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Simulation and Experimental Characterization of the Water Management for an Open-Cathode Direct Methanol Fuel Cell That ...

Permanent Link: http://ufdc.ufl.edu/UFE0043738/00001

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Title: Simulation and Experimental Characterization of the Water Management for an Open-Cathode Direct Methanol Fuel Cell That Utilizes a Liquid Barrier Layer
Physical Description: 1 online resource (145 p.)
Language: english
Creator: Kuo, Cheng-Chan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: crossover -- dmfc -- experiments -- hydrophobic -- liquid-barrier-layer -- multi-component-mass-transport -- open-cathode -- simulation -- stefan-maxwell -- water-balance
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Water management is a critical issue in a practical direct methanol fuel cell (DMFC) system. In this research, the interaction of water management and performance of a DMFC stack with an open-cathode design utilizing a liquid barrier layer (LBL) is studied. As compared to the traditional design, the novel DMFC stack investigated here has a passive water-recovery mechanism and eliminates the water collection and replenishment devices at the cathode exit, reducing the complexity and size of the system. However, water management of the new DMFC stack can impose significant operating constraints if the water balance is not well-controlled. The purpose of this research is to analytically and experimentally study the effects of change of the key parameters on the water balance of the novel DMFC stack. A model was developed to simulate the cell performance, rate of methanol crossover, and multi-component mass transport of the stack. A dimensionless water balance parameter, based on the conservation of mass of the water inside the stack, was also created to facilitate the study of the water balance of the stack. A water management map of the novel DMFC stack was created based on the developed. The modeling results were validated with the data from our experiments on this novel stack design. The results showed that the stack temperature dominates the control of water management of this DMFC stack design. Increases in the operating current density, solution molarity and the rate of methanol crossover favor the water recovery of the stack. However, the most effective way to change the stack from water-loss mode to water-recovery mode is to reduce the stack temperature. The results also showed that the novel DMFC stack (under the same material properties) could operate in water-recovery or water-neutrality mode only for stack temperatures of 50 oC or lower, when the current density was under the nominal design value of 150 mA/cm2. The developed model can simulate the trend of the cell performance and water management of the stack by varying the key parameters, such as stack temperature, solution molarity, and the porosity of the LBL. The modeling results showed that the LBL has a more significant effect on the water balance and cell performance than the CGDL does. By increasing the porosity of the LBL 30%, the cell performance is increased significantly but the vent rate of the water vapor is also increased, resulting in a water-loss mode. The results also showed that a decrease of the porosity of the LBL enhances the water recovery of the stack, but that the cell performance is degraded.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Cheng-Chan Kuo.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lear, William E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043738:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043738/00001

Material Information

Title: Simulation and Experimental Characterization of the Water Management for an Open-Cathode Direct Methanol Fuel Cell That Utilizes a Liquid Barrier Layer
Physical Description: 1 online resource (145 p.)
Language: english
Creator: Kuo, Cheng-Chan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: crossover -- dmfc -- experiments -- hydrophobic -- liquid-barrier-layer -- multi-component-mass-transport -- open-cathode -- simulation -- stefan-maxwell -- water-balance
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Water management is a critical issue in a practical direct methanol fuel cell (DMFC) system. In this research, the interaction of water management and performance of a DMFC stack with an open-cathode design utilizing a liquid barrier layer (LBL) is studied. As compared to the traditional design, the novel DMFC stack investigated here has a passive water-recovery mechanism and eliminates the water collection and replenishment devices at the cathode exit, reducing the complexity and size of the system. However, water management of the new DMFC stack can impose significant operating constraints if the water balance is not well-controlled. The purpose of this research is to analytically and experimentally study the effects of change of the key parameters on the water balance of the novel DMFC stack. A model was developed to simulate the cell performance, rate of methanol crossover, and multi-component mass transport of the stack. A dimensionless water balance parameter, based on the conservation of mass of the water inside the stack, was also created to facilitate the study of the water balance of the stack. A water management map of the novel DMFC stack was created based on the developed. The modeling results were validated with the data from our experiments on this novel stack design. The results showed that the stack temperature dominates the control of water management of this DMFC stack design. Increases in the operating current density, solution molarity and the rate of methanol crossover favor the water recovery of the stack. However, the most effective way to change the stack from water-loss mode to water-recovery mode is to reduce the stack temperature. The results also showed that the novel DMFC stack (under the same material properties) could operate in water-recovery or water-neutrality mode only for stack temperatures of 50 oC or lower, when the current density was under the nominal design value of 150 mA/cm2. The developed model can simulate the trend of the cell performance and water management of the stack by varying the key parameters, such as stack temperature, solution molarity, and the porosity of the LBL. The modeling results showed that the LBL has a more significant effect on the water balance and cell performance than the CGDL does. By increasing the porosity of the LBL 30%, the cell performance is increased significantly but the vent rate of the water vapor is also increased, resulting in a water-loss mode. The results also showed that a decrease of the porosity of the LBL enhances the water recovery of the stack, but that the cell performance is degraded.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Cheng-Chan Kuo.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lear, William E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043738:00001


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SIMULATIONANDEXPERIMENTALCHARACTERIZATIONOFTHEWATERMANAGEMENTFORANOPEN-CATHODEDIRECTMETHANOLFUELCELLTHATUTILIZESALIQUIDBARRIERLAYERByCHENG-CHANKUOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011Cheng-ChanKuo 2

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Tomylovingfather,mother,family,andfriendsfortheirlove,sacrice,andfullsupport 3

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ACKNOWLEDGMENTS Thecompletionofthisworkwouldnotbepossiblewithoutmycommittee,Dr.WilliamLear,Dr.OscarCrisalle,Dr.JamesFletcher,Dr.JacobChung,andDr.HerbertIngley.Theyhavesupported,guided,andencouragedmewithalotofpatiencethroughmyPh.D.program.Iwouldliketotakethisopportunitytoexpressmythankfulgratitudetothem.Iwouldliketothankmyfather,mother,littlebrotherandsisterfortheirfullsupportfrommyhomecountry,andmycousin,CindyWu,andherfamilyfortheirfullsupportandcompanyduringmyPh.D.life.Also,Iwouldliketothankmygroupmembers,SydniCredle,PraneethPillarisetti,AnupamPatil,RafeBiswas,ShyamPrasadMudiraj,WeiChen,LukeNeal,PhilBailey,JohnCrittenden,JasonHarrington,BenjaminSwanson,JasonCarryl,SabaRahmani,RussellBarton,AlexMossman,HenryVoss,andDr.Coxfortheirvaluablediscussionsandsuggestions,andmygoodfriends,FotouhAlRaqom,AyyoubMehdizadeh,BhupinderSingh,XunJia,AkikoHiramatsu,andJamesWangfortheircompanyandfulllmentofmyPh.D.life.Iwouldliketoexpressmydeepestgratitudetomylovelywife,I-Ching.Withouthersacriceandendlesslove,thisworkcouldnothavebeendone.Ialsowanttothankherforbringingmeacuteson,Wesley,andfortakingcareofhim.HisdailysmilehasbeenmymostpowerfulfuelduringmyPh.D.life. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 NOMENCLATURE ..................................... 11 ABSTRACT ......................................... 18 CHAPTER 1INTRODUCTION ................................... 20 1.1Background ................................... 20 1.1.1IntroductiontoDMFCStacks ..................... 21 1.1.2ComparisonsofDMFCtoPEMFC .................. 22 1.1.3AdvantagesofaDMFCforPortableElectronicDevices ....... 24 1.2ExistingDMFCTechnicalChallenges ..................... 25 1.2.1SlowKinetics .............................. 25 1.2.2MethanolCrossover .......................... 26 1.2.3WaterManagement .......................... 27 1.3MotivationsandObjectives .......................... 27 1.3.1Motivations ............................... 27 1.3.2Objectives ................................ 28 1.4OutlineofThisDissertation .......................... 28 2LITERATUREREVIEW ............................... 37 2.1DMFCModels ................................. 37 2.1.1EmpiricalModels ............................ 37 2.1.2AnalyticalModels ............................ 38 2.1.3NumericalModels ........................... 40 2.1.3.1Singlephasemodels .................... 41 2.1.3.2Two-phasemodels ...................... 42 2.1.4ReviewPapersforDMFCModels ................... 44 2.2BoundaryConditions .............................. 44 2.3ExperimentalWorks .............................. 45 2.3.1PolarizationCurve ........................... 45 2.3.2WaterBalance ............................. 45 2.4ConcludingRemarks .............................. 48 3SIMULATIONANDMODELING ........................... 49 3.1SimulationMethod ............................... 49 5

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3.2StackModel ................................... 50 3.2.1ModelFormulation ........................... 51 3.2.1.1Methanolmassbalance ................... 51 3.2.1.2Polarizationcurvemodel .................. 53 3.2.2ModelingImprovements ........................ 54 3.2.2.1Improvediterationalgorithm ................ 54 3.2.2.2Predictionofthemaximumcurrentdensity ........ 55 3.2.2.3SystematicmethodforestimatingOverpotentials ..... 56 3.2.2.4ConcentrationboundaryconditionsattheMEM/CCLinterface ........................... 59 3.2.3ModelValidation ............................ 60 3.3SpeciesMassTransportattheCathodeSide ................ 62 3.3.1ModelFormulation ........................... 62 3.3.2ModelDiscretization .......................... 64 3.3.3ModelValidation ............................ 66 3.3.4ModelingResults ............................ 66 3.4WaterBalanceoftheDMFCStack ...................... 67 3.4.1MassBalanceofWateroftheDMFCStack ............. 68 3.4.2WaterBalanceParameter ....................... 70 3.4.3WaterVaporMassTransportParameter ............... 71 3.5AnalysisoftheBulkConcentration ...................... 72 3.5.1ModelFormulations .......................... 72 3.5.2Results ................................. 74 4EXPERIMENTS ................................... 87 4.1ExperimentalDetails .............................. 88 4.1.1Apparatus ................................ 88 4.1.2ExperimentalConditions ........................ 89 4.1.3ExperimentalProcedure ........................ 89 4.2ErrorAnalysis .................................. 90 4.3PolarizationCurveMeasurements ...................... 92 4.3.1OverallPolarizationCurveMeasurements .............. 92 4.3.2SegregatedPolarizationCurveMeasurements ........... 92 4.4WaterBalanceMeasurements ........................ 93 4.4.1WaterBalanceParameter ....................... 93 4.4.2MassTransportResistanceofWaterVapor ............. 94 4.4.3MethanolCrossover .......................... 95 5RESULTS ....................................... 98 6CONCLUSIONSANDFUTUREWORK ...................... 119 APPENDIX ATOTALERRORANALYSISOFTHEWATERBALANCEEXPERIMENTS ... 122 6

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BMETHODSTOSOLVETHECOUPLEDREACTION-DIFFUSIONEQUATIONS 126 B.1Newman'sMethod ............................... 127 B.2FiniteVolumeMethod ............................. 132 B.3ode45Function ................................. 134 B.4Results ..................................... 135 REFERENCES ....................................... 140 BIOGRAPHICALSKETCH ................................ 145 7

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LISTOFTABLES Table page 3-1CorrectionofparametersinGarcaetal.[ 1 ] .................... 61 3-2Predictionsofmaximumcurrentdensityforfourbulkmethanolconcentrationvalues ......................................... 62 3-3Dimensionsofanodeandcathodeowchannels ................. 73 4-1Experimentalconditionsforcriticalparameters .................. 89 4-2Specicationsforthecomponentsinthefuelcellteststation ........... 91 5-1Parametervalues ................................... 99 8

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LISTOFFIGURES Figure page 1-1ComparisonofthetheoreticalefciencyofafuelcellandaCarnotengine. .. 30 1-2TheschematicofaDMFCstack. .......................... 31 1-3IllustrationofpolarizationcurvewithdifferentoverpotentialsforaDMFC. .... 32 1-4ComparisonsofthegravimetricenergydensityofDMFCsandLi-ionbatteriesversusoperationtimeperdutycycle. ........................ 33 1-5ComparisonsofthevolumetricenergydensityofDMFCsandLi-ionbatteriesversusoperationtimeperdutycycle. ........................ 34 1-6ConventionaldesignofaDMFCsystem. ...................... 35 1-7SimplieddesignofaDMFCsystemwithliquidbarrierlayer. .......... 36 3-1ThehierarchyoftheDMFCsystemlevelmodel. .................. 76 3-2SchematicofthesimulationdomainofaDMFC. ................. 77 3-3Validationofpolarizationcurvemodel ....................... 78 3-4ScanningelectronmicroscopeoftheCCL ..................... 79 3-5Validationofmulti-componentmasstransportmodel ............... 80 3-6DiagramillustratingwaterbalanceofaDMFCstack ............... 81 3-7KH2O,DO2H2O,CO2,andxO2asafunctionofLBLthickness,porosity,tortuosity,andaverageporesize. ................................ 82 3-8KH2O,DO2H2O,CO2,andxO2asafunctionofCGDLthickness,porosity,tortuosity,andaverageporesize. ................................ 83 3-9KH2O,DO2H2O,CO2,andxO2asafunctionofstacktemperature. ......... 84 3-10Molarityofmethanolsolutionalongowchannel ................. 85 3-11O2concentrationalongowchannel ........................ 86 4-1Diagramoftheexperimentalrig. .......................... 97 5-1Experimentalandmodelingresultsofsegregatedanode,cathode,andOhmicoverpotentialsat50C. ............................... 108 5-2Experimentalandmodelingresultsofpolarizationcurvesatstacktemperaturesof45C,50C,and60C. ............................. 109 5-3Temperatureeffectonthewatervapormasstransportparameter ........ 110 9

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5-4Experimentalandmodelingresultsofmethanolcrossoveratstacktemperaturesof45C,50C,55C,and60C. ......................... 111 5-5Therateofmethanolcrossoverduetodiffusionandelectro-osmoticforceat50C. ......................................... 112 5-6Waterbalanceoffuelcellstackatvarioustemperatures ............. 113 5-7Polarizationcurvesatvariousmolarities ...................... 114 5-8Methanolcrossoverrateatvariousmolarities ................... 115 5-9Watervapormasstransportparameteratvariousmolarities ........... 116 5-10Waterbalanceoffuelcellstackatvariousmolarities ............... 117 5-11Keyparametersoffuelcellstackasafunctionofporosity ............ 118 B-1Theschematicofthenodes. ............................ 136 B-2Dimensionlessreactionconcentrationasafunctionofpositionandcurrentdensity ........................................ 136 B-3Dimensionlessmatrixpotentialasafunctionofpositionandcurrentdensity .. 137 B-4Dimensionlessporepotentialasafunctionofpositionandcurrentdensity ... 137 B-5Reactionrateasafunctionofpositionandcurrentdensity ............ 138 B-6Comparisonsofresultsatcurrentdensityof0.2A=cm2 .............. 138 B-7Comparisonsofresultsatcurrentdensityof1.6A=cm2 .............. 139 10

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NOMENCLATURE aSpecicsurfaceareaofanodeelectrode,1=mhCGDLGridspaceofCGDL,mhLBLGridspaceofLBL,mjVolumetriccurrentdensity,A=m3kDimensionlessconstantintherateexpressionkGEffectivepermeability,m2_nCao,CO2RateofCO2owattheexitofcathodeowchannel,mol=s_nAni,H2ORateofthewaterowattheanodeinlet,mol=s_nAno,H2ORateofthewaterowattheanodeoutlet,mol=s_nAnH2ORateofwatergenerationduetousefulcurrentdensityattheanodeside,mol=s_nCaH2ORateofwatergenerationduetousefulcurrentdensityatthecathodeside,mol=s_nCCL!AFCH2ORateofwaterowbackfromCCLtoAFC,mol=s_nXOMeOHH2ORateofwatergenerationduetomethanolcorssover,mol=s_nrecycleH2ORateoftherecycled-waterowattheanodeside,mol=s_nventH2ORateofthevented-waterowatthecathodeside,mol=spArbitrarynon-zeronumberrAverageporesizetLBL+CGDLThicknessofLBLandCGDL,mvGGasphasevelocity,m=sxDistancefrominletofowchannel,m 11

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xiMolefractionofspecieszAbscissaAActivecellarea,m2CTotalconcentration,mol=m3CbBulkconcentrationofmethanol,mol=m3CCCLH2OConcentrationofwatervaporatCCL,mol=(m2s)CCFCH2OConcentrationofwatervaporatCFC,mol=(m2s)CAIConcentrationofmethanolatABL/ACLinterfacefromACLside,mol=m3CAIIConcentrationofmethanolatACL/MEMinterfacefromACLside,mol=m3CMIIIConcentrationofmethanolatMEM/CCLinterfacefromMEMside,mol=m3CAMeOHConcentrationofmethanoldistributionintheACL,mol=m3CBMeOHConcentrationofmethanoldistributionintheABL,mol=m3CMMeOHConcentrationofmethanoldistributionintheMEM,mol=m3CO2Concentrationofoxygen,mol=m3CO2,refReferenceconcentrationofoxygen,mol=m3Dei,jEffectivebinarydiffusioncoefcientofspecies,m2=sDAEffectivediffusioncoefcientofmethanolintheACL,m2=sDBEffectivediffusioncoefcientofmethanolintheABL,m2=sDeK,iEffectiveKnudsendiffusioncoefcientofspecies,m2=sDMEffectivediffusioncoefcientofmethanolintheMEM,m2=sECellvoltage,VFFaraday'sconstant,96485C=equiv 12

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IMeOH0,refExchangecurrentdensityofmethanol,A=m2IO20,refExchangecurrentdensityofoxygen,A=m2JcellCurrentdensity,A=m2JleakMethanolcrossovercurrentdensity,A=m2JmaxMaximumcurrentdensityatgivencondition,A=m2JxoMethanolcrossovercurrentdensity,A=m2KH2OMasstransportresistanceofwatervaporacrossLBLandCGDL,mm=sKIPartitioncoefcientatABL/ACLinterfaceKIIPartitioncoefcientatACL/MEMinterfaceLLengthofowchannel,mMMolarityofmethanolsolution,MNiMolaruxofspecies,mol=(m2s)NcellNumberofcellsNventH2OMolaruxofwatervaporfromCCLtoCFC,mol=(m2s)NAMeOHMethanolmolaruxatACL,mol=(m2s)NBMeOHMethanolmolaruxatABL,mol=(m2s)NMMeOHMethanolmolaruxatMEM,mol=(m2s)PsatSaturatedwatervaporpressure,PaPGGaseouspressuregradientacrossthelayers,PaRuUniversalgasconstant,8.314J=(molK)RHambRelativehumidityofambientTambAmbienttemperature,KTstackStacktemperature,K 13

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THHighertemperatureforCarnotefciencycalculation,KTLLowertemperatureforCarnotefciencycalculation,KUMeOHEquilibriumpotentialofmethanoloxidation,VUO2Equilibriumpotentialofoxygenreduction,VVcellCellvoltage,VWWidthofowchannels,mGreekAAnodictransfercoefcientCCathodictransfercoefcientWaterbalanceparameterAThicknessofACL,mBThicknessofABL,mCThicknessofCCL,mCGDLThicknessofCGDL,mMThicknessofMEM,mCGDLPorosityofCGDLLBLPorosityofLBLAAnodeoverpotential,VCCathodeoverpotential,VCarnotCarnotcycleefciencyAProductofAandACProductofCandCIonicconductivityofthemembrane,S=m 14

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Constantintherateexpression,mol=m3GDynamicviscosityofgases,N=(m2s)MultiplesofstoichiometryCGDLTortuosityofCGDLLBLTortuosityofLBLMeOHElectro-osmoticdragcoefcientofmethanolSuperscripteEffectiveAAnodecatalystlayerAnAnodeBAnodebackinglayerCaCathodeMMembraneMeOHMethanolSubscriptambAmbientbBulkconcentrationofmethanolchanFlowchanneliInletorSpeciesoOutletsatSaturationuUniversal 15

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xoMethanolcrossoverAAnodecatalystlayerorAnodeBAnodebackinglayerCCathodeCGDLCathodegasdiffusionlayerCarnotCarnotcycleGGasphaseIABL/ACLinterfaceIIACL/MEMinterfaceIIIMEM/CCLinterfaceIVCCL/CGDLinterfaceKKnudsenAbbreviationsABLAnodebackinglayerACLAnodecatalystlayerAFCAnodeowchannelAGDLAnodegasdiffusionlayerBLBackinglayerCCLCathodecatalystlayerCDMCatalyzeddiffusionmediumCFCCathodeowchannelCGDLCathodegasdiffusionlayerDMFCDirectmethanolfuelcell 16

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Eff(l)ConversionefciencyforhigherheatingvalueEff(g)ConversionefciencyforlowerheatingvalueGDLGasdiffusionlayerHHighertemperatureHHVHigherheatingvalueLLowertemperatureLBLLiquidbarrierlayerLHVLowerheatingvalueLWBLLiquidwaterbarrierlayerMMembraneMeOHMethanolMEAMembraneelectrodeassemblyMEMMembraneMFCMassowcontrollerMPLMicroporouslayerOCVOpencircuitvoltagePEMFCProtonexchangemembranefuelcellPTFEPolytetrauoroethyleneRSSRoot-sum-squareSLPMSpecicliterpermintueXOCrossover 17

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySIMULATIONANDEXPERIMENTALCHARACTERIZATIONOFTHEWATERMANAGEMENTFORANOPEN-CATHODEDIRECTMETHANOLFUELCELLTHATUTILIZESALIQUIDBARRIERLAYERByCheng-ChanKuoDecember2011Chair:WilliamE.LearMajor:MechanicalEngineeringWatermanagementisacriticalissueinapracticaldirectmethanolfuelcell(DMFC)system.Inthisresearch,theinteractionofwatermanagementandperformanceofaDMFCstackwithanopen-cathodedesignutilizingaliquidbarrierlayer(LBL)isstudied.Ascomparedtothetraditionaldesign,thenovelDMFCstackhasapassivewater-recoverymechanismandeliminatesthewatercollectionandreplenishmentdevicesonthecathodeside,reducingthecomplexityandsizeofthesystem.However,watermanagementofthenewDMFCstackcanimposesignicantoperatingconstraintsifthewaterbalanceisnotwell-controlled.ThepurposeofthisresearchistoanalyticallyandexperimentallystudytheeffectsofthechangeofthekeyvariablesonthewaterbalanceofthenovelDMFCstack.Amodelwasdevelopedtosimulatethecellperformance,rateofmethanolcrossover,andmulti-componentmasstransportofthenovelDMFCstack.Adimensionlesswaterbalanceparameter,,basedontheconservationofmassofthewaterinsidethestack,wasalsocreatedtofacilitatethestudyofthewaterbalanceofthestack.AwatermanagementmapofthenovelDMFCstackwascreatedbasedonthedevelopedmodel.Themodelingresultswerevalidatedwiththedatafromourexperimentsonthisnovelstackdesign.TheresultsshowedthatthestacktemperaturedominatesthecontrolofwatermanagementofthisDMFCstackdesign.Increasesinthe 18

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operatingcurrentdensityandtherateofmethanolcrossoverfavorthewaterrecoveryofthestack.However,themosteffectivewaytochangethestackfromwater-lossmodetowater-recoverymodeistoreducethestacktemperature.TheresultsalsoshowedthatthenovelDMFCstack(underthesamematerialproperties)couldoperateinwater-recoveryorwater-neutralitymodeonlyforstacktemperaturesof50Corlower,whenthecurrentdensitywasunderthenominaldesignvalueof150mA=cm2.Thedevelopedmodelcansimulatethetrendofthecellperformanceandwatermanagementofthestackbyvaryingthekeyvariables,suchasstacktemperature,solutionmolarity,andtheporosityoftheLBL.ThemodelingresultsalsoshowedthattheLBLhasamoresignicanteffectonthewaterbalanceandcellperformancethantheCGDLdoes.ByincreasingtheporosityoftheLBL30%,thecellperformanceisincreasedsignicantlybuttheventrateofthewatervaporisalsoincreased,resultinginawater-lossmode.TheresultsshowedthatadecreaseoftheporosityoftheLBLenhancesthewaterrecoveryofthestack,butthatthecellperformanceisdegraded. 19

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CHAPTER1INTRODUCTION 1.1BackgroundAfuelcellisadevicethatconvertschemicalenergydirectlytoelectricitybyelectrochemicalreactions.Comparedtoatraditionalheatengineelectricalgenerator,afuelcelleliminatesthemulti-stepprocessofconvertingchemicalenergytothermalenergy,convertingthermalenergytomechanicalenergy,andnallymechanicalenergytoelectricityenergy.Additionalenergyconversionprocessestendtoresultinlowerconversionefciencyduetoenergyconversionlossesforeachstep[ 2 ].Thetheoreticalefciencyofafuelcellishigherthanatraditionalheatengineatlowoperatingtemperature.Theenergyconversionefciencyisdenedastheratiooftheusefuloutputenergytotheinputenergyin[ 3 ].Inafuelcell,thegeneratedelectricity,whichisthedenedusefulenergy,isconvertedfromtheGibbsfreeenergyandtheinputenergyisfromtheenthalpyofthereactants.TheratiooftheGibbsfreeenergytotheenthalpyofthereactantsofafuelcellcanbeeasilyover80%,givingthemaximumefciency.Forinstance,theoverallchemicalreactionforadirectmethanolfuelcell(DMFC)isCH3OH(l)+3 2O2!CO2+2H2O(l) (1)At25C,thechangeinGibbsfreeenergyandtheenthalpyforthisreactionare-167.91kcal/moleand-182.61kcal/mole,respectively.TheratioortheenergyconversionefciencyoftheGibbsfreeenergytotheenthalpyis96.7%.Foraheatengine,themaximumefciencywhenoperatingbetweentwotemperaturesistheCarnotefciency:Carnot=1)]TJ /F4 11.955 Tf 14.03 8.09 Td[(TL TH 20

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Byvaryingthetemperature,thecomparisonofthetheoreticalefciencyofaprotonexchangemembranefuelcell(PEMFC)andaCarnotengineisshownasFigure 1-1 .Thetemperaturevariesfrom25Cto800C.NotethatEff(l)istheconversionefciencyforhigherheatingvalue(HHV)ascomparedtoEff(g),whichdenotesaslowerheatingvalue(LHV).ThedifferencebetweenHHVandLHVisthephaseoftheproductofthereactionandwateristheproductinaPEMFC.Iftheendproductisinliquidphase,HHVisgivenduetothesavingsofthelatentheatforliquid-gasphaseconversion.Ontheotherhand,LHVisobtainediftheendproductisingasphase.AsshowninFigure 1-1 ,thetheoreticalefciencyofthefuelcellishigherthanaCarnotenginebelow650CandtheefciencyofHHVcalculationishigherthanLHV. 1.1.1IntroductiontoDMFCStacksAliquid-feedDMFCisafuelcellthatfeedsmethanolandwatermixtureasthereactantattheanodeandoxygenorairatthecathode.ThechemicalreactionsattheanodeandcathodeareAnode:CH3OH+H2O!CO2+6H++6e)]TJ ET BT /F1 11.955 Tf 440.1 -383.15 Td[((1)Cathode:3 2O2+6H++6e)]TJ /F2 11.955 Tf 10.41 -4.94 Td[(!3H2O (1)At25C,thereversiblethermodynamicvoltageis0.046Vattheanodeand1.23Vatthecathode.Thisgivesthemaximumthermodynamicvoltage1.18Vfortheoverallreaction.TheelectrolyteusedinaDMFCisnormallyeitherpolymerorhydrocarbonmembrane.ThestructureofaDMFCstackiscomposedofbackingplates,gasdiffusionlayers,catalystlayers,andthemembrane.Thebackingplatesareusuallymadeofgraphiteduetogoodelectricalconductivityproperties.Afterthegasdiffusionlayer,catalystlayerandthemembranehavebeenintegratedtogether,theproductisreferred 21

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asthemembraneelectrodeassembly(MEA).Figure 1-2 showstheschematicofaDMFCstack.Ontheanodesideofthestack,amethanol/watermixtureisfedtotheowchannelsasthereactants.Themethanolmixturersttransportstothecatalystlayerviadiffusionandconvectionthroughthegasdiffusionlayer.Then,themethanolmixtureiselectrochemicallyreactedatthecatalystlayer.Asshowninequation( 1 ),theproductsarecarbondioxide(CO2),hydrogenprotons(H+),andelectrons.ThegeneratedCO2elutesattheanodeexitwiththecirculatedowandH+transportsthroughtheMEAandreactswiththeelectrons,whichowexternallytogenerateelectricity,andO2onthecathodesidetogeneratewater.Forareactiontotakeplaceonbothelectrodestogenerateelectricity,apotentialisneededtoovercometheenergybarriertomovethereactionfromequilibriumstate,whichisreferredasaoverpotential.Iftheappliedpotentialislessthanthenecessarypotentialtodrivethereaction,noreactionwillhappen.Ontheotherhand,oncetheappliedpotentialisovertheminimumpotentialtodrivethereaction,themoreoverpotentialappliedontheelectrodewillcausefasterreactionandhencegeneratemorecurrent.Therearethreetypesofoverpotentials:1)activationoverpotential,2)ohmicoverpotential,and3)concentrationoverpotential.Theactivationoverpotentialisthepotentialrequiredforthereactiontotakeplace.Theohmicoverpotentialisthepotentialduetotheinternalresistanceoftheelectrolyteaswellaselectrodesandtheconcentrationoverpotentialisthepotentiallossduethemasstransportlimitationofthereactants.Figure 1-3 illustratesthedifferentoverpotentialsandthenetvoltageofaDMFC. 1.1.2ComparisonsofDMFCtoPEMFCAlthoughthePEMFCandDMFChavemuchincommon,especiallytheMEAconstruction,therearesomemajordifferences.First,theanodereactantsareobviouslydifferent.ForaPEMFC,hydrogenisthesolereactantattheanode.Theuseof 22

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hydrogengiveshigherthermodynamicvoltage,1.229Vat25CthanaDMFC[ 2 ],1.214Vat25C.Thereactionkineticsforbothfuelcellsisalsodifferent.ThehalfchemicalreactionsattheelectrodeforaPEMFCare:Anode:H2!2H++2e)]TJ /F4 11.955 Tf -155.28 -76.66 Td[(Cathode:1 2O2+2H++2e)]TJ /F2 11.955 Tf 10.4 -4.94 Td[(!H2OADMFChasaslowerreactionkineticsattheanodecomparedtoaPEMFC[ 4 ].DuetotherelativelyslowanodereactionkineticsofaDMFC,thecatalystlayerdesignforthetwofuelcellsisalsodifferent.ForaDMFC,inordertohaveareasonablereactionrateattheanode,theplatinum(Pt)loadingisantheorderof2to4mg=cm2.InadditiontoPt,rutheniumisalsoaddedontheanodeelectrode.Theintermediateproductsfortheanodereactionsuchascarbonmonoxide(CO)canpoisontheanodecatalystlayer.Hence,thecatalystusedinaliquidDMFCisusually50/50platinum/ruthenium(Ru)alloyinsteadofpurePttopreventfrompoisoning.Theanodereactionproductsforthetwotypesoffuelcellsarealsonotthesame.AttheanodeofaDMFC,CO2isgenerated.thegenerationofCO2cancausesignicantproblemssuchasblockingtheporespacesoftheanodediffusionlayerandresultinginearliermasstransportlimitationforthemethanolmixture.Intermsofsystemdesign,thegeneratedCO2canalsocausecavitationofthefuelcirculationpumpifnotthoroughlyremoved.ThedissolvedCO2generatescarbonateacid(H2CO3)ifnotremoveddownstreamofthestack,potentiallydamagingtheothercomponents.ForaPEMFC,itisimportanttokeepthemembranewell-hydratedinordertomaintainhighhydrogenprotonconductivity.However,hydrationisnotaproblemforaDMFCbecauseoftheuseofamethanol/watermixtureasthereactant.Adifferentwater 23

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managementissue,maintainingproperanodeloopconcentration,willbediscussedinalatersection. 1.1.3AdvantagesofaDMFCforPortableElectronicDevicesADMFCisgenerallydeemedmoresuitableforportableelectronicapplicationsthantheothertypesoffuelcells.AlthoughthestructureofaDMFCMEAisverysimilartothatofaPEMFC,theDMFChassignicantsystemadvantages.TherstadvantageistheuseofliquidmethanoltofuelaDMFC.Liquidmethanoliseasiertostoreandtransportthanhydrogengas,whichisthefuelforaPEMFC.ThestorageofthefuelsignicantlyaffectsthecompactnessoftheDMFCrelativetothePEMFCbecauseofthehighvolumetricenergydensityofmethanol.Thevolumetricenergydensityofliquidmethanol,15.6MJ/L,ismuchhigherthanhydrogen,0.01MJ/L,adifferenceofthreeordersofmagnitudeunderstandardtemperatureandpressurecondition.HighervolumetricenergydensityofthefuelallowstheDMFCtobecomeapotentialcandidateforpoweringportableelectricdevices,suchaslaptopsandcellphones,providedthatthebalanceofplantcanalsobecompact.DependingonthecellefciencyoftheDMFC,thevolumetricandgravimetricenergydensityisalsosuperiortothatofaclassicalbatterypoweredsystemforlongoperationtimesperdutycycle.Figures 1-4 and 1-5 showthecomparisonsofthegravimetricandvolumetricenergydensityofDMFCstothatofLi-ionbatteriesasafunctionofoperatingtimes[ 5 ].Differenttypesofcommercially-availableDMFCproductsarecomparedtoastate-of-the-artLi-ionbattery.AlthoughthegravimetricenergydensityoftheDMFCislowerthanLi-ionbatteriesforshortoperatingtimes,DMFCswilleventuallysurpassthebatteriesforlongoperatingtimes.HigherefciencyDMFCssurpassbatteriesbeginningatsmalleroperationtimes.Figure 1-5 alsodemonstratesthesametrendforthevolumetricenergydensitycomparison.Liquidmethanolisamaturechemicalproductandeasilytoobtain,producedprimarilyfromsynthesisgas.Thechemicalreactionequationis 24

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CO+2H2!CH3OHThemethanecomponentinnaturalgasisthemajorfeedstock.ThesynthesisgascanbeobtainedviathesteamreformingreactionCH4+H2O!CO+3H2Inadditiontonaturalgas,coalandbiomassareincreasinglyusedasthefeedstocktoproducemethanol. 1.2ExistingDMFCTechnicalChallengesAlthoughaDMFChasstrongpotentialtopowerportableelectronicdevices,therearesomeexistingproblemsthathinderthedevelopmentofthistechnology,includingthenaturaloftheslowmethanoloxidationreactionmechanism,theundesirableeffectsofmethanolandwatercrossover,andanoften-observeddegradationofthecellperformancewithtime[ 6 ].Activeresearchanddevelopmentisongoinginordertoaddressthefollowingproblems. 1.2.1SlowKineticsAsmentionedabove,thereactionkineticsattheanodesideofaDMFCismoresluggishascomparedtoaPEMFC[ 4 7 ].Ameasurementtoevaluatethekineticsofthereactantonanelectrodeistheexchangecurrentdensity.Theexchangecurrentdensityisthereactionrateatwhichareactionproceedsatequilibrium[ 8 ].TheexchangecurrentdensityforhydrogenoxidationonaPtelectrodeat25Cisabout10)]TJ /F7 7.97 Tf 6.58 0 Td[(4A=cm2[ 3 ].Formethanoloxidation,theexchangecurrentdensityisabout10)]TJ /F7 7.97 Tf 6.58 0 Td[(5to10)]TJ /F7 7.97 Tf 6.59 0 Td[(6A=cm2[ 9 ],onetotwoordersofmagnitudelowerthanhydrogenoxidationinaPEMFC.ThelimitingcurrentdensityforaPEMFCisalsohigherthanforaDMFC,approaching1.5A=cm2;however,foraDMFC,thelimitingcurrentdensitycanonlyreach0.5A=cm2. 25

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Duetotheslowerreactionkinetics,highercatalystloadingisneededtoachievereasonableperformance.ForaPEMFC,theloadingofPtisaround0.2mg=cm2andtheloadingofPtforaDMFCisaround4mg=cm2.TheloadingofPtadverselyaffectsthemanufacturingcost. 1.2.2MethanolCrossoverMethanolcrossover[ 10 ]isthephenomenoninwhichasmallportionofthemethanolpermeatesfromtheanodesidetothecathodesideofaDMFC.Thedrivingforcesforthemethanolcrossoverarediffusion,duetotheconcentrationgradient,electro-osmoticdrag,duetohydrogenprotontransport,andconvection,duetothepressuregradientacrossthefuelcellstack.Ahigherrateofmethanolcrossoverresultsinalowercellperformancebecauseofthemixedpotential[ 11 12 ]andwaterooding[ 13 ]atthecathodeside,inadditiontosysteminefciencyduetodirectlossoffuel.Thewateroodingonthecathodesideoccupiestheporesinthediffusionlayerandhencerestrictstheoxygentransporttothereactionzone.Therearesomefeasiblesolutionsformethanolcrossover.Therstoneistoincreasethethicknessofthemembrane,suchasusingNaonR117toreplaceNaonR112.Theeffectsofchangingthemembranethicknessonthecellperformancecanbefoundin[ 14 ].Certainlytheincreaseofthemembranethicknesswillincreasetheinternalresistance.However,theoverallcellperformanceisimprovedbyreducingthemethanolcrossovereffects.Anothersolutiontolowermethanolcrossoveristoreducethemethanolconcentrationtofeedintothefuelcellstack.Typically,themethanolconcentrationusedinoperationisbetween0.5Mto1.0M.Theconsequenceofreducingmethanolconcentrationistheneedforhighersolutionowrates.Lowermethanolconcentrationsalsoinducesthemasstransportlimitationontheanodesidethanthatofhigherconcentrations. 26

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1.2.3WaterManagementThewatermanagementiscriticalfortheimplementationofDMFCsforportableelectronicdevices.IntheconventionaldesignofaDMFCsystem,aportionofthewatergeneratedonthecathodesideiscondensedattheexitofthecathodeairstream.Thereactantwaterisreplenishedintheanodeloopbypumpingthecollectedwaterfromthecathodeexit.Thetraditionaldesignofwatermanagementsystemcomplicatesthedesign,addssignicantsystemvolumeandweight,andaddstoparasiticpowerconsumption.Inadditiontotheundesirableadditiontothesystemvolumeandweight,watermanagementisalsoacriticalissueforsystemoperation.Foragiventankwithconstantvolumeofmethanol-watermixture,thehigherlossrateofwatercausesmorefrequentlyreplenishmentofwaterbythecustomers.Tocompensatethehighwaterlossrate,abiggermethanol-watermixturetankcanbesupplied,but,again,thisincreasesthesizeofaDMFCsystem. 1.3MotivationsandObjectives 1.3.1MotivationsADMFCisapotentialcandidateforportableelectronicdevices.However,slowreactionkinetics,methanolcrossover,celldegradation,andwatermanagementaretheexistingproblemstoberesolvedtocommercializethistechnology.Figures 1-6 and 1-7 showtwokindsofcathodedesigns,closed-cathodeandopen-cathode,ofaDMFCstack.Intheclosed-cathodedesign,thewaterstorage,condensing,andpumpingsystemsareneededonthecathodesidetorecirculatethewaterbacktotheanodeside.ThiscomplicatestheDMFCsystemdesignandmakesthesystembulky.Anopen-cathodedesigneliminatestheconventional,bulkywatermanagementsystem,astrongadvantageoftheopen-cathodeconguration.Theopen-cathodedesignofDMFCstacksisimplementedbytheadditionofahydrophobicliquidbarrierlayer(LBL)intheMEAbetweenthecathodecatalystlayer(CCL)and 27

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thecathodegasdiffusionlayer(CGDL).TheLBLallowspassivereplenishmentofthereactantwaterintheanodestreamviatransportthroughthemembrane,abetterwatermanagementsystem.Inthisresearch,thewatermanagementissueofaDMFCstackwithanopen-cathodedesignwasaddressed.InaDMFC,thewaterisconsumedontheanodesideduetotheelectrochemicalreaction.TomaintaintheDMFCstackinwaterbalance,theconsumedwaterneedstobereplenishedfromthewatergeneratedonthecathodeside.Inanopen-cathodedesign,theDMFCstackcanalsolosethegeneratedwatertotheopenowchannelonthecathodeside.ThewatergenerationandlossratedependontheoperatingconditionandmaterialusedinsidetheMEAs.Hence,thewatermanagementiscrucialtoaDMFCsystemwithanopen-cathodedesign.AgoodwatermanagementsystemcansimplifyandcompactthedesignofaDMFCsystem.ItcanalsoimprovetheparasiticpowerconsumptionandmolaritycontrolofaDMFCsystem 1.3.2ObjectivesTheobjectivesofthisresearcharetoexperimentallyandanalyticallyunderstandthewatermanagementofaDMFCsystemwithanopen-cathodeMEAdesign.Anintegratedmodelwillbecreatedtosimulatethepolarizationcurve,methanolcrossover,multi-componentmasstransportofthereactantspecies,andwaterbalanceoftheDMFCstack.Themodelingresultswillalsobevalidatedwiththeexperimentaldatabymeasuringthepolarizationcurve,watervaporandCO2concentrationatcathodeexit,andcertainphysicalquantitiesoftheDMFCsystem. 1.4OutlineofThisDissertationTheorganizationofthisdissertationisasfollows.Chapter 2 givesaliteraturereviewondifferentDMFCmodelsandexperimentalworks.WatermanagementofaDMFCisalsodiscussed.Chapter 3 presentsthesimulationandmodelingtechniqueusedtoconductthisresearch.Thevalidationsofthemodelswithexistingliteraturearealsopresented.InChapter 4 ,theexperimentsconductedwiththeopen-cathodeDMFCstack 28

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arepresented.Theerroranalysisandexperimentalconditionsaredescribed.Chapter 5 presentsmodelingandexperimentalresultstogetherandChapter 6 presentstheconclusionsandfutureworksneededtobedonetomakethisstudymorethoroughly.Inadditiontothebodychapters,Appendix A showsthetotalerroranalysisfortheexperimentsconductedinthisresearchandAppendix B demonstratesdifferentnumericaltechniquetosolvethenonlinearelectrochemistryequation. 29

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Figure1-1. ComparisonofthetheoreticalefciencyofafuelcellandaCarnotengine. 30

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Figure1-2. TheschematicofaDMFCstack. 31

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Figure1-3. IllustrationofpolarizationcurvewithdifferentoverpotentialsforaDMFC. 32

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Figure1-4. ComparisonsofthegravimetricenergydensityofDMFCsandLi-ionbatteriesversusoperationtimeperdutycycle. 33

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Figure1-5. ComparisonsofthevolumetricenergydensityofDMFCsandLi-ionbatteriesversusoperationtimeperdutycycle. 34

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Figure1-6. ConventionaldesignofaDMFCsystem. 35

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Figure1-7. SimplieddesignofaDMFCsystemwithliquidbarrierlayer. 36

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CHAPTER2LITERATUREREVIEWInthischapter,theliteraturereviewfordifferenttypesofDMFCmodelsandexperimentalworksispresented.Themodelsarecategorizedintoanalyticalmodelsandnumericalmodels.Innumericalmodels,theliteratureisfurtherbeingreviewedassingleandtwo-phasemodels.Themodelsreviewedaremainlyfocusingonthemasstransportfordifferentspecies.Hence,mostmodelsareassumedisothermal.Thisworkliststheliteraturethatimposedinappropriateboundaryconditionattheinterfacebetweenmembraneandcathodecatalystlayer.Thereasonstoconsidertheimposedboundaryconditioninappropriatearealsoaddressed.Attheendofthischapter,literaturereviewfortheexperimentalworks,especiallyinwatermanagement,ofDMFCsispresented. 2.1DMFCModelsInthepasttwodecades,theDMFCisanemergingtechnologyduetothebreakthroughinthematerial.SinceaDMFCisaverycomplicatedelectrochemistrysystemandsomein-situparametersarehardtomeasure,therearealotofmodelscreatedtounderstandandpredicttheperformanceofaDMFC.Themodelscanbebrieycategorizedintoempirical,analyticalandnumericalmodelsasfollows.Inthisstudy,wefocusedonreviewingthemasstransportphenomenaoftheliquid-feedmethanolfuelcell. 2.1.1EmpiricalModelsSrinivasanetal.[ 15 ]publishedanempiricalequationthatdescribedthelowandintermediatecurrentdensityregionofthepolarizationcurve.Themodelshowedgoodreproducibilitybelowtheintermediatecurrentdensityofthepolarizationcurve;however,failedatthehighcurrentdensityregion.Kimetal.[ 16 ]proposedanempiricalmodelforaPEMFCtottheexperimentaldataatdifferentoperationconditions.Anexponentialtermintheempiricalequationwasintroducedtofacilitatethepredictionatthemasstransportlimitationregion. 37

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Squadritoetal.[ 17 ]furtherdevelopedanempiricalequation,basedonKimetal.[ 16 ],bychangingthemasstransportlimitationterm.Althoughthemodeltwellwiththeexperimentaldata,itcannotdeneagoodmathematicalcorrelationbetweenthe,acorrelatingvariablefortheelectrodeproperties,andthephysicalpropertiesofthesystem.Argyropoulosetal.[ 18 ]proposedanalternativeempiricalequationforthedirectionmethanolfuelcell.Insteadofusingthreevariablesforthemasstransportlimitationterm[ 17 ],thismodelusedonlytwovariables.Theauthorspointedouttheshortagesoftheempiricalmodelweresensitivetotheinitialguessandnothavinguniquesolutioninmostcasessinceanonlinearequationwassolved.FraserandHacker[ 19 ]publishedamodiedempiricalmodel,especiallyforttingsmallcurrentdensityregion.Theinternalcurrentdensityandexchangecurrentdensitywereintroducedinthechargetransferoverpotentialtermtoeliminatethettingdiscrepancyatthelowcurrentdensityregion. 2.1.2AnalyticalModelsKulikovsky[ 20 ]proposedamodeltopredictthepolarizationcurve.Thismodelwasbasedontheanalyticalsolutionofthecatalystlayerpolarizationvoltage[ 21 ].Attingequationintermsofthekineticandtransportparameterswiththeprescribedexactsolutionwasusedforthederivationofthepolarizationcurvepredictionformula.Thecoefcientsofthettingequationwereextractedfromtheexperimentaldata[ 22 ].Theoptimalfeedmethanolconcentrationintheowchannelwasalsoderived.Garcaetal.[ 1 ]presentedasimpleone-dimensional(1D),isothermal,semi-analyticalmodel,whichwasdevelopedtodescribethemethanolconcentrationdistributioninadirectmethanolfuelcell.Multi-stepsmethanoloxidationmechanismandmixedpotentialeffectduetocrossovermethanolwereincludedinthemodel.Thismodelcanpredicttheamountofmethanolcrossover,methanolconcentrationdistributionandthepolarizationcurveasafunctionofcurrentdensity.Althoughthismodelcanbefastimplemented, 38

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aftercarefulexaminationofthemodel,theresultspublishedwerenotconvergedathighcurrentdensityregionandtheimposedboundaryconditionattheendofmembranewasnotappropriate.Guoetal.[ 23 ]publishedatwo-dimensional(2D)analyticalmodelofadirectmethanolfuelcellwasdeveloped;however,thismodelwasmorelikeasemi-2Danalyticalmodelsincetheaverageconcentrationandcurrentdensityofowchannelwereusedtoderivetheanalyticalsolutionofmasstransportperpendiculartothefeedowdirection.Theboundaryconditionattheendofmembranewassettozerotoo.Tafelkineticswasusedtosimulatethereactionmechanismsonbothelectrodes.Modelparameterswereextractedfromttingtheexperimentalresultsandtheminimumvelocityofinletair/oxygenstreamcanbepredictedbasedontherequirementofmassbalanceofoxygenreductionandtheoxidationofthecrossovermethanol.Scottetal.[ 24 25 ]publishedaonedimensionalmodel.Itwassolvedanalyticallytoexplorethecurrentandpotentialdistributioninaporouselectrode.Theeffectofthemasstransportlimitationinsidetheporeswasalsotakenintoaccount.Giventhesamecurrentdensity,theparametricstudyshowedthattheincreaseofcatalystsurfaceareaandmasstransportcoefcientresultsinalowerelectrodeoverpotential.However,Kulikovsky[ 26 ]pointedoutamajorerrorlaterafterthepublishedpaper.Inthissameyear,Scottetal.[ 24 25 ]publishedanotheronedimensionalmodelwithtwomechanismsofmethanoloxidation[ 27 28 ].Thekeyparameterswereextractedfromtheexperimentaldata.Parametricstudies,suchasmethanolconcentrationanddimensionlessmethanolpolarization,wereconductedtocharacterizethefuelcell.Themajordifferencebetweenthismodeland[ 24 ]wasthattheTafelkineticswasusedinsteadofusingthemulti-stepsmethanoloxidationmechanisms.Shivhareetal.[ 29 ]presentedananalyticalmodelforanodepolarizationcurve.Thekineticmodelwasbasedonthesurfacecoverageofintermediatesandthemodelingresultswerevalidatedwiththeexperimentalworks. 39

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Vera[ 30 ]publishedathree-dimensional(3D)/1DmodelforaDMFC.The3Dmodelfortheanodeowchannelandgasdiffusionlayerwascoupledwiththe1Dmasstransportmodelthroughthelayersinsidethestack.The3Dsimulationwassolvedbyacommercialcomputationaluiddynamics(CFD)package,FLUENT,forthemethanolconcentrationdistributionontheanodeside.Afterthelocalmethanolconcentrationandcellvoltagewereobtained,asetofanalyticalequationsasafunctionofmethanoluxandconcentrationwerereadytobesolved.Kareemulla[ 31 ]presenteda1D,singlephase,semi-analyticalmodelforaDMFC.Thismodelaccountedforthediffusionandconvectivetransportontheanodesideandonlydiffusiononthecathodeside.Themulti-stepmethanoloxidationmechanismwasusedfortheanode.TheTafeltypekineticswasusedtoaccountforthemixedpotentialphenomenononthecathodeside.Stefan-Maxwellequationswereusedinthismodeltodescribethediffusiontransportforthemulti-componentspecies.Theresultshowedthatthelimitingcurrentdensity,causedbymasstransportlimitation,wasinducedbyfeedinghighmethanolconcentrationandresultedinahighmethanolcrossoverratetodepletetheoxygenonthecathodeside.Mosquera[ 32 ]derivedtheanalyticalsolutionsforthemethanolconcentrationandprotoncurrentdensityintheanodecatalystlayerforaDMFC.TheanodeoverpotentialwasexpressedasafunctionofthecurrentbytheThielemodulus.Theresultshowedthattheanodeoverpotentialvariedquadraticallyathighcurrentdensity. 2.1.3NumericalModelsBesidestheanalyticalmodels,alotofcomplicatedmodelswerealsopresentedinthepastdecade.Aliquid-feedDMFCisaverycomplicatedelectrochemistrysystem.Itincludestwo-phasemasstransportphenomena,reactionkineticsinthecatalystlayer,microstructuresindifferentlayers,anddifferentowchannelgeometries,etc.Tocaptureallthosecomplicationinamodelisquietimpossible.Inthissection,modelsaccountedfordifferentphysicalphenomenaintheliteraturearereviewed. 40

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2.1.3.1SinglephasemodelsScottetal.[ 33 34 ]presentedsinglephasemodelstostudythelimitingmasstransportofmethanolontheanodeside.Thismodelwasusedtopredicttheeffectivemethanolconcentrationattheanodecatalystlayer.Baxteretal.[ 35 ]presentedanisothermal,steady-statemodelforaDMFC.Theyassumedtheanodestructureasthecatalystparticlescoatedwithathinion-selectivepolymerlayerandtheevolvedcarbondioxidewasalwaysdissolvedinthemixture.Theelectrolyteconductivityandthicknessoftheanodelayerwerefoundasimportantparametersathighcurrentdensity.MeyersandNewman[ 36 38 ]developedamodeltostudytheequilibriumofmulticomponentspeciesandtransportphenomenainaDMFCbasedontheconcentrated-solutiontheory.However,theconvectivedrivingforceforthespecieswasnotconsidered.NordlundandLindbergh[ 39 ]presentedanagglomeratemodeltostudytheinuenceoftheporousstructureofaDMFC.Theresultshowedthatthemasstransportlimitationontheanodesidecanbeneglectedexceptforlowmethanolconcentration.Modelingresultswerealsovalidatedwiththeexperimentaldata.JengandChen[ 40 ]presenteda1D,singlephaseDMFCmodel.TheTafelkineticswasusedtosimulatetheanodecatalystlayerreaction.Theresultsshowedtheprotonicconductivityhadmoreimpactonthereaction-ratedistributionthanthemethanoldiffusioncoefcient.However,thismodelsettheboundaryconditionattheendofthemembranetozerotosimplifythecalculation.Equation(8)inthispaperisalsoquestionable.Sincethemodelconsideredtheconvectiontermofmethanolmasstransport,themethanoluxintheanodediffusionlayerisafunctionofmethanolconcentration.Thegradientofthemethanolconcentrationintheanodediffusionlayerisnotaconstantand,therefore,themethanoluxshouldnotbeconstant. 41

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Saarinenetal.[ 41 ]publisheda3D,singlephasemodelforanairbreathingDMFC.Insteadofvalidatingthemodelingresultswiththepolarizationcurve,theycomparedtheirresultstothecurrentdensitydistributionmapstheymeasuredbefore. 2.1.3.2Two-phasemodelsMurgiaetal.[ 42 ]presenteda1D,two-phase,andsteadystatemodelforaDMFC.Theyintroducedananalyticaltreatmentforthenonlinearkineticequation.Theyalsoaccountedforthetwo-phasetransportphenomena,waterooding,condensationandevaporation,inthediffusionlayer.Kulikovsky[ 43 ]presenteda2Dmodelbasedonthemassandcurrentconservationequations.Theresultsshowedthatthemethanoltransportneartheowchannelwasmainlyduetopressuregradientanddiffusionwhileclosetothemembrane.Thehydraulicpermeabilityofthebackinglayerresultedintheinversepressuregradientasthevaluecomparabletothevalueatthemembraneandtheactivelayers.WangandWang[ 44 ]publishedatwo-phase,multi-componentmodelofaDMFC.Drivingforces,diffusionandconvection,weretakenintoaccountforbothgasandliquidphases.Themixed-potentialeffectsduetomethanolcrossoverandcapillaryeffectsonbothsideswerealsosimulatedinthismodel.Theresultsshowedthatthelimitingcurrentdensityincreasedwiththeincreaseofthefeedingmethanolconcentrationbelow1M;however,thecellpotentialdroppeddramaticallyastheincreaseofthemethanolconcentrationover2Mduetotheexcessivemethanolcrossover.Theexcessivemethanolcrossoveralsocausedtheoxygenmasstransportlimitation.Diviseketal.[ 45 ]developeda2D,two-phasemodelforaDMFC.Thewaterinthediffusionlayerwasmodeledbasedoncapillaryeffectsvaryingintheporespace.Thehydrophobicandhydrophiliceffectswereaccountedforbyintroducingarelationbetweencapillarypressureandsaturation.InsteadofusingTafelorButler-Volmerequationstodescribethereactionkinetics,multi-stepmethanoloxidationreactionequationswereused. 42

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YangandZhao[ 46 ]presenteda2D,two-phase,isothermalmodelforaliquid-feedmethanolfuelcell.Thismodelusedclassicalmulti-phasetheoryintheporouslayers.Amodieddrift-uxmodelandmist-owmodelwereutilizedfortheanodeandcathodeowchannel,respectively.Amicro-agglomeratemodelwasusedtomodelthecathodecatalystlayerforoxygentransport.GeandLiu[ 47 ]publisheda3D,isothermal,two-phaseowmodelforaliquidmethanolfuelcell.Theresultsshowedthatthesinglephasemodeloverpredictedthemethanolcrossoverrate.Italsoshowedthattheporosityofthediffusionlayeraffectsthecellperformance.LowporositylayerscannotremovethegeneratedCO2andblockedthemethanoltransporttothereactionzone.Thephenomenonwasthesameforwateroodingonthecathodeside.Casalegnoetal.[ 48 ]developedatwo-phase,1D+1DmodelforaDMFC.Themodelingresultswerevalidatedwith[ 49 ],composedof6polarizationcurves.Therewerefourttingsvaluesandsomeassumedparametersusedinthismodel.Theauthorsappliedstatisticmethodtocalibratethemodelswiththepublishedexperimentaldata.YanandJen[ 50 ]publisheda2D,two-phaseoemodelforaDMFC.Theyfoundoutthatallthegoverningequationwereinthesimilarformatandcanbesolvedasinonedomainwithouttheinterfaceboundaryconditions.Theeffectsofthekeyparameters,suchastemperature,pressure,andconcentration,onthecellperformancewerediscussed.Heetal.[ 51 ]presenteda2D,two-phaseowmodeltounderstandthemasstransfercharacteristicsofaDMFC.Thismodelinvestigatedtheliquid-gascounterconvectioneffectbysolvingthevelocityforliquidandgaseousphasesintheanodeporousstructure.Theresultsshowedthatthepropertiesandstructureoftheporouslayerarethemajorresistanceformethanoltransport.Increasingtheporesizeornumbersoftheporouslayerscanhaveabettercellperformance. 43

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Yeetal.[ 52 ]developeda2D,two-phasemodelforasemi-passiveDMFCwithanopencathodedesign.Theresultsshowedthatthemethanolcrossoverwasdominatedbydiffusionatlowcurrentdensitiesanddominatedbyelectro-osmoticforceathighcurrentdensities.Thewatergenerationonthecathodesidewasmainlyduetotheoxidationofthepermeatemethanolfromanodeatlowcurrentdensities.Athighcurrentdensities,watercrossovercontributedthemajorwateramountonthecathodeside.Yangetal.[ 53 ]presentedatwo-phasemasstransportmodeltopredicttheliquidwaterdistributioninthediffusionmedium.Insteadofapplyingacurrentindependentboundaryattheowchannelanddiffusionlayerinterface,asimpletheoreticalapproachcombinedwithanin-situmeasuredwater-crossoveruxwasproposedtodeterminethewatersaturationinthecatalystlayers. 2.1.4ReviewPapersforDMFCModelsWang[ 54 ]presentedareviewpaperaboutthefundamentalmodelingworksforfuelcells.AsetofgenericequationsforCFDandsolvingtechniqueswererstintroduced.Thereafter,differenttypesofmodelswerediscussed.Oliveiraetal.[ 55 ]presentedageneralreviewofDMFCmodels.TheauthorsdividedDMFCmodelsintoanalytical,semi-empirical,andmechanisticcategories.Themechanisticmodelswerefurthersub-categorizedintosingle-domain(unied)andmulti-domain(governingequationsfordifferentlayers)models.Aradarchartwasusedtodemonstratedifferentfeaturesofinterestsofmodels(dimensions,transportphenomena,methanolcrossover,thermalmanagement,etc.). 2.2BoundaryConditionsTheboundaryconditionofmethanolattheendofmembraneaffectstheoverallmodelingresults.Inordertoreducethecomplexityofthemodels,zeromethanolconcentrationassumptionwasappliedatthemembraneandcathodecatalystlayerinterface[ 1 30 33 40 56 ].Themajorreasonforthemtocomeupwiththezeromethanolconcentrationassumptionisthemethanoloxidationrateatthecathode 44

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catalystisfast.Themethanolconcentrationatthemembraneandcathodecatalystlayerinterfaceisverylow. 2.3ExperimentalWorksTheliteraturereviewsoftheexperimentalworksaremainlyfocusingonthewatermanagementofaDMFC.Theeffectofdifferentoperatingparametersonthecellperformanceisalsobrieyreviewed. 2.3.1PolarizationCurveGeandLiu[ 57 ]experimentallystudiedtheeffectsofthemajoroperatingparametersontheperformanceofaDMFC.Thestudiedparameterswerecelltemperature,methanolconcentration,anodeowrate,cathodeairowrate,andthehumidicationonthecathodeside.Theresultsshowedthatthecathodehumidicationhasminoreffectsonthecellperformance.Theresultalsoshowedthattheincreaseoftheairowrateorthepartialpressurecanreducethemethanolcrossoverrate. 2.3.2WaterBalanceThewatertransportthroughthemembraneisgovernedbythreemechanisms:electro-osmoticdrag,diffusionandconvection.Theelectro-osmoticdragismainlyduetohydrogenprotontransport.Thediffusionisgovernedbythewaterconcentrationgradientandtheconvectionisduetoahydraulicpressuregradientbetweenanodeandcathodeside[ 56 58 60 ].Asaforementioned,aDMFCsystemwithbetterwatermanagementcanhavebetterperformanceintermsoftheefciencyandruntime.Oneofthemethodstoimprovewaterrecoveryistoincreasethepressuregradientbetweenthecathodeandanodebychangingthecathodestructure.Peledetal.[ 61 ]proposedahydrophobicliquidwaterbarrierlayer(LWBL)bymodifyingthecathodestructureforaDMFC.Severalhydrophobiclayerswereappliedonbothsidesofthecurrentcollector.Thisstructurecreatedhighhydraulicpressureonthecathodeside.Thus,thepressuregradientbetweenanodeandcathodewasincreasedforwatertoowbackfromanodetocathode. 45

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Byapplyingthesamestructure,Blumetal.[ 62 ]wasabletoreducetheratioofwaterventedfromthecathodetothemethanolconsumedattheanodeby3.5times.Byoptimizingthewaterrecoverystructure,awater-neutraloperationcanbeachievedforlowconcentrationDMFCs.Luetal.[ 63 ]reportedanewdesignofMEAwiththecombinationofNaonR112andamicroporouslayer(MPL).ThehighlyhydrophobiccharacteristicoftheMEAhadthesamefeaturetocreatehydraulicpressureforwaterbackow.AnimprovedMEAwasfurtherdemonstratedbythesamegroup[ 64 65 ].Acatalyzeddiffusionmedium(CDM)wasappliedontheanode.TheCDMservedasamethanoldiffusionbarriertoreducethemethanolcrossoverrate.Xuetal.[ 66 ]experimentallystudiedtheeffectonthewatertransportandcellperformancebyvaryingthecontentofpolytetrauoroethylene(PTFE)loadinginthebackinglayer(BL)andMPL.ThecarbonloadingintheMPLonthecathodesidewasalsostudied.ItwasfoundthathighPTFEloadingintheBLcanresultinbadcellperformanceandunstabledischargingprocessduetotheincreaseoftheoxygentransportresistance.However,theincreaseofPTFEloadingintheMPLcaneffectivelyreducethewatersaturationlevelandfurtherreducetheoxygentransportresistance.TheoptimalPTFEloadingvalueintheMPLwas40wt%intheirwork.ThecarbonloadingintheMPLhadthesameeffectonwatercrossoverandoxygentransportresistance.Theoptimalcarbonloadingwasreportedas2.0mg=cm2.XuandZhao[ 58 ]alsopresentedanin-situmeasurementofwatercrossoverforaDMFC.However,thisworkwasdonebymeasuringthecollectionofwateratthecathodeexitandanalyticallydeterminedthewater-crossoverux.Theyinvestigatedtheeffectsofdifferentoperatingconditions,suchascelltemperature,membranethickness,methanolconcentration,etc.,anddesignparametersonthecellperformance.TheresultsshowedthatthecombinationofPTFE-treatedBLandhydrophobicMPLcansignicantlywatercrossoverandthusimprovethecellperformance.Themasstransport 46

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limitationoftheonlyPTFE-treatedBLwasmuchlowerthantheuntreatedBLbecauseoftheblockingofoxygentransportpassageintheBLbyTeonlm.Songetal.[ 67 ]studiedtheeffectsofchangingtheMEAstructureonthewatercrossover.TheresultsshowedthattheintroductionofaMPLinthecathodediffusionlayer,thereductionofthemembranethickness,andthedecreaseofhydrophobicityinthebackinglayercanreducethewatercrossover.AnadditionalMPLinthecathodediffusionlayercanincreasethehydraulicpressuredifferenceandresultedinthereductionofwatercrossover.Thereductionofthewatercrossovercanalsoreducethemethanolcrossoverrate.Tianetal.[ 68 ]presentedanin-situanalysisonthewatertransportofaDMFCunderdurabilitytest.TheirworkshowedthattheaccumulatedwaterinthecathodeGDLcausedtemporarycelldegradation.Thetemporarydegradationwaspartiallyrecoveredafterblowingdryairfor150hours.ThemicrostructureandhydrophobicpropertiesofthecathodeGDLbeforeandafterdurabilitytestwerealsocharacterizedandcompared.JiangandChu[ 69 ]experimentallystudiedthewatercrossoverofaDMFCstackunderconstantvoltage.Thecrossoverofwaterandmethanolwasdeterminedquantitativelybasedonthemassbalanceanalysis.Thestudyshowedthattheamountofwatercrossoverwasmuchhigherthanthewatergeneratedbymethanoloxidationreaction.Thewaterevaporationonthecathodesidewashighlydependentontheoperatingconditions.Thedisadvantageoftheirwaterbalancemeasurementwastheexperimentswereconductedunderconstantvoltageinsteadofconstantcurrentmode.Atconstantvoltagemode,thecurrentdensitycanuctuateandhenceaffectthewaterbalancemeasurement.PolyFuelInc.[ 70 71 ]demonstratedaDMFCsystemformobiledeviceswithahydrophobictreatedliquidbarrierlayer(LBL).TheLBLappliedonthecathodesidecreatesthehydraulicpressure,whichenableswatertoowbackfromcathodetoanode. 47

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Zhaoetal.[ 72 ]furthercategorizedthewaterrecoverymethodsintothreedifferentmethods,activewaterrecovery,passivewaterrecoverywithpuremethanol,andpassivewaterrecoverywithdilutemethanol. 2.4ConcludingRemarksLiteraturereviewedwasdoneinChapter 2 intermsofDMFCmodels,boundaryconditionimposition,andexperimentalworks.TheDMFCmodelswerefurtherreviewedasempirical,analyticalandnumericalmodels.Simpliedsinglephaseandtwo-phasemodelsweremainlyreviewedinnumericalmodelsection.Theimposedboundaryconditionattheinterfaceofthemembraneandcathodecatalystlayerwasalsobrieyreviewed.Theexperimentalworksweremainlyreviewedinthemethodsofreducingwatercrossover.Asshowninthemodelingreviewsection,therearedifferentadvantagesanddisadvantagesonthemodels.Itiseasiertoconstructtheempirical,butthemodelscannotbeusedforfurtherpredictions.TheanalyticalandsinglephasemodelscanbequicklysolvedandimplementedinaDMFCsystemdesign,butthesacriceofaccuracyareunavoidable.Thetwo-phaseowmodelsareabletocapturemorephysics,buthardertobeimplementedinasystemdesign.Inthiswork,asemi-analyticalmodel[ 1 ]ischosenasafoundationbecauseofthequickimplementationinasystemdesign.Somecritiquesarebroughtupaftercarefullyexaminedthemodel.Thefurtherimprovementsandapplicationsofthemodelsarepresentedinthefollowingchapter.Asaforementioned,watermanagementisanimportantissueforthesystemdesignofaDMFC.Themajorworksdoneintheliteratureweretryingtochangethepropertiesofthematerialtoincreasethehydrophobicityandthusreducethewatercrossover.Thefocusesofthoseliteratureswereallontheeffectsofthecellperformance;notmanyofthemdiscussedtheeffectsofcriticaloperatingparametersonthewatermanagement.Thisworkexperimentallyconcentratesontheeffectsofdifferentoperatingparametersonthewatermanagementandmethanolcrossover. 48

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CHAPTER3SIMULATIONANDMODELINGThesimulationofthewatermanagementofaDMFCstackwithanopen-cathodeMEAdesignwaspresentedinthischapter.TounderstandthewatermanagementofaDMFCsystemwithanopen-cathodeMEAdesign,thesimulationoftheuxandsourcetermsofwaterisrequiredandhenceacomprehensiveDMFCstackmodelisneededforthisstudy.ThecomprehensivemodelcreatedinthisworkshouldincludethesimulationsofthewatergenerationandlosstermsoftheDMFCstack.ThesourcetermsofwatergenerationincludeusefulelectrochemicalreactionandmethanolcrossoverinsidetheDMFCstack.Ontheotherhand,thesourcetermsofwaterlossconsistsofusefulelectrochemicalreactionandtheventofwatervaportocathodeowchannelduetotheopen-cathodedesignoftheDMFCstack.Themethanolcrossoverandtheventrateofwatervaporcoupledthestackperformance.ToaccuratelysimulatethewatermanagementoftheDMFCstack,acomprehensivemodelshouldcomprisestackperformance,multi-componentmasstransportandwaterbalanceoftheDMFCstack.Inthiswork,theDMFCmodelincludesthreeparts:1)Stackperformancemodel,2)multi-componentmasstransportmodelatcathodeside,and3)massconservationofwateroftheDMFCstackundervariousoperatingconditions.Thedetailsofeachpartweredemonstratedasfollows. 3.1SimulationMethodTheDMFCsimulationisprogrammedinacommercialsoftwarepackage,MATLABR[ 73 ].ThehierarchyoftheDMFCsystemlevelmodelisshownasFigure 3-1 .ThehierarchyseparatestheDMFCsystemintofuelsideandairside.Atthefuelside,thesimulatedcomponentsincludemethanolfeedpump,methanolwatermixturerecirculationpump,gasliquidseparator,methanolsensor,andthemethanolcartridge.Attheairside,afanservesasthefunctionforcoolingandprovidingairissimulated. 49

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Thestackmodelinthisworkmainlyfocusesonthestackperformance,multi-componentspeciesmasstransport,andthemassbalanceofwateroftheDMFCstack.Theequationsusedtodescribetheelectrochemicalreactioninthecatalystlayersandmulti-componentmasstransportweresolvednumerically.Thecreatedmodelswerevalidatedwiththepublishedresultsbeforemodications. 3.2StackModelThestackmodelinthisworkisbasedonthemodelproposedbyGarcaetal.[ 1 ].AschematicdiagramoftheDMFCconsideredconsistsofasequenceoffourlayersdeningtheMEA,conguredasindicatedinFigure 3-2 .Theintermediatelayersindicatedinthegurearetheanodecatalystlayer(ACL)andtheproton-exchangemembrane(MEM).Thethicknessofeachlayerisindicatedbythesubscriptedvariable.Ahorizontalcoordinate,z,spanstheMEA,rangingfromz=0correspondingtotheleft-mostboundaryoftheanodebackinglayer(ABL),toz=zIVcorrespondingtotheright-mostboundaryofthecathodecatalystlayer(CCL).Thedashedlinesindicatethemethanolconcentrationprolealongthezdirectionthatispredictedbythemodelingequations.TheconcentrationdiscontinuitiesatcoordinateszIandzIIareduetoclassicalpartition-coefcientdistributionsassociatedwithmass-transferequilibriumacrossaboundary.Thesteady-stateDMFCmodelproposedbyGarcaetal.[ 1 ]hasanumberofsignicantmerits.Inparticular,duetoitsabilitytoproduceanalyticalmethanolconcentrationexpressionsandsemi-analyticalcell-currentexpressions,itisahighlycompactmodel.Asaconsequenceitpossessestheadvantageofcomputationalsimplicity,allowingfastcomputationofpolarizationcurves,includingapredictionoftheeffectthatthebulk-fuelconcentrationintheanodechannelhasonthepolarizationcurve.Ourexperiencesuggeststhatthemodelisparticularlyadeptatpredictingpolarizationcurvesthatmatchtheohmicregimeoftheexperimentaldatareportedin[ 1 ]. 50

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Inspiteofitsnumerousadvantages,themodelin[ 1 ]hassomeshortcomingsthatneedtobeaddressedforthepurposeofstrengtheningitspredictivecapabilitiestobettermatchexperimentaldataoverawidercurrent-densityrange.Intheensuingsubsectionsweidentifymodelingissuesthatneedtoberevisited,andinthenextsectionweproposespecicmodelingmodicationsandpracticesthatcanovercomeinadequaciesofthemodel. 3.2.1ModelFormulation 3.2.1.1MethanolmassbalanceThemodelofGarcaetal.[ 1 ]providesamethanolconcentrationprolealongthez-coordinatedirectionatsteadystate,describedbytheanalyticexpressions CBMeOH=KICAI)]TJ /F4 11.955 Tf 11.96 0 Td[(Cb Bz+Cb(3)CAMeOH=Jcell 12FADAz2+C1z+C2and CMMeOH=KIICAIIB+A)]TJ /F4 11.955 Tf 11.96 0 Td[(z M+1(3)where C1=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(CAII)]TJ /F4 11.955 Tf 11.95 0 Td[(CAI A)]TJ /F9 11.955 Tf 13.15 8.09 Td[(Jcell(2B+A) 12FADA(3) C2=CAI)]TJ /F14 11.955 Tf 13.15 18.53 Td[()]TJ /F4 11.955 Tf 5.48 -9.69 Td[(CAII)]TJ /F4 11.955 Tf 11.95 0 Td[(CAIB A+JcellB(B+A) 12FADA(3) CAI=1 DBKI(ADMKII+MDA)+BDADMKIIADMKIIDBCb)]TJ /F9 11.955 Tf 13.15 8.09 Td[(JcellB 12F+MDADBCb)]TJ /F9 11.955 Tf 11.95 -.16 Td[((1+6MeOH)JcellB 6F(3) 51

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and CAII=M DBKI(ADMKII+MDA)+BDADMKIIDADBCb)]TJ /F10 11.955 Tf 11.95 0 Td[(ADBKI(1+12MeOH)Jcell 12F)]TJ /F10 11.955 Tf 9.3 0 Td[(BDA(1+6MeOH)Jcell 6F(3)Theseanalyticalsolutionsforthemethanolconcentrationproles( 3 )to( 3 )areobtainedbysolvingthefollowing1Dmaterial-balanceequationsformethanol,whichareapplicableundersteadystateconditionstothegeometryshowninFigure. 3-2 :dNBMeOH,z dz=0dNAMeOH,z dz=)]TJ /F4 11.955 Tf 9.3 0 Td[(j 6FdNMMeOH,z dz=0wherejisthevolumetriccurrentdensitymodeledintheformj=aIMeOH0,refkCAMeOH CAMeOH+eAAF=RuTeAAF=RuTproposedinMeyersandNewman[ 37 ].ThemolaruxesinABLandACLsectionsofFigure 3-2 obeyFick'slawandhencearerespectivelymodeledbyNBMeOH,z=)]TJ /F4 11.955 Tf 9.3 0 Td[(DBdCBMeOH dzandNAMeOH,z=)]TJ /F4 11.955 Tf 9.3 0 Td[(DAdCAMeOH dzThemethanolmolaruxinMEMlayerisaffectedbyeletro-osmoticdrag[ 74 ]inadditiontodiffusion,andhenceisexpressedas 52

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NMMeOH,z=)]TJ /F4 11.955 Tf 9.3 0 Td[(DMdCMMeOH dz+MeOHJcell F(3)Useequation( 3 ),itfollowsthatdCMMeOH dz=)]TJ /F4 11.955 Tf 9.29 0 Td[(KIICAII MFinally,theboundaryconditionsateachinterfacealongz-coordinateareproposedin[ 1 ]asfollows:z=8>>>><>>>>:0,CBMeOH=CbzI,CBI=KICAIz=8>>>><>>>>:zI,CAMeOH=CAIzII,CAMeOH=CAIIand z=8>>>><>>>>:zII,CMMeOH=KIICAIIzIII,CMMeOH0(3)whereKIandKIIarethepartitioncoefcientsthatrespectivelycharacterizethelocalequilibriumacrosstheACL/BLandACL/MEMinterfaces. 3.2.1.2PolarizationcurvemodelThepolarizationcurverelatingtheDMFCvoltageVcelltothecurrentJcellismodeledin[ 1 ],basedontheelectrochemical-potentialbalanceexpression Vcell=UO2)]TJ /F4 11.955 Tf 11.95 0 Td[(UMeOH)]TJ /F10 11.955 Tf 11.95 0 Td[(C)]TJ /F10 11.955 Tf 11.95 0 Td[(A)]TJ /F10 11.955 Tf 13.16 8.09 Td[(MJcell (3) 53

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whereinturnthecellcurrentJcelliscalculatedasthevolume-integralofthecurrentdensityovertheACLthroughtheexpression Jcell=ZB+ABaIMeOHo,refkCAMeOH CAMeOH+eAAF=RuTeAAF=RuTdz(3)Notethatthecellcurrentexpressiongiveninequation( 3 )canbecalculatedusingthepoint-wisemethanol-concentrationprolesgivenintheanalyticalexpressions( 3 )to( 3 ).However,becausetheseanalyticalconcentrationexpressionsarethemselvesfunctionsofthecellcurrent,werefertoequation( 3 )asasemi-analyticalequationsinceitrepresentsanequationthatisimplicitinthevariableJcell.Garcaetal.[ 1 ]proposedaniterativemethodforndingasolutiontotheimplicitequation( 3 ),withtheanticipationthatafteranumberofiterations,theprocedureconvergestoavalueofananodeoverpotentialvalueAthatmakestheleft-handsideofequation( 3 )equalitsright-handside.ThecathodeoverpotentialCinequation( 3 )isthenobtainedbysolvingforitfromtheTafelkinteicsequationatcathodeside Jcell+Jleak=IO2o,refCO2CCL CO2,refeCCF=RuT(3)where Jleak=6FNMMeOH,z(3)istheleakagecurrentdensityduetomethanolcrossoverfromanodesideandoxidizedinCCL. 3.2.2ModelingImprovements 3.2.2.1ImprovediterationalgorithmTheanodicoverpotentialA[ 1 37 ],whichisafunctionofthecurrentdensity,playsanimportantroleinthemodel'sabilitytorepresenttheelectrochemicalphenomenatakingplaceintheDMFC.ApracticalmethodwasproposedforestimatingAviaan 54

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improvediterativealgorithmforsolvingequation( 3 )comprisedofthefollowingsteps:(i)specifyingonevalueforthecurrentdensityJcellthatisofinterest,whilesatisfyingtheconstraintJcellJmax,whereJmaxisthemaximumcurrentdensitythatcanbeobtainedforagivenoperatingpoint,(ii)producingandinitialguessforAatthegivencurrentdensity,(iii)carryingouttheintegrationindicatedontheright-handsideofequation( 3 ),(iv)calculatingthedifferencebetweentheresultsoftheintegrationwiththevalueofthecurrentappearingontheleft-handsideofequation( 3 ),and(v)stoppingiftheabsolutevalueofthedifferencemeetsthefollowingconvergencecriterion Ccriteria=absJSpeciedcell)]TJ /F9 11.955 Tf 11.96 0 Td[(JIntegratedcell JSpeciedcell<10)]TJ /F7 7.97 Tf 6.58 0 Td[(4(3)wheretheabsoperationdesignatestheabsolutevalue.IftheconversioncriterionisnotsatisedinStep(v),thenthecurrentestimateforAisupdatedandSteps(iii)to(v)arerepeated.Theconvergencecriterion( 3 )iscalledforterminationwiththerelativeerrorincurrentdensityobtainedthecurrentAupdateislessthan1%.TheupdateatStep(v)canbedoneusinganystandardnumericalroot-ndingroutinesforsingle-variableproblems[ 75 ]. 3.2.2.2PredictionofthemaximumcurrentdensityStep(i)oftheAiterationschemedescribedinSection 3.2.2.1 involvesainitialguessofcurrentdensitythatsatisestheconstraintJcellJmax,whereJmaxisthelimitingcurrentoftheDMFC.TheconvergenceschemefailswhenthisconstraintisviolatedbecausethedifferencecalculatedinStep(iv)mayneverevolvetoreachanabsolutevaluesufcientlysmalltomeetthespeciedconvergencecriterion.Thislack-of-convergenceproblemcanbeavoidediftheusercanrstobtainareliableestimateofJmax,andhenceensurethesatisfactionoftheconstraintinStep(i)ofthealgorithm.Weidentifythevalueofthelimitingcurrentthatcanbepredictedbythemodelthroughequation( 3 ),takingadvantageofthefactthatJmaxcorrespondstoa 55

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situationofzero-voltagecurrentwheretheanodicoverpotentialAisinprincipleinnitelylarge.Equivalently,Jmax=limA!1Jcell.Furthermore,recognizingthattheratioAF=RTisapositivequantity,itfollowsthat limA!1CAMeOH eAAF=RuT+=(3)TakingthelimitasAtendstoinnityonbothsidesofequation( 3 )andusingtherelationship( 3 )yields Jmax=ZB+ABa IMeOHo,refkCAMeOHdz(3)Notethatthemethanolconcentrationintheanodecatalystlayer,denotedasinequation( 3 ),isafunctionofthecurrentdensityJcell.HenceaniterativealgorithmanalogoustothatdescribedinSection 3.2.2.1 ,isusedtosolveforJmaxusingequation( 3 ).Notethateveryiterationinvolvestheupdateofthevariables,C1andC2inequations( 3 )and( 3 ).Notethatwedeliberatelyintroducetheterminologymaximumcurrentdensityinsteadoflimitingcurrentdensitytohighlightthefactthattheformerisfoundbysolvingamodelingequationandishenceonlyanestimateofthelatter.Equation( 3 )alsoshowsthatthemaximumcurrentdensityisrelatedtothespecicsurfaceareaoftheanodecatalyst,exchangecurrentdensity,andthemethanolconcentrationproleinthecatalystlayer,etc.ThoseareindeedthedesignparametersavailabletoincreasethemaximumcurrentdensityinaDMFC. 3.2.2.3SystematicmethodforestimatingOverpotentialsTheanodicandcathodicoverpotentialsAandCandthetransfercoefcientsAandCinthecurrent-modelingequations( 3 )and( 3 )mustbespeciedbytheuser.Inprinciple,allfourparameterscanbeestimatedvialaboratoryexperimentscarriedoutundercloselycontrolledconditions.Inpracticalapplications,however, 56

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literaturesourcesareoftenunavailableandtheparametersmustbeestimatedbynumericalttoexperimentaldataobtainedfromaworkingfuelcell.ThetechniqueadoptedinGarcaetal.[ 1 ]foraddressingthisparameterestimationprobleminvolvesassumingapriorithevaluesofAandC,andthenproceedingtoidentifyvaluesofAandCthatwhenincorporatedintotheDMFCmodelproduceanacceptablematchtoexperimentalpolarization-curvedata.Whilethisproceduremaysucceedinyieldingagoodmodel-basedpolarization-curveprediction,itmayneverthelessfailtoidentifyreasonablevaluesfortheindividualoverpotentialparameters.Theproblemstemsfromthefactthatinthemodelingequation( 3 ),theanodicoverpotentialandmass-transfercoefcientalwaysappearintheformofaproductAA.Asaconsequence,agivennumericalvalueA=AAfortheproductthatleadstogoodcurrent-voltagepredictionscanunfortunatelyberealizedbyaninnitenumberofpairsof(A,A)values.Forexample,ifapairofvalues(A,A)yieldsauniqueproduct-valueA,allpairsofparametricestimates(pA,A=p),wherepisanarbitrarynon-zeronumber,yieldthesameproductAbecausethefactorpcancelsoutwhenbothparametersaremultipliedtoobtainA.ThesameargumentappliestotheproductCCappearinginequation( 3 ).Insummary,iftheaprioriestimatesofAandCareaffectedbyerror,theresultingestimatesfortheoverpotentialsAandCmaybehighlyuncertainduetotheerrorpropagationinducedbytheparameter-productconstraintdiscussed.Weproposeamorerobustandsystematicmethodforidentifyingbothoverpotentialsandbothtransfercoefcients.Tothatend,considertheparameter-productvariablesA=AAandC=CC 57

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sothatequations( 3 )to( 3 )canbewrittenintermofthenewvariablesas Vcell=UO2)]TJ /F4 11.955 Tf 11.95 0 Td[(UMeOH)]TJ /F10 11.955 Tf 11.96 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(1CC)]TJ /F10 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(1AA)]TJ /F10 11.955 Tf 13.15 8.09 Td[(MJcell (3) Jcell=ZB+ABaIMeOHo,refkCAMeOH CAMeOH+eAF=RuTeAF=RuTdz(3)and Jcell+Jleak=IO2o,refCO2 CO2,refeCF=RuT(3)Notethatequation( 3 )canbesolvedtoyield C=RuT Fln Jcell+Jleak IO2o,refCO2,ref CO2!(3)Theproposedproceduremakesuseofexperimentalpolarization-curvedata,andconsistsofthefollowingsteps:(i)solvefortheparameter-productAfromequation( 3 )asdescribedinSection 3.2.2.1 exceptthataninitialguessforAisgiveninsteadofA,(ii)solvefortheparameter-productCfromequation( 3 ),and(iii)solvefortheparametersAandCfromequation( 3 )usingtheleast-squaresproceduredescribedbelow.TheexecutionofStep(iii)takesadvantageofthefactthatthereciprocaltransfercoefcients)]TJ /F7 7.97 Tf 6.59 0 Td[(1Aand)]TJ /F7 7.97 Tf 6.59 0 Td[(1Cappearlinearlyinequation( 3 ).Thisallowstheefcientimplementationofaleast-squaresparameteridenticationprocedure.Firstconsiderthei-thexperimentalpointinagivenpolarizationcurve,denetheparameter-productvaluesA,iandC,iatthesamepoint,andintroducethequantityYi=)]TJ /F14 11.955 Tf 9.3 16.86 Td[(Vcell)]TJ /F4 11.955 Tf 11.96 0 Td[(UO2+UMeOH+MJcell isothatequation( 3 )canbewrittenasYi=)]TJ /F7 7.97 Tf 6.59 0 Td[(1AA,i+)]TJ /F7 7.97 Tf 6.59 0 Td[(1CC,iforagenerici-thexperimentalpoint.Next,repeatthisprocessforeachpolarizationcurvedatapoint, 58

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namelybysettingi=1,2,...,n,wherenisthetotalnumberofthedatapointsandthenrearrangingtheresultingequationsintovector-matrixnotationintheform 266666666664Y1...Yi...Yn377777777775=266666666664A,1C,1......A,iC,i......A,nC,n377777777775264)]TJ /F7 7.97 Tf 6.59 0 Td[(1A)]TJ /F7 7.97 Tf 6.59 0 Td[(1C375(3)equation( 3 )canbesolvedusingstandardleast-squaresregressiontechniques[ 75 ]thatyieldasasolutiontheoptimalreciprocaltransfercoefcients)]TJ /F7 7.97 Tf 6.59 0 Td[(1Aand)]TJ /F7 7.97 Tf 6.58 0 Td[(1C.Theestimatesforthetransfercoefcientssoughtcanthenbereadilyobtainedbyinvertingtherespectiveleast-squaressolutions. 3.2.2.4ConcentrationboundaryconditionsattheMEM/CCLinterfaceThissectioninvestigatestheeffectofmethanolconcentrationsspeciedattheMEM/CCLinterfacedenedatthecoordinatelocationz=zIII.Moreprecisely,insteadofassumingthatthemethanolconcentrationiszeroattheMEM/CCLinterface,aconstantvalueCMIIIisimposedsothattheboundaryconditionsinequation( 3 )aresubstitutedby z=8>>>><>>>>:zII,CMMeOH=KIICAIIzIII,CMMeOH=CMIII(3)Underthenewproblemformulationintroducedbyequation( 3 ),themethanolconcentrationprolegivenbyequation( 3 )adoptsthenewform CMMeOH=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(KIICAII)]TJ /F4 11.955 Tf 11.96 0 Td[(CMIIIzIII)]TJ /F4 11.955 Tf 11.96 0 Td[(z M+CMIII(3) 59

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Furthermore,theanalyticalsolutionsforthemethanolconcentrationattheABL/ACLandACL/MEMinterfacesgivenbyequations( 3 )and( 3 )respectivelybecome CAII=[DBKI(ADMKII+MDA)+BDADMKII])]TJ /F7 7.97 Tf 6.59 0 Td[(1MDADBCb)]TJ /F10 11.955 Tf 11.95 0 Td[(ADBKI(1+12MeOH)Jcell 12F)]TJ /F10 11.955 Tf 9.3 0 Td[(BDA(1+6MeOH)Jcell 6F+DMCMIII(ADBKI+BDA)(3)and CAI=1 DBKIDMCMIIIB M)]TJ /F4 11.955 Tf 13.15 8.08 Td[(DMKIICAIIB M+DBCb)]TJ /F9 11.955 Tf 13.15 8.08 Td[(JcellB(1+6MeOH) 6F(3)Giventhatthenewmethanolconcentrationinequation( 3 ),itfollowsthattheconcentrationgradientinMEMisconsequentlygivenby dCMMeOH dz=CMIII)]TJ /F4 11.955 Tf 11.96 0 Td[(KIICAII M(3)Hence,themethanolconcentrationgradientinequation( 3 )isreplacedbyequation( 3 ).NotethattheassumedboundaryvalueCMIIIadoptedinequation( 3 )effectivelyinuencestheconcentrationsCAIandCAII,asindicatedinequations( 3 )and( 3 ).NotealsothatwhenonesetsCMIII=0,equations( 3 )to( 3 )respectivelyreducetotheoriginalexpressions( 3 ),( 3 )and( 3 ),derivedbyGarcaetal.[ 1 ]. 3.2.3ModelValidationFigure 3-3 showstheexperimentaldataalongwithmodeledpolarizationcurves.Thecirclemarkersdenoteexperimentaldatareportedin[ 1 ].Thesolidcurvesrepresentpolarizationcurvepredictionsatthefourbulkmethanolcompositionsconsidered,andareproducedbyanimprovedmodelobtainedbyadoptingthemodicationsdescribedinSections 3.2.2.1 and 3.2.2.2 .Inparticular,thesolidlinesincludefullconvergence 60

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oftheimplicit-equationalgorithm.Thedashlinerepresentsthemodelingresultfor0.5Mwithoutreachingfullconvergence.ThemodelincorporatestheerratacorrectionsreportedinTable 3-1 .Aquantitativeanalysisshowsthattheerrata-freemodelproposedin[ 1 ]andusedtogeneratethepolarizationcurvesreportedinFigure 3-3 leadstoaworst-caseabsoluteerrorinexcessof100%withrespecttotheexperimentaldataforthecaseofabulkmethanolconcentrationof0.5M.AnotherquantitativeassessmentoftheextentofmodelingimprovementcanbemadefromFigure 3-3 byfocusingonthepredictionsofmaximumcurrentdensitiesforeachcaseofbulkmethanolconcentrationconsidered.Notethatthemaximumcurrentdensitypredictedbythemodelusingequation( 3 )canalsobeidentiedfromFigure 3-3 asthepointofintersectionofthehorizontalaxisandthesolid-linepolarization-curveforeachbulkconcentrationofmethanol.TheresultsarereportedinTable 3-2 ,wheretherstcolumnliststhebulkmethanolconcentration(Cb)andthesecondcolumnthecorrespondingmaximumcurrent(Jmax)extractedfromFigure 3-3 .ItisreasonabletoacceptthatthebestexperimentalestimateofthemaximumcurrentdensityJmaxaffordedateachbulkmethanolconcentrationvalueCbisgivenbytheexperimentaldatapointcorrespondingtothelargestcurrentdensityreported(i.e.theright-mostcirclemarkerineachpolarizationcurveofFigure 3-3 ).Usingthoseestimatesasareferencebasis,theresultsinTable 3-2 canbeusedtoreadilyestablishthattheimproved-modelpredictionsforJmaxmatchthebestexperimentalestimateswithadiscrepancynogreaterthan1%absoluteerroratallbulkmethanolconcentrations.This Table3-1. CorrectionofparametersinGarcaetal.[ 1 ] ParameterOriginalValueCorrectedValue DB,cm2=s8.7108.710)]TJ /F7 7.97 Tf 6.58 0 Td[(6KI0.81.25,mol=cm32.810)]TJ /F7 7.97 Tf 6.59 0 Td[(92.110)]TJ /F7 7.97 Tf 6.58 0 Td[(9 61

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analysislendsquantitativesupporttotheclaimofthattheimprovedmodelleadstomoreaccuratepredictions. 3.3SpeciesMassTransportattheCathodeSideThecathodesideoftheDMFCstackinvolvesmultispeciesmasstransport.OxygenneedstotransportthroughCGDLandLBLtoCCL,wherereactiontakingplace.Reaction( 1 )alsooccursatCCLduetothecrossoverofmethanolfromanodesideofthestack.Hence,fourspecies,N2,O2,CO2,andH2O,areinvolvedatthecathodeside.Thecrossovermethanolisassumedtobetotallyreacted.AmasstransportmodelbasedonStefan-Maxwellequationiscreatedtostudythemulticomponentspeciesmasstransportphenomenainthiswork. 3.3.1ModelFormulationThebasicformatoftheStefan-Maxwellequationsis rxi=Xi6=jxiNj)]TJ /F4 11.955 Tf 11.96 0 Td[(xjNi CDei,j(3)wherexidenotesthemolefractionofspeciesi,Niistheuxofspeciesi,Cdenotesthetotalconcentrationofallspecies,andDei,jistheeffectivediffusioncoefcientbetweenspeciesiandj.TheKnudsendiffusionisconsideredtomodelthemasstransportphenomenaintheLBL.TheLBLisahydrophobiclayerthatpreventsliquidwater,generatedattheCCL,fromventingoutthestack.TheKnudsendiffusionisconsideredtosimulatethe Table3-2. Predictionsofmaximumcurrentdensityforfourbulkmethanolconcentrationvalues Cb,MJmax,mA=cm2 0.51500.2620.1310.0516 62

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moleculardiffusioninvolvingintheLBLbecausetheporesizeiscomparabletothemeanfreepathofthespecies.Inadditiontothediffusionterms,theconvectiontermingasphasecanalsobetakenintoaccountbyincludingDarcy'slawvG=)]TJ /F4 11.955 Tf 11.1 8.08 Td[(kG GrPGwherekGistheeffectivepermeability.Theoverallequationusedtosimulatethemasstransportyields rxi=)]TJ /F4 11.955 Tf 17.37 8.09 Td[(xikG DeKiGrPG+Xi6=jxiNj)]TJ /F4 11.955 Tf 11.95 0 Td[(xjNi CDei,j)]TJ /F4 11.955 Tf 21.94 8.09 Td[(Ni CDeKi(3)Attherighthandsideof 3 ,thersttermsdenotesthemasstransportduetoconvection,thesecondtermsistheFicksdiffusion,andthelasttermistheKnudsendiffusion.Theconvectionterminequation( 3 )isnegligiblebasedonthepublishedresultsin[ 76 ].Theratioofthemaximumtotheminimummolecularmassesisabout2.44andtheinducederrorbyneglectingtheconvectivetermisabout1.5%.Afterthereachesthecathodecatalystlayerasagasphase,theamountofoxygeninvolvedinthereactionsitesisaffectedbytheamountofliquidwatergeneratedinthislayer.Figure 3-4 showstheCCLtakenbyacanningelectronmicroscopy.ItshowsthattherearebigcracksandsmallporesintheCCL.Ataconstantstacktemperature,theincreaseofthecurrentdensitywillincreasetheamountofliquidwatergeneratedinCCL.Ifthecatalystareallcoveredbytheliquidwithawaterlm,basedonthecalculationbyapplyingHenry'slaw,thedissolvedoxygenintheliquidwatercannotsustaintheuxoftheoxygenrequiredbythereactionasseenintheexperiments.Thecellvoltagedropsdowntozeroevenatthebeginningofthereaction.Oneofthepossiblemechanismsisthatthegeneratedliquidwaterllsinthebigcracksandoxygentransportstothereactionsitethroughthesmallporesorpassages.Astheincreaseof 63

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theoperatingcurrentdensity,morecracksorbiggerporesarelledwithliquidwaterandtheamountofoxygeninvolvedinthereactionareless.Weassumedtheamountoftheoxygeninvolvedinthereactionsitesisafunctionoftheuxoroperatingcurrentdensity,whichcanbeexpressedasCCCLO2=CCCL=LBLO2+NO2C3whereCCCL=LBLO2istheoxygenconcentrationattheinterfaceofCCLandLBL,NO2istheoxygenuxrequiredforthereactionandC3isanarbitraryconstantadjustedbasedonthemaximumcurrentdensityfromtheexperiments.ThepositivesignontherighthandsideofequationisbecauseNO2isnegativerelativetothedenedcoordinate. 3.3.2ModelDiscretizationThesimulationdomainincludestheCGDLandLBL.Becausethereisnoreactiontakingplaceatregions,thegoverningequationsareCrxi=Xi6=jxiNj)]TJ /F4 11.955 Tf 11.96 0 Td[(xjNi Dei,j)]TJ /F4 11.955 Tf 17.93 8.09 Td[(Ni DeKiandrNi=0whereNN2iszeroduetonoinvolvementinthereaction.NO2andNCO2canbecalculatedoncetheusefulandcrossovercurrentdensitiesarespecied.ApplyingconservationlawofthespeciesXixi=1andforwardnitedifference,thegoverningequationcanbeexpressedinmatrixform 64

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266664B)]TJ /F4 11.955 Tf 11.95 0 Td[(A)]TJ /F7 7.97 Tf 13.21 4.71 Td[(1 hC)]TJ /F4 11.955 Tf 11.96 0 Td[(AD)]TJ /F4 11.955 Tf 11.95 0 Td[(AG)]TJ /F4 11.955 Tf 11.96 0 Td[(FH)]TJ /F4 11.955 Tf 11.96 0 Td[(F)]TJ /F7 7.97 Tf 13.21 4.71 Td[(1 hI)]TJ /F4 11.955 Tf 11.96 0 Td[(FL)]TJ /F4 11.955 Tf 11.95 0 Td[(KM)]TJ /F4 11.955 Tf 11.96 0 Td[(KN)]TJ /F4 11.955 Tf 11.96 0 Td[(K)]TJ /F7 7.97 Tf 13.21 4.71 Td[(1 h377775266664xO2(j)xCO2(j)xH2O(j)377775+2666641 h0001 h0001 h377775266664xO2(j+1)xCO2(j+1)xH2O(j+1)377775=266664)]TJ /F9 11.955 Tf 9.3 0 Td[((A+E))]TJ /F9 11.955 Tf 9.3 0 Td[((J+F))]TJ /F9 11.955 Tf 9.3 0 Td[((O+K)377775whereA=NO2 DO2N2,B=)]TJ /F4 11.955 Tf 15.95 8.08 Td[(NCO2 DO2CO2)]TJ /F4 11.955 Tf 18.6 8.08 Td[(NH2O DO2H2O,C=NO2 DO2CO2,D=NO2 DO2H2O,E=NO2 DKO2,F=NCO2 DCO2N2,G=NCO2 DCO2O2,H=)]TJ /F4 11.955 Tf 18.78 8.09 Td[(NO2 DCO2O2)]TJ /F4 11.955 Tf 21.43 8.09 Td[(NH2O DCO2H2O,I=NCO2 DCO2H2O,J=NCO2 DKCO2,K=NH2O DH2ON2,L=NH2O DH2OO2,M=)]TJ /F4 11.955 Tf 18.49 8.09 Td[(NH2O DH2OCO2,N=)]TJ /F4 11.955 Tf 18.95 8.09 Td[(NO2 DH2OO2)]TJ /F4 11.955 Tf 21.6 8.09 Td[(NCO2 DH2OCO2,O=NH2O DKH2OTheboundaryconditionattheendofCGDLis266664100010001377775266664xO2(nj)xCO2(nj)xH2O(nj)377775=2666640.2100377775AttheinterfaceofCGDLandLBL,thegoverningequationsisrxijLBL=rxijCGDLwhichyields xi(j)]TJ /F9 11.955 Tf 11.95 0 Td[(2))]TJ /F9 11.955 Tf 11.95 0 Td[(4xi(j)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+3xi(j) 2hLBL=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(xi(j+2))]TJ /F9 11.955 Tf 11.96 0 Td[(4xi(j+1)+3xi(j) 2hCGDL(3) 65

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afterapplyingthree-pointforwardandbackwardnitedifferenceschemes.h1andh2arethegridspacesatLBLandCGDLrespectively.Equation( 3 )canbefurtherexpressedinmatrixformatforallspeciesinterestedandgive2666641 2hLBL0001 2hLBL0001 2hLBL377775266664xO2(j)]TJ /F9 11.955 Tf 11.95 0 Td[(2)xCO2(j)]TJ /F9 11.955 Tf 11.96 0 Td[(2)xH2O(j)]TJ /F9 11.955 Tf 11.95 0 Td[(2)377775+266664)]TJ /F7 7.97 Tf 16.52 4.71 Td[(2 hLBL000)]TJ /F7 7.97 Tf 16.51 4.71 Td[(2 hLBL000)]TJ /F7 7.97 Tf 16.51 4.71 Td[(2 hLBL377775266664xO2(j)]TJ /F9 11.955 Tf 11.96 0 Td[(1)xCO2(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1)xH2O(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1)377775+2666643(hLBL+hCGDL) 2hLBLhCGDL0003(hLBL+hCGDL) 2hLBLhCGDL0003(hLBL+hCGDL) 2hLBLhCGDL377775266664xO2(j)xCO2(j)xH2O(j)377775+266664)]TJ /F7 7.97 Tf 19.24 4.7 Td[(2 hCGDL000)]TJ /F7 7.97 Tf 19.23 4.71 Td[(2 hCGDL000)]TJ /F7 7.97 Tf 19.23 4.71 Td[(2 hCGDL377775266664xO2(j+1)xCO2(j+1)xH2O(j+1)377775+2666641 2hCGDL0001 2hCGDL0001 2hCGDL377775266664xO2(j+2)xCO2(j+2)xH2O(j+2)377775=266664000377775 3.3.3ModelValidationAternarygaseousdiffusionexample[ 77 78 ]waschosentovalidatethemodelingresultbasedonequation( 3 ).Aliquidmixtureofacetoneandmethanolevaporatesthroughastagnantlayerofairwassimulated.Figure 3-5 showsthemodelingresultwithadimensionlesslengthofevaporationandthemolefractionofeachspecies.Theparametersusedinthisexamplecanbefoundin[ 78 ]. 3.3.4ModelingResultsFigures 3-7 to 3-9 demonstratedthemodelingresultsofthemulti-componentmasstransportthroughtheCGDLandLBL.KH2OmeansthemasstransportresistanceofthewatervaporthroughtheLBLandCGDL.DO2H2Oistheeffectivebinarydiffusioncoefcientofoxygenandwatervapor.CO2andxO2aretheoxygenconcentrationandthemolefractionofoxygenattheendofLBL.Figure 3-7 showedtheresultsofKH2O,DO2H2O,CO2,andxO2asafunctionofLBLthickness,porosity,tortuosity,andaverage 66

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poresize.TheincreaseoftheLBLthicknessandtortuosityincreasethetransportresistanceofthespecies.HenceKH2O,CO2,andxO2increaseastheincreaseofthethickness.TheincreaseoftheporosityandporesizeoftheLBLreducesthetransportresistanceofthespecies.Inaddition,theeffectivediffusioncoefcientDO2H2OalsoincreaseswiththeincreaseofporosityandhenceKH2O,DO2H2O,CO2,andxO2areallincreasedwiththeincreaseofporosity.Figure 3-8 showedtheresultsofKH2O,DO2H2O,CO2,andxO2asafunctionofCGDLthickness,porosity,tortuosity,andaverageporesize.ThechangesofthematerialpropertiesofCGDLdonothavesignicanteffectonthemasstransportofthespecies.ThisisbecausetheporesizeoftheLBListhreeorderofmagnitudessmallerthantheporesizeoftheCGDL.ThematerialpropertiesoftheLBLhavedominanteffectsonthemasstransportofthespecies.Figure 3-9 showedtheeffectofthestacktemperatureontheKH2O,DO2H2O,CO2,andxO2.KH2OandDO2H2Oincreasewiththeincreaseofthestacktemperature.KH2OandDO2H2Odenotethetransportresistanceofwatervaporandisrelatedtotheeffectivediffusioncoefcientofwatervapor.Theeffectivediffusioncoefcientincreaseswiththetemperatureandhencelowerthetransportresistance.CO2,andxO2decreaseastheincreaseofthestacktemperature.Theincreaseofthetemperaturefavorsthediffusioncoefcientandthetotaluxforoxygentransportisspeciedbasedonthecurrentdensityandrateofmethanolcrossover.Inaddition,theincreaseofthetemperaturealsoadvantagesthegenerationofwatervapor.Theincreaseoftemperaturefrom40to60Cresultsin2.7timesofincreaseinwatersaturationpressure.BotheffectsresultthedecreaseoftheoxygenconcentrationattheendofLBL. 3.4WaterBalanceoftheDMFCStackFigure 3-6 showstheschematicofthewaterbalanceofaDMFCstackwithanopen-cathodedesign.Attheinletoftheanodeside,methanol/watermixtureowsintothestackandcomeswithlowerconcentrationofmethanol/watermixtureandbyproduct 67

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CO2duetoelectrochemicalreaction.Atthecathodesideofthestack,airowsinforusefulelectrochemicalreactionandoxidizedcrossovermethanolandowsoutwithdilutedoxygenandwatervapor.Thesourcetermsforwatergenerationarefromusefulelectrochemicalreaction( 1 )andoxidationofmethanolcrossover( 1 ).ThewaterlosstermsoftheDMFCstackincludesthewaterneededforusefulelectrochemicalreaction( 1 )andthewatervaporventingoutofthestacktocathodeowchannel.TheoverallmassbalanceofwaterinsidetheDMFCstackandtwowaterbalanceparametertoquantifythewaterbalancedesignofthestackaredescribedinthefollowingsections. 3.4.1MassBalanceofWateroftheDMFCStackAsshowninFigure 3-6 ,wedenedtwovariables,_nrecycleH2Oand_nventH2O,toquantifytherateoftherecycled-waterowattheanodesideandtherateofvented-waterowatthecathodesideofthestack.Therateoftherecycled-waterowcanbefurtherexpressedasthedifferenceoftherateofwaterowoftheanodeoutletandthatoftheanodeinletasindicatedby_nrecycleH2O=_nAno,H2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nAni,H2O (3)Thewaterbalanceattheanodesidecanbecharacterizedintheform_nAno,H2O=_nAni,H2O+_nCCL!AFCH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nAnH2O (3)where_nCCL!AFCH2Odenotesthetherateofwatergenerationduetousefulelectrochemicalreaction,_nCaH2O,andmethanolcrossover,_nXOMeOHH2O,lesstherateofwaterventatthecathodeside_nCCL!AFCH2O=_nCaH2O+_nXOMeOHH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nventH2O (3) 68

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_nAnH2Oinequation( 3 )indicatestherateofwaterconsumptionduetousefulelectrochemistryreaction( 1 )asgivenby_nAnH2O=NcellJcellA 6F_nAnH2Oalsodenotesthemethanolconsumptionrateforastoichiometryof1atagivencurrentdensityinmole=s.JcellistheoperatingcurrentdensityinmA=cm2,andAistheactivecellareaincm2.Therateofwatergenerationatthecathodeduetousefulelectrochemicalreactioninequation( 3 )isequaltoastoichiometryof3formethanolconsumption,asdictatedbytheelectrochemicalreaction( 1 )_nCaH2O=3_nAnH2OBasedontheoverallchemicalreaction( 1 )formethanolcrossoveratthecathodesideofaDMFC,anotherwatergenerationterminequation( 3 ),_nXOMeOHH2O,isequaltotwicethatofmethanolcrossoverrate.Hence,_nXOMeOHH2O=2_nXOMeOHwhere_nXOMeOHisthemethanolcrossoverrateinmole/s.Theventrateofwatervaporinequation( 3 )isassumedtobedrivenonlybytheconcentrationgradientofthewatervaporbetweentheCCLandtheCFCaccordingtheclassicalFickeanmodel_nventH2O=DeH2OCCCLH2O)]TJ /F4 11.955 Tf 11.96 0 Td[(CCFCH2O tLBL+CGDLA (3) 69

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ThewatervaporconcentrationintheCCLisassumedtobesaturated,henceCCCLH2O=Psat(Tstack) RuTstackInturn,thewatervaporconcentrationintheCFCcanbeexpressedasCCFCH2O=Psat(Tamb)RHamb RuTamb (3)whereRHambistheambientrelativehumidityandPsatisthesaturationpressureatagiventemperature.Notethatthewatervapordiffusioncoefcientinequation( 3 )istheeffectivediffusioncoefcientinm2=sfortheLBLandfortheCGDL,andthediffusionlengthisthetotalthicknessinmoftheLBLandCGDL. 3.4.2WaterBalanceParameterThewater-balanceparameter=_nrecycleH2O _nventH2Oisdenedastheratiooftherateofrecycledwaterattheanodesidetotherateofventedwateratthecathodeside.Therecycled-waterinthenumerator,_nrecycleH2O,canbeobtainedbysubstitutingequations( 3 )to( 3 ).Theventrateofwatervaporinthedenominator,_nventH2O,canalsobefoundbysubstitutingequations( 3 )to( 3 ).Thewater-balanceparameter,,canbefurthermanipulatedtoadoptsuccessiveformsgivenby 70

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=_nAno,H2O)]TJ /F9 11.955 Tf 13.46 0 Td[(_nAni,H2O _nventH2O=_nCCL!AFCH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nAnH2O _nventH2O=_nCaH2O+_nXOMeOHH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nventH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nAnH2O _nventH2O=_nCaH2O+_nXOMeOHH2O)]TJ /F9 11.955 Tf 13.45 0 Td[(_nAnH2O _nventH2O)]TJ /F9 11.955 Tf 11.95 0 Td[(1 (3)Thesignoftheright-handsideofequation( 3 )revealsthreepossiblescenariosforoperatingaDMFCstack:(1)waterrecoverymode(>0),(2)waterneutralitymode(=0),and(3)waterlossmode(<0).WhenisgreaterthanzerotheDMFCstackisinwaterrecoverymodebecausetherateofrecycledwaterislargerthanthewaterconsumptionrateintheDMFCstack.Whenislessthanzero,theDMFCstackisinwaterlossmodebecausetherateofwatergenerationisnotsufcienttocompensatefortherateofwaterlossesduetoreactionanddiffusionthroughtheliquidbarrier.AwaterrelloperationwouldbeneedediftheDMFCstackoperatesforasustainedperiodoftimeinthewaterlossmode.Finally,whenisequaltozerothewaterbalanceintheDMFCstackisinwaterneutralitymodebecausethewatergenerationrateisequaltotherateofwaterloss.Sincetheoperatingenvironmentsmaydiffersignicantly,itisimperativethattheDMFCdesignincludeappropriatepassivemechanismsoractivecontrolstrategiestotransferthesystembetweenthesethreemodesasneededtoensureoptimaloperation. 3.4.3WaterVaporMassTransportParameterThewatervapormasstransportparameterisintroducedKH2O=DeH2O tLBL+CGDL (3) 71

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tocharacterizethewatertransportintheMEA,andtoprovideaquantiablemetricusefulforengineeringthephysicalpropertiesoftheMEA'sconstitutivelayers.Theunitsofthewatervapormass-transportparameterareinmm=s.Asshowninequation( 3 ),inadditiontothewatervaporconcentrationgradientbetweentheCCLandCFClayers,thematerialpropertiesoftheLBLandCGDLalsoaffecttherateofmasstransportofthewatervaporfromCCLtoCFC.Usingthedenitiongiveninequation( 3 )anddividingbothsidesbytheactivecellarea,equation( 3 )canberewrittenintheequivalentformNventH2O=KH2O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(CCCLH2O)]TJ /F4 11.955 Tf 11.95 0 Td[(CCFCH2OwhereNventH2OdenotestheuxofthewatervaportransportfromtheCCLtoCFClayers.TheparameterKH2OcanbeinterpretedasthemasstransportresistancetowatervaportransferfromtheCCLtotheCFClayerscausedbytheLBLandCGDL.AhigherKH2OvaluemeansthatthereislowerresistanceforwatervaportotransportthroughtheLBLandCGDLlayers,andhenceresultsinmorewaterlossunderthesameoperatingconditions.AlowerKH2OforthematerialisneededtoreducethewatervaporlossesthroughtheLBLandCGDLlayers.ThenominalKH2Ovalueisbetween1.0to2.5mm=sat50C,dependingonthephysicalingredientsofthematerial.Notethat,asshowninequation( 3 ),KH2OisafunctionofDeH2O,whichinturnvarieswithtemperature.Hence,KH2Oisexpectedtochangewithstacktemperature. 3.5AnalysisoftheBulkConcentration 3.5.1ModelFormulationsOnthebothsidesofthestack,theconcentrationsofthereactantsvaryalongtheowchannels;however,thebulkconcentrationusedinthesimulationislocal,correspondingtoaparticularpointofthemeasurement.Anerroranalysisisdonein 72

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thissectiontoaccountfortheerrorscausedbyusingalocalconcentrationinsteadofintegratingtheconcentrationsalongthechannels.Assumingthattheratesofconsumptionofthereactantsalongtheowchannelwereconstantandthatthestackwasinwater-balancemode,theconcentrationofthereactantsattheanodeandthecathodealongtheowchannelcanbeexpressedas_nAnx,MeOH=_VAni,MeOHM)]TJ /F4 11.955 Tf 11.96 0 Td[(xWAnchanNAMeOH (3)and_nCax,O2=Pamb_VCai,O2 RuTstack)]TJ /F4 11.955 Tf 11.96 0 Td[(xWCachanNO2 (3)wherexisthedistancefromtheinletofthechannel,Wchanisthewidthoftheowchannel,andMisthemolarityofthesolution.Theconcentrationsattheoutletsoftheanodeandcathodecanbeobtainedbysubstitutingxwiththelengthofowchannels,Lchan,inequations( 3 )and( 3 ).Table 3-3 givesthedimensionsandowratesoftheanodeandcathodesideofthestack.Theowchannelattheanodesideisserpentine.Theowchannelsatthecathodesidearestraightandparallel.Assumingafactor,,bothontheanodeandcathoderespectivelytoaccountfortheadditionaluxofthereactantsinducedfrommethanolcrossover,equations( 3 )and( 3 )canbefurthermodiedas Table3-3. Dimensionsofanodeandcathodeowchannels AnodeCathode _Vi2mL/min/cell1SLPMDchan4E-4m1.2E-3mLchan1.0m1.9E-2mNchan154Wchan1E-3m1.5E-3m 73

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_nAnx,MeOH=_VAni,MeOHM)]TJ /F4 11.955 Tf 11.96 0 Td[(xWAnchanJcell 6Fand_nCax,O2=Pamb_VCai,O2 RuTstack)]TJ /F4 11.955 Tf 11.96 0 Td[(xWCachanJcell 4FWhenisequalto1,theuxofthereactantsaccountsonlyfortheusefulreaction.Whenisequalto1.3,thereis30%moremethanolandoxygenconsumptiontocompensatefortherateofmethanolcrossover. 3.5.2ResultsFigure 3-10 showsthemethanolconcentrationvariationalongtheanodeowchannelwithvaryingfrom1to2andcurrentdensityvaryingfrom0to150mA=cm2atmolaritiesof0.6M,0.8M,1.0M,and1.2M.Theanodeowratewassetataconstantvalueof2mL/minpercellforeachexperimentinthiswork.Theresultshowsthatwhenthestackisoperatedat0.6Mand150mA=cm2,theextremecaseofinterest,themethanolconcentrationdecreasesfrom0.6Mto0.52Matof1.Thisgivesanaverageerrorof6.38%when0.6Misusedasthebulkconcentrationinsimulation.Whenisequalto1.3,meaningthat30%moreofmethanolisconsumedtoaccountfortherateofmethanolcrossover,themethanolconcentrationdecreasesfrom0.6Mto0.5M,givinganaverageerrorof8.3%if0.6Misusedasthebulkconcentrationinsimulation.Formostoftheinterestedoperatingcurrentdensitiesandmolarities,theaverageerrorisbelow5%,whichisaroundthesameastheuncertaintymeasurementsofthewaterbalanceandmethanolcrossover.Figure 3-11 showstheoxygenconcentrationvariationalongthecathodeowchannelwithvaryingfrom1to2andcurrentdensityvaryingfrom0to150mA=cm2.Theairowratewassetataminimumvalueof1SLPMpercellforallexperiments. 74

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Theresultshowsthatwhenthecurrentdensityincreasesfrom0to150mA=cm2,theoxygenconcentrationdecreasesfrom21%Mto20.12%atof1.Whenisequalto1.5,theoxygenconcentrationfurtherdecreasesto19.68%,givingtheaverageerrorof3.11%when21%ofoxygenisusedasthebulkconcentrationinthesimulation.Again,thecalculatedaverageerrorisaroundthesameastheuncertaintymeasurementsofthewaterbalanceandmethanolcrossover. 75

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Figure3-1. ThehierarchyoftheDMFCsystemlevelmodel. 76

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Figure3-2. SchematicofthesimulationdomainofaDMFC. 77

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Figure3-3. Polarizationcurvesfordifferentmethanolconcentrationsafterconvergenceversustheexperimentaldata[ 1 ]. 78

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Figure3-4. ScanningelectronmicroscopeoftheCCL. 79

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Figure3-5. Validationofmulti-componentmasstransportmodel. 80

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Figure3-6. DiagramillustratingvariouswaterowratescontributingtowaterbalanceinaDMFCstack. 81

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Figure3-7. KH2O,DO2H2O,CO2,andxO2asafunctionofLBLthickness,porosity,tortuosity,andaverageporesize. 82

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Figure3-8. KH2O,DO2H2O,CO2,andxO2asafunctionofCGDLthickness,porosity,tortuosity,andaverageporesize. 83

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Figure3-9. KH2O,DO2H2O,CO2,andxO2asafunctionofstacktemperature. 84

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Figure3-10. Molarityofmethanolsolutionalongowchannelwithaowrateof2mL/minpercellasafunctionofandcurrentdensity. 85

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Figure3-11. O2concentrationalongowchannelwithairowrateof1SLPMpercellasafunctionofandcurrentdensity. 86

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CHAPTER4EXPERIMENTSThewatercrossoverissueisparticularlyimportantinDMFCswithanopen-cathodeconguration[ 56 58 59 ].InordertodeveloppracticalDMFCsforportableelectronicdevices,theoverallsystemneedstobecompact,astrongadvantageoftheopen-cathodeconguration.Inatraditionalclosed-cathodeDMFCsystemdesign,waterisextractedfromthecathodeexitairstreambycondensation.Thereactantwaterneededintheanodesideisreplenishedbypumpingthewatercollectedfromthecathodeexit.Thiswatermanagementsystemcomplicatesthedesign,addstotheparasiticpowerconsumption,andincreasesthesystemvolumeandweight.Anopen-cathodeDMFCdesigneliminatesthebulkywatermanagementsystembyallowingpassivereplenishmentoftheanodereactantwaterviatransportthroughthemembrane.ThisistypicallyimplementedbytheadditionofahydrophobicLBLintheMEAbetweentheCCLandtheCGDL.VariousmodicationsoftheMEAtoincorporateaLBLhavebeenproposedbyseveralresearchgroups[ 61 67 69 ].TheLBLimplementationinthisworkdifferentiatesitselffromliteratureprecedentsinthatitincludesmembranechannelsdesignedtoallowwatertoowtotheanode.Figure 1-2 showsthestructureoftheMEAusedinthiswork.ThehydrophobicLBLispositionedbetweenthecathodecatalystlayerandcathodeCGDL.TheLBLpreventsliquidwaterfromexposuretothecathodeairstream,sothatonlywaterinvaporphasecanventthroughthehydrophobicLBLandtheCGDL.SincetheliquidwatergeneratedbythereactionintheCCLisinhibitedfromreachingthecathodeairstream,aportionisdrivenbythepressure,throughthemembranechannels,totheanodesidewhereitservesasareagent.ThisworkinvestigatestheeffectoftheLBLonthecellperformanceasmeasuredthroughtheresultingpolarizationcurves.Theeffectsonthestackwaterbalanceobservedbyvaryingthetemperatureandthecurrentarealsodiscussed. 87

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4.1ExperimentalDetails 4.1.1ApparatusTheexperimentalrigforthisinvestigationisshowninFigure 4-1 .Ontheanodesideapistonpumpwasusedtodriveamethanol/watermixturefuelstream,andanexternalfuelheaterwithmaximumpoweroutputof470Wwasusedtoheat-conditionthemethanolsolutionbeforethestackfeedingport.MethanolofHPLCgradeanddeionizedwaterwitharesistivityof15M-cmwereusedtopreparethemethanolsolution.ThepreheatedtemperaturewasadjustedtoequalthestacktemperaturethroughthemanipulationofthepoweroutputbyaPIDtemperaturecontroller.Zero-gradecylinderairwassuppliedforthereactionatthecathodesideofthestack.Theairowrateonthecathodesidewasadjustedbyamassowcontrollerwitharangeupto50SLPM.Theairowratewasregulatedbythemassowcontrollertomaintainthetargetstacktemperaturedesiredforeachexperiment.TheairowattheexitofthestackwasbypassedthroughapathconnectedtoaLI-CORLI840,awaterandCO2concentrationanalyzer.Thebypassairow-ratewasadjustedbetween0.7to0.9SLPMbyaneedlevalvelocateddownstreamfromtheanalyzer.Afour-cellDMFCstackwithanactivecellareaof15.5cm2wasusedinthiswork.TheMEAsweremanufacturedbyourresearchteamatUniversityofNorthFloridaandconditionedatUniversityofFlorida.Acommercialelectronicloadregulatedtheoutputcurrentandvoltageofthestack.Theexperimentalrigincludedfourthermocouplesandonedifferentialpressuretransducer.Thethermocoupleswerepositionedattheanodeinletpre-heatertubing,theanodeowstackinlet,theanodeowstackoutlet,andthecathodeowstackoutlet.Thedifferentialpressuretransducerwasconguredtomonitorthepressuredifferenceacrossthecathodeowchannel.ThedatawereacquiredinrealtimebyacustomizedLabVIEWTMcodeanddataacquisitionsystemfromNationalInstruments. 88

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4.1.2ExperimentalConditionsThecriticalexperimentalparametersinthisstudywerethestacktemperature,thepreheattemperatureofanodesolution,theanodesolutionowrate,thecathodeairowrate,andtheoperatingcurrentdensity.Table 4-1 showsthevaluesofeachparameteradoptedinspecicexperiments.Thestacktemperaturesofexperimentalinterestareshowninthetable.Notethatthepreheattemperatureoftheanodesolutionwascontrolledtobethesameasthestacktemperature.Theanodesolutionowratewassetto2mL/minpercell,whichcorrespondstoastoichiometryof6.64at150mA=cm2.ThecathodeairowwasregulatedbyaPIDcontrollerinLabVIEWTMtoensurethatthestacktemperaturewasmaintainedatthespeciedsetpoint.Aminimumairowrateof0.75SLPMpercellwassettomakesuretherewasenoughairforreactionandpreventwatersaturationthatwouldotherwiseinterferewiththewaterconcentrationmeasurement.Theexperimentswererununderaconstantcurrent-densitymodeat20,40,60,80,100,120,and150mA=cm2.Forthemaximum-current-densityexperiments,theoperatingcurrentdensitieswereincreasedfrom0mA=cm2tothemaximumvalues,inthisfashionstillallowingtheaveragestackvoltagetoremainabove0.2Vandhencepreventingcatalyticdeterioration.ThemaximumcurrentdensitiesrealizedarelistedinSection 4.4 Table4-1. Experimentalconditionsforcriticalparameters ParametersRanges Stacktemperature45,50,55,60,70,80CCathodeairowrateAsneededforstackthermalbalanceAnodeowrate2.0mL/minpercellAnodesolutionpreheattemperatureEqualtothestacktemperatureCurrentdensity20,40,60,80,100,120,150mA=cm2 4.1.3ExperimentalProcedureForeachsetofexperiments,a1.0Mmethanolsolutionwaspreparedusingavolumetricaskwithanaccuracyof0.3mL.Aweightof32.04gofmethanol,withanaccuracyof0.03g,wasdilutedintheasktoreachthetargetconcentration.The 89

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initializationprotocolofeachexperimentincludedpre-conditioningthestacktemperatureat50C,andsettingtheanodeowratesetat2mL/minpercellandholdingtheowconditionsforatotalof5minutes.Afterthepreconditioningprocedure,thecurrentdensitywasincreasedstepwisefrom0to150mg=cm2.Ateachcurrentdensity,a20-minutecontinuoustestwasconductedtoensurethattheDMFCstackreachedsteadystate.Duringthetransitionbetweencurrentdensities,acatalystrejuvenationprocedurewascarriedouttoreversedegradationeffects.Thecathodeairowratewasregulatedasneededformaintainingthedesiredstacktemperature,whilestilldeliveringatleast0.75SLPMpercell.ThemeasuredparametersincludedthewatervaporandCO2concentrationsattheexitofthecathodeowchannel,theairowrateatthecathodeside,theexittemperatureoftheairowatthecathodeowchannel,theinletsolutiontemperatureattheanode,thestacktemperature,thecurrentdensity,andthecellvoltage.Attheendofeachexperiment,apolarization-curvemeasurementat50CwasconductedandcomparedwithanestablishedreferenceleveltomakesuretheDMFCstackhadnotexperienceddegradation. 4.2ErrorAnalysisForanexperimentalwork,theerroranalysisisveryimportantbecauseeveryinstrumenthasinherentaccuracy(biaslimit)andprecisionlimit.Thestandardsandguidelinesforuncertaintyanalysishavebeendevelopedfordecadesbyprofessionalsocietiesandinternationalorganizations[ 79 80 ].In1995,AmericanInstituteofAeronauticsandAstronautics(AIAA)furtherreportedastandarduncertaintyanalysismethodologyforthewindtunneldata[ 81 ].Inadditiontotheprofessionalsocietiesandinternationalorganizations,ColemanandSteele[ 82 83 ]alsodemonstratedtheapplicationoftotaluncertaintyanalysistotheengineeringandscienticmeasurements.Inthiswork,theoveralluncertaintyanalysiswerebasedontheaforementionedreferences.Theoveralluncertaintyanalysisincludedtheaccuracyofeachinstrument 90

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Table4-2. Specicationsforthecomponentsinthefuelcellteststation ComponentsSpecications TDIelectronicloadRBL488100-30-200,0.5%offullscale(F.S.)Agilentelectronicload6063B,0.15%10mA,constantcurrentmode0.12%120mV,constantvoltagemode0.8%200m,constantresistancemodeMKSmassowcontrolmeter(MFC)1179A,10000SCCM1.0%F.S.LOVERtemperaturecontroller16111-992,32Aseries,0.25%F.S.,1leastsignicantdigitGilsonRperistalticpumpMinipuls3,48rpm,0.2%F.S.SymmetryTMscalePR42,0.001%F.S.LI-CORRLI-840CO2:0-3000ppm,1.5%F.S.H2O:0-80ppt,1.5%ofF.S.ALICATTMScienticMFC50SLPM,(0.8%ofreading+0.2%F.S.)NI9219universalmoduleThermocouple125mV,0.1%F.S.Voltage15V,0.3%F.S.NI9211ThermocouplemoduleThermocouple80mV,0.05%F.S.NIcDAQ9178Analoginput:Samplerate:6.4MS/s(system)Timingaccuracy:50ppmofsamplerateAnalogoutput:Samplerate:1.6MS/s(system)Timingaccuracy:50ppmofsamplerateNIUSB-6218Analoginput:Samplerate:250kS/s(system)Timingaccuracy:50ppmofsamplerateAnalogoutput:Samplerate:250kS/s(system)Timingaccuracy:50ppmofsamplerate andtheprecisionerrorofthemeasurements.TheaccuracyofeachinstrumentusedinthisresearchislistedinTable 4-2 .Thetotaluncertaintyofanexperimentistheroot-sum-square(RSS)oftheaccuracyandprecisionerrors.ThedetailedproceduretoestimatethetotaluncertaintyisdemonstratedinAppendix A 91

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4.3PolarizationCurveMeasurements 4.3.1OverallPolarizationCurveMeasurementsApolarizationcurveiscomposedofthreemajorregions,knownastheactivation,ohmic,andconcentrationoverpotentials.Inthelowcurrent-densityregion,thepolarizationcurverepresentstheactivationoverpotentialoftheDMFC.Then,thecellvoltageexperiencesalineardecreaseastheincreaseofthecurrentdensity.Thepolarizationcurvecorrespondstotheohmicoverpotentialinthisregion.Theremaininghigh-currenttail-endcorrespondstotheconcentrationoverpotential.Atagivencurrentdensity,ahighercellvoltageindicateslowerimpedanceforthereactiontoovercome,andhencehighercellperformance.Inthiswork,asetofpolarizationcurvesweremeasuredtounderstandthecellperformance.Thestacktemperaturesrangedfrom45Cto80C.Theoperatingcurrentdensitywasincreasedfrom0mA=cm2tothemaximumcurrentdensitythatstillallowsthestackaveragevoltagetoremainabove0.2Vtopreventcausingdamagetothestack.Thepolarizationcurvesofdifferentmethanolconcentrations,0.6M,0.8M,1.0M,and1.2M,werealsotakentostudytheconcentrationeffectonthecellperformance. 4.3.2SegregatedPolarizationCurveMeasurementsAsmentioned,theoverallpolarizationcurveiscomposedofactivation,ohmic,andconcentrationoverpotentials.Theactivationoverpotentialcanfurtherbedistinguishedintoanodicoverpotentialandcathodicoverpotential.Inordertofurtherstudythereactionmechanismsonbothelectrodesandtheoverpotentialscausedbyinternalresistanceandmasstransportlimitation,themeasurementofeachoverpotentialisnecessary.Theexperimentaldataofeachoverpotentialcanalsobeusedtovalidatethemodelingresultsandstudytheeffectsofthekeyparametersonthereactionmechanisms.ThesegregatedpolarizationcurvemeasurementswereconductedbyusingSolartron1255BfrequencyresponseanalyzerandSolartron1480MultiStat.The 92

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anodepolarizationwasmeasuredwithhydrogenowingthroughthecathodesideofthestackasthereferenceelectrodeandmethanol-watermixtureowingthroughtheanodesideastheworkingelectrode.Theappliedvoltageweresetfrom0.01Vto0.55Vwithascanningrateof2mV/s.Thelimitationofthecurrentwassetas4Aforsafetyconsideration.Theinternalresistanceorimpedancemeasurementswerealsoconductedwiththesameinstruments.Thereactantsonbothelectrodeswerealsothesameastheanodepolarizationmeasurements.Thescanningrateofthefrequencyanalyzerwasfrom20KHzto0.01HzwithanACamplitudeof20mV.Aftertheanodepolarizationandinternalresistancedatawereobtained,thecathodicpolarizationcanbeextractedbyusingequation( 3 )withtheknownoverallpolarizationcurveandtheoreticalthermodynamicpotential. 4.4WaterBalanceMeasurementsTwocriticalparametersareintroducedtofacilitatethecharacterizationofthewaterbalance,namelythestackwater-balanceparameter,,andthewater-vapormasstransportparameter,KH2O.Thewater-balanceparameterallowsdeterminingiftheDMFCstackisoperatingunderawaterrecovery,waterneutrality,orwaterlossmode.Thewater-vapormasstransportparameterquantiesthedegreeofwater-vaportransportthroughtheLBLandtheCGDLtothecathodeowstream. 4.4.1WaterBalanceParameterFigure 3-2 showsaschematicproposedtoaccountforthewaterbalanceoftheDMFCstack.Firsttherecycledwatermolarowrateisdenedasthedifferencebetweentheanodeoutletandanodeinletowrates.Thenthevariable_nrecycleH2Oshownintheguredenotesthemolarowrateoftherecycledwaterattheanodesideofthestack.Inturn,thevariable_nventH2Orepresentsthemolarowrateoftheventedwateratthecathodesideofthestacktothecathodeowchannel(CFC).Thisnotationallowstheintroductionofthestackwater-balanceparameter 93

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=_nrecycleH2O _nventH2Odenedastheratiooftherecycledwaterattheanodesidetotherateofventedwaterfromthecathodecatalystlayertothecathodeowchannel(cf.Figure 3-2 ).AdetailedderivationofisgiveninSection 3.4.1 .TherearethreeDMFCoperatingmodesthatcanbedenedintermsofstackwater-balanceparameter:(1)waterrecovery(>0),(2)waterloss(<0),and(3)waterneutrality(=0).Whenisgreaterthanzero,theDMFCstackoperatesinwaterrecoverymodebecausetherateofwatergenerationduetomethanolcrossoverandelectrochemicalreactionisfasterthanthesumoftherateofwaterconsumptionforreactionandwaterlossinvaporphasetotheCFC.Conversely,whenislessthanzerothestackoperatesinwaterlossmodebecausetherateofwatergenerationinsidetheDMFCstackisnotsufcienttocompensateforthewaterlossesduetoreactionanddiffusiontotheCFC.Inpractice,ifoperationinwaterlossmodeismaintainedforasufcientlylongtime,awaterrellwouldbeneededtoensurethestackhassufcientwaterforsustainingthechemicalreactionattheanode.Finally,whenisequaltozero,theDMFCstackoperatesinawaterneutralitymode,indicatingthattherateofwatergenerationexactlycompensatesfortheoverallrateofwaterloss.ToensurethatapracticalDMFCsystemoperatesunderoptimalconditions,characterizationofthethreemodesofoperationviathestackwater-balanceparameterisofgreatimportance. 4.4.2MassTransportResistanceofWaterVaporWeintroducethewater-vapormasstransportparameterKH2O=DeH2O tLBL+CGDL (4) 94

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thatcharacterizeswatertransportthroughtheLBLandtheCGDL,andhenceprovidesthemeanstoquantifyaphysicalpropertyoftheliquidbarrierlayer.Thenumeratorinequation( 4 )istheeffectivewater-vapordiffusioncoefcientthroughthecombinedLBLandCGDLstructures,andhencecorrespondstotheproductofthenominaldiffusioncoefcient,porosity,andtortuosityofthematerial.WatervaportransportintheMEAisaffectedbythematerialpropertiesoftheLBLandCGDL,aswellasdifferenceinwater-vaporconcentrationbetweenCCLandCFC(alsoknownastheconcentrationgradient),whichservesasatransportdrivingforce.Hence,theuxofwater-vaportransportedfromtheCCLtotheCFCisgivenbyN"ventH2O=KH2O)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(CCCLH2O)]TJ /F9 11.955 Tf 11.95 0 Td[(CCFCH2O (4)whereKH2OcapturesthecombinedmaterialpropertiesoftheLBLandCGDLtransportmedia.Morespecically,KH2Orepresentstheresistancetowater-vaportransportfromtheCCLtotheCFC.ThevalueofKH2Oisaffectedbytheeffectivediffusivityofthecombinedlayers,fromwhichcanbefurtherinferredthattheresistancetowatervaportransportisinuencedbythetortuosityandporosityoftheopenchannelsinthecombinedlayers.AhighervalueofKH2Oindicatesthattheresistancetowater-vaportransportislower,andhencetherateofwatervaportransportishigher.AMEAdesignedwithhighervalueofKH2OmaynditeasiertooperateunderawaterlossmoderelativetoaMEAwithlowervalueofthetransportparameter.Inourexperimentalsystems,thevalueofKH2Oisnormallybetween1.0to2.5mm/sat50C,dependingontheamountofhydrophobicmaterialappliedtocreatetheliquidbarrierlayer. 4.4.3MethanolCrossoverMethanolcrossoveristhephenomenonwheremethanolfuelistransportedthroughthemembrane,fromtheanodetowardsthecathode,leadingtothereactionofmethanolandairatthecathodethroughthechemicalpathequation( 1 ).Inthiswork,methanol 95

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crossoverisassumedreactcompletelyatthecathodeside,anditisalsoassumedthattheresultingCO2gasdoesnotdiffusefromthecathodetotheanode.SincealloftheCO2generatedatthecathodesideofthestackisventedtothecathodeairstream,itfollowsthatthecrossoverrateofmethanolcanbedeterminedbymeasuringtheventingrateofCO2attheexitofthecathodeowchannel.Themethanolcrossoverrate(givenintermsofanequivalentcurrentdensityexpressedinmA=cm2)isgivenbyJxo=6F_nCao,CO2 ANcellwhere_nCao,CO2denotestherateofCO2owattheexitofCFC,Aistheactivecellarea,NcellisthenumberofcellscomprisingtheDMFCstackandFisFaraday'sconstant. 96

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Figure4-1. Diagramoftheexperimentalrig. 97

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CHAPTER5RESULTSThemodeldescribedinChapter 3 hasbeenimplementedinMATLABRandexercisedparametricallytoexplorearangeofconditionsofinterest.Table 5-1 liststhekeyparametersandresourcesusedinthiswork.Thetransfercoefcients,AandC,weredeterminedbytheexperiments.Theanodeandcathodepolarizationcurvesweremeasuredandthenequations( 3 )and( 3 )wereappliedtondoutthevalues.Theaverageporesize,porosity,andtortuosityoftheLBLwereadjustedtothesameorderofmagnitudeofthein-situKH2Ovalue.AftertheLBLweremanufactured,theex-situKH2Ovaluesweremeasuredexperimentally.AfterthehotpressingofthelayerstocreatetheMEAs,theKH2Ovaluewasmeasuredagainandreferredasanin-situvalue.ThethicknessoftheAGDLwasassumedtobe100m,thesameorderofmagnitudeasthethicknessoftheCGDL,whichwasmeasuredtobearound200m.ThemethanoldiffusioncoefcientDMinthemembranewasexperimentallymeasuredtobetheorderof10)]TJ /F7 7.97 Tf 6.58 0 Td[(11m2=s.Hence,thereferredfunctionofDM[ 84 ]wasadjustedtothesameorderofmagnitude.TheotherkeyparameterswereallfromtheliteratureasshowninTable 5-1 .Figure 5-1 showstheexperimentalandmodelingresultsoftheanodeoverpotential,cathodeoverpotential,ohmicoverpotential,andtheoverallpolarizationcurveat50C.Themaximumcurrentdensityatthegivenconditionisabout252mA=cm2.Thecellvoltageis0.399Vat120mA=cm2,whichyields0.742Wpercell.TheanodeoverpotentialandtheinternalresistanceweremeasuredbythemethoddescribedinSection 4.3.2 .NotethattheanodeoverpotentialwasIR-corrected,meaningthattheinternalresistanceduetothehydrogenprotontransportwithinthemembranewassubtractedfromtheanodeoverpotential.Thecathodeoverpotentialwascalculatedwithequation( 3 ),wheretheidealcellvoltageisestimatedas1.21V,with1.24Vatthe 98

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Table5-1. Parametervalues ParameterValueReference A0.8ExperimentC0.6-CGDL0.8UNFNWLBL0.285AssumedA3010)]TJ /F7 7.97 Tf 6.59 0 Td[(6mUNFNWB10010)]TJ /F7 7.97 Tf 6.58 0 Td[(6mAssumedC3010)]TJ /F7 7.97 Tf 6.59 0 Td[(6mUNFNWM2010)]TJ /F7 7.97 Tf 6.59 0 Td[(6m-2.810)]TJ /F7 7.97 Tf 6.58 0 Td[(3mol=m3Garcaetal.[ 1 ]CGDL1.5-LBL3AssumedMeOH1.2xMeOHUNFNWa1051=mGarcaetal.[ 1 ]k7.510)]TJ /F7 7.97 Tf 6.58 0 Td[(4-rCGDL100mAssumedrLBL50nmAssumedA15.510)]TJ /F7 7.97 Tf 6.59 0 Td[(4m2UNFNWDA2.810)]TJ /F7 7.97 Tf 6.58 0 Td[(9e2436(1=353)]TJ /F7 7.97 Tf 6.59 0 Td[(1=T)m2=sScottetal.[ 84 ]DB8.710)]TJ /F7 7.97 Tf 6.58 0 Td[(10m2=sGarcaetal.[ 1 ]DM0.710)]TJ /F7 7.97 Tf 6.58 0 Td[(10e2436(1=333)]TJ /F7 7.97 Tf 6.58 0 Td[(1=T)m2=sScottetal.[ 84 ],ExperimentF96485C=equivIMeOH0,ref94.25e73200=Ru(1=353)]TJ /F7 7.97 Tf 6.59 0 Td[(1=T)A=m2Parthasarathyetal.[ 85 ]IO20,ref4.222e35570=Ru(1=353)]TJ /F7 7.97 Tf 6.59 0 Td[(1=T)A=m2ExperimentKI1.25Garcaetal.[ 1 ]KII0.8-Ncell40UNFNWPamb101325PaRu8.31451J=(molK)RHamb0 cathodesideand0.03Vattheanodesideofthestack.Theresistivityis0.298)]TJ /F4 11.955 Tf 12.13 0 Td[(cm2withastandarddeviationlessthan1%ofthemeasuredvalue.Theanodeoverpotentialandcathodeoverpotentialincreasewithincreasingcurrentdensity.Theoverpotentialcanbeviewedastheenergyrequiredforthereactiontotakeplace.Togeneratehighercurrentdensity,thedeviationsoroverpotentialsfromtheequilibriumstatusofbothelectrodesincreaseastheincreaseofcurrentdensity.Theanodeandcathodeoverpotentialswerecalculatedwithequations( 3 ) 99

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and( 3 ).Themodelingresultshowedthatthecathodicoverpotentialincreasednonlinearlyattheconcentrationoverpotentialregion.Asshowninequation( 3 ),theratioofoxygenattheCCLtotheambientaffectsthecathodicoverpotential.Attheconcentrationoverpotentialregion,theoxygenratioismuchlowerduetotheneedofhigherconcentrationgradientforhigherreactionrate.Thedecreaseoftheoxygenratioresultsintheincreaseofthecathodicoverpotential.Figure 5-2 showstheexperimentalandmodelingresultsofthecellvoltageasafunctionofcurrentdensitywithstacktemperaturesof45C,50C,60C,70C,and80C.Theexperimentswereruninconstant-currentmodewitha1.0Mfeedmethanolsolution.Theoperatingcurrentdensitywasincreasedfrom0mA=cm2tothemaximumcurrentdensitythatstillallowsthestackaveragevoltagetoremainabove0.2Vtopreventcausingdamagetothestack.Themaximumcurrentdensityreachedforeachofthevestacktemperaturesinvestigatedwas230,220,210,170,and135mA=cm2,respectively.Asexpected,thecellhasbetterperformanceathighertemperaturebecausehighertemperatureacceleratesthereactionsaswellasenhancingthemasstransportgenerally.AsshowninFigure 5-2 ,intheactivationandohmicoverpotentialregionsthecellperformanceincreasedasthestacktemperatureincreased.However,intheconcentrationoverpotentialregionthecellperformancedoesnotalwaysincreasewithincreasingstacktemperature.Notethateventhoughtthestackperformanceincreasedwhenthestacktemperaturewasincreasedfrom45to50C,thereisareversalinthispatternstartingat60Cwhereaperformancedegradationrelativetothelowertemperatureswasobserved.Morespecically,notethatatatemperatureof60Candacurrentofapproximately185mA=cm2thevoltageofthepolarizationcurvedroppedbelowthatofthe50Cpolarizationcurveandathighercurrentitfurtherdroppedbelowthe45Cpolarizationcurve.Hence,intheconcentrationoverpotentialregion,atemperatureincreaseto60Ccausedadropinperformance.Withstacktemperatures 100

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at70and80C,thepolarizationcurvesrevealedasimilarvoltagedropstartingat150and100mA=cm2,respectively,andcrossedthepolarizationcurvescorrespondingtolowertemperatures.Theinterpretationoftheseresultsinvolvesrecognizingthatintheconcentrationoverpotentialregionthecellperformanceislimitedbythemasstransportofreactants.Infact,thereactionrateinthehighcurrent-densityregionisveryhigh,andthereforesustainedoperationrequiresahighertransportrateofneededreactantstothecatalysissites,inparticularahigherrateofoxygentransportbecomesnecessary.Duetotheopen-cathodedesignandtheimplementationoftheLBL,atthecathodesideofthestack,watervaporandairtransportcompetewitheachotherforavailablevoidspace(channels,capillaries,etc.).Asmentioned,theLBListopreventthelossofliquidwaterfromthefuelcellstack.TheLBLhashighlyhydrophobiccharacteristics,enablingthepassivewaterrecoveryinsidethestack.TheliquidwatergeneratedatthecathodecatalystlayercanonlytransportthroughtheLBLandCGDLinvaporform,consistentwiththeexperimentalobservationthatnoliquidwaterexitsthecathodesideofthestack.Duetothedramaticincreaseofthesaturationvaporpressureofwaterincreaseswiththeincreaseofthestacktemperature,atasufcientlyhightemperaturetheuxofwatervaportransportedfromtheCCLtotheCFCexperiencesadramaticincrease,occupyingalargefractionofthevoidspaceavailableforgasdiffusion,hencedeprivingoxygenmoleculesfromaccessingatransportspacelargeenoughtosatisfythereaction'sreagent-ratedemandsatthecathode.Inaddition,duetotwoprimaryreasons,thewatervaporuxgiveninequation( 4 )experiencesadramaticincreaseatsufcientlylargetemperatures.First,thewater-vaporconcentrationgradientappearingontherighthandsideofequation( 4 )increasesinanonlinearfashionathighertemperatures.Infact,at80Cthewater-vaporpressureisapproximately4.9timeshigherthanat45C.Second,thewatervapormasstransportparameterappearingasafactorontherighthandsideofequation( 4 )alsoincreases 101

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becausetheeffectivediffusioncoefcientofwatervapor(seenumeratorofequation( 4 )becomeslargerwithtemperature,aswillbeshowninFigure 5-3 .ItisthenconcludedthatathighcurrentdensitiestheuxofwatervaportransportedfromtheCCLtotheCFCcanexperiencearelativelylargeincreaseathighertemperatures,leadingtothereductionofdiffusionchannelsforoxygenmoleculesandcausingtheobservedperformancedegradationintheconcentrationoverpotentialregionofthepolarizationcurve.Figure 5-3 showshowthewatervapormasstransportparameterKH2Ovarieswiththeoperatingcurrent-densityatstacktemperaturesof45,50,and60C.TheaverageexperimentalKH2Ovaluesatthosethreetemperaturesare1.4194,1.4686,and1.5238mm/s,respectively.ThegureshowsthatKH2Oincreaseswithanincreaseinstacktemperature.Theinterpretationoftheseresultsisthatathigherstacktemperatureswatervaporistransportedmoreeasilyduetothecorrespondingincreaseinitslumpedeffectivediffusioncoefcient.Hence,thenumeratorinequation( 4 )increasesasafunctionoftemperature,andconsequently,thewatervapormasstransportparametercorrespondinglyincreases,consistentwiththeobservationsofFigure 5-3 .Fromapracticalperspective,theresultsindicatethatsinceahigherKH2Ovaluecorrespondstolowerresistancetowater-vaportransport,underthoseconditionsthestackmayneedtooperateatlowertemperaturestoreducetheuxofwatervaportransportedfromtheCCLtotheCFCandhencepreservewater-balanceneutrality.However,sinceitisoftendesiredtooperatethestackathighertemperatures(wherehigherefcienciescanbeobtained),itmaybeofinteresttomanufacturestackswithstructuresalowerKH2Ovalue(introducingsmallerporesizes,orincreasingtheporetortuosity,forexample),hencereducingwaterlossratesathigheroperatingtemperatureandthusallowingthepreservationofwaterneutrality.Thislatterdesignoptioninvolvesanoptimizationcompromisegiventhatthesmallerporesizesandothermaterial 102

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featuresthatlowerthevalueofthewatervapormasstransportparameterunfortunatelyalsotendtocauseanundesirableincreaseinoxygen-diffusionresistance.Figure 5-4 showstherateofmethanolcrossoverJxo,varieswiththecurrentdensityforstacktemperaturesof45,50,and55C.Themeasurementerrorsarehigheratlowercurrentdensitybecausethecontrolofthestacktemperatureisharderatlowcurrentdensitywithoutaninternalheater.ThemethanolcrossoverrateisexpressedinmA=cm2becauseitiseasiertoquantifyandcomparewiththeoperatingcurrentdensity.Comparingthemethanolcrossoverratesatthecurrentinvestigated,itisobservedthatat45,50,and55Cthemethanolcrossoverratedecreasesfrom55.29to40.52mA=cm2,61.17to45.06mA=cm2,and70.80to53.97mA=cm2,respectively.Themethanolcrossoverrateissimulatedwithequations( 3 ),( 3 ),( 3 )and( 3 ).Equation( 3 )showsthatthedrivingforcesofthemethanolcrossoverarethemethanolconcentrationgradientacrossthemembraneandtheelectro-osmoticforcefromthehydrogenproton.Thematerialproperties,suchasmethanoldiffusioncoefcientsandthethicknessofeachlayer,canalsoaffecttherateofmethanolcrossover.Astheincreaseofthecurrentdensity,moremethanolisconsumedattheACLduetotheusefulelectrochemicalreaction.Hence,theconcentrationofmethanolattheinterfaceofACLandMEMaswellastheconcentrationgradientacrossthemembraneislower.Thisconsequentlyreducesthemethanolcrossoverrate.Althoughtheincreaseofthecurrentdensityalsoincreasestheamountofhydrogenprotonsortheelectro-osmoticforce,therateofmethanolcrossoverisdominatedbymethanoldiffusion.Figure 5-5 showsthemodelingresultoftherateofmethanolcrossoverduetodiffusionandelectro-osmoticforceat50C.Therateofmethanolcrossoverisalwaysdominatedbydiffusion.Itisworthtomentionthattheelectro-osmotictermistheproductoftheelectro-osmoticdragcoefcientandthecurrentdensityasshowninequation( 3 ).Theelectro-osmoticdragcoefcientisproportionaltothemethanolmolefractionattheinterfaceofACLandMEM.The 103

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methanolmolefractionattheinterfaceofACL/MEMisalsoafunctionofthecurrentdensity.Hence,theproleoftherateofmethanolcrossoverduetotheelectro-osmoticforceisparabolic.Atagivencurrentdensity,themethanolcrossoverrateincreaseswiththeincreaseofthestacktemperature.Forexample,themethanolcrossoverrateincreasesfrom42.51mA=cm2to57.59mA=cm2atthecurrentdensityof120mA=cm2whenthestacktemperatureincreasesfrom45Cto55C.WhentheDMFCoperatesatahigherstacktemperature,theresultingmethanolcrossoverrateishigher.Thereasonforthisobservationisthat,forxedcurrentdensity,atahigherstacktemperaturetherateofmasstransportofmethanolisfasterthanatalowertemperature.Theincreaseofmethanolcrossoveralsoaffectsthewaterbalanceofthestack.Crossovermethanolisasourceforwatergenerationinsidethestack.Athigherstacktemperaturesthewatergenerationrateinsidethestackishigherduetohighermethanolcrossoverrate;however,theoverallwaterbalancedecreasesathigherstacktemperatureaswillbeshowninFigure 5-6 .Infact,at150mA=cm2,anincreaseofstacktemperaturefrom45Cto55Ccanchangeoperationsfromwaterrecoverymodetowaterlossmode.Asmentionedbefore,thestacktemperatureincreasealsoraisestherateofwatervaporventingoutofthestack.Althoughthewatergenerationrateisincreasedduetotheincreaseofmethanolcrossoverrate,thewaterlossrateduetotheincreaseofstacktemperaturestilldominatestheoverallwaterbalanceofthesystem.Figure 5-6 showsthestacktemperatureeffectonthedimensionlesswater-balanceparameterdenedinSection 4.4.1 .Thestacktemperaturewassetto45,50,55,and60Candthemaximumoperatingcurrentdensitywas150mA=cm2.At45C,increasesfrom-0.62to0.29overtheoperatingrange.At50C,increasesfrom-0.67to0.01.At55C,increasesfrom-0.69to-0.18.At60C,increasesfrom-0.72to-0.33. 104

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Recallthatwhileisgreaterthanzero,theDMFCisrunninginwaterrecoverymode,addingtowaterinventoryintheanodeloop.Conversely,theDMFCisoperatinginwaterlossmodewhileislessthanzero,andwaterneutralitymodewhileisequaltozero.FromthedataofFigure 5-6 ,itisconcludedthatonlystacktemperaturesof45and50ChavethepossibilitytooperatetheDMFCunderwaterrecoverymode(i.e.,withawaterbalanceparametervaluegreaterthanzero),akeysystemcapability.Thewater-neutralityconditionoccursatabout101mA=cm2ifthestacktemperatureis45Candatabout150mA=cm2ifthestacktemperatureis50C.Thewaterbalanceparameterbearsfurtherexaminationfromapracticalperspective.Thesourcetermsthatcontributetowatergenerationaretheelectrochemicalreactionandtheoxidationofcrossovermethanol.Thewatersink(loss)termiswaterventingthroughLBLandCGDLstructurestothecathodeairstream.Byincreasingtheoperatingcurrentdensitythereactionrateisalsoincreased,andthereforemorewaterisgeneratedinsidethestack.AsshowninFigure 5-4 ,anincreaseofcurrentdensityalsodecreasesthemethanolcrossoverrate,reducingtherateofwatergeneration.However,therateofdecreaseofthemethanolcrossoverratecausedbyraisingthecurrentdensityisdominatedbytherateofwatergenerationthroughtheincreaseincurrentdensity.Thiscancompensateforthelossofwatergenerationduetothedecreasedmethanolcrossover.Byincreasingthestacktemperature,boththewatervaportransportparameterKH2Oandthewatervaporpressureareincreased,resultinginmorewaterlossesbecauseoftheincreaseofwatervaportransportux.Themodelingandexperimentalresultsshownabovewereallat1Mmethanol-watermixturesolution.Inordertotestthecapabilityofthecreatedmodel,themodelingresultswithdifferentmolaritiesareshowninFigures 5-7 to 5-10 .Figure 5-7 showsthecellvoltageasafunctionofcurrentdensityatmolaritiesof0.6M,0.8M,1.0M,and1.2M,coveringmostoftheoperationrangeofthepracticalsystem.Asshowninthegure,themodelcanpredicttheexperimentalresultsverywell.Theerrorofthemodelingresults 105

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iswithin1%.Inaddition,theresultsshowthatthechangeofthemolarityaffectsthecellperformance.Forexample,thecellvoltageof0.6M,0.8M,1.0M,and1.2Mat150mA=cm2are0.369V,0.368V,0.361V,and0.352V,respectively.Thepredictedmaximumcurrentdensityof0.6M,0.8M,1.0M,and1.2Mare210.32mA=cm2,276.55mA=cm2,252.41mA=cm2,and238.49mA=cm2.Basedonthemodelingresults,thesuggestedmolaritytooperatetheDMFCat50Ctohaveabetterperformancewillbeat0.8Mbecauseofthecellvoltageandthemaximumcurrentdensitycanbereached.Figure 5-8 showstherateofmethanolcrossoverasafunctionofcurrentdensityatmolaritiesof0.6M,0.8M,1.0M,and1.2Mat50C.Asexpected,themoreconcentratedofthemethanol-watermixturehasahigherrateofmethanolcrossover.Forexample,themethanolcrossoverrateincreasesfromaround40mA=cm2to80mA=cm2whenthemolarityincreasesfrom0.6Mto1.2M.Themodelingresultsdonotttheexperimentaldataquitewellat0.6Mand1.2M.ThisisbecausethosetwosetofexperimentswereconductedwhentheDMFChadexperienceddegradation,causingahigherrateofmethanolcrossover.Figure 5-9 showsthewatervapormasstransportparameter,KH2O,asafunctionofcurrentdensityatmolaritiesof0.6M,0.8M,1.0M,and1.2Mat50C.Asmentioned,KH2Oisatemperaturedependentparameter.Becausetheexperimentalandmodelingresultsareatconstanttemperature,KH2Odoesnotvaryverymuch.TheaveragevalueofKH2Oisabout1.45mm/s,whichisthein-situmeasuredvalueaftertheMEAwasmanufactured.Figure 5-10 showsthewaterbalanceparameter,,asafunctionofcurrentdensityatmolaritiesof0.6M,0.8M,1.0M,and1.2Mat50C.Asshowninthegure,at150mA=cm2,increasesfrom-0.122to0.106,turningtheDMFCstackfromwaterlossmodetorecoverymode,whenthemolarityincreasesfrom0.6Mto1.2M.AsshowninFigure 5-8 ,themoreconcentratedmethanolsolutioninducesahigherrateofmethanol 106

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crossover.Recallthattherateofmethanolcrossoverisoneofthesourcetermsofwatergeneration.Hence,atconstantcurrentdensityandstacktemperature,themoreconcentratedmethanolsolutionfavorswaterbalanceoftheDMFCstack.Inadditiontoexercisingthemodeltostudythekeyparametersofinterest,thismodelisalsocapableofchangingthepropertiesofthematerial.Forexample,Figure 5-11 showstheeffectsonthecellperformancebychangingoftheporosity,,oftheLBL30%referredtothebaseline,0.285.Bedecreasingto0.2,theDMFCstackhasadramaticincreaseonthewaterbalanceparameter,.Thestackcanreachwaterrecoverymodeat10mA=cm2,nevertheless,thecellperformanceissacriced.Themaximumcurrentdensitycanbeachievedisonly50mA=cm2.Thereasonforthedecreaseofthecellperformanceisbecausetheoxygenishardertotransporttothereactionzonewhentheporosityissmaller.ThedenitionofKO2hereissimilartoKH2O.ThehighervalueofKO2meansthelowerresistanceforoxygentotransporttothereactionzone.Asshowninthegure,theaverageKO2decreasesfrom1.061to0.376whendecreasesfrom0.285to0.2.ThedramaticdecreaseofKO2greatlyincreasestheoxygentransportresistance,resultingalowercellperformance.Incontrast,thecellperformancecanbeincreasedbytheincreaseofto0.37butthewaterbalanceofthestackissacricedduetothedecreaseofthewatervaportransportresistance,resultingahigherventingrateofwatervapor.Inpractice,theporositycanbecontrolledbyamountofhydrophobicinkappliedtotheCGDL.AhighporositylayercanbeobtainedbypaintingfewertimesofthehydrophobicinkontheCGDLand,viceversa,alowporositylayercanbeachievedbypaintingmoretimesofthehydrophobicinkontheCGDL. 107

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Figure5-1. Experimentalandmodelingresultsofsegregatedanode,cathode,andOhmicoverpotentialsat50C. 108

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Figure5-2. Experimentalandmodelingresultsofpolarizationcurvesatstacktemperaturesof45C,50C,and60C. 109

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Figure5-3. Experimentalandmodelingresultsofthewatervapormasstransportresistanceatstacktemperaturesof45C,50C,and60C. 110

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Figure5-4. Experimentalandmodelingresultsofmethanolcrossoveratstacktemperaturesof45C,50C,55C,and60C. 111

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Figure5-5. Therateofmethanolcrossoverduetodiffusionandelectro-osmoticforceat50C. 112

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Figure5-6. Experimentalandmodelingresultsofwaterbalanceparameteratstacktemperaturesof45C,50C,and60C. 113

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Figure5-7. Experimentalandmodelingresultsofcellvoltageatmolaritiesof0.6M,0.8M,1.0M,and1.2M. 114

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Figure5-8. Experimentalandmodelingresultsofmethanolcrossoverrateatmolaritiesof0.6M,0.8M,1.0M,and1.2M. 115

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Figure5-9. Experimentalandmodelingresultsofwatervapormasstransportparameteratmolaritiesof0.6M,0.8M,1.0M,and1.2M. 116

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Figure5-10. Experimentalandmodelingresultsofwaterbalanceparameteratatmolaritiesof0.6M,0.8M,1.0M,and1.2M. 117

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Figure5-11. Keyparametersoffuelcellstackasafunctionofporositywith30%changeofbaselinecase. 118

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CHAPTER6CONCLUSIONSANDFUTUREWORKThewaterbalanceandperformanceofanin-houseDMFCstackwithanopen-cathodedesignutilizingaliquidbarrierlayerwascharacterizedexperimentallyandanalytically.Asetofexperimentswasdesignedandconductedtomeasurethecellperformance,therateofmethanolcrossover,andthekeyparametersrelatingtowaterbalanceofthestack.Inaddition,aDMFCmodelwascreatedandexercisedtostudytheeffectofthekeyparameters,suchasstacktemperature,solutionmolarity,andporosityoftheLBL,onthecellperformanceandwaterbalanceofthestack.Themodelingresultswerealsovalidatedwithexperimentaldata.Anewdimensionlesswaterbalanceparameterhasbeendenedinordertoquantifytheperformanceinthatarea.TheresultsshowedthatthewaterbalanceoftheDMFCstackishighlydependentontheoperatingtemperature.TheexistingDMFCstackcouldoperateinwater-recoveryorwater-neutralitymodeonlyforstacktemperaturesof50Corlower,whenthecurrentdensitywasunderthenominaldesignvalueof150mA=cm2.Anincreaseofthestacktemperatureresultsinanincreaseoftherateofwaterloss.TheresultssuggestthatthemosteffectivewaytocontroltheDMFCstackfromwater-lossmodetowater-recoverymodeistoreducethestacktemperature.Accordingtothemodel,therateofmethanolcrossoverdecreaseswithanincreaseoftheoperatingcurrentdensitybecausemoremethanolisconsumedattheanodecatalyst,reducingthemethanolconcentrationgradientwhichisthedrivingforceofmethanolcrossover.Themethanolcrossoverratealsoincreaseswithincreasingstacktemperaturebecausehighertemperaturefavorsthetransportofmethanol.Themethanolconcentrationgradientdominatestherateofmethanolcrossoverascomparedtotheelectro-osmoticforce.Theelectro-osmoticdragisproportionaltotheproductofthemethanolconcentrationattheanodecatalystandtheoperatingcurrentdensity.Inthelowcurrentdensityregion,themethanolconcentrationishigh;ontheotherhand, 119

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themethanolconcentrationislowathighcurrentdensity,resultinginaquadraticproleoftheelectro-osmoticdrivingforce.Thedevelopedmodelwasalsousedtopredicttheeffectsofchangingthesolutionmolaritiesonthecellperformance,methanolcrossover,andwaterbalanceofthestack.TheresultsshowedthattheDMFChasbestperformanceatthesolutionmolarityof0.8M.Thecellperformanceandmaximumcurrentdensitystarttodecreasewhenthesolutionmolarityincreasesfrom0.8to1.2Mbecauseoftheincreaseofthemethanolcrossover,causingmixedpotentialontheelectrode.However,theincreaseofsolutionmolarityenhancesthewaterrecoverybecauseoftheincreaseofthemethanolcrossoverrate.Therateofmethanolcrossoverandoperatingcurrentdensityalsoaffectthestackwaterbalance.Theresultsshowedthatincreasesofeitherofthosephysicalvariablesresultsingreaterliquidwaterrecovery.However,thewaterbalanceofthestackislesssensitivetothosevariablesthantothestacktemperature.Themodeldevelopedinthisworkisabletosimulatethemasstransportlimitationregionofthepolarizationcurvewiththeincorporationofamulti-componentmasstransportmodelinthecathode.Thein-houseDMFCstackpolarizationcurveexperiencesasharpdropathigherstacktemperatureandthemodelisabletocapturethisphenomenon,duetothecounter-diffusionofwatervaportotheowchannelandoxygentothereactionzone.ThemodelingresultsalsoshowedthattheLBLhasamoresignicanteffectonthewaterbalanceandcellperformancethantheCGDLdoes.ByincreasingtheporosityoftheLBL30%,thecellperformanceisincreasedsignicantlybuttheventrateofthewatervaporisalsoincreased,resultinginawater-lossmode.TheresultsalsoshowedthatadecreaseoftheporosityoftheLBLenhancesthewaterrecoveryofthestack,butthatthecellperformanceisdegraded.Inadditiontothevariablesstudiedinthiswork,thismodelcanalsobeusedtodesignaDMFCsystemorstackbychangingtheotherkeyvariables,forexample,the 120

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electrodeproperties:transfercoefcient,thicknessofeachlayer,catalystsurfaceareaperunitvolume,porosityandtortuosity,etc.,owchanneldesign:channellength,depthandwidth,etc.,andenvironmentalcondition:ambienthumidity,oxygenconcentration.ThusthemodelissuitableforoptimizationoftheoverallsystembasedondesignchoicesfortheindividuallayerswithintheMEAandonsystemchoicessuchasanodeloopmolarityandstoichiometry.Thereareafewsuggestionstomakeforfollow-onwork.1)Extendthemodeltobetwo-dimensional.Apreliminarycalculationshowedthatthedecreaseoftheoxygenmolefractionalongtheowchannelisabout2.08%whenthecurrentdensityisat150mA=cm2.Also,ontheanodeside,theerrorduetothedecreaseofmethanolconcentrationalongtheowchannelcanbeashighas6.38%whenthecurrentdensityisat150mA=cm2.Thesimulationofthischangeonthecellperformancealongtheowchannelshouldincreaseaccuracyincomparingexperimentalresultstosimulationresults.2)Usethemodelforanoptimizationstudy:exercisethemodeltodeterminetheoptimizedparameters,perhapsbycouplingtoanexistingoptimizer.3)Includedegradationmechanismsinthemodel.TheDMFCdegradeswithoperatingtime;itisusefultobuildamodeltopredictthedegradationeffectsonthecellperformance.Theactivesurfaceareaofthecatalystparticlesmaybeakeyparametercontributingtothedegradationmechanism.4)Simulatethestructureoftheliquidbarrierlayerinmoredetail.Themorphologyoftheliquidbarrierlayeraffectsthemasstransportofthespeciesaswellasthewaterbalanceandcellperformance.Theporesizedistributionandtortuosityofthislayercanbeobtainedbyapplyingimagingprocessingtechnique.5)Designacontrolalgorithmbasedonthewaterbalancemodelandimplementsitintoapracticalsystem. 121

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APPENDIXATOTALERRORANALYSISOFTHEWATERBALANCEEXPERIMENTSThetotalerroranalysisofthewaterbalanceexperimentsisprimarilybasedonthemethodologypublishedby[ 81 83 86 ].Thedetailedderivationscanalsobeseenintheabovereferences.Theapplicationoftheuncertainanalysismethodtothewaterbalanceexperimentconductedinthisworkisdemonstratedstepbystepasfollows.Inthewaterbalanceexperiment,sixindividualvariables:CO2concentrationattheexitofcathode,waterconcentrationattheexitofcathode,airowrateattheinletofcathode,currentdensity,stacktemperature,andambienttemperaturearemeasured.Fromthosemeasurements,fourimportantparameters,CO2owrateattheexitofcathode,watervaporowrateattheexitofcathode,methanolcrossovercurrentdensity,andwatervapormasstransportparameterareabletobeobtainedby_nCao,CO2=)]TJ /F9 11.955 Tf 6.98 -9.68 Td[(_nCai,O2+_nCao,N2)]TJ /F9 11.955 Tf 13.45 0 Td[(_nCaO2X 106+0.5X)]TJ /F9 11.955 Tf 11.95 0 Td[(103Y (A)_nCaO2=NcellJcellA 4F10)]TJ /F7 7.97 Tf 6.59 0 Td[(3 (A)_nCao,H2O=Y X_nCao,CO2103 (A)Jxo=6F_nCao,CO2 NcellA103 (A)KH2O=N"ventH2O1 CCCLH2O)]TJ /F4 11.955 Tf 11.96 0 Td[(CCFCH2O107 (A) 122

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CCCLH2O=Psat(Tstack) RuTstack (A)CCFCH2O=Psat(Tamb)RHamb RuTamb (A)ThetotaluncertaintyUrcomprisesofbiaslimit,Br,andprecisionlimit,Pr.Thesquare-sum-root(SSR)ofthetotaluncertaintyisexpressedasU2r=B2r+P2r (A)wherethesubscriptrdenotestheresultoftheparameters.Asmentionedbefore,theparametersofinterestinthisexperimentsare _nCao,CO2, _nCao,H2O,Jxo,andKH2O.TheSSRofbiaslimit,Br,canbefurthergivenbyB2r=JXi=12iB2i+2J)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=1JXk=i+1ikBik (A)Here,irepresentsthesensitivityoftheindividualvariables,Biisthemagnitudeoftheresolutionofeachinstrument,andBikmeanstheinteractionbetweentheindividualvariables.Theparametersofinterestasafunctionoftheindividualvariablesarelistedas _nCao,CO2=(X,Y,Jcell,_nCai,Air) (A) _nCao,H2O=(X,Y,Jcell,_nCai,Air) (A) 123

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Jxo=(X,Y,Jcell,_nCai,Air) (A)KH2O=(X,Y,Jcell,TStack,Tamb,_nCai,Air) (A)andthesensitivitiesoftheindividualvariablescanbeobtainedbyi=@r @Xi (A)Forexample,thebiaslimitfortheCO2owrateattheexitofcathodecanbeexpressedasB2 _nCao,CO2=2XB2X+2YB2Y+2JcellB2Jcell+2XYBXBY (A)andthesensitivitiesoftheindividualvariablesareX=@ _nCao,CO2 @X=(_nCai,Air)]TJ /F5 7.97 Tf 13.15 5.11 Td[(NcellJcellA10)]TJ /F6 5.978 Tf 5.76 0 Td[(3 4F)(106)]TJ /F9 11.955 Tf 11.96 0 Td[(103Y) (106+0.5X)]TJ /F9 11.955 Tf 11.96 0 Td[(103Y)2 (A)Y=@ _nCao,CO2 @Y=103X(_nCai,Air)]TJ /F7 7.97 Tf 13.15 5.11 Td[(NcellJcellA10)]TJ /F6 5.978 Tf 5.76 0 Td[(3 4F) (106+0.5X)]TJ /F9 11.955 Tf 11.96 0 Td[(103Y)2 (A) 124

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Jcell=@ _nCao,CO2 @Jcell=)]TJ /F9 11.955 Tf 9.3 0 Td[(X(NcellJcellA10)]TJ /F6 5.978 Tf 5.76 0 Td[(3 4F) (106+0.5X)]TJ /F9 11.955 Tf 11.95 0 Td[(103Y) (A)_nCai,Air=@ _nCao,CO2 @_nCai,Air=X (106+0.5X)]TJ /F9 11.955 Tf 11.96 0 Td[(103Y) (A)wherethebiaslimitmagnitudesforeachinstrumentareshowninTable 4-2 .TheprecisionlimitisestimatedbyP2 _nCao,CO2=(2S _nCao,CO2=p M)2 (A)whereMandS _nCao,CO2representsdegreeoffreedomandstandarddeviationofthetests,respectively.Theestimationisassumedtheresultsfollownormaldistributionand95%condenceinterval.Afterthebiaslimitandprecisionlimitareobtained,theSSRofthetotaluncertaintyisestimatedbyU2 _nCao,CO2=B2 _nCao,CO2+P2 _nCao,CO2 (A)Thesameproceduresalsoapplytotheotherparametersofinterestforthetotaluncertaintiesestimateandtheformulasareasfollows.Thepreliminaryanalysisshowsthattheprecisionlimitmainlycontributetothetotaluncertainty.Thecontributiontothetotaluncertaintyofbiaslimitislessthan2%.Thetotaluncertaintyfortheexperimentalsetupis10to15%. 125

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APPENDIXBMETHODSTOSOLVETHECOUPLEDREACTION-DIFFUSIONEQUATIONSForanelectrochemicalsystem,asetofdifferentialequationsareusuallyobtainedbasedonthematerialandenergybalanceofthesystem.Theoriginalsetofreaction-diffusionequationsisusuallynonlinearandproperlinearizationofthesetofnonlinearequationsisoftenneeded.In1968,Newman[ 87 ]rstpublishedanumericaltechniqueforsolvingthenonlinear,coupledordinarydifferentialequations(ODEs).ThismethodinvolvestheproperlinearizationoftheoriginalnonlinearsetofcoupledODEsbyatrialsolutionandputthesetoflinearizedequationsintoanitedifferenceform.ThesolutionofthealgebraequationscanalsobeobtainedbyNewman'sBNAD(J)algorithm.AfurtherdetailedexplanationofthisnumericaltechniquecanalsobefoundinAppendixCin[ 74 ].InordertopromoteNewman'snumericaltechnique,White[ 88 ]furtherpublishedanalternativederivationofNewman'sworkforsolvingtheboundaryvalueproblems.In1991,FanandWhite[ 89 ]presentedamodiedBAND(J)algorithm,calledMBAND(J),todealwiththemulti-regionelectrochemicalsystems.TheMBAND(J)algorithmcanmaintaintheaccuracytoorderh2andsuccessfullyreducethecomputationtimevetimesthantheBNAD(J)algorithm.Duanetal.[ 90 ]furtherappliedNewman'smethodtomodelthecouplednonlineartwopointboundaryproblemsinthecatalystlayerofafuelcell.ThenitevolumemethodproposedbyPatankar[ 91 ]isalsowidelyusedinthecommercialpackagesforsolvingthecouplednonlinearreactiondiffusionproblems.ThecouplednonlinearODEscanalsobesolvedbytheode45functioninMATLABR[ 73 ].Here,howtoapplytheNewman'stechnique,nitevolumemethod,andode45functiontosolvetheoftenencounteringnonlinearBulter-Volmerequationinanelectrochemicalsystemisdemonstrated.Thecomparisonofthemodelingresultsisalsoshowninthissection.Thesameexampleistakenas[ 90 ]forareference. 126

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B.1Newman'sMethodAttheanodesideofaprotonexchangemembranefuelcell,theelectrochemicalreactioncanbeexpressedasH2!2H++2e)]TJ /F1 11.955 Tf -268.63 -40.8 Td[(andtheuxofthehydrogentothereactionzoneisNH2=)]TJ /F4 11.955 Tf 9.3 0 Td[(DH2dCH2 dX (B)Basedonthematerialbalanceofthereactiondiffusionprocess,thechangeofthehydrogenuxisequaltotherateofreaction)]TJ /F4 11.955 Tf 10.5 8.09 Td[(dNH2 dX=RH2 (B)wherethereactionrateisgivenbyButler-VolmerequationRH2=ak0fCH2exp[(1)]TJ /F10 11.955 Tf 11.95 0 Td[()nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.95 0 Td[(E0)])]TJ /F4 11.955 Tf 11.95 0 Td[(CH+exp[)]TJ /F10 11.955 Tf 9.3 0 Td[(nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.95 0 Td[(E0)]g (B)Substituteequations( B )and( B )intoequation( B )yieldsDH2dCH2 dX=ak0fCH2exp[(1)]TJ /F10 11.955 Tf 11.96 0 Td[()nF RuT(s)]TJ /F10 11.955 Tf 11.95 0 Td[(e)]TJ /F4 11.955 Tf 11.96 0 Td[(E0)])]TJ /F4 11.955 Tf 11.96 0 Td[(CH+exp[)]TJ /F10 11.955 Tf 9.3 0 Td[(nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.96 0 Td[(E0)]g (B)Forthesolidmatrixandtheelectrolyte,theelectronicandioniccurrentdensityareexpressedas 127

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is=)]TJ /F10 11.955 Tf 9.3 0 Td[(ds dXandie=)]TJ /F10 11.955 Tf 9.3 0 Td[(de dXbasedonOhm'slaw.TheratesofchangeoftheelectronicandioniccurrentarealsoproportionaltothereactionrategivenbyBulter-Volmerequationonavolumetricbasisandcanbeexpressedasd2s dX2=nFak0fCH2exp[(1)]TJ /F10 11.955 Tf 11.95 0 Td[()nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.96 0 Td[(E0)])]TJ /F4 11.955 Tf 11.95 0 Td[(CH+exp[)]TJ /F10 11.955 Tf 9.3 0 Td[(nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.96 0 Td[(E0)]g (B)d2e dX2=)]TJ /F4 11.955 Tf 9.3 0 Td[(nFak0fCH2exp[(1)]TJ /F10 11.955 Tf 11.95 0 Td[()nF RuT(s)]TJ /F10 11.955 Tf 11.96 0 Td[(e)]TJ /F4 11.955 Tf 11.95 0 Td[(E0)])]TJ /F4 11.955 Tf 11.95 0 Td[(CH+exp[)]TJ /F10 11.955 Tf 9.3 0 Td[(nF RuT(s)]TJ /F10 11.955 Tf 11.95 0 Td[(e)]TJ /F4 11.955 Tf 11.95 0 Td[(E0)]g (B)TheboundaryconditionsareAtX=0,CH2=C0H2,ds dX=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(Icell ,e=0 (B)AtX=L,dCH2 dX=0,ds dX=0,de dX=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(Icell (B)Giventhedimensionlessvariables 128

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c0=CH+ C0H2,c1=CH2 C0H2,c2=nF RuTs,c3=nF RuTex=X L,2=ak0L2 DH2,2s=an2F2k0L2C0H2 RuT,2e=an2F2k0L2C0H2 RuTU=nF RuTE0,ks=nFL RuTIcell,ke=nFL RuTIcellequations( B )to( B )becomed2c1 dx2=2[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.59 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U))]TJ /F4 11.955 Tf 11.96 0 Td[(c0e)]TJ /F15 7.97 Tf 6.59 0 Td[((c2)]TJ /F5 7.97 Tf 6.59 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U)] (B)d2c2 dx2=2s[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.59 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U))]TJ /F4 11.955 Tf 11.96 0 Td[(c0e)]TJ /F15 7.97 Tf 6.59 0 Td[((c2)]TJ /F5 7.97 Tf 6.59 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U)] (B)d2c3 dx2=)]TJ /F9 11.955 Tf 9.3 0 Td[(2e[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.59 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U))]TJ /F4 11.955 Tf 11.95 0 Td[(c0e)]TJ /F15 7.97 Tf 6.59 0 Td[((c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.59 0 Td[(U)] (B)andtheboundaryconditions( B )and( B )areAtx=0,c1=1,dcs dx=)]TJ /F4 11.955 Tf 9.29 0 Td[(ks,c3=0 (B)Atx=1,dc1 dx=0,dc2 dx=0,dc3 dx=)]TJ /F4 11.955 Tf 9.3 0 Td[(ke (B)Tosolveequations( B )to( B )withNewman'smethod,therststepistoapproximatethesecondorderderivativeswiththreepointcentraldifference. 129

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c1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(2c1(j)+c1(j+1)=h22[c1(j)e(1)]TJ /F15 7.97 Tf 6.58 0 Td[()(c2(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c3(j))]TJ /F5 7.97 Tf 6.59 0 Td[(U))]TJ /F4 11.955 Tf 11.96 0 Td[(c0e)]TJ /F15 7.97 Tf 6.58 0 Td[((c2(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c3(j))]TJ /F5 7.97 Tf 6.58 0 Td[(U)] (B)Thethreepointcentraldifferencehastheaccuracytotheorderofh2.Wecanfurtherexpressequation( B )asafunction,F1,j,andconducttheTaylorseriesexpansionofittotheaccuracyO(h2)F1,j=c01(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.95 0 Td[(2c01(j)+c01(j+1))]TJ /F4 11.955 Tf 11.96 0 Td[(h22[c01(j)e(1)]TJ /F15 7.97 Tf 6.58 0 Td[()(c02(j))]TJ /F5 7.97 Tf 6.58 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.58 0 Td[(U))]TJ /F4 11.955 Tf 11.95 0 Td[(c0e)]TJ /F15 7.97 Tf 6.59 0 Td[((c02(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.59 0 Td[(U)]+@F1,j @c1,j)]TJ /F7 7.97 Tf 6.59 0 Td[(10c1(j)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(c01(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+@F1,j @c1,j0c1(j))]TJ /F4 11.955 Tf 11.96 0 Td[(c01(j)+@F1,j @c1,j+10c1(j+1))]TJ /F4 11.955 Tf 11.96 0 Td[(c01(j+1)+@F1,j @c2,j0c2(j))]TJ /F4 11.955 Tf 11.96 0 Td[(c02(j)+@F1,j @c3,j0c3(j))]TJ /F4 11.955 Tf 11.96 0 Td[(c03(j)andthiscanbeexpressedasamatrixform1000BBBB@c1,j)]TJ /F7 7.97 Tf 6.59 0 Td[(1c2,j)]TJ /F7 7.97 Tf 6.59 0 Td[(1c3,j)]TJ /F7 7.97 Tf 6.59 0 Td[(11CCCCA+)]TJ /F9 11.955 Tf 9.3 0 Td[((2+h22r0))]TJ /F4 11.955 Tf 9.3 0 Td[(h22rc0h22rc00BBBB@c1,jc2,jc3,j1CCCCA+1000BBBB@c1,j+1c2,j+1c3,j+11CCCCA=h22r0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22rc0h22rc00BBBB@c01,jc02,jc03,j1CCCCA+h22rj0,0
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r0=e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c02(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.59 0 Td[(U)rj0=c01(j)e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c02(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.59 0 Td[(U))]TJ /F4 11.955 Tf 11.96 0 Td[(c0e)]TJ /F15 7.97 Tf 6.58 0 Td[((c02(j))]TJ /F5 7.97 Tf 6.58 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.58 0 Td[(U)rc0=(1)]TJ /F10 11.955 Tf 11.95 0 Td[()c01(j)e(1)]TJ /F15 7.97 Tf 6.58 0 Td[()(c02(j))]TJ /F5 7.97 Tf 6.58 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.58 0 Td[(U)+c0e)]TJ /F15 7.97 Tf 6.59 0 Td[((c02(j))]TJ /F5 7.97 Tf 6.59 0 Td[(c03(j))]TJ /F5 7.97 Tf 6.59 0 Td[(U)Afterapplyingthesameproceduretoequations( B )and( B ),acompactmatrixformcanbeobtainedA(j)C(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+B(j)C(j)+D(j)C(j+1)=G(j),16j6N)]TJ /F9 11.955 Tf 11.96 0 Td[(1 (B)whereA(j)=266664100010001377775B(j)=266664)]TJ /F9 11.955 Tf 9.3 0 Td[((2+h22r0))]TJ /F4 11.955 Tf 9.3 0 Td[(h22rc0h22rc0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22sr0)]TJ /F9 11.955 Tf 9.3 0 Td[((2+h22src0)h22src0h22er0h22erc0)]TJ /F9 11.955 Tf 9.3 0 Td[((2+h22erc0)377775C(j)=c1,jc2,jc3,jTD(j)=266664100010001377775G(j)=266664)]TJ /F4 11.955 Tf 9.3 0 Td[(h22r0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22rc0h22rc0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22sr0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22src0h22src0h22er0h22erc0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22erc0377775266664c01,jc02,jc03,j377775+266664h22rj0h22srj0)]TJ /F4 11.955 Tf 9.3 0 Td[(h22erj0377775 (B) 131

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Theboundaryconditionsatx=0andx=1alsocanbearrangedinasimilarformafterapplyingthreepointforwardandbackwardnitedifferenceapproximationtohavetheaccuracytotheorderofh2B(j)C(j)+D(j)C(j+1)+XC(j+2)=G(j),j=0YC(j)]TJ /F9 11.955 Tf 11.95 0 Td[(2)+A(j)C(j)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+B(j)C(j)=G(j),j=N (B)whereB(0)=266664100030001377775,D(0)=2666640000)]TJ /F9 11.955 Tf 9.3 0 Td[(40000377775,X=266664000010000377775,G(0)=26666412hks0377775,Y=266664100010001377775,A(N)=266664)]TJ /F9 11.955 Tf 9.3 0 Td[(4000)]TJ /F9 11.955 Tf 9.3 0 Td[(4000)]TJ /F9 11.955 Tf 9.3 0 Td[(4377775,B(N)=266664300030003377775,G(N)=26666400)]TJ /F9 11.955 Tf 9.3 0 Td[(2hke377775 (B)Then,thecompactmatrixcombinedwithequations( B )and( B )canbesolvednumericallybyguessingtrialinitialvaluesandtheconvergedresultscanbeobtainedafterseveraliterations.Theconvergedresultscanbefurthervalidatedwiththeboundaryconditions( B )and( B ). B.2FiniteVolumeMethodAlthoughthenalformatofthealgebraexpressionsisverysimilartoNewman'smethodbynitevolumemethodinthiscaseafterthelinearizationofthenonlineartermsandtheapproximationofthederivativesinequations( B )to( B ),thenitevolumemethodhasitsmerittotreatthenonlineartermasasourceterm.Basedonthismerit,auniversalformatofthediffusionreactionequationcanbewrittenfordifferentlayers 132

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intheMEAs.ThismeritalsoallowstheexpansionofthesimulationdomaintomultidimensionalmorefeasiblethanNewman'smethod.Figure B-1 illustratesthenodesassignmentforthenitevolumemethod.Letusrstrewriteequations( B )to( B )asd2c1 dx2)]TJ /F9 11.955 Tf 11.96 0 Td[(2[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U))]TJ /F4 11.955 Tf 11.95 0 Td[(c0e)]TJ /F15 7.97 Tf 6.58 0 Td[((c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U)]=d2c1 dx2+S1 (B)d2c2 dx2)]TJ /F9 11.955 Tf 11.96 0 Td[(2s[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U))]TJ /F4 11.955 Tf 11.95 0 Td[(c0e)]TJ /F15 7.97 Tf 6.58 0 Td[((c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U)]=d2c2 dx2+S2 (B)d2c3 dx2+2e[c1e(1)]TJ /F15 7.97 Tf 6.59 0 Td[()(c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U))]TJ /F4 11.955 Tf 11.95 0 Td[(c0e)]TJ /F15 7.97 Tf 6.58 0 Td[((c2)]TJ /F5 7.97 Tf 6.58 0 Td[(c3)]TJ /F5 7.97 Tf 6.58 0 Td[(U)]=d2c3 dx2+S3 (B)Toapproximatethederivativesinequations( B )to( B )bynitevolumemethod,therststepistointegratethederivativeswithinacontrolvolumeZewd2c1 dx2dx=dc1 dxe)]TJ /F4 11.955 Tf 14.34 8.09 Td[(dc1 dxw=cE1)]TJ /F4 11.955 Tf 11.96 0 Td[(cP1 xe)]TJ /F4 11.955 Tf 13.15 8.09 Td[(cP1)]TJ /F4 11.955 Tf 11.96 0 Td[(cW1 xw=cW1 xw)]TJ /F9 11.955 Tf 11.95 0 Td[((1 xe+1 xw)cP1+cE1 xe=c1,j)]TJ /F7 7.97 Tf 6.59 0 Td[(1 xw)]TJ /F9 11.955 Tf 11.96 0 Td[((1 xe+1 xw)c1,j+c1,j+1 xe (B)andtheintegrationofthenonlinearsourcetermisZewS1dx= S1x (B)Thensettheaverage S1asalinearfunctionbyTaylorseriesexpansion 133

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S1=SC1+SP1,1cP1+SP1,2cP2+SP1,3cP3=(2r0c01,j+2rc0c02,j)]TJ /F9 11.955 Tf 11.95 0 Td[(2rc0c03,j)]TJ /F9 11.955 Tf 11.96 0 Td[(2rj0)+()]TJ /F9 11.955 Tf 9.3 0 Td[(2r0)c1,j+()]TJ /F9 11.955 Tf 9.29 0 Td[(2rc0)c2,j+2rc0c3,j=2r02rc0)]TJ /F9 11.955 Tf 9.3 0 Td[(2rc00BBBB@c01,jc02,jc03,j1CCCCA)]TJ /F9 11.955 Tf 11.95 0 Td[(2rj0+)]TJ /F9 11.955 Tf 9.3 0 Td[(2r0)]TJ /F9 11.955 Tf 9.3 0 Td[(2rc02rc00BBBB@c1,jc2,jc3,j1CCCCA (B)Aftersubstitutingequations( B )to( B )intoequation( B ),asimilarformasequation( B )canbeobtained1 xw000BBBB@c1,j)]TJ /F7 7.97 Tf 6.58 0 Td[(1c2,j)]TJ /F7 7.97 Tf 6.58 0 Td[(1c3,j)]TJ /F7 7.97 Tf 6.58 0 Td[(11CCCCA+)]TJ /F9 11.955 Tf 9.3 0 Td[(((1 xe+1 xw)+x2r0))]TJ /F9 11.955 Tf 9.3 0 Td[(x2rc0x2rc00BBBB@c1,jc2,jc3,j1CCCCA+1 xe000BBBB@c1,j+1c2,j+1c3,j+11CCCCA=)]TJ /F9 11.955 Tf 9.3 0 Td[(x2r0)]TJ /F9 11.955 Tf 9.3 0 Td[(x2rc0x2rc00BBBB@c01,jc02,jc03,j1CCCCA+x2rj0,0
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B.4ResultsThesamesimulatedresultsasshownin[ 90 ]aredemonstratedinFigure B-2 toFigure B-5 .Figure B-2 toFigure B-5 demonstratethereactantconcentration,electronicpotential,ionicpotential,andreactionrateasafunctionofpositionandcurrentdensity,separately.c1isthedimensionlessreactantconcentration,c2representsthedimensionlesspotentialinmatrix(electronicpotential),c3isthedimensionlesspotentialinpore(ionicpotential),andk1isadimensionlessvariablewhichisproportionaltotheoperatingcurrentdensity.AsshowninFigure B-2 ,theconcentrationofthereactantdecreaseswithrespecttopositionduetotheelectrochemicalreaction.Thedecreasingmagnitudeofreactantconcentrationissmalleratloweroperatingcurrentdensitythanthehigheronesincethereactionrateislower.Figure B-3 showsthepotentialinthematrix(solidphase)asafunctionofpositionandoperatingcurrentdensity.BasedonOhm'slaw,thepotentialgradientisbiggerathigheroperatingcurrentdensitythanatthelowercurrentdensity.Figure B-4 alsoshowsthesameresult.Notethat,theanodiccurrentisdenedaspositiveandthereforetheelectronstransportingdirectionisfromrighttoleft.Figure B-5 furthershowsthereactionrateasafunctionofpositionandoperatingcurrentdensity.Thereactionrateishigherathighercurrentdensity.Amoredetaileddiscussionoftheresultsisnotthepurposehereandcanbefoundin[ 90 ].Figure B-6 andFigure B-7 alsoshowthecomparisonsofthesimulatedresultswithaforementionedmethodsat200mA=cm2and1600mA=cm2,respectively.Theresultsfromdifferentnumericalmethodsagreewitheachotherwithin1%error. 135

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FigureB-1. Theschematicofthenodes. FigureB-2. Dimensionlessreactionconcentrationasafunctionofdimensionlessvariablex(position)andk1(currentdensity). 136

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FigureB-3. Dimensionlesspotential(matrix)asafunctionofdimensionlessvariablex(position)andk1(currentdensity). FigureB-4. Dimensionlesspotential(pore)asafunctionofdimensionlessvariablex(position)andk1(currentdensity). 137

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FigureB-5. Reactionrateasafunctionofdimensionlessvariablex(position)andk1(currentdensity). FigureB-6. Comparisonsofresultsobtainedbydifferentnumericaltechniquesatoperatingcurrentdensityof0.2A=cm2. 138

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FigureB-7. Comparisonsoftheresultsbydifferentnumericaltechniquesatoperatingcurrentdensityof1.6A=cm2. 139

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BIOGRAPHICALSKETCH Cheng-Chan(CC)KuowasborninTainanCity,Taiwanin1979.HeearnedhisBachelorofScienceandMasterofSciencedegreesbothintheDepartmentofAeronauticsandAstronauticsEngineeringatNationalChengKungUniversityinTaiwanin2001and2003,respectively,undertheadvisingofDr.Muh-RongWang.AfterservinginAirForceforalmostoneandahalfyears,hestartedtopursuehisdoctoraldegreeinMechanicalEngineeringattheUniversityofFlorida(UFL),undertheguidanceofDr.WilliamE.LearandDr.JamesFletcher,in2006.Hemarriedhiswife,I-ChingHsueh,beforehewenttoUFL.Duringthisstudy-abroad,hislovelyson,WesleyWisdomKuo,wasbornin2009.HeisinterestedinenergyrelatedengineeringandwouldliketokeepcontributinghisknowledgelearnedatUFLtomaketheworldabetterplaceforlifeaftergraduation. 145