must be provided to avoid excessive pol arization losses. For high current density operation, very large plates, or when air is used as an oxidant, alternate designs have been proposed based upon the serpentine design. Figure 1.3. A serpentine flow field design [2] Figure 1.4. A parallel flow field design [2]

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13 Several continuous flow channels can be used to limit the pressure drop and reduce the amount of power used for pressu rizing the air thr ough a single serpentine channel. This design allows no stagnant area formation at the cathode surface due to water accumulation. The reactant pressure drop through the channels is less than the serpentine channel, but still an important parameter to consider. Figure 1.5. Multiple serpentine flow channel design [2] The reactant flow for the interdigitated flow field design is parallel to the electrode surface. Often, the flow channels ar e not continuous from the plate inlet to the plate outlet. The flow cha nnels are dead-ended, which fo rces the reactant flow, under pressure, to go through the porous reactant layer to reach the flow channels connected to

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14 the stack manifold. This design can remove wate r effectively from the electrode structure, which prevents flooding and enhances performan ce. The interdigitated flow field enables the gas to be pushed into th e active layer of th e electrodes where forced convection avoids flooding and gas diffusion limitations This design is sometimes noted in the literature as outperforming conve ntional flow field designs, es pecially on the cathode side of the fuel cell. The interdigitat ed design is shown in Figure 1.6. Figure 1.6. Interdigitated flow channel design [2] 1.1.5 Stack Design and Configuration In the traditional bipolar stack design, the fuel cell stac k has many cells in series, and the cathode of one cell is connected to the anode of the next cell. The MEAs, gaskets, bipolar plates and end plates ar e the typical layers of the fuel cell. The stack is clamped by bolts, rods, or another pressure device to cl amp the cells together. For an efficient fuel

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15 cell design, the following should be considered: 1. Fuel and oxidant should be uniformly distributed through each cell, and across their surface area. 2. The temperature must be uniform throughout the stack. 3. The membrane must not dry out or become flooded with water. 4. The resistive losses should be kept to a minimum. 5. The stack must be properly seal ed to ensure no gas leakage. 6. The stack must be sturdy and able to withstand the necessary environments it will be used in. The most common fuel cell configura tion is shown in Figure 1.7. Each cell (MEA) is separated by a plate with flow fiel ds to distribute the fuel and oxidant. The majority of fuel cell stacks are of this confi guration regardless of fu el cell size, type or fuel used. Figure 1.7. Typical fuel cell stack configur ation (a two-cell stack) [2]

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16 Fuel cell performance is dependent upon th e flow rate of the reactants. Uneven flow distribution can result in uneven perfor mance between cells. Reactant gases need to be supplied to all cells in the same stack through common manifolds. Some stacks rely on external manifolds, while others use an inte rnal manifold system. One advantage of an external manifold is its simplicity, which allo ws a low pressure drop in the manifold, and permits good flow distribution between cells. A disadvantage is that the gas may flow in cross flow, which can cause uneven temperat ure distribution over th e electrodes and gas leakage. One, of the most common methods, is ducts formed by the holes in the separator plates that are aligned once the stack is assemble d. An example of this type of manifold is shown in Figure 1.8. Stack Inlet Stack Outlet Figure 1.8. A Z-type manifold [4] 1.1.6 Operating Conditions There is a wide range of operating conditions that can be used for PEM fuel cells. The range of operating conditions and the opt imal conditions are summarized in Table 1.2. The fuel cell performance is determined by the pressure, temperature, and humidity based upon the application requirements, a nd can often be improved (depending upon

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17 fuel cell type) by increasing the temperatur e, pressure, humidity and optimizing other important fuel cell variables. The ability to increase these variables is applicationdependent, since system issues, weight and co st play important factors when optimizing certain parameters. Table 1.2 Operating conditions of PEMFCs in literature ________________________________________________________________________ Operating Parameter Range of Conditions Optimal Conditions ________________________________________________________________________ Temperature 20– 90 C 60– 80C Pressure 1 – 3 atm 2 – 3 atm Humidity 50 – 100 % RH 100 % RH Oxidant Air or O2 O2 ________________________________________________________________________ The range of temperatures in the literatu re for PEM fuel cells are 20 – 90 C, and it is well known that higher temperatures result in better fuel cell performance. The polymer membrane that is used for the majo rity of PEMFCs limits the upper temperature to below the glass transition temperature of the polymer. In addition, proton conductivity of the membrane is affected by the water content in the membrane; therefore, the temperature is also limited by the amount of liquid water content in the membrane. However, it may not be advantageous for th e fuel cell system design to require high operating temperatures. The pressure range for most PEMFCs in literature is from 1 – 3 atm. Fuel cells that operate at 3 atm require additional equipment to regulate and monitor the pressure. Consequently, it may not be advant ageous to run the fuel cell system above

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18 ambient pressure. The relative humidity shoul d be monitored since it changes daily under ambient conditions. The humidity, pressure, temperature, and hydroge n and oxidant flow rates should all be monitored and contro lled depending upon ambient conditions and system requirements. 1.1.7 Polarization Curves The traditional measure of characterizing a fuel cell is through a polarization curve – which is a plot of cell potential versus current density. This I-V curve is the most common method for characterizing and compari ng fuel cell efficiency to other published data. The polarization curve illustrates the voltage-cur rent relationship based upon operating conditions such as temperature, hum idity, applied load, and fuel/oxidant flow rates. Figure 1.9 shows a typical polarization curve for a single PEM fuel cell, and the regions of importance.

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19 Figure 1.9 Example of a PEMFC polarization curve [4] As shown in Figure 1.9, the polarization curv e can be divided in to three regions: 1. the activation over potential region, 2. the ohmic overpotential region, and 3. the concentration overp otential region. In the activation overpotential region voltage losses occur when the electrochemical reactions are slow in being driven from equilibrium to produce current. The reduction of oxygen is the electrochemical re action that is responsible for most of the activation overpotential. As the PEM fuel cell produces more current, the activation losses increase at a slower rate than the ohm ic losses. The ohmic overpotential is due to the resistance of the transport of charged species in the polymer electrolyte membrane, catalyst and gas diffusion la yers and bipolar plates.

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20 The concentration overpotential is due to mass transport limitations; the rates of the electrochemical reactions within the catalyst layers are hindered by a lack of reactants. The mass transport limitations ar e due to both diffusiona l limitations in the electrode backing layer and water flooding in the cathode catalyst layer. At high current densities, the amount of li quid water produced in the ca thode catalyst layer becomes greater than the amount of wate r that can be removed from the flow in the gas channels. 1.2 Previous Modeling Approaches Mathematical models provide detailed in formation about the processes occurring within the fuel cell. The processes include mass, momentum, species, energy and charge transport, and can be described mathematica lly by using the cont rol volume approach commonly used in engineering sciences. Th e model developed in this dissertation provides a good balance between micro-scale and macro-scale models. In micro-scale models, transport phenomena is commonly modeled at the molecular level, and macroscale models look at the overall system complexity to predic t certain variables, without considering the molecular effects. Many of the micro-scale models (such as the interactions between the ion, wa ter and polymer molecules) ar e impractical for the entire PEM fuel cell stack since the number of com putations required creates long computation times. Therefore, the mathematical models reviewed in this chapter are macro-scale models. Many of the molecular interactions have been simplified, for example, using diffusion coefficients to represent the inte ractions between molecules. There are two main classifications of macro-scale ma thematical models: (1) An MEA centered approach, and (2) an along-the-channel approach.

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21 The MEA-centered approach considers the membrane to be the most important aspect of the fuel cell, and models this layer in detail, while making simplifying assumptions for the other layers. The alongthe-channel model concentrates on modeling the flow channels coupled with the processes that occur with in the MEA. This approach uses many of the same equations as the memb rane-centered approach it is based upon. 1.2.1 MEA-Centered Approach Most of the modeling efforts that use th e MEA-centered approach are variations or combinations of the two original models: the models of Bernardi and Verbrugge [5, 6] and Springer et al. [7, 8]. Both of these models made steady-state and isothermal assumptions. The species transport was assumed to be one-dimensional through the MEA, and transport in the gas channels was one-dimensional along the channel. In the channel portion of the model of Be rnardi and Verbrugge [5, 6], no pressure drop was assumed, and the species transport was through convection only. The electrode layers assumed no pressure drop, and the sp ecies transport was through diffusion only. The Stefan-Maxwell equations were used to describe the diffusive fluxes, and the conservation of momentum equation was written as Darcy's law. Charge transport was modeled using Ohm’s law. The polymer electrolyte layer consisted of a porous network of channels, and was assumed to be fully hydrated. The ion trans port was governed by the Nernst-Plank equation, and the liquid wate r transport was described by Schlogl’s equation. The catalyst layers were considered to be porous media, with the diffusion of the reactant gases characteri zed by Fick’s law. The oxida tion of hydrogen in the anode

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22 catalyst layer, and the reducti on of oxygen in the cathode cata lyst layer, were modeled using the Butler-Volmer equation. Bernardi and Verbrugge [5, 6] assumed that the water and charge transport in the polymer electrolyte membrane was constant. However, the water content in a fuel cell membrane is not constant during the produc tion of current. In a ddition, the protonic conductivity is highly depende nt upon water content. The other pioneering fuel cell model is by Sp ringer et al [7], which included the modeling of variable membrane hydration. A semi-empirical governing equation is used, which consists of a Fickian equation combined with an osmotic drag coefficient. The diffusional velocity depends upon a potential gradient, and is a function of membrane hydration. The water diffusion co efficient, electroosmotic co efficient and the electrical conductivity are all depende nt upon membrane hydration, which was found to be a function of the relative humidity of the ga ses. The gas flow channels and the gas diffusion media were modeled in a similar ma nner as the Bernardi and Verbrugge [5, 6] models. However, the modeling of the ca talyst layers was simplified, and the electrochemical reactions were assumed to occur at the catalyst/gas diffusion media interface. Most other fuel cell models in the li terature are based upon the Bernardi and Verbrugge [5, 6] and Springer et al. [7, 8] approaches. The Be rnardi and Verbrugge [5, 6] used an oxygen reduction rate constant for the exchange current density several times larger than the experimental value in order to obtain agreement with the experimental data. Weisbrod et al. [9] combined the deta iled catalyst layer model of Bernardi and Verbrugge [5, 6] with the vari able hydration membrane model of Springer et al. [7, 8].

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23 Gloaguen and Durand [10] were able to impr ove this by assuming th at the catalyst layer consists of a solid matrix with void space occupied by reactant gas. Eikerling and Kornyshev [11] modeled the cathode catalyst layer with high and lo w overpotentials, and developed solutions for poor electrical conductivity and poo r oxygen transport. Marr and Li [12] used the membrane mode l of Bernardi and Verbrugge [5, 6], and improved the gas flow channel and catalyst la yer formulations. The pressure in the flow channels was allowed to vary with the a ssumption of one-dimensional pipe flow. The average concentration going to the gas diffu sion media was assumed to differ from the average concentrations in the bulk flow of th e channel. The average concentrations at the interface were calculated using a log mean concentration relationship. Marr and Li [12] also used the basic catalyst layer model of Bernardi and Verbrugge [5, 6], but occupied the void space of the catalys t layer with polymer electr olyte and liquid water. Baschuk and Li [13] allowed the void space to be occupied by gaseous reactants, liquid water, and polymer electrolyte by vary ing a parameter called the degree of water flooding to simulate the concentration overpot ential region of the polarization curve. Twophase flow was added to the model by Pisa ni et al. [14]. This model used the liquid water governed by Darcy’s law. The permeabilit y of the electrode backing and catalyst layers was dependent upon the liquid water saturation. Heat transfer in PEM fuel ce lls is of interest since h eat is produced due to the exothermic reaction in the catalyst layers. In addition, the water management of a PEM fuel cell is coupled with the thermal management In order to model th e heat transfer in a PEM fuel cell, the conservation of energy must be applied to the fuel cell. The model of Bevers et al. [15] and Wohr et al. [16] included mass, species, momentum and energy

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24 transport in the gas diffusion, catalyst and membrane layers. Although the temperature was allowed to vary within the fuel cell, the temperature of the gases/fluid in the solid and void space were assumed to be equal. Th e Dusty Gas model was used to describe the mass, momentum and species transport for th e reactants in the ga s diffusion and catalyst layers. The flow of the gaseous reactants and liquid water were coupled with porosity, since the presence of liquid water decr eases the available pore volume. The electrochemical reactions were modeled us ing the Butler-Volmer equation and heat generation due to entropy changes and charge transfer resistance, or reversible and irreversible heat generation, were included. The transfer of water and protons in the polymer electrolyte layer was modele d with the Stefan-Maxwell equation. A non-isothermal model was also deve loped by Rowe and Li [16], and was similar to the models developed by Bevers et al. [15] and Wohr et al. [17] in that the gas/fluid and solid temperatures were assume d to be equal. However, this model also included mass and species transport in a sim ilar manner to the Bernardi and Verbrugge [5, 6] models. In the one-dimensional models described t hus far, the gas flow in the channels and gas diffusion media was solved separatel y, and the water produced in the PEM fuel cell was removed by the flow channels. Reactan t depletion along the channels also affects the electrochemical reactions in the catalyst layers. Fuller an d Newman [18] modeled this interaction between the gas flow channels a nd the MEA. The variati on in temperature and reactant concentration was integrated along th e gas flow channel, and combined with the MEA model. The Fuller and Newman [18] model assumed no pressure drop, and the species transport in the gas diffusion and catalyst layers was assumed to be through

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25 diffusion only. However, this model differed from both Bernardi and Verbrugge [5, 6] and Springer et al. [7, 8] in modeling transpor t in the polymer electrolyte. Concentration solution theory was used to model the wate r and proton transport. The conservation of energy was applied by assuming that sections of the MEA were of uniform temperature, and the temperature was varied along the fl ow direction of the gas flow channel. Nguyen and White [19] also developed a quasitwo-dimensional, PEM fuel cell model. This model was similar to Nguyen and White [19] except that the polymer electrolyte membrane layer was modeled usi ng the variable hydration model of Springer et al. [7, 8], and the catalyst layer was considered to be an interface. Thirumalai and White [20] added pressu re drop to the model assuming that gas flow channels could be m odeled as a pipe network. Yi and Nguyen [21] further developed the model by allowi ng the bipolar plate, MEA and the gas flow within the channels to have different te mperatures. van Bussel et al. [22] developed a transient, quasi-two-dimensional model, based on the one -dimensional model of Springer et al. [7, 8] Another method of modeling the MEA with th e gas flow channels is to model the MEA in a multi-dimensional manner, and si mulate variations along the channel as boundary conditions. Singh et al. [23] developed a two-dimensional model using the same approach as Bernardi and Verbrugge [5, 6]. Kazim et al. [24] applied the conservation of mass, momentum and species for modeling the cathode backing layer. The catalyst layer was assumed to be an in terface, and the conservation of momentum was expressed in the form of Darcy's law. Bradean et al. [25] extended this model by including the conservation of energy.

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26 Two phase flow has also been modeled using a quasi-two-dimensional approach. He et al. [26] modeled the cathode backing layer of a PEM fuel cell with the catalyst layer is considered to be a surface, and the e ffect of the gas flow channels were included as a boundary condition. The conservation of ma ss, momentum, and species were applied to both the liquid and the gas phases and then solved separately. The conservation of momentum was expressed by Darcy’s law for th e liquid and gas phase. The definition of capillary pressure was used so that the liq uid phase velocity was proportional to the gas phase velocity and the gradient of satura tion. The mass transport between the liquid and gas phases was expressed by an interfacial s ource term that was proportional to the water vapor partial pressure and the liquid water saturation pressure. Natarajan and Nguyen [27] also devel oped a two-phase, two-dimensional model of the cathode electrode backing layer, wh ich was extended to a quasi-three-dimensional model in Natarajan and Nguyen [28]. The gas flow in the channels was incorporating by assuming it was one dimensional along the fl ow direction. This was used as boundary conditions for the 2-D analysis. 1.2.2 Channel-Centered Approach Since the MEA-centered approach does not solve the Navier-Stokes equations, the transport in the gas flow channels cannot be fully coupled with the MEA processes. Therefore, the channel-centered approach was initiated by three research groups: the University of Miami, Pennsylvania State University, and the University of South Carolina. In the channel-cente red approach, the governing eq uations for the entire fuel cell are discretized with the finite volume method.

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27 The channel-centered approach started with the model of Gurau et al. [29] at the University of Miami. The Gurau et al. [29] model was a single-phase, two-dimensional model that included the gas flow channels, el ectrode backing layers catalyst layers, and polymer electrolyte membrane layer. Th e model was united since the equations representing the conservation of mass, momentum, species, and energy in each layer had the same general form, and differed through the source terms. Schlogl’s equation was used to model the transport of liquid wate r for the polymer electrolyte membrane, and this model was similar to the model of Bern ardi and Verbrugge [ 5, 6]. The gas diffusion media was modeled using a generalized Darc y’s equation, and the catalyst layer was assumed to consist of a solid matrix with void space filed with the polymer electrolyte membrane. Ohm’s law was used to model the current flow, and the electrical conductivity in the membrane was allowed to vary with membrane hydration using the model of Springer et al. [7, 8] Fick’s law was used to mode the diffusional flux of each species. Zhou and Liu [30] extended the two-di mensional model of Gurau et al. [28] into three-dimensions, while You and Liu [31] developed a two-phase, isothermal, twodimensional model of the cathode gas flow cha nnels, electrode backing layer and catalyst layer. The channel-centered approach at the Univ ersity of South Caro lina started with a three-dimensional, single-phase model. Th e commercial CFD software FLUENT was used to create the model which included the conservation of mass, momentum, and species for the gas flow channels, gas diffu sion media, catalyst layers and the polymer electrolyte membrane. The model of Springer et al. [6, 7] was used to model the water and current transport in the polymer electrolyt e membrane layer, and Fick’s law was used

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28 to describe the diffusional fl ux. The catalyst and polymer electrolyte membrane layers were modeled as surfaces since the water tran sport, current flow and reaction rate was not allowed to vary. Shimpalee and Dutta [32] added the conservation of energy, and then time dependence in Shimpalee et al. [33, 34] Two-phases were added to the three dimensional model in Shimpalee et al. [35], and the interfacial mass transfer rate was proportional to the difference between the wate r vapor partial pressu re and the saturation pressure. The Pennsylvania State University resear ch group began their channel-centered approach with a two-phase, two-dimensiona l model of the cathode flow channel and diffusion media. The catalyst layer was treate d as a surface, and m odeled with a boundary condition. The conservation of mass, momentum and species were applied to both the gas and liquid phases, and then added t ogether. Darcy’s law was used for the conservation of momentum in th e cathode electrode backing laye r, and the velocity of the liquid water was found to be a function of the capillary pr essure and gr avitational body force. The capillary pressure was a function of the saturation of the liquid water in the electrode backing void space. Um et al. [36] presented a single phase, isothermal, two-dimensional, transient model using a similar formulation to Gurau et al. [29] and then extended to three dimensions in Um et al. [36]. Wang and Wang [37] and Wang and Wang [38] have recently presented a single phase model that uses the membrane water transport equations of Springer et al. [7, 8]. Th e recent models do not assume that the catalyst and polymer electrolyte layers are one-di mensional, but use the procedure introduced by Kulikovsky [39] to couple the gas phase and membrane water transport.

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29 Kulikovsky [39] created a three-dimensiona l model of the flow channel and gas diffusion media, and then coupled this with one-dimensional model of the transport in the catalyst layers and polymer electrolyte. The Sp ringer et al. [7] model was used to model the transport in the polymer membrane, and the gas transport in the catalyst layer was assumed to be from Knudsen diffusion only. The water flux in the catalyst layer was due to a gradient in the gas phase water concen tration, and the hydration of the membrane, which were related through th e hydration versus relative hum idity curves of Springer et al. [7]. Siegel et al. [40] solved the gas phase and liquid water transp ort separately, and coupled them with an interf acial mass transfer term that was analogous to Newton's law of cooling for convective heat transfer. Siegel et al. [40] assumed that the void space of the catalyst layer was filled with both gas a nd polymer electrolyte. The membrane model of Springer et al. [7, 8] was used to desc ribe the water and current transport in the polymer electrolyte, and the conservation of mass, momentum and species was applied to the gas phase. Berning et al. [41] devel oped a three dimensional, si ngle-phase fuel cell model that included the gas flow channels, electr ode backing layers, and polymer electrolyte membrane layer; the catalyst layers were treated as interfaces in a similar manner as Shimpalee et al. [32]. The conservation of mass, momentum, species and energy was used, and the conservation of energy for th e gas and solid phases were considered separately, therefore, the temperatures of th e gas and solid phases could differ. The heat transfer through the solid and gas phases were modeled with a convective heat transfer coefficient.

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30 1.3 Summary and Comparison of PEM Fuel Cell Mathematical Models Fuel cell models must be robust and accurate and be able to provide solutions to fuel cell problems quickly. A good model s hould predict fuel cell performance under a wide range of fuel cell operating conditions. Even a modest fuel cell model will have large predictive power. A few important paramete rs to include in a fuel cell model are the cell, fuel and oxidant temperat ures, the fuel or oxidant pres sures, the cell potential, and the weight fraction of each reactant. Some of th e parameters that must be solved for in a mathematical model are shown in Figure 1.10. Figure 1.10 Parameters that need to be solv ed in a mathematical model [4]

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31 The necessary improvements for fuel cell performance and operation demand better design, materials, and op timization. These issues can only be addressed if realistic mathematical process models are available. Table 1.3 shows a summary of equations or characteristics of fuel cell m odels presented in Section 1.2. Table 1.3 Comparison of the characteristics of recent mathematical models ________________________________________________________________________ Model Characteristic Description/Equations ________________________________________________________________________ No. of Dimensions 1, 2 or 3 Mode of Operation Dynamic or Steady-State Phases Gas, Liquid or a Combination of Gas & Liquid Kinetics Tafel-Type Expre ssions, Butler-Volmer Equations, Or Complex Kintics Equations Mass Transport Nernst-Plank + Schogle, Stafan-Maxwell Equation, Or Nernst-Plank + Drag Coefficient ________________________________________________________________________ Most models in the early 1990s were 1-D, models in the late 1990s to early 2000s were 2-D, and more recently there have b een a few 3-D models for certain fuel cell components. Although 2-D and 3-D models woul d seem to have more predictive power than 1-D models, most of them in the liter ature use the same e quations and methodology of a 1-D model, but apply it to 3 dimens ions. As shown in Table 1.3, most published models have steady-state volta ge characteristics and conc entration profiles, and the electrode kinetic expressions are simple Tafel-type expres sions. Some models use Butler-

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32 Volmer–type expressions, or more realistic, complex multi-step reaction kinetics for the electrochemical reactions. It is well known that there are two phases ( liquid and gas) that coexist under a variety of operating conditions. Inside the cathode structure, water may condense and block the way for fresh oxygen to reach the catalyst layer. However, most published models only examine a single phase. An important feature of each model is the mass transport descriptions of the anode, cathode, and electrolyte. Simple Fi ck diffusion models or Nerst-Planck mass transport expressions are often used. The conve ctive flow is typical ly calculated from Darcy’s law using different formulations of the hydraulic permeability coefficient. Some models use Schlogl’s formulations for conve ctive flow instead of Darcy’s law, which also accounts for electroosmotic flow, and can be used for mass transport inside the PEM. Another popular type of mass tr ansport description is the Maxwell-Stefan formulation for multi-component mixtures. This has been used for gas-phase transp ort in many models, but this equation would be better used for liquid-vapor-phase mass transport. A very simple method of incorporating electroosmotic flow in the membrane is by applying the drag coefficient model, which a ssumes a proportion of water and fuel flow to proton flow. The swelling of polymer me mbranes is modeled through empirical or thermodynamic models for PEM fuel cells. Most models assume a fully hydrated membrane. In certain cases, the water uptake is described by an em pirical correlation, and in other cases a thermodynamic model is us ed based upon the change of Gibbs free energy inside the PEM based upon water content. A model is only as accurate as its as sumptions allow it to be. The assumption needs to be well understood in order to understand the model’s limitations and to

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33 accurately interpret its results. Common assu mptions used in fuel cell modeling are: 1. Ideal gas properties 2. Incompressible flow 3. Laminar flow 4. Isotropic and homogeneous electrolyt e, electrode, and bipolar material structures 5. A negligible ohmic potential drop in components 6. Mass and energy transport is modele d from a macro-perspective using volume-averaged conservation equations These concepts can be applied to all pol ymer membrane-based fuel cell types, regardless of the fuel cell geometry. Even simple fuel cell models will provide tremendous insight into determining why a fu el cell system performs well or poorly. The physical phenomenon that occurs inside a fuel cell can be represented by the solution of the equations presented throughout this disser tation, and are discussed in Chapters 2 – 8. 1.4 Dissertation Objectives and Outline The performance of a PEM fuel cell is a ffected by the processes occurring within each layer of the cell. Due to the thinness of the layers, in-situ measurements are difficult to obtain, therefore, mathematical mode ling has become necessary for a better understanding and optimization of PEM fuel cells. Therefore, the objective of this dissertation is to develop a transient, two-pha se model of a PEM fuel cell, which differs from most published previous models in several respects:

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34 1. A fully integrated transient heat and mass transfer model that includes all layers in the fuel cell stack. 2. The model uses Fick’s law for all types of mass transport in the MEA layers. This allows an accurate pred iction of mass transport for a vast range of operating conditions (20 – 90 C). 3. Water uptake by the membrane is accounted for by an empirical model first developed by Spri nger et al. [7, 8]. 4. A complete energy balance is incl uded to account for heat conduction, convection and production. 5. A complete transient mass balance model for all layers is included in the model. 6. Pressure drops throughout the fuel cell are included. 7. Two phases are modeled in the anode and cathode layers. 8. Butler-Volmer type rate descriptio ns will be used for both electrode reactions. A comprehensive general engineering formula tion is developed that can be used as a starting point for all mathematical models for PEM and other types of low-temperature fuel cells. The numerical solution of the fo rmation is developed using MATLAB to take advantage of the built-in ordinary differentia l equations solvers. The numerical results from the simulation of the physical and chem ical phenomena within the PEM fuel cell are provided. The general formulation is comprehens ive because it include s phenomena in all layers of a PEM fuel cell. The engineering model includes a control volume analysis of

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35 each node in all of the layers within the fuel cell stack. Energy, mass and charge balances were created and pressure drops were calculated for each control volume. In addition to the complete fuel cell mode l developed in this dissertation, a model for calculating optimal torque of the fuel cell stack was developed. To validate these models, four fuel cell stacks were cons tructed. The stacks had active areas of 16 cm2, 4 cm2 and two had 1 cm2. Six different sets of flow field plates were constructed for the 1 cm2 stacks to be able to compare both m acro and micro-sized fuel cell stacks. Chapter 2 summarizes the general theory for PEM fuel cell models that currently exist in the literature. The heat transfer por tion of the mathematical model is included in Chapter 3, and the mass and pressure portion is discussed in Chapter 4. Chapter 5 is devoted to the membrane portion of the mode l. The bolt torque model is presented in Chapter 6, and the fabrication of micro fuel cells is presented in Chapter 7. Chapter 8 review the results of the mathematical model. A summary and suggestions for future work are given in Chapter 9.

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36 2 GENERAL THEORY AND EQUATIONS One of the reasons why fuel cell modeling is important is to determine why the actual voltage of a fuel cell is different th an the thermodynamically predicted theoretical voltage. As explained by thermodynamics the maximum possible cell potential is achieved when the fuel cell is operate d under the thermodynamically reversible condition. This can be described as the net out put voltage of a fuel cell, which is the reversible cell potential minus the irreversible potential at a certain current density [42]: irrev revV V V (4) where r revE V is the maximum (reversible) voltage of the fuel cell, and irrevV is the irreversible voltage loss (overpot ential) occurring at the cell. The actual open circuit voltage of a fuel cell is lower than the theoretical model due to reaction, charge and ma ss transfer losses. As descri bed in Section 1.8 and shown in Figure 2.1, the performance of a polarization curve can be broken into three sections: (1) activation losses, (2) ohmic losses, and (3) mass transport losses. Therefore, the operating voltage of the cell can be represen ted as the departure from ideal voltage caused by these polarizations [42]: conc ohmic act rV V V E V (5) where V is the cell potential, rE is the thermodynamic poten tial or Nernst voltage, actV

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37 is the voltage loss due to activation polarization, ohmicV is the voltage loss due to ohmic polarization and concV is the voltage losses due to concentration polarization. The explanation of the terms in equation 5 and Figure 2.1 stems from the detailed study of different disciplines. The Nernst voltage comes from thermodynamics, activation losses are described by electrochemistry, charge transport examines ohmic losses and concentration losses can be explained by ma ss transport. Activation and concentration polarization occurs at both the anode and cathode, while the ohmic polarization represents resistive losses throughout the fuel cell. Current Density (A/cm2)Cell Potential (V) Ideal Voltage of 1.2 V Open circuit loss due to fuel crossover Rapid drop due to activation losses Linear drop due to ohmic losses Mass transport losses at high current densities 0.250.500.751.001.25 0 0.25 0.50 0.75 1.00 1.25 Figure 2.1. Hydrogen–oxygen fuel cell polarizati on curve at equilibrium [4] Activation losses mainly occur when the electrochemical reactions are slow in being driven from equilibrium to produ ce current. The reduction of oxygen is the

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38 electrochemical reaction that is responsible for most of th e activation overpotential. As the PEM fuel cell produces more current, the activation losses increase at a slower rate than the ohmic losses. Ohmic losses are due to the movement of charges from the electrode where they are produced, to the load where they are consumed. The two major types of charged particles are electrons an d ions, and both electronic and ionic losses occur in the fuel cell. The electronic loss between the bipolar, cooli ng and contact plates ar e due to the degree of contact that the plates make with each other. Ionic transport is far more difficult to predict and model than the fuel cell electron transport. The ionic charge losses occur in the fuel cell membrane when H+ ions travel thro ugh the electrolyte. Concentration losses are due to react ants not being able to reach the electrocatalytic sites, and can significantly affect fuel cell performance. These mass transport losses can be minimized by optimiz ing hydrogen, air and wate r transport in the flow field plates, gas diffusion layer and catalyst layers. This chapter explains the theory and equations relevant to the study of th ese potential losses through explanation of thermodynamics, electrochemistry, charge transp ort and mass transport in relation to fuel cells and the work presented in this study. 2.1 Thermodynamics As shown in Figure 2.1, the thermodynami c potential is the highest obtainable voltage for a single cell. The Nernst equati on gives the ideal open circuit potential, and provides a relation between the ideal standard potential for the cell reaction, and the ideal equilibrium potential at the partial pre ssures of the reactants and products. The

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39 relationship between voltage and temperat ure is derived by taking the free energy, linearizing about the standard conditions of 25 C, and assumi ng that the enthalpy change H does not change with temperature [43]: nF S T H nF G Erxn r (6) ) 25 ( ) 25 ( T nF S T dT dE Er (7) where Er is the standard-state reversible voltage, and rxnG is the standard free-energy change for the reaction. The change in entr opy is negative; theref ore, the open circuit voltage output decreases with increasing temper ature. The fuel cell is theoretically more efficient at low temperatures as shown in Figure 2.2. However, mass transport and ionic conduction is faster at higher temperatures and this more than offsets the drop in opencircuit voltage. 300 400 500 600 700 800 900 1000 1100 1200 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Temperature (K)Nernst Voltage (Volts) Figure 2.2. Nernst voltage as a function of temperature [4]

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40 In the case of a hydrogen–oxygen fuel cell under standard-state conditions: H2 (g) + O2 (g) H2O ( H = –285.8 KJ/mol; G = –237.3 KJ/mol) (8) V mol C mol mol KJ EO H229 1 / 485 96 2 / 3 2372 2/ (9) At standard temperature and pressure, this is the highest voltage obtainable from a hydrogen–oxygen fuel cell. Most fuel cell react ions have voltages in the 0.8 to 1.0 V range. To obtain higher voltages, several cells have to be connected together in series. For nonstandard conditions, the reversible volta ge of the fuel cell may be calculated from the energy balance between the reactants and the products [44]. The theoretical potential for an electrochemical reaction can be expressed by the Nernst equation [43]: 2 / 12 2 2lnO H O H ra a a nF RT E V (10) where V is the actual cell voltage, Er is the standard-state reversible voltage, R is the universal gas constant, T is the absolute te mperature, n is the number of electrons consumed in the reaction, and F is Faraday’s constant. Figures 2.3 and 2.4 illustrate the Nernst voltage as a function of the activity of hydrogen and oxygen.

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41 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.92 0.94 0.96 0.98 1 1.02 1 04 Activity of HydrogenNernst Voltage (Volts) Figure 2.3. Nernst voltage as a func tion of activity of hydrogen 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.975 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1 025 Activity of OxygenNernst Voltage (Volts) Figure 2.4. Nernst voltage as a func tion of activity of oxygen

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42 At standard temperature and pressure, th e theoretical potenti al of a hydrogen–air fuel cell can be calculated as follows [43]: V mol C K mol J Er219 1 21 0 1 1 ln ) / ( 6485 9 2 15 298 )) /( ( 314 8 229 12 / 1 (11) The potential between the oxygen cathode wher e the reduction occurs and the hydrogen anode at which the oxidation occurs will be 1. 219 volts at standard conditions with no current flowing. By assuming the gases are ideal (the ac tivities of the gases are equal to their partial pressures, and the activity of the water phase is equal to unity), equation 10 can be written as [43]: i v i rip p nF RT E V0ln (12) The form of the Nernst equation that is relevant for this study is: 2 / 1 ,2 2 2* ln 2 2O H O H k liq f rp p P F RT F G E (13) where liq fG, is the free-energy change for the reaction, R is the universal gas constant, T is the absolute temperature, F is Faraday’s constant, O HP2 is the partial pressure of water, 2Hp is the partial pressure of hydrogen and 2Op is the partial pressure of oxygen. The saturation pressure of wa ter can be calculated by [19]: 3 7 2 5* 10 4454 1 10 1837 9 02953 0 1794 2 log2c c c O HT T T P (14) where cT is the temperature in C.

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43 The partial pressure of hydrogen is [19]: OH K H HPTi P p2 2 2)))/(*653.1exp(/(*5.03341 (15) The partial pressure of oxygen can be obtained by [19]: OH K air OPTi Pp2 2)))/(*192.4exp(/(3341 (16) Equation 13 can be used to obtain the th ermodynamically reversible voltage at a temperature T. Further details for the parameters in the above equations and thermodynamic discussions can be found from various books [43, 45]. 2.2 Voltage Loss Due to Activation Polarization Activation polarization is the voltage ove rpotential required to overcome the activation energy of the electroc hemical reaction on the catalytic surface [5]. This type of polarization dominates losses at low curre nt density, and measures the catalyst effectiveness at a given temperature. This is a complex three-phase interface problem, since gaseous fuel, the solid metal catalyst, and electrolyte must all make contact. The catalyst reduces the height of the activation ba rrier, but a loss in vol tage remains due to the slow oxygen reaction. The total activation polarization overpotentia l is 0.1 to 0.2 V, which reduces the maximum potential to less than 1.0 V even under open-circuit conditions [42]. Activation overpotential expressions can be derived from the ButlerVolmer equation. The activation overpotential increases with current density and can be expressed as [46]:

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44 cath o anode o cathact anodeact acti i nF RT i i nF RT V ln ln, (17) where i is the current density, and i0, is the reaction exchange current density, n is the number of exchange protons per mole of reactant, F is Faradays constant, and is the charge transfer coefficient used to describe the amount of electrical energy applied to change the rate of the electrochemical react ion [47]. The exchange current density, io is the electrode activity for a particular reaction at equilibri um. In PEMFC, the anode io for hydrogen oxidation is very hi gh compared to the cathode io for oxygen reduction, therefore, the cathode co ntribution to this polarization is often neglec ted. Intuitively, it seems that the activation pol arization should increase linea rly with temperature based upon Equation 17; however, the purpose of increasing temp erature is to decrease activation polarization. In Figure 2.2, incr easing the temperature would cause a voltage drop within the activatio n polarization region. The exchange current density measures th e readiness of the electrode to proceed with the chemical reaction. It is a function of temperature, catalyst loading, and catalyst specific surface area. The higher the exchange cu rrent density, the lower the barrier is for the electrons to overcome, and the more activ e the surface of the el ectrode. The exchange current density can usually be determined e xperimentally by extrapol ating plots of log i versus act to act = 0. The higher the exchange current density, the better is the fuel cell performance. The effective exchange current density at any temperat ure and pressure is given by the following equation [46]: ref r ref r r cc refT T RT E P P Laii1 exp00 (18)

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45 where i0ref is the reference exchange current density per unit catalyst surface area (A/cm2), ac is the catalyst specific area, Lc is the catalyst loading, Pr is the reactant partial pressure (kPa), Pr ref is the reference pressure (kPa), is the pressure coefficient (0.5 to 1.0), Ec is the activation energy (66 kJ/mol for O2 reduction on Pt), R is the gas constant [8.314 J/(mol*K)], T is the temperature, K, and Tref is the reference temperature (298.15 K). The activation losses as a function of ex change current density are shown in Figure 2.5. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 g Current density (A/cm2)Activation Loss (Volts) io=10-8 io=10-6 io=10-4 io=10-2 Figure 2.5. Effect of the exchange current density on the activation losses [4] If the currents are kept low so that the surface concentrations do not differ much from the bulk values, the Butler-Volmer equation can be written as [46]:

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46 RT nF i RT nF i ianode act cath act 0 0exp exp (19) where i is the current density per unit catalyst surface area (A/cm2), i0 is the exchange current density per unit ca talyst surface area (A/cm2), act is the activation polarization (V), n is the number electr ons transferred per reaction ( ), R is the gas constant [8.314 J/(mol*K)], and T is the temperature (K). Th e transfer coefficient is the change in polarization that leads to a cha nge in reaction rate for fuel cells is typically assumed to be 0.5. Figure 2.6 illustrates the affects of transfer coefficient on the activation losses. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 Current density (A/cm2)Activation Loss (Volts) alpha = 0.3 alpha = 0.7 alpha = 0.5 Figure 2.6. Effect of the transfer coeffi cient on the activation losses [4] The Butler-Volmer equation is valid for both the anode and cathode reaction in a fuel cell. It states that the current pro duced by an electrochemical reaction increases

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47 exponentially with activation overp otential [42]. This equation also says that if more current is required from a fuel cell, volta ge will be lost. The Butler-Volmer equation applies to all single-step reactions, and can also be used for multi-step approximations with some modifications to the equation. If the exchange current de nsity is low, the kinetics become sluggish, and the activation overpotential will be larger for any particular net current. If the exchange current is very large, the system will suppl y large currents with insignificant activation overpotential. If a system has an extremely small exchange current density, no significant current will flow unless a large activation overp otential is applied. The exchange current can be viewed as an “idle” current for charge exchange across the interface. If only a small net current is drawn from the fuel cell, only a tiny overpotential will be required to obtain it. If a net current is required that ex ceeds the exchange current, the system has to be driven to deliver the charge at the re quired rate, and this ca n only be achieved by applying a significant overpotentia l. When this occurs, it is a measure of the systems ability to deliver a net current with significant energy loss. In this study, the activation losses are estimated using the Butler-Volmer equation, and can be expressed as [46]: cath act anode act actV, (20) where the activation losses fo r the anode are [46, 48]: anodei S a i) 1 (2 1 (21) anode act c anode act a ref H H anodeRT F RT F p p i, 2 2exp exp (22)

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48 and the activation losses for the cathode are [46, 48]: cathodei S a i) 1 (2 1 (23) cath act c ref O O cathodeRT F p p i, 2 2exp (24) The Butler-Volmer activation losses are illustrated by Figure 2.7. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.21 -0.2 -0.19 -0.18 -0.17 -0.16 -0.15 -0.14 -0.13 -0.12 -0.11 Cell Current (A/cm2)Voltage (Volts) Figure 2.7. Butler-Volmer activation losses [4] 2.3 Voltage Loss Due to Charge Transport Every material has an intrinsic resistance to charge flow. The material’s natural resistance to charge flow causes ohmic polarization, which results in a loss in cell voltage. All fuel cell components contribute to the total electrical resi stance in the fuel cell, including the electrolyte, the catalyst layer, the gas diffusion layer, bipolar plates,

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49 interface contacts and terminal connections. The reduction in voltage is called “ohmic loss”, and includes the electronic (Relec) and ionic (Rionic) contributions to fuel cell resistance. This can be written as [42]: ) (ionic elec ohmic ohmicR R i iR V (25) Rionic dominates the reaction in Equation 25 because ionic transport is more difficult than electronic charge transport. Rionic represents the ionic resistance of the electrolyte, and Relec includes the total electrical re sistance of all other conductive components, including the bipolar plat es, cell interconnects, and contacts. The material’s ability to support the flow of charge through th e material is its conductivity. The electrical resist ance of the fuel cell compon ents is often expressed in the literature as conductance ( ), which is the reciprocal of resistance [49]: ohmicR1 (26) where the total cell resistance (Rohmic) is the sum of the electronic and ionic resistance. Resistance is characteristic of the size, shape and properties of the material, as expressed by Equation 27 [49]: cond condA L R (27) where Lcond is the length or thickness (cm) of the conductor, Acond is the cross-sectional area (cm2) of the conductor, and is the electrical conductivity (ohm 1 cm 1). The current density, i, (A/cm2), can be defined as [42]: cellA I i (28)

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50 The total fuel cell ohmic losses can be written as: cath ionic anode ohmicA L R A L iA R iA (29) where L can either be the length or thickness of the material, or the total “land area”. The first term in Equation 29 applies to the anode the second to the elec trolyte and the third to the cathode. In the bipolar plates, the “land area” can vary depending upon flow channel area. As the land area is decreased, the contact resistance increases since the land area is the term in the denomina tor of the contact resistance: contact contact contactA R R (30) where contactA equals the land area. Therefore, with increasing land area, or decreasing channel area, the contact resistance losses will decrease and the voltage for a given current will be higher. This concept is illustrated in Figure 2.8.

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51 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Cell Current (A/cm2)Voltage (Volts) 0.25 0.50 0.75 1.0 1.25 1.50 1.75 Land to Channel Ratio, 1:__ Figure 2.8. Cell voltage and current density based upon land to channel [4] One of the most effective ways for reducing ohmic loss is to either use a better ionic conductor for the electr olyte layer, or a thinner electrolyte layer. Thinner membranes are also advantageous for PEM fuel cells because they keep the anode electrode saturated through “back” diffusion of water from the cathode. At very high current densities (fast fluid flows), mass tran sport causes a rapid drop off in the voltage, because oxygen and hydrogen simply cannot diffuse through the electrode and ionize quickly enough, therefore, products cannot be moved out at the necessary speed [42]. Since the ohmic overpotential for the fuel ce ll is mainly due to ionic resistance in the electrolyte, this can be expressed as [4, 42]: i A iA iRcell cell ohmic ohmic (31)

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52 where Acell is the active area of the fuel cell, is the thickness of th e electrolyte layer, and is the conductivity. As seen from Equa tion 31 and Figures 2.9 and 2.10, the ohmic potential can be reduced by using a thinner electrolyte layer, or using a higher ionic conductivity electrolyte. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 gyy Cell Current (A/cm2)Voltage (Volts) 20 30 40 50 60 70 80 Increasing Thickness Figure 2.9. Cell voltage and current density due to electrolyte thickness (microns) [4]

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53 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y Current Density (A/cm2)Ohmic Loss (V) L=0.0025 L=0.0050 L=0.0075 L=0.001 L=0.015 Increasing Electrolyte Thickness Figure 2.10. Ohmic loss as a function of electrolyte thickness (cm) [4] 2. 4 Voltage Loss Due to Mass Transport As described in Section 2.1, concentrat ion affects fuel cell performance through the Nernst equation since the thermodynamic volta ge of the fuel cell is determined by the reactant and product concentrations at the catalyst sites [43]: i iv ts reac v products ra a nF RT E Vtanln (32) In order to calculate the incremental volta ge loss due to reactant depletion in the catalyst layer, the changes in Nernst potential using cR values instead of cR 0 values are represented by the fo llowing [42, 43, 46]: Nernst r concE E V (33)

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54 i r r concC nF RT E C nF RT E V1 ln 1 ln0 (34) i concC C nF RT V0ln (35) where Er is the Nerst voltage using C0 values, and ENernst is the Nernst voltage using Ci values. The ratio i/iL can be expressed as [42]: 01 C C i ii L (36) Therefore, the ratio C0/Ci (the concentration at the b acking/catalyst layer interface can be written as [2, 42]: i i i C CL L i 0 (37) Substituting equation 37 into 35 yields [42, 46]: i i i nF RT VL L concln (38) This expression is only valid for i < iL. Concentration also affects fuel cell pe rformance through reaction kinetics. The reaction kinetics is dependent upon the reactant and product concentra tions at the reaction sites. As mentioned previously, the reaction kinetics can be described by the ButlerVolmer equation [42, 46]: RT nF c c RT nF c c i icath act P P anode act R R/ ) 1 ( exp ) / exp(, 0 * 0 0 (39) where cR and cP are arbitrary concentrations and i0 is measured as the reference reactant

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55 and product concentration values cR 0* and cP 0*. In the high current-density region, the second term in the Butler-Volmer equation dr ops out and the expression then becomes: ) / exp(* 0 0RT nF c c i iact R R (40) In terms of activati on over voltage using Rc instead of 0 Rc [42, 46]: * 0 R R concc c nF RT V (41) The ratio can be written as: i i i c cL L R R 0 (42) The total concentration loss can be written as [42, 46]: i i i nF RT VL L conc 1 1 (43) Fuel cell concentration loss (or mass transpor t loss), may be expressed by the equation [42, 46]: i i i c VL L conc ln (44) where c is a constant, and can have the approximate form [42, 46]: 1 1nF RT c (45) Actual fuel cell behavior fr equently has a larger valu e than what the Equation 45 predicts. Due to this, c is often obtained em pirically. The concentra tion loss appears at high current density, and is severe. Signifi cant concentration losses limit fuel cell performance.

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56 In this study, the mass transport losses can be calculated using the following equation [4, 46]: i i i nF RT i i i nF RT Vanode L anode L cath L cath L conc , ,ln ln (46) where Li is the limiting current density, expre ssed by the following equation [4, 46]: ) (2 1C C nFD iAB L (47) where ABD is the diffusion coefficient, C is the concentration, and is the thickness.

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57 3 HEAT TRANSFER MODEL There are many areas of fuel cell technology that need to be improved in order for it to become commercially viable. Among thes e areas, the temperature of the fuel cell layers and the heat transfer through the stack are very importa nt for optimal performance. Temperature in a fuel cell is not always uni form, even when there are constant mass flow rates in the channels. Uneven fuel cell stac k temperatures are a result of water phase change, coolant temperature, air convection, the trapping of water, and heat produced by the catalyst layer. Figure 3.1 illustrates the heat generation from the catalyst layer for a PEM fuel cell. The membrane has to be ad equately hydrated in order for proper ionic conduction through the fuel cell. If the fuel cell is heated too much, the water in the fuel cell will evaporate, the membrane will dry out, and the performance of the fuel cell will suffer. If too much water is produced on th e cathode side, water removal can become a problem, which affects the overall cell heat dist ribution. This ultimately leads to fuel cell performance losses. In addition, the existen ce of phase change, and the combination of fuel cell phenomena in the stack complicate the heat transfer analysis. In order to precisely predict temperature-dependent para meters and rates of reaction and species transport, the heat distribution throughout th e stack needs to be determined accurately. Both detailed experimentation and modeling are needed to optimize the stack design and the electrochemical performance.

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58 Electrolyte H+ H+ H+ eeeee ee-Catalyst Layer Catalyst Layer Gas Diffusion Layer O2 -O2 Anode Gas Channel Cathode Gas Channel Cathode Plate Anode Plate Fuel Fuel Oxidant Oxidant Electrical Output Heat Figure 3.1. Illustration of a polymer electrolyte membrane (PEM) fuel cell with heat generation from the catalyst layers [2] The thermal model developed in this dissertation includes the computation of energy balances and thermal resistances defi ned around the control volumes in each fuel cell layer to enable the study of the diffusion of heat thr ough a particular layer as a function of time or position. 3.1 Model Development A 1-D transient numerical model is develope d for predicting the heat transfer and temperature distribution through the layers of a fuel cell stack. The numerical model consists of the calculation of both conductiv e and convective heat transfer. The energy balances for each layer include the thermal resi stance, the heat generated by the fuel cell

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59 reactions, the heat flows from the nodes on the left and right sides, and the heat loss by the fuel cell gases, liquids and the surroundi ngs. Conductive heat tran sfer occurs in the solid and porous structures, and convective heat transfer occurs between the solid surfaces and gas streams. Heating and cooli ng of the stack was examined to determine the accuracy of the model for predicting heati ng of the fuel cell catalyst layers, and the effect of running coolant through different portions of the fuel cell. The motivation of this work was to build a transient model that can be used to examine the effects of thermal diffusion, catalyst heating, me mbrane hydration, and material design and selection for a fuel cell stack. 3.1.1 Background and Modeling Approaches Heat transfer in fuel cell stacks have been studied in the literature during the last decade. The majority of the existing fuel cell st ack models in the lite rature investigate the heat transfer in the stack during steady-stat e conditions [50,51,52, 53], conduct or include heat transfer in a very crude manner, such as using the overall fuel cell stack as the control volume [51, 54]. There ar e very few studies that have used the fuel cell layers or smaller nodes to analyze the heat transfer ; however, these are typically steady-state models, and there has not been any experime ntal validation of these models. Maggio et al. [50], Park and Li [52] and Zong et al. [ 53] focused on the fuel cell cooling and flow field plate layers, and the heat transfer to the gases, but did not include the effects that the other layers may have had on the temperature distribution in the stack Zhang et al. [51] focused on a simple stack thermal model, and incorporated it into a system with thermal model of the balance of plant components. Sundaresan and Moore [55] have presented a

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60 zero-dimensional thermal layered model to an alyze cold start behavior from a sub-zero environment. This model focuses on cold-start conditions, and each layer only as a single-point temperature, which limits the data that the model can predict. Shan et al. [56] and [57] developed a transient stack system model to study the effect of varying load on the start-up during normal opera ting conditions. Khandelwal et al. [58] presented a transient stack model for cold-start analysis using a layered model. However, this model did not provide any experimental data like mo st thermal models in the literature. In addition, there are currently no thermal models that study the heat distribution through a single fuel cell in order to obtain information about the behavior of the catalyst, membrane and gas diffusion layers, and thei r effect on surrounding flow field layer temperatures. 3.1.2 Methodology In establishing the methodology for the heat transfer calculations, two important factors should be considered. The fuel cell st ack layers are made up of varying materials, each with a different thermal conductivity. Th ere is strong potential for axial conduction through the flow field channel plates, gas di ffusion media and catalyst layers. Some of the layers, such as the end plates, gaskets and terminals act as extended heat transfer surfaces, and other layers have a large area th at is in direct contact with the fuel (hydrogen), the oxidant (air) and water. Due to the simultaneous coupled conduction and convection within the channels and other layers, conjugate effects must be addressed. Therefore, the heat transfer analysis is conducted by analyzing the fuel cell stack by layer. Appendix G provides the detailed pr ocedure employed for the heat transfer

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61 calculations discussed in this se ction. The five main steps in the segmental heat transfer analysis are: 1. Definition of the layers and nodes 2. Definition of the boundary conditions 3. An energy balance computation for each node. 4. Definition of the thermal resistance for each potential heat flow path. 5. Calculation of heat transfer coefficients. 6. Calculation of additional parameters such as the heat generated by the catalyst layers. The following subsections describe each of th e above steps in the nodal heat transfer analysis. 3.2 Definitions of Segments and Nodes Figure 3.2 shows a schematic of the PEMFC stack, and the grid structure used in the fuel cell thermal analysis model. The s ections of the geomet ry under consideration vary depending upon fuel cell stack layers and construction. The main layers under consideration in this model are the end plates gaskets, terminals, flow field plates, gas diffusion media, catalyst and membrane layers The flow field plate layers are subdivided into two separate layers due to part of the layer containing both conductive and convective heat transfer, and the other part only contai ning conductive heat transfer. Although only a small percentage of the total layer area in the end plates, gaskets and contact layers has gas or liquid flow, conduc tion and convection is both assumed to be the modes of heat transfer.

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62 In the actual calculations conducted with the mathematical model, the number of segments is specified by the user, and was vari ed from 1 to 60 segments for each layer for the outputs of this study. Figure 3.2. Schematic of the PEMFC stack and the nodes used for model development. For the uniform distribution of nodes that is shown in Figure 3.2, the location of each node (xi) is:

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63 (1) for1.. 1ii x LiN N (48) where N is the number of nodes used for the simulation. The distance between adjacent nodes (x) is: 1 L x N (49) Energy balances have been defined around each node (control volume). The control volume for the first, la st and an arbitrary, internal node is shown in Figure 3.2, and explained in further detail in Appendix E. 3.2.1 Boundary Conditions The next step in the analysis is to de termine each layer, the hydrogen, air and water temperatures. The initial conditions for th is problem are that all of the temperatures at t = 0 are equal to Tin. ,1 for 1...iinTTiN (50 ) Note that the variable T is a one-dimensional array.

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64 3.2.2 Model Assumptions The following assumptions were made for the stack thermal model. 1. The heat transfer in the stack is one-dimensional (x-direction). 2. All material thermal properties are constant over the temperature range considered (20 to 80 C). 3. For the MEA layers, only the active area was included in the model. The materials surrounding the MEA were not included in the model. 3.3 Energy Balances and Thermal Resistances for Each Fuel Cell Layer This section illustrates the energy balances for each layer. Each fuel cell layer requires a unique energy balance because there are different thermal resistances, materials, and phases in each layer. Energy ba lances and thermal resistances are created for the end plate, contacts, flow field, gas diffusion, and catalyst a nd membrane layers in Sections 3.3.1 to 3.35. 3.3.1 End Plates, Contacts and Gasket Materials The end plate is typically made of a meta l or polymer material, and is used to uniformly transmit the compressive forces to the fuel cell stack. The end plate must be mechanically sturdy enough to support the fu el cell stack, and be able to uniformly distribute the compression forces to all of the major surfaces of each layer within the fuel cell stack. Depending upon the stack design, there also may be contact and gasket layers in the fuel cell stack. The gasket layers help to prevent gas leaks and improve stack

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65 compression. The contact layers or current coll ectors are used to collect electrons from the flow field plate and ga s diffusion layer (GDL) [4]. Depending upon the stack design, one or more of fuels may enter the end plates, and although the area of the fuel flow is small in these layers, both conduction and convection are both considered modes of heat transfer. Often one side of each of these layers is exposed to an insulating material (or the ambient envir onment), and the other side is exposed to a conduc tive current collector plate or insulating material. An illustration of the energy balance is shown in Figure 3.3 [4]. Solid Material HH2,i+1 qsurrqi+1Ti HH2,i HH2Ov,iHH2Ol,iHH2Ov,i+1HH2Ol,i+1 End Plate Energy Balance TsurrT2 qsurrqi+1 x/2 dU/dt Ti-1Ti+1 qi+1 x dU/dt TN-1TN+1 qi-1 x/2 dU/dt TiTN T1 qfqi+1qi-1 Figure 3.3. End plate energy balance

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66 The general energy balance for the solid portion of the end plate, contact and gasket layers can be written as [59]: 1 1 ,) (i i i s i tot mix pq q dt dT cp x A n c i Ol H i Ov H i HH H H, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 (51) where is the density, Ais the area, x is the thickness of node i, cpis the specific heat capacity of the layer, 1iq and 1iq are the heat flows from the left and right nodes, and f iq, is the heat flow from the gases/fluids. Th e derivative on the left side is the rate of change of control volume temperature (dt dTi/ ). 3.3.1.1 Thermal Resistances As shown in Figure 3-2, the heat flow for the first node takes into account the heat from the surrounding environment and the he at flow from the right node [59]. ) (i surr i surr surrT T A U q (52) If the heat is coming from the surroundings, the overall heat transfer coefficient can be calculated by [60]: surr i i surrh k x U 1 1 (53)

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67 where x is the thickness of node i, k is the thermal conductivity of node i and surrh is the convective loss from the stack to the air. The heat flow from the left node is: ) (1 1 1i i i iT T U q (54) where 1iU is the overall heat transf er coefficient for the left node, A is the area of the layer and T is the temperature of the node. Th e overall heat transfer coefficient for the heat coming from Layer 1 is [60]: 1 1 1 11 i i i i i i iA k x A k x U (55) The conduction from the adjacent node can be expressed as [59]: ) (1 1 1 1 i i i i iT T A U q (56) The overall heat transfer coefficient fo r the heat coming from node i+1 is [60]: i i i i ik x k x U 1 1 11 (57)

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68 3.3.1.2 Heat Flow From Fluid/Gases in the Layer to the Solid The conduction thermal resistance for the heat flow from the center of the gas channels to the center of the plate layer is a combination of two th ermal resistances: the conduction resistance from the center of the gas channels to the interface, and the resistance from the interface to the plate surf ace. The heat flow from the fuel cell layers to the gases/fluids based upon the total c onduction thermal resistan ce is given by [59]: ) (, , f i i f i f iT T U q (58) where iT is the temperature at node i, f iT, is the temperature of th e gases/fluid at node i, and f iU, is the overall heat transfer coefficient, which can be expressed as: void f s i i i f iA h A k x U1 1, (59) where ix is the thickness of the solid po rtion of the layer at node i, ik is the thermal conductivities of the solid and gases respectively, fh is the convective heat transfer coefficient and s iA, and voidA is the area of the solid and gases respectively. The area of solid portion of the layer is: void s iA A A (60) And the channel area is calculated by: chan chan voidL w A (61) where chanw is the width of the channel, and chanL is the length of the channel.

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69 3.3.2 Flow Field Plate In the fuel cell stack, the flow field plat es separate the reactant gases of adjacent cells, connect the cells electrically, and act as a support structure. Th e flow field plates have reactant flow channels on both sides, forming the anode and cathode compartments of the unit cells on the opposi ng sides of the flow field pl ate. Flow channel geometry affects the reactant flow veloci ties, mass transfer, and fuel cell performance. Flow field plate materials must have high conductivity a nd be impermeable to gases. The material should also be corrosion-resistant and chemi cally inert due to the presence of reactant gases and catalyst. An illustration of th e energy balance is shown in Figure 3.4.

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70 Solid Materialqi-1,f HH2,i+1 HH2_outqi+1,fqi-1qi+1qresiTi HH2,i HH2Ov,iHH2Ol,iHH2Ov,i+1HH2Ol,i+1HH2Ov_outHH2Ol_out Solid Materialqi+1,fCathode Flow Field Energy Balance qi-1,fqi+1qi-1qres,iTi HN2,i+1 HH2Ov,i+1HH2Ol,i+1HH2Ov_outHH2Ol_out HO2,i+1 HN2,iHH2Ov,iHH2Ol,iHO2,iHN2_outHO2_out Anode Flow Field Energy Balance Control Volume for Node 1 x/2 dU/dt Control Volume for Node i Ti-1Ti+1 x dU/dt Control Volume for Node N TN-1TN+1 x/2 dU/dt TiTN T1 qi+1qi-1qi-1qi+1qi+1qi-1Ti-1Ti+1 Figure 3.4. Anode and cathode flow fi eld plate energy balance The flow field plate has both conductive a nd convective heat tr ansfer due to the gas channels in the plate. The total area of th e flow field plate that has channels affects the heat transfer of the overall plate; ther efore, this is accounted for by calculating the effective cross-sectional area for conduction he at transfer, A1R, which represent the area of the solid material in contact with the pr evious and next node. The equation for heat transfer in the anode flow fiel d plate can be written as [59]: iOvHiHiresfififiii i si totmixpHHqqqqqq dt dT cpxAnc,2 ,2 ,,1 ,111 ,) ( 1,2 1,2 1,2 ,2 iOlH iOvH iHiOlHH HHHoutOlH outOvH outHH HH_2 _2 _2 (62)

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71 where is the density of the layer,A is the area of the layer, cp is the specific heat capacity of the layer, 1 iq is the gas heat flow from the previous node, R iq, 1 is the heat flow from the previous node to the solid material, 1 iq is the gas heat flow from the next node, R iq, 1 is the heat flow from the ne xt node to the solid material, i resq, is the heat generation in the layer due to electrical resistance, and iH is the enthalpy of species i coming into or out of the current node. The derivative on the left side is the rate of change of control volume temperature (dt dTi/). The heat flows coming from the right and left layer will transfer a different amount of heat from the layer to the solid and gas flow in the channels. The area of the flow field layers for axia l heat flow through the plate is given by the following equation: chan chan voidL w A (63) where chanw is the width of the flow channel, and chanL is the total length of the flow channel in the layer. The heat flows are written similarly to Equations 52 through 57 both the anode and cathode flow field plates. Fo r the anode shown in Figure 3.4, the heat flow from the previous layer to the channels is: ) (1 1 1 1 i i i i iT T A U q (64) where 1iAis the area of the channels. The heat fl ow from the previous node to the solid material is:

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72 ) (1 1 1 i i f i f iT T U q (65) where R iA, 1 is the area of the solid. The heat flow from the next node to the channels is: ) (1 1 1 1 i i i i iT T A U q (66) The heat flow from the next node to the solid material is: ) (, 1 1 1 i f i f i f iT T U q (67) where Avoid is the area of the channe ls in the plate, and A1R is the area of the solid material. The enthalpy of each gas or liquid flow into or out of the layer can be defined as: i i i iT h n H (68) where iH is the enthalpy of the stream entering or leaving the layer, in is the molar flow rate of species i, ih is the enthalpy of species i at the temperature of the node (iT). The overall heat transfer coefficient term for the previous node can be calculated as [60]: s i i i s i i i iA k x A k x U, 1 1 1 11 (69)

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73 The overall heat transfer coefficient term for the heat flow from fl uid/gases in left node: void i f i s i i i f iA h A k x U, 1 1 , 11 1 (70) The overall heat transfer coefficient fo r the heat coming from node i+1 is [60]: s i i i s i i i iA k x A k x U, 1 1 1 11 (71) The overall heat transfer coefficient term for the heat flow from flui d/gases in right node: s i i i void i f i f iA k x A h U, 1 1 11 1 (72) 3.3.3 Anode/Cathode Gas Diffusion Layer The gas diffusion layer (GDL) is located between the flow field plate and the catalyst layer. This layer allows the gases a nd liquids to diffuse through it in order to reach the catalyst layer. Th e GDL has a much lower thermal conductivity than the flow field plates and other metal components in the fuel cell; therefore, it partially insulates the heat-generating catalyst layers When modeling the heat tran sfer through this layer, the solid portion has conductive heat transfer, a nd the gas/liquid flow has advective heat transfer. An illustration of the ener gy balance is shown in Figure 3.5.

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74 Solid Materialqi-1,f HH2,i+1 HH2_outqi+1,fqi-1qi+1qresiTi HH2,i HH2Ov,iHH2Ol,iHH2Ov,i+1HH2Ol,i+1HH2Ov_outHH2Ol_out Solid Materialqi+1,fCathode Flow Field Energy Balance qi-1,fqi+1qi-1qres,iTi HN2,i+1 HH2Ov,i+1HH2Ol,i+1HH2Ov_outHH2Ol_out HO2,i+1 HN2,iHH2Ov,iHH2Ol,iHO2,iHN2_outHO2_out Anode Flow Field Energy Balance Control Volume for Node 1 x/2 dU/dt Control Volume for Node i Ti-1Ti+1 x dU/dt Control Volume for Node N TN-1TN+1 x/2 dU/dt TiTN T1 qi+1qi-1qi-1qi+1qi+1qi-1Ti-1Ti+1 Figure 3.5. GDL energy balance Heat is generated in the GDL due to oh mic heating. Since the GDL has high ionic conductivity, ohmic losses are negligible compared with the catalyst and membrane layers. The overall energy balance equati on for the anode GDL can be written as: iOlHiOvHiHires ifii i totmixpHHHqqqq dt dT cpxAnc,2 ,2 ,2 ,1 ,11 ,) ( 1,2 1,2 1,2 iOlH iOvH iHH HH (73) The overall energy balance equation fo r the cathode GDL can be written as: iOlHiOvHiHiresfiii i totmixpHHHqqqq dt dT cpxAnc,2 ,2 ,2 ,111 ,) ( 1,2 1,2 1,2 iOlH iOvH iHH HH (74)

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75 3.3.4 Anode/Cathode Catalyst Layer The anode and cathode catalyst layer is a porous layer made of platinum and carbon. It is located on either side of th e membrane layer. When modeling the heat transfer through this layer, the solid portion has conduc tive heat transfer, and the gas/liquid flow has advective heat transfer. Figure 3.6 show s the energy balance of the catalyst layer. Porous Material qi+1 HH+,i+1qi-1 HH2,iHH2Ov,iHH2Ol,i HH2Ov,i+1HH2Ol,i+1 Porous Material Cathode Catalyst Energy Balance qi+1HO2,i+1qi-1 HH+,iHH2Ov,iHH2Ol,i HH2Ov,i+1HH2Ol,i+1 HN2,i+1Anode Catalyst Energy Balance Control Volume for Node 1 x/2 dU/dt Control Volume for Node i Ti-1Ti+1 x dU/dt Control Volume for Node N TN-1TN+1 x/2 dU/dt TiTN Tiqres,iqint,iTiqres,iqint,iqi+1qi-1qi+1qi-1qi+1qi-1T1Ti-1Ti+1 Figure 3.6. Catalyst energy balance

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76 The overall energy balance equation fo r the anode and cathode energy balance can be written [4, 59]: i Ol H i Ov H i H i i res i i i tot mix pH H H q q q q dt dT cp x A n c, 2 2 2 int, 1 1 ,) ( 1 2 1 2 1 2 i Ol H i Ov H i HH H H (75) The heat generation in the catalyst layer is due to the electrochemical reaction and voltage overpotential. The heat generation term in the cataly st layer can be written as [58]: act i i inF S T x i q int, (76) where iT is the local catalyst temperature, i is the current density, ix is the node thickness, n is the number of elect rons, F is Faraday’s constant, S is the change in entropy and act is the activation overpot ential. The entropy change at standard state with platinum catalyst is taken as S = 0.104 J mol 1 K 1 for the anode, and S = 326.36 J mol 1 K 1 for the cathode. The activ ation over-potential (act ) was calculated using the Butler-Volmer equation. 3.3.5 Membrane The PEM fuel cell membrane layer is a persulfonic acid layer that conducts protons, and separates the anode and cathode compartments of a fuel cell. The most commonly used type is DuP ont’s Nafion membranes. Th e dominant mode of heat transfer in the membrane is conduction. An illustration of the energy balance is shown in Figure 3.7.

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77 Figure 3.7. Membrane energy balance The overall energy balance e quation can be written as: i Ol H i Ov H i H i i res i i i tot mix pH H H q q q q dt dT cp x A n c, 2 2 int, 1 1 ,) ( 1 2 1 2 1 i Ol H i Ov H i HH H H (77) Note that the heat generation term in the membrane consists of Joule heating only.

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78 3.4 Heat Generated by Electrical Resistance The rate at which energy is created by passing current, i, through a medium of electrical resistance is [43, 49]: R i qi res 2 (78) If the layer material is ohmic, the resistance can be found by [49]: i i i resA x R (79) If the layer conducts electr icity (such as the contact layer), then there is an additional heat generation in node i (i resq,) due to electrical resistance, which can be calculated as: i i i res i resA x i q, 2 (80) where i is the current density, A is the area of the layer, i res is the specific resistance of the material,ix is the thickness of the layer and t is the amount of time that the current is flowing (sec). There is no heat generated in the end plate, contact or gasket layers. However, in some fuel cell stack designs, th e end plate may be heated; therefore, an additional heat generation term would need to be added to the model formulation.

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79 3.5 Heat Transfer to Gases The conduction thermal resistance for the he at flow from the center of the plate layer to the center of the gas channels is a combination of two thermal resistances: the conduction resistance from the center of th e plate surface to th e interface, and the resistance from the interface to the center of the gas channels. The channel energy balance is shown in Figure 3.8. HH2,i+1 HH2,i HH2Ov,iHH2Ol,iHH2Ov,i+1HH2Ol,i+1 qs qsurr qi+1Channel Energy Balance Figure 3.8. Energy balance for channels or vo id space in the fuel cell layers The overall channel energy balan ce equation can be written as: s i i i f i tot mix pq q q dt dT n c, 1 1 ,) ( i Ol H i Ov H i HH H H, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 (81) The heat flow from the fuel cell layer node s to the center of th e channel is based upon the total conduction therma l resistance is given by:

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80 ) (, , i s i s i s iT T U q (82) where iT is the temperature at node i, s iT, is the temperature of the solid at node i, and s iU, is the overall heat transfer coe fficient, which can be expressed as: void f s i i i s iA h A k x U1 1, (83) where ix is the thickness of the solid po rtion of the layer at node i, ik is the thermal conductivities of the solid and gases respectively, fh is the convective heat transfer coefficient and s iA, and voidA is the area of the solid and gases respectively. The area of solid portion of the layer is: void s iA A A (84) And the channel area is calculated by: chan chan voidL w A (85) where chanw is the width of the channel, and chanL is the length of the channel.

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81 3.6 Convective Heat Transfer Coefficient The calculation of the heat transfer coefficien t is critical for obtaining a precise heat transfer model. In order to obtain the convectiv e heat transfer coefficient, the procedure is as follows [61]: 1. Calculate the fluid properties in cluding the viscosity and thermal conductivity. 2. Calculate the Reynold’s number from the flui d properties and duct geometry. 3. Calculate the flow regime from the Reynold’s number. 4. Calculation of the Nusselt number and c onvective heat transfer coefficient. The properties of the gases are needed to evaluate the convective heat transfer coefficient at each wall. To calculate the dyna mic viscosity of the components in a gas stream as a function of temperat ure, a fifth order polynomial is used with the constants in Table 3.1: 6 1 ,1000n n nT a (86)

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82 Table 3.1 Polynomial coefficients for ca lculating dynamic viscosity ________________________________________________________________________ Constant Hydrogen Oxygen Water ________________________________________________________________________ A 15.553 -169.18 -6.7541 B 299.78 889.75 244.93 C -244.34 -892.79 419.50 D 249.41 905.98 -522.38 E -167.51 -598.36 348.12 F 62.966 221.64 -126.96 G -9.9892 -34.754 19.591 ________________________________________________________________________ A similar expression is used for thermal conduc tivity with the constants in Table 3.2 [61]: 6 1 ,1000 01 0n n n kT a k (87) Table 3.2 Polynomial coefficients for ca lculating thermal conductivity ________________________________________________________________________ Constant Hydrogen Oxygen Water ________________________________________________________________________ A 1.5040 -0.1857 2.0103 B 62.892 11.118 -7.9139 C -47.190 -892.79 419.50 D 249.41 -7.3734 35.922 E -31.939 -4.1797 35.993 F 11.972 1.4910 -18.974 G -1.8954 -0.2278 4.1531 ________________________________________________________________________

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83 Since the model presented in this study predicts the temperatures and compositions locally, at each point in the ce ll, the evaluation of the heat transfer coefficients must include a dependence on pos ition and composition in side the cell. The Nusselt number is typically calculated from correlations fitted to empirical data, and most of these studies give average values for Nu along the whole duct, and only a few of them are applicable to local studies [61]. 3 / 2 3 / 21 ) 1 (Pr ) 8 / ( 7 12 1 Pr ) 1000 )(Re 8 / ( L D f f Nuh (88) 64 1 ln(Re) 79 0 1 f (89) Gnielinski’s equation is used to evaluate Nu, and it is applicable to Re > 2300, 0.5 < Pr < 2000 and L > Dh. In the literature, si mpler equations are often used such as Colburn’s, which is valid for Re > 10,000, 0. 7 < Pr < 160 and L > 10Dh. This correlation is easier to evaluate, but can l ead to errors as high as 20% [59]. In addition, many of the flows within the cell are from 2300 to 10,000, and the values from this equation are significantly higher than when using Gnielinski’s equation. The convective heat transfer coefficient is evaluated directly from the value of Nu using the following equation [61]: hD k Nu h (90) where hD is calculated at the axial position. Th e literature shows a s light underestimation

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84 of Nu, however, the error is very small, and does not substantially increase the uncertainty in the value of th e heat transfer coefficients. Liquid or gas flow confined in channels can be laminar, turbulent, or transitional and is characterized by an important di mensionless number known as the Reynold’s number (Re). This number is the ratio of the inertial forces to viscous forces and is given by [2, 46]: v D Dch m ch m Re (91) where m is the characteristic velocity of the flow (m/s), Dch is the flow channel diameter or characteristic length (m), is the fluid density (kg/m3), is the fluid viscosity (kg/(m*s)), and is the kinematic viscosity (m2/s). When Re is small (< 2000), the flow is laminar. When Re greater than 4000, the flow is turbulent. When Re is between 2000 and 4000, it is know to be in the “transitional” range, where the flow is mostly laminar, with occasional bursts of irregular behavior. The flow in fuel cell channels usually falls in the laminar flow regime. The velocity (m/s) in a fuel cell channe l near the entrance of the cell is [59]: ch in HA v v_ 2 where 22 1 r Ach (92) where r is the radius of the flow channel. The specific heat capacity (J/molK) of hydrogen and oxygen were obtained from the shomate equations NIST chemistry webbook [62]:

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85 2 3 2/ * t E t D t C t B A cp (93) The enthalpy of each gas (J/mol) can be calculated by [62]: F t E Dt Ct Bt At h 4 3 23 2 (94) where t is given by: 1000, i fT t (95) where, A, B, C, D, and E can be obt ained from Table 3.3, and t is T/1000. Table 3.3 Polynomial coefficients for calculating speci fic heat capacity and fo rmation enthalpies ________________________________________________________________________ Constant Hydrogen Oxygen Water Vapor Liquid Water (T=298-1000k) (T=298-6000K) ________________________________________________________________________ A 33.066178 29.659 30.09200 -203.6060 B -11.363417 6.137261 6.832514 1523.290 C 11.432816 -1.186521 6.793435 3196.413 D -2.772874 0.095780 -2.534480 2474.455 E -0.158558 -0.219663 0.082139 3.855326 F -9.980797 -9.861391 -250.8810 -256.5478 ________________________________________________________________________ The specific heat capacity of the mi xture can be calculated by [45]: j p j i p i mix pc x c x c, , (96)

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86 4 MASS, CHARGE AND PRESSURE DROP MODEL Mass, charge and pressure drop phenomena are all important when characterizing fuel cell performance. The fuel cell must be supplied continuously with fuel and oxidant, and product water must be removed continually to insure proper fuel and oxidant at the catalyst layers to maintain high fuel cell e fficiency. High fuel a nd oxidant flow rates sometimes insure good distribution of reactants, but if the flow rate is too high, the fuel may move too fast to diffuse through the GDL a nd catalyst layers. If it is too low, the fuel cell will loose efficiency. Mass transport in the fuel cell GDL and catalyst layers are dominated by diffusion due to th e tiny pore sizes of these laye rs (2 to 10 microns). In a flow channel, the velocity of the reactants is usually slower near the walls; therefore, this aids the flow change from convective to diffusive. The pressure drop of the mixture gas in th e fuel cell flow channels have rarely been considered in the fuel cell literature. However, in industrial design, it is a very important characteristic because it directly affects the efficien cy of a fuel cell system, and is directly related to the selection of th e system pump. In addition, since increased pressures within the fuel cell increase the overa ll fuel cell performance, it is very helpful to know the local pressures inside the fuel cell to better optimize th e fuel cell design. A schematic of convective and diffusive mass tran sport in the fuel cell layers is shown in Figure 4.1.

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87 Figure 4.1. Fuel cell layers (flow field, gas diffusi on layer, and catalyst layer) that have convective and diffusive mass transport [4] The transport of charges is also very im portant since efficient charge transport ensures the highest possible electricity produced by the fuel cell stack. The two major types of charged particles are electrons and ions, and both electronic and ionic losses occur in the fuel cell. The el ectronic loss between the bipola r, cooling and contact plates are due to the degree of cont act that the plates make w ith each other due to the

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88 compression of the fuel cell stack. The ionic losses occur in the membrane; therefore, ensuring optimal ionic transport is critical for good fuel cell performance. A charge balance only needs to be conducted on the layer if it conducts electrons. The general mass balance equations presented in this chapter are used both for the outlet and inlet of each fuel cell stack layer. For the end plates, gaskets, contacts, and flow field plate layers, the mo le fractions are determined using the saturation pressure equations. In the MEA layers (the GDL, cat alyst and membrane layers), the same mass balance equations are used. However, more sophisticated methods of determining the mole fractions or concentrations are used due to diffusive tr ansport in these layers. These are then substituted into the overall mass ba lance equation to obtain the rate of mass accumulation. An illustration of the mass, energy and charge balances in a layer are shown in Figure 4.2.

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89 TLayer2 System Border (Control Volume)QLayer1Layer 1Layer 2Layer 3 TLayer1TLayer3 QLayer3 mH2_2 mH2_1mH2Ov_1mH2Ol_1 mH2Ov_2mH2Ol_2 Charge from Layer 3 Charge to LoadNon-electrically conductive layer Electrically conductive layer Electrically conductive layer Figure 4.2. Mass, energy and charge balance around a layer 4.1 Methodology In establishing the methodology for the ma ss and charge transfer, and pressure drop model, there are several importa nt factors should be considered: 1. Mass and species conservation 2. Momentum and pressure across each layer 3. Pressure drop

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90 Appendix H provides the detailed procedur e employed for the mass, charge and pressure drop calculations discussed in this se ction. The five main steps in the analysis are: 1. Definition of the layers and nodes 2. Definition of the boundary conditions 3. A mass balance computation for each node. 4. A pressure drop calculation as a function of x. 5. Calculation of additional parameters such as concentration and relative humidity. The following subsections describe each of the above steps in the nodal layer computation. 4.2 Definitions of Segments and Nodes Figure 4.3 shows a schematic of the PEMFC stack, and the grid structure used in the fuel cell model. In the act ual calculations conducted with the mathematical model, the number of segments is specified by the user and was varied from 1 to 60 segments for the membrane layer for the outputs of this study.

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91 Figure 4.3. Slices created for mass, charge a nd pressure drop portion of the model For the uniform distribution of nodes that is shown in Figure 4.3 the location of each node (xi) is: (1) for1.. 1ii x LiN N (97) where N is the number of nodes used for the simulation. The distance between adjacent nodes ( x) is: 1 L x N (98)

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92 4.3 Boundary Conditions The model solves for the concentration of water, potential, temperature and pressure simultaneously. In order to solve for these transient variables, initial and boundary conditions are required. At x = 0, four boundary conditions are necessary to fully specify the problem. These are: For the left boundary: ) ( ) (1 1 2 2 i m i O H i m i O Hx c x c (99) ) ( ) (1 1 i i i ix T x T (100) ) ( ) (1 1 , i i m i i mx x (101) ) ( ) (1 1 , i i tot i i totx P x P (102) For the right boundary: ) ( ) (1 1 2 2 i m i O H i m i O Hx c x c (103) ) ( ) (1 1 i i i ix T x T (104) ) ( ) (1 1 , i i m i i mx x (105) ) ( ) (1 1 , i i tot i i totx P x P (106)

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93 4.4 Model Assumptions The following assumptions were made fo r the mass, charge and pressure drop portion of the model: 1. All material thermal properties are constant over the temperature range considered. 2. The gases/fluid in each laye r have ideal gas behavior. 3. The gas diffusion media is composed of void space and carbon fibers. 4. The catalyst layer is composed of carbon powder, platinum and void space, and its physcial structure is as sumed to be composed of spherical agglomerates. 5. The electrochemical reaction occu rs in the catalyst layer. 6. The transport of the reactants from the gas channels to the catalyst layer occurs only by diffusion to the agglomerate surface. 4.5 General Mass Balance Equations In order to predict accurate hydrogen, oxygen and water mixture compositions throughout the fuel cell stack, accurate ma ss balances are required. Mass balance equations are used both for the outlet and in let of each fuel cell stack layer. The mass balances for the end plate la yer are shown in Figure 4.4.

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94 HH2,i+1 HH2,iHH2Ov,iHH2Ol,iHH2Ov,i+1HH2Ol,i+1 Channel Mass Balance Figure 4.4. Mass balance illustration for the channels or void space in the fuel cell layers Based upon the assumption that the mixture is regarded as an ideal gas, the volumetric flow rate is first converted to a molar flow rate using the ideal gas law [63]: in in in in totRT P n (107) where in totn_ inlet molar flow rate,inP inlet pressure,in inlet volumetric flow rate,inT inlet temperature, and is the R ideal gas constant. For transient mass balances, the total molar accumulation totn can be written as [63]: 1 , i tot i tot totn n dt dn (108) where i totn, is the total molar flow rate of mixture into the control volume, and 1 i totn is the total molar flow rate of mixture out of the control volume. The rate of H2 accumulation is: 1 1 2 , 2 2) ( i tot i H i tot i H tot Hn x n x n x dt d (109)

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95 where i Hx, 2 is the hydrogen mole fraction into the control volume, and 1 2 i Hx is the hydrogen mole fraction out of the control volume. The rate of O2 accumulation is: 1 1 2 , 2 2) ( i tot i O i tot i O tot On x n x n x dt d (110) where i Ox, 2 is the oxygen mole fraction into the control volume, and 1 2 i Ox is the oxygen mole fraction out of the control volume. The rate of H2O accumulation is: 1 1 2 , 2 2) ( i tot i O H i tot i O H tot O Hn x n x n x dt d (111) where i O Hx, 2 is the hydrogen mole fraction into the control volume, and 1 2 i O Hx is the hydrogen mole fraction out of the control volume. In order to calculate the mole fraction of the water vapor going into the fuel cell stack, the first step is to calculate the vapor pressure of the inlet water vapor, i O Hp, 2[63]: ) (, 2 f i sat in i O HT P p (112) where ) (, f i satT P is saturation pressure at the ga s/fluid temperatur e at node i and in is the inlet humidity of the gas stream. Humidity is the ratio of the mass of the vapor in one unit mass of vapor-free gas. The humidity depends upon the partial pressu re of the vapor in the mixture [64]. ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H (113) whereO HM2 molecular weight of water, 2HM molecular weight of hydrogen and i totP, total pressure at node i.

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96 The mole fraction of the water vapor is [64]: O H H O H i Ov HM H M M H x2 2 2 21 (114) The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 (115) The mole fraction of the liquid water in the fuel and oxidant streams entering the fuel cell stack is assumed to be zero: 0, 2i Ol Hx (116) Liquid water is included for all other nodes by calculating the molar flow rate for water condensation and evaporation using the following equation [53, 65]: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n (117) where cd channel depth (m),cw is the channel width (m) and ck is the evaporation and condensation rate constant (s-1). The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2 (118) The total mole fraction of water is: i tot i O H i O Hn n x, 2 2 (119) The mole fraction of hydrogen is: i O H i Hx x, 2 21 (120)

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97 The molar flow rate of hydrogen is: i tot i H i Hn x n, 2 2 (121) Total flow rate out of the layer is: 1 2 1 2 1 i O H i H i totn n n (122) In order to present the state of water va por and liquid water, the relative humidity (RH) and relative water content are defined as follows [53, 65]: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH (123) Relative water content [53, 65]: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW (124) 4.6 Pressure Drop The pressure drop of the gas mixture in the fuel cell literature has rarely been considered. However, in industrial design a nd practice, it is a significant parameter simply because it directly affects system efficiency. In a typical flow channel, the gas moves from one end to the other at a certain mean velocity. The pressure difference between the inlet and outlet drives the fluid flow. By increasing the pressure drop between the outlet and inlet, the velocity is increased. The flow through bipolar plate ch annels is typically laminar, and proportional to the flow rate. The velocity (m/s) in a fuel cell cha nnel near the entrance of the cell is [59]: ch in chanA v (125)

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98 where chA is the cross-sectional area of the channel (m2), and in inlet volumetric flow rate (m3/s). The pressure drop can be approximated using the equations for incompressible flow in pipes [46]: 2 22 2v K v D L f dx dPL H chan tot (126) where f is the friction factor, Lchan is the channel length, m, DH is the hydraulic diameter, m, is the fluid density, kg/m3, v is the average velocity, m/s, and KL is the local resistance. The hydraulic diameter for a circular fl ow field can be defined by [46, 59]: cs c i HP A D 4, (127) where Ac is the cross-sectional area, and Pcs is the perimeter. In this work, the flow field channels are rectangular, and the inlet cha nnels through the plates are circular. For rectangular channels, the hydraulic diam eter can be defined as [46, 59]: c c c c i Hd w d w D 2, (128) where wc is the channel width, and dc is the depth.

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99 The channel length can be defined as [2, 46]: ) (, L c ch i cell chanw w N A L (129) where Acell is the cell active area, Nch is the number of parallel channels, wc is the channel width, m, and wL is the space between channels, m. The friction factor can be defined by [46, 59]: Re 56 if (130) Liquid or gas flow confined in channels can be laminar, turbulent, or transitional and is characterized by an important di mensionless number known as the Reynold’s number (Re). This number is the ratio of the inertial forces to viscous forces and is given by [46, 59]: v D Dch m ch m i Re (131) where m is the characteristic velocity of the flow (m/s), Dch is the flow channel diameter or characteristic length (m), is the fluid density (kg/m3), is the fluid viscosity (kg/(m*s)), and is the kinematic viscosity (m2/s). When Re is small (< 2000), the flow is laminar. When Re greater than 4000, the flow is turbulent, which means that it has random fluctuations. When Re is between 2000 and 4000, it is know to be in the “transitional” range, where the flow is mostly laminar, with occasional bursts of irregular behavior. It is found that rega rdless of channel size or flow velocity, f Re = 16 for

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100 circular channels. The flow in fuel cell channe ls usually falls in the laminar flow regime with low reactant pressures. The total outlet pressure (P a) of each node is obtained by subtracting the pressure drop at the control volume inlet from the pre ssure at the inlet of the control volume [53, 65]: dx dx dP P Px tot i tot i tot 0 1 (132) 4.7 Charge Transport Most models neglect conductivity calculations, since most metallic and carbonbased fuel cell layers have good conductivity. However, a rigorous model should include this calculation since it can become a limiti ng factor due to geom etry or composition. Ohm’s law can be used to take this into account [48]: 0 1i x (133) where 1 and 0 are the volume fraction and electric al conductivity, respectively. All electrochemically conductive la yers in the fuel cell (besid es the MEA layers) will use Equation 133.

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101 4.8 Flow Field Plate Layers The transient mass balance equations, for the anode and cathode flow field plates, are similar to Equations 108 111, except that there is an additional term for the mass flows leaving the stack. The mass flow s are illustrated in Figure 4.5. Figure 4.5. Cathode flow field plat e mass/charge balance For transient mass balances, the total mo lar accumulation can be written as [63]: 2 _ tot out tot in tot totn n n dt dn (134) The rate of H2 accumulation is: 2 2 2 _ 2 _ 2 2) (tot H out tot out H in tot in H tot Hn x n x n x n x dt d (135)

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102 The rate of H2O accumulation is: 2 2 2 _ 2 _ 2 2) (tot O H out tot out O H in tot in O H tot O Hn x n x n x n x dt d (136) The rate of O2 accumulation is: 2 2 2 _ 2 _ 2 2) (tot O out tot out O in tot in O tot On x n x n x n x dt d (137) 4.8.1 Diffusive Transport From the Flow Fiel d Channels to the Gas Diffusion Layer As shown in Figure 4.6, the reactant is supplied to the flow channel at a concentration 0C, and it is transported from the flow channel to the concentration at the electrode surface sC through convection. The rate of mass transfer is then [1, 4]: ) (0 s m iC C h A m (138) where iA is the electrode surface area, and mh is the mass transfer coefficient.

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103 Figure 4.6. Entire channel as the control volume fo r reactant flow from the flow channel to the electrode layer [4] The value of mh is dependent upon the wall conditi ons, the channel geometry, and the physical properties of species i and j. The mass transfer coefficient, mh, can be found from the Sherwood number [1, 4]: h j i mD D Sh h, (139)

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104 where Sh is the Sherwood number, hD is the hydraulic diameter, and j iD, is the binary diffusion coefficient for species i and j gi ven in Appendix B. The Sherwood number depends upon channel geometry, a nd can be expressed as [1, 4]: k D h Shh H (140) where Sh = 5.39 for uniform surface mass flux (m = constant), and Sh = 4.86 for uniform surface concentration (Cs = constant). The concentrations are calculated at the node inlet [59]: f i i tot i O H i O HRT P x C, , 2 2 (141) f i i tot i H i HRT P x C, , 2 2 (142) The outlet average concentration [1, 4]: i m H m i H i Hbu x h C C, 2 1 2exp (143) The average limiting current density is [1, 4]: 1 2 2 1 2 2lni H i H i H i H m LC C C C nFh i (144)

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105 4.8.2 Calculation of Pressure Drop The flow through flow field plate channels is typically laminar, and proportional to the flow rate. The pressure drop in the flow field and cooling layers are calculated using the same equations in addition to an equation for the increase/decrease in channel width. The initial volumetric flow rate is first calculated for the number of inlet channels. The velocity (m/s) in the entrance of the flow field layer is [59]: ch i tot i f i tot iN P R T n v , ,* (145) The velocity is than calculated in each of the channel inlets using [63]: ch in chanA v (146) where in inlet volumetric flow rate (m3/s), and chA is the cross-sectional area of the channel (m2). Often, after the reactant flow enters the entrance channe l, the flow rate changes because the channel increases or decreases in cross-sectional area. Th e molar flow rate in each channel is calculated us ing the ideal gas law [63]: R T P v nlay f lay chan chan* *, (147)

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106 In the anode and cathode flow field plates there are two outlets : the outlet at the end of the flow channels that lead to the next layer and the outlet from the flow channels into the gas diffusion media. To calculate th e flow rate from the channels to the GDL layer, the total channel length is calculated using the following equation [46]: ) (, L c ch i cell chanw w N A L (148) where cellA is the cell active area, chN is the number of channels, cw is the channel width, m, and Lw is the space between channels, m. The hydraulic diameter for the rectangular flow channels is estimated using the hydraulic diameter equation for a rectangular flow field [46]: c c c c i Hd w d w D 2, (149) where wc is the channel width, and dc is the depth. The Reynold’s number at the channel exit can be written as [46, 59]: v D Dch m ch m i Re (150) where m is the characteristic velocity of the flow (m/s), Dch is the flow channel diameter or characteristic length (m), is the fluid density (kg/m3), is the fluid viscosity (kg/(m*s)), and is the kinematic viscosity (m2/s).

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107 The friction factor is calculated using th e formula for rectangular channels [46, 59]: Re 56if (151) The pressure drop can be approximated using the equations for incompressible flow in pipes [46]: 2 22 2v K v D L f dx dPL H chan tot (152) where f is the friction factor, Lchan is the channel length, m, DH is the hydraulic diameter, m, is the fluid density, kg/m3, v is the average velocity, m/s, and KL is the local resistance. The velocity going to the GDL layer is then calculated using the following equation [66, 67]: i tot i i i mP x k u, (153) where k is the permeability (m2), is the viscosity (Pa-s), x is the thickness of node i (m), and i totP, is the change in total pressure (Pa). 4.9 Anode/Cathode Diffusion Layer The same mass balance equations are used for the anode and cathode GDL layer, except the mass flow rates are obtained from the gas concentr ations calculated using a

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108 derivation based upon Fick’s law that is s hown in Appendix F. The overall mass and charge balances are illu strated in Figure 4.7. Figure 4.7. GDL mass/charge balance The overall mass balances for the GDL layer are calculated in the same manner as described in Sections 4. 5 – 4.8. The pre ssure drop is calculated using Darcy’s law: x A k Pi i i i tot (154) where is the viscosity, is the volumetric flow rate, k is the permeability, A is the cross-sectional area (m2) is the void space, and x thickness of node i (m). The electrochemical reaction in the catalyst layer can l ead to reactant depletion, which can affect fuel cell performance through concentration losses. In turn, the reactant depletion will also cause activation losses. The difference in the catalyst layer reactant

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109 and product concentration from the bulk values determines the extent of the concentration loss. The average outlet concentration can be calculated as shown in Equation 155 [1, 4]: i m H m i H i Hbu x h C C, 2 1 2exp (155) where Hx is the height of gas diffusion layer, i mu, is the velocity of mixture (m/s), b is the distance between flow channe ls and gas diffusion layer and i HC, 2 is the concentration of hydrogen at node i. Using Fick’s law, the diffusional trans port through the gas diffusion layer at steady-state is [1, 4]: i i H i H j i i i Hx C C D A n ) (1 2 2 1 2 (156) where Ci is the reactant concentration at the GDL/catalyst interface, and ix is the gas diffusion layer thickness, and j iD, is the effective diffusion coefficient for the porous GDL, which is dependent upon the bulk diffusi on coefficient D, and the pore structure. Assuming uniform pore size, and the gas diffusi on layer is free from flooding of water, eff j iD, can be defined as [66]: 2 / 3 ,j i eff j iD D (157) where is the electrode porosity.

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110 Since the GDL layer is made of carbon, a ch arge transport relation is required. In order to account for porosity and tortuosity the Bruggeman correction is used. Ohms law is again used for charge transport [48]: 0 51 1 1i x (158) where 1 and 0 are the volume fraction and electrical conductivity, respectively. The Bruggeman correction is used in Equation 158 to account for porosity and tortuosity. Since the GDL is often coated with Tefl on to promote hydrophobicity, carbon is the conducting phase and the Teflon is insulating. 4.10 Anode/Cathode Catalyst Layer The catalyst layer contains many phases: liquid, gas and different solids. Although various models have different equatio ns, most of these are derived from the same governing equations, regardless of the effects being modeled. In most cases, the anode reaction can be described by a ButlerVolmer type expression, except for those which use a fuel other than pure hydrogen. In these cases, the platinum catalyst becomes poisoned. The carbon monoxide adsorbs to the electrocatalytic site s and decreases the reaction rate. There are models in the litera ture that account for this by using a carbon monoxide site balance and examining the reaction steps involv ed. For the cathode, a Tafel-type expression is comm only used, due to the slow ki netics of the four-electron transfer reaction.

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111 The mass and charge transpor t in the catalyst layer are interdependent, therefore, they are calculated together. Figure 4.8 show s the overall mass and charge balances for the anode and cathode catalyst layers. Figure 4.8. Catalyst layer mass/charge balances The gaseous species in the anode catalyst layer are hydrogen and water. The gases are transported through the porous catalyst layer primarily through diffusion. The diffusive flux can be derived using Fick’s la w. The agglomerate structure for the catalyst layer was proposed by Ridge et al. [68] and has recently gained support through microscopy observations [68, 40]. Several mode ls have assumed that the catalyst layer has a spherical agglomerate structure, a nd several studies have proved that this assumption provides a better fit with e xperimental results [68, 40, 69]. Since the cathode catalyst layer is modelle d using an agglomerate approach, the kinetics expression for the tota l cathodic reaction rate per un it volume of electrode can be

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112 written as [40, 69]: E i a icathode cath 2 1 (159) where 2 1a is the specific interfacial area per unit volume of the catalyst layer, and cathodei is the transfer current fo r the oxidation reduction reacti on. The solution of the mass conservation equation in spheri cal agglomerate yields an analytical expression for the effectiveness factor, which is the mass tran sfer and reaction with in each agglomerate [70,71]: L L LE 3 1 3 tanh 1 1 (160) where L is the Thiele modulus for the spherica l agglomerate, and can be expressed as [70,71]: eff agg O c agg LD k r, 23 (161) where aggr is the radius of the spherical agglomerate, which can be determined by [70,71]: agg agg aggS V r3 (162) and k is a rate constant given by [70]: ) ( exp ) 1 ( 42 0 2 1 ORR c ref O cl v ref cRT F c F i a k (163)

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113 where 21a is the interfacial area between the electrically conducting and membrane phase with no flooding, refi0is the exchange current density for the reaction, ca is the cathodic transfer coefficient, ORR is the cathode overpotential, and the refere nce concentration is that concentration in the agglomerate that is in equilibrium with the reference pressure [70, 71]: aggO ref O ref OHpc,22 2 (164) where aggOH,2 is Henrys constant for oxygen in th e agglomerate. If external mass transfer limitations can be neglected, then the surface concentration can be set equal to the bulk concentration, which is assumed unifo rm throughout the cataly st layer in simple agglomerate models. The local overpotential, ORR can be defined as [70, 71]: ionel ORR (165) The porosity of the catalyst layer, i.e. the sp ace that is not occupied by the solid space, can be calculated using [70]: cl S cl v1 (166) This is the volume fraction of macro-por es for oxygen transport. The solid phase volume fraction can be calculated knowing the amounts of platinum and carbon in the catalyst layer [70, 71]: L m CPt CPtPt C Pt cl S / / 1 1 (167)

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114 where Pt and C are the platinum and carbon densities, C Pt/ is the platinum to carbon ratio, Ptm is the platinum loading and L is the catalyst layer thickness. With the assumption that the catalyst laye r is made of spherical agglomerates, the number of agglomerates per unit vol ume, n, can be written as [70]: 33 / 4agg cl Sr LH n n (168) Many models use catalyst loading, which is defined as the amount of catalyst in grams per geometric area of the fuel cell f ace. If a turnover frequency is desired, the reactive surface area of platinum can be used. This is related to the radius of the platinum particle, which assumes a roughness factor that is experimentally inferred using cyclic voltammetry measuring the hydrogen adsorption. These variables are used to calculate the specific interfacial area be tween the electrocatalyst and the electrolyte [40, 69, 70]: L A m aPt Pt2 1 (169) where L is the thickness of the catalyst layer. PtA is the active surface ar ea of platinum in the catalyst layer, which can be determ ined with an empirical formula [70]: 5 5 2 5 3 510 5950 1 ) ( 10 0153 2 ) ( 10 5857 1 ) ( 10 2779 2 C Pt C Pt C Pt APt (170) where C Ptis the ratio of platinum catalyst and carbon powder. The cell current versus the effectiveness f actor is illustrated in Figure 4.9, and the superficial flux density of hydrogen is shown in Figure 4.10.

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115 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cell Current (A/cm2)Effectiveness Factor Figure 4.9. Cell current versus effectiveness factor The hydrogen anode reaction can be written as [48, 64]: E i a ih 2 1 2 1 2 (171) E i S a F Nanode G H) 1 ( 2 12 1 ,2 (172) anode act c anode act a ref H H anodeRT F RT F p p i_ 2 2exp exp (173) The liquid water cathode catalyst reac tion can be written as [48, 64]: E i S a F Ncathode L O H) 1 ( 4 12 1 2 (174) cath act c ref O O cathodeRT F p p i_ 2 2exp (175)

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116 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 Cell Current (A/cm2)Flux density of H2 (mol/cm2-s) Figure 4.10. Superficial flux density of hydrogen [4] The mass flow through the GDL layer is calculated in the same manner as described in Sections 4.5 – 4.8. However, th e mass balances also n eed to include a term for the consumption of hydrogen or oxygen, and the water generated in the cathode catalyst layer [48, 64]: nF iA ni H, 2 (176) where i is the nominal cell current density, A is the cross-sectional area, F is Faraday’s constant, and n is 2 for the anode and 4 for the cathode (for the number of protons and electrons transferred). The pressure drop is calculated using Darcy’s law: x A k Pi i i i tot (177)

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117 where is the viscosity, is the volumetric flow rate, k is the permeability, A is the cross-sectional area (m2) is the void space, and x thickness of node i (m). As in the other layers, Ohms law is used to calculate the potential [48]: 0 51 1 1i x (178) where 1 and 0 are the volume fraction and electrical conductivity, respectively. The mass balances for the reactants should ta ke into account the reaction and the mass transport.

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118 5 POLYMER ELECTROLYTE MEMBRANE MODEL In proton exchange membrane fuel cells (P EMFC), the fuel travels to the catalyst layer, and is decomposed into protons (H+) and electrons. The electrons travel to the external circuit to power the load, and the hydrogen protons tr avel through the electrolyte until they reach the cathode to combine with oxygen to form water. The electrolyte layer is essential for a fuel cell to work properly. The PEMFC electrolyte must provide high ionic conductivity, present an adequate barrier to the re actants, be chemically and mechanically stable, have low electronic conductivity, be easily manufactured and preferably low-cost. Ionic transport in polymer electrolytes follows the e xponential relationship [42]: kT Eae T/ 0 (179) where 0 represents the c onductivity at a reference state, and Ea is the activation energy (eV/mol). As seen in Equation 179, the conductivity increases exponentially with increasing temperature. The ch arged sites in the polymer ha ve the opposite charge of the moving ions, and provide a temporary resting place for the ion. Ions are transported through the polymer membrane by hitching ont o water molecules that move through the membrane. As mentioned previously, Nafion is a persulfonated polytetrafluoroethylene (PTFE)-based polymer which has high conduc tivity, and is current ly the most popular membrane used for PEM fuel cells. Nafion has a similar structure to Teflon, but includes

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119 sulfonic acid groups (SO3 –H+) that provide sites for proton tr ansport. Figure 5.1 shows an illustration of the chemical structure of Nafion. Figure 5.1. Illustration of the chemi cal structure of Nafion [4] Nafion has to be fully hydrated with wa ter in order to have high conductivity. Hydration can be achieved by humidifying the ga ses, or through fuel cell design to allow product water to hydrate the membrane. In the presence of water, the protons form hydronium complexes (H3O+), which transport the protons in the aqueous phase. When the Nafion is fully hydrated, the conductivity is similar to liquid electrolytes. The polymer electrolyte membrane c ontains water and hydrogen protons, therefore, the transfer of the water and protons transfer are important phenomena to

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120 investigate [6,7,8,72 75]. In addition to species transfer, the primary phenomena investigated inside the membrane are energy transfer and poten tial conservation [20]. For water transport, the principle driving forces modeled are a convective force, an osmotic force (i.e. diffusion), and an electric force [6 ,7,8,19,20,22]. The first of these results is from a pressure gradient, the second from a c oncentration gradient, a nd the third from the migration of protons from anode to cathode and their effect (dra g) on the dipole water molecules. Proton transport is described as a protonic current and consists of this proton driven flux and a convective flux due to the pressure driven fl ow of water in the membrane [6,7,8,19,20,22]. Figure 5.2 illustrates the tran sport phenomena for the protons taking place within the membrane. Electrolyte Layer Catalyst Layer (Carbon supported catslyst) Gas Diffusion Layer (Electrically conductive fibers)Hydrogen electrons eeH+ Water AnodeCathode H+ Potential driven H+ (drag force on H2O) H2O and H+ convection O2N2H2O diffusion Figure 5.2. Membrane transport phenomena [4]

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121 The dry membrane absorbs water in orde r to solvate the aci d groups. The initial water content is associated strongly with th e sites, and the addition of water causes the water to become less bound to the polymer and in turn, the water droplets to aggregate. The water clusters eventually grow and fo rm interconnections with each other. These connections create “water channels,” are transitory, and have hydrophobicities comparable to that of the matrix. A trans port pathway forms when water clusters are close together and become linked. Th is percolation phenomenon occurs around = 2. The next stage occurs when a complete clus ter-channel network has formed. In the last stage, the channels are now filled with liquid, and the uptake of the membrane has increased without a change in the chemical potential of water. This phenomenon is known as Schroeder’s paradox. An illustra tion of the water uptake of the Nafion membrane is shown in Figure 5.3.

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122 Figure 5.3. A pictorial illustration of th e water uptake of Nafion [4] 5.1 Model Development Proton exchange membrane fuel cell (PEM FC) models are nece ssary to predict fuel cell performance in order to optimize pe rformance to help reduce development costs and time. Water management is critical for e fficient fuel cells due to its large effect on ohmic and mass-transport overpotentials, operating conditions and membrane electrode assembly design. Since the membrane is the key element in a fuel cell, a lot of attention has been focused on it in terms of modeling. In the literature, there are both macroscopic and microscopic models. The microscopic models fo cus on single ions and pore-level effects,

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123 and the macroscopic models are typically more empirical and focus on the transport phenomena. Although the microscopic models re veal valuable information about what occurs in the membrane, they are generally too complex to use in an overall fuel cell model. The membrane system is assumed to consist of three main components: the membrane, protons and water. The membrane model presented in this dissertation is a compact model that was integrated into the overall fuel cell stack model, and can simultaneously calculate the temperature, pressure, water concentration a nd potential at a user-s pecified number of positions through the membrane. 5.1.1 Background and Modeling Approaches Most membrane models in the literature have been isothermal, and therefore, unsuitable for water and heat management st udies. A relatively small number of models include noniosthermal effects [17, 18, 19, 77, 78], and typically, the ones that do focus on modeling multiple fuel cell layers, with simplifying assumptions for the membrane layer. Transient models examine changes in potential and transport phenomena (flow rates, water production and current density). These models are aimed at examining different load requirements. Most models do not examine transients due to the computational cost and complexity. Some codes in the literature can take on the order of tens of minutes in certain circumstances [78]. One of the first models to examine transients in PEM fuel cells is a stack –lev el model by Amphlett et al. [79]. This is an empirical model that examines temperature a nd gas flow rates. There have been some more complex transient models that have exam ined the behavior of water content in the

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124 membrane that have demonstrated the effect s of the membrane drying out [22, 80]. Other transient models have either not included liq uid water, do not report transient results or focuses mainly on water transport in the gas diffusion layers [15, 17, 27, 36, 77, 79, 81]. There are no results reported in the literature that simulta neously show the temperature, potential, water concentration and pressure profile in the membrane based upon varying current densities, temperatures and pressure gradients. Verbrugge and Hill [72] and Bernardi and Verbrugge [6] developed a steadystate, isothermal, one-dimensional model fo r the electrochemical performance in a PEMFC. They claim that the li quid and gas pressure evolve se parately in the GDL layer, which implies that they are not at equilibrium with each other. This model only applies to fully hydrated membranes, and the drag flux due on the water molecules is not taken into account. Springer, Zawodzinski and Gottesfeld [7] pr esented a 1-D, steady-state isothermal model of a PEMFC with emphasis on wa ter transport phenomena through a Nafion membrane. An improved model with a detaile d treatment of ion transport and ionic conductivity in the catalyst and backing layer was developed in [8]. This model predicted the mass transport limitations at high current densities. In [73], Springer, Zawodzinski, Wilson and Gottesfeld provide experimental and theoretical results for unsteady-state effects in a 1-D isothermal PEMFC stack. Th ey use a frequency diagram to quantify the specific influences of severa l sources of losses such as activity in the cathode and conductivity of the catalyst layer and the membrane. Weisbrod, Grot, and Vanderborgh [74] de veloped a through the electrode model to predict fuel cell performance as a function of water balance in the channels, and across

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125 the membrane. The model predicts the influe nce of both the catalyst layer thicknesses, and its Platinum catalyst loading. Nyguyen and White [19] developed a 1-D, steady-state water and heat management model for PEMFCs. This model does not study the details of the membrane and the catalyst layers separately since it models that entire electrode as one unit. It does steady the effect of humidificat ion levels and their effect on fuel cell performance. This model was enhanced in [75], with the addition of a linear model for the membrane, and then a 2-D, steady-state model for multispecies transport in the electrodes. This model studies the effect of an inte rdigitated gas distri butor on PEMFC performance. However, it was unable to predict the effect of liquid wa ter within the system. Thirumalai and White [20] used the model developed in [75] to predict the operating parameters, flow field design and gas manifold geometry on the performance of the fuel cell stack. Van Bussel, Koene and Mallant [22] cr eate a 2-D dynamic model, with a 1-D model through the membrane. Th e model is based upon the work of Springer et al. [7], but uses experimental data from Hinatsu, Mizhuta and Takenaka [82]. The model showed that current density can vary strongly along the gas channels especially when operating with dry gases. Gurau, Kakac, and Lui [76] develope d a 2-D non-isothermal model. They considered the gas channel, and the diffuse r-catalyst layer a single entity. The model shows a non-uniform, reactant distribution has an important impact on the current density distribution. This model is based upon an infini tely thin catalyst layer, which is unable to predict the voltage due to transport limitations in the catalyst layer.

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126 Fuller and Newmann [18] and Weber and Newmann [83] developed a steadystate, 2-D model for the membrane el ectrode assembly. Unlike other models, concentration solution theory was used. They argued that water was produced in the gaseous phase at the catalyst surfaces. Their model is va lid as long as there is no condensation within th e catalyst layer. However, expe rimental evidence implies that liquid water forms as a result of the electrochemical reac tion at the anode and cathode catalyst layers. 5.1.2 Methodology In establishing the methodology for the membrane model, there are several important factors should be considered: 1. Mass and species conservation 2. Conservation of energy 3. Momentum and pressure across the membrane. Proton and water transport is simultaneous ly coupled in the polymer membrane layer, and conjugate effects must be a ddressed. Appendix J pr ovides the detailed procedure employed for the calculations discusse d in this section. The five main steps in the proton exchange membrane analysis are: 1. Definition of the layers and nodes 2. Definition of the boundary conditions 3. A mass and energy balance computation for each node. 4. Calculation of additional para meters such as conductivity.

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127 The following subsections describe each of the above steps in the nodal membrane computation. 5.2 Definitions of Segments and Nodes Figure 5.4 shows a schematic of the PEMFC stack, and the grid structure used in the fuel cell membrane model. In the actual calculations conducted with the mathematical model, the number of segments is specified by the user, and was varied from 1 to 60 segments for the membrane layer for the outputs of this study. Figure 5.4. Slices created for 1-D membrane model

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128 For the uniform distribution of nodes that is shown in Figure 5.4, the location of each node (xi) is: (1) for1.. 1ii x LiN N (180) where N is the number of nodes used for the simulation. The distance between adjacent nodes (x) is: 1L x N (181) Energy balances have been defined around each node (control volume). The control volume for the first, la st and an arbitrary, internal node is shown in Figure 5.4. 5.3 Boundary Conditions The model solves for the concentration of water, potential, temperature and pressure simultaneously. In order to solve for these transient variables, initial and boundary conditions are required. At x = 0, four boundary conditions are necessary to fully specify the problem. These are: For the left boundary: ) ( ) (1 1 2 2 i m i O H i m i O Hx c x c (182) ) ( ) (1 1 i i i ix T x T (183) ) ( ) (1 1 , i i m i i mx x (184)

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129 ) ( ) (1 1 , i i tot i i totx P x P (185) For the right boundary: ) ( ) (1 1 2 2 i m i O H i m i O Hx c x c (186) ) ( ) (1 1 i i i ix T x T (187) ) ( ) (1 1 , i i m i i mx x (188) ) ( ) (1 1 , i i tot i i totx P x P (189) 5.3.1 Model Assumptions The following assumptions were made for the membrane model: 1. Water diffusion perpendicular to the membrane surface (membrane thickness is much smaller than the channel length). 2. All material thermal properties are constant over the temperature range considered (20 to 80 C). 3. For the MEA layers, only the active area was included in the model. The materials surrounding the MEA were not included in the model. 4. The gases/fluid in the membra ne have ideal gas behavior. 5.4 Mass and Species Conservation In polymer electrolyte membrane fuel cel ls, the two important fluxes or material balances are the proton flux a nd the water flux. The membrane needs to stay hydrated in

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130 order to ionically conduct hydrogen; therefore, the water profile must be calculated in the electrolyte. One of the main reasons water co ntent varies in Nafion is because the protons usually have one or more water molecules associated with them This phenomenon is called the electroosmotic drag (ndrag), which is the number of water molecules accompanying the movement of each proton [7, 8, 73]: 22 5.232/ SOOH dragn (190) where dragn is the electroosmotic drag (usu ally between 2.5 +/ 0.2), and is the water content (which ranges from 0 to 22 water molecules per sulfonate group, and when = 22, Nafion is fully hydrated). The relationshi p between water activity on the faces of the membrane and water content can be described by [7, 8, 73, 81, 84]: 84] [81,K 353 =at 1.14 168.103.0 8] [7,K 303 =at 36 85.398.17043.0 ),(3 2 3 2 w w w w w w ifwa aa aa a Ta(191) Since in the study, the con cept of nonisothermal conditions are of interest, the expression for membrane water content needs to be modified to take into account the temperature variation on the polymer memb rane as proposed by Yi et al. [84]: 50 303 ))303,()353,(()303,(),(, if w w w ifwT a a aTa (192) This concepts of water uptake ( ) and water activity (wa), and the influence on cell potential and current density is de monstrated in Figures 5.5 and 5.6.

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131 Lambda vs. a (activity) 0 100 200 300 400 500 600 700 00.511.522.533.5 a (activity of water)Lambda Figure 5.5. Lambda ( ) versus activity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Cell Current (A/cm2)Voltage (Volts) 0.5 1.0 1.5 2.0 2.5 3.0 Water Activity in the Electrolyte Figure 5.6. Cell voltage and current dens ity based upon electrolyte RH

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132 The relation for the water activity within the membrane is given by the reciprocal of the sorption curve. As with the water vapo r activity at the interfaces, the results from Springer et al. [7] for water va por activity in Nafion 117 at 30 C is given by [7, 8, 73]. 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2/ / / / 3 / 1 2 3 / 2 5 / 4 3 / 2 1 3 / 1 2 / 5 / 4 3 / 2 18 16 3 8 16 14 0021 9 7143 0 14 2160 797 216 2160 134183 216 2160SO O H SO O H SO O H SO O H SO O H SO O H SO O H SO O H SO O H SO O Hc c c c c c c c c c a (193) where 1c is 410 41956 2cis 310 39968 1 3c is 610 82482 3 4c is 310 51739 2 and 5c is 610 19904 4 The water drag flux from the anode to the cathode with a net current i is [7, 8, 73]: F i n Jdrag drag O H2 2,2 (194) where drag O HJ,2 is the molar flux of water due to the electroosmotic drag (mol/scm2), and j is the current density of the fuel cell (A/cm2). The electroosmotic drag moves water fr om the anode to the cathode, and when the water builds up at the cat hode, some water travels back through the membrane. This is known as back diffusion, and it usually happens because the amount of water at thecathode is many times greater than at the anode. The water back-diffusion flux can be determined by [42]:

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133 dz d D M Jm dry ion backdiffus O H ,2 (195) where dry is the dry density (kg/m3) of Nafion, Mn is the Nafion e quivalent weight (kg/mol), Dis the water diffusivity and z is the direction thr ough the membrane thickness. The total amount of water in the membrane is a combination of the electroosmotic drag and back diffusion, and can account for by the following equation [42]: F i n x c D Jx drag m O H T O cH M O H 2 2 2, (196) where dragn is the measured drag coefficient,xiis the protonic current in the x direction, F is Faradays constant, 3 2/SO O H is the water content (molH2O/molSO3-), m dry is the dry membrane density (kg/m3), T O cHD,2 is the diffusion coefficient and m M is the membrane molecular mass (kg/mol). Many different values for the diffusion co efficients have been reported in the literature.T O cHD,2 is the diffusion coefficient wh ich includes a correction for the temperature and for the water content it is expr essed in a fixed coordinate system with the dry membrane by [7, 8]: 2 / 1 303 1 2416 ,108 9 78 81 173 2 2a a a e D DSO O H T I O cH (197) where a is the activity of water, and 'D(m2/s) is the diffusion coefficient measured at constant temperature and in coordinates moving with the swelling of the membrane. 'D

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134 has been added to the above equation to en sure that water contents below 1.23 do not result in negative diffusion coefficients. 'Dat 30 C is written as [7, 8]: 14 6 6 23 1 23 1 10 1625 2 10 5625 2 10 5 9 10 75 7 10 642276 23 2 3 2 3 2 3 2 3 2 3 2/ / / 10 / 11 11 / 11 / 13 SO O H SO O H SO O H SO O H SO O H SO O HD (198) When modeling the polymer electrolyte me mbrane, a typical assumption is that the concentration of pos itive ions is fixed by electroneut rality, which means that a proton occupies every fixed SO3charge site. The charge sites are assumed to be distributed homogeneously throughout the membrane, which results in a constant proton concentration in the membrane. A flux of prot ons, thus, results from a potential gradient and not a concentration gradie nt. In addition, the number of protons that can be transported is only one, which helps to simplify the governing transport equations. Now, due to the assumption of electroneutrality and the homogeneous distribution of charge sites, the mass conservation of protons simplifies to: 0 x cH (199) 0 t cH (200) Thus, as soon as a current exists, the membrane is charged; and the concentration of protons remains constant. The charge of th e protons equals that of the fixed charges. The diffusive molar flux for the protons,HJ, can, therefore, be written as [7,8]: x c D RT F Jm H H H (201)

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135 Combining this diffusive flux with the c onvective flux results in the total molar flux for the hydrogen protons, i.e.: m H H Hu c J N (202) 5.5 Charge Transport The equation for the proton potential is derived from Ohm’s law. The electroneutrality assumption allows the total molar proton flux to be related directly to current density and results in the fi rst term. The second term containing mu represents the convective flux of protons. Combined they result in the following equation [7,8]: m H m m mu c F i x (203) where is the conductivity of the membrane. The conductivity of a membrane is highly dependent upon the structure and water cont ent of the membrane. The amount of water uptake in the membrane also depends upon the membrane pre-treatment. For example, at high temperatures, the water uptake by the Nafion membrane is much less, due to changes in the polymer at high temperatures. Springer et al. [7,8,73] correlated the ionic conductivity ( )(in S/cm) to water content and temperature with the following relation [7, 8]: T m me1 303 1 1268 303 (204) with m303, the conductivity of the membrane at 303 K given by [7, 8]: 1 326 0 5139 03 2 3 2/ / 303 SO O H SO O H mfor (205)

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136 Since the conductivity of Nafion can cha nge depending upon water content, the resistance of the membrane changes with water saturation. The total resistance of a membrane (Rm) is found by integrating the local resistance over the membrane thickness [7, 8]: mt mz dz R0) ( (206) where tm is the membrane thickness, is the water content of membrane, is the conductivity (S/cm) of the membrane. Figures 5.7 and 5.8 show the correlation between membrane thickness and water content, and membrane thickness and local conductivity. 0 0.002 0.004 0.006 0.008 0.01 0.012 7 8 9 10 11 12 13 14 15 1 6 Membrane Thickness (cm)Water Content(H2O/SO3) Figure 5.7. Membrane thickness and water content

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137 0 0.002 0.004 0.006 0.008 0.01 0.012 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Membrane Thickness (cm)Local Conductivity(S/cm) Figure 5.8. Membrane thickness an d local conductivity 5.6 Pressure in the Polymer Membrane Most models in the literature assume only concentration gradients, and not pressure gradients [7]. A pressure drop can occur if the anode and cathode pressure are different. The pressure in the membrane layer was calculated based upon the pressures and concentrations on the feed and permeate side as shown in Figure 5.9. The average membrane pressure was obtained by subtracting the pressure on the anode minus the cathode side.

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138 Figure 5.9. Pressure profile for transpor t through polymer membrane The mixture pressure gradient is assume d to behave linearly between the anode and cathode interfaces so that the total pressure at node i i totP, (Pa) is [6]: i i tot i tot i totx P P P 1 1 , (207) where x is the thickness of node i (m), 1 ,i totPand 1 ,i totPare the pressures at the anode/membrane and cathode/membrane interf ace. At the interface with the anode catalyst layer, the mixture pre ssure is assumed equal to that of the gas pressure under the assumption that no liquid is present. At the cathode catalyst interface, it is assumed that the mixture pressure can be approximated by a linear relation and the liquid pressure, weighted by the saturation ratios (the volume ratio of liquid wa ter to gaseous water in the pores of the catalyst layer). For the results ge nerated, the saturation ratio was set to zero; therefore there was no effect of liquid pressure on the pressure gradient.

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139 5.7 Momentum Equation For the water, protons and gases mixture, the momentum equation takes the form of the generalized Darcy relation [66, 67]: i tot i i i mP x k u, (208) where k is the permeability (m2), is the viscosity (Pa-s), x is the thickness of node i (m) and i totP, is the change in total pressure (Pa) with respect to x. 5.8 Gas Permeation The membrane should theoretically be imperm eable to reactant sp ecies in order to prevent mixing. However, due to the membrane ’s porous structure, its water content and solubility of hydrogen and oxygen in wate r, some gas does permeate through the membrane. Permeability is a product of diffusivity and solubility [46, 85]: 2 2 2H H HS D P (209) 2 2 2O O OS D P (210) The solubility of hydrogen in Nafion was shown to be SH2 = 2.2 x 10-10 mol-cm3 Pa-1, and is independent of temperature and diffusivity. The hydrogen diffusivity can be calculated as follows [46, 85]: i f HT D, 22602 exp 0041 0 (211) where i fT, is the temperature of gas/fluid mixture in the membrane. The oxygen solubility is a function of te mperature, and is given by the following expression [46, 85]:

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140 i f OT S, 12 2666 exp 10 43 7 (212) The oxygen diffusivity (cm2 s-1) can be calculated from [46, 85]: i f OT D, 22768 exp 0031 0 (213) Hydrogen has an order of magnitude higher permeability than oxygen in Nafion. The oxygen and hydrogen permeability can then be used to calculate the hydrogen and oxygen permeation rates [46, 85]: i i tot i H i Hx P A P n 2 2 (214) i i tot i O i Ox P A P n 2 2 (215)

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141 6 BOLT TORQUE MODEL There are many steps involved in the manu facturing of a fuel cell stack. One of these steps is the hot pressing of the polym er electrolyte membrane to the two gas diffusion layers (GDLs). This creates a three layer laminate membrane electrode assembly (MEA). Other steps involve the mach ining or etching of the end plates, bipolar plates and cooling plates, and the sizing of the gaskets, contacts and MEA surrounds. After all of the fuel cell components have b een manufactured, they are stacked together and clamped using a clamping mechanism such as bolts. The contact resistance, mass and charge transfer between the electrolyte membrane and the GDL is very good due to the fusion of the three layers [86] In contrast, the remaining layers are separated until they are clamped together using bolts or some other type of cl amping device. Therefore, the interfacial resistances between the remaining layers are significant. The contact, cooling and bipolar plates are clam ped together, and since the Poisson’s ratio and Young's Modulus of the bipolar, cooling and contact layers are similar, and the surface roughness can be considered negligible, the contact re sistance between these layers is small when the stack is clamped together. The interf ace that is most affected by the clamping pressure is the GDL and bipolar plate interface. The material properties of these adjoining layers are extremely different, and since the GDL layer is porous, it is highly sensitive to the clamping pressure. Not only does the GDL thickness change with clamping pressure,

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142 but the change in thickness also affects the porosity and permeab ility of the GDL. The bolt torque, clamping force, contact resistance and permeability of the GDL all affect the electrochemical performance of a PEM fu el cell by influencing the ohmic and masstransport polarizations inside of the fuel cell [86]. In order to predict the optimal clamping pressure, and ultimately, the ideal bolt torque, a MATLAB program was created to calcu late the force required for optimal GDL compression and assembly force of the stack. The program is illustrated in Figure 6.1. Parameters These parameters are specified for each layer: (1) Thickness (mm) (2) Modulus of elasticity in Tension (MPa) (3) Youngs Modulus (N/mm2) (4) Poissons ratio Constants (5) No. of Bolts (6) Bolt diameter (mm) (7) Bolt hread root diameter (mm) (8) Thread Pitch (9) Pitch diameter (mm) (10) Bolt head diameter (mm) (11) Thickness of Bolt Head (mm) (12) Bolt length (between bolt head & nut) (mm) (13) Outer diameter of annulus seating face (mm) (14) Inner diameter of annulus seating face (mm) (15) Nut thickness (mm) (16) Bolt clearance hole (mm) (17) Modulus of elasticity in tension of bolt material (MPa) (18) Force for optimal GDL compression (N) (19) Diameter of active area of material (mm) (20) Total elastic compression (microns) (21) friction coefficient in seating face of head (nut) of the bolt (22) Coefficient of tightness (23) friction coefficient in he thread (24) Interface area (mm) Calculate the force required for optimal GDL compression Inputs: 6, 7, 9, 10, 11, 12, 15, 17 Inputs: 3, 4, 18, 19, 20 Calculate the bolt, head, shaft & nut stiffness Calculate he stiffness of each fuel cell layer Inputs: 1, 2, 16 Calculate total stiffness of clamped fuel cell layers Inputs: 5 Calculate stiffness of the group of surcharged parts of the stack Calculate the stiffness of the relieved parts Part of force relieving he clamped parts Bolt seating coefficent Assembly force of the stack Tightening torque Average interface contact pressure Inputs: 13, 14, 16, 21 Inputs: 22 Inputs: 8, 9, 23 Inputs: 5, 24 Figure 6.1. Flow chart of bolt torque model

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143 6.1 The Mechanics of Bolted Joints Materials bolted together withstand mo ment loads by clamping the surfaces together, where the edge of the part acts as a fulcrum, and the bolt acts as a force to resist the moment created by an external force or moment. Figure 6.2 shows forces exerted by the clamped materials (fuel cell layers) on a clamping bolt and nut. The forces exerted by the tightening bolts are due to the bolt material properties, the properties of the materials being clamped together and the torque applied to the bolts. Figure 6.2. The forces exerted by the clamped mate rials (fuel cell laye rs) on the bolt and nut

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144 Materials that are held together by a bolt are subjected to a force a distance away from the center of stiffness of a bolt pattern. In order to determine the optimal torque for a bolt, a maximum bolt force is typically calculated based upon the maximum amount of stress and force that can be applied to the bol t, and the “joint” which is the fuel cell stack in this case. The optimal torque is found by calculating the force that can be applied to joint until the force on the joint is lost. When the joint starts to leak, at which the bolts break, the total stress in a bolt when the joint begins to leak, and the percent of maximum stress that can be used by the bolt head. A ssuming each of N bolts is a distance from the bolt pattern’s center of stiffne ss, each bolt has the same force and there is a coefficient of friction between the bolted members [87, 88]. Tightening the bolts stretches the bolts a nd compresses the stack materials. If an external force is applied to the stack, the optim al torque usually means that the stack stays compressed. This ensures proper stiffness and fatigue life of the stack. Figure 6.3 shows how the region under a bolt head acts like a spring.

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145 Fuel cell stack materials F FF F F Force from materials Force from bolts Force from bolts Force from materials Figure 6.3. The forces exerted by the clamped materials and bolt Of the energy created by the bolt force, about 50% of the energy goes to friction under the bolt head, 40% goes to friction in the threads and about 10% goes to create tension in the threads [88, 89] The rotation of the bolt head relative to the parts being bolted together is a good measure of tension in the bolt. As shown in Figure 6.4, there is a strain or stress cone under the bolt head that project from 30 to 45 degrees from the vertical, and 45 is most commonly used for bolt torque calculations [88, 89].

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146 Figure 6.4. Compressive stiffness zones underneath a bolt head in a fuel cell stack When determining a fuel cell stack desi gn and the optimal clamping pressure, two questions need to be answered: 1. How much tension does the clamping de vice (a bolt in this case) actually create? 2. What is the optimal tightness for ideal permeability through the MEA layers? 3. What is the ideal tightness to minimize contact resistance? 4. What effect do all the fuel cell layers have on ideal tightness? All of these propertie s need to be considered when trying to determine the optimal clamping pressure for a stack. The traditiona l method of determining the ideal clamping pressure is to just take the fuel cell prototype into the lab, and obtain I-V curves for the fuel cell stack to determine the optimal clamping pressure. However, this can be very time-consuming and unrealistic for real wo rld applications since the stacks can be

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147 extremely large with greater than 200 cells, and often multiple stacks with varying number of cells are rapidly be ing prototyped. In addition, new stack configurations from very large to very small scal e, and clamping methods are be ing created where it may be more convient to calculate the ideal clamping pressure and bolt torque. In bolt science, the optimal torque can be found by calculating the forces that the bolt can withstand, along with the stiffne ss of the materials being clamped, and the desired tightness that the clamped materials require. The numerical model of the ideal tightening torque originally proceeded in this direction, but these forces overestimated the required torque for a fuel cell stack because they were based upon the amount of stress that the bolt material could handle. For a fuel cell stack, the bolt ma terial can handle more force than the fuel cell stack needs for optim al performance. Therefore, in order to calculate the ideal torque for a fuel cell stack, the effects of compression of the GDL and the channel land area had to be added to the existing model. 6.2 Calculating the Force Required on the Stack for Optimal Compression of the GDL The contact resistance and GDL perm eability is governed by the material properties of the contacting GDL and bipolar pl ate layers. The contact resistance is most reliant on the contact between these layers. The contact resistance between the catalyst, gas diffusion and membrane laye rs is low because they are fused together. The contact resistance between the bipolar plates a nd the gas diffusion media can vary depending upon the land to channel area, the GDL por osity after compression. The important aspects for calculating the optimal bolt tor que and clamping pressure are as follows:

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148 1. The Poissons’ ratios and Young’s Moduli have large differences (a hard material with a soft material) 2. The GDL layer is porous, and the permeability has been reduced due to the reduction in pore vo lume or porosity, and 3. Part of the GDL layer blocks the flow channels that are in the bipolar plate creating less permeability through the GDL as the compression increases. The original clamping pressure model di d not take this into effect, and only calculated the optimal torque on the bolts based upon the forces that the bolt could withstand, and the stiffness of the materials. The tightening torque calculations predicted the optimal torque for the tightness of the bolts due to the stiffness of the bolt and materials being clamped together. However, it did not take into account the optimal tightness of the bolt (pressure on the stack) for optimal co mpression of the GDL. If the GDL is not adequately compressed, the fuel cell gases may leak, and therefore, will not be able to react inside the fuel cell. In a ddition, the contact resistance will be high due to inadequate contact between the GDL and the ot her fuel cell layers. Therefore, a relation had to be included for the id eal GDL compression thickness. Herzian compression effects are used to determine the compression of the GDL and bipolar plate materials. Th e calculations assume that th e surfaces in contact are not perfectly smooth (which is not th e case as presented in [90, 91]), that the elastic limits of the materials are not exceeded, that the ma terials are homogenous and that there are no frictional forces within the contact area. The actual variation due to the frictional effects from non-smooth surfaces lead to compression effects differing from these calculations by 5%. These formulas are sufficiently precise for use with this tightening torque model.

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149 The compression formula for two spheres in contact is [92]: 3 / 1 2 1 3 / 2 2 1 3 / 2 3 / 21 1 ) ( 2 3 ) ( D D V V F x h (216) where ) ( x h is the total elastic compression at the point of contact of two bodies ( m), measured along the line of applied force, F is the total applied force, D is the diameter of the active area of the material (width of MEA), and [92] E V ) 1 (2 (217) where is Poisson’s ratio, a nd E is the Young’s modulus. As noted by Nitta et al. [93], the change in thickness of the GDL caused by compression is mainly attributed to the lo ss of pore volume, which affects the mass and charge transport through the GDL. The gas permeability decr eased non-linearly when the thickness of the GDL was decreased by compression. The permeability was reduced by one order of magnitude when th e GDL was compressed to 250 m from the initial thickness of 380 m. These results agree with Mathias et al [94], who determined the inplane permeability to be in the range from 5 x 10-12 when Toray paper was compressed to 75% of the initial thickness. The compression of the GDL leads to loss of pore volume; therefore, porosity can be correlated di rectly with compressed GDL thickness. As shown in Figure 6.5 which was adap ted from [93], both the in-plane and through-plane conductivities increase as the compressed thickne ss of the GDL was decreased. The conductivities have a linear dependence on the GDL compressed thickness. This may be due to the reduced porosity of the GDL, which leads to shorter distances between conductive carbon fibers and better c ontacts between the fibers.

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150 Conductivity and Permeability as a Function of GDL Compressed Thickness 0 1000 2000 3000 4000 5000 6000 400350300250200150 GDL thickness (um)Conductivity (S/m0 5E-12 1E-11 1.5E-11 2E-11 2.5E-11 3E-11Permeability (m^2 ) In-plane conductivity (S/m) Through-plane conductivity (S/m) Permeability (m^2) Figure 6.5. Conductivity and permeability as a function of GDL compressed thickness [93] Using the intersection of the in-plane, through-plane and gas permeability from Figure 6.5, a compressive GDL thickness of 325 m was assumed to be an ideal compression for optimal GDL conductivity and permeability. The force in equation 216 was calculated based upon a compression of 75 m (assuming that the GDL has a 400 m thickness), the diameter of the MEA and th e part of the bipolar plate contacting the GDL (the channel area), and the properties of the bipolar plate and GDL materials. This force was used as part of the ideal comp ression force for the bolt-torque model. From the data from Figure 6.5, a third degree polynomial fit was made with the least square sum method to th e permeability data, and the fo llowing function results [93]:

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151 3 2 3 7 11) ( 754 2 ) ( 10 484 1 ) ( 10 760 2 10 700 1 ) ( x h x h x h x k (218) The GDL in and through plan e conductivities were modeled as a linear fits from the experimental data, and can be written as [93]: ) ( 10 159 1 6896 ) (7 ,x h xx GDL (219) ) ( 10 385 8 3285 ) (6 ,x h xy GDL (220) 6.3 The Stiffness of Bolted Joints In order to accurately determine the id eal clamping pressure (tightening torque) for a fuel cell stack, the stiffness of the mate rials between the bolts has to be estimated. The stiffness of the materials includes the co mpressive stiffness of the materials under the bolt head in series with the stiffness of the physical interface, which increases with pressure, and the stiffness of the threaded ma terial. Some of the dimensions used in the bolt and layer stiffness calculatio ns are shown in Figure 6.6.

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152 Figure 6.6. Dimensions used in the bolt and layer stiffness calculations In order to determine the stiffness of th e cone-like section under the bolt head, the first step is to calculate the stiffness of each layer of the fuel cell stack [94]: 2cos 2 * 4bore layer bolthead bolthead layer layer layerd h d d E h k (221) where layerk is the stiffness of the fuel cell layer (such as the end plat e or bipolar plate), layerh is the thickness of that particular layer, layerE is the modulus of elasticity in tension (MPa) of the material, boltheadd is the diameter of the bolt head, is the effective cone angle and bored is the clearance hole diameter. The stiffness of the bolt, head, shaft and nut are all calculated in a similar fashion. The tensile stiffness of the bolt shaft is [94]:

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153 bolt bolt dia bolt boltshaftL E d k2 _2 (222) where dia boltd_ is the bolt diameter, boltE is the Young’s Modulus of the bolt, and boltL is the bolt length. The shear stiffness of the bolt head is [94]: ) 2 ln( ) 1 (bolt bolt bolthead boltheadv E h k (223) where boltheadh is the thickness of the bolt head, boltE is the Young’s Modulus of the bolt, and boltv is the Poisson’s ratio of the bolt. The shear stiffness in the nut is [94]: ) 2 ln( ) 1 (bolt bolt nut nutv E h k (224) The total stiffness of the stack is [94, 95]: i i layer stackk N k1 ,1 (225) where N is the number of bolts in the stack. The stiffness of the bolt shaft in tension, and the head and nut (if a nut is used) in shear, a ll act in series, so thei r stiffness combine to give the total stiffness of the bolt [94, 95]: nut bolthead boltshaft boltk k k N k 1 1 1 (226) As the stack thickness incr eases, the length of the bo lt to pass through the stack thickness also increases, so the bolt stiffness decreases in a linear fashion. On the other hand, the diameter of the strain cone increase s, which offsets much of the height increase, and the stack stiffness decreases far mo re slowly than that of the bolt.

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154 The ratio of flange to bolt stiffness is [94, 95]: bolt stack b sk k k (227) The total stiffness can be expressed by [94, 95]: stack bolt totk k k (228) 6.4 Calculating the Tightening Torque The stiffness of the group of surcha rged parts of the stack is [96]: stack boltk n k c ) 1 ( 11 (229) where n is the coefficient of implementation of the operational force (0.5). The resulting stiffness of the group of relieved parts of the stack is [96]: n k cstack2 (230) The part of the operational force relieving the clamped parts is [96]: ) ( *2 1 2 2c c c F F (231) where F is the force required for the ideal compression of the GDL by 75 microns. The bolt seating coefficient is calculated by [96]: 2 *1seat seat c seatDi De m m (232) where seatDe is the outer diameter of the seating face, and seatDi is the inner diameter of the seating face and mc is the friction coefficient in seat ing face of head (nut) of the bolt. The assembly force of the st ack can be calculated by [96]:

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155 05 0 *0 2 0 T aF F F q F (233) where qa is the desired coefficient of tightness, and TF0 is the change of force required due to the heating of the connection. TF0 was assumed to be zero for all of the calculations since the stacks used for valida ting the model were all air-breathing fuel cell stacks tested at room temperature. Th e bolt seating is ca lculated by [96]: 0 1* F m Mseat seat (234) The tightening torque is then [96]: i pitch pitch i pitch pitch seat pitchm thr d m d thr M d F M * * * *0 (235) where 0F is the assembly force of the stack, pitchd is the pitch diameter, pitchthr is the thread pitch, im is the friction coefficient in thread (0.15). 6.5 Relating Torque to the Total Clamping Pressure Applied to the Stack The average interface contact pressure,avgP ,can be calculated by dividing the total clamp force (product of the number of bolts, a nd the individual bolt clamp force) with the interface contact area, intA [97]: int 0A F N Pavg (236) where intA is the land area of the flow field plate. The average contact pressure is a linear function of bolt torque.

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156 6.6 Torque Tightening Parameters Many simulations were performed in orde r to estimate the tightening torque for several fuel cell stacks. In or der to calculate the stiffness of each fuel cell layer and the total stack stiffness, the materials and their applicable properties ar e listed in Tables 6.1 to 6.3 for each fuel cell stack. Fuel cell stack #1 has an active area of 16 cm2, had stainless steel bipolar plates and had end plates of 10 mm in thickness. Fuel cell stack #2 is similar in construction, with an active area of 4 cm2. The end plates were 8 mm in thickness, and the flow fields were made of 2 separate layers: one Nylon mesh and one stainless steel mesh. Stack #3 had a slightly different constructi on than the other two stacks with aluminum end plates and Delrin bipolar plates.

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157 Table 6.1 Material properties used for material stiffn ess and compression calculations for stack #1 Fuel Cell Layer/ Material Thickness (mm) Modulus of elasticity in Tension (MPa) Young’s Modulus (N/mm2) Poisson’s ratio Polycarbonate end plate 10 2,896 2,200 0.37 Gasket: Black Conductive Rubber 1 2 100 0.48 SS Flow field plate 0.5 206,000 200,000 0.31 Carbon Cloth 0.4 2 300 0.4 Nafion 0.05 2 236 0.487 Carbon Cloth 0.4 2 300 0.4 SS Flow field plate 0.5 206,000 200,000 0.31 Gasket: Black Conductive Rubbe r 1 2 100 0.48 Polycarbonate end plate 10 2,896 2,200 0.37

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158 Table 6.2 Material properties used for material stiffn ess and compression calculations for stack #2 Fuel Cell Layer/Material Thickness (mm) Modulus of elasticity in Tension (MPa) Young’s Modulus (N/mm2) Poisson’s ratio Polycarbonate end plate 8 2,896 2,200 0.37 Gasket: Black Conductive Rubbe r 1 2 100 0.48 Nylon Flow field plate 0.2 4,067 7,000 0.41 SS Flow field plate 0.1 206,000 200,000 0.31 Carbon Cloth 0.4 2 300 0.4 Nafion 0.05 2 236 0.487 Carbon Cloth 0.4 2 300 0.4 SS Flow field plate 0.1 206,000 200,000 0.31 Nylon Flow field plate 0.2 4067 7,000 0.41 Gasket: Black ConductiveRubber 1 2 100 0.48 Polycarbonate end plate 8 2,896 2,200 0.37

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159 Table 6.3 Material properties used for material stiffn ess and compression calculations for stack #3 Fuel Cell Layer/Material Thickness (mm) Modulus of elasticity in Tension (MPa) Young’s Modulus (N/mm2) Poisson’s ratio Aluminum end plate 6 70,000 0.35 62,052.8 Gasket: Silicon Rubber 1 320,000 0.48 2 Delrin Flow field plate 1 3,100 0.35 3,300 Carbon Cloth 0.4 300 0.4 2 Nafion 0.05 236 0.487 2 Carbon Cloth 0.4 300 0.4 2 Delrin Flow field plate 1 3,100 0.35 3,300 Aluminum end plate 6 70,000 0.35 620,52.8 In order to calculate the bolt stiffness, th e bolt parameters for each stack are listed in Table 6.1. Each stack used different bolts Stack #1 and #2 used stainless steel bolts, and stack #3 used Nylon. The lengths, diamet ers and other characteristics of the bolts varied, as shown in Table 6.4.

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160 Table 6.4 Bolt properties used for bolt stif fness and torque calculations ________________________________________________________________________ Property Stack 1 Bolts Stack 2 Bolts Stack 3 Bolts ________________________________________________________________________ Number 4 4 4 Material SS316 SS316 Nylon Hex Key Size 5/32” 3/32” 3/32” Bolt Diameter mm 4.826 2.18 2.18 Bolt Thread Root 3.451 1.60 1.60 Thread Pitch 1.058 0.45 0.45 Pitch Diameter mm 4.139 1.89 1.89 Bolt Head Diameter 8 5 2.5 Thickness of Head 5 2.5 2 Bolt Length 25 23 25 Outer DiameterAnnulus Seating 7.925 5 3.175 Inner DiameterAnnulus Seating 5.232 3 2.3876 Nut Thickness 3 2 1.59 Bolt Clearance 5.232 4 2.38 ________________________________________________________________________ 6.7 Electrochemical Performance of PEM Fuel Cell Stacks Three single cell, air breathing fuel ce ll stacks were assembled for fuel cell I-V tests with different tightening torques. Five -layered MEAs are used, which are composed of Nafion 112, GDL of carbon cloth material and 1 mg/cm2 of Pt loading on both anode and cathode. The active fuel cell area for stack #1 is 16 cm2, 4 cm2 for stack #2 and 1 cm2 for stack #3. Each stack was constructed di fferently, with different fuel cell layers, thickness and used different types of clampi ng bolts. The torque was measured using a Precision Instruments Dial I ndicating ” torque driver with a range of 0 48 oz/in with

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161 hex head adapters to fit the fuel cell stack bolts. The single cell fuel cell stacks are shown in Figure 6.7. Figure 6.7. Fuel cell stack sizes that were tested (a) 16 cm2, (b) 4 cm2, and (c) 1 cm2 active areas Cell performance tests are c onducted with 0.5 to 1.0 sta ndard cubic centimeter per minute (SCCM) of hydrogen from an electrolyz er, with no additional humidification. All tests are taken at 25 C and ambient pressure I–V curves of these cell performance tests with various tightening torques are plotted in Figures 6.8 through 6.10. Figure 6.8 shows the polarization curves of the current of th e PEM fuel cell under five different clamping pressures. The current is dynamically stable for four of the five clamping pressures. The lowest clamping pre ssure of 28 oz-in displayed the worst I-V

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162 performance, due to mass transfer limita tions and higher contact resistance. The polarization curves continuously increase until a torque of 36 oz-in is reached. As the torque continues to increase to 44 oz-in, th e polarization curves again begin to decrease. 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.21.4Current Density (A/cm^2)Voltage (V ) 28 oz-in (0.20 N-m) 32 oz-in (0.23 N-m) 36 oz-in (0.25 N-m) 40 oz-in (0.28 N-m) 44 oz-in (0.31 N-m) Figure 6.8. Polarization curves with tightening torques of 28 oz -in to 44 oz-in for stack #1 The material and bolt properties from Tables 1 and 2 were entered into the numerical model for stack #1, and the optimal force, pressure and tightening torque was calculated. The results ar e shown in Table 6.5.

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163 Table 6.5 Calculated force, tightening torque and contact pressure for stack #1 ________________________________________________________________________ Parameter Value ________________________________________________________________________ Total Force on the Stack 310.8 N Tightening Torque 36.35 oz-in (0.257 N-m) Average Interface Contact Pressure 0.194 MPa (1.94 bar) ________________________________________________________________________ The values in Table 6.5 show that th e calculated optimal tightening torque matches the tightening torque associated with the best fuel cell I-V curve in Figure 6.9. Figure 6.9 displays the performance curv es for fuel cell stack #2 with five different clamping pressures. Again, the polarization curves reflect the effect of the interfacial electrical resistan ce, mass transfer and optimal clamping pressure on the fuel cell stack. As seen in Figure 34, the fuel ce ll performance appears to be the poorest with the 6 oz-in clamping pressure. Compression with a torque of 10 oz-in shows the best performance curve. As the torque increased fr om 10 to 14 oz-in, the fuel cell performance decreased as the mass transfer is hindered due to the decreased porosity of the GDL layer.

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164 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.7 Current Density (A/cm^2)Voltage (V ) 6 oz-in (0.4 N-m) 8 oz-in (0.6 N-m) 10 oz-in (0.7 N-m) 12 oz-in (0.8 N-m) 14 oz-in (1.0 N-m) Figure 6.9. Polarization curves with ti ghtening torques of 6 oz-i n to 14 oz-in for stack #2 The numerical model for tightening torque was run for stack #2, and the optimal force, pressure and torque calculations are show n in Table 6.6. Like fuel cell stack #1, the calculated optimal tightening torque matches the torque associated with the best fuel cell performance.

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165 Table 6.6 Calculated force, tightening torque and contact pressure for stack #2 ________________________________________________________________________ Parameter Value ________________________________________________________________________ Total Force on the Stack 205.9 N Tightening Torque 10.6 oz-in (0.7 N-m) Average Interface Contact Pressure 0.129 MPa (1.29 bar) ________________________________________________________________________ As shown in Figure 6.10, the case of 4 oz-in compression showed the best polarization curve. As with the previous polar ization curves for the other fuel cell stacks, the lowest torque showed a poor polarizati on curve in comparison with the polarization curve obtained with the optimal torque. It se ems to be difficult to achieve more than 40 mA cm 2 of current density with a compression of 6 oz-in due to the mass-transfer limitation. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 020406080 Current Density (mA/cm^2)Voltage (V ) 1 oz-in (0.01 N-m) 4 oz-in (0.03 N-m) 6 oz-in (0.04 N-m) Figure 6.10. Polarization curves with tightening tor ques of 1 oz-in to 6 oz-in for stack #3

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166 The numerical model was again run to obt ain the tightening torque for stack #3, and the optimal force, pressure and torque calculations are show n in Table 6.7. Once again, the calculated optimal tightening tor que matches the torque associated with the best fuel cell performance in Figure 6.10. Table 6.7 Calculated force, tightening torque and contact pressure for stack #3 ________________________________________________________________________ Parameter Value ________________________________________________________________________ Total Force on the Stack 126.4 N Tightening Torque 4.8 oz-in (0.3 N-m) Average Interface Contact Pressure 0.079 MPa (0.79 bar) ________________________________________________________________________ Therefore, it can be concluded that the numerical model does a good job of estimating the tightening torque for a fuel cel l stack using clamping bolts ( 2%). When polarization curves are obtained with the tightening torque values lower than the predicted value, the poor performance in co mparison with the performance obtained with the optimal torque can be attributed to mostly high contact resistance. Since the polarizations curves generally have the same shape at slightly lower tightening torques, the ohmic polarization seems to be dominating the losses. If the torque is well below the calculated value, concentration (mass transpor t) losses are also seen in the polarization curve as with the 28 oz-in in Figure 6.8. When polarization curves are obtained with the tightening torque values highe r than the predicted value, the poor performance in

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167 comparison with the performance obtained with the optimal torque can be attributed to mostly high mass transfer resistance. This is very obvious in Figure 6.10 with 6 oz-in where the fuel cell I-V curve drops abruptly after the activation polar ization part of the polarization curve. The effect of changing the clamping pre ssure on the performance of a PEM fuel cell has been investigated numerically a nd experimentally. A numerical model was developed with four major parts: the stiffness of the stack materials, stiffness of the bolts, ideal compression of the GDL, and finally th e tightening torque. The compression of the GDL, and the effects of contact electrical re sistance and limited mass transfer affects is estimated and taken into consideration in the numerical model. A Herzian equation is used for predicting the optimal force on the GDL layer based upon ideal gas permeability and GDL contact resistance. The torque is us ed as an indirect means of measuring the stack clamping pressure, and has a direct e ffect on fuel cell stack performance. The experimental validation consiste d of experimentally examinin g the effect of the clamping pressure on the electro-physic al properties on three differe nt free-convection PEM fuel cell stacks. As the stack material stiffness, bolt material, or GDL compression changed, the resulting fuel cell polarization curve ch anged. Results show that the numerical calculations agree well ( 2%) with the fuel cell stack torque tests. It is further shown that low tightening torque results in a high interf acial resistance between the bipolar plate and the gas diffusion layer that reduces the electr ochemical performance of a PEM fuel cell. In contrast, high tightening torque reduces the contact resistance between the graphite plate and the gas diffusion layer, but mean while narrows down the diffusion path for mass transfer from gas channels to the cat alyst layers. The model and experimental

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168 validation verified the correct balance of obtaining a tightening torque based upon stack and bolt stiffness, contact resistance and mass transfer limitations within a fuel cell stack.

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169 7 DESIGN AND FABRICATION OF MICRO FUEL CELL STACKS An understanding of how the design a nd manufacturing processes influence performance variables is critical in order to successfully design new fuel cells. There have been numerous design variables examined in this dissertation, and some of the most important ones include flow channel geomet ry, catalyst particle size and shape, and electrolyte thickness. Studying the fuel cell microstructure is very important for optimizing fuel cell electrical behavior, however it is even more important for micro fuel cells since surface characteristics begin to dominate over bulk effects [98, 99]. The flow field plates are one of the mo st important components of the fuel cell stack. The flow field plates distribute the fu el across the electrode surface, remove liquid water, conduct electricity and mechanically stabilize the fuel ce ll membrane electrode assembly (MEA). The traditional materials used for these plates include stainless steel or graphite, aluminum or nickel. The processe s commonly used to produce the flow field design are CNC (computer numerical contro l) machining, injection molding and stamping. These materials and processes are not suitable for MEMS-based (microelectro mechanical) fuel cell systems. Typical materi als that have been used for MEMS fuel cells, in the literature, are silicon wafers, carbon paper, PD MS (polydimethylsiloxane), SU-8 (EPON SU-8 epoxy resin from Shell Ch emical), copper and stainless-steel metal foils [98].

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1707.1 Background and Approaches Several studies have investigated the creation of microchannels using MEMS techniques in the literature. Flow channel depths ranging fr om 50 to 200 m were created in a silicon wafer in Lee et al. [100]. The mi cro fuel cell produced a current density of 50 mA/cm2, which is typical performance for a micro fuel cell. However, the fuel cell performance could have been improved if a non-corrosive metallic layer was applied to the silicon flow field plate to increase the c onductivity. Yu et al. [101] had developed a reactive ion etching (RIE) proce ss on silicon wafers with a 2 00 m flow channel depth. A conductive metal (0.5–1.5 m Au, Cu or Ti) was sputtered on the surface of the silicon wafer. The results showed that the micfeatur es created on the sili con-based flow field plates would provide more uniform dist ribution of fuels under the same operating conditions of gas pressure and flow rate over traditional flow field plates. Schmitz et al. [102] applied MEMS fabrication processes to create the flow field plates. The current density could have been higher if the copper flow channels (35 m) were deeper, and the glass fiber substrate was more conductive. O’Hayre et al. [103] designed a 16-cell PEM fuel cell in a 3.5 in.2 glass fiber composite plate, which had an open circuit voltage of 12 V for a 3C battery. However, there was sti ll a large contact resi stance, and the glass fiber substrate did not have the required st acking pressure, and there was large contact resistance. The feature sizes for flow channels in the literature range from 100 m x 200 m x 20 m to 500 m x 500 m to 750 mm x 750 mm x 12.75 mm, with many lengths, widths and depths in between with various rib widths [98, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113]. Intuitively, it seems that fuel cell performance should improve as the

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171 channel feature size decreases and gas flow ve locity increases, since the increased flow velocity enhances mass transport. Yet, one of the disadvantages of the smaller feature size is the increased pr essure drop in the flow channels. Although there has been a lot of speculation in the literature regarding the dimensions that should give the best performance, the entire range of channel width and depth dimensions has not been experimentally compared. The viewpoints re garding the performance of microchannels conflict mainly in the 20 m to 500 m range For example, in [104] it mentions that better performance is gained between feature sizes of 483 m – 99 m, but the pressure losses under 200 m are so large that it nega tes the effect of dow n-scaling [104]. In [105], when the channel dept h was decreased from 1 mm to 300 m, the power density performance increased by 71.9%. When the fl ow field channel depth was further reduced to 100 m, the performance decreased by 8.6% [98, 105]. 7.2 Design and Production of the Micro Fuel Cell Stack Two micro fuel cell stacks were designed for this study, and are illustrated in Figure 7.2. Each stack was 25.4 mm x 25.4 mm x 14.7 mm, and the dimensions of the fuel cell components are given in Table 7.1. One stack used polyvinyl chloride (PVC) end plates, and the other used Delrin end plates, due to the low-cost, and commercial availability of the materials. Thin silicon gask ets were used to prevent gas leakage, and a contact layer was created by depositing a gold layer on the sides of the end plates that were in contact with the flow field plates. Six different 1 cm2 flow field patterns were fabricated with various serpentine channel sizes ranging from 1000 m to 20 m in width and depth. The channel dimensions are shown in Table 7.2, and were chosen to give a

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172 comparison of the range of micro to MEMS-sized channels in order to compare the effect of the overall performance due to the change in flow field dimensions. Table 7.1 Prototype stack dimensions ________________________________________________________________________ Stack Dimensions Inches Millimeters ________________________________________________________________________ End Plate 0.250 6.350 Gasket 0.004 0.102 Silicon Flow Channel Plate 0.016 0.400 Gasket 0.004 0.102 MEA (Fuel Cell) 0.040 1.016 Silicon Flow Channel Plate 0.016 0.400 End Plate 0.250 6.350 Total Thickness 0.58 14.72 ________________________________________________________________________ The flow field plates were made from 400 m thick, 4 silicon wafers. Two flow field plates for a single cell had a total cell area of 6.45 cm2 and a reaction area of 1 cm2. A deep reactive-ion etching (DRIE) fabricati on process was used for the fabrication of micro flow fields in the sili con wafer for the 200 m – 20 m depth. In order to compare the silicon DRIE fabricated flow field plates with conventional machining and dimensions, four additional plates made of Delrin were made using traditional CNC machining process. The micro-sized flow fi elds had channel dimensions of 500 m and 1000 m. As shown in Table 7.2, the width a nd depth of the flow channels ranged from 1000 m – 20 m, and the channel length range was from 7.8 – 8.0 mm. The width of the

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173 ribs also ranged from 1000 m – 20 m, with a consistent channel area of 50% (channel to rib ratio of 1:1) [98]. Table 7.2 Flow field plate channel dimensions No. Channel Width (microns) Channel Depth (microns) Rib (Gap) Width (microns) % of Active Area that is Channels No. of Channels & Ribs Channel Length 1 1000 1000 1000 50.0% 4 7 2 500 500 500 50.0% 8 7.5 3 200 200 200 50.0% 20 7.8 4 100 100 100 50.0% 40 7.9 5 50 50 50 50.0% 80 8.0 6 20 20 20 50.0% 200 8.0 The serpentine flow field design was chosen because it has been shown to perform the best in several MEMS fuel ce ll studies [104, 105], and it had to be easily compared with other micro-sized channel stud ies in the literature. One advantage of the serpentine flow path is that it reaches a la rge portion of the active area of the electrode by eliminating areas of stagnant flow. The flow field plates were coated with gold in order to promote conductivity and reduce contact resistance. The openings in the inlet and outlet of the gas channel and end plates were made much larger than the flow field channel dimensions in order to fit standard connectors for gas flow into the st ack. Figure 7.1 illustrate s the single cell stack assembly. The flow chart of research methodology is presented in Figure 7.2.

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174 MEA Gasket Anode flow field plate Cathode flow field plate End plate End plate Cathode flow field pattern Anode flow field pattern Figure 7.1 Single cell design and its components [98]

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175 Flow field design Creation of bipolar plates DRIE process CNC machining process Design fuel cell stack Select materials Creation of end plates Size gaskets, contacts, gas fittings and connectors 50 –200 um channels 500 -1000 um channels Stacking and assembly of fuel cell components Fuel cell testing Performance analysis SEM and profilometer scans Gold plating for contact Figure 7.2. Flow chart of research methodology [98]

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1767.3 Microchannel Fabrication Process The first step in the microchannel fabrication process is depositing a 2 m thick PECVD SiO2 layer on both sides of the Si wafer. The front side was patterned using the channel mask and Futurex PR2000 photoresist. The exposed SiO2 was partially etched by RIE for 10 minutes. Next, Shipley 1813 photoresist was manually placed over the alignment marks, and then baked for 1 minute at 90C. The remaining SiO2 was etched off by RIE for 60 minutes. The photoresist wa s then stripped off using acetone/methanol. The wafer was then put into DRIE, and etched (~ 1 m/min) to the desired depth of the channels [98]. After the microchannels were created, th rough-holes were then made in the same silicon wafer in order for the silicon flow field plate to be placed into the fuel cell stack. PR2000 was spun onto the back side of the wafer, and then RIE of SiO2 was performed for 70 minutes. The through-holes were created with a through-wafer DRIE process. The last step for creating the th rough holes was stripping off the oxide layer using BOE. A layer of Ti/Au 300nm/1m was then sputtered on the wafer from the channel side (front side) to prevent corrosion and improve conduc tivity [98]. The processes used to create the flow field pattern are presented in Figure 7.3, and further detail s of the fabrication process can be found in [114].

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177 Figure 7.3 Flow chart of the RIE process used for the creation of the flow field plates [98]

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1787.3.1 The Two Stage DRIE Process Two iterations of the etching process are conducted in order to create the micro flow fields and through holes in the silicon fl ow field plates. The first iteration of the etching process created the main flow fiel d channel pattern, a nd the second iteration created the through holes for the gas inlet, outlet and through bolts. Figure 7.4 shows the main flow channels. Figure 7.5 shows the th rough hole with the mi cro flow channels. Figure 7.4. Micro flow field channels in silicon flow field plate Figure 7.5. Through-hole added to micro flow field channels in silicon flow field plate

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179 The surface profile and depth of the flow channel were obtained using scanning electron microscopy (SEM) and profiler scans. Figure 7.6 compares the micro flow field channels at 20 m, 50 m and 200 m. The SE M photos demonstrate th e precision of the micro channel structure create d by the DRIE process [98]. a) b) c) Figure 7.6. SEM images of micro flow field ch annels and through holes, (a) 20 m, (b) 50 m, and (c) 200 m width channels 7.3.2 Single Cell Fuel Cell Stack Performance Tests The two single cell, air breathing fuel ce ll stacks had an active fuel cell area of 1 cm 1 cm, and was comprised of a 5-la yered MEA made of Na fion 112, carbon cloth and 1 mg/cm2 of Pt loading on both the anode and cathode. The same MEA and stack is used with the different micr o flow field plates (20 m – 200 m flow channels). A second stack was assembled for the 500 m a nd 1000 m channel flow field plates [98]. The single cell fuel cell stack s are shown in Figure 7.7.

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180 Figure 7.7 Prototypes of the single ce ll fuel cell stacks [98] Cell performance tests are taken at 25 C and ambient pressure with 0.5 standard cubic centimeter per minute (SCCM) of hydr ogen from an electrolyzer, with no additional humidification. I – V curves of these cell perfor mance tests are plotted in Figures 7.8 and 7.9. The 1000 m and 500 m flow channels had the worst cell performance characterized by low current de nsities, high contact resistance and poor mass transfer [98].

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181 0 0.2 0.4 0.6 0.8 1 1.2 050100150200 Current Density (mA/cm^2)Voltage (V) 1000 um 500 um 200 um 100 um 50 um 20 um Figure 7.8. I–V curve of the cell pe rformance tests [98] 0 10 20 30 40 50 60 70 80 050100150200Current Density (mA/cm^2)Power Density (mW/cm^2 1000 um 500 um 200 um 100 um 50 um 20 um Figure 7.9. Fuel cell power density curves for 20 1000 m channel widths and depths [98]

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182 The single fuel cell stack was designed as a smaller version of a traditional fuel cell to enable comparison with both larger co mmercial fuel cells, and with other MEMS fuel cells in the lite rature. The channel and rib dimensi ons selected for this study were used to determine the optimal flow channel di mensions for a MEMS fuel cell. Some of the benefits of the MEMS flow regime in clude laminar flow, higher velocities, rapid diffusion, low leakage, surface effects, good fl ow control and very small dead volumes. A major advantage for MEMS fuel cells is th at many of the layers can be applied through sputtering (or some other MEMS -based process). The layers can be extremely thin, which will make the future stacks lighter and less co stly, but will allow the fuel cell to maintain high current densities. When de signing MEMS fuel cells, some of the issues that may be encountered are surface roughness, uneven topography, bubbles and flooding in flow channels [98]. The flow field channels increase in pe rformance with the decrease in channel width, depth and rib size, which is the space between flow channels. The 20 m flow channel width, depth and rib size outperformed all other channel sizes in terms of power density and current density. In the activation polarization dom inated region (~ 0.8 – 1.0 V), all of the activation voltage losses were a bout the same for all of the fuel cell tests conducted. Since the same fuel cell MEA wa s used, the electrode kinetics should be similar, and therefore, th e activation voltage losses should be similar [98]. In the ohmic polarization dominated re gion (~ 0.5 – 0.8 V), the 20 m flow channel width, depth and rib size had superi or performance in terms of voltage and current density. Since the majority of the ohmic resistance in fuel cells is the electrolyte, and the same MEA was used, the difference in ohmic resistance is due to the difference

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183 in width, depth, rib size, the number of cha nnels and the percent ch annel area. As shown in Table 7.2, the percent area of channel and rib space (50 %) is consistent for all of the flow field plates. The decreased contact resist ance for the flow field plates with the 20 m dimensions may be due to the gas diffusi on media protruding into the flow channel. This provides greater surface area of the GDL la yer in contact with the flow field plates. The concentration polarization dominated re gion (~ 0 – 0.5 V) displays the most notable difference between polarization curves fo r the dimensions of the flow field plates. As the channel width and depth decreases from 1000 to 20 m, the velocity and pressure drop increase rapidly. The large increase in pressure drop is counteracted by the rapid increase in velocity. Although the channel to rib ratio is identi cal for all of the flow field plates created (1:1), the decrease in rib size may aid in better overall reactant flow through the gas diffusion media since the “voi d space” between channels is decreased. In addition, since the depth of the 20 m is s ubstantially less than the other depths, the stagnant flow region at the interface be tween the channel and gas diffusion media encompasses a larger portion of the flow ch annel. Also, if the gas diffusion media is protruding into the channels, this stagnant flow region may encompa ss a large portion of the channel, and therefore, the flow in th e channel enters the diffusive regime with greater ease than in larger channels where the flow has to convert from convective to diffusive [98]. Although the performance of the MEMS fuel s cells presented in this dissertation performed better than most other MEMS fuel cells currently in the literature, the performance is still poor in comparison to convectional fuel cells where the current density typically reaches 1 – 1.5 A/cm2 (0.5 – 8.0 A/cm2 for free-convection fuel cells).

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184 One of the issues with MEMS fuel cells is that liquid water dropl ets generated at the cathode can block a flow channel entirely. Thes e blockages can lead to reactant starvation at the cathode, which not only affects the concentration polarization region of the polarization curve, but also affects the fuel cell perfor mance through reaction kinetics (the activation polarization region) due to the dependence upon the reactant and product concentrations at the reaction sites. In addi tion, when the reactants are deficient at the reactant sites, this genera tes less charge, therefore, th e amount of charge that is transported through the cell is reduced, wh ich contributes to the ohmic polarization dominated region. The combination of these voltage losses creates a total polarization curve with poor performance in compar ison to traditional fuel cells [98].

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185 8 FUEL CELL MODEL RESULTS A mathematical model can help the fuel ce ll engineer to design a better fuel cell through an understanding of the physical phenomena occurring within the PEM MEA layers. This is important because the direct measurement of concentrations and velocities within a fuel cell is currently unavailable due to the thinness of, and the bonding between, the MEA layers. Therefore, a transient 1-D mass, heat, pressure and membrane model was created in MATLAB to study the transpor t phenomena, and this chapter highlights some of the processes that the current model illustrates. In order to examine these processes occu rring within the PEM fuel cell, design parameters were taken from several actual PE M fuel cell stacks, and necessary constants were taken from the literature, and are noted in Appendix A. The model considers mass and energy balances, heat generation equatio ns at the anode and cathode catalyst layer, and pressure losses throughout the fuel cell stack. The model was coded to allow the user to divide each fuel cell layer into smaller nodes along the x-axis if specified. Unlike most published models, this model includes all of the layers in the fuel cell stack, including the end, flow field and cooling plates, terminal s, the gas diffusion layers (GDL), catalyst layers and membrane. Many of the variables in the model were put into arrays to make the code cleaner, and to reduce the number of lines in the code. The numerical code allows the discretization of each of the layer into smaller control volumes. The temperatures were assumed to be at the center of each node, and the mass flow rates,

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186 pressure drop, velocity and charge transpor t was defined at the boundaries of each control volume as illustrated in Figure 8.1. The set of equations were put into matrix form, and solved simultaneously using MATLAB’s ode45 ordinary differential equation solver. ode45 is based on an explicit Runge-Kutta (4 ,5) formula, the Dormand-Prince pair. It computes y(tn) in one step, and needs only th e solution at the immediately preceding time point, y(tn-1). T1 Bipolar PlateRest of stack End Plate Control Volume for Node 1 T1T3 qlhsqrhsx/2 dU/dt Control Volume for Node i Ti-1Ti+1 qlhsqrhsx dU/dt Control Volume for Node N TN-1TN+1 qlhsqrhsx/2 dU/dt TiTN T2TN+1 m m m m m m m m m m Figure 8.1. Schematic of the PEMFC stack and its components for model development

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187 The numerical code that was develope d for this study has approximately 3500 lines. Lines 11 – 117 initialize all of the consta nts used in the model, and the lines 118 – 225 initialize all of the parameters in form of vectors, which are listed in Appendix A. Lines 235 – 253 assign the layer numbers, and the number of user-specified nodes in each layer. The x coordinates for each node ar e then calculated assuming a uniform distribution. Lines 235 – 253 incl ude code that specifies skip ping the layers that do not repeat in each cell (such as the end plates), and assign coordinates to all of the nodes for each layer in the fuel cell stack. Lines 256 318 calculate or specify the initial pressures, temperatures, velocities, molar flow rates and potentials for the simulation program. The state variable matrix is formed in lin es 324 – 341, and this is passed to the fuel cell function, which calculates the change in temper ature, pressure, velocity, molar flow rates, and potentials with respect to tim e using the MATLAB’s ode45 solver. In the fuel cell function, the components of stat e vector are separated on lines 349 – 355. The vectors are initial ized for all of the output s on lines 362 – 389 and 434 – 500. The Prandtl numbers are calculated on lin es 508 – 518 to obtain the heat transfer coefficients. The mass transfer section range s from lines 520 – 1593, the pressure drop section is in lines 1595 -2317, the temper ature section spans lines 2319 – 2762 and the potential section makes up lines 2764 – 2997. The rate change equations for the molar flow rates, pressures, velocities, potentia ls, and temperatures are on lines 2999 – 3089. The remainder of the code creates the plots that are automatically generated while the program is running. The mass flow, pressure, temperature portions of the model will be discussed in more detail throughout this chapter. An overall diagram of the MATLAB simulation program is illustrated in Figure 8.2.

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188 Parameter Vectors These parameters are specified for each layer: (1) Number of slices (2) Density (kg/m3) (3) Area (m) (4) Area of void (m2) (5) Channel Area (m2) (6) Thickness (m) (7) Thermal Conductivity (W/m-K) (8) Specific Re sistance (ohm-m) (9) Specific heat capacity (J/kg-K) (10) Channel radius (m) (11) Channel width (m) (12) Channel depth (m) (13) Channel lengh (m) (14) No. of Bends (15) No. of Channels Assign layer numbers, and slices to each layer, specify x coordinates based upon the uniform distribution of nodes Constants (16) Iniial stack temperature (K) (17) Hydrogen temperature (K) (18) Air temperature (K) (19) Convective loss to air from the stack (20) Ambient temperature (K) (21) Current (A) (22) Humidity (23) Activaion overpotenial (V) (24) Entropy change for anode (25) Entropy change for cathode (26) Volumetric flow rate of hydrogen (m3/s) (27) Volumetric flow rate of air (m3/s) (28) Viscosity of hydrogen (Pa-s) (29) Viscosity of air (Pa-s) (30) Hydrogen pressure (Pa) (31) Air pressure (Pa) (32) Density of hydrogen (kg/m3) (33) Density of air (kg/m3) Inputs: 1, 6 Calculate Slice Thicknesses: Get derivatives between xs dx(layer) Compute heat transfer coefficients for he left and right sides of each layer U(layer) Iniialize temperature, total mass flow, pressure and velocity vectors, and specify the initial values T(layer), n_tot(layer), P(layer) and u_m(layer) Inputs: 7, 19 Inputs: 16, 20, 26, 27, 30, 31 Call fuel cell function to calculate dTdt, dndt and dPdt Combine into rate change of T, n_tot and P: dTdt = (Q_left + Q_right + Q2 + H) ./ (mass + thmass) dndt(oulet(i)) = dndt(oulet(i)) + n_outlet(i) -n_tot(oulet(i)) dPdt(outlet(i)) = dPdt(outlet(i)) + P_outlet(i) -P(oulet(i)) Calculate the change in T, n_tot and P using ode45 (ODE solver based upon Runge-Kutta (4,5) formula) Form state variable matrix Separate components of state vector Calculate inlet and outlet molar flows Inputs: 17, 18, 22, 30, 31, 32, 33 Calculate molar flows of protons, oxygen and water in the catalyst layers Calculate velocity Calculate pressure drop Calculate the right and left heat flows for each layer (Q_left + Q_right), energy mass terms (mass), thermal mass (thmass), layer-specific heat flows (Q2) and enthalpies (H) Inputs: 2, 3, 4, 8, 9, 21, 23, 24, 25 Use calculated T, P and mole fractions of H2 and O2 to calulate charge generated in catalyst layers Polarization curve Inputs: 5, 6, 10, 11, 12, 13, 14, 15, 17, 18, 28, 29, 30, 31 Figure 8.2. Overall diagram of MATLAB code created Since the hydrogen flow rate into the fuel cell enters the stack from one end, and the oxygen enters from the other end, this creates a challenge when creating an overall fuel cell stack model. Figure 8.3 shows a diagra m of the order of fuel cell layers in the stack, the directions of the flow into each layer, and the asso ciated layer numbers used for the model. To better understand the outputs disc ussed in this chapter, the layer and flow numbering are shown in Figures 8.3 and 8.4. Both the layer a nd the flows into each layer are numbered from left to right.

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189 Figure 8.3. Illustration of fuel cell stack layer numbering

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190 Figure 8.4. Schematic of the numbering of laye rs and flows for the PEMFC model 8.1 Heat Transfer Portion of th e Overall Fuel Cell Stack Model A numerical code was deve loped to investigate the effect of various stack materials and operating parameters on fuel cell heat transfer behavior. The energy balances and thermal resistance equations fo r each layer are integrated simultaneously using MATLAB’s ode45 function. Arrays were created for the node temperatures, thermal resistances, heat tran sfer coefficients, heat fl ows, Nusselt numbers, specific heats, thermal conductivities and enthalpies of each node or layer. The stack dimensions and other parameters used in the simu lations are summarized in Appendix G. As mentioned previously, th e initiation of the variable s, and initial temperature parameters are given in lines 267 – 279, 313 – 319, 366 -382, 403 – 428, 456 – 482, 508 – 518 in the overall model code. Beginning w ith line 2319, the ohmic heating, thermal resistance, enthalpies, specific heats, viscosities, and ther mal conductivities for the nodes in each layer are calculated. In addition, the thermal resist ances for the solid portion of

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191 the layer to the gas/liquid por tion of each layer are computed A summary of the thermal portion of the code in shown in Figure 8.5. Figure 8.5. Temperature portion of overall model The heat distribution in stacks with at least 20 cells shows an almost identical distribution with stacks of la rger size. Therefore, it was f ound that stacks with at least 20 cells were adequate in simulating stacks of 100 cells or more. Since the minimum number of cells is a strong function of end plate and stack design, the result s presented in this section is for a generic stack, and will not be applicable for all stack configurations. Figures 8.6 and 8.7 shows a typical temper ature distribution thr ough a 20 cell and 250 cell stack with an initial hea ting of the stack to 333 K, a current density of 0.6 A/cm2, and reactant gas pr essure of 3 atm.

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192 0 0.05 0.1 0.15 0.2 326 328 330 332 334 t = 300 position (m)Temperature (K) Figure 8.6. Temperature distribution in a 20 cell fuel cell stack, a) surface plot of the temperature distribution as a function of position and time, (b ) temperature distribution at t = 300s

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193 0 0.5 1 1.5 2 326 328 330 332 334 t = 300 position (m)Temperature (K) Figure 8.7. Temperature distribution in a 250 cell fu el cell stack, (a) surface plot of the temperature distribution as a function of position and time, (b ) temperature distribution at t = 300 s

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1948.1.1 Temperature Distribution of Various Stack Sizes The minimum number of cells that can be used to simulate a larger stack is influenced by the stack and end plate design. Figure 8.8 illustrates a comparison of the temperature distribution of a 5, 10, 20, 50 and 100 cell stacks. Due to the number of cells in the 20, 50 and 100 cell stacks, the temperatur e distribution in the ce nter cells for the 20 and 50 cell stacks were almost identical at vary ing times for the heating in the cell layers, which indicates that the 20 cell stack is ade quate for studying the te mperature distribution and other stack transport phenomena.

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195 a) 0 0.02 0.04 0.06 330.8 331 331.2 331.4 331.6 331.8 t = 58.7879 59.3939 60 Position (m)Temperature (K)b) 0 0.02 0.04 0.06 0.08 0.1 330.8 331 331.2 331.4 331.6 331.8 t = 59.3939 60 Position (m)Temperature (K) c) 0 0.05 0.1 0.15 0.2 330.8 331 331.2 331.4 331.6 331.8 t = 58.7879 59.3939 60 Position (m)Temperature (K)d) 0 0.1 0.2 0.3 0.4 330.8 331 331.2 331.4 331.6 331.8 t = 58.7879 59.3939 60 e) 0 0.2 0.4 0.6 0.8 330.8 331 331.2 331.4 331.6 331.8 t = 59.3939 60 Figure 8.8. Temperature distribution at the end of 60 seconds for (a) 5 (b) 10 (c) 20 (d) 50 and (e) 100 cell stacks

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196 8.1.2 Stack Temperature Distribution Over Time Figure 8.9 shows the effect of time on th e stack temperature distribution. As expected, as the time increased, the catalyst layers become hotter, and the entire stack heats up due to catalytic heating. There is a significant increase in heating from t = 10 s to t = 600 sec. The local heating in the anode a nd cathode catalyst la yers increases from 331.3 K to 331.3 K after 10 s. By 60 s, the local heating of the catalys t layers ranges from 331.6 to 331.8 K, and at 600 s, the local heating in these layers has increased approximately 7 K. The catalytic heating in th e cells of the fuel cell stack can present a challenge for fuel cell designers. However, th ese local temperatures are unable to be accurately measured within the fuel cell stack. The fuel cell re searcher is able to measure the temperature of the bipolar plates instea d in order to obtain an idea of the heat generated by the ca talytic heating.

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197 a) b) c) d) e) Figure 8.9. Temperature distribution at different times (a) 10 (b) 30 (c) 60 (d) 300 and (e) 600 seconds

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198 8.1.3 Temperature Distribution in a Single Cell When the number of nodes are increased significan tly for each layer, the temperature variation in the graphs become mi nimized because the heat is transferred to the previous and next nodes, and the effect of the heat/coo ling is shown in the overall cell or stack temperature distribution. However, th e local heating from th e catalyst layers are still very obvious in the graphs and there is little change in the magnitude of the heating of the catalyst layers. Increasing the number of nodes per layer is very important as the layer thickness increases. This enables the heat to diffuse through each node more quickly, and be transferred to the next node fast er, which crates a realistic result. Figure 8.10 illustrates the temperature distribution in a single fuel cell with 1, 10, 32 and 64 nodes per layer.

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199 (a) 0 0.5 1 1.5 2 2.5 x 10-3 295.1 295.2 295.3 295.4 295.5 t = 300(b) 0 1 2 3 4 x 10-4 295.1 295.2 295.3 295.4 295.5 295.6 t = 300 (c) 0 2 4 6 x 10-4 295 295.2 295.4 295.6 295.8 296 t = 300(d) 0 0.2 0.4 0.6 0.8 1 x 10-4 295.1 295.2 295.3 295.4 295.5 t = 300 Figure 8.10. Temperature distribution through a single fuel cell, with using a (a) 1, (b) 10, (c) 32 and (d) 64 nodes per layer 8.1.4 Variation of Operating Current Density Figure 8.11 shows the stack temperature dist ribution for current densities i = 0.1, 0.6 and 1.0 A/cm2 respectively. After 300 seconds, the temperature increased from 334 K to 336 K for a current density of 0.1 A/cm2, it increased an extra degree for a current density of 0.6 A/cm2 and it increased to 342 K for a current density of 1.0 A/cm2. The asymmetric stack distribution can be attributed to the different heat source term on the

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200 anode and cathode sides [115]. Similar stack te mperature distributions were also achieved by Khandelwal et al. [115] and Shan and Choe [116]. In certain stack designs, it may be advantageous to use the fact that there is rapi d catalytic heating at cu rrent densities of 1.0 A/cm2. Some of the heat generated by the ca talyst layers can be removed by the reactant gases or by the coolant. The effect of inle t gas temperature and c oolant temperature is explained in Sections 8.1.5. 0 0.05 0.1 0.15 0.2 334 334.5 335 335.5 336 336.5 t = 300 Position (m)Temperature(K)b) 0 0.05 0.1 0.15 0.2 334 335 336 337 338 t = 300 Position (m)Temperature (K) c) 0 0.05 0.1 0.15 0.2 334 336 338 340 342 344 t = 300 Position (m)Temperature (K) Figure 8.11. Stack temperature profile for base conditions at various time for (a) i = 0.1 A/cm2 (b) i = 0.6 A/cm2 (c) i = 1.0 A/cm2

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2018.1.5 Effect of the Inlet Ga s and Coolant Temperatures The effect of inlet gas temperature on th e solid portion of each layer of the fuel cell stack is shown in Figure 8.12. Heati ng the anode gas will help to reduce the temperature difference between the anode a nd cathode side due to the unbalanced heat generation in the electrodes. Heating the ca thode gas may also be useful to enhance product water uptake to help minimize the water flooding in the cathode, and to help enhance mass transport. Heat lo ss to the reactant gas or cool ant can be reduced by either increasing the inlet gas flow temperature or reducing the gas flow rate. As expected, the gas temperature profile is similar to the stack temperature profile. The temperature of the gases rises very slowly in comparison with th e temperature of the stack due to the heat capacity and thermal conductivity of the gases. 298 318 338 358 378 12345678910111213 Fuel cell layer No.Temperature (K) T, 0.1 A/cm2 Tf, 0.1 A/cm2 T, 0.6 A/cm2 Tf, 0.6 A/cm2 T, 1.0 A/cm2 Tf, 1.0 A/cm2 Figure 8.12. Stack gas temperature profile fo r base conditions at 1200 s for (a) i = 0.1 A/cm2 (b) i = 0.6 A/cm2, and (c) i = 1.0 A/cm2

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202 Figure 8.13 shows the effect of heating th e fuel cell stack layers on the inlet gas temperature from 60 s to 1200 s. As the stack heats up due to catal ytic heating, the gas/fluid temperature also heat s up. The gas/fluid temperatur e enters the stack at 298 K, and the stack is heated and maintained at 353 K. 298 318 338 358 378 12345678910111213 Fuel cell layer No.Temperature (K) T, 60 s Tf, 60 s T, 300 s Tf, 300 s T, 600 s Tf, 600 s T, 1200 s Tf, 1200 s Figure 8.13. Effect of heating the fuel cell stack layers on the inlet gas temperature Figure 8.14 illustrates the effect of heating the inlet gases to 353 K, and the effect on the stack temperature initially at 298 K. Of course, this is hi ghly dependent on stack design, and Figure 8.14 illustrates a single cell st ack, therefore, it is more difficult to heat the stack with the gases due to the larg e amount of stack volume that is solid.

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203 298 318 338 358 378 12345678910111213 Fuel cell layer No.Temperature (K) T, 60 s Tf, 60 s T, 300 s Tf, 300 s T, 600 s Tf, 600 s T, 1200 s Tf, 1200 s Figure 8.14. Effect of heating the inlet gas temperat ure on the temperature of the fuel cell stack Figure 8.15 illustrates the comparison of the stack temperature with inlet gas and coolant temperature of 298 K and 288 K respec tively. As the coolant temperature, in layers 3 and 11, is lowered from 298 K to 288 K, the effect of the coolant temperature on the inlet gases is minimal. However, the effect on maintaining a more uniform stack temperature is very obvious. The heating by th e catalyst in layers 6 and 8 is minimized after 1200 s by the coolant in layers 3 and 11.

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204 298 318 338 358 378 12345678910111213 Fuel cell layer No.Temperature (K) T, 1200 s Tf, 1200 s T coolant 15 C Tf coolant 15 C Figure 8.15. Comparison of the effect of c oolant on the stack temperature Figure 8.16 illustrates the effect of temperature on relative humidity on the single fuel cell stack for the temp erature results presented by Figure 8.13. The anode side relative humidity is beginning to decrease due to the electrochemical reaction, since for every mole of hydrogen that is removed: two moles of water are also removed. In the cathode channel, the relative humidity of stream is equal to 1.0. This is due to the fact that the water is produced con tinually, therefore, the water content continually increases. The mass flow rates and mole fractions of water, hydrogen and oxygen will be discussed in more detail in section 8.2.

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205 0.995 0.996 0.997 0.998 0.999 1 12345678910111213 Fuel cell layer No.Relative Humidity 60 s 300 s 600 s 1200 s Figure 8.16. Relative humidity of the gas streams in the fuel cell stack 8.2 Mass and Charge Transfer and Pressure Drop Portion of the Overall Fuel Cell Stack Model The numerical code was further developed to incorporate the effects of mass and charge transfer and pressure drop in order to study the fu el cell behavior. The mass and charge balances, and pressure drop mathema tical equations for each layer are solved simultaneously in MATLAB. An array was created for the molar flow rates, mole fractions, concentrations, humidities, pre ssure drops, resulting pressures, hydraulic diameters, Reynold’s numbers and potenti als for each node or layer. The stack dimensions and other parameters used in th e simulations are summarized in Appendix A. In the code, lines 430 1592, the mole frac tions, molar flow rate s, concentrations and humidity’s are calculated fo r the nodes for each fuel cell layer. The pressure drops for each layer are calculated on lines 1595 to 2316. The velocities, hydraulic diameters,

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206 Reynold’s numbers, friction fact ors and the change in pressu re with respect to x are calculated for each node. The charge transfer portion begins on line 2769, and includes the calculation of current dens ities in the anode and cathode catalyst layer, the potential losses due to activation polarization, ohmi c polarization and concentration losses. A charge balance is also included for each layer. A summary of the mass and pressure portion of the code is shown in Figure 8.17. Figure 8.17. Mass transfer and pressure drop portion of the model 8.2.1 Total Mass Flow Rates The molar flow rate of hydrogen and oxyge n through the end plates, terminals and gasket layers are the largest due to th e large pipe diameter. The hydrogen and oxygen

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207 flow rate decreases th rough the flow channels due to th e branching of th e inlet channel into several channels. The hydrogen and oxygen flowrate changes as it goes through the GDL, and catalyst layers due to the pore si zes. The flow through the membrane in the base case is just due to permeability and wate r concentration. The total mass flow rates, for a 20 cell fuel cell stack, are shown in Figure 8.18.

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208 (a) (b) 0 0.05 0.1 0.15 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 t = 300 Molar flow rate (mol/s)position (m) Figure 8.18. Mass flow rates through a 20 cell fuel cell stack, (a) surface plot of the mass flow rate distribution as a function of position and time, (b ) mass flow distribution at t =300 s

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209 Figure 8.19 compares the mass flow rates of the fuel cell layers at 1, 2 and 3 atm. The flow rate decreases from the flow field la yers (1 and 7) because only a small fraction of the total flow rate enters the GDL layers. The remainder of the flow rate exits the flow field plates to the manifold. As mentioned previously, the decrease and increase of the mass flow rates in the GDL, catalyst and membra ne layers is due to the changes in pore sizes of each layer. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 1234567 Fuel cell layer No.Molar flow rate (mol/s) Total molar flow, 3 atm Total molar flow, 2 atm Total molar flow, 1 atm Figure 8.19. Comparison of total mass flow rates with pressures of 1, 2 and 3 atm 8.2.2 Pressures Through Fuel Cell Stack The pressures of hydrogen and oxygen thr ough the layers of a 20 cell fuel cell stack, and a single cell fuel ce ll stack for the base case shown in Appendix L, are shown in Figures 8.20 and 8.21 for T = 298 K, P = 3 atm with a current density of 1.0 A/cm2.

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210 The pressure drop of hydrogen and oxygen through the end plates, terminals, and gasket layers is minimal due to the short pipe le ngth. The hydrogen and oxygen pressure drop is substantial through the flow channels due to the small channel diam eter, channel length, number of bends, and number of channels The hydrogen and oxygen pressure decreases even further as the gases pass through the GDL and catalyst layers due to the small pore sizes. The pressure in the membrane is dependent upon the pressure at the anode catalyst/membrane and cathode catalyst/mem brane interfaces, and displays a similar distribution as previously shown in Figure 5.9.

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211 (a) (b) 0 0.05 0.1 0.15 0.2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 x 105 t = 300 position (x)Pressure (Pa) Figure 8.20. Pressure distribution through a 20 cell fuel cell stack, (a) surface plot of the pressure distribution through a 20 cell st ack as a function of position and time, (b) pressure distribution at t = 300 s

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212 (a) 0 100 200 300 0 2 4 6 8 x 10-3 3.2 3.25 3.3 3.35 3.4 3.45 x 105 time (s) Pressure distribution through fuel cell stack (Pa) position (m)Pressure (Pa) 0 1 2 3 4 5 6 7 x 10-3 3.2 3.25 3.3 3.35 3.4 3.45 x 105 t = 296.9697 position (m)Pressure (Pa) Figure 8.21. Pressure distribution through a single cell fuel cell stack, (a) surface plot of the pressure distribution through a singl e cell stack as a function of position and time, (b) pressure distribution at t = 300 s

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213 Figures 8.22 and 8.23 compare the pressure drop for the base case fuel cell stack at pressures of 1, 2 and 3 atm. It is wellknown that higher pressure s lead to higher fuel cell performance. However, there is a greate r effect on fuel cell performance between 1 and 2 atm than between 2 and 3 atm. This e ffect becomes more obvious at higher current densities because the higher pressures of th e reactants will bring more water into the channel. As a result, the membrane is better hydrated and the speed of chemical reaction increases. Therefore, the fuel cell can gene rate more power under the high flow pressure. However, whether to use the high pressure in a fuel cell system depends on the tradeoff between fuel cell performance improvement, and cost to store and distribute the compressed gas.

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214 (b) (c) Figure 8.22. Pressure distribution for a 20 cell fuel cell stack with in itial pressure of (a) 3 atm, (b) 2 atm, and (c) 1 atm

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215 0 0.5 1 1.5 2 2.5 3 3.5 4 1234567 Fuel cell layer No.Pressure (Pa, x 10^5) P = 3 atm P = 2 atm P = 1 atm Figure 8.23. Pressure distribution through a single cell fuel cell stack 8.2.3 Velocity Distribution Through the Fuel Cell Stack The velocities of hydrogen and oxygen in the end plate layers are the largest due to the pressure and pipe diameter. The hydrogen and oxygen velocity increases in the flow channels due to the decrease in flow diameter. When the molar flow reaches the outlet of the flow channels, the velocity then decreases because the outlet channel of the flow field plate widens. The hydrogen and oxygen velocity is slightly higher as it goes into the GDL. The velocity leaving the GDL an d catalyst layers increase again due to the small pore diameters in these layers. The ve locity through the membrane varies based upon the pressure differential and flow rate. Figu re 8.24 illustrates the velocity profile of a 20 cell stack, and Figure 8.25 shows a surface plot of the velocity profile in the flow field, gas diffusion, catalyst and memb rane layers of a single fuel cell.

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216 (a) (b) 0 0.05 0.1 0.15 0.2 -1 0 1 2 3 4 t = 300 position (m)Velocity (m/s) Figure 8.24. Velocity distribution through a 20 cell fuel cell stack, (a) surface plot of the velocity distribution through a 20 cell st ack as a function of position and time, (b) velocity distribution at t = 300 s

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217 (a) 0 2 4 6 8 10 0 2 4 6 x 10-3 0 2 4 6 8 10 time (s) Velocity through fuel cell stack (m/s) position (m)Velocity (m/s) (b) 0 1 2 3 4 5 6 x 10-3 0 2 4 6 8 10 t = 9.49495 9.59596 9.69697 9.79798 9.89899 1 0 position (m)Velocity (m/s) Figure 8.25. Velocity profile in the flow field, gas diffusion, catalyst and membrane layers of a single fuel cell, (a ) surface plot as a function of position and time, (b) velocity distribution at t = 10 s

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218 The velocity in the gas diffusion layers is approximately two orders of magnitude smaller than in the gas flow channels. The velocity in the diffusion media is smaller than in the flow channels because it has a much higher resistance to flow due to the small pores in this layer. The change in porosity fr om the GDL (0.55) to the catalyst layer (3.0) results in an increase in velocity. If the de nsity of the gas phase is constant across the interface between the two layers, the velocities can be related by [117]: catalyst GDL layer GDL in the magnitude Velocity layer catalyst in the magnitude Velocity (237) Therefore, the magnitude of velocity in the ca talyst layer should be about twice the value of that in the electrode backing layer. This is in agreement with the results shown in Figures 8.26 and 8.27. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1234567 Fuel cell layer No.Velocity (m/s) P = 3 atm P = 2 atm P = 1 atm Figure 8.26. Velocity of a single cell

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219 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Anode GDLAnode catalyst Polymer membrane Cathode catalyst Cathode GDL Fuel cell layerVelocity (m/s) P = 3 atm P = 2 atm P = 1 atm Figure 8.27. Velocity of the MEA layers at different pressures 8.2.4 Hydrogen Transport For the gas phase species, it is assumed th at convection is the dominant mode of mass transport in the end plate, terminal, gask et and flow field laye rs, and diffusion is the dominant mode of transport in the GDL, cat alyst and membrane laye rs. The direction of diffusional flux generally moves from the anode flow field to the anode catalyst layer, where the hydrogen is consumed. However, some of the hydrogen diffusional flux also flows in the opposite direction than the total hydrogen mass flux, and the convective velocity. Since the electrochemical reaction re quires hydrogen to be supplied to the anode catalyst layer, diffusion hinders the electro-che mical reactions. At low pressures, such as 1 atm, the mole fractions of hydrogen begi n to decrease in the anode GDL due to hydrogen consumption in the anode catalyst layer. Therefore, it seems as though the

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220 amount of hydrogen diffusing into the catalyst layer could be limiting the electrochemical reactions. As the pressure is increased to 3 atm, the hydroge n mole fraction in the anode flow field, GDL and catalyst layer is consis tent, which indicates that enough hydrogen is being supplied to the anode catalyst layer. Figure 8.28 illustrates the increase in mole fraction from the gas flow channe ls to the anode catalyst layer. 0.986 0.99 0.994 0.998 Anode flow fieldAnode GDLAnode catalystHydrogen mole fraction P = 1 atm P = 2 atm P = 3 atm Figure 8.28. Hydrogen mole fraction in the anode gas flow channel, electrode backing layer and catalyst layer Figure 8.29 again shows that there is a significant decrease of hydrogen mole fraction in the anode catalyst layer to the anode gas flow channel at P = 1 atm. The mole fraction in the anode gas flow channels stay s nearly constant, and increases at the GDL/ flow channel interface. As the current dens ity increases, there was no noticeable change in hydrogen mole fraction as shown by Figure 8.29.

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221 0.986 0.99 0.994 0.998 Anode flow fieldAnode GDLAnode catalystHydrogen mole fraction 0.1 A/cm2 1.0 A/cm2 Figure 8.29. Hydrogen mole fraction due to the vary ing current density in the anode gas flow channel, GDL layer and catalyst layer The concentration of hydrogen also increa ses as shown in Figures 8.30 and 8.31. Although hydrogen is consumed, the mole fraction increases. This increase is due to the electrochemical reactions since for every mole of hydrogen that is removed; two moles of water are also removed. The hydrogen mole fraction (2Hx) will be positive if it is greater than 0.053 according to the following equation: O H H H HM M M x2 2 2 22 (238)

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222 0 5 10 15 20 25 30 35 40 45 1234567 Fuel cell layer No.Concentration (mol/cm3) Hydrogen, T = 353 K Hydrogen, T = 333 K Hydrogen, T = 303 K Oxygen, T = 353 K Oxygen, T = 333 K Oxygen, T = 303 K Figure 8.30. The concentration of hydr ogen in the anode gas fl ow channel, electrode backing layer and catalyst layer 0 20 40 60 80 100 120 140 160 1234567 Fuel cell layer No.Concentration (mol/cm3) Hydrogen, 3 atm Hydrogen, 2 atm Hydrogen, 1 atm Oxygen, 3 atm Oxygen, 2 atm Oxygen, 1 atm Figure 8.31. Hydrogen and oxygen concentration in the MEA fuel cell layers

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2238.2.5 Oxygen Transport In the cathode catalyst layer, water is produced, and oxygen is consumed. The oxygen travels from the flow channel to the cath ode catalyst layer. It is assumed that the transport of oxygen from the gas flow cha nnels to the reaction sites in the cathode catalyst layer is by diffusion. The oxygen mole fr action at a pressure of 1 atm is lower in the flow field and cathode catalyst layers as shown in Figure 8.32. As the pressure increases to 2 and 3 atm, the oxygen mole frac tion begins to become more uniform in the cathode flow field layer, GDL and catalyst la yers. This again illustrates that with lower pressure, the decrease in oxygen concentrati on hinders the electroch emical reaction -which is proportional to oxygen concentration. 0.9 0.92 0.94 0.96 0.98 1 Cathode catalystCathode GDLCathode flow fieldOxygen mole fraction P = 1 atm P = 2 atm P = 3 atm Figure 8.32. The mole fraction of oxygen in the anode gas flow channel, gas diffusion layer and catalyst layer

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224 Figure 8.33 illustrates the mole fraction of oxygen with varying current density with P = 1 atm and T = 298 K. As expected the lowest current density of 0.1 A/cm2 has the highest oxygen concentration in the cathode catalyst layer, a nd the highest current density of 1.0 A/cm2 has the lowest oxygen mole frac tion in the cathode catalyst layer due to the greater consumption of oxygen. 0.9 0.92 0.94 0.96 0.98 1 Cathode catalystCathode GDLCathode flow fieldOxygen mole fraction 0.1 A/cm2 0.6 A/cm2 1.0 A/cm2 Figure 8.33. The mole fraction of oxygen in the cathode gas flow channel, gas diffusion layer and catalyst layer 8.2.6 Water Transport Water exists in both the ga s and liquid phase throughout the fuel cell stack. Due to the electrochemical reactions, water is consum ed in the anode catalys t layer and produced in the cathode catalyst layer. The water in th e anode catalyst layer is primarily from the

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225 humidity in the hydrogen inle t stream. The water flux in the polymer membrane is primarily due to water generated by the cathode catalyst layer. Figure 8.34 shows the mole fraction of wa ter for the flow field and MEA layers with varying current densities at T = 298 K a nd P = 1 atm. As seen experimentally, the largest amount of water (mole fraction of 0.092) is with the highest cu rrent density of 1.0 A/cm2, and the mole fraction of water decreases with the decrease in current density. This is due to the fact that a great er amount of water is generate d with a higher current density according to Faraday’s law (equation 133). 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1234567 Fuel cell layer No.Water mole fraction 0.1 A/cm2 0.6 A/cm2 1.0 A/cm2 Figure 8.34. Effect of current dens ity on water mole fraction Figure 8.35 shows the water mole frac tion over a total tim e of 1200 s with a current density of 1.0 A.cm2, T = 303 K and P = 3 atm. At time = 10 s, the water mole fraction is 0.011, and the mole fraction increa ses to 0.019 at t = 1200 s. The water mole

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226 fraction is also increasing in the anode fl ow field and GDL, and the cathode flow field layers. This is due to the water traveling from the cathode catalsyt layer and accumulating in the flow field and GDL layers. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 1234567 Fuel cell layer No.Water mole fraction 10 s 60 s 300 s 600 s 1200 s Figure 8.35. Effect of time on water mole fraction Approximately 25% of the water consumed by the anode catalyst layer reaction comes from the cathode catalyst layer. Ideally, the wa ter produced in the cathode catalyst layer should provide 100% of the water needed by the anode catalyst layer since this would eliminate the need to have fully hydrated reactants. However, in practice, the reactants must be fully humidified in order to adequately hydrate the membrane. Water concentration as a function of time is illustrated in Figure 8.36.

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227 (a) 0 0.1 0.2 0 50 100 0 0.5 1 1.5 x t = 58.1818 58.7879 59.3939 60 t Water concentration (mol/cm3) (b) 0 0.1 0.2 0 200 400 600 0 0.5 1 1.5 x t = 600 t Water Concentration (mol/cm3) Figure 8.36. Water concentration as a function of tim e at 3 atm and i = 1 A/cm2, (a) 60 s and (b) 600 s. Figure 8.37 shows the distribution of wate r concentration at the different inlet flow temperatures at P = 1 atm and a current density of 1.0 A/cm2. It is found that the local water activities in the membrane are less than 1.0 when the inlet flow temperatures

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228 are 303 and 313 K. When the stack and gas temp erature is lower, the saturation pressure will drop and the water activities will incr ease. For the cases with higher inlet temperature, such as, 333 and 343 K, the gase s carry more water vapor into the channel, and the water activity in the membrane will be greater than 1.0. When the water activities are large, the membrane conductivity change s will be small. This is because the membrane is well hydrated, and the speed of electrochemical reacti on is faster. As a result, more oxygen is consumed and the par tial pressure of oxyge n decreases quickly. 0 0.5 1 1.5 2 2.5 3 3.5 4 1234567 Fuel cell layer No.Water concentration (mol/cm3) T = 353 K T = 333 K T = 303 K Figure 8.37. The concentration of water in the anode gas flow channel, electrode backing layer and catalyst layer

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229 Figure 8.38 illustrates the water concentrati on in each fuel cell layer with varying pressures. The water transport across the pol ymer electrolyte layer is driven by a water concentration gradient. The amount of water contained in the gas phase and electrolyte can be characterized by the membrane activity and water uptake, as described in Chapter 5. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1234567 Fuel cell layer No.Water concentration (mol/cm3) Water, 3 atm Water, 2 atm Water, 1 atm Figure 8.38. Water concentration as a function of pressure Figure 8.39 shows the hydrogen, oxygen and water concentration at 3 atm at T = 298 K, and the current density is 0.1 A/cm2. The hydrogen and oxygen concentration decreases slightly from the flow field to the GDL layers, and then again slightly from the GDL to the catalyst layer. The water mole fraction increases from Figure 8.39 to Figure

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230 8.40 from approximately 0.1 mol/cm3 to 1.2 mol/cm3 with a current density of 0.1 A/cm2 to 1.0 A/cm2. 0 0.1 0.2 0 200 400 0 50 100 150 x t = 300 t Hydrogen concentration (mol/cm3) 0 0.1 0.2 0 200 400 0 50 100 150 x t = 300 t Oxygen concentration (mol/cm3) 0 0.1 0.2 0 200 400 0 0.05 0.1 0.15 0.2 x t = 300 t Water concentration (mol/cm3) Figure 8.39. Hydrogen, oxygen and water concentration at 3 atm, i = 0.1 A/cm2

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231 0 0.1 0.2 0 200 400 0 50 100 150 x t = 300 t Hydrogen Concentration (mol/cm3) 0 0.1 0.2 0 200 400 0 50 100 150 x t = 300 t Oxygen concentration (mol/cm3) 0 0.1 0.2 0 200 400 0 0.5 1 1.5 x t = 300 t Water concentration (mol/cm3) Figure 8.40. Hydrogen, oxygen and water concentration at 3 atm, i = 1 A/cm2 8.3 Membrane Portion of the Overall Fuel Cell Stack Model The membrane is treated differently than the other layers in the numerical code because the transport phenomena are different due to the membrane properties. Lines 1464 – 1592 calculate the mass flow through the membrane, which includes the calculation of the amount of water in the memb rane (water activity), the water uptake, the amount of hydrogen and oxygen that diffuse d into the membrane, and the hydrogen,

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232 oxygen and water concentrations. The pressure and velocity is calculated in 2301 2306, and the membrane temperature is calculated on lines 2765 2762. The potential is based on water content, and is calculated on lin es 2827 2848. A summary of the membrane portion of the code is shown in Figure 8.41. Specify x coordinates based upon the uniform distribution of nodes from 0 to the membrane hickness. Constants (1) Number of slices (2) Thickness (m) (3) Temperature at the interface (K) (4) Temperature gradient (K) (5) Potential (V) (6) Pressure (Pa) (7) Water concentration (mol/cm3) (8) Mesh size (9) ideal gas constant (J/K-mol) (10) Molecular weight of hydrogen (kg/mol) (11) Mass conservation for protons (mol/m3) (12) Molecular weight of water (kg/ mol) (13) Viscosity of hydrogen (kg/ms) (14) Viscosity of water (kg/ms) (15) Gravitational constant (m/s2) (16) Degrees (17) Current density (A/cm2) (18) Faradays constant (C/mol) (19) Density of membrane (kg/m3) (20) Specific heat of membrane (J/ kgK) (21) Specific heat of water (J/kgK) (22) Water flux (23) Proton Diffusivity (cm2/s) Inputs: 1, 2 Boundary conditions Inputs: 3, 4, 5, 6, 7, 8 Call memmode function to calculate dydx Initialize left hand side (lhs) mass matrix, and right hand side (rhs) non differential terms. Calculate the initial molar velocity (Darcys Law) Calculate proton potential, energy conservation, diffusive flux for water & hydrogen and velocity of the mixture. Calculate initial conditions (water activity, water uptake and conductivity) Combine into rate change of y (invert mass matrix): dydx = lhs/rhs Calculate the change in T, P, water concentration and potential using bvp4c (ODE solver based upon Runge-Kutta (4,5) formula) Inputs: 6, 7, 10, 11, 12, 15, 16 Inputs: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 Inputs: 3, 7, 9 meminit function perform linear interpolation between boundary values memboundary function Boundary condition residual Inputs: 3, 4, 5, 6, 7 Inputs: 3, 4, 5, 6, 7 Figure 8.41. Flow chart of membrane model

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2338.3.1 Effect of Current Density As the current density increases, protons navigate from the anode where they are produced, to the cathode where they are cons umed. As protons migrate, water molecules are dragged through the membrane. The concen tration in the membrane changes with time with an applied current density. The solid lines show the water concentration with the specified applied current density. The water concentration on the anode side becomes lower with increased current density. The num ber of water molecules on the cathode side also is higher with the increased current dens ity. In addition, the overall water content in the membrane is lower with higher current dens ity due to an increase d number of protons dragging more water molecules out of the me mbrane. These phenomena are illustrated in Figure 8.42.

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234 3.4 3.6 3.8 4 4.2 4.4 4.6 00.0560.1130.1690.2260.2820.3390.3950.4520.508 Membrane position (x e-4 m)Water concentration (mol/cm3) i = 0.1 A/cm2 i = 0.6 A/cm2 i = 1.0 A/cm2 Figure 8.42. Effect of current density on wa ter concentration (a) 0.1 A/cm2 (b) 0.9 A/cm2 (c) comparison of 0.1 A/cm2, 0.5 A/cm2 and 0.9 cm2 8.3.2 Effect of Temperature Figure 8.43 shows how the concentrati on varies with temperature in the membrane. As the membrane temperature incr eases, the water concentration across the membrane becomes more uniform – even with high current densities. This indicates the membrane conductivity is better with increase d temperatures – as long as the membrane can maintain adequate hydration. The ohmic heating results in a very small temperature increase across the membrane from the ini tial conditions. As the temperatures become higher, convective transport e ffects begin to dominate. The ohmic heating still heats up the membrane slightly, however, the convect ive effects dominate and the temperature decreases across the membrane.

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235 3.4 3.6 3.8 4 4.2 4.4 4.6 00.060.110.170.230.280.340.40.450.51 Membrane position (x e-4 m)Water concentration (mol/cm3) i = 1.0 A/cm2, T = 353 K i = 1.0 A/cm2, T = 333 K i = 1.0 A/cm2, T = 303 K i = 0.1 A/cm2, T = 303 K i = 0.1 A/cm2, T = 333 K i = 0.1 A/cm2, T = 353 K Figure 8.43. Effect of temperature on water concentration (a) 353 K (b) 323 K (c) comparison of 343 K, 348 K, 353 K and 358 K 8.3.3 Effect of Water Activity at th e Catalyst/Membrane Interfaces Figures 8.44 ad 8.45 illustrates the effect of water activity at the catalyst layer/ membrane interfaces with the water concentr ation across the polymer membrane. If the water activity is 1.0 at the catalyst/membra ne interface, the water concentration through the membrane is very uniform. As the water activity at the cathode catalyst interface decreases, the water concentr ation on the anode side decr ease, which means that the membrane conductivity decreases. This same phenomena resulted regard less of the initial membrane concentration.

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236 0.5 0.6 0.7 0.8 0.9 1 1.10 0 0 5 64 0.1129 0 1 6 93 0.2258 0.28 2 2 0 3 3 87 0.3951 0 4 5 16 0.50 8Membrane position (x e-4 m)Water concentration (mol/cm3) a = 1 a = 0.5 a = 0.1 Figure 8.44. Water concentration in the membrane with varying water activity at the membrane/cathode catalyst layer interface 4 4.1 4.2 4.3 4.4 4.5 4.60 0. 0 564 0.1129 0 .1 693 0. 2 258 0. 2 822 0. 3 387 0.3 9 51 0. 4 516 0. 50 8Membrane position (x e-4 m)Water concentration (mol/cm3) a = 1 a = 0.5 a = 0.1 Figure 8.45. Water concentration in the membrane with varying water activity at the membrane/cathode catalyst layer interface

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2378.4 Electron Transport Electrons are produced in the anode catal yst layer, consumed in the cathode catalyst layer, and transported in the solid phase The electronic current density is zero in the membrane because it is electronically in sulative. The current density in the anode catalyst layer is much faster than the reac tion at the cathode catalyst layer. Since the oxygen reduction reaction is slower, it requires a larger surface area for th e reaction than the cathode catalyst layer. The solid potential dist ribution for t = 300 s is illustrated in Figure 8.46. 0 0.05 0.1 0.15 0.2 0.25 1234567 Fuel cell layer No.Potential (V) i = 0.1 A/cm2 i = 0.6 A/cm2 i = 1.0 A/cm2 Figure 8.46. The solid phase potential in the PEM fuel cell The electronic current density is relatively constant in the gas flow channels and gas diffusion layers. The potenti al varies in each layer ba sed upon the area of the solid

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238 portion of the layer, such as the channel and void space, the thickness and the intrinsic resistivity of the layer. 8.5 Overall Fuel Cell Model Validation A 16 cm2 single cell, air breathing fuel cell st ack was used for additional fuel cell I-V tests. Five-layered MEAs are used, wh ich are composed of Nafion 112, GDL of carbon cloth material and 1 mg/cm2 of Pt loading on both anode and cathode. Cell performance tests are conducted with 0.5 to 1.0 standard cubic centimeter per minute (SCCM) of hydrogen from an electrolyzer, with no additional humidification. All tests are taken at 25 C and ambient pressure. I–V curves of these cell performance tests are shown in Figure 8.47, and compar ed with the model results.

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239 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Cell Current (A/cm2)Voltage (V) Model Experiment Figure 8.47. Comparison between fuel cell model and experiments at 298 K and 1 bar Several more IV tests were performed with different fuel cell stack temperatures. As shown in the Figure 8.48, the model results agree well with the actual results obtained with the experiments.

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240 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Cell Current (A/cm2)Voltage (V) 327 K model 327 K experiment 324 K model 324 K experiment 315 K model 315 K experiment 306 K model 306 K experiment Figure 8.48. Comparison between fuel cell model and experiments at various temperatures The overall fuel cell model for predic ting electrochemical performance was created and validated using a 16 cm2 fuel cell stack. A numeri cal model included energy, mass and charge balances for each fuel cell layer. In order to precisely model the electrochemical reactions, an agglomerate cat alyst layer was included in the model using porous electrode equations. In addition, an empirical membrane model correlating water content and conductivity was inte grated into the model. The experimental validation consisted of experimentally examining the IV curves of the PEM fuel cell stack.

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241 9 SUMMARY AND FUTURE WORK The PEM fuel cell consists of several layers where several processes occur simultaneously in the same layer. In the flow field plates, reactant gas flows in the channels, while current flows in the solid portion of the layer. The gas diffusion media also have flow through the porous media, while transporting electrons through the material. The acidic polymer electrolyte layer has both positive ions and water flowing in through the polymer. Like the gas diffusion medi a, the catalyst laye r had reactant gases flowing through the porous structure, while transporting electrons to the gas diffusion layer. In addition, the electro-chemical re actions convert the react ants directly into electrical energy. Heat and water are also produced in this layer. The pro cesses that occur in thee layers are complicat ed by the thinness of the la yers, high temperatures and pressures, and the presence of two phases. Th e direct measurement of these properties are currently unavailable, therefore, mathematical modeling is needed to help provide insight into the phenomena that is occurring within the fuel cell. There has been an increased interest in modeling fuel cel ls during the last decade. A lthough these model are very helpful in trying to understand th e transport phenomena that is occurring in the fuel cell, it is difficult to understand how all of th e operating variables, such as pressure, temperature, humidity and load requires ar e affecting the transport phenomena within the fuel cell, and how these transport pro cesses can be improved with new designs.

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242 When considering the formulation of this model, the fuel cell was first considered to be composed of several phases: a multi-component gas phase which includes hydrogen, oxygen, water and sometimes nitrogen and carbon dioxide or carbon monoxide. The liquid phase consists of water, which is produced at the cathode catalyst layer, and is also entering the fuel cell in the reactant streams, the so lid portion consists of the layer materials: the end plate, gasket, terminal, gas diffusion media material, catalyst layer material – which is made of carbon and platinum and the polymer electrolyte membrane. The conservation of mass, mome ntum, energy and charge transport was applied to each node of each la yer in the form of traditiona l engineering mass, energy and charge balances. The effect of pressure drop was also included in the model. To accomplish the objectives described in th is dissertation, detailed models were required for each of the various fuel cell laye rs. The model developed for this dissertation is complex enough to handle all of the im portant governing phenomena, but remains simple enough to run in a realistic amount of time. Part of the overall model included a detailed model of the membrane which accoun ts for many of the effects experimentally observed. It bridges the gap of many models currently in the literature, and allows one to understand how all of the fuel cell parameters a ffect each other. In this research, both a model for the single PEM fuel cell and th e PEM fuel cell stack was developed in MATLAB. The solution of the numerical model emphasized many of the important processes that occur within the PEM fuel cell. Due to the nature of the electrochemical reactions, the hydrogen and water were removed from the gas phase at a ratio of 1:2, which resulted in the hydrogen concentration increa sing in the catalyst layers although it was

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243 being consumed. Water was transported through a ll of the regions of the fuel cell because it is present in both the gas and liquid phase Water was consumed in the anode catalyst layer and produced in the cathode catalyst layer. Most of the water consumed in the anode catalyst layer was obtained from the a node gas flow channel, while a large portion of the water produced in the cathode catal yst layer exited the fuel cell through the cathode gas flow channel. However, some of the water produced in the cathode catalyst layer traveled through the polym er electrolyte layer. The re lative humidity in the gas phase on the cathode side of the cell was great er than 100%. On the anode side of the cell, the relative humidity was below 100% in the catalyst layers although the reactant flows through the anode gas flow channels were fully humidified. Therefore, these simulations suggest that both liquid water flooding and membrane dehydration could occur simultaneously. The reaction rate distributions in the anode and cathode catalyst layers illustrate the importance of the mass tran sport on the conversion of chem ical energy to electrical energy in the fuel cell. In the cathode catalys t layer, the reactant gas transport, and the amount of water produced affect ed the reaction rate. At the anode-side, hydrogen seemed to be aided by convection, whic h influenced the reaction rate. The higher the current density, the more wa ter was driven from the anode to the cathode, and out of the membrane. A positive pressure gradient from the anode to the cathode could be used to drive water toward th e anode side – which is more likely to dry out. The effect of the water flux into and out of the membrane illustrated that if too much water flows into the membrane, flooding may occur, whereas, if too much water is removed from the membrane, drying may o ccur. These results seem obvious, but the

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244 model allows these phenomena and their effect on temperature and voltage to be studied, and quantified. The model is also capable of predicting transient water, concentration voltage and temperature profiles for transien t boundary conditions. This capability will prove useful when attempting to develop a co ntrol strategy for the fuel cell, and when investigating highly transien t processes such as fuel cell startup on a vehicle. The bulk gas phase flow acted to hinder the transport of oxygen from the cathode gas flow channels to the cathode catalyst laye r. As a result, the c oncentration of hydrogen increased in the anode catalyst layer, but decreased in the cathode catalyst layer. Water was transported in both the gas phase and as a liquid phase in the polymer electrolyte. Due to the high conductivity of the solid phase, the potential remained relatively constant in the fuel cell layers. The potential in the electrolyte is influenced purely by the water content of the membrane. Therefore, it is important that wa ter concentration and ion transport is coupled in the polymer model. Since the humidification of both the anode and cathode sides of the PEM fuel cell are important, the temperature throughout the fu el cell is also very important. Injecting liquid water into the anode channel inle t may be useful for improving fuel cell performance improvement. The optimal amount of liquid water could be determined by using the model and running simulations. Heat can be either added or removed from the fuel cell stack by adjusting the temperature of the reactant gases. However, the fuel cell engineer must take into consideration what the additional equipment cost will be for cooling or heating the fuel cell in this ma nner. Decreasing the cooling temperature may be helpful in improving fuel cell performa nce. For many stack designs, it may be advantageous to thermally isolate the fuel cell stack end plates due to the loss of heat at

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245 this point in the fuel cell stack. To obtain a uniform heat distributi on within the fuel cell stack, it may be useful to h eat the bipolar plates, but depending upon the stack design, this may be difficult to implement compared with just heating the end plates. Heating the anode side slightly higher th an the cathode-side may be a good option to ensure uniform heat distribution in a fuel cell stack. The results of this dissertat ion research suggest several areas of future research. For the heat transfer analysis, it is important to consider the heat transfer in 2-D and 3-D to obtain realistic results. Although both th e gaseous and liquid phase of water was studies in this model, there was no relations hip introduced between the two phase for the porous GDL and catalyst layers. One option would be the introduc tion of a simple capillary pressure equation to relate the two phases. In addition, the velocity was calculated for the mixture, but it would be mo re accurate to calcul ate the gas and liquid phase velocity separately. The simulation based on this model can be used to analyze water transport across the membrane, the water phase change effect the pressure varia tion along the channel and the energy balance. It can also be used to predict the characteristics of the flows inside the channel and analyze the factors th at affect the fuel cell performance. The overall simulations demonstrated that optim al performance in PEMFCs is a balance between different phenomena. Optimizati on of the right operating conditions and structural properties depends upon the quantific ation of this interp lay. The optimization that can be accomplished with the mode l are almost endless and depend on the phenomenon being studied.

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259 [111] Li, W., et al. (2003). Preparation and characterization of multiwalled carbon nanotube-supported platinum for cathode catal ysts of direct methanol fuel cells. Journal of Physical Chemistry B107:6292 – 6299. [112] Lu, Q., Wang, C.Y., Yen, T.J. a nd Zhang, X. (2004). Development and Characterization of a Silicon-based Micro Direct Methanol Fuel Cell. Electrochimica Acta 49:821 – 828. [113] Lee, S., Chen, Y., Huang,C. (2005). Elec troforming of metallic bipolar plates with micro-featured flow field. Journal of Power Sources 145:369 – 375. [114] Agarwal, R., Samson, S., Kedia, S. and Bhansali, S. (2007). Fabrication of Integrated Vertical Mirror Surfaces and Transparent Window for Packaging MEMS Devices. Journal of Microel ectromechical Systems, S. 16(1):122-129. [115] Khandelwal, M., Lee, S. and Mench, M.M.(2007). One-dimensional thermal model of cold-start in a polymer electrolyte fuel cell stack. Journal of Power Sources, 172:816 – 830. [116] Shan, Y. and Choe, S.Y. (2006). Modeli ng and simulation of a PEM fuel cell stack considering temperature effects. Journal of Power Sources, 158:274 – 286. [117] Baschuk, J. J. (2006). Comprehensive, consistent and systematic approach to the mathematical modeli ng of PEM fuel cells [PhD. Dissertation]. University of Waterloo:Waterloo, ON.

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260 APPENDICES

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261 Appendix A Fuel Cell Layer Parameters Used for Model Table A.1 Parameters used for the end plate layers Variable Notation Description Value Units Plate area Aaend End plate area [118] 0.007225 m2 Plate width End plate width [118] 0.085 m Material N/A Clear PVC [118] N/A N/A Thickness thickaen d Thickness [118] 0.01 m Conductivity kaen d Conductivity [119] 0.32 W/mK Density a_end Density [120] 1740 kg/m3 Heat Capacity cpaen d Heat Capacity [120] 1460 J/kgK Specific Resistance res Specific Resistance 0 Ohmm Coolant radius r Inlet channel radius [118] 0.002 m Coolant length L Channel length [118] 0.01 m Coolant crosssectional area Ac Channel cross-sectional area [118] r2 = 1.256e-5 m2 Coolant perimeter Pcs Channel perimeter [118] 2 r = 0.01256 m Reactant channel radius r Inlet channel radius 0.004 m Reactant channel length L Channel length 0.01 m Reactant channel cross-sectional area Ac Channel cross-sectional area r2 = 5.024e-5 m2 Reactant channel perimeter Pcs Channel perimeter 2 r = 0.02512 m This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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262 Appendix A (Continued) Table A.2 Parameters used for the anode end plate Variable Notation Description Value Units H2 Temperature in HT_ 2 Initial hydrogen temperature 298 K Volumetric flow rate in Hv_ 2 Volumetric flow rate per cell [118] 1.25e-7 m3/sec Humidity in H_ 2 Humidity 1 N/A Pressure in HP_ 2 Hydrogen pressure 344,737.864Pa Hydrogen density in H_ 2 Hydrogen density @ room temp [120] 0.08988 kg/m3 Hydrogen molecular weight 2Hmw Hydrogen molecular weight 0.0020159 kg/mol Hydrogen viscosity in Hmu_ 2 Hydrogen viscosity 8.76e-6 Pa-s Thermal conductivity in Hk_ 2 Hydrogen thermal conductivity [120] 0.165 W/mK Specific heat capacity in Hcp_ 2 Hydrogen specific heat capacity [120] 14,160 J/kg-K This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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263 Appendix A (Continued) Table A.3 Parameters used for the cathode end plate Variable NotationDescription Value Units Oxidant Temperature in OT_ 2 Initial oxygen temperature 298 K Volumetric flow rate in Ov_ 2 Volumetric flow rate [118] 1.25e-7 m3/sec Humidity in O_ 2 Humidity 1 N/A Pressure in OP_ 2 Oxygen pressure 344,737.864Pa Hydrogen density in O_ 2 Oxygen density @ room temp 1.429 kg/m3 Hydrogen molecular weight 2Omw Oxygen molecular weight 0.032 kg/mol Hydrogen viscosity in Omu_ 2 Oxygen viscosity 20.18e-6 Pa-s Thermal conductivity in Ok_ 2 Oxygen thermal conductivity 0.024 W/mK Specific heat capacity in Ocp_ 2 Oxygen specific heat capacity 920 J/kg-K This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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264 Appendix A (Continued) Table A.4 Parameters used for the current collector Variable Notation Description Value Units Plate area Aaend Current collector area [118] 0.001289 m2 Material N/A Aluminum 7015 or 6061 N/A N/A Thickness thickaen d Thickness [118] 0.001 m Conductivity kaen d Conductivity 250 W/mK Density a_end Density 2720 kg/m3 Heat Capacity cpaen d Heat Capacity 950 J/kgK Specific Resistance res Specific Resistance 2.65e-8 Ohm-m Channel radius r Inlet channel radius [118] 0.002 m Channel length L Channel length [118] 0.01 m Channel crosssectional area Ac Channel crosssectional area [118] r2 = 1.256e-5 m2 Channel perimeter Pcs Channel perimeter [118] 2 r = 0.01256 m Coolant radius r Inlet channel radius 0.004 m Coolant length L Channel length 0.01 m Coolant channel cross-sectional area Ac Channel crosssectional area r2 = 5.024e-5 m2 Coolant channel perimeter Pcs Channel perimeter 2 r = 0.02512 m This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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265 Appendix A (Continued) Table A.5 Parameters used for the flow field layers Variable Description Value Units Total plate area Total plate area [118] 0.003025 m2 Active plate area Area of plate that has channels [118] 0.001 m2 Material Material [ 118] Graphite N/A Thickness Thickness [118] 0.0033 m Conductivity Conductivity [120] 10 W/mK Density Density [120] 1400 kg/m3 Heat Capacity Heat Capacity [120] 935 J/kgK Specific Resistance Specific Resistance 1e-4 Ohmm Total Length Total Channel length [118] 0.426 m “U” bends in channel No. “U” bends in channel [118] 12 N/A Avg Bends Average No. of “L” bends [118] (includes “U” bends) 24 N/A Length of straight sections Length of straight channel sections [118] 0.0325 m No. of channels No. of channels [118] 13 N/A Channel depth Channel depth [118] 0.0015 m Channel width Channel width [118] 0.0015 m Channel area Channel area lw = 6.39e-004 m2 Perimeter Channel Perimeter 0.00471 m This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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266 Appendix A (Continued) Table A.6 Parameters used for cooling channels Variable Description Value Units Length Total Channel length 0.426 m “U” bends in channel No. “U” bends in channel 12 N/A Avg Bends Average No. of “L” bends (includes “U” bends) 24 N/A Length of straight sections Length of straight channel sections 0.0325 m No. of channels No. of channels 13 N/A Channel depth Channel depth 0.0015 m Channel width Channel width 0.0015 m Channel area Channel area lw = 6.39e-004 m2 Perimeter Channel Perimeter 0.00471 m This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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267 Appendix A (Continued) Table A.7 Parameters used for surroundings Variable Description Value Units Outside Temperature Ambient temperature 298 K Outside pressure Ambient pressure 101,325 Pa Heat coefficient Convective loss from stack 17 W/K Table A.8 Parameters used for hydrogen, oxygen and water Variable Hydrogen Air Water Temperature of gas or liquid going into stack (K) 298 298 298 Humidity of gas or liquid going into stack 0.5 0.5 N/A Pressure of gas going into stack (Pa) 101,325.01 101,325.01 N/A Volumetric flow rate of gas or liquid going into stack (m3/s) 1.7e-8 1e-8 N/A Molecular weight (kg/mol) 1e-3 (8e-3 Viscosity (Pa-s) 8.6e-6 (98.8e-7 kg/ms) 8.6e-6 (8.91e-4 kg/ms) Density (kg/m3) 972 1.3 Thermal Conductivity (W/m-K) 0.165 0.223 Specific heat capacity (J/kg-K) 300 1005 4190

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268 Appendix A (Continued) Table A.9 Parameters used for GDL layer Variable Notation Description Value Units Layer area Aaend GDL area [118] 0.001 m2 Material N/A Car bon cloth N/A N/A Thickness thickaen d Thickness [118] 0.0004 m Conductivity kaen d Conductivity 0.42 W/mK Density a_end Density 450 kg/m3 Heat Capacity cpaen d Heat Capacity 710 J/kgK Specific Resistance res Specific Resistance [121] 1e-4 Ohm-m This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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269 Appendix A (Continued) Table A.10 Parameters used for the catalyst layers Variable Description Value Units Layer area Catalyst area [118] 0.001 m2 Material Platinum/carbon N/A N/A Thickness Thickness [121] 1.5e-3 cm Conductivity Thermal Conductivity [120] 0.27 W/mK Density Density [121] Pt: 21.5 C: 2.0 g/cm3 Heat Capacity Heat Capacity [120] 710 J/kgK Electrical conduc tivity Electrica l conductivity [121] 32.64 S/cm Anode transfer coefficient Anode transfer coefficient 1 Cathode transfer coefficient Cathode transfer coefficient [121] 0.61 Henry’s constant Henry’s constant [121] 3.1664e10 Pa-cm3/mol Platinum loading Platinum loading [121] 0.4 mg/cm2Pt/C ratio Pt/C ratio [121] 0.28 No. of aggregates No. of aggregates [121] 4 Aggregate thickness Aggregate thickness [121] 80 nm Aggregate radius Aggregate radius [121] 1 m Anode entropy change Anode entropy change [120] 0.104 J/mol-K Cathode entropy change Cathode entropy change [120] -326.36 J/mol-K This parameter was an actual measurement from a fuel cell stack, or it was assumed.

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270 Appendix A (Continued) Table A.11 Parameters used for the membrane layer Variable Description Value Units Initial proton concentration Initia l proton concentration 1.2e-3 mol/m3 Proton diffusivity Proton diffu sion coefficient 4.5e-5 cm2/s Density of membrane Density of membrane 2,000 kg/m3 Molecular weight of membraneMolecular weight of membrane1.1 Kg/mol SO3Specific heat of membrane Specifi c heat of membrane 852.63 J/kgK Permeability Permeability of membrane 1.8e-18 m2 Initial saturation ratio Initial saturation ratio 0.02 N/A

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271 Appendix B Diffusion Coefficients Table B.1 Values for the various gas phase coefficients Property Value Hydrogen/water diffusion coefficient (bar cm2/s) [122]: 3342 2,255.146 2470.0 T pDOHH Air/water diffusion coefficient (bar cm2/s) [122]: 3342 2,42.299 2599.0 T pDOHair Oxygen/water diffusion coefficient (bar cm2/s) [122]: 3342 2,283.323 3022.0 T pDOHO

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272 Appendix C Derivation of Overa ll Heat Transfer Coefficient T3q” T2T4 q” T1T5 1/U1/U 1/U k2k3k4 x2 x3 x4 h1h5 Figure C.1. Schematic for overall heat tran sfer coefficient derivation An arbitrary temperature profile, and the th ermal resistances for the heat transfer through three nodes is shown in Figure C1 The nodes that define the resistance boundaries have been placed at the center of each section. This method was selected in order to obtain the average temperature in each node. The parameters k and t are the thermal conductivities and laye r thickness respectively.

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273 Appendix C (Continued) Each control volume has conductive heat transfer with each adjacent node in addition to energy storage: LHSRHSdU qq dt (239) Each term in Equation 239 must be ap proximated. The conduction terms from the adjacent nodes are modeled as: ) ( "1 2 2i iT T x k q (240) ) ( "1 3 3i iT T x k q (241) Add the heat flux equations together: ) ( "3 2 3 3 2 2T T k x k x q (242) The heat overall heat transfer coefficient is: 3 3 2 21k x k x U (243)

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274 Appendix C (Continued) For heat transfer on a node from the surroundings and the next node: ) ( "1 2 2i iT T x k q (244) ) ( "1iT T h q (245) Add the heat flux equations together: ) ( 1 "2 1 2 2T T k x h q (246) The heat overall heat transfer coefficient is: h k x U1 12 2 (247)

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275 Appendix D Control Volume Energy Rate Balance The conservation of energy for a control volume can be introduced by Figure, which shows a system with a fixed quantity of matter, mm that occu pies different regions at time t, and a later time t + t. At time, t, the energy of the system can be expressed as: i i i i cvgz V u m t E t E2 ) ( ) (2 (248) where ) ( t Ecv is the sum of the internal, kinetic and gravitational potential energies of the mass contained within the control volume at time t. The specific energy of the mass, im is i i igz V u 22. In the time interval, t, all mass in region i crosses the control volume boundary, and the system at this time can be expressed as: e e e e cvgz V u m t t E t t E2 ) ( ) (2 (249)

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276 Appendix D (Continued) i igz V ui 22im iz e e egz V u 22em ez Figure D.1. Illustration of the control volume conservation of energy principle The mass and energy within the control volume may have changed over the time interval, and the masses mi and me are not necessarily the same. The closed system energy balance can be applied: WQtEttE )() ( (250) Introducing and the overall energy balance equation: WQgz V umtEgz V umttEi i ii cv e e ee cv 2 )( 2 )(2 2 (251)

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277 Appendix D (Continued) Rearranging: e e eei i ii cv cvgz V umgz V umWQtEttE 2 2 )()(2 2 (252) After dividing each term by th e time interval, and taking the limit of each term as t approaches zero, we obtain: e e eei i ii cvgz V umgz V umWQ dt dE2 22 2 (253) The term dt dEcv, represents the total energy associ ated with the control volume at time, t, and can be written as a volume integral: v v cvdVgz V uedVtE2 )(2 (254) The terms accounting for energy transfers accompanying mass flow and flow work at inlets and outlets can be expr essed as shown in the following form: i A e e e i A i i i vVdAgz V h VdAgz V h WQedV dt d 2 22 2 (255)

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278 Appendix D (Continued) Since all of the kinetic and potential en ergy effects can be ignored, the energy balance can be reduced to: i e e i i ih m h m W Q dt dU (256) The internal energy of the system is the su m of the internal energies of the species in the mixture: ) (i i i iT u m U (257) If the specific heat c, is taken as a constant, then iu can be expressed as: ) (1 i i iT T c u (258) The energy balance of a mixture in a control volume can now be written as: i e e i i i i ih m h m W Q c m c m c m dt d ) ... (2 2 1 1 (259)

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279 Appendix E Energy Balances Around Each Node Energy balances have been define d around each node (control volume). The control volume for the first, last and an arbitrary, internal node is shown in Figure E.1. Figure E.1. Schematic of the PEMFC stack and th e nodes used for model development

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280 Appendix E (Continued) Each control volume has conductive heat transfer with each adjacent node in addition to energy storage: LHSRHSdU qq dt (260) Each term in Equation 258 must be calculat ed. The conduction terms from the adjacent nodes are modeled as: 1 ii LHSkATT q x (261) 1 ii RHSkATT q x (262) where A is the area of the plate. The rate of energy storage is the product of the time rate of change of the nodal temp erature and the thermal ma ss of the control volume: idT dU Axc dtdt (263) Substituting Equations 257 through 260 leads to: 11iiii ikATTkATT dT Axc dtxx (264)

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281 Appendix E (Continued) Solving for the time rate of the temperature change: 11 22 for 2...1i iiidT k TTTiN dtxc (265) The control volumes on the edges must be trea ted separately because they have a smaller volume and experience different energy transfers. The control volume for the node located at the outer surfaces (node N) provides the energy balance: LHSconvdU qq dt (266) or 12NN N f NkATT dT Axc hATT dtx (267) Solving for the time rate of temperature change for node N: 1 222N NNfNdT kh TTTT dtcxxc (268)

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282 Appendix E (Continued) Note that the equations provi de the time rate of change for the temperature of every node given the temperatures of the node s. The energy balance for each control volume provides an equation for the time rate of change of the temperature in terms of the temperature. Therefore, the energy bala nce written for each c ontrol volume has a set of equations for the time rate of change. The temperature of each node is a function both of position (x) and time (t). The index that specifies the node’s position is i where i = 1 corresponds to the adiabatic plate and i = N corresponds to the surface of the pl ate. A second index, j, is added to each nodal temperature in order to indicate the ti me (Ti,j); j = 1 corresponds to the beginning of the simulation and j = M corresponds to the end of the simulation. The total simulation time is divided into M time steps; most of the techniques discussed here will divide the simulation time into time steps of equal duration, t: 1simt M (269) The time associated with any time step is: 1 for 1...jtjtjM (270)

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283 Appendix F Derivation of Mass Transport in the Flow Channels and Through the Porous Media [4] Mass transport in the fuel cell flow struct ures is dominated by convection and the laws of fluid dynamics since the flow channe ls are macroscale (usual ly in millimeters or centimeters). The mass transport of the fuel cell electrodes occur on a microscale and are dominated by diffusion. Convection is stirring or hydrodynamic tran sport. Fluid flow generally occurs because of natural convection, which is th e movement of the fluid due to density gradients. Forced convection is characterized by laminar or turbulent flow and stagnant regions. The convective forces that dominate mass transfer in th e flow channels are imposed by the fuel, while the oxidant flow rates are imposed by the user. High flow rates can ensure a good distribution of reactants but may cause other problems in the fuel cell stack, such as high pressures, fuel cell membrane rupture, and many others. The diffusive forces that occur in the electrode/catalyst layer are shielded from the convective forces in the flow channels. The ve locity of the reactants tends to slow down near the gas diffusion/catalyst layers where th e diffusion regime of the reactants begins. Figure F1 illustrates convective flow in the reactant flow channel and diffusive flow through the gas diffusion and catalyst layers.

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284 Appendix F (Continued) Figure F.1. Fuel cell layers (flow field, gas diffu sion layer, catalyst layer) that have convective and diffusive mass transport F.1. Convective Mass Transport From Flow Channels to Electrode As shown in Figure F.1, the reactant is supplied to the flow channel at a concentration C0, and it is transported from the flow channel to the concentration at the electrode surface Cs through convection. The rate of mass transfer is then: ) (0s m elecC C h A m (271) where Aelec is the electrode surface area, and hm is the mass transfer coefficient.

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285 Appendix F (Continued) The value of hm is dependent upon the channel geometry, the physical properties of species i and j, a nd the wall conditions. Hm can be found from the Sherwood number: h j i mD D Sh h, (272) Sh is the Sherwood number, Dh is the hydraulic diameter, and Dij is the binary diffusion coefficient for species i and j. The Sh erwood number depends upon channel geometry, and can be expressed as: k D h Shh H (273) where Sh = 5.39 for uniform surface mass flux (m = constant)., and Sh = 4.86 for uniform surface concentration (Cs = constant). F.2 Diffusive Mass Transport in Fuel Cell Electrodes As shown in Figure F.1, the diffusive fl ow occurs at the electrode backing and catalyst layer, where the mass transfer occurs at the micro level. The electrochemical reaction in the catalyst layer can lead to reac tant depletion, which can affect fuel cell performance through losses due to reactan t depletion (as predicted by the Nernst equation) and activation losses. To determine the size of the concentration loss, the amount the catalyst layer reactant and product concentrations differ from the bulk values needs to be found.

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286 Appendix F (Continued) The rate of mass transfer by diffusion of the reactants to the catalyst layer (m) can be calculated as shown in equation 274: dx dC D m (274) where D is the bulk diffusion coefficient a nd C is the concentr ation of reactants. Using Fick’s law, the diffusional transport th rough the electrode b acking layer at steadystate is: i s eff elecC C D A m (275) where Ci is the reactant concentration at th e backing layer/catalyst interface, and is the electrode-backing layer thickness, and Deff is the effective diffusion coefficient for the porous electrode backing layer, which is de pendent upon the bulk diffusion coefficient D, and the pore structure. Assuming uniform pore size the backing layer is free from flooding of water or liq uid electrolyte, Deff can be defined as: 2 / 3D Deff (276) where is the electrode porosity. The total resist ance to the transport of the reactant to the reaction sites can be expressed by combining Equations 275 and 276: elec eff elec m iA D A h C C m10 (277) where elec mA h 1 is the resistance to the convective mass transfer, and elec effA D L is the resistance to the diffusional mass transf er through the electro de backing layer.

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287 Appendix F (Continued) When the fuel cell is turned on, it begins producing electricity at a fixed current density i. The reactant and product concentrations in the fu el cell are constant. As soon as the fuel cell begins producing current, the electrochemical re action leads to the depletion of reactants at the catalyst layer. The fl ux of reactants and products will match the consumption/depletion rate of reactants and products at the catalyst layer as described by the following equation: elecA m nF i (278) where i is the fuel cell’s operating current de nsity, F is the Faraday constant, n is the number of electrons transferred per mol of reactant consumed, and m is the rate of mass transfer by diffusion of reactants to the cat alyst layer. Substituting Equation 277 into 278 yields: eff m iD h C C nF i10 (279) The reactant concentration in the backi ng layer/catalyst interf ace is less than the reactant concentration supplied to the flow channels, which depends upon i, and Deff. The higher the current density, the worse the concentration losses will be. These concentration losses can be improved if the diffusion layer thickness is reduced, or the effective diffusivity is increased. The limiting current density of the fuel ce ll is the point where the current density becomes so large the reactant concentration falls to zero. The limiting current density (iL)

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288 Appendix F (Continued) of the fuel cell can be calculated if the minimum concentration at the backing catalyst layer interface is Ci = 0 as follows: eff m LD h C nF i10 (280) When designing a fuel cell, the limiti ng current density can be increased by ensuring that C0, is high, which is accomplished by designing good flow structures to evenly distribute the reacta nts, and ensuring that Deff is large and is small by optimizing fuel cell operating conditions (such as temper ature, pressure), electrode structure and flooding, and diffusion layer thickness. The typical limiting current density is 1 to 10 A/cm2. The fuel cell will not be able to produce a higher current density than its limiting current density. However, other types of losses may limit the fuel cell voltage to ze ro before the limiting current density does. F.3 Convective Mass Transport in Flow Structures Fuel cell flow structures are designed to di stribute reactants across a fuel cell. The typical fuel cell has a series of small flow fiel ds to evenly distribute reactants, and to keep mass transport losses to a minimum. The ne xt couple of sections demonstrate the derivations for the mass trans port in the flow channels.

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289 Appendix F (Continued) F.3.1 Mass Transport in Flow Channels The mass transport in flow channels can be modeled using a control volume for reactant flow from the flow channel to th e electrode layer as shown in Figure F.2. Figure F.2. Control volume for reactant flow from the flow channel to the electrode layer The rate of convective mass transfer at the electrode surface (sm) can be expressed as: ) (s m m sC C h m (281)

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290 Appendix F (Continued) where mC is the mean concentration of the reactant in the flow channel (averaged over the channel cross-section, and decreases along the flow direction, x), and sC is the concentration at the electrode surface. As shown in Figure F.2, the reactan t moves at the molar flow rate, m m cv C A at the position x, where Ac is the channel cross-sectional area and mv is the mean flow velocity in the flow channel. This can be expressed as: elec s m m cw m v C A dx d (282) where elecw is the width of the electr ode surface. If the flow in the channel is assumed to be steady, then the velocity is constant and the concentration is constant, then: flow m s mw v m C dx d (283) The current density is small (i < 0.5 iL), it can be assu med constant. Using Faraday’s law, nF i ms and integrating: x w v nF i x C x Cflow m in m m ) ( ) (, (284) where in mC, is the mean concentration at the flow channel inlet. If the current density is large (i > 0.5 iL), the condition at th e electrode surface can be approximated by assuming the c oncentration at the surface (Cs) is constant. This can be written as follows:

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291 Appendix F (Continued) s m flow m m s mC C w v h C C dx d (285) After integrating from the channel inlet to location x in the flow channel, equation becomes: flow m m in s m s mw v x h C C C C exp (286) At the channel outlet, x = H, and equation becomes: flow m m s in m s out mw v H h C C C C exp, (287) where out mC, is the mean concentration at the flow channel outlet. A simple expression can be derived if the en tire flow channel is assumed to be the control volume as shown in Figure F.3: ) ( ) (out in elec flow m s out in elec flow m sC C w w v m C C w w v m (288)

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292 Appendix F (Continued) Figure F.3. Entire channel as the control volume fo r reactant flow from the flow channel to the electrode layer If Cs is constant, substituting for elec floww w : lm m sC Ah m (289) where out in out in lmC C C C C ln (290)

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293 Appendix F (Continued) The local current density corresponding to the rate of mass transfer is: flow m m s m mw v x h C C nFh x i exp (291) The current density averaged over the electrode surface is: lm mC nFh i (292) The limiting current density when Cs approaches 0 is: flow m m in m m Lw v x h C nFh x i exp, (293) out in out in m LC C C C nFh i ln (294) Both the current density and limiting curre nt density decrease exponentially along the channel length.

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294 Appendix G Heat Transfer Model Table G.1 Heat transfer equations for the e nd plate, manifold and gasket layers Main Parameters Equations Inputs: density (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat (J/Kg-K) x mole fraction totn total molar flow rate of mixture (mol/s) iT temperature of the node (K) k thermal conductivity of node i (W/mK) surrh convective loss from the stack to the air Calculated: mix pc, specific heat of mixture (J/Kg-K) surrU overall heat transfer coefficient for the surroundings 1iU overall heat transfer coefficient for the left node surrq heat flow from the surroundings 1iq heat flow from the left node iH enthalpy of component i 1 1 ,) (i i i s i tot mix pq q dt dT cp x A n c i Ol H i Ov H i HH H H, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Specific heat of mixture: j p j i p i mix pc x c x c, , Heat flow from surroundings: ) (,i surr s i surr surrT T A U q surr i i surrh k x U 1 1 Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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295 Appendix G (Continued) Table G.1 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) surrh convective loss from the stack to the air chanw channel width (m) chanL channel length (m) Calculated: f iU,overall heat transfer coefficient from the fluid 1iU overall heat transfer coefficient for the right node 1iq heat flow from the right node f iq, heat flow from the gases/fluids Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases to solid ) (, ,f i i f i f iT T U q void f s i i i f iA h A k x U 1 1, Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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296 Appendix G (Continued) Table G.2 Gas temperature calculations for the e nd plate, manifold and gasket layers Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/KgK) totn total molar flow rate of mixture (mol/s) s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iT, temperature of gas/fluid mixture iH enthalpy of component i s i i i f i tot mix pq q q dt dT n c, 1 1 ,) ( i Ol H i Ov H i HH H H, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Specific heat of mixture: j p j i p i mix pc x c x c, , Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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297 Appendix G (Continued) Table G.2 (continued) Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/KgK) totn total molar flow rate of mixture (mol/s) chanw channel width (m) chanL channel length (m) in molar flow rate of component i (mol/s) Calculated: s iU, overall heat transfer coefficient from solid to gases/fluid s iq, heat flow from the solid portion of the layer s iA, solid area of layer (m2) voidA void area of layer (m2) iH enthalpy of component i iT temperature of the node (K) Heat flow from solid to fluid/gases: ) (, ,i s i s i s iT T U q void f s i i i s iA h A k x U 1 1, Enthalpies of each gas/liquid flow: i i i iT h n H Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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298 Appendix G (Continued) Table G.3 Heat transfer coefficient for the e nd plate, manifold and gasket layers Main Parameters Equations Inputs: m is the characteristic velocity of the flow (m/s) is the fluid density (kg/m3) is the fluid viscosity (kg/(m*s) Pcs is the perimeter cd channel depth (m) cw channel width (m) chA cross-sectional area of the channel (m2) Pr is the Prandtl number L length of channel at node i (m) Calculated: hD is the hydraulic diameter (m) Nu Nusselt number iRe Reynold’s number at node i f friction factor h convective heat transfer coefficient Calculate Reynold’s number: v D Dh m h m i Re Hydraulic diameter for a circular flow field: cs ch i hP A D 4, Hydraulic diameter for a rectangular flow field: c c c c i hd w d w D 2, Nusselt number: 3 / 2 3 / 21 ) 1 (Pr ) 8 / ( 7 12 1 Pr ) 1000 )(Re 8 / ( L D f f Nuh The friction factor can be defined by: 64 1 ln(Re) 79 0 1 f The convective heat transfer coefficient is: hD k Nu h

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299 Appendix G (Continued) Table G.4 Heat transfer calculations for the flow field plate layers Main Parameters Equations Inputs: density (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node i (W/m-K) totn total molar flow rate of mixture (mol/s) Calculated: f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iq, 1 heat flow from fluid/gases in left node 1iq heat flow from the left node 1iq heat flow from the right node f iq, heat flow from the gases/fluids mix pc, specific heat of mixture (J/Kg-K) iH enthalpy of component i iT temperature of the node (K) f i i i i s i tot mix pq q q dt dT cp x A n c, 1 1 1 ,) ( i Ol H i Ov H i H i res f i f iH H H q q q, 2 2 2 , 1 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1 i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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300 Appendix G (Continued) Table G.4 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) i res, resistivity of solid portion of node i i current density (A/m2) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node 1iU overall heat transfer coefficient for the right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node 1iq heat flow from the right node f iq, heat flow from the gases/fluids i resq, heat flow due to ohmic heating Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from fluid/gases to solid ) (, , f i i f i f iT T U q void f i s i i i f iA h A k x U, ,1 1 Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A Ohmic heating: i i i res i resA x i q, 2

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301 Appendix G (Continued) Table G.5 Gas temperature calculations for the flow field plate layers Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/Kg-K) totn total molar flow rate of mixture (mol/s) s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node f iT, temperature of gas/fluid mixture iH enthalpy of component i f i f i i i i tot mix pq q q q dt dT n c, 1 1 1 1 ,) ( i Ol H i Ov H i H f iH H H q, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Specific heat of mixture: j p j i p i mix pc x c x c, , Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1

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302 Appendix G (Continued) Table G.5 (continued) Main Parameters Equations Inputs: s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient from fluid/gases f iq, 1 heat flow from fluid/gases in right node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iT, temperature of gas/fluid mixture Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from solid to fluid/gases: ) (, , i s i s i s iT T U q void f s i i i s iA h A k x U 1 1,

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303 Appendix G (Continued) Table G.5 (continued) Main Parameters Equations Inputs: chanw channel width (m) chanL channel length (m) Calculated: f iT, temperature of gas/fluid mixture iH enthalpy of component i s iA, solid area of layer (m2) voidA void area of layer (m2) Enthalpies of each gas/liquid flow: f i i i iT h n H, Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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304 Appendix G (Continued) Table G.6 Heat transfer coefficient for the flow field plate layers Main Parameters Equations Inputs: m is the characteristic velocity of the flow (m/s) is the fluid density (kg/m3) is the fluid viscosity (kg/(m*s) Pcs is the perimeter cd channel depth (m) cw channel width (m) chA cross-sectional area of the channel (m2) Pr is the Prandtl number L length of channel at node i (m) Calculated: hD is the hydraulic diameter (m) Nu Nusselt number iRe Reynold’s number at node i f friction factor h convective heat transfer coefficient Calculate Reynold’s number: v D Dh m h m i Re Hydraulic diameter for a circular flow field: cs ch i hP A D 4, Hydraulic diameter for a rectangular flow field: c c c c i hd w d w D 2, Nusselt number: 3 / 2 3 / 21 ) 1 (Pr ) 8 / ( 7 12 1 Pr ) 1000 )(Re 8 / ( L D f f Nuh The friction factor can be defined by: 64 1 ln(Re) 79 0 1 f The convective heat transfer coefficient is: hD k Nu h

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305 Appendix G (Continued) Table G.7 Heat transfer equations for the gas diffusion layers Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node i (W/m-K) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node iH enthalpy of component i iT temperature of the node (K) i res i f i i i tot mix p H q q q q dt dT cp x A n c, 1 1 1 ,) ( 1 2 1 2 1 2 2 2 i Ol H i Ov H i H i Ol H i Ov HH H H H H Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1 i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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306 Appendix G (Continued) Table G.7 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) i res, resistivity of solid portion of node i i current density (A/m2) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node 1iU overall heat transfer coefficient for the right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node 1iq heat flow from the right node f iq, heat flow from the gases/fluids i resq, heat flow due to ohmic heating Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from fluid/gases to solid ) (, , f i i f i f iT T U q void f i s i i i f iA h A k x U, ,1 1 Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A Ohmic heating: i i i res i resA x i q, 2

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307 Appendix G (Continued) Table G.8 Gas temperature heat transfer equations for the gas diffusion layers Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/Kg-K) totn total molar flow rate of mixture (mol/s) s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node f iT, temperature of gas/fluid mixture iH enthalpy of component i f i f i i i i tot mix pq q q q dt dT n c, 1 1 1 1 ,) ( i Ol H i Ov H i H f iH H H q, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Specific heat of mixture: j p j i p i mix pc x c x c, , Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1

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308 Appendix G (Continued) Table G.8 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node 1iU overall heat transfer coefficient for the right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node 1iq heat flow from the right node Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from solid to fluid/gases: ) (, , i s i s i s iT T U q void f s i i i s iA h A k x U 1 1,

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309 Appendix G (Continued) Table G.8 (continued) Main Parameters Equations Inputs: chanw channel width (m) chanL channel length (m) Calculated: f iT, temperature of gas/fluid mixture iH enthalpy of component i s iA, solid area of layer (m2) voidA void area of layer (m2) Enthalpies of each gas/liquid flow: f i i i iT h n H, Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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310 Appendix G (Continued) Table G.9 Heat transfer equations for the catalyst layers Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node i (W/m-K) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node iH enthalpy of component i iT temperature of the node (K) i res i f i i i tot mix p H q q q q dt dT cp x A n c, 1 1 1 ,) ( 1 2 1 2 1 2 2 2 i Ol H i Ov H i H i Ol H i Ov HH H H H H Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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311 Appendix G (Continued) Table G.9 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) i res, resistivity of solid portion of node i i current density (A/m2) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node f iq, heat flow from the gases/fluids i resq, heat flow due to ohmic heating Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from fluid/gases to solid ) (, , f i i f i f iT T U q void f i s i i i f iA h A k x U, ,1 1 Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A Ohmic heating: i i i res i resA x i q, 2

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312 Appendix G (Continued) Table G.10 Gas temperature heat transfer equations for the catalyst layers Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/Kg-K) totn total molar flow rate of mixture (mol/s) s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node f iT, temperature of gas/fluid mixture iH enthalpy of component i f i f i i i i tot mix pq q q q dt dT n c, 1 1 1 1 ,) ( i Ol H i Ov H i H f iH H H q, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1 i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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313 Appendix G (Continued) Table G.10 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node 1iU overall heat transfer coefficient for the right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node 1iq heat flow from the right node f iq, heat flow from the gases/fluids Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from solid to fluid/gases: ) (, , i s i s i s iT T U q void f s i i i s iA h A k x U 1 1, Enthalpies of each gas/liquid flow: i i i iT h n H Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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314 Appendix G (Continued) Table G.11 Heat transfer equations for the membrane layer Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node i (W/m-K) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node iH enthalpy of component i iT temperature of the node (K) i i res i i i tot mix pq q q q dt dT cp x A n cint, 1 1 ,) ( 1 2 1 2 1 , 2 2 i Ol H i Ov H i H i Ol H i Ov HH H H H H Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1 i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1 Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11

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315 Appendix G (Continued) Table G.11 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) i res, resistivity of solid portion of node i i current density (A/m2) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node f iq, heat flow from the gases/fluids i resq, heat flow due to ohmic heating Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from fluid/gases to solid ) (, , f i i f i f iT T U q void f i s i i i f iA h A k x U, ,1 1 Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A Ohmic heating: i i i res i resA x i q, 2

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316 Appendix G (Continued) Table G.11 (continued) Main Parameters Equations Inputs: mix pc, specific heat of mixture (J/Kg-K) totn total molar flow rate of mixture (mol/s) s iA, solid area of layer (m2) voidA void area of layer (m2) k thermal conductivity of node I (W/m-K) x thickness of the node (m) Calculated: 1iU overall heat transfer coefficient for the left node 1iU overall heat transfer coefficient for the right node f iU, 1 overall heat transfer coefficient for the gases/fluids in the left node 1iq heat flow from the left node 1iq heat flow from the right node s iq, heat flow from the solid portion of the layer f iq, 1 heat flow from fluid/gases in left node f iT, temperature of gas/fluid mixture iH enthalpy of component i f i f i i i i tot mix pq q q q dt dT n c, 1 1 1 1 ,) ( i Ol H i Ov H i H f iH H H q, 2 2 2 1 2 1 2 1 2 i Ol H i Ov H i HH H Hout Ol H out Ov H out HH H H_ 2 2 2 Specific heat of mixture: j p j i p i mix pc x c x c, , Heat flow from left node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in left node: ) (1 1 1i i f i f iT T U q void i f i s i i i f iA h A k x U, 1 1 , 11 1

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317 Appendix G (Continued) Table G.11 (continued) Main Parameters Equations Inputs: density of the layer (kg/m3) s iA, solid area of layer (m2) voidA void area of layer (m2) x thickness of the node (m) cp specific heat of layer (J/Kg-K) iT temperature of the node (K) k thermal conductivity of node I (W/m-K) chanw channel width (m) chanL channel length (m) Calculated: f iU, 1 overall heat transfer coefficient for fluid/gases in right node 1iU overall heat transfer coefficient for the right node f iU, overall heat transfer coefficient for fluid/gases to solid f iq, 1heat flow from fluid/gases in right node 1iq heat flow from the right node f iq, heat flow from the gases/fluids Heat flow from right node: ) (1 1 1 i i i iT T U q s i i i s i i i iA k x A k x U, 1 1 1 11 Heat flow from fluid/gases in right node: ) (, 1 1 1i f i f i f iT T U q s i i i void i f i f iA k x A h U, 1 1 11 1 Heat flow from solid to fluid/gases: ) (, , i s i s i s iT T U q void f s i i i s iA h A k x U 1 1,

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318 Appendix G (Continued) Table G.11 (continued) Main Parameters Equations Inputs: chanw channel width (m) chanL channel length (m) Calculated: f iT, temperature of gas/fluid mixture iH enthalpy of component i s iA, solid area of layer (m2) voidA void area of layer (m2) Enthalpies of each gas/liquid flow: f i i i iT h n H, Area of solid portion of the layer: void s iA A A Channel area: chan chan voidL w A

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319 Appendix H Mass Transfer Analysis Table H.1 Mass transfer equations for the end plate, manifold and gasket layers Main Parameters Equations Inputs: in totn_ inlet molar flow rate (mol/s) inP inlet pressure (Pa) in inlet volumetric flow rate (m3/s) inT inlet temperature (K) R ideal gas constant (m3-Pa/K-mol) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen Convert volumetric flow rate to molar flow rate: in in in in totRT P n Total molar accumulation: 1 , i tot i tot totn n dt dn The rate of H2 accumulation is: 1 1 2 , 2 2) ( i tot i H i tot i H tot Hn x n x n x dt d The rate of H2O accumulation is: 1 1 2 , 2 2) ( i tot i O H i tot i O H tot O Hn x n x n x dt d The rate of O2 accumulation is: 1 1 2 , 2 2) ( i tot i O i tot i O tot On x n x n x dt d

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320 Appendix H (Continued) Table H.2 Mole fraction calculations for the end plate, manifold and gasket layers Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) in inlet humidity of the gas stream O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, Total pressure at node i (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) R ideal gas constant (m3-Pa/K-mol) f iT, temperature of gas/liquid mixture at node i (K) x thickness of node i (m) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate Calculate the vapor pressure of the inlet water vapor: ) (, 2f i sat in i O HT P p Calculate humidity: ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H The mole fraction of the water vapor is: O H H O H i Ov HM H M M H x2 2 2 21 The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 Water condensation and evaporation: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2

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321 Appendix H (Continued) Table H.2 (continued) Main Parameters Equations Inputs: R ideal gas constant (m3-Pa/K-mol) ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) i totP, Total pressure at node i (Pa) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen RH Relative humidity RW Relative water content f iT, temperature of gas/liquid mixture at node i (K) The total mole fraction of water is: i tot i O H i O Hn n x, 2 2 The mole fraction of hydrogen is: i O H i Hx x, 2 21 The molar flow rate of hydrogen is: i tot i H i Hn x n, 2 2 Total flowrate out of the layer is: 1 2 1 2 1 i O H i H i totn n n Relative humidity: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH Relative water content: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW

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322 Appendix H (Continued) Table H.3 Mass transfer calculations for the flow field layers Main Parameters Equations Inputs: in totn_ inlet molar flow rate (mol/s) inP inlet pressure (Pa) in inlet volumetric flow rate (m3/s) inT inlet temperature (K) R ideal gas constant (m3-Pa/K-mol) Calculated: totn total molar flow rate of mixture 2 totn total molar flow rate leaving the plate, and going back to the manifold 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen Convert volumetric flow rate to molar flow rate: in in in in totRT P n Total molar accumulation: 2 1 , tot i tot i tot totn n n dt dn The rate of H2 accumulation is: 2 2 2 1 1 2 , 2 2) (tot H i tot i H i tot i H tot Hn x n x n x n x dt d The rate of H2O accumulation is: 2 2 2 1 1 2 , 2 2) (tot O H i tot i O H i tot i O H tot O Hn x n x n x n x dt d The rate of O2 accumulation is: 2 2 2 1 1 2 , 2 2) (tot O i tot i O i tot i O tot On x n x n x n x dt d

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323 Appendix H (Continued) Table H.3 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) in inlet humidity of the gas stream O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, Total pressure at node I (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen Calculate the vapor pressure of the inlet water vapor: ) (, 2f i sat in i O HT P p Calculate humidity: ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H The mole fraction of the water vapor is: O H H O H i Ov HM H M M H x2 2 2 21 The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 The molar flow rate for water condensation and evaporation is: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2

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324 Appendix H (Continued) Table H.3 (continued) Main Parameters Equations Inputs: i totP, Total pressure at node i (Pa) hD hydraulic diameter (m) Sh Sherwood number i mu, velocity of mixture (m/s) b distance between flow channels and gas diffusion layer Hx height of gas diffusion layer Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen mh mass transfer coefficient j iD, diffusion coefficient (m2/s) out iC, Outlet average concentration f iT, temperature of gas/liquid mixture at node i (K) The total mole fraction of water is: i tot i O H i O Hn n x, 2 2 The mole fraction of hydrogen is: i O H i Hx x, 2 21 The molar flow rate of hydrogen is: i tot i H i Hn x n, 2 2 The concentrations are calculated at the node inlet: f i i tot i O H i O HRT P x C, , 2 2 f i i tot i H i HRT P x C, , 2 2 Mass transfer coefficient: h j i mD D Sh h,

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325 Appendix H (Continued) Table H.3 (continued) Main Parameters Equations Inputs: i totP, Total pressure at node i (Pa) hD hydraulic diameter (m) Sh Sherwood number Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen mh mass transfer coefficient j iD, diffusion coefficient (m2/s) out iC, Outlet average concentration Outlet average concentration for hydrogen: i m H m i H i Hbu x h C C, 2 1 2exp Average limiting current density: 1 2 2 1 2 2lni H i H i H i H m LC C C C nFh i Outlet molar flow: ) (1 2 2 1 2 i H i H m i i HC C h A n Total flowrate out of the layer is: 1 2 1 2 1 i O H i H i totn n n The total mole fraction of hydrogen is: 1 1 2 1 2 i tot i H i Hn n x

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326 Appendix H (Continued) Table H.3 (continued) Main Parameters Equations Inputs: i totP, Total pressure at node i (Pa) ) (, f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) i Ov Hn, 2 molar flow rate of water vapor totn total molar flow rate of mixture i O Hn, 2 total water molar flow rate Calculated: O Hx2 mole fraction of water RH Relative humidity RW Relative water content The total mole fraction of water is: 1 1 2 1 2 i tot i O H i O Hn n x Relative humidity: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH Relative water content: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW

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327 Appendix H (Continued) Table H.4 Mass transfer calculations for the gas diffusion layers Main Parameters Equations Inputs: i totP, Total pressure at node i (Pa) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen Total molar accumulation: 1 , i tot i tot totn n dt dn The rate of H2 accumulation is: 1 1 2 , 2 2) ( i tot i H i tot i H tot Hn x n x n x dt d The rate of H2O accumulation is: 1 1 2 , 2 2) ( i tot i O H i tot i O H tot O Hn x n x n x dt d The rate of O2 accumulation is: 1 1 2 , 2 2) ( i tot i O i tot i O tot On x n x n x dt d

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328 Appendix H (Continued) Table H.4 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) in inlet humidity of the gas stream O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, Total pressure at node I (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) Calculated: totn total molar flow rate of mixture O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate f iT, temperature of gas/liquid mixture at node i (K) Calculate the vapor pressure of the inlet water vapor: ) (, 2f i sat in i O HT P p Calculate humidity: ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H The mole fraction of the water vapor is: O H H O H i Ov HM H M M H x2 2 2 21 The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 The molar flow rate for water condensation and evaporation is: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2

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329 Appendix H (Continued) Table H.4 (continued) Main Parameters Equations Inputs: i totP, total pressure at node i (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) x thickness of node i (m) i mu, velocity of mixture (m/s) b distance between flow channels and gas diffusion layer Hx height of gas diffusion layer is the electrode porosity Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen i O HC, 2 Concentration of water at node i i HC, 2 Concentration of hydrogen at node i Li average limiting current density eff j iD, effective diffusion coefficient j iD, diffusion coefficient f iT, temperature of gas/liquid mixture at node i (K) The concentration at the node inlet is: f i i tot i H i HRT P x C, , 2 2 Outlet average concentration for hydrogen: i m H m i H i Hbu x h C C, 2 1 2exp Average limiting current density: i i H j i Lx C nFD i 1 2 Outlet molar flow: i i H i H j i i i Hx C C D A n ) (1 2 2 1 2 Effective diffusion coefficient: 2 / 3 ,j i eff j iD D Total flowrate out of the layer is: 1 2 1 2 1 i O H i H i totn n n

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330 Appendix H (Continued) Table H.4 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node I (Pa) i totP, Total pressure at node I (Pa) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen RH Relative humidity RW Relative water content i O HC, 2 Concentration of water at node i i HC, 2 Concentration of hydrogen at node i f iT, temperature of gas/liquid mixture at node i (K) The total mole fraction of hydrogen is: 1 1 2 1 2 i tot i H i Hn n x The total mole fraction of water is: 1 1 2 1 2 i tot i O H i O Hn n x Outlet concentration of water: f i i tot i O H i O HRT P x C, 1 2 1 2 Relative humidity: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH Relative water content: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW

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331 Appendix H (Continued) Table H.5 Mass transfer calculations for the catalyst layers Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node I (Pa) in inlet humidity of the gas stream O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, Total pressure at node I (Pa) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen Total molar accumulation: 1 , i tot i tot totn n dt dn The rate of H2 accumulation is: 1 1 2 , 2 2) ( i tot i H i tot i H tot Hn x n x n x dt d The rate of H2O accumulation is: 1 1 2 , 2 2) ( i tot i O H i tot i O H tot O Hn x n x n x dt d The rate of O2 accumulation is: 1 1 2 , 2 2) ( i tot i O i tot i O tot On x n x n x dt d Calculate the vapor pressure of the inlet water vapor: ) (, 2 f i sat in i O HT P p Calculate humidity: ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H

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332 Appendix H (Continued) Table H.5 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, Total pressure at node i (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) Calculated: totn total molar flow rate of mixture O Hx2 mole fraction of water 2 Ox mole fraction of oxygen H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i O HC, 2 Concentration of water at node i i HC, 2 Concentration of hydrogen at node i f iT, temperature of gas/liquid mixture at node i (K) The mole fraction of the water vapor is: O H H O H i Ov HM H M M H x2 2 2 21 The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 The molar flow rate for water condensation and evaporation is: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2 The concentrations at the node inlet: f i i tot i O H i O HRT P x C, , 2 2 f i i tot i H i HRT P x C, , 2 2

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333 Appendix H (Continued) Table H.5 (continued) Main Parameters Equations Inputs: i totP, Total pressure at node I (Pa) is the electrode porosity x thickness of node i (m) i mu, velocity of mixture (m/s) b distance between flow channels and gas diffusion layer Hx height of gas diffusion layer is the electrode porosity Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen i O HC, 2 Concentration of water at node i i HC, 2 Concentration of hydrogen at node i Li average limiting current density eff j iD, effective diffusion coefficient j iD, diffusion coefficient Outlet average concentration for hydrogen: i m H m i H i Hbu x h C C, 2 1 2exp Average limiting current density: i i H j i Lx C nFD i 1 2 Outlet molar flow: i i H i H j i i i Hx C C D A n ) (1 2 2 1 2 Effective diffusion coefficient: 2 / 3 ,j i eff j iD D Total flowrate out of the layer is: 1 2 1 2 1 i O H i H i totn n n The total mole fraction of hydrogen is: 1 1 2 1 2 i tot i H i Hn n x

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334 Appendix H (Continued) Table H.5 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) i totP, Total pressure at node i (Pa) Calculated: totn total molar flow rate of mixture O Hx2 mole fraction of water i Ov Hn, 2 molar flow rate of water vapor i O Hn, 2 total water molar flow rate RH Relative humidity RW Relative water content i O HC, 2 Concentration of water at node i f iT, temperature of gas/liquid mixture at node i (K) The total mole fraction of water is: 1 1 2 1 2 i tot i O H i O Hn n x Outlet concentration of water: f i i tot i O H i O HRT P x C, 1 2 1 2 Relative humidity: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH Relative water content: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW

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335 Appendix I Pressure Drop Analysis Table I.1 Pressure drop calculations for the end plate, termi nal and gasket layers Main Parameters Equations Inputs: in inlet volumetric flow rate (m3/s) chA cross-sectional area of the channel (m2) f is the friction factor fluid density (kg/m3) v average velocity (m/s) KL local resistance Pcs perimeter cd channel depth (m) cw channel width (m) chN number of parallel channels density (kg/m3) i totP, total pressure at node i (Pa) Calculated: iv velocity (m/s) totn total molar flow rate of mixture 1iP pressure drop Lchan channel length (m) HD hydraulic diameter or characteristic length (m) The velocity (m/s) in the fuel cell channel near the entrance of the cell is: ch in iA v Pressure drop: 2 22 2 1v K v D L f PL H chan i Hydraulic diameter for a circular flow field: cs ch i HP A D 4, For rectangular channels the hydraulic diameter is: c c c c i Hd w d w D 2,

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336 Appendix I (Continued) Table I.1 (continued) Main Parameters Equations Inputs: HD hydraulic diameter or characteristic length (m) fluid density (kg/m3) fluid viscosity (kg/(m*s)) m characteristic velocity of the flow (m/s) kinematic viscosity (m2/s) Calculated: i totP, total pressure at node i (Pa) iRe Reynold’s number if friction factor The channel length can be defined as: ) (, L c ch i cell chanw w N A L The friction factor can be defined by: Re 56 if Reynold’s number: v D DH m H m i Re Pressure at outlet node: dx dx dP P Px tot i tot i tot 0 1

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337 Appendix I (Continued) Table I.2 Pressure drop calculations for the flow field layers Main Parameters Equations Inputs: in inlet volumetric flow rate (m3/s) chA cross-sectional area of the channel (m2) if friction factor fluid density (kg/m3) v average velocity (m/s) KL local resistance Pcs perimeter cd channel depth (m) cw channel width (m) Acell cell active area (m2) chN number of parallel channels Lw space between channels (m) density (kg/m3) i totP, total pressure at node i (Pa) Calculated: iv velocity (m/s) totn total molar flow rate of mixture 1iP pressure drop Lchan channel length (m) HD hydraulic diameter or characteristic length (m) iRe Reynold’s number if friction factor The velocity (m/s) in the entrance of the flow field layer is: ch i tot i f i tot iN P R T n v , ,* The velocity (m/s) in each fuel cell channel is: ch in chanA v where 22 1 r Ach Molar flow rate in each channel: in in in in totRT P n Pressure drop: 2 22 2v K v D L f dx dPL H chan tot Hydraulic diameter for a circular flow field: cs c i HP A D 4,

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338 Appendix I (Continued) Table I.2 (continued) Main Parameters Equations Inputs: if friction factor fluid density (kg/m3) v average velocity (m/s) KL local resistance Pcs perimeter cd channel depth (m) cw channel width (m) Acell cell active area (m2) chN number of parallel channels Lw space between channels (m) density (kg/m3) i totP, total pressure at node i (Pa) m characteristic velocity of the flow (m/s) kinematic viscosity (m2/s) Calculated: iv velocity (m/s) totn total molar flow rate of mixture 1iP pressure drop Lchan channel length (m) HD hydraulic diameter or characteristic length (m) For rectangular cha nnels, the hydraulic diameter is: c c c c i Hd w d w D 2, The channel length can be defined as: ) (, L c ch i cell chanw w N A L The friction factor can be defined by: Re 56 if Reynold’s number: v D Dch m ch m i Re Pressure at outlet node: dx dx dP P Px tot i tot i tot 0 1

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339 Appendix I (Continued) Table I.3 Pressure drop calculations for the gas diffusion layers Main Parameters Equations Inputs: viscosity (Pa-s) volumetric flow rate (m3/s) k permeability (m2) A cross-sectional area (m2) void fraction x thickness of node i (m) k permeability (m2) viscosity (Pa-s) x thickness of node i (m) i totP, change in total pressure (Pa) Calculated: i totP, change in total pressure (Pa) i mu, velocity of mixture (m/s) Pressure drop: x A k Pi i i i tot Volumetric flow rate: i tot i f i tot iP R T n v, ,* Velocity of the mixture in the membrane: i tot i i i mP x k u,

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340 Appendix I (Continued) Table I.4 Pressure drop calculations for the catalyst layers Main Parameters Equations Inputs: viscosity (Pa-s) volumetric flow rate (m3/s) k permeability (m2) A cross-sectional area (m2) void fraction x thickness of node i (m) k permeability (m2) viscosity (Pa-s) x thickness of node i (m) i totP, change in total pressure (Pa) Calculated: i totP, change in total pressure (Pa) i mu, velocity of mixture (m/s) Pressure drop: x A k Pi i i i tot Volumetric flow rate: i tot i f i tot iP R T n v, ,* Velocity of the mixture in the membrane: i tot i i i mP x k u,

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341 Appendix J Polymer Membrane Layer Table J.1 Polymer electrolyte membrane layer mass balance equations Main Parameters Equations Inputs: in totn_ inlet molar flow rate (mol/s) inP inlet pressure (Pa) in inlet volumetric flow rate (m3/s) inT inlet temperature (K) R ideal gas constant (m3-Pa/K-mol) ) (, f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) Calculated: totn total molar flow rate of mixture i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen Convert volumetric flow rate to molar flow rate: in in in in totRT P n Total molar accumulation: 1 , i tot i tot totn n dt dn The rate of H2 accumulation is: 1 1 2 , 2 2) ( i tot i H i tot i H tot Hn x n x n x dt d The rate of H2O accumulation is: 1 1 2 , 2 2) ( i tot i O H i tot i O H tot O Hn x n x n x dt d The rate of O2 accumulation is: 1 1 2 , 2 2) ( i tot i O i tot i O tot On x n x n x dt d

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342 Appendix J (Continued) Table J.2 Calculation of mole fractions and molar flow rates for the PEM layer Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) in inlet humidity of the gas stream O HM2 molecular weight of water (kg/mol) 2 HM molecular weight of hydrogen (kg/mol) i totP, total pressure at node i (Pa) cd channel depth (m) cw channel width (m) ck evaporation and condensation rate constant (s-1) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen i O Hp, 2 vapor pressure of the inlet water vapor H is the humidity i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate Calculate the vapor pressure of the inlet water vapor: ) (, 2f i sat in i O HT P p Calculate humidity: ) (, 2 2 2 2 i O H i tot H i O H O Hp P M p M H The mole fraction of the water vapor is: O H H O H i Ov HM H M M H x2 2 2 21 The molar flow rate of water vapor is: i tot i Ov H i Ov Hn x n, 2 2 The molar flow rate for water condensation and evaporation is: ) (, 1 1 2 1 2 f i sat i tot i tot i Ov H f i c c c i Ol HT P P n n RT d w k n The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2

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343 Appendix J (Continued) Table J.2 (continued) Main Parameters Equations Inputs: ) (,f i satT P saturation pressure at the gas/fluid temperature at node I (Pa) i totP, Total pressure at node I (Pa) Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water i Ov Hn, 2 molar flow rate of water vapor i Ol Hn, 2 molar flow rate of liquid water i O Hn, 2 total water molar flow rate i Hn, 2 molar flow rate of hydrogen RH Relative humidity RW Relative water content The total molar flow rate of water is: i Ol H i Ov H i O Hn n n, 2 2 2 The total mole fraction of water is: i tot i O H i O Hn n x, 2 2 The mole fraction of hydrogen is: i O H i Hx x, 2 21 The molar flow rate of hydrogen is: i tot i H i Hn x n, 2 2 Total flowrate out of the layer is: 1 2 1 2 1 i O H i H i totn n n Relative humidity: ) (, 1 1 2 f i sat i tot i tot i Ov HT P P n n RH Relative water content: ) (, 1 1 2 f i sat i tot i tot i O HT P P n n RW

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344 Appendix J (Continued) Table J.3 Diffusive flux and potential relations for the PEM layer Main Parameters Equations Inputs: wa water activity R ideal gas constant (m3-Pa/K-mol) ) (,f i satT P saturation pressure at the gas/fluid temperature at node i (Pa) i totP, Total pressure at node i (Pa) i O HC, 2 Concentration of water at node i m dry dry membrane density (kg/m3) m M membrane molecular mass (kg/mol) 1c – 5c constants for the activity of water molecules Calculated: totn total molar flow rate of mixture 2 Hx mole fraction of hydrogen O Hx2 mole fraction of water 2 Ox mole fraction of oxygen water uptake ionic conductivity xm proton potential Calculate water uptake: K 353 = at 1 14 16 8 10 3 0 K 303 = at 36 85 39 8 17 043 0 ) (3 2 3 2 w w w w w w i f wa a a a a a T a 50 303 )) 303 ( ) 353 ( ( ) 303 ( ) (, i f w w w i f wT a a a T a Calculate ionic conductivity: T 1 303 1 1268 exp ) 00326 0 005139 0 ( Proton potential: m H m m mu c F i x Diffusion coefficient: 2 / 1 303 1 2416 ,108 9 78 81 173 2 2a a a e D DSO O H T i O cH 14 6 6 23 1 23 1 10 1625 2 10 5625 2 10 5 9 10 75 7 10 642276 23 2 3 2 3 2 3 2 3 2 3 2/ / / 10 / 11 11 / 11 / 13 SO O H SO O H SO O H SO O H SO O H SO O HD

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345 Appendix J (Continued) Table J.4 Pressure, velocity and diffusive fl ux equations for the PEM layer Main Parameters Equations Inputs: i O cHD,2 diffusion coefficient HD proton diffusion coefficient m O Hc2 water concentration Hc proton concentration m potential F Faraday’s constant R ideal gas constant k permeability (m2) viscosity (Pa-s) x thickness of node i (m) i totP, change in total pressure (Pa) i totP, total pressure at node i (Pa) Calculated: M O HJ2 diffusive molar flux for water HJ diffusive molar flux for protons i mu, velocity of mixture (m/s) 1 i totP pressure at outlet node (Pa) Diffusive molar flux for water: F i n x c D Jx drag m O H T O cH M O H 2 2 2, Diffusive molar flux for protons: x c D RT F Jm H H H Velocity of the mixture in the membrane: i tot i i i mP x k u, Pressure at node i: i i tot i tot i totx P P P 1 1 , Pressure at outlet node: dx dx dP P Px tot i tot i tot 0 1

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346 Appendix J (Continued) Table J.5 Gas permeation equations for the PEM layer Main Parameters Equations Inputs: f iT, temperature of gas/liquid mixture at node i (K) Calculated: mP permeability 2 OS oxygen solubility 2 HD hydrogen diffusivity (cm2 s-1) 2 OD oxygen diffusivity (cm2 s-1) i On, 2 oxygen molar flow rate (mol/s) i Hn, 2 hydrogen molar flow rate (mol/s) Permeability: S D Pm Oxygen solubility i f OT S, 12 2666 exp 10 43 7 Oxygen diffusivity: i f OT D, 22768 exp 0031 0 Hydrogen diffusivity: i f HT D, 22602 exp 0041 0 Hydrogen molar flow rate: i i tot i H i Hx P A P n 2 2 Oxygen molar flow rate: i i tot i O i Ox P A P n 2 2

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347 Appendix K Parameters for 16 cm2 Fuel Cell Stack Table K.1 Material properties used for the anode layers of the 16 cm2 fuel cell stack Fuel Cell Layer Material Thickness (m) Area (m2) Area of void (m2) Density (kg/m2) Thermal Conductivity (W/m-K) Specific heat capacity (J/kg-K) Specific Resistance (ohm-m) End plate Polycarbonate 0.01 0.0064 0 1300 0.2 1200 0 Gasket Black Conductive Rubber 0.001 0.0017040 1400 1.26 1000 0 Flow field plate SS 0.0005 0.0033850.0016925 8000 16 500 7.2e-7 Diffusion media Carbon Cloth 0.0004 0.0016 0.00128 2000 65 840 0.000014 Catalyst Pt/C 0.000065 0.0016 0.00112 387 0.2 770 0.000014 Membrane Nafion 0.00005 0.0016 0 1740 0.21 1100 0.1

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348 Appendix K (Continued) Table K.2 Material properties used for the cathode layers of the 16 cm2 fuel cell stack Fuel Cell Layer Material Thickness (m) Area (m2) Area of void (m2) Density (kg/m2) Thermal Conductivity (W/m-K) Specific heat capacity (J/kg-K) Specific Resistance (ohm-m) Catalyst Pt/C 0.000065 0.0016 0.00112 387 0.2 770 0.000014 Diffusion media Carbon Cloth 0.0004 0.0016 0.00128 2000 65 840 0.000014 Flow field plate SS 0.0005 0.0033850.0016925 8000 16 500 7.2e-7 Gasket Black Conductive Rubber 0.001 0.0017040 1400 1.26 1000 0 End plate Polycarbonate 0.01 0.0064 0 1300 0.2 1200 0 Hydrogen 0.090 0.165 14,160 Air 1.30 0.0223 1005

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349 Appendix L Typical Outputs for Each Fuel Cell Layer Table L.1 Typical outputs of th e anode end plate, terminal and cooling channel layer after 30 sec Fuel cell layer Inlet Outlet Left end plate Flow rate: 0.1628 mol/s Pressure: 344,737.864 Pa Velocity: 3.6079 m/s Temperature: 313 K Flow rate: 0.1628 mol/s Pressure: 344,737.864 Pa Velocity: 3.6079 m/s Temperature: 313 K Terminal/gasket layers Flow rate: 0.1628 mol/s Pressure: 344,737.864 Pa Velocity: 3.6079 m/s Flow rate: 0.1628 mol/s Pressure: 344,734.864 Pa Velocity: 0.6882 m/s Cooling channel layer Flow rate: 0.0779 mol/s Pressure: 202,650.02 Pa Velocity: 1.9808 m/s Temperature: 326.5 K Flow rate: 0.0779 mol/s Pressure: 70,100 Pa Velocity: 1.9808 m/s Temperature: 326.5 K

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350 Appendix L (Continued) Table L.2 Typical outputs of th e anode flow field and GDL layers after 30 sec Fuel cell layer Inlet Outlet Anode Flow Field In each inlet Flow rate: 0.0203 mol/s Pressure: 344,737.864 Pa Velocity: 0.0032 m/s In each of the channels Flow rate: 0.0051 mol/s Pressure: 344,080 Pa Velocity: 113.34 m/s Pressure at the end of the flow channels (due to pressure drop): 240,720 Pa Temperature: 331 K In each outlet Flow rate: 0.0203 mol/s Pressure: 344,737.864 Pa Velocity: 0.0032 m/s Going to GDL layer Flow rate: 0.0021 mol/s Pressure: 240,720 Pa Velocity: 0.0083 m/s Temperature: 331 K Anode GDL Flow rate: 0.0021 mol/s Pressure: 240,720 Pa Velocity: 0.0083 m/s Temperature: 331 K Flow rate: 0.0021 mol/s Pressure: 239,630 Pa Velocity: 0.3492 m/s Temperature: 331 K

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351 Appendix L (Continued) Table L.3 Typical outputs of the anode catalyst and membrane layers after 30 sec Fuel cell layer Inlet Outlet Anode Catalyst Flow rate: 0.0021 mol/s Pressure: 239,630 Pa Velocity: 0.3492 m/s Temperature: 331 K Flow rate: 0.0010 mol/s Pressure: 225,880 Pa Velocity: 0.5741 m/s Temperature: 331 K Membrane (Nafion 115) Total hydrogen flow rate into membrane due to permeability Flow rate: 9.7707e-007 mol/s Total oxygen flow rate into membrane due to permeability Flow rate: 1.1293e-007 mol/s Temperature: 331 K Pressure: 14,930 Pa Total flow rate in anode catalyst layer Flow rate: 0.0021 mol/s Total flow rate in cathode catalyst layer Flow rate: 0.0015 mol/s

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352 Appendix L (Continued) Table L.4 Typical outputs of th e cathode catalyst and GDL layers after 30 sec Fuel cell layer Inlet Outlet Cathode Catalyst Flow rate: 0.0021 mol/s Pressure: 265,750 Pa Velocity: 0.4458 m/s Temperature: 331 K Flow rate: 0.0015 mol/s Pressure: 226,450 Pa Velocity: 0.5194 m/s Temperature: 331 K Cathode GDL Flow rate: 0.0021 mol/s Pressure: 270,890 Pa Velocity: 0.0026 m/s Temperature: 331 K Flow rate: 0.0021 mol/s Pressure: 265,750 Pa Velocity: 0.4458 m/s Temperature: 331 K

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353 Appendix L (Continued) Table L.5 Typical outputs of th e cathode flow field layer after 30 sec Inlet Outlet In each inlet Flow rate: 0.0145 mol/s Pressure: 344,737.864 Pa Velocity: 0.0023 m/s In each of the channels Flow rate: 0.0036 mol/s Pressure: 344,400 Pa Velocity: 80.97 m/s Pressure at the end of the flow channels (due to pressure drop): 270,890 Pa Temperature: 331 K In each outlet Flow rate: 0.0145 mol/s Pressure: 344,737.864 Pa Velocity: 0.0023 m/s Going to GDL layer Flow rate: 0.00094 mol/s Pressure: 270,890 Pa Velocity: 0.0026 m/s

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354 Appendix L (Continued) Table L.6 Typical outputs of th e cathode end plate, terminal and cooling layers after 30 sec Fuel cell layer Inlet Outlet Cooling channel layer Flow rate: 0.0779 mol/s Pressure: 202,650.02 Pa Velocity: 1.9808 m/s Temperature: 330 K Flow rate: 0.0779 mol/s Pressure: 70,100 Pa Velocity: 1.9808 m/s Temperature: 330 K Manifold layer Flow rate: 0.1159 mol/s Pressure: 344,737.864 Pa Velocity: 2.5687 m/s Temperature: 326.5 K Flow rate: 0.1159 mol/s Pressure: 344,734.864 Pa Velocity: 0.4900 m/s Temperature: 326.5 K Right end plate Flow rate: 0.1159 mol/s Pressure: 344,737.864 Pa Velocity: 2.5687 m/s Temperature: 313 K Flow rate: 0.1159 mol/s Pressure: 344,737.864 Pa Velocity: 2.5687 m/s Temperature: 313 K

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ABOUT THE AUTHOR Colleen S. Spiegel, received a BSChE and MSChE in chemical engineering from the University of South Florida in 2001 a nd 2003 respectively. She has been an R&D manager, chemical engineer and engineering co nsultant for a total of eight years, and her expertise is in the areas of design and mode ling. Mrs. Spiegel has worked in several areas of research and process development and was instrumental in establishing new ideas for several companies. Mrs. Spiegel is the author of “Des igning and Building Fuel Cells” (McGrawHill), and “PEM Fuel Cell Modeling and Simu lation with MATLAB” (Elsevier Science). She is the founder of Clean Fuel Cell Energy, LLC, which offers fuel cell components to engineers and scientists globally. She is a me mber of the American Institute of Chemical Engineers (AICHE), the Instit ute of Electrical & Electroni cs Engineers (IEEE) and the National Association of Science Writers (NASW).