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Stationary Hemispherical Electrode under Submerged Jet Impingement and Validation of Measurement Model Concept for Imped...


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STATIONARYHEMISPHERICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENTANDVALIDATIONOFMEASUREMENTMODELCONCEPTFORIMPEDANCESPECTROSCOPYByPAVANKUMARSHUKLAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA 2004

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ACKNOWLEDGMENTSIwouldliketosincerelythankmyadvisor,ProfessorMarkE.Orazem,forhistechnicalguidancethroughoutmygraduateresearch.Iwillalwaysbegratefulforhispatienceandunderstanding.IwouldliketopayspecialthankstoDr.OscarCrisalleforhisexpertiseandguidanceintheparameterestimationtheoryandpro-gramming.Histimeandeffortwereinvaluabletome,especiallyinunderstandingtransferfunctionmethodology.IwouldliketoexpressmygratitudetoDr.GertNelissanforhishardworkinnumericalsimulations.Hissimulationresultsbuttressedmyworkalongsidemyexperiments.IgratefullyacknowledgeOLISystems,Inc.AmericanWay,MorrisPlains,NJUSA,www.olisystems.com foruseoftheirCorrosionAnalyzer1.3software.Iwouldliketoexpressmyappreciationtothemembersofmycommittee,Dr.JasonButler,Dr.AnujChauhanandDr.DarrylButt,fortheircontributionsinmydissertationdefense.Iwouldliketothankthepreviousandpresentmembersoftheelectrochemi-calengineeringgroup,MikeMembrino,ChenChenQui,KerryAllahar,NellianPerez-Garcia,andVickyHuang.Iwasfortunatetobeamemberofthisgroup.Iwouldliketoexpressmyheartfeltthanksandgratitudetomyfamilymem-bersandfriendswhohavealwaysencouragedandfacilitatedmyacademicpur-suits.IwouldliketomentionmyhighschoolphysicsteacherMr.GyanendraSharma,whoseloveforphysicsinspiredmetobecuriousaboutscience. ii

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TABLEOFCONTENTSpage ACKNOWLEDGMENTS :::::::::::::::::::::::::::::ii LISTOFTABLES ::::::::::::::::::::::::::::::::::vii LISTOFFIGURES :::::::::::::::::::::::::::::::::x ABSTRACT :::::::::::::::::::::::::::::::::::::xxiCHAPTER 1INTRODUCTION ::::::::::::::::::::::::::::::1 1.1HistoryofElectrodeSystems ...................... 3 1.2MeasurementModelConcept ...................... 4 1.3ScopeandStructureoftheThesis .................... 6 2HYDRODYNAMICMODELSFORASTATIONARYELECTRODEUN-DERSUBMERGEDJETIMPINGEMENT :::::::::::::::::9 2.1SchematicIllustrationoftheSystem .................. 9 2.2GoverningEquations ........................... 10 2.3PotentialFlowCalculation ........................ 12 2.4BoundaryLayerFlowCalculation ................... 14 2.4.1SolutionMethod ......................... 17 2.4.2Results ............................... 18 2.5BoundaryLayerSeparation ....................... 21 2.6NumericalSimulation .......................... 22 2.6.1GoverningEquations ....................... 23 2.6.2NumericalMethod ........................ 25 2.6.3SimulationResults ........................ 25 2.7Summary .................................. 26 3CONVECTIVE-DIFFUSIONMODELSFORASTATIONARYHEMISPHER-ICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENT ::::29 3.1GoverningEquations ........................... 29 3.2SolutionMethodandResults ...................... 32 3.3MassTransferLimitedCurrent ..................... 33 iii

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3.4NumericalSimulations .......................... 36 3.5Conclusion ................................. 39 4HYDRODYNAMICANDMASS-TRANSFERMODELSFORAROTAT-INGHEMISPHERICALELECTRODE :::::::::::::::::::41 4.1SchematicIllustrationoftheSystem .................. 41 4.2HydrodynamicModel .......................... 42 4.2.1GoverningEquations ....................... 43 4.2.2Results ............................... 46 4.2.3FluidFlowattheCorner ..................... 47 4.3MassTransfer ............................... 48 4.4Summary .................................. 53 5CURRENTANDPOTENTIALDISTRIBUTIONATAXISYMMETRICELEC-TRODES ::::::::::::::::::::::::::::::::::::54 5.1Introduction ................................ 54 5.2DevelopmentofMathematicalModel ................. 55 5.2.1Hydrodynamics .......................... 56 5.2.2MassTransfer ........................... 57 5.2.3ElectrodeKinetics ......................... 59 5.2.4ConcentrationOverpotential .................. 59 5.2.5SolutionPotentialinOuterRegion ............... 60 5.2.6ElectrodePotential ........................ 62 5.3DimensionlessQuantities ........................ 62 5.4CalculationProcedure .......................... 66 5.4.1Diskelectrode ........................... 66 5.4.2Hemisphericalelectrode ..................... 68 5.5CurrentDistributionatDiskElectrode ................. 70 5.5.1PrimaryDistribution ....................... 70 5.5.2SecondaryCurrentDistribution ................ 71 5.5.3TertiaryCurrentDistribution .................. 72 5.6CurrentDistributionatHemisphericalElectrode ........... 76 5.6.1PrimaryDistribution ....................... 76 5.6.2SecondaryDistribution ..................... 78 5.6.3TertiaryDistribution ....................... 78 5.7CurrentDistributionontheRotatingHemisphericalElectrode ... 88 5.7.1GoverningEquations ....................... 89 5.7.2NumericalProcedure ....................... 89 5.7.3Results ............................... 91 5.8Summary .................................. 92 6VALIDATIONOFTHEMEASUREMENTMODELCONCEPT :::::96 6.1Introduction ................................ 96 6.2DenitionofErrors ............................ 99 6.3EquivalenceofMeasurementModels .................. 99 6.4Kramers-KronigRelations ........................ 101 iv

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6.5ComplexNonlinearLeast-squareRegression ............. 102 6.5.1SolutionMethod ......................... 106 6.5.2ConvergenceCriterion ...................... 107 6.5.3WeightingStrategy ........................ 108 6.5.4ComputerProgramImplementation .............. 110 6.5.5CondenceInterval ....................... 110 6.6Method ................................... 111 6.7Results ................................... 112 6.7.1EvaluationofStochasticErrors ................. 112 6.7.2EvaluationofBiasErrors .................... 119 6.8Conclusions ................................ 130 7ELECTROCHEMICALMEASUREMENTSOFOXYGENREDUCTIONATNICKELELECTRODE ::::::::::::::::::::::::::::131 7.1ReactionMechanismofOxygenReduction .............. 132 7.2Experimental ............................... 134 7.3PolarizationMeasurements ....................... 139 7.4ImpedanceMeasurements ........................ 144 7.5MeasurementModelAnalysis ...................... 145 7.5.1DeterminationofStochasticErrorStructure .......... 148 7.5.2Kramers-KronigConsistencyCheck .............. 152 7.6ProcessModel ............................... 158 7.7Summary .................................. 162 8ELECTROCHEMICALMEASUREMENTSOFFERRICYANIDEREDUC-TIONATNICKELELECTRODE ::::::::::::::::::::::163 8.1Introduction ................................ 163 8.2ExperimentalMethod ........................... 164 8.3ExperimentalResults ........................... 165 8.3.1Steady-StateMeasurement ................... 165 8.3.2ImpedanceMeasurement .................... 166 8.4MeasurementModelAnalysis ...................... 170 8.4.1DeterminationofErrorStructure ................ 170 8.4.2Kramers-KronigConsistencyCheck .............. 172 8.5SurfaceAnalysisforDiskElectrode ................... 173 8.6OpticalMicrographsoftheHemisphericalElectrode ......... 176 8.7ThermodynamicAnalysis ........................ 180 8.8Discussion ................................. 181 8.9Conclusions ................................ 184 9CONCLUSIONS :::::::::::::::::::::::::::::::186 10SUGGESTEDFUTURERESEARCH ::::::::::::::::::::188 v

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APPENDIX AHYDRODYNAMICEQUATIONSINSERIESEXPANSION :::::::189 A.1OrdinaryDifferentialEquationsforH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1 ...... 189 A.2SolutionsofH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 ................... 195 A.3ExtrapolationofFiniteDifferenceValuesforF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 ......... 199 BSOLUTIONOFCONVECTIVE-DIFFUSIONEQUATIONFORINFINITESCHMIDTNUMBER :::::::::::::::::::::::::::::204 CBOUNDARY-LAYERPROGRAMLISTING ::::::::::::::::208 C.1ProgramListing .............................. 208 C.1.1MainProgram ........................... 208 C.1.2MainSubroutines ......................... 209 C.1.3IncludeFiles ............................ 233 DPROGRAMLISTINGFORCONVECTIVEDIFFUSIONCALCULATIONS 247 D.1ProgramListing .............................. 247 D.1.1MainProgram ........................... 247 D.1.2MainSubroutines ......................... 248 D.1.3IncludeFiles ............................ 273 EPROGRAMLISTINGFORCALCULATINGTHECURRENTDISTRIBU-TIONATTHESTATIONARYHEMISPHERICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENT ::::::::::::::::::::281 E.1ProgramListing .............................. 281 E.1.1MainProgram ........................... 281 E.1.2MainSubroutines ......................... 283 E.1.3IncludeFiles ............................ 288 E.1.4InputFile ............................. 289 FPROGRAMLISTINGFORCALCULATINGTHECURRENTDISTRIBU-TIONATTHEROTATINGHEMISPHERICALELECTRODE ::::::290 F.1ProgramListing .............................. 290 F.1.1MainProgram ........................... 290 F.1.2MainSubroutines ......................... 292 F.1.3IncludeFiles ............................ 308 F.1.4InputFile ............................. 311 REFERENCES :::::::::::::::::::::::::::::::::::312 BIOGRAPHICALSKETCH ::::::::::::::::::::::::::::321 vi

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LISTOFTABLESTable page 2.1SeriesexpansioncoefcientsF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1intheequations 2-34 and 2-35 forH;andF;at=0. ........... 21 2.2SeriesexpansioncoefcientsF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1intheequations 2-34 and 2-35 forH;andF;at=0. ........... 22 3.1Calculatedvaluesforcoefcients01;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and02;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1usedinequation 3-14 formass-transfer-limitedcurrentdistribution. ... 33 3.2Physicalpropertiesoftheelectrolyteusedinthenumericalsolutionofequation 3-21 ............................. 38 4.1F02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1coefcientsintheseriesexpansionofequa-tions 4-16 and 4-15 forH;andF;at=0.ThethirdcolumninthetableliststhevaluesreportedbyBarciaetal.andthefourthcolumnliststhevaluescalculatedusingthecontinuityequation. .................................. 47 4.2Calculatedvaluesforcoefcientsusedinequation 4-29 forcalcu-latingmass-transfer-limitedcurrentdistribution. ........... 50 5.1Calculatedvaluesforuniformityparameterdiskseeequation 5-55 ,iavg=ilim,andir=0=iavgforthecurrentdistributionspresentedinFigures 5-3 and 5-4 .ThevaluesofJandNwas5and125,respec-tively. ................................... 76 5.2Valuesofuniformityparameterhsseeequation 5-60 forforsta-tionaryThecalculatedvaluesareforhemisphericalelectrodeunderjetimpingement.thecurrentdistributionspresentedinFigure 5-7 .ParameterJwasxedat5. ....................... 86 5.3Valuesofiavg=ilimavgforstationaryhemisphericalelectrodeunderjetimpingement.Thecalculatedvaluesareforthecurrentdistribu-tionspresentedinFigure 5-7 .ParameterJwasxedat5. ..... 87 vii

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5.4Valuesofhsseeequation 5-60 forstationaryhemisphericalelec-trodeunderjetimpingement.Thecalculatedvaluesareforthecur-rentdistributionspresentedinFigure 5-10 .ParameterNwasxedat20. .................................... 87 5.5Valuesofiavg=ilimavgforstationaryhemisphericalelectrodeunderjetimpingement.Thecalculatedvaluesareforthecurrentdistribu-tionspresentedinFigure 5-10 .ParameterNwasxedat20. .... 87 5.6Valuesofiavg=ilimavgandhsfortherotatinghemisphericalelec-trode.Thecalculatedvaluesareforthecurrentdistributionspre-sentedinFigure 5.14a .ParametersNandJwasxedat125and5,respectively. .............................. 94 6.1ModelparametersforthetofaVoigtmeasurementmodeltoimpeda-ncescans#1,#5,and#25presentedinFigure 6-3 ........... 128 6.2ModelparametersforthetofaTransferfunctionmeasurementmodeltotheimpedancescan#1,#5,and#25presentedinFigure 6-3 129 7.1ChemicalcompositionofNickel270 .................. 131 7.2Propertiesofoxygensaturated0.1MNaClat25oC. .......... 138 7.3SpeciesconsideredincalculationofthePourbaixdiagrampresentedasFigure 7-8 ............................... 142 7.4Computedvaluesofhydrodynamicconstantaforthediskelectrode. 144 7.5Calculatedvaluesofhydrodynamicsconstantaforthehemispheri-calelectrode. ................................ 144 7.6Experimentalconditionsforimpedancescanofoxygenreductionatdiskandhemisphericalelectrode .................... 145 7.7Modelparametersoferrorstructurefordifferentexperimentalcon-ditionsondiskelectrode. ......................... 151 7.8Modelparametersoferrorstructurefordifferentexperimentalcon-ditionsonhemisphericalelectrode. ................... 151 7.9ModelparametersforthetofaVoigtmeasurementmodeltoimag-inarypartofrstimpedancescansatdiskelectrode.Thejetvelocityforthissetofexperimentswasat1.99meter/sec. ........... 153 viii

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7.10ModelparametersforthetofaVoigtmeasurementmodeltoimag-inarypartofrstimpedancescansathemisphericalelectrode.Thejetvelocityforthissetofexperimentswasat3.98meter/sec. .... 158 7.11EstimatedmodelparametersofaCPEequivalentcircuitmodeltoimpedancedatacollectedatthediskelectrode.Reportedparame-tersvaluesareaverageofsevenreplicatespectrumcollectedatanexperimentalcondition. ......................... 160 7.12EstimatedmodelparametersofaCPEequivalentcircuitmodeltoimpedancedatacollectedatthehemisphericalelectrode.Reportedparametersvaluesareaverageofsevenreplicatespectrumcollectedatanexperimentalcondition. ...................... 160 8.1Electrolytepropertiesusedinexperiments. .............. 164 8.2Calculatedvaluesofsolutionresistanceforprimarycurrentdistri-bution,RPsol,usingelectricalconductivitiesofelectrolytelistedintable 8.1 ................................... 165 8.3Modelofobtainederrorstructureforimpedancespectraondiskandhemisphericalelectrode. ...................... 170 8.4ModelparametersforthetofaVoigtmeasurementmodeltoimag-inarypartofrstimpedancescansatdiskandhemisphericalelectrode 173 8.5SpeciesconsideredincalculationofthePourbaixdiagrampresentedasFigure 8-15 ............................... 183 8.6Theboundarylayerpointofseparationatthestationaryhemispher-icalelectrode. ............................... 184 ix

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LISTOFFIGURESFigure page 2-1Schematicillustrationofastationaryhemisphericalsubmergedim-pingingjetelectrodesystem. ....................... 10 2-2Computedowtrajectoriescorrespondingtothepotentialowso-lution,givenasequation 2-12 ,forthehemisphericalelectrodesub-jectedtoasubmergedimpingingjetsystemwith)]TJ/F22 11.955 Tf 9.298 0 Td[(c)]TJ/F20 7.97 Tf 6.586 0 Td[(1r)]TJ/F20 7.97 Tf 6.587 0 Td[(20asapa-rameter. .................................. 14 2-3Distributionofthedimensionlesspressuregradientgivenasequa-tion 2-15 ................................. 15 2-4Schematicdiagramofgridforcalculationdomain.Histhespacingbetweenadjacentnodes. ......................... 18 2-5DimensionlessradialandcolatitudefunctionsH1andF1asafunctionofseeequations 2-24 and 2-25 ............ 19 2-6Calculateddimensionlesssurfaceshearstressasafunctionofan-gle.Solidlinesrepresenttheresultforthestationaryhemisphereundersubmergedjetimpingementanddashedlinesrepresenttheresultfortherotatinghemisphere. ................... 23 2-7Schematicrepresentationofthesimulatedowgeometry.Thedi-mensionsaregiveninunitsofm.Thearrowrepresentsthegeneraldirectionofow,andthecylindricalelectrodeislocatedattheorigin. 24 2-8FluidstreamlinesinthevicinityoftheelectrodeforaninletReynoldsnumberof1,100.Thecolormapindicatethepressuredistribution.Theradialdimensionisgiveninunitsofm. .............. 26 2-9FluidstreamlinesinthevicinityoftheelectrodeforaninletReynoldsnumberof11,000.Thecolormapindicatethepressuredistribution.Theradialdimensionisgiveninunitsofm.Figure 2.9b providesanenlargedimageoftherecirculationshowninFigure 2.9a ... 27 x

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3-1Calculatedmass-transferlimitedcurrentdensityforahemispher-icalelectrodesubjectedtoasubmergedimpingingjet.Solidlinesrepresentresultsforthestationaryelectrode,andthedashedlinesrepresentresultsfortherotatinghemisphericalelectrode.aContri-butiontoequation 3-17 foraninniteSchmidtnumber;bContri-butiontoequation 3-17 providingcorrectionforaniteSchmidtnumber. ................................... 35 3-2Reactantconcentrationdistributionasafunctionofdistancefromelectrodesurface,obtainedthroughnumericalsimulationofequa-tion 3-21 .Thebluelinecorrespondstozeroconcentrationontheelectrodesurface,whereasredcorrespondstothebulkreactantcon-centration.Theradialdimensionisgiveninunitsofcm.Thesesim-ulationswereperformedforRe=11300inthenozzle. ........ 38 3-3Calculatedmass-transfer-limitedcurrentdensityfordifferentReynoldsnumberattheinletofthenozzle.Theverticaldashlineat62isthepointofboundarylayerseparation.ThephysicalpropertiesoftheelectrolyteusedinthesimulationsarelistedinTable 3.2 ...... 39 4-1SchematicsillustrationofRotatingHemisphericalElectrode. .... 41 4-2ShearStressDistributionattheelectrodesurface.Solidlinerepre-sentresultsofBarciaetal.,thedashedlinerepresenttheresultsofChin,andthedottedlinerepresenttheresultofManohar. ..... 46 4-3Atwodimensionaldepictionofboundarylayerattheintersectionofelectrodeandinsulatingplane. .................... 48 4-4Calculatedmass-transferlimitedcurrentdensityforarotatinghemi-sphericalelectrode.aContributiontoequation 4-32 foraninniteSchmidtnumber;bContributiontoequation 4-32 providingcor-rectionforaniteSchmidtnumber. ................... 52 4-5Relativeerrorinmass-transfer-limitedcurrentgivenbyexpressions 4-36 and 4-37 asafunctionofSchmidtnumber. .......... 53 5-1Schematicsillustrationofanaxisymmetricbodyinacurvilinearco-ordinatesystem.Thehorizontaldashlinerepresentstheaxisofsymmetry,andtheuideldisassumedtobesymmetricaroundthisaxis. .................................. 56 5-2Primarycurrentdistributionatthediskelectrode.Thevalueoflocalcurrentapproachestoinnityasr=r0!1 ............... 72 xi

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5-3Calculatedcurrent,concentration,andsolutionpotentialdistribu-tionatthediskelectrode.ThesimulationsweredoneforJ=5,N=125,andC=0.5to0.9inincrementalstepsof0.1.ai=ilimasafunctionofr=r0.bDimensionlessconcentrationdistributionasafunctionofr=r0.cDimensionlesssolutionpotentialattheelec-trodesurfaceasafunctionofr=r0. ................... 74 5-4Calculatedcurrent,concentration,andsolutionpotentialdistribu-tionatthediskelectrode.ThesimulationsweredoneforJ=5,N=125,andC=0.4,0.3,0.2,0.1,0.05.ai=ilimasafunctionofr=r0.bDimensionlessconcentrationdistributionasafunctionofr=r0.cDimensionlesssolutionpotentialattheelectrodesurfaceasafunctionofr=r0. ............................. 75 5-51)]TJ/F22 11.955 Tf 11.955 0 Td[(ir=0=iavgasafunctiondiskfordifferentvaluesofC. ..... 77 5-6i=iavgasafunctionr=r0fordifferentvaluesofC. ......... 77 5-7Calculatedcurrentdistributionasafunctionatthestationaryhemi-sphericalelectrodeundersubmergedjetimpingement.Thesimula-tionweredonefordifferentvaluesofpoleconcentrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5. ........ 80 5-8Calculatedconcentrationprolecorrespondingtothecurrentdistri-butionpresentedinFigure 5-7 asafunctionatthestationaryhemi-sphericalelectrodeundersubmergedjetimpingement.Thesimula-tionweredonefordifferentvaluesofpoleconcentrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5. ........ 81 5-9Calculatedvaluesofsolutionpotentialcorrespondingtothecur-rentdistributionpresentedinFigure 5-7 asafunctionofatthestationaryhemisphericalelectrodeundersubmergedjetimpinge-ment.Thesimulationweredonefordifferentvaluesofpolecon-centrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5. ................................. 82 xii

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5-10Calculatedcurrentdistributionasafunctionofatthestationaryhemisphericalelectrodeundersubmergedjetimpingement.ThesimulationweredoneforN=20,anddifferentvaluesofpolecon-centrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1. 83 5-11Calculatedconcentrationprolecorrespondingtothecurrentdistri-butionpresentedinFigure 5-10 asafunctionofatthestationaryhemisphericalelectrodeundersubmergedjetimpingement.ThesimulationwerecarriedoutforN=20,anddifferentvaluesofpoleconcentrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1. ............................ 84 5-12CalculatedvaluesofsolutionpotentialcorrespondingtothecurrentdistributionpresentedinFigure 5-10 asafunctionofatthesta-tionaryhemisphericalelectrodeundersubmergedjetimpingement.ThesimulationwerecarriedoutforN=20,anddifferentvaluesofpoleconcentrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1. ............................ 85 5-13CurrentdistributioncalculationspresentedinthepaperbyNisanciogluandNewman.aFigure6ofthepaperbyNisanciogluetal.bFig-ure2ofthepaperbyNisanciogluetal. ................ 88 5-14Calculatedvaluesofcurrentdistribution,concentrationdistribu-tion,surfaceoverpotential,andconcentrationoverpotentialasafunc-tionofattherotatinghemisphericalelectrode.ThelinesinblackcolorcorrespondstothecalculationsforinniteSchmidtnumber,andlinesinbluecolorcorrespondstocalculatedresultswithSc=1000.0.ThesecalculationswereperformedforJ=5andN=125.aCurrentdistributionasafunctionof,bDimensionlessCon-centrationdistributionasafunctionof,cDimensionlesssurfaceoverpotentialasafunctionof,dDimensionlessconcentrationoverpotentialasafunctionof. ..................... 93 5-15CalculateddimensionlessSolutionpotentialalongtheelectrodesur-faceasafunctionof.Theresultscorrespondstothecurrentdistri-butionsgiveninFigure 5.14a .aDimensionlesssolutionpotentialwithoutSchmidtnumbercorrection,bDimensionlesssolutionpo-tentialwithSchmidtnumbercorrection. ................ 94 6-1AschematicrepresentationofaVoigtelementmeasurementmodel. 97 xiii

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6-2Impedancespectraobtainedforthereductionofferricyanideonaplatinumrotatingdiskelectrode. .................... 111 6-3CurrentmeasurementsbeforeandaftertheimpedancescansshowninFigure 6-2 .Thedatasetssingledoutforerroranalysisarehigh-lighted. ................................... 112 6-4RelativedeparturesfromthemeanvaluefortherstfourspectragiveninFigure 6-2 :arealpartandbimaginarypartoftheimpeda-nce. ..................................... 115 6-5Residualerrorsforthetofatransfer-functionmeasurementmodel,equation 6-3 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. ............. 116 6-6ResidualerrorsforthetofaVoigtmeasurementmodel,equation 6-1 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. .................. 117 6-7StandardDeviationsforthedatapresentedinFigure 6-2 ,obtainedfromtheresidualerrorspresentedinFigures 6-5 and 6-6 .ThedashedlinerepresentstheresultsobtainedfortheKramers-Kronig-consistentdatainset2and3. ............................ 118 6-8Relativedeparturesfromthemeanvalueforthesecondfourspec-tragiveninFigure 6-2 :arealpartandbimaginarypartoftheimpedance. ................................. 120 6-9Residualerrorsforthetofatransfer-functionmeasurementmodel,equation 6-3 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. ............. 121 6-10ResidualerrorsforthetofaVoigtmeasurementmodel,equation 6-1 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. .................. 122 6-11StandardDeviationsforthedatapresentedinFigure 6-2 :aresultsobtainedfromtheresidualerrorspresentedinFigures 6-9 and 6-10 ,andbresultsobtainedfromtheresidualerrorsforDataset3.ThedashedlinerepresentstheresultsobtainedfortheKramers-Kronig-consistentdatainset2and3. ...................... 123 xiv

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6-12ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFig-ure 6-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ...................... 125 6-13Residualerrorsforthetofatransfer-functionmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFigure 6-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ...................... 127 7-1Experimentalsetupusedforthestudyofoxygenreductionreaction. 136 7-2Schematicdiagramofimpingingjetelectrochemicalcell.aLayoutofthecellwithitscomponent.bImportantcelldimensions. .... 137 7-3Imageofthehemisphericalelectrodeduringthepolarizationmea-surementofoxygenreductionreaction. ................ 139 7-4Polarizationcurvefortheoxygenreductionreactioncollectedatthediskelectrode.Thesolidlinecorrespondstoaverageuidjetveloc-ityof1.99m/s,dashlinecorrespondsto2.99m/s,anddottedlinecorrespondsto3.98m/s. ........................ 140 7-5Polarizationcurvefortheoxygenreductionreactioncollectedatthehemisphericalelectrode.Thesolidlinecorrespondstoaverageuidjetvelocityof1.99m/s,dashlinecorrespondsto2.99m/s,anddot-tedlinecorrespondsto3.98m/s. .................... 140 7-6Diffusionlimitedcurrentforoxygenreductionin0.1MNaClasafunctionofsquarerootofthejetvelocityforNi270diskelectrode.Thedashedlineisalinearttothedatapoints. ............ 141 7-7Diffusionlimitedcurrentforoxygenreductionin0.1MNaClasafunctionofsquarerootofthejetvelocityforNi270hemisphericalelectrode.Thedashedlineisalinearttothedatapoints. ..... 142 xv

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7-8Thepotential-pHdiagramofnickelinoxygensaturatedsodiumchloridesolution.Thepotentialisreportedwithrespecttostan-dardhydrogenelectrodeSHE.TheverticaldashlinecorrespondstopHof0.1Msodiumchloridesolution.Thisdiagramwasgener-atedbycomputersoftwareCorrosionAnalyzer1.3Revision1.3.33.OLISystems,Inc.Theactivityofnickelionswasassumedtobe1:010)]TJ/F20 7.97 Tf 6.587 0 Td[(6M. ................................ 143 7-9Firstimpedancescancollectedduringthestudyofoxygenreduc-tionatthediskelectrodeundersubmergedjetimpingement.Theimpedancespectrumwerecollectedfordifferentjetvelocitiesandbiaspotential. ............................... 146 7-10Firstimpedancescancollectedduringthestudyofoxygenreduc-tionatthehemisphericalelectrodeundersubmergedjetimpinge-ment.Theimpedancespectrumwerecollectedfordifferentjetve-locitiesandbiaspotential. ........................ 146 7-11Collectedimpedancespectrumforjetvelocityof2.99m/sandbiaspotentialof-0.540V.aComplexplaneplot;Realandimaginaryimpedancearenormalizedwithsurfacearea;bRealandimaginaryimpedanceasafunctionoffrequency. ................. 147 7-12StandardDeviationsofstochasticerrorsfortheimpedancedatacol-lectedondiskelectrode.ArepresentativerstscanoftheanalyzeddataispresentedinFigure 7-9 .Theresultsarepresentedfordiffer-entjetvelocitiesandappliedbiaspotentials.aValuesofbiaspo-tentialswasselectedtoprovidetheaveragecurrentlevelataboutquarterofmass-transfer-limitedcurrent;bValuesofbiaspoten-tialswasselectedtoprovidetheaveragecurrentlevelatabouthalfofmass-transfer-limitedcurrent. .................... 149 7-13StandardDeviationsofstochasticerrorsfortheimpedancedatacol-lectedondiskelectrode.ArepresentativerstscanoftheanalyzeddataispresentedFigure 7-10 .Theresultsarepresentedfordifferentjetvelocitiesandappliedbiaspotentials.aValuesofbiaspotentialswasselectedtoprovidetheaveragecurrentlevelataboutquarterofmass-transfer-limitedcurrent;bValuesofbiaspotentialswasselectedtoprovidetheaveragecurrentlevelatabouthalfofmass-transfer-limitedcurrent. ......................... 150 xvi

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7-14ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftheimpedancespectrumpresentedinFigure 7-9 byopencircles.attotheimaginarypart,wheredashedlinesrep-resentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceinter-valsfortheestimatedparameters. ................... 154 7-15ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftheimpedancespectrumpresentedinFigure 7-9 byhalflledcircles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructurede-terminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ................. 155 7-16ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftheimpedancespectrumpresentedinFigure 7-10 byopentraingles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructurede-terminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ................. 156 7-17ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFig-ure 7-10 byinvertedhalflledtriangles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticer-rorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condencein-tervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. .... 157 7-18ACPEequivalentcircuitmodelttotheimpedancedatacollectatthejetvelocityof2.99m/s.Thebiaspotentialwassetat-0.540V.aColpmexplaneplotofthettothedata;bRealandimaginaryresidualerrorsasafunctionoffrequency. ............... 161 8-1Polarizationcurveofnickeldiskelectrodeinthesolutionof1.0MNaOH,0.005MK3FeCN6andK4FeCN6.Theaverageuidveloc-ityinthejetwas1.99meter/second. .................. 166 xvii

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8-2Impedancespectraobtainedforthereductionofferricyanideonanickeldiskelectrodeundersubmergedjetimpingement.Theav-erageuidvelocityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelec-trolyteforthissetofexperimentsconsistedof1.0MNaOH,0.005MK3FeCN6andK4FeCN6. ....................... 167 8-3CollectedImpedancespectraforthereductionofferricyanideonanickelhemisphericalelectrodeundersubmergedjetimpingement.Theaverageuidvelocityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelectrolyteforthissetofexperimentsconsistedof1.0MNaOH,0.005MK3FeCN6andK4FeCN6. ...................... 167 8-4Complex-planeplotsofimpedanceobtainedonthediskelectrode.at=60s;andbt=1;860s. ...................... 168 8-5Complex-planeplotsofimpedanceobtainedonthehemisphericalelectrode.at=60s;andbt=1;860s. ................ 169 8-6StandardDeviationsforthedatapresentedinFigure 8-2 .Thesolidlinerepresentsthettotheerrorstructure. .............. 171 8-7StandardDeviationsforthedatapresentedinFigure 8-3 .Thesolidlinerepresentsthettotheerrorstructure. .............. 171 8-8ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFig-ure 8-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ...................... 174 8-9ResidualerrorsforthetofaVoigtmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFig-ure 8-3 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters. ...................... 175 xviii

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8-10Topviewofthediskelectrodeafterimpedanceexperiments.aUndisturbedimageofelectrode.bImageobtainedaftertherightsideofdiskelectrodewascleanedwithsandpapertohighlightthecontrastbetweenmetalsurfaceanddeposits. ............. 176 8-11ScanningElectronspectroscopyofofadiskelectrodeafterimmer-sionintheelectrolytesupportedby1.0MNaOH. .......... 177 8-12EnergyDispersiveSpectroscopyEDSanalysisofadiskelectrodeafterimmersionintheelectrolytesupportedby1.0MNaOH. ... 177 8-13Imagesofhemisphericalelectrodeafterimpedanceexperiments. .. 178 8-14Sideviewofthehemisphericalelectrodeafterwashingitwithdeion-izedwater. ................................. 180 8-15Thepotential-pHdiagramfornickelinwatercontainingsodiumhydroxide,potassiumferricyanide,potassiumferrocyanide,anddis-solvedoxygen.ThepotentialisreportedwithrespecttostandardhydrogenelectrodeSHE.TheverticaldashedlinesrepresentthepHofelectrolytesolutionusedinthepresentstudy.Thelineontheleftcorrespondstoasolutioncontaining0.1MNaOH,0.005MK3FeCN6andK4FeCN6,andthelineontherightcorrespondsto1.0MNaOH,0.005MK3FeCN6andK4FeCN6.ThisdiagramwasgeneratedusingCorrosionAnalyzer1.3Revision1.3.33byOLISystems,Inc.Theactivityofnickelionswasassumedtobe110)]TJ/F20 7.97 Tf 6.586 0 Td[(6M. 182 8-16CollectedImpedancespectrumforthereductionofferricyanideonanickeldiskelectrodeundersubmergedjetimpingement.Theav-erageuidvelocityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelectrolyteforthisexperimentconsistedof0.1MNaOH,0.005MK3FeCN6andK4FeCN6.Therepresentedimpedancespectrumwascollectedafter5hoursofimmersionoftheelectrodeintheelectrolyte. .... 184 A-1Calculatedprolesofdimensionlessradialandcolatitudefunctionsintheexpansionof 2-24 and 2-25 .aH1andF1asafunc-tion,bH3andF3asafunction,cH5andF5asafunction,anddH7andF7asafunction. .......... 196 A-2Calculatedprolesofdimensionlessradialandcolatitudefunctionsintheexpansionof 2-24 and 2-25 .aH9andF9asafunc-tionof,bH11andF11asafunctionof,cH13andF13asafunctionof,anddH15andF15asafunctionof. ... 197 xix

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A-3Calculatedprolesofdimensionlessradialandcolatitudeintheex-pansionof 2-24 and 2-25 .aH17andF17asafunctionof,bH19andF19asafunctionof,cH21andF21asafunctionof,anddH23andF23asafunctionof. ...... 198 A-4Calculatedprolesofdimensionlessradialandcolatitudeintheex-pansionof 2-24 and 2-25 .aH25andF25asafunctionof,bH27andF27asafunctionof ................. 199 A-5FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1at=0fordifferentgridspacing.aF01asafunctionofH2,bF03asafunctionofH2,cF05asafunctionofH2,anddF07asafunctionofH2. ............................ 200 A-6FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1at=0fordifferentgridspacing.aF09asafunctionofH2,bF011asafunctionofH2,cF013asafunctionofH2,anddF015asafunctionofH2. ........................ 201 A-7FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1at=0fordifferentgridspacing.aF017asafunctionofH2,bF019asafunctionofH2,cF0210asafunctionofH2,anddF023asafunctionofH2. ........................ 202 A-8FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1at=0fordifferentgridspacing.aF025asafunctionofH2,bF027asafunctionofH2 ......................... 203 xx

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophy STATIONARYHEMISPHERICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENTANDVALIDATIONOFMEASUREMENTMODELCONCEPTFORIMPEDANCESPECTROSCOPYByPavanKumarShuklaAugust2004 Chair:MarkE.OrazemMajorDepartment:ChemicalEngineeringInterpretationofelectrochemicalimpedancemeasurementsrequiresanade-quateunderstandingofelectrodesurfacephenomena,currentdistribution,andstochasticerrorstructure.Mostimportantly,nonuniformcurrentdistributionsob-fuscateimpedanceanalysisusingregression.Traditionalelectrodesystemssuchastherotatingdiskelectrodehaveanonuniformcurrentdistribution;therefore,useoftherotatingdiskelectrodeisnotsuitableforimpedancestudiesathighcur-rentlevels.Inthiswork,astationaryhemisphericalelectrodeundersubmergedjetimpingementissuggestedtobeanalternative.Primaryandsecondarycurrentdistributionsonstationaryhemisphericalelectrodesystemareuniform,increasingthelikelihoodofuniformtertiarycurrentdistribution.Moreover,electrochemicalprocessescanbemonitoredinsituonastationaryhemisphericalelectrode.Inthepresentwork,ahydrodynamicmodelwasdevisedusingtheboundarylayerthe-oryandcomparedtothecomputationaluiddynamicmodeldevelopedatVrijeUniversiteitBrussel,Belgium.Bothmodelspredictedaseparationofboundary xxi

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layeratthestationaryhemisphericalelectrodeundersubmergedjetimpingement.Thehydrodynamicmodelresultswereusedtoobtainasolutionofconvective-diffusionatthemass-transferlimitingcondition.Calculationsforsteady-statecurrentandpotentialdistributionbelowthemass-transferlimitedcurrentwereperformedtoobtaintheconditionsforuniformcurrent.Reductionofoxygenandferricyanidewerestudiedonboththediskandthehemisphericalelectrodeunderjetimpingement.Theobjectivewastounderstandthedifferencesinimpedanceresponseofthediskandthehemisphericalelec-trodes.Repeatedimpedancesmeasurementswereconductedonbothelectrodesystems.Theimpedanceanalysisofferricyanidereductionshowedtheevidenceofboundarylayerseparationatthehemisphericalelectrode.Asystematicstudywasundertakentoevaluatethemeasurementmodelap-proachforassessingtheerrorstructureofelectrochemicalimpedancemeasure-ments.Theremainingquestionwaswhethertheerrorstructureobtainedwiththismodelwasapropertyofthemeasurementordependedonthearbitraryselectionofameasurementmodel.TransferfunctionandVoigt-elementbasedmodelswereusedtoassesstheerrorstructureofimpedancemeasurements.Inspiteofdiffer-encesinthettingerrorsandnumbersofparametersneededfortheregression,thevaluesforthefrequency-dependentstochasticerrorswerefoundtobeinde-pendentofthemeasurementmodelused.Theseresultsconrmthemeasurementmodelapproachforerroranalysis.Thecondenceintervalsfortheparameteres-timatesdifferedforthetwomodels.TheVoigt-elementbasedmodelwasfoundtoprovidethetightestcondenceintervalsandwasmoresuitedforevaluationofconsistencywiththeKramers-Kronigrelations. xxii

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CHAPTER1INTRODUCTIONAccuratedeterminationofphysicalpropertiesforelectrochemicalsystemsre-mainsachallenge.ElectrochemicalimpedanceSpectroscopyEISprovidesaframe-workwherebydifferentphysicalpropertiesofthesystemcanbeestimatedsimul-taneouslyevenforcomplexsystems.Interpretationofelectrochemicalmeasure-mentsisfacilitatedwhenexperimentsareconductedunderwell-denedandeas-ilycharacterizedowconditions.ExperimentalsystemssuchastherotatingdiskelectrodeRDE1andthestationarydiskelectrodeunderasubmergedimping-ingjet2,3havebeenemployedextensivelyinelectrochemicalinvestigations.Therotatingandimpingingjetdiskelectrodegeometriesareattractivebecauseanac-curatesolutionisavailableforconvectivediffusion,andthecurrentdistributionisuniformatthemass-transfer-limitedconditions.Experimentalinvestigationsofelectrochemicalreactionmechanisms,however,arenotgenerallyconductedundermass-transferlimitations.Thecurrentandpo-tentialdistributiononadiskelectrodebelowthemasstransferlimitedcurrentisnotuniform,4andithasbeenshownthatneglectofthenonuniformcurrentdis-tributionintroduceserrorinestimationofkineticparametersfromsteadystateDCmeasurements.7Eventhemostcompleteexpressionsavailableforconvectivediffusionimpeda-nceonarotatingdiskelectrode10,11oronadiskelectrodeunderasubmergedimpingingjet12assumethatthesystemmaybetreatedashavingauniformcur-rentdistribution.NumericalcalculationspresentedbyAppelandNewman13andDurbhaetal.14illustratedtheinuenceofanon-uniformcurrentdistribution 1

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2 ontheimpedanceresponse.Orazemetal.15suggestedthatthediscrepancybe-tweenexperimentalmeasurementsandadetailedmathematicalmodelcouldbeattributedpartiallytotheinuenceofthenon-uniformcurrentdistributionbelowthemass-transfer-limitedvalue.ThisclaimwasdiscussedfurtherbyOrazemandTribollet.11Matosetal.16havedemonstratedexperimentallythattheimpedanceresponseonadiskelectrodewassignicantlydifferentthanthatonarotatinghemisphereelectrodeRHE,forwhichtheprimarycurrentandpotentialdistri-butionsareuniform.Currentmathematicalmodelsfortheimpedanceofadiskelectrodewithnonuni-formcurrentdistribution13,14aretoocomplexforregressionanalysis.Thepre-ferredapproachforexperimentalinvestigationofelectrodekineticsistousege-ometriesforwhichmass-transferiswell-denedandcurrentdistributionisuni-formattheexperimentalcondition.Therotatinghemisphericalelectrode,intro-ducedbyChin,17hasauniformprimarycurrentdistributionandwouldthereforebeasuitablecongurationforexperimentsconductedunderconditionssuchthatthecurrentdistributionisnotinuencedbythenon-uniformaccessibilitytomasstransfer.NisanciogluandNewman18demonstratedthatcurrentdistributionintheRHEisuniform.Thisconditionofuniformityisachievedwhenthetotalcurrentissmallerthan68percentoftheaveragemass-transfer-limitedvalue.ArenedmathematicalmodelfortheconvectivediffusionimpedanceofaRHE,developedbyBarciaetal.,19providedanexcellentmatchtoexperimentalimpedancemea-surementsconductedundertheseconditions.Systemsthatemployastationaryelectrodefacilitateuseofinsituobservationorsurface-analysistechniques.Orazemetal.,12forexample,usedinsituvideomicroscopytoobtainimagesofadiskelectrodeunderasubmergedimpingingjet.Theseimageswerethenusedtointerpretimpedancemeasurementsintermsof

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3 viscoelasticpropertiesofcorrosionproductlms.20Experimentsusingscanningellipsometryonadiskelectrodeunderasubmergedimpingingjetwereemployedtodistinguishbetweentheinuenceofconvectivediffusionandhydrodynamicshear.21FlowchannelexperimentshavebeenemployedbyAlkireandCangellari22toillustratetheroleofcurrentdistributiononformationofsaltlms.Todate,noexperimentalsystemexistsintheliteratureexhibitingauniformprimarycurrentdistribution,astationaryelectrodeamenabletoinsituobserva-tion,andwelldenedowcharacteristicsallowingcontrolofconvectivediffu-sion.Theobjectiveofthepresentworkwastodevelopthehydrodynamic,convec-tivediffusionandcurrentdistributioncalculationsforastationaryhemisphericalelectrodesubjectedtoasubmergedimpingingjet.Theuseofastationaryelec-trodewasintendedtofacilitateinsituobservationofelectrodeprocesses,andthehemisphericalelectrodegeometrywasintendedtoensurethattheprimaryandsecondarycurrentdistributionswouldbeuniform.23Thepresentworkprovidesafoundationforthedesignofelectrodesystemsandfordevelopmentofmodelsfortheimpedanceresponse.1.1HistoryofElectrodeSystemsGeometriessuchasdisks,spheres,andcylindershavebeenwidelyexploredinuidmechanics,heat,andmass-transferstudies.Theideatoemployadiskgeom-etryasanelectrochemicalexperimentaltoolwasenvisionedafterLevich24treatedtheconvectivediffusionproblematarotatingdiskelectrode.Levichshowedthatthesurfaceofarotatingdiskhasuniformmass-transferforthelimitingconditions.Foralongtimeitwasassumedthatthecurrentbelowthemass-transferlimitedvalueisalsouniformfortherotatingdiskelectrode.Rotatingcylindersweresuggestedtobeanalternativetothediskelectrodeformass-transferresearchasreviewedbyEisenbergetal.25Thediskelectrodehas

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4 beenwidelyusedinnumerousstudiesbecauseofitssimpledesignandeaseofoperation.Thesurfaceofthediskelectrodecanbeeasilypolishedandreusedwithoutlosingitsgeometricfeatures.Riddiford26providedadetailedaccountoftheevolutionofthediskelectrodeanditsuseinelectroanalyticalstudies.New-man's27treatmentofmass-transfercoupledwithpotentialdistribution,andelec-trodekineticsfortherotatingdiskelectrodeshowedthatthecurrentdistributionattheelectrodesurfaceishighlynonuniformevenatcurrentlevelsslightlybelowthemass-transferlimitedvalue.Chin17proposedtherotatingsphericalelectrodetobeanalternativetothediskelectrodeforhigh-ratedepositionanddissolutionstudies.Matloszandcowork-ers28proposedahybridelectrodegeometrywithcentraldiskandasurroundinghemisphericalelectrode.Theresultingelectrodewassubsequentlycalledadisk-hemisphericalelectrode.Thegeometricfeaturesofthesystemallowedthepri-marycurrentdistributiontobeniteattheedgeoftheelectrode.Madoreetal.29suggestedacylindricalhullelectrode.Theycalculatedtheprimarycurrentdis-tributionforthesystemwithdifferentcellparameters.Dinanetal.30proposedarecessedrotatingdiskelectrode.Thisgeometryprovidedauniformcurrentdis-tribution;however,uniformaccessibilitytomass-transferwaslostduetoitsgeo-metricfeature.1.2MeasurementModelConceptMeasurementmodelconceptwasrstintroducedbyAgarwaletal.31Themethodologywasdevisedforthefollowingreasons. 1. Toestimatestochasticerrorstructureofelectrochemicalimpedancespec-troscopydata,and 2. tocheckforconsistencyofimpedancedatawithKramers-Kronigrelations.

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5 TheproposedmeasurementmodelconsistedofVoigtelementsandasolutionre-sistanceconnectedinserieswitheachother.Agarwaletal.31showedtheapplica-bilityofmeasurementmodeltovariousimpedancedata.TheVoigtmeasurementmodelwithsufcientparameterswasabletottheimpedancedatawithinthenoiselevel.Later,Agarwaletal.32devisedamethodtolterthereplicationerrorsofimpedancedatainordertodistinguishbetweenstochasticerrorsanddeter-ministicerrors.Inthesubsequentpaper,Agarwaletal.33showedtheapplicabilityofVoigtmeasurementmodeltoassesstheconsistencyofimpedancedatawithKramers-Kronigrelations.Pauwelsetal.34haverecentlyproposedatransferfunctionbasedmeasurementmodel.InlightofPauwels'smodel,themeasurementmodelconceptwasreeval-uatedforestimationofstochasticerrors.ThepurposeofthisworkwastoanswerthequestionwhethertheerrorstructureobtainedwithVoigtmodelwasaprop-ertyofthemeasurementordependedonthearbitraryselectionofameasurementmodel.Bothmeasurementmodelswereappliedtoestimatestochasticerrorsintheimpedancemeasurementscollectedattherotatingdiskelectrodeofferricyanidereduction.Furthermore,thesamedatasetwasalsoanalyzedforKramers-Kronigconsistencycheckusingthetwomeasurementmodels.Theestimatederrorstruc-turewasfoundtobeindependentofchoiceofmeasurementmodeleventhoughtransferfunctionmodelrequiredfewerparameterstottheimpedancedata.Thecondenceintervalsfortheparameterestimatesdifferedforthetwomodels.TheVoigt-elementbasedmodelwasfoundtoprovidethetightestcondenceinter-vals.Asaresult,Kramers-KronigconsistencycheckwasmoresensitiveforVoigtelementbasedmeasurementmodel.

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6 1.3ScopeandStructureoftheThesisThestructureofthethesiscanbedividedintothreeparts.TherstpartofthethesisispresentedinChapters 2 3 4 ,and 5 .Thispartdealswiththehydrody-namicmodels,convectivediffusionmodels,andcurrentdistributionscalculationsforsubmergedstationaryhemisphericalelectrodeunderjetimpingementandro-tatinghemisphericalelectrode.ThesecondpartofthethesispresentsastudyofmeasurementmodelconceptsforthreedifferentmeasurementmodelsinChap-ter 6 .ThethirdpartdealswithexperimentalinvestigationoftwoelectrochemicalsystemsinChapters 7 and 8 .Areadercangothroughtherstandthirdpartofthisthesisexclusivelywithoutlosingthecontinuity.Thesecondpartcanbereadindependently.Chapter 2 providesarigoroustreatmentofuidmechanicsforstationaryelec-trodeunderjetimpingement.Twohydrodynamicmodelsweredevelopedforthesystem.Therstmodelwasdevelopedusingboundarylayertheory,andthegoverningequationsweresolvedbyaseriesmethod.Themodelpredictedtheseparationofboundarylayeratanangleof54.8fromthepole.However,thismodelisvaliduptothepointofboundarylayerseparation,afterwhichtheuidmechanicsbecomesundenedintheregionbeyondseparation.Acom-putationaluiddynamicmodelCFD,developedbyDr.GertNelissenatVrijeUniversiteitBrussel,Belgium,wasusedtoidentifytheuidmechanicsovertheentireelectrodesurface.TheCFDmodelpredictedvortexformationinthesep-aratedpartoftheboundarylayer.Theangleofseparationwaspredictedtobe62bytheCFDmodel.Asolutionofconvective-diffusionequationisprovidedinChapter 3 .Asolutionofconvective-diffusionwasdevelopedwithseriesmethodwhichpredictedthemass-transfer-limitedcurrentuntiltheboundarylayersepa-rationpoint.AcomplementaryCFDmodelofconvective-diffusion,developedby

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7 Dr.GertNelissenatVrijeUniversiteitBrussel,Belgium,solvedthegoverningequationovertheentiresurface.Chapter 4 presentsareviewofthehydrodynamicsandthemass-transferforarotatinghemisphericalelectrode.Thegoverningequationsweresolvedusingtheseriessolution.TheobjectiveofthischapterwastoprovideacorrectiontothesolutiongivenbyBarciaetal.19Ageneralizedmathematicalmodeltoobtainthecurrentandpotentialdistri-butionataxisymmetricelectrodeswasdevelopedinChapter 5 .Themodelwasthenappliedtocalculatethedistributionatthesubmergedstationarydiskandhemisphericalelectrodeunderjetimpingement.Anumericalcalculationproce-durewasdevelopedtosolvethegoverningequations.Amodiedmathematicalmodelwasalsodevelopedtoobtainthecurrentandpotentialdistributionattherotatinghemisphericalelectrode.Thismodelaccountedforcorrectioninthemass-transfertotheelectrodeduetoanitevalueoftheSchmidtnumber.Analgorithmwasdevelopedtosolvethegoverningequations.Chapter 6 reviewsthemeasurementmodelconceptforestimationofstochasticerrorsinimpedancespectroscopydata.Thechapterpresentsthethreedifferentmodels.Impedancedatacollectedattherotatingdiskelectrodesforferricyanidereductionwereanalyzedforstochasticerrors.ThedatawerealsoanalyzedforconsistencywithKramers-Kronigrelations.Chapter 7 presentsexperimentalstudyofoxygenreductionatthenickelelec-trode.Electrochemicalmeasurementswereperformedatthediskandhemispher-icalelectrodes.Repeatedimpedancespectrumwerecollectedatdifferentexperi-mentalconditions.Chapter 8 providesanexperimentalstudyofferricyanidereductionatthediskandhemisphericalelectrodeundersubmergedjetimpingement.Impedance

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8 measurementswerecarriedoutatelectrodesmadeofnickel.Analysisofimpeda-nceatthestationaryhemisphericalelectrodeprovidedanevidenceofboundarylayerseparation.ConclusionsfromthisworkareprovidedinChapter 9 ,andsuggestionsforfutureresearchisinChapter 10

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CHAPTER2HYDRODYNAMICMODELSFORASTATIONARYELECTRODEUNDERSUBMERGEDJETIMPINGEMENTThischapterpresentsadetaileddescriptionofthehydrodynamicsofastation-arysubmergedhemisphericalelectrodeundersubmergedjetimpingement.Theelectrodeisamenabletoinsituobservationandhasauniformprimaryandsec-ondarycurrentdistributionbelowthemass-transfer-limitedvalue.Thepresentworkisintendedtoprovideafoundationforthedesignofstationaryhemispher-icalelectrodesystemsandfordevelopmentofsteadystatemasstransferandter-tiarycurrentdistributioncalculationsforthesystem.2.1SchematicIllustrationoftheSystemAschematicillustrationofastationaryhemisphericalelectrodeunderasub-mergedimpingingjetispresentedinFigure 2-1 ,whereahemisphericalelectrodeprotrudesoutofaplanarinsulatingsurfaceandanozzleisplacedabovethehemi-sphere.Thecenterofnozzleisaxisymmetricwiththehemisphere.Thewholesystemissubmergedinanaqueouselectrolyteassumedtohaveuniformuidproperties.Thedimensionsofthenozzlearesufcientlylargeanditsplacementissufcientlyapartfromtheelectrodesuchthatoweldoftheuidcomingoutnozzlecanbedescribedasbeingapotentialowwithuniformaxialveloc-ity.AdetaileddescriptionofoweldgeneratedfromthenozzlecanbefoundinthepaperbyScholtzandTrass.35Thewallsoftheenclosurewereassumedtobesufcientlydistantthattheydonotinuencetheowpatternsneartheelec-trodesurface.Asphericalpolarcoordinatesystemisemployedtodescribethesystem,whererrepresenttheradialoutwarddirection,representsthecolatitude 9

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10 Figure2-1:Schematicillustrationofastationaryhemisphericalsubmergedim-pingingjetelectrodesystem. direction,andisalongthebodyoftherevolution.Thecorrespondinguideldvelocitycomponentsarevr,v,andv,respectively.2.2GoverningEquationsThesteady-stateuidowaroundhemisphereundersubmergedjetimpinge-mentcanbetreatedbydividingtheoweldintotworegions:theouterorpo-tentialowregion,whereinertialforcesdominate,andtheinnerorboundarylayerregion,whereviscousandinertialforcesareofthesamemagnitude.TheuidowintheboundarylayerregionisdescribedbyNavier-Stokesandmass-conservationequations,andtheowinthepotentialowregionisdescribedbymass-conservationonly.Howarth36rstderivedthegoverningequationsforuidowaroundarotat-ingsphereinthesphericalcoordinatesystem.TheseequationsarevalidforhighReynoldsnumbers,whichcorrespondstoahighrotationspeedofthesphere.Theequationsforuidowintheboundarylayercanbemodiedbysettingthecomponentofuidvelocityequaltozero.Thisisavalidassumption,becausetheuidowaroundthishemisphereisaxisymmetric.Underassumptionofconstantuidproperties,theequationsgoverningathinboundarylayeronan

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11 axisymmetricbodyofrotation37,36areconservationofmomentuminthecolati-tudedirectionv r0@v @+vr@v @r=)]TJ/F15 11.955 Tf 15.59 8.088 Td[(1 r0@ps @+@2v @r2-1andconservationofmass1 r0@v @+@vr @r+v r0cot=0-2Theunderlyingassumptionsinderivingtheaboveequations 2-1 and 2-2 are: 1. Theuidowintheboundarylayerislaminar,andthegradientsofallquan-titiesarelargeinthedirectionnormaltothesurface:however,theirtangen-tialgradientsarerelativelysmall. 2. Themomentumowinther-directionismuchsmallerthanthe-direction.Fromthisassumption,itcanbeconcludedthatthepressuregradientinther-directionvanishes. 3. Thethicknessofthemomentumboundarylayer,0,ismuchsmallerthantheradiusofthehemisphere.Thegoverningequationforthepotentialowregionis1 r@ @r)]TJ/F22 11.955 Tf 5.479 -9.684 Td[(r2vr+1 sin@ @sinv=0-3whichisthecontinuityequationintheouterowregion.Equations 2-1 2-2 ,and 2-3 completethedescriptionofuidowaroundthestationaryhemisphere.Theobjectiveistodeterminetheuidoweldwithintheboundarylayer.Thesolutionprocedureprogressedintwostages.First,followingtheusualbound-arylayerdevelopmentforforcedow,37asolutionwasobtainedforthepotential

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12 owregion.Thepotentialowsolutionprovidedthepressuredistributionovertheelectrodesurfaceandthefar-eldboundaryconditionsneededforsolutionoftheboundarylayerequations.Second,theboundarylayerequationweresolvedusingseriesexpansiondiscussedbyBarciaetal.19fortherotatinghemisphericalelectrode.2.3PotentialFlowCalculationThevelocitypotentialsatisesLaplace'sequation,whichcanbewritteninsphericalpolarcoordinatesas@ @rr2@ @r+1 sin@ @sin@ @=0-4wheretheradialcomponentoftheuidvelocityisgivenbyvr=)]TJ/F22 11.955 Tf 10.494 8.087 Td[(@ @r-5theangularorcolatitudecomponentoftheuidvelocityisgivenbyv=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(1 r@ @-6andrandaretheradialandangularcomponents,respectively.Theno-penetrationboundaryconditionscanbeexpressedas@ @r;==2=0-7fortheinsulatingplaneandas@ @rr=r0;=0-8fortheelectrodesurface.Asymmetryconditionforthecenterlinecanbeexpressedas@ @r;=0=0-9

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13 Undertheassumptionthattheowcanbeconsideredtobeofuniformvelocitytowardstheinsulatingplaneandthepresenceofhemispheredoesnoteffecttheuideldfarawayfromtheelectrode,thevelocitypotentialshouldapproachanasymptoticbehaviorandcanbeexpressedasjr!1;=cr2 2)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(3cos2)]TJ/F15 11.955 Tf 11.956 0 Td[(1-10wherecisahydrodynamicconstant.Equation 2-10 waspreviouslyappliedinthedevelopmentofthepotentialowsolutionforasubmergedjetimpingementontoaatdisk.38Thus,useofequation 2-10 constitutesastatementthat,farfromtheelectrode,theinuenceoftheshapeofthehemisphericalelectrodeshoulddiminish.Thesolutionofequation 2-4 subjectedtoboundaryconditions 2-7 to 2-10 isgivenby=)]TJ/F22 11.955 Tf 9.299 0 Td[(cr021 2r2 r20+1 3r30 r3)]TJ/F15 11.955 Tf 5.479 -9.684 Td[(3cos2)]TJ/F15 11.955 Tf 11.955 0 Td[(1-11withthecorrespondingstreamfunctiongivenby=)]TJ/F22 11.955 Tf 9.299 0 Td[(cr02r3 r30)]TJ/F22 11.955 Tf 13.151 8.087 Td[(r20 r2sin2cos-12Computedvaluesforowtrajectories,givenas)]TJ/F22 11.955 Tf 9.298 0 Td[(c)]TJ/F20 7.97 Tf 6.587 0 Td[(1r)]TJ/F20 7.97 Tf 6.586 0 Td[(20,arepresentedinFigure 2-2 asafunctionofdimensionlesspositionscaledbythehemisphereradiusr0.TheboundarylayercalculationpresentedinthesubsequentsectionemploysthepressuregradientobtainedfromBernoulli'sequationp+1 2v2=constant-13wherethevelocityisgivenbythepotentialowsolution.Thus,giventhatvrjr0=0and,fromequations 2-11 and 2-6 ,thatvjr0=5cr0 2sin-14

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14 Figure2-2:Computedowtrajectoriescorrespondingtothepotentialowsolu-tion,givenasequation 2-12 ,forthehemisphericalelectrodesubjectedtoasub-mergedimpingingjetsystemwith)]TJ/F22 11.955 Tf 9.299 0 Td[(c)]TJ/F20 7.97 Tf 6.586 0 Td[(1r)]TJ/F20 7.97 Tf 6.587 0 Td[(20asaparameter. thepressuregradientattheelectrodesurfacecanbeexpressedas)]TJ/F15 11.955 Tf 23.295 8.088 Td[(1 c2r20@ps @=25 4sin-15ThedimensionlesspressuregradientalongtheelectrodesurfaceisgiveninFig-ure 2-3 asafunctionofcolatitudeangle.Thedimensionlesspressuregradientchangessignatapositionof==4.As-showninthesubsequentsection,thereversalofthepressuredrivingforceforowinducesseparationofthevelocityboundarylayer.2.4BoundaryLayerFlowCalculationThesolutiontechniqueemployedtosolveequations 2-1 and 2-2 closelyfol-lowscloselythedevelopmentpresentedbyBarciaetal.19fortherotatinghemi-sphericalelectrode.Equations 2-1 and 2-2 canbeconvenientlywrittenindimensionlessformbyintroducingdimensionlessvariable,H;,F;.Thedimensionlessmomen-tumandcontinuityequationsare:1 4F;@F; @)]TJ/F22 11.955 Tf 11.955 0 Td[(H;@F; @=sin 4+1 2@2F; @2-16

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15 Figure2-3:Distributionofthedimensionlesspressuregradientgivenasequation 2-15 and1 2@F; @)]TJ/F15 11.955 Tf 11.956 0 Td[(2@H; @+1 2F;cot=0-17respectively,wherethepressuregradientwasintroducedfromequation 2-15 ,isthedimensionlessradialpositiongivenintermsofthehydrodynamicconstantaas=r a r)]TJ/F22 11.955 Tf 11.955 0 Td[(r0-18H;isthedimensionlessradialvelocity,suchthatvr=)]TJ/F15 11.955 Tf 9.299 0 Td[(2p aH;-19andF;isthedimensionlesscolatitudevelocity,suchthatv=ar0 2F;-20Theno-slipboundaryconditionsattheelectrodesurfaceforthecolatitudeandradialvelocitycomponentscanbeexpressedasF;j=0=0-21

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16 andH;j=0=0-22respectively.TheconditionthattheowmustapproachthepotentialowsolutionfarfromthesurfaceisexpressedbyF;j!1=sin2-23Comparisonbetweenequations 2-23 and 2-14 revealsthata=5c,whichpro-videsthattheboundarylayerequationscorrespondingtoajetimpinginguponaplanarsurfacearerecoveredfor=0.FollowingHowarth,36H;andF;canbeexpandedintermsofandasH;=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(2H2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-24andF;=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-25respectively.Thesintermarisinginequation 2-16 fromthecolatitudepres-suregradientcanbeexpandedassin=4)]TJ/F15 11.955 Tf 13.15 8.088 Td[(43 3!3+45 5!5)]TJ/F15 11.955 Tf 13.15 8.088 Td[(47 7!7++)]TJ/F15 11.955 Tf 9.299 0 Td[(1n+12n)]TJ/F20 7.97 Tf 6.587 0 Td[(1 n)]TJ/F15 11.955 Tf 11.955 0 Td[(1!-26andthecottermappearinginequation 2-17 canbeexpandedascot=1 )]TJ/F22 11.955 Tf 13.187 8.088 Td[( 3)]TJ/F22 11.955 Tf 13.747 8.088 Td[(3 45)]TJ/F15 11.955 Tf 13.747 8.088 Td[(25 945)-222(-27Thenumberoftermsintheseriesncanbearbitrarilyselectedtoachieveadesiredlevelofaccuracy.Inthepresentwork,thenumberoftermsintheexpansions 2-24 to 2-27 waslimitedton=14becausetermsofhigher-orderintheexpansion 2-26 forthecolatitudepressuregradientwerenegligiblysmallascomparedtothelargesttermsinequation 2-16 .Introductionofequations 2-24 2-27 into

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17 equations 2-16 and 2-17 ,andcollectingthetermsofgivenordersofyieldsaseriesof28coupledordinarydifferentialequationsforH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1.Forexample,theequationforH1andF1wereobtainedbycollectingthetermsoforderofinthemomentumbalance.Itisrepresentedas1 4F21)]TJ/F22 11.955 Tf 11.955 0 Td[(H1dF1 d=1+1 2d2F1 d2-28Similarly,collectingthetermsoforder0inthecontinuityequationyields1 2F1=dH1 d-29Thehigherordertermsofinthemomentumandthecontinuityequations,listedinAppendix A ,givethegoverningequationsforH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1.Theno-slipboundaryconditionattheelectrodesurfaceforvrandvisrelatedtoH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1bythefollowingH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=F2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=0:0at=0-30andthefar-eldboundaryconditionforvyieldsF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(11=)]TJ/F15 11.955 Tf 9.298 0 Td[(1i)]TJ/F20 7.97 Tf 6.587 0 Td[(122i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 i)]TJ/F15 11.955 Tf 11.956 0 Td[(1!-31Thus,thegoverningordinarydifferentialequationsforH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1withboundaryconditions 2-30 and 2-31 describetheuidowwithintheboundarylayer.Thesolutionprocedureisdescribedinthenextsection.2.4.1SolutionMethodTheabovesetofordinarydifferentialequationsweresolvedusingtheBANDalgorithmintroducedbyNewman.39Theboundarycondition 2-31 forthecolat-itudevelocity,i.e.,F2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=1wasappliedat=40:0.ThecalculationdomainwasdividedintoagridofNnodes.ThenodeswerespacedatdistanceofHwitheachother.ThemomentumequationswerediscritizedatnodeJaspresentedin

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18 Figure2-4:Schematicdiagramofgridforcalculationdomain.Histhespacingbetweenadjacentnodes. Figure 2-4 .ThecorrespondingcontinuityequationswerediscritizedathalfpointbetweennodeJandJ)]TJ/F15 11.955 Tf 11.955 0 Td[(1.ThediscretizationprocedureensuresthattheorderofresultingequationsateachnodeisofH2.ThedisctitizedformofcontinuityandmomentumequationsforH1andF1equations 2-28 and 2-29 aregivenasG=F1J+1)]TJ/F15 11.955 Tf 11.955 0 Td[(2F1J+F1J)]TJ/F15 11.955 Tf 11.955 0 Td[(1 2H2+1+H1JF1J+1)]TJ/F22 11.955 Tf 11.955 0 Td[(F1J)]TJ/F15 11.955 Tf 11.955 0 Td[(1 2H)]TJ/F22 11.955 Tf 10.494 8.088 Td[(F1J2 4-32andG=2H1J)]TJ/F22 11.955 Tf 11.955 0 Td[(H1J)]TJ/F15 11.955 Tf 11.955 0 Td[(1 H)]TJ/F15 11.955 Tf 13.15 8.088 Td[(F1J+F1J)]TJ/F15 11.955 Tf 11.955 0 Td[(1 2-33whereGandGaretheresidualsforthemomentumandthecontinuityequa-tionatnodeJ.TheBANDalgorithmsolvesequations 2-33 and 2-32 withboundaryconditions 2-30 and 2-31 suchthattheresidualsGandGareeffectivelyequaltozerowithinthespeciedtoleranceateachnode.ThesameprocedurewasfollowedforrestoftheequationslistedinAppendix A .AFOR-TRANcodewasusedtosolvetheequations.ListingofthecodewithitsmainprogramandsubroutinesispresentedinAppendix C .2.4.2ResultsThesolutionsobtainedforH1andF1asafunctionofarepresentedinFigure 2-5 .TheresultsareinagreementwiththoseobtainedbyHomann40fordiskelectrodeundersubmergedjetimpingement.TheobtainedsolutionforH3,F3,H5,F5,:::,F27arepresentedinAppendix A.2

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19 Figure2-5:DimensionlessradialandcolatitudefunctionsH1andF1asafunctionofseeequations 2-24 and 2-25 ThevelocitydistributionneartheelectrodesurfacecanbeapproximatedbyTaylor'sseriesexpansionsforF;andH;asF;="14Xi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1#+1 2"14Xi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1F002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#2-34andH;=1 2"14Xi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1H002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#2+1 6"14Xi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1H0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#3-35wheretherstandsecondtermsofequation 2-34 representtherstandthesecondorderderivativeofvwithrespectto,respectively.Similarly,therstandsecondtermsofequation 2-35 representthesecondandthethirdorderderivativeofvrwithrespectto,respectively.Therstorderderivativeofvrwithrespecttoiszerobecauseofnopenetrationcondition.Velocityexpansions 2-34 and 2-35 provideaconvenientwaytorepresenttheuidoweldwithintheboundarylayer.Later,thecoefcientofthevelocityoweldareutilizedinthesolutionof

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20 convective-diffusionequation.Thisisavalidapproachforconvective-diffusionprocesseswithlargeScnumber,becausethemasstransferboundarylayerismuchthinnercomparetothemomentumboundarylayer,andtheuidvelocitiesvandvrcanbeapproximatedwithaquadraticandthird-degreepolynomialinwithinthemasstransferboundarylayer.ThegradientexpressionsF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1attheelectrodesurfacewerecalculatedfromtheobtainedsolutionusingthreepointforwarddifferencemethod.41Inordertominimizetheinuenceofnite-differenceerrorsonevaluationofF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1,thedifferentialequationswereapproximatedtotheorderofthesquareofthemesh-sizeH,andthenumericalvalueswereobtainedbyextrapolationtozeromeshsize.PlotsF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1vsH2aregiveninAppendix A .ThenumberofdigitsgiveninTable 2.1 areconsistentwiththestandarddeviationobtainedthroughtheregressionprocedure.TwomethodscanbeemployedinobtainingH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1.Therstmethodutilizesthesolutionofordinarydifferentialequationsanddifferenceschemes.Thesec-ondmethodestimatesH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1bysubstitutingthecalculatedvaluesofF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1inthecorrespondingcontinuityequations.Mostimportantly,thesecondmethodreduceserrorsintheevaluationoftheexpressionsH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1byensuringthattheresidualsformass-balancearezeroattheelectrodesurface.ThehigherorderexpressionsF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereobtainedbysubstitutingtheno-slipconditioninthemomentumequation 2-16 .Thus,theexpressionforsecondderivativeofF;attheelectrodesurfaceisgivenby@2F; @2=0=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(sin4 2-36AftersubstitutionoftheseriesexpansionforF;equation 2-25 andfurtherexpansionofsin4intermsof,valuesofF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereobtained.TheexpressionsH0002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1wereobtainedbydoubledifferentiatingthethecontinuityequation 2-16

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21 Table2.1:SeriesexpansioncoefcientsF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1intheequations 2-34 and 2-35 forH;andF;at=0. i F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 H002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1 1 2.6238754 1.31193772 -3.99262600 -4.211282293 1.71640917 2.892755504 -0.41810335 -0.958447855 0.014014149 0.062315385 6 -0.0325842274 -0.09730865797 -0.025404936 -0.0861294598 -0.02354329 -0.091867539 -0.0221291 -0.097459910 -0.02122090 -0.10411441 11 -0.02068333 -0.1118529912 -0.02043571 -0.1207596913 -0.02042491 -0.1309315014 -0.02061600 -0.14248419 andsubstitutingthesecondderivativeofF;at=0.Thesubstitutionyields:@3H; @3=0=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(cos4 2)]TJ/F15 11.955 Tf 13.151 8.088 Td[(cos2cos2 2-37Again,aftersubstitutionoftheseriesexpansionfortheseriesexpansionforH;seeequation 2-24 andfurtherexpansionoftrigonometricfunctionsintermsof,thevaluesofF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereobtained.ThecalculatedvaluesforcoefcientsF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1fori=1,:::,14aregiveninTable 2.1 .Similarly,valuesofF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH0002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1areprovidedinTable 2.2 .2.5BoundaryLayerSeparationBoundarylayerseparationtakesplaceatthelocationwherethenormalderiva-tiveofthecolatitudevelocity,i.e.,@v @r0;6=0,hasavalueequaltozero.Thus,bound-arylayerseparationisobservedatthevalueofwherethedimensionlessshearstress,B=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-38

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22 Table2.2:SeriesexpansioncoefcientsF002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1intheequations 2-34 and 2-35 forH;andF;at=0. i F002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 H0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1 1 -2 -12 16=3 11=23 -64=15 -41=64 512=315 161=455 -1024=2853 -641=630 6 8192=155925 2561=141757 -32768=6081075 -133=60758 262144=638512875 81922=425675259 -262144=10854718875 -163841=127702575010 2097152=1856156927625 93623=13956067125 11 -8388608=19489647700625 -2621441=928078463812512 67108864=49308808782358125 146654=1499203672312513 -134217728=3698160658676859375 -5991863=2113234662101062514 1073741824=1298054391195577640625 335544322=48076088562799171875 hasavalueequaltozero.Thevalueofatwhichseparationwascalculatedde-pendedslightlyonthenumberoftermsretainedintheseriesexpansion.Thepointofseparationreachedavalueof54:8degreesforn=14.AplotofBispresentedinFigure 2-6 asafunctionof,showingclearlythepointofboundarylayerseparation.ThecorrespondingresultobtainedbyBarciaetal.,19andrepro-ducedinthepresentwork,fortherotatinghemisphereispresentedinFigure 2-6 toprovidecomparison.Theuiddynamicscalculationsofarotatinghemispheredoesnotpredictboundarylayerseparation,althoughasmallregionofistheo-reticallypossiblenearthesingularitywheretheelectrodecontactstheinsulatingplane.2.6NumericalSimulationInthepresentsection,computationaluidmechanicscalculationswereper-formedforthestationaryhemisphericalelectrodeunderjetimpingement.Thehydrodynamicmodeldevelopedhereinvolvedsimultaneousnumericalsolution

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23 Figure2-6:Calculateddimensionlesssurfaceshearstressasafunctionofangle.Solidlinesrepresenttheresultforthestationaryhemisphereundersubmergedjetimpingementanddashedlinesrepresenttheresultfortherotatinghemisphere. oftheNavier-Stokesandthecontinuityequations.Numericalsolutionofthegov-erningequationswasdevelopedbyDr.GertNelissen,VrijeUniversiteitBrussel,Belgium.2.6.1GoverningEquationsAsshowninFigure 2-7 ,atwo-dimensionalcylindricalcoordinatesystemwasemployedtodescribethesystem.Ther-coordinatecorrespondedtothehorizontalaxis,andz-coordinatewasassumedintheverticaldirection.Inthisrepresenta-tion,thethirddimension,i.e.,-coordinate,wasassumedtobearoundtheverticalz-coordinate.Theowwassymmetricinthe-coordinate,thereforethevelocitycomponentandthederivativeofquantitiesinthedirectionweresubstitutedbyzerointhegoverningequations.ThemathematicaldevelopmentandnumericalapproachusedinthepresentworkaredescribedbyNelissenetal.42Theowwasassumedtobesteadystate,andtheuidwasassumedtobeincompressible.Thus,

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24 Figure2-7:Schematicrepresentationofthesimulatedowgeometry.Thedimen-sionsaregiveninunitsofm.Thearrowrepresentsthegeneraldirectionofow,andthecylindricalelectrodeislocatedattheorigin. conservationofmomentuminr,andz-coordinatescouldbeexpressedbyvr@vr @r+vz@vr @z=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(@p @r+@ @r1 r@rvr @r+@2vr @z2-39andvr@vz @r+vz@vz @z=)]TJ/F22 11.955 Tf 10.541 8.087 Td[(@p @z+1 r@ @rr@vz @r+@2vz @z2-40whereistheuiddensity,pisthepressure,isthemolecularviscosityoftheuid.Fortheincompressibleuid,conservationofmassisrepresentedby1 r@rvr @r+@vz @z=0-41Underturbulentconditions,ReynoldsaveragedNavierStokesRANSequationswereused.42Theboundaryconditionsforequations 2-39 2-40 ,and 2-41 werethatno-slipandno-penetrationconditionsappliedatsolidsurfaces,andthatauni-formuidvelocityprolewasimposedattheinlettothesystem

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25 correspondingtothenozzle.Inaddition,areferencepointforthepressurewaslocatedattheoutletofthesystem.Theboundaryconditionsforequations 2-39 2-40 ,and 2-41 are: 1. Thenoslipboundaryconditioni.e.,vr=vz=0attheelectrodesurface. 2. Imposeduidvelocityprolefromtheinlet.Inthiscase,uideldwasassumedtobeemanatingwithaconstantvelocityacrossthenozzle. 3. Areferencepointforthepressurei.e.,p=pref=0wasassumedtobelocatedattheoutletofthesystem.2.6.2NumericalMethodThepartialdifferentialequations 2-39 2-40 ,and 2-41 weresolvedwithresidualdistributionmethod.43Thediscretizationwasdoneonthegridsoftri-anglesinthegeometricdomainofinterest.TheLax-Wendroff43schemewasap-pliedtotheconvectiontermsofthemomentumbalances.Theviscoustermsweretreatedinastandardniteelementmanner.Thenumericalschemeprovidedasecondorderaccuracy.Theresultingnon-linearsetofequationsweresolvedbyusingtheNewton-Raphsonmethod,withexplicitcalculationofthejacobianma-trices.AnincompleteLUpreconditionsgmreswasusedtoapproximatethesolu-tionofthelinearsystem.Thegridintheboundarylayerregimecontainedatleasttenelementsinthedirectionnormaltotheelectrode.2.6.3SimulationResultsThecalculatedresultspresentedherecorrespondtotwodifferentinletowrates.Fluidpropertieswereassumedtobethoseofwaterat25C.Simulationscor-respondingtoaninletReynoldsnumberof1,100arepresentedinFigure 2-8 .Theowintheinletregionislaminar.Thefalsecolorimagesindicatethatthepressureneartheelectrodeislargeatthestagnationpoint=0,decreasesintheregion

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26 Figure2-8:FluidstreamlinesinthevicinityoftheelectrodeforaninletReynoldsnumberof1,100.Thecolormapindicatethepressuredistribution.Theradialdimensionisgiveninunitsofm. of==4,andincreasesneartheelectrode-insulatorinterface==2.Theadversepressuregradientseenforangleslargerthat==4inducesaboundarylayerseparation,justaspredictedbypotentialowcalculations.Theuideldshowsacirculationzonestartingatanangleofabout62whichisslightlylargerthanthevalueof54.8obtainedbytheboundary-layerhydrodynamicmodel.TheoweldforaninletReynoldsnumberof11,000ispresentedinFigure 2-9 .Inthiscase,theinletowisturbulent.Theresultsagainshowthatacirculationzoneisformedatananglenear62.Figure 2.9b providesanenlargedimageoftherecirculationzoneshownneartheelectrode-insulatorinterfaceinFigure 2.9a .2.7SummaryThischapterhaspresentedtwohydrodynamicmodelsforuidowaroundastationarysubmergedhemisphereunderjetimpingement.Therstmodelwasbaseduponsemi-analyticalsolutionofthemomentumandmassconservationequation.Theresultsofthecalculationshowaformationofboundarylayerseparationforthesystem.Thepointofboundarylayerseparationwaspredictedtooccurat54.8.Thesecondmodelnumericallysolvedthemomentumandthe

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27 a bFigure2-9:FluidstreamlinesinthevicinityoftheelectrodeforaninletReynoldsnumberof11,000.Thecolormapindicatethepressuredistribution.Theradialdimensionisgiveninunitsofm.Figure 2.9b providesanenlargedimageoftherecirculationshowninFigure 2.9a

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28 continuityequationswithoutinvokinganyapproximations.Thismodelalsosuc-cessfullyshowtheformationofvortexinseparatedpartoftheboundarylayer,andthepointofboundarylayerseptationwaspredictedat62.Theresultsofthehydrodynamicmodelsareusedinsubsequentchapters.

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CHAPTER3CONVECTIVE-DIFFUSIONMODELSFORASTATIONARYHEMISPHERICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENTThischapterprovidesadetaileddescriptionofconvective-diffusionprocessestakingplaceintheboundarylayerofahemisphericalelectrodeundersubmergedjetimpingement.Asolutionforconvective-diffusionequationofthesystemisprovidedinthischapter.Theobtainedsolutionprovideaframeworkforfurtherinvestigationofimpingingjethemisphericalelectrode.3.1GoverningEquationsUndertheassumptionsthatthePecletnumberislargeandthattheconcen-trationofthereactantcRissmallwithrespecttothesupportingelectrolyte,thesteady-stateconvectivediffusionequationwithintheboundarylayercanbewrit-tenasvr@cR @r+v r0@cR @=DR@2cR @r2-1wherevrandvaretheradialandcolatidutevelocitycomponentsofuidoweldwithintheboundarylayer.TheirexpressionshavebeenobtainedfromtheuidmechanicsdevelopmentpresentedinChapter 2 .Thefollowingsassumptionsweremadeinderivingequation 3-1 : 1. theuidowintheboundarylayerislaminarandnormalderivativeofallquantitiesaremuchlargercomparetothetangentialderivatives, 2. thediffusionofreactantcRinthetangentialdirectiondirectionismuchsmallercomparetothenormaldirectionattheelectrodesurface,and 29

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30 3. thethicknessofthemasstransferboundarylayer,m,isinnitelysmallcom-paretotheradiusofthehemisphere.Itisimportanttonotethatequation 3-1 isonlyvalidintheunseparatedpartofthemomentumboundarylayerattheelectrodesurface.Theobjectivehereisthecalculationofthemass-transfer-limitedcurrentdistri-butionattheelectrodesurface;thus,theboundaryconditionsforequation 3-1 aregivenascRjr=r0;=c0-2cRjr=1;=c1-3andcRjr;=0=c1-4TheconcentrationcRcanbeexpandedasaseriesfunctionof,,andSc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3ascR)]TJ/F22 11.955 Tf 11.955 0 Td[(c1 c0)]TJ/F22 11.955 Tf 11.955 0 Td[(c1=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(21;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1+Sc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(22;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-5suchthatthersttermoftheexpansionprovidesthesolutionundertheassump-tionthattheSchmidtnumberScisinnitelylarge,andthesecondtermprovidesacorrectionforanitevalueofSc.ThecharacteristicdimensionlessdistanceformasstransfercanbedenedtobeZ=Sc1=3-6whichaccountsforthedifferenceinscalebetweentheconvectionandmasstrans-ferboundarylayerthicknesses.Fourteencoupledordinarydifferentialequationsfor1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1;1and2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1wereobtainedthroughfollowingsteps:

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31 1. SubstitutionofvrandvintermsofaTaylorseriesdimensionlessvelocitiesF;andH;fromequation 2-34 and 2-35 withn=14intoequation 3-1 2. introductionofdimensionlessconcentrationfromequationandscaleddis-tanceZfromequation 3-5 and 3-6 ,respectively,and 3. collectionoftermscorrespondingto2iand2iSc)]TJ/F20 7.97 Tf 6.586 0 Td[(1=3.Thederivedequationscanbewritteningeneralformasfollowing:d21;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ2)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 2H001Z2d1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ)]TJ/F15 11.955 Tf 11.955 0 Td[(2iF0iZ1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Z=Z2n=iXn=11 2H002n+1d1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ+Zn=i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Xn=12nF02i)]TJ/F23 7.97 Tf 6.586 0 Td[(n+11;2n+1Z -7 andd22;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ2)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 2H001Z2d2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ)]TJ/F15 11.955 Tf 11.955 0 Td[(2iF0iZ1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Z=Z3n=iXn=01 2H0002n+1d2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z dZ+Z2n=i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Xn=0i)]TJ/F22 11.955 Tf 11.956 0 Td[(nF002n+11;2i)]TJ/F23 7.97 Tf 6.587 0 Td[(n+1Z+Z2n=iXn=11 2H002n+1d2;2i)]TJ/F23 7.97 Tf 6.587 0 Td[(n+1Z dZ+Zn=i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Xn=12i)]TJ/F22 11.955 Tf 11.956 0 Td[(nF02n+12;2i)]TJ/F23 7.97 Tf 6.587 0 Td[(n+1Z -8 whereequationsets 3-7 and 3-8 represent14ordinarydifferentialequationsfor1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1,respectively.

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32 Theboundaryconditionsfor1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1are1;1=1:0 -9 1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=0:02;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=0:0atZ=0,and1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=0:0 -10 2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=0:0atZ=1.Theanalyticsolutionsof1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andanumericalsolutionsof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1areprovidedinthenextsection.3.2SolutionMethodandResultsTheequationset 3-7 correspondingto1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1withboundaryconditions 3-9 and 3-10 weresolvedanalytically.Fori=1,thesolutioncorrespondstothediskelectrodeundersubmergedjetimpingement,44anditisgivenas1;1Z=1:0)]TJ/F15 11.955 Tf 11.955 0 Td[(0:8500069ZZ0exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3dZ-11Fori>1,thegeneralsolutionoftheequationset 3-7 isgivenby1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Z=exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3j=i)]TJ/F20 7.97 Tf 6.586 0 Td[(1Xj=1)]TJ/F22 11.955 Tf 5.479 -9.684 Td[(jZ3j)]TJ/F20 7.97 Tf 6.587 0 Td[(2-12wherevaluesofjwerededucedbysubstitutionofequation 3-12 intoequation 3-7 .Thecompleteexpressionsof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1for2i14aregiveninAppendix B .Theequations 3-8 for1i14weresolvednumericallyusingtridiagonalBANDalgorithmdescribedbyNewman.39Theexpressionsinvolving1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1in

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33 Table3.1:Calculatedvaluesforcoefcients01;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and02;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1usedinequa-tion 3-14 formass-transfer-limitedcurrentdistribution. i 01;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 02;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1 1 -0.8500077 0.07190992 0.5456994 -0.08504493 0.1954955 -0.01939524 0.4638516 -0.02036335 0.3719751 -0.0182829 6 0.3247121 -0.01721547 0.2949598 -0.01664298 0.2761729 -0.01641069 0.2647235 -0.016436610 0.2585353 -0.0166741 11 0.2563565 -0.017095412 0.2574086 -0.017684913 0.2612022 -0.018434714 0.2674345 -0.0194908 equationsset 3-8 weresubstitutedwiththeiranalyticalexpressionsasprovidedinAppendix B .ThenumericalsimulationwereperformedfordifferentvaluesofgridspacingH.Thequantitiesofinterestformass-transfer-limitedcurrentarerstderivativesof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1withrespecttoZattheelectrodesurface.Inordertominimizetheinuenceofnite-differenceerrors,thedifferentialequationswereapproximatedtotheorderofthesquareofthemesh-size,andthenumericalvaluesof02;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereobtainedbyextrapolationtozeromeshsize.Calculatedvaluesfor01;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and02;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1areprovidedinTable 3.1 .3.3MassTransferLimitedCurrentTheuxattheelectrodesurfaceisgivenbyNR=)]TJ/F22 11.955 Tf 12.487 0 Td[(DR@cR @rr=r0-13

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34 whichcanbeevaluatedintheformofamass-transfer-limitedcurrentdensityintermsofthedimensionlessvariablesintroducedaboveasilim=nFc1)]TJ/F22 11.955 Tf 11.956 0 Td[(c0DR r0Sc1=3Re1=2"nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(201;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1+Sc)]TJ/F20 7.97 Tf 6.587 0 Td[(1=3nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(202;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1!#-14wheretheReynoldsnumberReisdenedtobeRe=ar20 -15Equation 3-14 canbeexpressedintermsofacharacteristicnumberN=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(nFc1)]TJ/F22 11.955 Tf 11.955 0 Td[(c0DR r0Sc1=3Re1=2-16asilim N=+Sc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3-17whereisthemass-transfer-limitedcurrentdensityforaninniteSchmidtnumber,andisthecorrectiontoaccountforthenitevalueoftheSchmidtnumber.Thus,=)]TJ/F23 7.97 Tf 17.356 14.944 Td[(nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(201;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-18and=)]TJ/F23 7.97 Tf 17.356 14.944 Td[(nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(202;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-19ThecalculatedvaluesforandarepresentedinFigures 3.1a and 3.1b ,respectively,asfunctionsofcolatitudeangle.Theboundary-layersolutionisvalidonlyuptothepointofboundarylayerseparation.TheresultobtainedforthestationaryhemisphericalelectrodeisinstarkcontrasttothatobtainedbyBarciaetal.19fortherotatinghemisphericalelectrode,alsoshowninFigure 3-1 ,whichdoesnotshowsuchaboundarylayerseparation.ForthestationaryelectrodeinFigure 3.1a ,theregionofcirculationisrepresentedbyauniformextensionofthevalueofatthepointofseparation.

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35 a bFigure3-1:Calculatedmass-transferlimitedcurrentdensityforahemisphericalelectrodesubjectedtoasubmergedimpingingjet.Solidlinesrepresentresultsforthestationaryelectrode,andthedashedlinesrepresentresultsforthero-tatinghemisphericalelectrode.aContributiontoequation 3-17 foraninniteSchmidtnumber;bContributiontoequation 3-17 providingcorrectionfora-niteSchmidtnumber.

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36 Workisneededtoprovideamorecorrectestimationofthemass-transferrateinthisregion.Apreliminaryapproachcouldbetoassumeauniformvalueforamass-transfercoefcientkMT.Thus,withintheregionofcirculation,ilimcanbeexpressedasilim=nFkMTc1)]TJ/F22 11.955 Tf 11.956 0 Td[(c0-20Integrationofequations 3-14 and 3-20 overtheelectrodesurfaceisrequiredtoobtainavalueforthetotalcurrentwhichisaccessiblefromexperimentalmeasure-ment.3.4NumericalSimulationsTheaforementionedmodelgivesanexpressionofconvective-diffusionpro-cessesuptothepointofboundary-layerseparation.Numericalsolutions,how-ever,wereusedtoobtainasolutionofconvective-diffusionovertheentireelec-trodesurface.NumericalsimulationofthegoverningequationwasperformedbyDr.GertNelissen,VrijeUniversiteitBrussel,Belgium.Theconvective-diffusionmodelinthecylindricalcoordinatesystemisgivenbyvr@cR @r+vz@cR @z=DR1 r@ @rr@cR @r+@2cR @z2-21wherevrandvzaretheuidvelocitycomponentcalculatedbythecomputationaluiddynamicmodelinthepreviouschapter,cRistheconcentrationofreactingspecies,andDRisthemoleculardiffusivityofthereactingspecies.Fluidowwasassumedtobelaminarinthemasstransferboundarylayer.Thetransportofthereactantduetoelectriceldinequation 3-21 hasbeenneglectedbyassumingthepresenceofexcesssupportingelectrolyte.Thesystemisassumedtooperateunderisothermalconditions.Attheelectrodesurface,theboundaryconditionsforequation 3-21 were:cR=0-22

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37 andinthebulkcR=c1-23Equation 3-21 wassolvedwithboundaryconditions 3-22 and 3-23 usingnu-mericalschemepresentedinthepreviouschapter.Toensurepositivity,thecon-vectiontermintheconvective-diffusionequationwerediscretizedusingtheN-scheme.45,42Standardniteelementdiscretizationwasappliedtothediffusionterm.Innumericalsimulations,thediameterofelectrodeandnozzlewasxedtobe1/4inch,andnozzlewasassumedtobeplacedat5.0cmfromtheelectrode.Thephys-icalpropertiesoftheelectrolyteusedinthesimulationsarelistedinTable 3.2 .TheconcentrationdistributionofreactantasafunctionofdistancefromtheelectrodesurfaceisshowninFigure 3-2 .Directlyontheelectrodesurfacethereactantcon-centrationiszero,whichisdepictedbytheblueline.Thearearepresentedbytheredcorrespondstothebulkreactantconcentration.Abovetheelectrodesurface,theconcentrationdistributionofreactantexhibitaverysharpgradient,showninFigure 3-2 bymarkedchangeincolor.However,atthepointofseparation=62,thecolorvariationbecomeswidersignalingadropinconcentrationgradient.Thisconrmsthatthemass-transferisminimalatthepointofboundarylayersepara-tion.Theexpressionofmass-transfer-limitedcurrentisgivenby:ilim=nFDR@cR @r-24Thelimitingcurrentdensityvectorattheelectrodesurfacewascalculatedusingthesimulation.TheparametersusedinsimulationsaregiveninTable 3.2 .TheresultsofsimulationsarepresentedinFigure 3-3 .SimulationweredonefortwoReynoldsnumberofuidinthenozzle.AthighRe,thecurrentishigher.CurrentreachesaminimumatthepointofboundarylayerseparationforbothRenumbers.Thecurrentintheseparatedzoneishigherthanatthepointofseparationforbothcases.

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38 Figure3-2:Reactantconcentrationdistributionasafunctionofdistancefromelec-trodesurface,obtainedthroughnumericalsimulationofequation 3-21 .Thebluelinecorrespondstozeroconcentrationontheelectrodesurface,whereasredcorre-spondstothebulkreactantconcentration.Theradialdimensionisgiveninunitsofcm.ThesesimulationswereperformedforRe=11300inthenozzle. Table3.2:Physicalpropertiesoftheelectrolyteusedinthenumericalsolutionofequation 3-21 Property Value /m2sec)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1:010)]TJ/F20 7.97 Tf 6.587 0 Td[(6DR/m2sec)]TJ/F20 7.97 Tf 6.587 0 Td[(1 7:010)]TJ/F20 7.97 Tf 6.587 0 Td[(10c1/molm)]TJ/F20 7.97 Tf 6.586 0 Td[(3 100:0

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39 Figure3-3:Calculatedmass-transfer-limitedcurrentdensityfordifferentReynoldsnumberattheinletofthenozzle.Theverticaldashlineat62isthepointofboundarylayerseparation.ThephysicalpropertiesoftheelectrolyteusedinthesimulationsarelistedinTable 3.2 3.5ConclusionSteadystatemass-transferwasobtainedforthesystem.Twomodelswerepre-sented.Therstmodelwasbaseduponsemi-analyticalseriesexpansionmethod.ThemodelsusedtheuidvelocityeldsobtainedinChapter 2 .Thesecondmodelmadeuseofanumericalschemetosolvetheconvective-diffusionequa-tion.Bothmodelpredictedanitevalueofmass-transfer-limitedcurrentatthepointofboundarylayerseparation.Thesecondmodelwasabletocalculatethemass-transferintheseparatedregionoftheboundary.Themass-transferisin-creasedintheseparatedregionwithaminimumatthepointofseparation.Thisresultisanalogoustotheheat-transferinsphere,whereheat-transferisenhancedintheseparatedpartoftheboundarylayer.46Animplicationofthiscalculationisthatthecurrentcanbeassumedtobeconstantintheregionofboundarylayer

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40 separation.Thisapproximationwillbeusedinthecalculationofthecurrentdis-tributionforthesysteminChapter 5

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CHAPTER4HYDRODYNAMICANDMASS-TRANSFERMODELSFORAROTATINGHEMISPHERICALELECTRODEThehemisphericalelectrodeunderjetimpingementrepresentsamodicationtotherotatinghemisphericalelectrodedescribedinthiswork.Thepresentworkprovidesareviewofprevioushydrodynamicandthesteady-statemass-transfermodelsofarotatinghemisphericalelectrodesystem.NewcalculationresultsarebeingpresentedalongsideresultsofBerciaetal.19andsubsequentlycompared.Thisstudyprovidesacorrectiontothepublishedwork.194.1SchematicIllustrationoftheSystemTherotatinghemisphericalelectrodewasrstsuggestedbyChin17asanalter-nativetorotaingdiskelectrodeforstudyofelectrochemicalsystems.Theadvan-tageofthehemisphereoverrotatingdiskisthatthecurrentdistributionremainsuniformevenatlargefractionofmasstransferlimitedcurrent.18Thisuniformityofcurrentcanbeexploitedtostudyelectrochemicalprocessesathighercurrentdensity.AschematicofthesystemispresentedinFigure 4-1 .InFigure 4-1 ,ahemisphericalelectrodeisxedinainsulatingmaterialwithanogapatthepointofintersectionwiththeinsulatingplane.Thesystemisrotated Figure4-1:SchematicsillustrationofRotatingHemisphericalElectrode. 41

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42 intheelectrolyticsolutionintheazimuthaldirection.Thesphericalpolarcoordi-natesystemisusedtodescribethesystem,whereristheradiallyoutwardnormaldirectionwithoriginatthecenterofhemisphere,iscolatitudedirection,andistheazimuthaldirection;andvr,v,andvarethecomponentsofuidvelocityeldinr,,anddirection,respectively.4.2HydrodynamicModelThehydrodynamicmodeloftherotatinghemispherehasbeenaddressedinliteraturebyseveralworkers.19,17,47Ithasbeenbasedonthehydrodynamicmodelofarotatingsphereinaquiescentuid.Howarth36rstintroducedtheproblemofarotatingspherein1951.Heprovidedasolutionofthemodelequationsusingapolynomialseriesexpansionforvr,v,andvintermsof.Hissolutionwaslim-itedtothersttwotermsintheexpansion.Healsosuggestedthattheuidowneartheequatorcannotbedescribedbyboundarylayerequations.Nigam48sug-gestedadifferentformofseriesexpansionthanthatofHowarth.36Heproposedthatthevelocityexpansionsintermsoftrigonometricfunctionsofforvelocitycomponents,andprovidedasolutionforrstthreetermsofvr,v,andv.Hiscalculationssuggestedthattheboundarylayerremainsintactattheequator;thus,boundarylayerequationsadequatelydescribetheownearequator.Stewartson49statedthattheuidowatequatorisoutwardalongtheequatorialplane;there-fore,boundarylayerwillbreakdownneartheequator.Hesuggestedthatthethicknessoftheregion,whereboundarylayerassumptionsfail,iswithinO1=2distanceoftheequator.Banks50improvedthesolutionbysolvingforcoefcientsintheseriesexpansionuptofourterms.HisseriesexpansionswerebaseduponHowarth's36model.Manohar,51ontheotherhand,solvedtheboundarylayerequationsusinganitedifferencetechnique.HereportedabetterconvergenceofsolutionthanthatofHowarth's36seriesmethod.

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43 Chin17,47treatedthehydrodynamicsoftherotatinghemisphericalelectrodeRHE,asdescribedinFigure 4-1 ,likethatofrotatingsphericalelectrode.HeusedHowarth'smethodofseriesexpansionforvelocitycomponentsandlimitedittofourterms.Morerecently,Barciaandcoworkers19extendedtheseriesexpansionuptotenterms.Inclusionofadditionaltermswasneededtoobtaintheaccuracyneededforimpedancecalculations.ThenextsectionrevisitsthehydrodynamicmodeloftheRHE.Thiswasmoti-vatedbytheobservationthattheresultsprovidedbyBarciaetal.19donotsatisfythecontinuityequationattheelectrodesurface.Theresultspresentedinthischap-terprovideacorrectiontotheresultsofBarciaandcoworkers.194.2.1GoverningEquationsThegoverningequations,36whichdescribetheuidmotionwithinthebound-arylayerofRHE,arewrittenas1 r0@v @+@vr @r+v r0cot=0-1v r0@v @+vr@v @r)]TJ/F22 11.955 Tf 13.151 9.321 Td[(v2 r0cot=@2v @r2-2v r0@v @+vr@v @r+vv r0cot=@2v @r2-3whereequation 4-1 isthecontinuityequation,andequations 4-2 and 4-3 arethemomentumbalancesinanddirections,respectively.Theunderlyingas-sumptionisthatderivativesofquantitieswithrespecttovanishandthereisnoimposedpressuregradient.Theboundaryconditionsforequations 4-1 4-3 arevr=v=0;v=!r0sinatr=r0-4andv!0;v!0atr!1-5

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44 wherer0istheradiusoftheelectrode,and!isitsrotationspeed.Dimensionlessvariablesforr,vr,v,andvcanbegivenby=r r)]TJ/F22 11.955 Tf 11.956 0 Td[(r0-6vr=p !H;-7v=r0!F;-8andv=r0!G;-9wheredimesionlessquantitiesH;,F;,andG;aredenedaspolyno-mialseriesexpansionwithrespecttosuchthatH;=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1H2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-10F;=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-11G;=nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1G2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-12wherenisthenumberoftermsincludedintheexpansion.Substitutionofequa-tions 4-6 to 4-12 intoequations 4-1 4-2 ,and 4-3 yields3nordinarydiffer-entialequationsforH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1,F2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1,andG2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1.TheboundaryconditionsforthederivedequationsareH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=F2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=0;G2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=)]TJ/F15 11.955 Tf 9.299 0 Td[(1i+1 i)]TJ/F15 11.955 Tf 11.955 0 Td[(1!-13at=0andF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(11!0;G2i)]TJ/F20 7.97 Tf 6.587 0 Td[(11!0-14TheTaylorseriesexpansionofH;andF;closetotheelectrodesurfaceisgivenby:H;=1 2"i=nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1H002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#2+1 6"i=nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1H0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#3-15

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45 andF;="i=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1#+1 2"i=nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(1F002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1#2-16andthedimensionlessshearstressattheelectrodesurfaceisgivenby:B=i=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(1F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-17ThedimensionlessshearstressderivedbyBarciaetal.,19Chin,47andManohar51,52areplottedinFigure 4-2 .Barciaandcoworkers19solutionispresentedbyasolidline,whereasthoseofChin47andManohar51,52arepresentedbydashedanddot-tedlines,respectively.Newman52obtainedanexpressionfordimensionlessshearstressbyttingapolynomialintotheresultsofManohar.51ItisrewrittenasB=0:51023)]TJ/F15 11.955 Tf 11.955 0 Td[(0:18088193)]TJ/F15 11.955 Tf 11.955 0 Td[(0:04408sin3-18Newman52consideredManohar's51solutiontobemoreaccuratethanthatofChin47forlargeangle.AsseeninFigure 4-2 ,theBpresentedbydottedlinehasthelowestvalueoftheatlargeangles.TheobjectiveofthehydrodynamicscalculationsistoestimatethevaluesofF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1.Thesecoefcientsareuseddirectlyinthesteady-statemasstransfer,currentdistribution,andconvectivediffusionimpedancecalculations.ThevaluesofF002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH0002i)]TJ/F20 7.97 Tf 6.587 0 Td[(153areobtainedbysubstitutingboundarycon-ditions.TheyaregivenbyF002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1=)]TJ/F15 11.955 Tf 9.298 0 Td[(1i22i)]TJ/F20 7.97 Tf 6.586 0 Td[(2 i)]TJ/F15 11.955 Tf 11.955 0 Td[(1!-19H0002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1=8><>:2ifi=13)]TJ/F20 7.97 Tf 6.587 0 Td[(1i+122i)]TJ/F21 5.978 Tf 5.756 0 Td[(3 i)]TJ/F20 7.97 Tf 6.587 0 Td[(2!ifi>1-20ThevaluesofF002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH0002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1aredeterminedfromaboveequation,whereas,coefcientexpressionsF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1areobtainedfromthenumericalsolu-tionofgoverningequationsdescribingtheboundary-layerhydrodynamicmodel.

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46 Figure4-2:ShearStressDistributionattheelectrodesurface.SolidlinerepresentresultsofBarciaetal.,thedashedlinerepresenttheresultsofChin,andthedottedlinerepresenttheresultofManohar. 4.2.2ResultsBarciaetal.19solvedtheordinarydifferentialequationsforhydrodynamicswithn=10usingBAND39,23algorithm.TheyshowedthattheircalculationswereinagreementwiththatofChin17,47forn=4.However,itwasfoundthatthesubstitutionofF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1inthecontinuityequationsfori>4doesnotyieldthereportedvalues19of1 2H002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1.Forexample,thevalueof1 2H009isreportedas0:3097810)]TJ/F20 7.97 Tf 6.586 0 Td[(3,whereascontinuityequationyieldsavalueof)]TJ/F15 11.955 Tf 9.299 0 Td[(0:2076310)]TJ/F20 7.97 Tf 6.587 0 Td[(3.Theerrorinthevaluesofcoefcientswillhavealargerimplicationinestimationofmass-transfer-limitedcurrentandconvective-diffusionimpedance.ItisassumedinthisworkthatthevaluesofcoefcientsF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1reportedBarciaetal.19aresuf-cientlyaccurate.Thecorrespondingvaluesof1 2H002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1arecalculatedfromcon-tinuityequations.TheyaretabulatedinTable 4.1 .Incomparingthevaluesof1 2H002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1fromTable 4.1 ,itisobservedthatthevaluesdifferforn>5thanthatof

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47 Table4.1:F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andH002i)]TJ/F20 7.97 Tf 6.587 0 Td[(1coefcientsintheseriesexpansionofequations 4-16 and 4-15 forH;andF;at=0.ThethirdcolumninthetableliststhevaluesreportedbyBarciaetal.andthefourthcolumnliststhevaluescalculatedusingthecontinuityequation. i F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1 2H002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1reportedbyBarciaetal.,19 1 2H002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1 0.51023 -0.51023 -0.510232 -0.22128 0.52761 0.527603 0.20711E-1 -0.93344E-1 -0.93344E-14 -0.18905E-2 0.90951E-2 0.90951E-25 -0.11499E-4 0.30978E-3 -0.20763E-3 6 -0.41534E-4 0.21299E-3 0.23024E-37 -0.11468E-4 0.70451E-4 0.71604E-48 -0.18727E-5 0.12325E-4 0.12434E-49 -0.89351E-6 0.75296E-5 0.75405E-510 -0.21900E-6 0.20009E-5 0.20020E-5 Barcia'setal.194.2.3FluidFlowattheCornerTheuidmotionofatthecorneri.e.,attheintersectionofhemisphericalelec-trodeandinsulatingplane,isdiscussedhere.Existingliteraturehasoverlookedthisdetail.Theproblemhasbeenignoredintheliterature,however,theremaybesomecompellingimplicationofthisissue.Forexample,Stewartson49suggestedthatboundarylayerequationsarenotvalidwithinO1=2distanceoftheequator.ThiswouldmeanthatforaqueouselectrolytesthedistancewouldbeO:1cm,whichrepresentsaconsiderableportionofaO:635cmdiameterelectrode.Onotherhand,Nisancioglu18suggestedthatthethicknessoftheregionwherebound-arylayerfailsisoftheorderofO)]TJ/F20 7.97 Tf 9.554 -4.977 Td[(1 RewhereReynoldsnumberReisdenedasRe=!r20 -21Thus,thethicknessoftheregioncanbereducedbyhighrotationspeedoftheelectrode.ItisimportanttonotethatNisancioglu's18aswellasStewartson's49statementsareforrotatingsphericalelectrodes.FluidmotioninRHEwillbemore

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48 Figure4-3:Atwodimensionaldepictionofboundarylayerattheintersectionofelectrodeandinsulatingplane. complicatedthanthatthatofrotatingsphericalelectrode.Aschematicrepresen-tationoftheuidmotionintheboundarylayernearthecornerisshowninFig-ure 4-3 .Asuidmovesalongtheboundarylayeronthesphericalsurfaceoftheelectrode,itwouldencountera90:0changeindirectionasitapproachestheatinsulatingplanewhereuidwillmovealongtheplaneintheoutwardrdirection.Thebulkofaforementionedparagraphdescribesarotatinghemisphericalelec-trode.However,foreaseofoperation,insulatingplanecanalsorotatewiththeelectrode.Asaresult,theuidowwillbefurtheracceleratedalongtheinsulat-ingplaneduetoitsrotation,causingtheboundarylayeratthecornertoshrink.Thisproblemrequiresfurtherinvestigation,anditissuggestedasfuturework.4.3MassTransferThesteadystateconvective-diffusionequationgoverningthemasstransferintheboundarylayercanbewrittenasvr@cR @r+v r0@cR @=DR@2cR @r2-22wherecRisthereactingspecies,vrandvaretheradialandcolatidutevelocitycomponentsofuidoweldwithintheboundarylayer,andDRisthediffusivity

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49 ofthereactingspecies.Theobjectiveistocalculatethesteady-statemass-transfer-limitedcurrentdistributionattheelectrodesurface,utilizingtheuidvelocitycoefcientsreportedinthiswork.Theboundaryconditionsforequation 4-22 aregivenascRjr=r0;=c0-23cRjr=1;=c1-24andcRjr;=0=c1-25AsstatedinChapter 3 ,cRcanbeexpandedasafunctionsof,and,andSc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3as:cR)]TJ/F22 11.955 Tf 11.955 0 Td[(c1 c0)]TJ/F22 11.955 Tf 11.955 0 Td[(c1=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(21;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1+Sc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(22;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-26suchthatthersttermoftheexpansionprovidesthesolutionundertheassump-tionthattheSchmidtnumberScisinnitelylargeandthesecondtermprovidesacorrectionforanitevalueofSc.ThecharacteristicdimensionlessdistanceformasstransfercanbedenedasZ=Sc1=3-27whichaccountsforthedifferenceinscalebetweentheconvectionandmasstrans-ferboundarylayerthicknesses.Tencoupledordinarydifferentialequationsfor1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1obtainedbyfollowingsteps: 1. SubstitutionofvrandvwithdimensionlessvelocitiesH;andF;givenbyequations 4-15 and 4-16 ,respectivelywithn=10. 2. RepresentingcRwiththedimensionlessconcentrations1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1giveninequation 4-26

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50 Table4.2:Calculatedvaluesforcoefcientsusedinequation 4-29 forcalculatingmass-transfer-limitedcurrentdistribution. i 01;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 02;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1 -0.62045 0.184902 0.12831 -0.43440E-13 0.34750E-2 -0.15360E-24 0.14694E-2 -0.55010E-35 0.35468E-3 -0.19855E-3 6 0.10783E-3 -0.56713E-47 0.34006E-4 -0.17257E-48 0.99642E-5 -0.60689E-59 0.32183E-5 -0.19643E-510 0.10249E-5 -0.66980E-6 3. SubstitutionofrwithZasgivenbyequation 4-27 4. Theabovethreesubstitutionsweremadeinequation 4-22 .Thetermscor-respondingto2iand2iSc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3werecollectedtogetthegoverningequations.TheobtainedequationsweresolvedusingNewman'sBAND54algorithm.Theob-tainedresultsintheformofrstderivativesof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zand2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1ZatZ=0aretabulatedinTable 4.2 .Thevalueof01;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1and02;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1arenotinagree-mentwiththosecalculatedbyBarciaetal.19fori>4.TheconcentrationuxattheelectrodesurfaceisgivenbyNR=)]TJ/F22 11.955 Tf 12.487 0 Td[(DR@cR @rr=r0-28whichcanbeevaluatedintheformofamass-transfer-limitedcurrentdensityintermsofthedimensionlessvariablesintroducedaboveasilim=nFc1)]TJ/F22 11.955 Tf 11.956 0 Td[(c0DR r0Sc1=3Re1=2"nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(201;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1+Sc)]TJ/F20 7.97 Tf 6.587 0 Td[(1=3nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(202;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1!#-29wheretheReynoldsnumberReisdenedtobeRe=!r20 -30

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51 Equation 4-29 canbeexpressedintermsofacharacteristicnumberN=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(nFc1)]TJ/F22 11.955 Tf 11.955 0 Td[(c0DR r0Sc1=3Re1=2-31asilim N=+Sc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3-32whereisthemass-transfer-limitedcurrentdensityforaninniteSchmidtnumber,andisthecorrectiontoaccountforthenitevalueoftheSchmidtnumber.Bothandaregivenby=)]TJ/F23 7.97 Tf 17.356 14.944 Td[(nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(201;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-33and=)]TJ/F23 7.97 Tf 17.356 14.944 Td[(nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(202;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1-34ThecalculatedvaluesforandarepresentedinFigures 4.4a and 4.4b ,respectively,asfunctionsof.TheaverageofmassTransferlimitedcurrentisgivenby:Ilim N= 2Z0sind+Sc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3 2Z0sind-35Uponintegratingsinandsinwithrespect,oneobtains:Ilim N=0:457101)]TJ/F15 11.955 Tf 11.955 0 Td[(0:27896Sc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3-36whereasBarciaetal.,19reportedIlimasIlim N=0:456361)]TJ/F15 11.955 Tf 11.955 0 Td[(0:28002Sc)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 3-37ForSc=1000,therelativeerrorbetweenthetwoexpressionswouldbeabout0:18%.AgraphofrelativeerrorasafunctionofScnumberispresentedinFigure 4-5 .TherelativeerrorisminimumforinniteScnumber,andincreaseasvalueofScnumberisdecreased.

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52 a bFigure4-4:Calculatedmass-transferlimitedcurrentdensityforarotatinghemi-sphericalelectrode.aContributiontoequation 4-32 foraninniteSchmidtnum-ber;bContributiontoequation 4-32 providingcorrectionforaniteSchmidtnumber.

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53 Figure4-5:Relativeerrorinmass-transfer-limitedcurrentgivenbyexpressions 4-36 and 4-37 asafunctionofSchmidtnumber. 4.4SummaryFollowingBarciaandcoworker,19thehydrodynamicandconvective-diffusionmodelsforarotatinghemisphericalsystemwaspresentedinthischapter.ThesecondorderradialvelocitycoefcientsH002i)]TJ/F20 7.97 Tf 6.586 0 Td[(1werecalculatedsuchthatresid-ualsofthecontinuityequationattheelectrodesurfaceisequaltozero.Thecal-culatedvelocitycoefcientswerethenutilizedtoobtainanexpressionofmass-transferlimitedcurrent.Therelativeerrorinthemass-transfer-limitedcurrentpresentedinthischapterandtotheonepresentedbyBarciaandcoworkers19isabout0.18%foravalueofSc=1000.Anaccuratedevelopmentofconvective-diffusionimpedancecanbeachievedusingthehydrodynamicmodelpresentedhere.

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CHAPTER5CURRENTANDPOTENTIALDISTRIBUTIONATAXISYMMETRICELECTRODESInChapters 2 and 3 ,hydrodynamicandconvectivediffusionmodelswerede-velopedforastationarysubmergedhemisphericalelectrodeunderjetimpinge-ment.Thischapterdiscussesthecurrentandpotentialdistributionofastationaryhemisphericalelectrode.Thecurrentdistributionforadiskelectrodewasalsocalculatedandcomparedtothehemisphericalelectroderesults.Theshearstress,obtainedbythesemi-analyticalhydrodynamicmodelinChapter 2 ,wasusedinthecurrentdistributioncalculationsforthestationaryhemisphericalelectrode.Ageneralizedaxisymmetricmodelforcurrentdistributionisdevelopedinthischapter.Themodelcandescribethecurrentdistributionforthediskandthehemi-sphericalgeometries.Calculationsforarotatinghemispherearealsoperformedinthischapter.Thesecalculationsaccountedforcorrectionsinmass-transferduetoniteSchmidtnumber.TheshearstressobtainedinChapter 4 wasusedincaseofrotatinghemisphericalelectrodesystem.5.1IntroductionCurrentdistributionplaysanessentialroleinelectrochemicalfabricationtech-nologiesandininterpretationofelectrochemicalprocesses.55Signicantefforthasbeenmadeondevelopmentofnewrotatingelectrodedesignstoobtaintheidealbalancebetweenuniformcurrentdistributions,well-denedmasstransfer,andeaseofsurfacecharacterization.56Theneedtocouplesurfacecharacteriza-tionwithelectrochemicalmeasurementsonuniformcurrentdistributionswasad-dressedbyMatloszandcoworkersetal.,28whousedaremovablediskelectrode 54

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55 insertedinarotatinghemisphericalelectrode.Theatdiskwassuitableforex-situsurfaceanalysis,andnumericalsimulationswereusedtoidentifyconditionsunderwhichtheinuenceoftheatsurfaceonthecurrentdistributioncouldbeneglected.Dinanetal.30proposedarecessedrotatingdiskelectrodethatwouldprovideauniformcurrentdistributionbycompromisingtheuniformaccessibilityoftherotatingdisk.Theadvantageofusingadiskelectrodeunderjetimpingementisthat,solongasthedisklieswithinthestagnationregionofow,anaccuratesolutionisavailableforconvectivediffusion,andthecurrentdistributionisuniformundermass-transfer-limitedconditions.Thecurrentandpotentialdistributiononadiskelectrodebelowthemass-transfer-limitedcurrentisnotuniform,5andithasbeenshownthatneglectofthenonuniformcurrentdistributionintroduceserrorines-timationofkineticparametersfromsteady-statemeasurements.7Similarerrorsareobservedwhenimpedancemeasurementsareinterpretedundertheassump-tionofauniformcurrentdistribution.13,14Stationaryelectrodesareattractivebecausetheycaneasilybeadaptedtouseofin-situobservationorsurfaceanalysistechniques.Theobjectiveofthisstudywastounderstandtheinuenceofuidmechanics,convective-diffusion,andelec-triceldbelowthemass-transfer-limitedcurrentforsubmergedstationarydiskandhemisphericalelectrodesunderjetimpingement.Thecurrentdistributionattherotatinghemisphericalelectrodewasalsoexplored.Thecalculationswereac-countedfortheeffectofniteSchmidtnumberonmass-transfer.5.2DevelopmentofMathematicalModelAtwo-dimensionalmathematicalmodeldescribingcurrentandpotentialdis-tributionwasdevelopedforaxisymmetricbodiesinacurvilinearcoordinatesys-tem.Examplesofaxisymmetricelectrochemicalsystemsarethediskand

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56 Figure5-1:Schematicsillustrationofanaxisymmetricbodyinacurvilinearcoor-dinatesystem.Thehorizontaldashlinerepresentstheaxisofsymmetry,andtheuideldisassumedtobesymmetricaroundthisaxis. hemisphericalelectrodeswithsymmetricuidoweld.Aschematicrepresenta-tionofanaxisymmetricsystemincurvilinearcoordinatesisillustratedinFigure 5-1 .Thegeometryoftheaxisymmetricbodiescanbedescribedbyx,y,andRxwherexisanarclengthmeasuredalongameridiansectionfrompointofstagnation,yisperpendiculartosurface,andRxistheradiusofthesectionofthebodyperpendiculartoitsaxisofsymmetry.Inthisdevelopment,theuidoweldisassumedtobesymmetricwithrespecttotheyaxisatx=0.5.2.1HydrodynamicsThevelocitycomponentsoftheuideldintheboundarylayeraregivenbyvxandvy.vxisassumedtobeknownandisrepresentedasvx=xy-1wherexisthetangentialshearstress,obtainedbysolvingtheNavier-Stokesequationsalongwiththecontinuityequation.Thecontinuityequationforincom-pressibleuidwithintheboundarylayerofaxisymmetricowcanberepresentedby1 Rx@Rxvx @x+@vy @y=0-2

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57 Substitutionofvxfromequation 5-1 intoequation 5-2 yieldsthefollowingex-pressionforvy.vy=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(1 2y21 RxdxRx dx-3Hence,vxandvycanbeexpressedintermsoftangentialshearstressincurvilinearcoordinatesaspresentedbyequations 5-1 and 5-3 ,respectively.5.2.2MassTransferTheconvectivediffusionequationinthecurvilinearcoordinatesystemwithintheboundarylayercanbewrittenas:vx@cR @x+vy@cR @y=DR@2cR @x2-4wherecRistheconcentrationofreactingspecies,andDRisthediffusivityofthereactingspecies.Equation 5-4 isvalidunderthefollowingassumptions. 1. ThePecletnumberislargeanddiffusionintheydirectioncanbeneglected. 2. ConcentrationofthereactantcRissmallwithrespecttothesupportingelec-trolyte.Asaresult,themigrationofcRduetotheelectriceldisnegligible.Theobjectivehereistosolveequation 5-4 alongwiththefollowingboundaryconditions:cR=cRxaty=0-5andcR=c1aty!1-6wherec1isthebulkconcentrationofcR.However,itisconvenienttosolveequa-tion 5-4 usingtheLighthilltransformation57andthentransformingthesolutionforboundaryconditions 5-5 and 5-6 .TheboundaryconditionsforLighthilltransformationsaregivenascR=c0aty=0-7

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58 wherec0istheuniformsurfaceconcentrationofthereactantalongtheelectrodesurface,cR=c1aty!1-8andcR=c1atx=0-9Thesolutionofequation 5-4 usingtheLighthill'stransformationcanberepre-sentedbyNx=DRc1p Rxx )]TJ/F28 11.955 Tf 9.307 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 39DRxR0Rxp Rxxdx1=3-10whereNxisuxofthereactanttotheelectrode.Thecorrespondingmasstrans-ferlimitedcurrentisgivenbyix=nFDRc1)]TJ/F22 11.955 Tf 11.955 0 Td[(c0 )]TJ/F22 11.955 Tf 11.955 0 Td[(tRsR)]TJ/F28 11.955 Tf 9.307 9.683 Td[()]TJ/F20 7.97 Tf 6.675 -4.976 Td[(4 3p Rxx 9DRxR0Rxp Rxxdx1=3-11whereFistheFaraday'sconstant,nisthenumberofelectronstakingpartinthereductionofreactant,tRisthetransferencenumberofthespecies,andsRisthestoichiometriccoefcientofthereactant.Duhamel'stheorem58wasusedtotransformequation 5-11 fornonuniformsurfaceconcentrationgivenbyequation 5-5 andbulkconcentrationconditiongivenbyequation 5-6 .Theresultingintegralequationcanbeexpressedasix=)]TJ/F22 11.955 Tf 9.299 0 Td[(nFDRp Rxx )]TJ/F22 11.955 Tf 11.955 0 Td[(t+sR)]TJ/F28 11.955 Tf 9.306 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 326664xZ0dcx dxx=x0dx0 9DRxR0Rxp Rxxdx1=3 -12 +cRx=0)]TJ/F22 11.955 Tf 11.955 0 Td[(c1 9DRxR0Rxp Rxxdx1=337775

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59 Equation 5-12 providesasolutionofthemasstransfertotheelectrodeundernonuniformsurfaceconcentration.Thisapproachhasseveraladvantages,whichwillbeelucidatedlater.5.2.3ElectrodeKineticsThecurrentgeneratedduetoelectrode-reactantchargetransfercanbede-scribedempiricallybytheButler-Volmerequation.Thisequationrelatesthesur-faceoverpotentialstothecurrentbyix=i0cRx c1expazFsx RT)]TJ/F22 11.955 Tf 11.955 0 Td[(exp)]TJ/F22 11.955 Tf 10.494 8.087 Td[(czFsx RT-13wherei0istheexchangecurrentdensityforthebulkconcentrationofthereactant,aandcaretheanodicandcathodicchargetransfercoefcients,respectively,Risthegasconstant,Tisthetemperatureofthesystem,andZ=)]TJ/F22 11.955 Tf 9.299 0 Td[(z+z)]TJ/F22 11.955 Tf 7.085 1.793 Td[(=z+)]TJ/F22 11.955 Tf 12.125 0 Td[(z)]TJ/F15 11.955 Tf 7.085 1.793 Td[(forabinarysalt,andZ=)]TJ/F22 11.955 Tf 9.299 0 Td[(nforareactantwithexcesssupportingelectrolyte.ThetermcRx c1providesacorrectiontotheexchangecurrentdensityforsur-faceconcentrationofthereactant,wheretheconstantdependsonthekineticmechanismofthereaction.5.2.4ConcentrationOverpotentialTheconcentrationgradientacrossthemass-transferboundarylayerleadstoaconcentrationoverpotential.Ageneralformofconcentrationoverpotentialcanbeexpressedascx=RT ZFlncRx c1+tR1)]TJ/F22 11.955 Tf 13.15 8.088 Td[(cRx c1-14wherecxistheconcentrationoverpotentialasafunctionofx.Inthedevelop-mentofequation 5-14 ,theconcentrationvariationisassumedtobelinearwithinthediffusionlayer.Furthermore,theconductivityvariationsofelectrolyteareneg-ligible.

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60 5.2.5SolutionPotentialinOuterRegionIftherearenoconcentrationvariationandelectrolyteiselectricallyneutral,thepotentialofthesolutionsinthediffusepartofthesolutioncanbedescribedbyLaplace'sequationr2=0-15andthecurrentowingintheelectrolytesolutionisgivenbyi=)]TJ/F22 11.955 Tf 9.299 0 Td[(r-16whereistheelectricalconductivityoftheelectrolyte.Understeadystate,thecurrentcalculatedfromequation 5-16 wouldbalancethecurrentfromelectrodekineticsandmass-transport.Asolutionofequation 5-14 forthediskandhemi-sphericalgeometriesisdiscussedinthefollowingsections.DiskElectrodeAdiskelectrodeofradiusr0embeddedininnitelylargeinsu-latingplaneisconsideredhere.Thepotentialfarfromthediskcanbeassumedtobeequaltozero,i.e.,=0atz2+r2!1-17wherezandraretheaxialandcylindricalcoordinates,respectively.Thecurrentontheinsulatingplaneisequaltozero,hence,equation 5-16 yields@ @z=0atz=0;r>r0-18ThesolutionofLaplace'sequationsatisfyingaboveboundaryconditionscanbeexpressedintherotationalellipticalcoordinatesystem.Thecoordinatesystemisdenedasz=r0;r=r0p +2)]TJ/F22 11.955 Tf 11.955 0 Td[(2whereandarethecoordinateaxesfortherotationalellipticalsystem.Thelocalpotentialofthesolutionpotentialinrotationalellipticalcoordinatesystemcanbe

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61 expressedby=RT ZF1Xn=0BnP2nM2n-19Intheabsenceofconcentrationvariation,thepotentialisrelatedtocurrentden-sityiaccordingtoequation 5-16 ;hence,thecurrentattheelectrodecanalsobegivenbyi=)]TJ/F22 11.955 Tf 9.298 0 Td[(@ @zz=0=)]TJ/F22 11.955 Tf 15.23 8.088 Td[( r0@ @=0=)]TJ/F22 11.955 Tf 15.521 8.088 Td[(RT r0ZF1Xn=0BnP2nM02n-20ThecoefcientBnarecalculatedbyapplyingtheorthogonalitypropertyofLeg-endrepolynomials.Thus,Bn=r0ZF M2n0RT1Z0iP2nd-21Anexplicitexpressionforlocalsolutionpotentialasafunctionofpositionisob-tainedbysubstitutionofBnintoequation 5-19 .HemisphericalElectrodeAhemisphericalelectrodeofradiusr0embeddedinainnitelyinsulatingplaneisconsideredhere.Sphericalpolarcoordinatesad-equatelydescribethesystem.Thepotentialfarawayfromtheelectrodecanbeassumedtobeequaltozero.Thus,thepotentialboundaryconditionatr!1isgivenby=0-22Thecurrentattheinsulatingplaneisequaltozero;hence@ @=0at= 2-23Theaboveconditionwillalsobevalidat=0tosatisfytheconditionofsymmetry.ThesolutionofLaplace'sequation 5-15 ,subjectedtoboundaryconditions 5-22 and 5-23 ,canbeexpressedas=RT ZF1Xn=0BnP2ncosr0 r2n+1-24

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62 Applicationofequation 5-16 relatesthecurrentattheelectrodetothepotentialaccordingtoi=)]TJ/F22 11.955 Tf 9.299 0 Td[(@ @rr=r0=r0ZF RT1Xn=0BnP2ncosr0 r2n+1-25andthecoefcientBnareobtainedbyapplyingtheorthogonalitypropertyofLeg-endrepolynomials.Thus,theexpressionforBnisgivenbyBn=)]TJ/F22 11.955 Tf 10.494 8.087 Td[(ZF RTn+1 n+1 2Z0iP2ncossind-26Anexplicitexpressionforlocalsolutionpotentialasafunctionofpositionisob-tainedbysubstitutionofBninequation 5-24 .5.2.6ElectrodePotentialTheelectrodepotentialVwithrespecttoareferenceelectrodecanbeparti-tionedasV)]TJ/F15 11.955 Tf 11.955 0 Td[(ref=s+c+0-27whererefisthepotentialofareferenceelectrode,0isthesolutionpotentialneartheelectrodesurface,sandcarethesurfaceandconcentrationpotentials,respectively.Undertheassumptionthatthethicknessofthediffuselayerisnegli-gible,0canbeassumedtobetheelectrolytesolutionpotentialalongtheelectrodesurface.5.3DimensionlessQuantitiesThediskelectrodecanbeeasilydescribedincylindricalcoordinates.Thecurvi-linearcoordinaterelatestothecylindricalcoordinatesystemthroughthefollow-ingequations:x=r;y=z;Rx=r

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63 whererandzarethecylindricalcoordinateaxes.Thedimensionlessshearstressxisgivenbyx=r=cha3=2h 1=2r-28wherechisaconstant,obtainedbysolvingtheNavier-StokesandtheContinuityequationsforthediskelectrode,andahisthehydrodynamicconstant.Fortherotatingdiskelectrode,ahisreplacedbydiskrotationspeed,!.Thevalueofchis0:51023forarotatingdiskelectrode,and0:36023foradiskelectrodeundersubmergedjetimpingement.Aftersubstitutionofabovequantitiesinequation 5-12 ,anexpressionforcur-rentatthediskelectrodeisgivenasidiskr=)]TJ/F22 11.955 Tf 9.299 0 Td[(nFDch1=3 )]TJ/F22 11.955 Tf 11.955 0 Td[(tRsR)]TJ/F28 11.955 Tf 9.307 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 3 9D1=3r ah 24rrZ0dcr drr=xdr r3)]TJ/F22 11.955 Tf 11.955 0 Td[(x31=3 -29 +c1)]TJ/F22 11.955 Tf 11.955 0 Td[(cR#Adimensionlesscurrentidisk,alongwithadimensionlessconcentrationandpa-rameterNaregivenbythefollowingi=ir0ZF RT-30Cx=cRx c1-31N=)]TJ/F22 11.955 Tf 16.13 8.088 Td[(nZF2Dc1 RT)]TJ/F22 11.955 Tf 11.955 0 Td[(t+r r20ah 9DR1=3-32whereNismeasureofthemass-transferresistancetotheohmicresistance.Aftersubstitutionofabovequantitiesintoequation 5-29 ,thedimensionlesscurrentatthediskelectrodecanbewrittenasidiskr=Nch1=3 )]TJ/F28 11.955 Tf 9.307 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.976 Td[(4 324rrZ0dC drr=xdx r3)]TJ/F22 11.955 Tf 11.955 0 Td[(x31=3+C)]TJ/F15 11.955 Tf 11.955 0 Td[(135 -33 Equation 5-33 providesaconvenientmethodtocalculatetheatthediskelectrodesurfacebyspecifyingparameterNandsurfaceconcentrationdistribution.

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64 Forthehemisphericalelectrodesystem,thecurvilinearcoordinatesx,y,andRxarerelatedtothespherical-polarcoordinatesbyx=r;y=r;Rx=r0sinandthecorrespondingshearstressxisexpressedasx==a3=2hr0 1=2B-34AnanalyticalexpressionofBforstationaryhemisphericalelectrodeunderjetimpingementhasbeenobtainedinChapter 2 .Similarly,BfortherotatinghemisphericalelectrodehasbeenderivedinChapter 4 .Aftersubstitutionofabovecoordinaterelationshipsanddimensionlessquantitiesi,C,andNintoequation 5-12 ,anexpressionfordimensionlesscurrentisgivenby:ihemisphere=Np sinB )]TJ/F28 11.955 Tf 9.307 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 32666664Z0dC dx=0d0 R0sinp sinBd!1=3 -35 +C)]TJ/F15 11.955 Tf 11.955 0 Td[(1 R0sinp sinBd1=3377775Equation 5-35 providesaconvenientwaytocalculatethecurrentdistributionforagivensurfaceconcentrationdistributionatthehemisphericalelectrode.Equa-tions 5-33 and 5-35 canalsousedtocalculatesurfaceconcentrationdistributionforagivencurrentdistributionattheelectrodesurface.ThedimensionlessquantitiesJ,Es,Ec,andEaredenedas:J=i0r0ZF RT;Es=ZFs RT;Ec=ZFc RT;E=Es+EcwhereJrepresentstheratioofohmicresistanceofelectrolytetokineticresistance,Esisthedimensionlesssurfaceoverpotential,Ecisthedimensionless

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65 concentrationoverpotential,andEisthetotaldimensionlessoverpotential.Withtheintroductionofabovementioneddimensionlessquantities,equation 5-14 forconcentrationoverpotentialandequation 5-13 forelectrodekineticscanbeasrewrittenasEcx=Ex)]TJ/F22 11.955 Tf 11.955 0 Td[(Esx=logCx+tR)]TJ/F22 11.955 Tf 11.956 0 Td[(Cx-36andix=JC[expaEs)]TJ/F15 11.955 Tf 11.956 0 Td[(exp)]TJ/F22 11.955 Tf 9.298 0 Td[(cEs]-37respectively.Equations 5-36 and 5-37 canbecombinedtoeliminatethesurfaceoverpotentialEs.Theresultingequationcanbeexpressedasfollowingix=JC)]TJ/F23 7.97 Tf 6.587 0 Td[(aexpaE)]TJ/F22 11.955 Tf 11.956 0 Td[(atR)]TJ/F22 11.955 Tf 11.956 0 Td[(C)]TJ/F22 11.955 Tf 11.956 0 Td[(C+cexp)]TJ/F22 11.955 Tf 9.299 0 Td[(cE+ctR)]TJ/F22 11.955 Tf 11.955 0 Td[(C-38Equation 5-38 providesaconvenientwaytocalculatecurrentdistribution.Itsusefulnesswillbeelucidatedinthesubsequentsection.Thedimensionlesssolutionpotentialisdenedas=ZF RT-39Equation 5-19 canberecastusingthedimensionlesssolutionpotentialas=1Xn=0BnP2nM2n-40wherecoefcientsBnaredenedintermsofdimensionlesscurrentdistributionidiskasBn=1 M2n01Z0idiskP2nd-41Similarly,dimensionlesssolutionpotentialforhemisphericalelectrodecanbeex-pressedas=1Xn=0BnP2ncosr0 r2n+1-42

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66 wherecoefcientsBnaredenedintermsofdimensionlesscurrentdistributionihemisphereasBn=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(4n+1 2n+1 2Z0ihemisphereP2ncossind-43andthedimensionlesselectrodepotentialVcanbeexpressedbyV=Es+Ec+0-44Theobjectiveistoobtainthecurrentprolebelowthemass-transfer-limitedcur-rent.Thisinvolvesasimultaneoussolutionofmass-transport,electrodekinetics,andcalculationofelectriceldintheelectrolyte.Thedimensionlessformofgov-erningequationsdescribingmass-transport,electrodekinetics,andsolutionpo-tentialprovideaconvenientwaytocalculatethecurrentdistribution.Thedimen-sionlessparameterJandNaretheonlyvariablespresentinthedimensionlessformofthegoverningequations.Therefore,byalteringJandN,acurrentdistri-butionprolecanbeobtainedforthediskandthehemisphericalelectrode.5.4CalculationProcedureTheAlgorithmforcalculatingthecurrentandpotentialdistributionisoutlined.Theprocedurepresentedisapplicableforsubmergedelectrodesystemsunderjetimpingement.Thecalculationprocedurefordiskelectrodeispresentedrst.Sincethehydrodynamicmodelpredictstheseparationofboundarylayeratthehemi-sphericalelectrode,amodiedalgorithmispresentedinthesubsequentsubsec-tion.ThemathematicalmodelwasprogrammedusingFORTRANwithdoubleprecisionaccuracy.TheprogramlistingisgiveninAppendix E .5.4.1DiskelectrodeThefollowingprocedurecalculatedthecurrentdistributionatthediskelec-trode.

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67 1. ValuesofJandNwereassigned,andCwasassumedtohaveavaluebe-tween0.0and1.0atthecenterofthedisk.Alternatively,valuesofelectrodepotentialorcurrentlevelcanalsobechosen.Thisaddsanextrastepinthecalculationprocedure,whichiteratesontheCatthecenteroftheelec-trode. 2. Ther=r0domainwasdiscretizedinirregularlyspacedgridasoutlinedbyAcrivosetal.59 3. Thevalueofcurrentatr=0wascalculatedbyidisk=)]TJ/F15 11.955 Tf 10.494 8.087 Td[(1:57886437117488)]TJ/F22 11.955 Tf 11.955 0 Td[(CN )]TJ/F28 11.955 Tf 9.306 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 3-45Thisexpressionwasderivedbytakingthelimitofequation 5-33 atr=0. 4. ThevaluesofsurfaceoverpotentialEsandconcentrationoverpotentialEcwerecalculatedatr=r0=0usingequations 5-36 and 5-37 ,respectively. 5. Asaninitialguess,thevaluesofEs,Ec,andidiskalongeachpointattheelectrodesurfacewereassumedtobethesameasatr=r0=0.ThevaluesCr=r0wereobtainedfromequation 5-35 usingthemethoddevisedbyAcrivosetal.59 6. Thecurrentdistributionwascalculatedusingequation 5-38 inthedis-cretizeddomain. 7. ThecoefcientsBnforsolutionpotentialwerecalculatedfromequation 5-43 .Thenumberoftermsinthesummationwerelimitedto51.Additionaltermsinequation 5-43 didnotimprovethecalculatedsolutionpotential. 8. Thesolutionpotentialadjacenttotheelectrodesurfacewasobtainedbyequation 5-42

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68 9. Thevalueoftheelectrodepotentialatr=r0=0wasobtainedusingequation 5-44 10. AnewoverpotentialdistributionEwascalculatedusingEr=r0=Er=r0+V)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F22 11.955 Tf 11.955 0 Td[(Er=r0-46wherecanhaveavaluebetween0and1.Inthisprocedure,avalueof0:05waschosen. 11. TherelativepercentagedifferenceofcoefcientB0wasusedasterminationcriterion.TheB0representtheaveragedimensionlesscurrentattheelec-trodesurface.IfB0;new)]TJ/F22 11.955 Tf 12.369 0 Td[(B0;old=B0;oldwasfoundtobelessthan1:0)]TJ/F20 7.97 Tf 6.586 0 Td[(6,cal-culationwasterminated;otherwise,thecalculationprocedurewasrepeatedstartingfromstep 6 to 10 12. Thecalculatedcurrent,potential,andconcentrationdistributionwerewrit-tentothele.5.4.2HemisphericalelectrodeThealgorithmforhemisphericalelectroderequiredmodicationduetobound-arylayerseparation.Atthepointofseparation,thevalueofBiszero.ThemethoddevisedbyAcrivosetal.59predictsthatthecurrentwillalsobezeroatthepointofseparation.However,thenumericaldifcultycanbeavoidedbytermi-natingthecalculationsjustbeforethepointofseparation.Thecurrentdistributioncalculationswereperformedupto54.4,justbeforethepointofboundarylayerseparationpredictedtooccurat54.8bytheboundarylayertheory.Thefollowingmodiedprocedurewasusedtocalculatethecurrentdistributionatthehemi-sphericalelectrode.

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69 1. ValuesofJandNwereassigned,andCwasassumedtohaveavaluebe-tween0.0and1.0at=0.Valuesofelectrodepotentialorcurrentlevelcanalsobechosen.Thisaddsanextrastepinthecalculationprocedure,whichiteratesontheCat=0. 2. Thedomainwasdiscretizedfrom0to54.4inanirregularlyspacedgridasoutlinedbyAcrivosetal.59 3. Thevalueofcurrentat=0wascalculatedbyihemisphere=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(1:57886437117488)]TJ/F22 11.955 Tf 11.955 0 Td[(CN )]TJ/F28 11.955 Tf 9.306 9.684 Td[()]TJ/F20 7.97 Tf 6.675 -4.977 Td[(4 3-47Thisexpressionwasderivedbytakingthelimitofequation 5-35 at=0. 4. ThevaluesofsurfaceoverpotentialEsandconcentrationoverpotentialEcwerecalculatedat=0usingequations 5-36 and 5-37 ,respectively. 5. Asaninitialguess,thevaluesofEs,Ec,ihemisphereateachnodewasassumedtobesameasat=0.Theconcentrationdistributionwasobtainedusingequation 5-35 withthemethoddevisedbyAcrivosetal.59 6. Thecurrentdistributionwasobtainedusingequation 5-38 inthediscretizeddomain.Thecurrentbeyondthepointofboundarylayerseparationwasas-sumedtobeuniformlyconstant,andwasassignedtheobtainedvalueat=54:40. 7. ThecoefcientsBnforsolutionpotentialwerecalculatedusingequation 5-43 .Thenumberoftermsinthesummationwerelimitedto51. 8. Thesolutionpotentialadjacenttotheelectrodesurfacewasobtainedusingequation 5-42

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70 9. Thevalueoftheelectrodepotentialwasobtainedfromequation 5-44 at=0. 10. AnewoverpotentialdistributionEwascalculatedusingE=E+V)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F22 11.955 Tf 11.955 0 Td[(E-48wherecanhaveavaluebetween0and1.Inthisprocedure,avalueof0:02wasselected. 11. TherelativepercentagedifferenceofcoefcientB0wasusedasterminationcriterion.TheB0representtheaveragedimensionlesscurrentattheelec-trodesurface.IfB0;new)]TJ/F22 11.955 Tf 11.048 0 Td[(B0;old=B0;oldwasfoundtobelessthan1:0)]TJ/F20 7.97 Tf 6.587 0 Td[(6,calcu-lationswasterminated,otherwise,thecalculationprocedurewasrepeatedstartingfromstep 6 to 10 .5.5CurrentDistributionatDiskElectrodeTheprimary,secondary,andtertiarycurrentdistributionatthediskelectrodearepresentedinthissection.Thesimulationresultsforcurrentdistributionbelowthemass-transfer-limitedcurrentarediscussedbelow.5.5.1PrimaryDistributionIfconcentrationsareuniformandtheelectrodereactionsarefast,thenEsandEccanbesetequaltozerointhegoverningequations.Asaresult,thesolutionpotentialadjacenttotheelectrodewillbeequaltotheelectrodemetalpotentialandwillhaveauniformvalue.Thisconditionissatisedbyequation 5-40 forn=0,andtheresultingdistributionistheprimarycurrentandpotentialdistribution.ThesolutionpotentialPisgivenbyP=P01)]TJ/F15 11.955 Tf 13.758 8.088 Td[(2 tan)]TJ/F20 7.97 Tf 6.587 0 Td[(1-49

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71 whereP0isthesolutionpotentialattheelectrodesurface.ThesuperscriptPstandsforprimarydistribution.Thecurrentdistributionatthedisksurfacewasevaluatedfromequation 5-20 i=)]TJ/F22 11.955 Tf 9.299 0 Td[(@P @zz=0=2P0 p r20)]TJ/F22 11.955 Tf 11.955 0 Td[(r2-50thetotalcurrentisI=2r0Z0irdr=4r0P0-51andtheresistanceisRP=P0 IP=1 4r0-52Forconvenience,equation 5-50 isrecastintermsofaveragecurrentasi iavg=0:5 r 1)]TJ/F28 11.955 Tf 11.955 13.271 Td[(r r02-53wheretheaveragecurrentisdenedtobeiavg=I r20=4P0 r0-54Agraphofi=iavgasafunctionofr=r0ispresentedinFigure 5-2 .Thecurrentisfairlywellbehavednearthecenteroftheelectrode,butitapproachesinnityattheedgeoftheelectrode.Asaresult,theprimarycurrentdistributionishighlynon-uniformforthediskelectrode.5.5.2SecondaryCurrentDistributionAsecondarycurrentdistributionisaoutcomeofthebalancebetweenelectrodekineticrateandOhm'slaw.Forthiscase,theLaplace'sequationforsolutionpo-tentialissolvedwiththeButler-Volmerequationforelectrodekinetics,whichactsasaboundaryconditionforthecurrentattheelectrodesurface.ThecurrentgivenbyOhm'slawisequatedtothecurrentgeneratedattheelectrodesurfaceduetocharge-transferreactions.Thesecondarycurrentdistributionatthediskelectrode

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72 Figure5-2:Primarycurrentdistributionatthediskelectrode.Thevalueoflocalcurrentapproachestoinnityasr=r0!1 hasbeendiscussedindetailbyNewman.23Thekineticsfactorlimitsthevalueofcurrentattheelectrodeedge.ThenalvalueofcurrentdistributiondependsontheparameterJ.NewmanhasshownthatthecurrentdistributionbecomesuniformatJ=0:1.5.5.3TertiaryCurrentDistributionThecurrentdistributionbelowthemass-transfer-limitedvaluewasobtainedatthediskelectrodeusingthenumericalalgorithmpresentedintheprevioussection.Theuidmechanicscoefcientchwasusedforthediskelectrodeun-derjetimpingement.NumericalsimulationswerecarriedoutforJ=5andN=125.ThevaluesofCwereselectedfrom0.05to0.9.Thevaluesofa,c,,andtRwerekeptxedat0.5.Aplotofi=ilimdistributionispresentedinFigure 5.3a .Thenonuniformbehaviorofthecurrentcanbeseenasr=r0ap-proaches1.0.Thecurrenthasamaximumvalueatr=r0=1forallCexceptat0.5.Forthiscondition,themaximumoccursbeforetheperipheryoftheelec-trode.ThisobservationisconsistentwithresultsreportedbyDurbha.6Itcan

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73 beexplainedasfollowing:Asunreactedreactantmaterialmovesalongtheelectrosurfaceinthemass-transferboundarylayer,theconcentrationgradient,i.e.,@C=@zbuildsup.Thiscauses@C=@ztobehigherbeforetheperiphery.Therefore,amax-imumincurrentisseenbeforer=r0=1.Thedimensionlessconcentrationofthereactantattheelectrodesurfaceasafunctionofr=r0ispresentedinFigure 5.3b .Thedimensionlesssolutionpotential0isplottedinFigure 5.3c .ThesolutionpotentialdistributionbecomesuniformforC=0.9.AparallelsetofcurrentdistributioncalculationsforC=0.4,0.3,0.2,0.1,and0.05arepresentedinFigure 5.4a .ThecorrespondingconcentrationdistributionisshowninFigure 5.4b .ThedimensionlesssolutionpotentialalongtheelectrodesurfaceispresentedinFigure 5.4c .Thefollowingexpressionwasusedtoquantifytheuniformityofcurrentdis-tributiondisk=vuuuuut r0R0i ir=0)]TJ/F15 11.955 Tf 11.955 0 Td[(12rdr r0R0rdr-55wherediskrepresentstheuniformityparameterforcurrentdistributionatthediskelectrode.Thequantitydiskwascalculatedforthecurrentdistributionde-scribedinFigures 5-3 and 5-4 .TheobtainedvaluesarelistedinTable 5.1 .ThevalueofdiskisminimumforCequalto0.05,andmaximumfor0.7.Theval-uesofiavg=ilimandir=0=ilimarealsogiveninTable 5.1 .Agraphof1)]TJ/F22 11.955 Tf 12.683 0 Td[(ir=0=ilimasafunctionofdiskfordifferentvalueofCispresentedinFigure 5-5 .Thisplotshowsamonotonicallyincreasingrelationshipbetween1)]TJ/F22 11.955 Tf 10.894 0 Td[(ir=0=ilimanddiskuptoC=0.7.Theratioi=iavgasafunctionofr=r0fordifferentvaluesofCispresentedinFigure 5-6 .ThecurrentdistributionismostuniformforC=0.05asseeninFigure 5-6 ,andtheaveragecurrentisabout98.9%ofmass-transfer-limitedcurrent.Thediskis0.021forC=0.05.Thisvalueofdiskischosen

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74 a b cFigure5-3:Calculatedcurrent,concentration,andsolutionpotentialdistributionatthediskelectrode.ThesimulationsweredoneforJ=5,N=125,andC=0.5to0.9inincrementalstepsof0.1.ai=ilimasafunctionofr=r0.bDimen-sionlessconcentrationdistributionasafunctionofr=r0.cDimensionlesssolutionpotentialattheelectrodesurfaceasafunctionofr=r0.

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75 a b cFigure5-4:Calculatedcurrent,concentration,andsolutionpotentialdistributionatthediskelectrode.ThesimulationsweredoneforJ=5,N=125,andC=0.4,0.3,0.2,0.1,0.05.ai=ilimasafunctionofr=r0.bDimensionlessconcentra-tiondistributionasafunctionofr=r0.cDimensionlesssolutionpotentialattheelectrodesurfaceasafunctionofr=r0.

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76 Table5.1:Calculatedvaluesforuniformityparameterdiskseeequation 5-55 ,iavg=ilim,andir=0=iavgforthecurrentdistributionspresentedinFigures 5-3 and 5-4 .ThevaluesofJandNwas5and125,respectively. C disk iavg=ilim ir=0=iavg 0.00 0.000 1.000 1.0000.05 0.021 0.989 0.9600.10 0.044 0.975 0.9220.20 0.097 0.937 0.8540.30 0.161 0.879 0.7960.40 0.235 0.797 0.7520.50 0.312 0.690 0.7250.60 0.376 0.563 0.7110.70 0.392 0.422 0.7100.80 0.282 0.266 0.7570.90 0.238 0.128 0.782 asconditionofuniformity.Therefore,currentdistributionswithdisk<0.021areuniform.5.6CurrentDistributionatHemisphericalElectrodeTheprimary,secondary,andtertiarycurrentdistributionatthestationaryhemi-sphericalelectrodeundersubmergedjetimpingementarediscussedinthissec-tion.5.6.1PrimaryDistributionIfnoconcentrationvariationsexistinthesystemandreactionkineticsisnotalimitingfactor,thesolutionpotentialisaresultofequation 5-24 equationforn=0.ItisgivenbyP=P0r0 r-56andthecurrentisgivenbyi=)]TJ/F22 11.955 Tf 9.299 0 Td[(@ @rr=r0=P0 r0-57ThetotalcurrenttothehemisphereisexpressedasI=2r0P0-58

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77 Figure5-5:1)]TJ/F22 11.955 Tf 11.955 0 Td[(ir=0=iavgasafunctiondiskfordifferentvaluesofC. Figure5-6:i=iavgasafunctionr=r0fordifferentvaluesofC.

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78 Consequently,theresistancetotheowofcurrentisRpsol=P0 I=1 2r0-59TheRpsolprovideavalueofsolutionresistancetothecurrentowfortheprimarycurrentdistribution.5.6.2SecondaryDistributionWhentherateofmass-transfertotheelectrodeisinnite,andslowelectrodekineticsistakenintoaccount,theresultingcurrentcalculationiscalledassec-ondarydistribution.Intheabsenceofmass-transferresistance,thesecondarycurrentdistributioninhemisphericalgeometrywouldyieldauniformcurrentthroughouttheelectrodesurface.ThiscanbeeasilydeducedbysolvingtheButler-VolmerequationfortheelectrodekineticswiththeLaplace'sequationfortheso-lutionpotentialsimultaneously.5.6.3TertiaryDistributionThecurrentdistribution,dimensionlesssurfaceconcentration,andsolutionpo-tentialbelowthemass-transferlimitedconditionswereobtainedforthestationaryhemisphericalelectrodeundersubmergedjetimpingement.SeveralsimulationswerecarriedoutforvariousvaluesJ,N,andC.TheparameterCwasvar-iedbetween0.5and0.9inincrementalstepsof0.1foreachvalueofJandN.Thevaluesofa,c,,andtRwerekeptxedat0.5.ThecalculatedcurrentdistributionsforfourvaluesofN,50,20,and5,andC=0.5,0.6,0.7,0.8,0.9,andforaxedvalueJ=5aregiveninFigure 5.7a .ComparisonofFigures 5.7a to 5.7d showthatthedistributionofcurrentbecomesfairlyuniformforthepoleconcentrationofC=0.9.SimulationresultsinFigure 5.7a forC=0.5,0.6,0.7,and0.8displayamaximumincurrent.Themaximumisobservedatthevaluesbetweenthepoleandthepointofboundary

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79 layerseparation.ThisfeatureofthecurrentdistributionstartstodiminishasthevalueofNischangedtoalowerquantity.Furthermore,themaximumincurrentdisappearsforN=5asshowninFigure 5.7d .ThedimensionlessconcentrationasafunctionofforparametersJ=5andN=125,50,20,and5areshowninFigure 5-8 .TheresultscorrespondtothecurrentdistributioninFigure 5-7 .ThedimensionlesssolutionpotentialalongtheelectrodesurfaceasafunctionofforJ=5andN=125,50,20,and5arepresentedinFigure 5-9 .ThecurrentdistributionresultsforN=20andJ=100,10,1,and0.1arepresentedinFigures 5.10a to 5.10d .ThecurrentdistributionremainsuniformforC=0.9regardlessofJvalues,asseeninFigure 5.10d .SimulationresultsinFigure 5.10a forC=0.5,0.6,0.7,and0.8,J=100,andN=20displayamaximumincurrentatthevaluesbetweenpoleandpointofboundarylaterseparation.ThisfeatureofcurrentdistributiondiminishesforJ=10,andN=20asshowninFigure 5.10b ,andremainsunchangedasvaluesofJarefurtherdecreased.ThecurrentdistributionforparametersN=20andJ=1;0:1areshowninFigures 5.10c and 5.10d ,respectively.ThedimensionlessconcentrationasafunctionforN=20andJ=100,10,1,and0.1arepresentedinFigure 5-11 .TheresultsinFigure 5-11 correspondstothecurrentdistributioninFigure 5-10 .ThedimensionlesssolutionpotentialalongtheelectrodesurfaceasafunctionforN=20andJ=100,10,1,and0.1aregiveninFigure 5-12 .Thefollowingexpressionwasusedtoquantifytheuniformityofcurrentdis-tribution:hs=vuuut R 20i i)]TJ/F15 11.955 Tf 11.955 0 Td[(12sind R 20sind-60wherehsrepresentstheparameterofuniformityforthecurrentdistributionat

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80 a b c dFigure5-7:Calculatedcurrentdistributionasafunctionatthestationaryhemi-sphericalelectrodeundersubmergedjetimpingement.ThesimulationweredonefordifferentvaluesofpoleconcentrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5.

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81 a b c dFigure5-8:Calculatedconcentrationprolecorrespondingtothecurrentdistribu-tionpresentedinFigure 5-7 asafunctionatthestationaryhemisphericalelec-trodeundersubmergedjetimpingement.ThesimulationweredonefordifferentvaluesofpoleconcentrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5.

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82 a b c dFigure5-9:CalculatedvaluesofsolutionpotentialcorrespondingtothecurrentdistributionpresentedinFigure 5-7 asafunctionofatthestationaryhemispher-icalelectrodeundersubmergedjetimpingement.ThesimulationweredonefordifferentvaluesofpoleconcentrationsC,andparametersJandN.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=125andJ=5,bN=50andJ=5,cN=20andJ=5,anddN=5andJ=5.

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83 a b c dFigure5-10:Calculatedcurrentdistributionasafunctionofatthestationaryhemisphericalelectrodeundersubmergedjetimpingement.ThesimulationweredoneforN=20,anddifferentvaluesofpoleconcentrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1.

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84 a b c dFigure5-11:Calculatedconcentrationprolecorrespondingtothecurrentdistri-butionpresentedinFigure 5-10 asafunctionofatthestationaryhemisphericalelectrodeundersubmergedjetimpingement.ThesimulationwerecarriedoutforN=20,anddifferentvaluesofpoleconcentrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1.

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85 a b c dFigure5-12:CalculatedvaluesofsolutionpotentialcorrespondingtothecurrentdistributionpresentedinFigure 5-10 asafunctionofatthestationaryhemispher-icalelectrodeundersubmergedjetimpingement.ThesimulationwerecarriedoutforN=20,anddifferentvaluesofpoleconcentrationsCandparametersJ.Theverticaldashlinerepresentthepointofboundarylayerseparation.aN=20andJ=100,bN=20andJ=10,cN=20andJ=1,anddN=20andJ=0:1.

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86 Table5.2:Valuesofuniformityparameterhsseeequation 5-60 forforstation-aryThecalculatedvaluesareforhemisphericalelectrodeunderjetimpingement.thecurrentdistributionspresentedinFigure 5-7 .ParameterJwasxedat5. hs C N=5 N=20 N=50 N=125 0.5 0.164 0.123 0.189 0.2580.6 0.122 0.078 0.158 0.2630.7 0.087 0.055 0.067 0.1120.8 0.061 0.025 0.023 0.0410.9 0.040 0.014 0.008 0.008 thehemisphericalelectrode.Thequantityhswascalculatedforthecurrentdis-tributionpresentedinFigures 5-7 and 5-10 .ThevaluesofhsareprovidedinTable 5.2 forJ=5andN=5,20,50,and125.ItisobservedthathsreachesaminimumforC=0.9andamaximumatforC=0.5.ForinbetweenvaluesofC,thevalueofhsdecreasesmonotonicallyforincreasingC.AsseeninFigure 5-7 ,thecurrentprolesdisplayuniformityforallNandC=0.9.Thedis-tributionstartstobecomenonuniformatC=0.8,andnon-uniformityincreaseforincreasingC.Usingtheconditionofuniformityforthediskelectrode,disk=0.021,hs=0.02becomestheconditionofuniformcurrentdistributionforthehemisphericalelectrode.Bytheapplicationoftheuniformitydenition,thecurrentdistributionpresentedinFigure 5-7 isuniformforC=0.9andN=20,50,and100.Theratioiavg=ilimavgrepresentstheaveragecurrentwithrespecttototalmass-transfer-limitedcurrentattheelectrodesurface.Thevaluesofiavg=ilimavgforJ=5andN=5,20,50and125aregiveninTable 5.3 .Theaveragecurrentlevelisabout25%ofmass-transferlimitedcurrentforC=0.9.Aone-to-onecom-parisonofTable 5.2 and 5.3 showsthatthecurrentdistributionbecomeuniformatabout25%ofmass-transfer-limitedcurrent.Similarly,thecalculatedvaluesofhsforN=20andJ=0.1,1,10and100are

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87 Table5.3:Valuesofiavg=ilimavgforstationaryhemisphericalelectrodeunderjetimpingement.ThecalculatedvaluesareforthecurrentdistributionspresentedinFigure 5-7 .ParameterJwasxedat5. iavg=ilimavg C N=5 N=20 N=50 N=125 0.5 0.831 0.892 0.896 0.9020.6 0.743 0.840 0.867 0.8520.7 0.618 0.713 0.774 0.7750.8 0.451 0.511 0.537 0.5440.9 0.242 0.268 0.275 0.282 Table5.4:Valuesofhsseeequation 5-60 forstationaryhemisphericalelectrodeunderjetimpingement.ThecalculatedvaluesareforthecurrentdistributionspresentedinFigure 5-10 .ParameterNwasxedat20. hs C J=0:1 J=1 J=10 J=100 0.5 0.121 0.121 0.126 0.1730.6 0.075 0.075 0.085 0.1650.7 0.043 0.043 0.052 0.1420.8 0.028 0.035 0.031 0.1010.9 0.021 0.019 0.017 0.067 givenTable 5.4 .Thecorrespondingvaluesofiavg=ilimavgarelistedinTable 5.5 .AsseeninFigure 5-10 ,thecurrentdistributionsexhibituniformityforC=0.9forallvaluesofJ.However,thedenitionofuniformityindicatesthatthecurrentproleisuniformforJ=0.1,1,and10. Table5.5:Valuesofiavg=ilimavgforstationaryhemisphericalelectrodeunderjetimpingement.ThecalculatedvaluesareforthecurrentdistributionspresentedinFigure 5-10 .ParameterNwasxedat20. iavg=ilimavg C J=0:1 J=1 J=10 J=100 0.5 0.892 0.892 0.892 0.8940.6 0.837 0.839 0.842 0.8430.7 0.708 0.709 0.719 0.7360.8 0.507 0.499 0.516 0.5350.9 0.267 0.267 0.269 0.277

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88 a bFigure5-13:CurrentdistributioncalculationspresentedinthepaperbyNisanciogluandNewman.aFigure6ofthepaperbyNisanciogluetal.bFigure2ofthepaperbyNisanciogluetal. 5.7CurrentDistributionontheRotatingHemisphericalElectrodeNisanciogluandNewman18haveaddressedtheissueofcurrentdistributiononarotatingsphericalelectrode.Theyreportedthatcurrentdistributionbecomesuniformatabout68%ofmass-transfer-limitedcurrentatthehighrotationspeed.ThecalculatedcurrentdistributionbyNisanciogluetal.ispresentedinFigure 5.13b .Inthisgure,thevalueofNisinnite,andthecurrentdistributioninonlycontrolledbythemass-transfertotheelectrode.AnothersetofcurrentdistributioncalculationsbyNisanciogluetal.arepresentedinFigure 5.13b .ThesecalculationweredoneforJ=0andN=10.Asthevalueoftotalaveragecurrentincreases,thecurrentdistributionbecomesmorenonuniform.AsseeninFigure 5.13b ,thecurrentdistributionismostuniformat26.6%ofmass-transfer-limitedcurrent.NisanciogluandNewmanrestrictedtheircalculationsforcaseofinniteSchmidtnumber.Inthepresentwork,currentdistributioncalculationshavebeenperformedforniteSchmidtnumber.Thegoverningequations,calculationalgo-rithm,andresultsarepresentednext.

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89 5.7.1GoverningEquationsThemethodologysuggestedbyNisancioglu18toformulateconvective-diffusionintermsofaSchlichtingintegral23seeequation 5-35 cannotbeappliedtoac-countforniteScnumber.However,aseriessolutionforreactantconcentration,whichexplicitlyaccountforScnumberismoreappropriate.ThismethodologyissimilartotheonepresentedinChapter 3 equation 3-5 ,wheredimension-lessconcentrationwasexpandedintwoterms.Followingthat,thedimensionlessreactantconcentrationasafunctionandycanbeexpressedasc1)]TJ/F22 11.955 Tf 11.955 0 Td[(cR c1=nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(21;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1y+Sc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3nXi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(22;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1y!-61wherethersttermontherighthandsideprovidesthesolutioninnitelylargeSchmidtnumberSc,andthesecondtermprovidesacorrectionforanitevalueofSc.Thecorrespondingdimensionlesscurrentattheelectrodesurfaceisgivenbyi=91=3NnXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(201;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1+Sc)]TJ/F21 5.978 Tf 7.782 3.259 Td[(1 3nXi=12i)]TJ/F20 7.97 Tf 6.587 0 Td[(202;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1!-62whereprimedenotestherstorderderivativewithrespecty.Thenumberoftermsinthesummationswerelimitedtoten.Aftersubstitutionsofdimensionlessconcentrationinconvective-diffusionequation,tenordinarydifferentialequationsfor1;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1werederived.TheobtainedequationsweresolvedusingtheBANDalgorithmwithappropriateboundaryconditions,andtheirsolutionyieldedthecurrentdistributionattheelectrodesurface.ThegoverningequationsforkineticsisrepresentedbyButler-Volmer,whereas,thesolutionspotentialisrepresentedbyLaplace'sequation.5.7.2NumericalProcedureAnalgorithmforcalculatingthecurrentdistribution,concentrationdistribu-tion,andsolutionpotentialalongtheelectrodesurfaceisoutlinedbelow.The

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90 algorithmwascomputationallyintensiveastwentyordinarycoupleddifferentialequationsweresoledforeachiteration. 1. ValuesofJandNwereassigned,Cwasassumedtohaveavaluebetween0.0and1.0at=0.Valuesofelectrodepotentialorcurrentlevelcanalsobechosen.Thisaddsanextrastepinthecalculationprocedure,whichiteratesontheCatthecenteroftheelectrodetoreachtheassignedvalueoftheelectrodepotentialorthespeciedcurrentlevel. 2. Asaninitialguess,amonotonicallydecreasingpositiveconcentrationprolewasassumedasshownbelow:C=C)]TJ/F22 11.955 Tf 11.955 0 Td[(a2-63ThesmallvalueofawaschosensothatCremainspositiveoverentireelectrodesurface. 3. Thedomainwasdiscretizedfrom0to90inequallyspacedgrid. 4. Aninitialcurrentdistributionwasobtainedbysolvingtheconvectivediffu-sionequationsfor1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1.Thedifferentialequationsweresolvedfordifferentmeshsizes.Therstderivativesof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1and2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1werecal-culatedbyextrapolationtozeromeshsize. 5. ValuesofsurfaceoverpotentialEs,andconcentrationoverpotentialEcwereobtainedateachnodeindomain. 6. Thesolutionpotentialadjacenttotheelectrodesurfaceforthecalculatedcurrentinthestep 4 wasdetermined. 7. AnewoverpotentialdistributionateachwasEwascalculatedusingthefollowingexpression:E=E+V)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F22 11.955 Tf 11.956 0 Td[(E-64

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91 wherecanhaveavaluebetween0and1.Inthisprocedure,asmallvalueof0:02wasselected. 8. Thesurfaceconcentrationdistributionusingequation 5-38 bytheNewton-Raphson60methodwasdetermined. 9. Apolynomialinwasttedtotheconcentrationproleobtainedinthepre-viousstep.ThefunctionalformofthepolynomialisgivenbyC=1)]TJ/F28 11.955 Tf 11.955 20.443 Td[(10Xi=12i)]TJ/F20 7.97 Tf 6.586 0 Td[(21;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1!-65where1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1arethecoefcientsofthepolynomialattheelectrodesur-face.Thevaluesofthesecoefcientsactasboundaryconditionsforconvective-diffusionequationsfor1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1r.Itshouldnotedthattheregressedvaluesofthecoefcients1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1mustbestatisticallysignicant.Ifthecondenceintervalsof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1includedzero,1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1weresettozero. 10. Thecurrentdistributionwasobtainedbysolvingconvective-diffusionequa-tionsfor1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1rand2;2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1r,andsolutionpotentialwascalculatedalongtheelectrodesurfacefortheobtainedcurrentdistribution. 11. TherelativepercentagedifferenceofcoefcientB0wasusedasterminationcriterion.TheB0representtheaveragedimensionlesscurrentattheelec-trodesurface.IfB0;new)]TJ/F22 11.955 Tf 11.738 0 Td[(B0;old=B0;oldwaslessthan1:0)]TJ/F20 7.97 Tf 6.586 0 Td[(6,calculationswerestopped,otherwise,thecalculationprocedurewasrepeatedstartingfromstep 5 to 10 .AFORTRANimplementationofthealgorithmispresentedinAppendix F .5.7.3ResultsThecurrentdistributionwascalculatedforJ=5,andN=125.ThevalueofSchmidtnumberwasxedat1000.Thecalculatedresultsarepresentedin

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92 Figure 5-14 .TheblackcoloredlinesintheFigure 5-14 correspondstocalcula-tionwithinniteScnumber,andgreycoloredlinescorrespondstoSc=1000.ThecurrentdistributionispresentedinFigure 5.14a fordifferentvaluesofthepoleconcentrationsC.TheSchmidtnumbercorrectionlowersthevalueofcur-rentalongthesurfaceasshowninFigure 5.14a .Thecorrespondingdimension-lesssurfaceconcentrationdistributionispresentedinFigure 5.14b .Thecon-centrationproleisslightlyhigherwithSchmidtnumbercorrection.Thedimen-sionlesssurfaceoverpotential,andconcentrationoverpotentialarepresentedinFigures 5.14c and 5.14d ,respectively.Thecalculateddimensionlesssolutionpo-tentialalongtheelectrodesurfaceforinniteScnumberispresentedinFigure 5.15a .TheresultswithSccorrectionaregiveninFigure 5.15b .TheseresultsshowthattheScnumbercorrectionlowersthecurrentalongtheelectrodesurface.TheeffectofSccorrectionismoresignicantathighercurrentlevels.ValuesofhsarelistedinTable 5.6 .UniformityparameterhsmonotonicallydecreasesforincreasingvaluesofC.TheconditionofuniformityisachievedatC=0.7forbothsimulationwithScnumbercorrectionandwithSc=1.Valuesofiavg=ilimavgarelistedinTable 5.6 .iavg=ilimavgisabout0.4forC=0.7,hence,currentdistributionattherotatinghemisphericalelectrodebecomesuniformatabout40%ofmasstransferlimitedcurrent.5.8SummaryAgeneralizedmathematicalmodelforcalculatingcurrentandpotentialdistri-butionforbothstationarydiskandhemisphericalelectrodesunderjetimpinge-mentwasdeveloped.Forhemisphericalelectrode,thecurrentdistributionintheseparatedpartoftheboundarylayerforhemisphericalelectrodewasapproxi-matedwithauniformcurrent.Thecurrentvalueatthepointofseparationwas

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93 a b c dFigure5-14:Calculatedvaluesofcurrentdistribution,concentrationdistribution,surfaceoverpotential,andconcentrationoverpotentialasafunctionofattherotatinghemisphericalelectrode.Thelinesinblackcolorcorrespondstothecal-culationsforinniteSchmidtnumber,andlinesinbluecolorcorrespondstocal-culatedresultswithSc=1000.0.ThesecalculationswereperformedforJ=5andN=125.aCurrentdistributionasafunctionof,bDimensionlessConcen-trationdistributionasafunctionof,cDimensionlesssurfaceoverpotentialasafunctionof,dDimensionlessconcentrationoverpotentialasafunctionof.

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94 a bFigure5-15:CalculateddimensionlessSolutionpotentialalongtheelectrodesur-faceasafunctionof.TheresultscorrespondstothecurrentdistributionsgiveninFigure 5.14a .aDimensionlesssolutionpotentialwithoutSchmidtnumbercorrection,bDimensionlesssolutionpotentialwithSchmidtnumbercorrection. Table5.6:Valuesofiavg=ilimavgandhsfortherotatinghemisphericalelectrode.ThecalculatedvaluesareforthecurrentdistributionspresentedinFigure 5.14a .ParametersNandJwasxedat125and5,respectively. Sc=1 Sc=1000 C hs iavg=ilimavg hs iavg=ilimavg 0.5 0.035 0.668 0.035 0.6670.6 0.023 0.537 0.023 0.5360.7 0.016 0.404 0.016 0.4030.8 0.011 0.271 0.012 0.2700.9 0.007 0.135 0.007 0.135

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95 assumedtobeapplicableintheseparatedpartoftheboundarylayer.Aniterativecalculationalgorithmwasdevisedforsolvingthegoverningequationsforcurrentdistribution.Theresultspresentedindicatethatthecurrentdistributionbecomesuniformatabout25%ofmass-transferlimitedcurrent.Likewise,anothermathematicalmodelwaspresentedtoestimatetheeffectofScnumbercorrectiononthecurrentdistributionattherotatinghemisphericalelec-trode.Again,thegoverningequationsweresolvedwithaniterativescheme.TheresultsshowthatcurrentdistributionwieldsasignicanteffectinthepresenceofniteScnumber.

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CHAPTER6VALIDATIONOFTHEMEASUREMENTMODELCONCEPTThischapterpresentsacomparativestudyofdifferentformsofmeasurementmodelstoestimatestochasticerrorcontributionintheimpedanceandtoassesconsistencyofthedatawithKramers-Kronigrelations.Voigtelement,Constantphaseelement,andtransferfunctionbasedimpedancemodelswereconsideredasthecandidatemeasurementmodels.Thedevelopmentofregressionprocedure,keyconcepts,andestimationofstochasticerrorswithmeasurementmodelsispre-sentedinthischapter.AcomparativeKramers-KronigconsistencychecksoftheimpedancedataweredonewithVoigtelementandtransferfunctionbasedmea-surementmodels.6.1IntroductionInterpretationofspectroscopydataisfacilitatedbyaquantitativeanalysisofthemeasurementerrorstructure.61Knowledgeoftheerrorstructurehasbeenshowntoallowenhancedinterpretationoflightscatteringmeasurementsintermsofparticlesizedistributionandparticleclassication.62,63Theerroranalysisap-proachhasbeensuccessfulforlightspectroscopymeasurementsbecausetheshorttimerequiredforsuchmeasurementsminimizesnonstationarycontributionstoreplicatedspectra.Inaddition,opticalmeasurementsgenerallysatisfytheKramers-Kronig64,65relations,thusallowingextensionofthemeasurablerangeofspectra.63Incontrast,thestochasticcontributiontotheerrorstructureofelectrochemicalimpedancespectroscopymeasurementscannotbeobtaineddirectlyfromrepli-catedmeasurementsbecausetheinherentlynon-stationarycharacterofelectro-chemicalsystemsintroducesanon-negligibletime-varyingbiascontributionto 96

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97 Figure6-1:AschematicrepresentationofaVoigtelementmeasurementmodel. theerror.66Inaddition,thecomparativelylongtimerequiredforimpedancemea-surementsfrequentlyintroducesnon-stationarycontributionstoimpedancespec-trawhichviolatetheKramers-Kronigrelations.Agarwaletal.31haveintroducedameasurementmodelasameanstoidentifytheerrorstructureofimperfectlyreplicatedimpedancedata.Agarwaletal.pro-posedthatageneralizedVoigtmodelcouldbeusedtolterthereplicationerrorsofimpedancedatainordertodistinguishbetweenstochasticorrandomerrorsanddeterministicerrorscaused,forexample,bysystematicchangesinsystemproperties.ThemeasurementmodelproposedbyAgarwaletal.31wascomposedofVoigtelementsinserieswithasolutionresistance,asshowninFigure 6-1 ,i.e.,Z=R0+KXk=1Rk 1+j!k-1Withasufcientnumberofparameters,theVoigtmodelwasabletoprovideastatisticallysignicantttoabroadvarietyofimpedancespectra.31AsthemeasurementmodelwaschosentobeconsistentwiththeKramers-Kronigrelations,Agarwaletal.31showedthatitcouldalsobeusedtoallowacheckofconsistencyofdatawithoutexplicitintegrationoftheKramers-Kronigrelations.Auniquefeatureoftheapproachwasthattheweightingstrategymadeuseofthemeasuredstochasticerrorstructure;thustheevaluationofconsistencywasconductedwithinthecontextofanoverallerroranalysis.Theconceptofusingcircuitmodelstocheckforconsistencyofdatawasdescribedmuchearlier

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98 byBrachmanandMacdonald67,68andlaterbyBoukampandMacdonald69andbyBoukamp.70Theseapproaches,however,didnotincludeacomprehensiveerroranalysis.TheconceptproposedbyAgarwaletal.31wasthatthemeasurementmodelconsistsofasuperpositionofline-shapeswhichcanbearbitrarilychosensubjecttotheconstraintsthattheparameterestimateswerestatisticallysignicantandthatthemodelsatisedtheKramers-Kronigrelations.Therefore,othertransferfunctionmodelscanbeconsidered.Forexample,equation 6-1 canbegeneralizedtoallowConstant-Phase-Elementbehavior,i.e.,Z=R0+NXk=1Rk 1+j!1)]TJ/F23 7.97 Tf 6.586 0 Td[(kk-2ThismeasurementmodelisKramers-Kronigconsistentprovidedkliesbetweenzeroandone.71Pauwelsetal.34haveproposedthatatransferfunctionformulationZ=PMk=0bkj!nk PPm=0amj!nm-3thatmaybeamoreparsimoniousmodelforcertainclassesofimpedancemeasure-ments.Pauwelsetal.34notedinparticularthat,forn=1=2,equation 6-3 yieldsafrequencydependencethatmaybeparticularlyusefulformodelingdatainu-encedbydiffusionprocesses.Ifn=1,equation 6-3 showsafrequencybehaviorsimilartothatofaVoigtmodelequation 6-1 .Theintroductionofnewmeasurementmodelsintroducesthepotentialtoeval-uatesomeofthekeyassumptionsofthemeasurementmodelapproach.Theobjectiveofthepresentworkwastoascertainthatthestochasticerrorstructureobtainedusingthemeasurementmodelparadigmisindependentoftheformofmeasurementmodelused.ThesecondobjectivewastoassessthesuitabilityofthedifferentmeasurementmodelsforassessingconsistencyofdatawiththeKramers-Kronigrelations.

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99 6.2DenitionofErrorsTheerrorsinanimpedancemeasurementcanbeexpressedintermsofthedifferencebetweentheobservedvalueZ!andamodelvaluebZ!as"res!=Z!)]TJ/F28 11.955 Tf 14.045 3.022 Td[(bZ!="t!+"bias!+"stoch!-4where"resrepresentstheresidualerror,"t!isthesystematicerrorthatcanbeattributedtoinadequaciesofthemodel,"bias!representsthesystematicexperi-mentalbiaserrorthatcannotbeattributedtomodelinadequacies,and"stoch!isthestochasticerrorwithexpectationEf"stoch!g=0.Adistinctionisdrawn,followingAgarwaletal.,31betweenstochasticerrorsthatarerandomlydistributedaboutameanvalueofzero,errorscausedbythelackoftofamodel,andexperimentalbiaserrorsthatarepropagatedthroughthemodel.Theexperimentalbiaserrors,assumedtobethosethatcauselackofconsistencywiththeKramers-Kronigrelations,72,65,64maybecausedbynonsta-tionarityorbyinstrumentalartifacts.Theproblemofinterpretationofimpeda-ncedataisthereforedenedasconsistingoftwoparts:oneofidenticationofexperimentalerrors,whichincludesassessmentofconsistencywiththeKramers-Kronigrelations,andoneoftting,whichentailsmodelidentication,selectionofweightingstrategies,andexaminationofresidualerrors.Theerroranalysisprovidesinformationthatcanbeincorporatedintoregressionofprocessmodels.6.3EquivalenceofMeasurementModelsThethreemeasurementmodelsdifferwitheachotherintermsoftheirparam-eters.However,undercertainconditionsaparametricequivalencecanbeestab-lishedbetweenthemodels.Forexample,Constant-Phase-ElementCPEisagen-eralizedformofVoigtelement;hence,fork=0,CPEmeasurementmodelissame

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100 asVoigtmodel.AsimilarequivalencecanbeformedbetweenConstant-Phase-Elementbasedandtransferfunctionbasedmeasurementmodels.ForN=M=Pand=n=0:5intheequation 6-2 and 6-3 ,followingrelationshipsbetweentheparametersofbothmeasurementmodelscanbededuced:a0=1a1=nXi=1ia2=NXi=1NXj=1iji6=ja3=NXi=1NXj=1NXk=1ijki6=j6=k...b0=NXi=1Rib1=R0a1+NXi=1NXj=1Riji6=jb2=R0a2+NXi=1NXj=1NXk=1Rijki6=j6=kb3=R0a3+NXi=1NXj=1NXk=1NXl=1Rijkli6=j6=k6=l...ThisanalysishasbeenlimitedtoN=3.Theaboverelationshipsshowthatthetransferfunctionbasedmeasurementmodelparameterscanbeexpressedintermsofconstant-phase-elementmeasurementmodel.Thus,thetransferfunctionbasedmeasurementmodelisaspecialcaseofConstant-Phase-elementbasedmeasure-mentmodel.

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101 6.4Kramers-KronigRelationsElectrochemicalimpedancemeasurementsareconductedunderthefollowinggeneralassumptions73aboutthesystem 1. Linear:Thesystemislineararoundthesteadystateatwhichperturbationsignalsareapplied.Asaresult,theimpedanceisindependentofmagnitudeofperturbationandresponse. 2. Stable:Thesteady-statepositionofthesystemdoesnotdriftwithtime,andresponseoftheappliedperturbationsremainbounded. 3. Causal:Theresponseofthesystemisonlyduetotheperturbationsappliedi.e.,theresponsecannotprecedetheappliedinputperturbationsignal.TheKramers-Kronigtransformsprovidethemathematicalequalities,bywhichtheaboveconditionscanbetestedforacollecteddataset.Thetransformscanbewrittenasfollowing:Zr!)]TJ/F22 11.955 Tf 11.955 0 Td[(Zr1=2 Z10xZjx)]TJ/F22 11.955 Tf 11.955 0 Td[(!Zj! x2)]TJ/F22 11.955 Tf 11.955 0 Td[(!2dx-5orZr!)]TJ/F22 11.955 Tf 11.955 0 Td[(Zr=2! Z10h! xZjx)]TJ/F22 11.955 Tf 11.955 0 Td[(!Zj!i1 x2)]TJ/F22 11.955 Tf 11.955 0 Td[(!2dx-6andZj!=)]TJ/F28 11.955 Tf 11.291 16.857 Td[(2! Z10Zrx)]TJ/F22 11.955 Tf 11.955 0 Td[(!Zr! x2)]TJ/F22 11.955 Tf 11.955 0 Td[(!2dx-7whereZr,Zjaretherealandimaginarypartoftheimpedance,respectively,and!isthefrequencyofmeasurement,andxistheintegrationvariable.Relationsgivenbyequations 6-5 and 6-6 areforpredictingrealpartofimpedanceusingimaginarypart.Ifthehighfrequencyasymptoteisknown,equation 6-5 isap-plicable,andifzerofrequencyasymptoteofrealpartisknown,equation 6-6 is

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102 asuitabletransformtouse.Similarly,imagerypartoftheimpedancecanbepre-dictedwithequation 6-7 .Thus,forgivenarealpartofaimpedancespectrum,theimaginarypartcanbepredictedandviceversa.Theserelationshipscanalsobeusedtochecktheconsistencyofthespectrumwithaforementionedassumptionsaboutthesystem.However,astraightforwardapplicationsofthetransformsisnotfeasibleduetolimitedfrequencyrangeofEISexperiments.Byapplyingthemeasurementmodelswhichareconsistentwiththetransforms,theassumptionscanbevalidatedfortheelectrochemicalsystemunderstudy.6.5ComplexNonlinearLeast-squareRegressionTheparametersofthemeasurementmodel,forgivenEISdata,arecalculatedusingregression.Sinceregressioninvolvesanonlinearmodelwithrealandimag-inaryparttoadatasetwithcomplexnumbers,themethodisknownasComplexNonlinearLeast-SquareCNLSregression.Toshowthekeyconceptsofthere-gressionprocedure,theVoigtelementmeasurementmodelisconsidered.Themodelimpedance^Z!canbewrittenas^Z!=^Zr!+j^Zj!-8where^Zr!and^Zj!aretherealandcomplexpartsof^Z!,respectively.^Zr!and^Zj!canbeexpressedintermsofR0,R1,1;:::,Rk,kasfollowing:^Zr!=R0+kXi=1Ri +2i!2-9and^Zj!=kXi=1)]TJ/F22 11.955 Tf 22.745 8.087 Td[(Rii! +2i!2-10R0,R1,R1;:::,Rk,andkcanbedeterminedbyCNLSregression.ThismethodCNLSwasrstdevelopedbySheppardetal.74,75whoappliedCNLStopermit-tivitymeasurements.Macdonaldetal.76werethersttoapplyCNLStoEISdata.

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103 TheresidualsumofsquaresforCNLSiswrittenasSSE=nXi=1Zri)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zri2 2ri+nXi=1Zji)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zji2 2ji-11where=26666666666666664R0R11...Rkk37777777777777775nisthenumberofdatapoints,ZriandZjiaretherealandimaginarypartofthemeasuredimpedanceattheithfrequencyand2riand2jiarethevariancesoftherealandimaginarypartofthemeasurement,respectivelyattheithfrequency.TheobjectiveistocalculatetheparametersetsuchthatSSEisgloballymini-mized.Forthesakeofconvenience,SSEcanberewrittenasSSE=2nXi=1Zi)]TJ/F15 11.955 Tf 14.44 3.022 Td[(^Zi2 2i-12whereZi=Zri,^Zi=^Zriand2i=2riifinandZi=Zji,^Zi=^Zjiand2i=2jiifi>n.MinimizationofSSErequiresthatthederivativesofSSEwithrespecttodifferentparametersshouldbeequaltozero.Thisconditiongives@SSE @R0=@SSE @R1=@SSE @1==@SSE @k=0-13ThepartialderivativesofSSEwithrespecttodifferentparametersarewritten

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104 as:@SSE @R0=2nXi=1Zi)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zi 2i@^Zi @Z0@SSE @R1=2nXi=1Zi)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zi 2i@^Zi @R1@SSE @1=2nXi=1Zi)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zi 2i@^Zi @1 -14 ...@SSE @k=2nXi=1Zi)]TJ/F15 11.955 Tf 14.439 3.022 Td[(^Zi 2i@^Zi @kTheabovesetofequations 6-14 arerewritteninthematrixform.Matrixalgebra,then,facilitateseaseofsolvingequations 6-14 simultaneously.266666666666666641 1@^Z1 @R01 2@^Z2 @R01 2n@^Z2n @R01 1@^Z1 @R11 2@^Z2 @R11 2n@^Z2n @R11 1@^Z1 @11 2@^Z2 @11 2n@^Z2n @1...1 1@^Z1 @Rk1 2@^Z2 @Rk1 2n@^Z2n @Rk1 1@^Z1 @k1 2@^Z2 @k1 2n@^Z2n @k3777777777777777526666666666666664Z1)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z1 1Z2)]TJ/F20 7.97 Tf 8.326 2.014 Td[(^Z2 2Z3)]TJ/F20 7.97 Tf 8.326 2.014 Td[(^Z3 3...Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1 2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1Z2n)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2n 2n37777777777777775=0-15Inequation 6-15 ,^ZiisfurtherexpandedusingaTaylorseriesexpansionabouttheparameterset.Secondandhigherordertermsaredroppedfromtheexpansion.Thisgives^Zi=^Zi+@^Zi @R0R0;new)]TJ/F22 11.955 Tf 11.955 0 Td[(Z0+@^Zi @R1R1;new)]TJ/F15 11.955 Tf 11.955 0 Td[(1+@^Zi @11;new)]TJ/F22 11.955 Tf 11.955 0 Td[(1++@^Zi @RkRk;new)]TJ/F22 11.955 Tf 11.955 0 Td[(Rk+@^Zi @kk;new)]TJ/F22 11.955 Tf 11.956 0 Td[(k -16

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105 Substitutionof^Ziinequation 6-15 fori=1,...,2ngives266666666666666641 1@^Z1 @R01 2@^Z2 @R01 2n@^Z2n @R01 1@^Z1 @R11 2@^Z2 @R11 2n@^Z2n @R11 1@^Z1 @11 2@^Z2 @11 2n@^Z2n @1...1 1@^Z1 @Rk1 2@^Z2 @Rk1 2n@^Z2n @k1 1@^Z1 @k1 2@^Z2 @k1 2n@^Z2n @k377777777777777752666666666666666426666666666666664Z1)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z1 1Z2)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2 2Z3)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z3 3...Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1)]TJ/F20 7.97 Tf 8.326 2.014 Td[(^Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1 2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1Z2n)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2n 2n37777777777777775)]TJ/F28 11.955 Tf -334.985 -99.405 Td[(26666666666641 1@^Z1 @R01 1@^Z1 @R11 1@^Z1 @11 1@^Z1 @Rk1 1@^Z1 @k1 2@^Z2 @R01 2@^Z2 @R11 2@^Z2 @11 2@^Z2 @Rk1 2@^Z2 @k1 3@^Z3 @R01 3@^Z3 @R11 3@^Z3 @11 3@^Z3 @Rk1 3@^Z3 @k...1 2n@^Z2n @R01 2n@^Z2n @R11 2n@^Z2n @11 2n@^Z2n @Rk1 2n@^Z2n @k377777777777526666666666666664R0;new)]TJ/F22 11.955 Tf 11.955 0 Td[(Z0R1;new)]TJ/F22 11.955 Tf 11.955 0 Td[(R11;new)]TJ/F22 11.955 Tf 11.955 0 Td[(1R2;new)]TJ/F22 11.955 Tf 11.955 0 Td[(R2...k;new)]TJ/F22 11.955 Tf 11.955 0 Td[(k3777777777777777537777777777777775=0-17LetA=26666666666641 1@^Z1 @R01 1@^Z1 @R11 1@^Z1 @11 1@^Z1 @Rk1 1@^Z1 @k1 2@^Z2 @R01 2@^Z2 @R11 2@^Z2 @11 2@^Z2 @Rk1 2@^Z2 @k1 3@^Z3 @R01 3@^Z3 @R11 3@^Z3 @11 3@^Z3 @Rk1 3@^Z3 @k...1 2n@^Z2n @R01 2n@^Z2n @R11 2n@^Z2n @11 2n@^Z2n @Rk1 2n@^Z2n @k3777777777775-18anderr=26666666666666664Z1)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z1 1Z2)]TJ/F20 7.97 Tf 8.326 2.014 Td[(^Z2 2Z3)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z3 3...Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2n)]TJ/F21 5.978 Tf 5.756 0 Td[(1 2n)]TJ/F21 5.978 Tf 5.757 0 Td[(1Z2n)]TJ/F20 7.97 Tf 8.326 2.015 Td[(^Z2n 2n37777777777777775-19

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106 Aanderraresubstitutedinequation 6-17 .ThisgivesAT[err)]TJ/F22 11.955 Tf 11.956 0 Td[(A[new)]TJ/F22 11.955 Tf 11.955 0 Td[(]]=0-20Rearrangementoftheequation 6-20 fornewyieldsnew=+ATA)]TJ/F20 7.97 Tf 6.587 0 Td[(1ATerr-21Asuccessivesolutionofaboveequationprovideaparametervectornew,whereresidualsumofsquareSSEnewisminimum.6.5.1SolutionMethodEstimationofwasobtainedbytheLevenbergMarquardtmethod.77,78ThemethodinterpolatesbetweenGradientmethod79andSteepestdecentmethod.79TheHessianmatrixHusedinsteepestdecentmatrixisequaltoATA.Mar-quard80suggestedanalternativecomputationschemetoensurethatinverseoftheHessianmatrixalwaysexistsandH)]TJ/F20 7.97 Tf 6.587 0 Td[(1isapositivedenite.HeproposedthatHshouldbedenedas:H=ATA+I-22whereisapositivenumber.AftersubstitutionofHinequation 6-20 inplaceofATAyieldsnew=+H)]TJ/F20 7.97 Tf 6.586 0 Td[(1ATerr-23Thecalculationof,whichminimizesSSE,involvestherepeatedcalculationofHandnewusingequation 6-22 and 6-23 .Thestepsfollowedinthecalcu-lationofparametersetwere: 1. Theinitialvalueofparametersetwaschosen. 2. A,errandATAwerecalculated.

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107 3. Avalueof=lowwasselected,andHwascalculatedusingequation 6-22 4. Avalueofnewwasgottenbyequation 6-23 andSSEwascalculated. 5. IfSSEnewSSE,thevaluewassettohigh,Hwascalculatedusingequation 6-22 .Thecalculationswereagainrepeatedstartingfromstep 4 .Thevaluesoflowandhighcanbearbitrarilychosen.Suggestedvaluesoflowandhighare1:010)]TJ/F20 7.97 Tf 6.586 0 Td[(3and1:0103,respectively.6.5.2ConvergenceCriterionThereconvergencecriteriawereusedforLevenbergMarquardtmethod.Thesecriteriaweretestedinthestep 5 ofthecalculationprocedure.Theyweretestedintheorderlistedbelow. 1. IftherelativeerrorinSSEi.e.,SSEnew)]TJ/F23 7.97 Tf 6.586 0 Td[(SSE :010)]TJ/F21 5.978 Tf 5.756 0 Td[(20+SSEwasfoundlessthanpre-scribedvalue,thesecondconvergencecriterionwastested.1:010)]TJ/F20 7.97 Tf 6.586 0 Td[(20wasaddedinthedenominatortoavoidanumericalsingularitywhenSSE=0.0 2. Thesecondconvergencecriterionistherelativeerrorintheparameterseti.e.,Ifnew)]TJ/F23 7.97 Tf 6.586 0 Td[( :010)]TJ/F21 5.978 Tf 5.756 0 Td[(20+islessthanprescribedvalue,calculationswerestopped. 3. Ifthenumberofiterationsexceededthemaximumallocatednumber,calcu-lationswerestopped.

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108 6.5.3WeightingStrategyWeightingisanessentialcomponentofnonlinearregression,becausetheef-fectofnoisydataonparameterestimationcanbereducedbyassigningthemlessweight.Asoundweightingstrategyyieldsabetterestimateofparametersetwithsmallercondenceinterval.IntherstapplicationofCNSLtoEISdata,MacdonaldandGarber76usedtheunityweightingi.e.,r=j=1.TheyalsosuggestedthatstandarddeviationofmeasurementXcanbewrittenasx=ajXjn-24whereaisaproportionalityconstantandtheexponentnsatises1n)]TJ/F15 11.955 Tf 21.918 0 Td[(4.Theauthorstestedtheaboveweightingschemefordifferentvaluesofnandfoundthatcondenceintervalsofparametersetweresmallestwhen)]TJ/F20 7.97 Tf 10.494 4.707 Td[(1 2n)]TJ/F15 11.955 Tf 21.918 0 Td[(1.SeveralotherweightingstrategieshavebeenproposedafterMacdonaldetal.76Zoltowski81suggestedamodulusweightingscheme.Heproposedthatr!=j!=ajZ!j-25whereaisaproportionalityconstant.Hearguedthatthemodulusweightingschemeisappropriatewhenerrorsinrealandimaginaryimpedancearecorre-lated.Boukamp84supportedtheideaofZoltowskiandusedmodulusweightingschemeinhiscalculations.TheideaofmodulusweightingwasrefutedbyMac-donaldandPotter.85Theystatedthaterrorsinrealandimaginarypartoftheimpedancearecorrelatedbythemeasuringapparatus.Theysuggestedthepro-portionalweightingschemePWT.InPWT,thestandarddeviationsofrealandimaginarypartwereassumedtobeproportionaltothemagnitudeofthemeasure-mentofthatcomponent.Twootherweightingstrategies,whichcouldbereducedtoproportionalweightingscheme,werealsosuggestedbyMacdonaldandPotter.TherstwascalledVWTandit'sformisdescribedinequation 6-24 .Thesecond

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109 wastermedasCWT.ItcanbewrittenasXk=1+BwjXkjn-26whereBwisaproportionalityconstant.ForBw=0,equation 6-26 yieldsunityweighting.ForlargeBwandn=1,itconvergestoproportionalweighting.An-otherweightingschemewastermedasresidualiterationweightingRWT85inwhicheachtermintheobjectivefunctionisweightedbytheresidualerrorbe-tweenmodelpredictionandobservedvalueatkthfrequencyatlastiteration.Agarwaletal.32tookadifferentapproachfortheestimationofstandarddevi-ation.Theycarriedoutseveralrepeatedmeasurementforstationaryandpseudo-stationarysystemsApseudostationarysystemdoesnotchangemuchoverascanbutitmaychangeconsiderablyoveralongperiodoftime..Theyobservedthatthestandarddeviationsofrealandimaginarypartsarefunctionsoffrequency,andtheirmagnitudesaresameforagivenfrequency.Theyproposedthatr=j==jZjj+jZrj+jZj2 Rm-27whereZrandZjaretherealandimaginarypartofimpedanceZand;;andRmareconstantswhicharedeterminedexperimentally.Amodicationtoequa-tion 6-27 wassuggestedbyOrazemetal.86toincludeanadditionalparameter.Themodiederrorstructuremodelwassuggestedasr=j==jZjj+jZrj+jZj2 Rm+-28whereistheadditionalparameter.Later,Orazemetal.,87renedtheerrorstruc-turebaseduponfurtherexperimentalobservation.Theirrenedmodelwaspro-posedasr=j==jZjj+jZr)]TJ/F22 11.955 Tf 11.955 0 Td[(R1j+jZj2 Rm+-29whereR1istheadditionalconstant.Equation 6-29 isapplicableforsystemswhereelectrolyteresistancearemuchhighertotheimpedanceofsystem.Durbha

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110 etal.88gaveananalyticprooffortheequalityofvarianceforrealandimaginarypartoftheimpedance.6.5.4ComputerProgramImplementationAFORTRANcodewaswrittentoimplementtheregressionofthemeasure-mentmodels.ThesoftwarepackagetitledAlgorithm71789wasdownloadedfromNetlibRepositorylocatedatOakRidgeNationalLaboratory.AusagesummaryofthesoftwarecanbefoundinareportbyGay.90Thesoftwarecomprisesofseveralsubroutines.ThemainsubroutineofthesoftwareiscalledDGLGB.Theinputtothesubroutineconsistsofdataset,thefunctionalformthemeasurementmodel,parameter'slowerandupperbound,andgradientsoftheinputmeasure-mentmodelwithrespecttoitsparameters.Theoutputfromthecodecontainsregressedmodelparameters,thevariance-covariancematrix,andthelevelofcon-vergence.AgraphicalinterfacecalledMMToolBoxoftheFORTRANcodewasdevelopedin-houseusingVISUALBASIC.6.5.5CondenceIntervalTheperformanceoftheweightingstrategycanbeseenthroughtheestimatedparametervalueandtheircondenceinterval.Ifthecondenceintervali.e.,pa-rametervalue2doesnotcontainzero,itcanbesaidwith95:4%probabilitythattheestimatedparameterisnotequaltozero.Thevarianceforithparameterisgivenby2ithparameter=iis2whereiiisithdiagonalelementofthematrixATAands2=SSE N)]TJ/F20 7.97 Tf 6.587 0 Td[(2k)]TJ/F20 7.97 Tf 6.586 0 Td[(1forkVoigtelements.Thismethodofcalculatingcondenceintervalisstillamatterofre-search.Sincethemeasurementmodelwaslinearisedaroundparameterset,theabovecalculationproceduregivesthecondenceintervalsforlinearisedmodelparameterset.

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111 Figure6-2:Impedancespectraobtainedforthereductionofferricyanideonaplat-inumrotatingdiskelectrode. 6.6MethodTheuseofmeasurementmodelsforanalysisoferrorsrequiresreplicatemea-surementoftheimpedanceresponseusingthesamemeasurementfrequenciesforeachreplicate.Thediscussionhereiscenteredonaseriesofimpedancemeasure-mentsreportedbyOrazemetal.15forreductionofferricyanideonaplatinumdiskrotatingat120rpm.Theelectrolyteconsistedof0.01MK3FeCN6and0.01MK4FeCN6in1MKCl.Thetemperaturewascontrolledat25.00.1C.Themea-surementswereconductedunderpotentiostaticregulation.Fortheexperimentsdescribedhere,thepotentialwassetatavalueforwhichthecurrentmeasuredwas1/4ofthemass-transfer-limitedcurrent.Asubsetoftheimpedanceresults,consistingoftherstfourscanscollected,thesecondfourscansandthenalfourscans,arepresentedinFigure 6-2 .Thesequenceofmeasurementsindicatesthattherewasasubstantialchangefromonemeasurementtoanother.Thislackofreproducibility,initself,raisesthequestionofwhethereachindividualmeasure-mentwascorruptedbynonstationaryphenomena.Theissueofwhetheranindi-vidualmeasurementwascorruptedbynonstationaryphenomenaisaddressedinasubsequentsectionbyevaluationofconsistencywiththeKramers-Kronigrela-tions.

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112 Figure6-3:CurrentmeasurementsbeforeandaftertheimpedancescansshowninFigure 6-2 .Thedatasetssingledoutforerroranalysisarehighlighted. ThecurrentmeasuredbeforeandaftereachscanisreportedinFigure 6-3 .Thesignicantshiftinmeasuredcurrentbeforeandaftertherstscantakensuggeststhattherstmeasurementmayhavebeeninuencedbychangesinsystemprop-erties.TheDCcurrentvaluesdonotshedlightonsubsequentmeasurements.6.7ResultsThesubsetsselectedforinterpretationbytheVoigtandtransfer-functionmea-surementmodelsareindicatedinFigure 6-3 .Eachofthreesetsweretreatedasreplicatedmeasurements.Theobjectiveofthisselectionwastodeterminethein-uenceofthenonstationarityevidentintherstscanontheerrorstructureob-tainedthroughthemeasurementmodelapproach.Thesubsequentsetsoffourarenotreplicatedinthesensethattherearesystematicdifferencesbetweenthescans,buteachscanwillbefoundtosatisfytheKramers-Kronigrelations.6.7.1EvaluationofStochasticErrorsToeliminatethecontributionofthedriftfromscantoscan,theVoigtandtransfer-functionmeasurementmodelswereregressedtoeachscanusingthemax-

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113 imumnumberofparametersthatcouldberesolvedfromthedata.Thisvaluewasdeterminedbycalculatingthecondenceintervalforeachparameterusingtheassumptionthattheregressioncouldbelinearizedaboutthetrialsolution.Un-dertheseconditions,thestandarddeviationsforparameterestimatescouldbedeterminedusingthestandardequationsforlinearregression.Forthedetermi-nationoferrorstructure,modulusweightingwasemployed.InthesubsequentevaluationofconsistencywiththeKramers-Kronigrelations,theweightingwasbasedontheerrorstructuredeterminedinthisstep.FollowingAgarwaletal.,32thevarianceofrealandimaginaryresidualerrorscanbeobtainedasafunctionoffrequency,i.e.,2Zr!=1 N)]TJ/F15 11.955 Tf 11.955 0 Td[(1NXk=1"res;Zr;k!)]TJET1 0 0 1 386.427 451.193 cmq[]0 d0 J0.478 w0 0.239 m5.478 0.239 lSQ1 0 0 1 -386.427 -451.193 cmBT/F22 11.955 Tf 386.427 444.611 Td[("res;Zr!2-30whereNisthenumberofreplicates,"res;Zr;krepresentstheresidualerroratfre-quency!forscank,obtainedfromitsuniquemodel,and "res;Zrrepresentsthemeanvaluefortheresidualerrorsatfrequency!.Asimilarexpressionisusedfortheimaginarypartoftheimpedance.Equation 6-30 canprovideagoodestimateforthevarianceofstochasticerrorsunderthefollowingsetofassumptions: 1. Themodelparametersaccountforthedriftfromonescantoanother. 2. Thefrequency-dependentsystematicerrorsassociatedwiththelackoftareunchangedfromonescantoanother.Thisassumptionisjustiedonlyifthesamenumberofstatisticallysignicantparametersareusedforeachscanandifthesamefeaturesareevidentinthesuccessiveimpedancescans. 3. Thesystematicerrorsassociatedwithinstrumentartifactsareunchangedfromonescantoanother.ThisassumptionisjustiedundertheconditionsthatAssumption 2 isjustied.

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114 4. Thesystematicerrorassociatedwithnonstationarybehaviorisunchangedfromonescantoanother.Theobjectiveofthepresentworkwastoexaminethevalidityofassumption 1 byseeingwhethertheerrorstructureobtainedisdependentonthemeasurementmodelused.Andingthattheerrorstructuredependsonthemodelusedwouldunderminetheverypremiseonwhichthemeasurementmodelapproachisbased.Afterassumption 1 ,assumption 4 representsthethemostseriousrestric-tiontothemeasurementmodelapproachforestimationoferrorstructure.Itcanbeanticipatedthattheinuenceofnonstationaritymaybelargestfortherstofasequenceofimpedancemeasurements.UnderconditionsthatthedatasetsdonotsatisfytheKramers-Kronigrelations,themeasurementmodelapproachmayover-estimatethestandarddeviationofstochasticerrors.TheincorrecterrorstructuremaybiastheregressionusedtocheckconsistencywiththeKramers-Kronigrela-tions.ThisdifcultycanbeaddressedinaniterativeapproachinwhichthedataidentiedasbeinginconsistentwiththeKramers-Kronigrelationsareremovedfromthedatasetusedtoobtainthestochasticerrorstructureestimate.Inanycase,theestimateobtainedusingthemeasurementmodelapproachwillbemoreaccu-ratethanwouldbeobtainedbydirectcalculationofthestandarddeviation.Thus,thevarianceoftherealandimaginaryresidualerrorsprovidesagoodestimateforthefrequency-dependentvarianceofthestochasticnoiseinthemeasurement.Thedeviationofindividualspectrafromthemeanvaluefordataset#1,therstfourspectra,ispresentedinFigure 6.4a and 6.4b forrealandimaginarypartsoftheimpedance,respectively.Directcalculationofthestandarddeviationfordataset#1clearlyincludesasignicantcontributionfromsystematicchangesbetweenspectra,and,thus,signicantlyoverestimatesthestandarddeviationofstochasticerrors.

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115 a bFigure6-4:RelativedeparturesfromthemeanvaluefortherstfourspectragiveninFigure 6-2 :arealpartandbimaginarypartoftheimpedance.

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116 a bFigure6-5:Residualerrorsforthetofatransfer-functionmeasurementmodel,equation 6-3 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. Thetransfer-functionmeasurementmodelwasregressedtoeachofthefourspectrawhichcompriseDateSet#1usingthemaximumnumberofparametersthatcouldbeobtainedwith95.4%condenceintervalsthatdidnotincludezero.Themodelrequired11parameters.TherelativeresidualerrorsfortheseregressionsarepresentedinFigure 6.5a and 6.5b ,respectively,forrealandimaginarypartsoftheimpedance.Theresidualerrorsweresmallerthan0.8per-centfortherealpartoftheimpedanceandsmallerthan3percentfortheimaginarypart.TheresultspresentedinFigure 6.5b showthattherelativeresidualerrors

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117 a bFigure6-6:ResidualerrorsforthetofaVoigtmeasurementmodel,equation 6-1 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance. fortheimaginaryimpedanceoftherstspectrummeasureddonotoverlaytheresidualerrorsforsubsequentspectra.Theseresultssuggestthattherstmea-surementwassubjecttobiaserrorsthatwerenotevidentinsubsequentspectra.TheVoigtmeasurementmodelwasappliedaswelltoeachofthefourspectrathatcompriseSet#1.TenVoigtelements,or21parameterswereused.Theresult-ingrelativeresidualerrorsarepresentedinFigures 6.6a and 6.6b forrealandimaginarypartsoftheimpedance,respectively.Theresidualerrorswerecompara-bletothoseshowninFigure 6-5 .Theresidualerrorsweresmallerthan0.6percent

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118 Figure6-7:StandardDeviationsforthedatapresentedinFigure 6-2 ,obtainedfromtheresidualerrorspresentedinFigures 6-5 and 6-6 .ThedashedlinerepresentstheresultsobtainedfortheKramers-Kronig-consistentdatainset2and3. fortherealpartoftheimpedanceandsmallerthan3percentfortheimaginarypart.InagreementwithFigure 6.5b ,theresultspresentedinFigure 6.6b showthattherelativeresidualerrorsfortheimaginaryimpedanceoftherstspectrummeasureddonotoverlaytheresidualerrorsforsubsequentspectra.Thestandarddeviationofthestochasticpartofthemeasurementswasesti-matedbycalculatingthestandarddeviationsoftheresidualerrorspresentedinFigures 6-5 and 6-6 .Theresults,presentedinFigure 6-7 ,showthattheerrorstructureobtainedusingthetransfer-functionandVoigtmeasurementmodelsareinfullagreement.Thestandarddeviationsforrealandimaginarypartsoftheimpedanceareequalatlowfrequencies,butathigherfrequencies,thestandarddeviationfortheimaginarypartismuchlargerthanthatseenfortherealpartoftheimpedance.TheobservedinequalityofstandarddeviationsisinconictwiththeobservationthatthevarianceofstochasticerrorsofrealandimaginarypartsoftheimpedanceareequalwhenobtainedusinginstrumentationbasedonFourieranalysis.91Thus,itislikelythatbiaserrorsnotlteredbythemeasurement

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119 modelstrategycontributedtotheerrorstructure.Thissuggestionissupportedbytheobservationthattheerrorsarelargerthantheerrorsrepresentedbythedashedlinethatwereestimatedfromsubsequent,morestationary,setsofdata.TheprincipalresultshowninFigure 6-7 isthat,inspiteofthedifcultiescausedbyunlteredbiaserrors,theerrorstructureobtainedusingthetwomeasurementmodelsareinfullagreementatallfrequencies.AparalleltreatmentwasmadeforthesecondfourspectrawhichcompriseSet#2.AsshowninFigure 6-8 ,signicantsystematicdifferencesareseen,eventhougheachscanmightbeassumedtobemorestationarythanthoseinSet#1.Therelativeresidualerrors,showninFigure 6-9 forthetransfer-functionmea-surementmodelandinFigure 6-10 fortheVoigtmeasurementmodel,arenowcloselygroupedforeachofthespectraanalyzed.ThetrendingshowninFigures 6-9 and 6-10 indicatethatthemeasurementmodelsdonotdescribecompletelythephysicsoftheexperimentalsystem.Thestandarddeviationoftheresidualerrorsobtainedusingthetransfer-functionandVoigtmeasurementmodels,asshowninFigure 6.11a ,areinagreement.TheerrorstructureobtainedforSet#3andpresentedplottedinFigure 6.11b showsasimilaragreementbetweenresultsobtainedusingthetransfer-functionandtheVoigtmeasurementmodels.Thearbitrarychoiceofmeasurementmodeldoesnotchangethecharacteristicsoftheestimatedstochasticerrorstructure.Thisworkservestovalidatethefoun-dationofthemeasurementmodelapproachforerroranalysisofnon-replicatedmeasurements.6.7.2EvaluationofBiasErrorsInprinciple,sincetheVoigtmeasurementmodelisitselfconsistentwiththeKramers-Kronigrelations,theabilitytotthismodeltowithinthenoiselevelofthemeasurementshouldindicatethatthedataareconsistent.Arenedap-

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120 a bFigure6-8:RelativedeparturesfromthemeanvalueforthesecondfourspectragiveninFigure 6-2 :arealpartandbimaginarypartoftheimpedance.

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121 a bFigure6-9:Residualerrorsforthetofatransfer-functionmeasurementmodel,equation 6-3 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance.

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122 a bFigure6-10:ResidualerrorsforthetofaVoigtmeasurementmodel,equation 6-1 ,totheimpedancedatapresentedinFigure 6-2 :arealpartandbimaginarypartoftheimpedance.

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123 a bFigure6-11:StandardDeviationsforthedatapresentedinFigure 6-2 :aresultsobtainedfromtheresidualerrorspresentedinFigures 6-9 and 6-10 ,andbresultsobtainedfromtheresidualerrorsforDataset3.ThedashedlinerepresentstheresultsobtainedfortheKramers-Kronig-consistentdatainset2and3.

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124 proachwasdevelopedtoresolvetheambiguitythatexistswhenthemodeldoesnotprovideagoodttothedata.ThelackoftofthemodelcouldbeduetocausesotherthaninconsistencywiththeKramers-Kronigrelations.Forexample,thenumberoffrequenciesmeasuredmightbeinsufcienttoallowregressionwithasufcientnumberofVoigtparameters,thenoiselevelofthemeasurementmightbetoolargetoallowregressionwithasufcientnumberofVoigtparameters,ortheinitialguessesforthenon-linearregressioncouldbepoorlychosen.Whileinprincipleacomplextofthemeasurementmodelcouldbeusedtoassesstheconsistencyofimpedancedata,sequentialregressiontoeithertherealortheimaginaryprovidesgreatersensitivitytolackofconsistency.Theoptimalapproachistotthemodeltothecomponentthatcontainsthegreatestamountofinformation.Ourworksuggeststhatimaginarypartoftheimpedanceismuchmoresensitivetocontributionsofminorlineshapesthanistherealpartoftheimpedance.Typically,moreVoigtlineshapescanberesolvedwhenttingtotheimaginarypartoftheimpedancethancanberesolvedwhenttingtotherealpart.Thesolutionresistancecannotbeobtainedbyttingthemeasurementmodeltotheimaginarypartoftheimpedance.Thesolutionresistanceistreatedasanarbi-trarilyadjustableparameterwhenttingtotheimaginarypartoftheimpedance.TheapplicationofmeasurementmodelstoassessconsistencywiththeKramers-KronigrelationsisdemonstratedfortherstscanshowninFigure 6-2 .Ameasure-mentmodelwasttotheimaginarypartofthespectrumusingtheexperimentallydeterminederrorstructuretoweighttheregression.ThenumberofVoigtelementswasincreaseduntilthemaximumnumberofstatisticallysignif-icantparameterswasobtained.ThettotheimaginarypartisshowninFigure 6.12a ,wheredottedlinesrepresentthe2boundforthestochasticerrorstruc-turedeterminedintheprevioussection.Thecorrespondingpredictionofthereal

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125 a bFigure6-12:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftherstimpedancespectrumpresentedinFigure 6-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheesti-matedparameters.

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126 partisgiveninFigure 6.12b wheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.ThesystematicdepartureoftherealpartfromitsexpectedvalueevidentinFigure 6.12b issymptomaticofinconsistencywiththeKramers-Kronigrelations;however,onlyafewdatapointsatlowfrequencyfelloutsidethecondenceintervalforthemodel.Regressiontothereduceddatasetprovidesanincreaseinthenumberofparametersthatcouldberesolved,conrmingthedeterminationthatthedatawascorruptedbynon-stationaryphenomena.ApplicationoftheVoigtmeasurementmodeltoallotherspectrarevealedthatthedatawereconsistentwiththeKramers-Kronigrelationsatallfrequencies.Asimilaranalysiswasperformedforthesamedatasetusingthetransfer-functionmeasurementmodel.ThettotheimaginarypartisshowninFigure 6.13a ,wheredottedlinesrepresentthe2boundforthestochasticerrorstruc-turedeterminedintheprevioussection.ThesignicanttrendingapparentinFigure 6.13a reectsthesmallernumberofparametersthatcouldberesolved.ThecorrespondingpredictionoftherealpartisgiveninFigure 6.13b wheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.ThecondenceintervalshowninFigure 6.13b forthetransfer-functionmeasurementmodelwasmuchlargerthanthatestimatedfortheVoigtmeasurementmodelandshowninFigure 6.12b .Thus,theregressionprocedurecouldnotbeusedtojustifyeliminatingdatabasedonfailuretoconformtotheKramers-Kronigrelations.AnappreciationforthedifferencesbetweentheVoigtandtransfer-functionmeasurementmodelscanbeobtainedbyexaminationofthemodelparameters

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127 a bFigure6-13:Residualerrorsforthetofatransfer-functionmeasurementmodeltotheimaginarypartoftherstimpedancespectrumpresentedinFigure 6-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.

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128 Table6.1:ModelparametersforthetofaVoigtmeasurementmodeltoimpeda-ncescans#1,#5,and#25presentedinFigure 6-3 Variable Spectrum#1 Spectrum#5 Spectrum#25 1/10)]TJ/F20 7.97 Tf 6.586 0 Td[(6s 2:740:16 2:650:14 1:620:29R1/ 0:56400:0056 0:55260:0051 0:6100:058 2/10)]TJ/F20 7.97 Tf 6.586 0 Td[(5s 1:8370:055 1:930:10 5:960:13R2/ 1:90560:0912 1:350:16 8:230:44 3/10)]TJ/F20 7.97 Tf 6.586 0 Td[(5s 4:821:87 15:930:96 1:230:08R3/ 2:280:07 1:890:08 0:4960:018 4/10)]TJ/F20 7.97 Tf 6.586 0 Td[(5s 21:240:12 4:630:12 61:283:95R4/ 1:350:037 5:690:09 2:560:08 5/10)]TJ/F20 7.97 Tf 6.586 0 Td[(4s 9:710:56 7:620:39 1:2830:055R5/ 1:8800:066 1:9620:053 6:460:38 6/10)]TJ/F20 7.97 Tf 6.586 0 Td[(3s 4:080:24 3:650:17 2:900:21R6/ 3:230:14 3:310:11 3:270:14 7/10)]TJ/F20 7.97 Tf 6.586 0 Td[(2s 1:590:11 1:5380:073 1:2470:075R7/ 5:850:23 6:200:20 6:060:21 8/10)]TJ/F20 7.97 Tf 6.586 0 Td[(2s 5:780:42 6:360:33 5:180:27R8/ 10:310:49 12:060:52 11:60:4 9/s 0:20290:0092 0:2210:010 0:19550:0070R9/ 20:380:52 19:260:48 20:770:41 10/s 1:1200:007 1:1270:007 1:1420:006R10/ 123:50:50 123:220:52 131:010:41 Re/ 7:57xed 7:57xed 6:9857xed andassociatedstandarddeviations.Thestandarddeviationswerecalculatedun-dertheassumptionthatthenonlinearregressioncouldbelinearizedaboutthetrialsolution.Thus,thestandardequationscouldbeusedforcalculationofthestandarddeviationofparameterestimates.60TheVoigtmeasurementmodelparametersobtainedbyregressiontotheimag-inarypartoftheimpedancearepresentedinTable 6.1 alongwiththestandarddeviations.Theelectrolyteresistancewasassumedtobeaconstantintheregres-sionasthemodelfortheimaginarypartoftheimpedanceisindependentofso-lutionresistance.ThestructureoftheVoigtmodelissuchthattheVoigtelementshavetheirmostsignicantcontributionnearitscharacteristicfrequency.Thus,the

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129 Table6.2:ModelparametersforthetofaTransferfunctionmeasurementmodeltotheimpedancescan#1,#5,and#25presentedinFigure 6-3 Variable Spectrum#1 Spectrum#5 Spectrum#25 b0/106 4:55210:6943 3:27920:8019 5:94251:9083b1/106s1=2 3:43340:8660 2:41700:9536 6:12232:7290b2/106s 3:96120:1323 2:78200:1719 2:97810:6399b3/105s3=2 3:88340:0945 3:30970:0556 2:59320:0385b4/102s2 10:5750:1556 6:54480:0976 4:91860:1359b5/s3=2 6:9962xed 6:9962xed 6:9962xed a0/104 2:53010:4343 1:78600:4858 3:27251:1080a1/104s1=2 2:12780:2252 1:43230:2487 1:78340:5899a2/104s 3:51380:8719 2:49230:9571 6:11192:5412a3/104s3=2 3:13550:0719 2:08170:0297 1:16960:0139a4/s2 104:293:1515 48:0162:0385 24:1222:0288a5/s3=2 1:0xed 1:0xed 1:0xed condenceintervalatlowfrequenciesisinuencedmostbythecondenceinter-valsoftheVoigtelementswiththelargesttimeconstants.TheresultsshowninTable 6.1 indicatethatthecondenceintervalsforparameterestimateswereverytightfortheVoigtelements9and10,whichhadthelargesttimeconstants.Sim-ilarresultswereseenforthe5thand25thimpedancescansmade,forwhichtheKramers-Kronigrelationswerefoundtobesatised.Thecorrespondingparameterestimatesforthetransfer-functionmeasurementmodelarepresentedinTable 6.2 .Parametersa5andb5wereconstantswithintheregression.Thedominanttermsinthemodelatthelow-frequencylimitareb0anda0,whichhavelargecondenceintervals.Thestandarddeviationrepre-sentedalargeportionoftheparametervalue;thus,thecondenceintervalforthemodelatlowfrequencieswaslarge.Thecondenceintervalathighfrequenciesisdominatedbythehigherorderterms,whichhaveamuchtightercondenceinter-val.Thesmallercondenceintervalathighfrequency,evidentinFigure 6.13b ,suggeststhatthetransfer-functionmeasurementmodelmaybemoresensitiveto

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130 high-frequencyKramers-Kroniginconsistenciesthatcouldarisefrominstrumentartifacts.6.8ConclusionsThedevelopmentofdifferentformsofmeasurementmodelsforimpedancehasallowedexaminationofkeyassumptionsonwhichtheuseofsuchmodelstoassesserrorstructurearebased.Thestochasticerrorstructuresobtainedusingthetransfer-functionandVoigtmeasurementmodelswereidentical,evenwhennon-stationaryphenomenacausedsomeofthedatatobeinconsistentwiththeKramers-Kronigrelations.Asreportedintheliterature,34thetransfer-functionmeasurementmodelcouldprovideanadequatettoimpedancedatawithasmallernumberofparametersthanwasobtainedusingtheVoigtmeasurementmodel.Thesuitabilityofthemea-surementmodelforassessmentofconsistencywiththeKramers-Kronigrelations,however,wasfoundtobemoresensitivetothecondenceintervalfortheparam-eterestimatesthantothenumberofparametersinthemodel.AtightercondenceintervalwasobtainedforVoigtmeasurementmodel,whichmadetheVoigtmea-surementmodelamoresensitivetoolforidenticationofinconsistencieswiththeKramers-Kronigrelations.Thedevelopmentofdifferentformsofmeasurementmodelsforimpedancehasallowedexaminationofkeyassumptionsonwhichtheuseofsuchmodelstoassesserrorstructurearebased.Thestochasticerrorstructuresobtainedusingthetransfer-functionandVoigtmeasurementmodelswereidentical.ThesuitabilityofthemeasurementmodelforassessmentofconsistencywiththeKramers-Kronigrelations,however,wasfoundtobemoresensitivetothecondenceintervalfortheparameterestimatesthantothenumberofparametersinthemodel.

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CHAPTER7ELECTROCHEMICALMEASUREMENTSOFOXYGENREDUCTIONATNICKELELECTRODEThischapterpresentstheexperimentalinvestigationofoxygenreductionreac-tiononcommerciallypurenickel.Nickelisknowntohaveexcellentcorrosionre-sistanceinneutralandalkalinesaltsolutions.94Thecompositionofcommerciallypurenickel,alsoknownasNickel270,electrodeislistedinTable 7.1 .Thecalcula-tionsinChapter 5 showthatcurrentdistributionshouldbecomeuniformatabout25%ofthemass-transferlimitedcurrentonahemisphericalelectrode;whereas,itremainsnon-uniformevenforalowerpercentagevalueoftotaltransport-limitedcurrentonadiskelectrodeinsubmergedimpingingjetsystem.TheobjectiveofthisstudyistounderstandthedifferenceinresponseofElectrochemicalimpeda-ncespectroscopyEISexperimentsatthediskandhemisphericalelectrodegeom-etryinsubmergedimpingingjetsystemforoxygenreductionreaction,thereby,delineatingtheeffectofelectrodegeometryandcurrentdistributionbyEISexper-iments. Table7.1:ChemicalcompositionofNickel270 Element percentagecomposition Carbon 0.006Si 0.001Cu 0.001Ti 0.001Zr 0.001Ni 99.99 131

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132 7.1ReactionMechanismofOxygenReductionThereductionofoxygenisanimportantreactioninelectrochemicalsystemsdealingwithcorrosionofmetalsinaqueousandopenenvironments.95,96Itisalsoasourceofenergygenerationinpolymerelectrolytebasedfuel-cell97aswellsolidoxidefuelcell.98Themechanismofthisimportantreactionisasourceofdebateinliterature.Inneutralaqueouselectrolytesolutions,theoxygenreductionreactionsisassumedtofollowmainlytwoparallelreactionpaths.96Therstreactionpathcanberepresentedas:O2+2H2O+4e)]TJ/F25 11.955 Tf 10.405 -4.936 Td[(!4OH)]TJ/F19 11.955 Tf 128.812 -4.936 Td[(-1whereonemoleofO2isreducedtofourmolesofOH)]TJ/F19 11.955 Tf 11.069 -4.338 Td[(ionsbycombiningwithtwomolesofwater.Thispathisknownasthefourelectronreactionpath.Aparallelcompetingreactionmechanismalsooccurssimultaneously.Itcanberepresentedintwostepsasfollowing:O2+2H2O+2e)]TJ/F25 11.955 Tf 10.406 -4.936 Td[(!2H2O2+2OH)]TJ/F19 11.955 Tf 105.004 -4.936 Td[(-2whereonemoleofO2isrstreducedtotwomolesofH2O2andonemoleOH)]TJ/F19 11.955 Tf 7.085 -4.339 Td[(.Thisreactionisfollowedby:2H2O2+4e)]TJ/F25 11.955 Tf 10.405 -4.936 Td[(!4OH)]TJ/F19 11.955 Tf 140.573 -4.936 Td[(-3wherehydrogenperoxideisfurtherreducedtoOH)]TJ/F19 11.955 Tf 10.074 -4.339 Td[(ions.Theevidenceofsecondaryreactionpathsdescribedabovehasbeenwidelyreportedintheliterature.Thereexistsalargebodyofexperimentalworkinliter-atureonthereactionmechanismofoxygenreduction.MostoftheexperimentalworkhasbeenreportedonaRotating-RingdiskelectrodeRRDEsystem,99whichconsistsofadiskelectrodeandaringelectrode.Theringelectrodeencirclesthediskelectrode,andthwelectrodesareseparatedbyinsulatingmaterial.There-actionproductsformedatthediskelectrodeelectrodearecarriedtotheringby

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133 convection,wheretheycanbedetectedbyapplyinganappropriatedpotentialtothering.Kingetal.100studiedtheoxygenreductionreactiononcopperinaneutralNaClelectrolytesolutionusingtheRRDE.Theyconcludedthatoxygenisreducedviaaseriesofpathwayswherehydrogenperoxideisinvolvedasanadsorbedspecies.However,theformationofperoxideisdependentuponelectronicstateoftheelectrodepotential.Athighelectronicstates,i.e.,morecathodicelectrodepotentials,lessperoxidewillbeproduced.AsimilarstudywasconductedbyJo-vancicevicetal.101onironelectrode.Theyconsideredvepossiblemechanismsforthisreaction.TheyconcludedthatO2reductionproceedsviafour-electronpathwaywithlittleH2O2asanintermediateonthebareiron.Anastasijevicetal.102presentedamathematicalapproachtoaccountforthenumberofelectrontransfersbaseduponexperimentalresults.Intheirrstpa-per,104theyanalyzedtheexperimentalresultsofO2reductiononagoldelec-trode.Themathematicalmodelwasmainlydevelopedforarotatingdiskelec-trode.Theiranalysissuggestedthatfourelectronstepsdominateathigherca-thodicpotential;however,atlowcathodicpotentials,themodelsuggestedthatthetwoelectronstepispredominant.Insubsequentpapers,102,103averygeneralkineticmodelofO2reductionwaspresentedfortheRRDE.Inthismodel,allelec-trochemicalreactionswereaccountedasatmostone-electronexchangeprocesses.Theircalculationresultsconcludedthatupto90%ofcurrentisproducedbyfour-electronstepandtheH2O2stepaccountedfor2-4%.Theremaininguxwasas-sumedtobeduetotheintermediate.Hsuehetal.105studiedtheeffectofelectrolytepHonoxygenreductionviaH2O2formation.Theyconcludedthatamajority80)]TJ/F15 11.955 Tf 11.955 0 Td[(97%ofthereductiontakeplaceviafour-electronreductionpath,andtheamountofH2O2producedisindepen-dentoftheelectrolytepH.Morerecently,Vukmirovicetal.106studiedtheoxygen

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134 reductionreactiononcopperina0:1MNa2SO4electrolytesolutionasafunctionofpH.Theirexperimentalresultssuggestedthatthefour-electronmechanismispredominantoncopperinthealkalinemediumoverentirepHrange.SomeworkhasbeenreportedforO2reductiononnickelelectrode.Mostno-table,Shumilovaetal.107studiedtheoxygenreductionreactiononnickelelectrodeinalkalinemedium.Theyconcludedthatoxygenreductionviatheintermediatehydrogenperoxideformationtothatofoxygenreductiondirectlytohydroxyliondependsonthesurfacestateofthenickelelectrodeanditspotential.Inasub-sequentstudy,Batotgkyetal.108triedtoquantifythereactionrateandquantityproduced.Inthepresentwork,itisassumedthattheoxygenreductionoccursviafourelectronstep.Totheauthor'sknowledge,thisisthersttimetheEISoftheO2reactionreactionisbeingreported.7.2ExperimentalAdiskelectrodeandahemisphericalelectrodeof1 4inchdiameter,embeddedinnonconductingAcrylic,weremadewithaNickel270rod.Thediskelectrodewasmechanicallypolishedwith2400siliconcarbidesandpaperand1200gridemorycloth.Analuminaslurrypolishingpowder0.05mindeionizedwaterwasusedduringthemechanicalpolishingwithemorycloth.Then,theelectrodewaswassubjectedtoultrasoundcleaningina1:1solutionofdeionizedwaterandethylalcohol.Thehemisphericalelectrodewasmechanicallypolishedbyaconvexcavityofemoryclothwithaluminapowderpolishingslurry.Itwasalsosubjectedtoultrasoundcleaningbeforeexperiments.Theelectrolytesolutionof0.1MNaClwasmadewithpuriedwaterandsodiumchloridesalt.ThewaterwaspuriedwithaBarnSteadE-puresystem.Thepuricationprocessyieldedthewaterwithelectricalresistivitygreaterthan

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135 16Mcm.ThesodiumchloridesaltwasprocuredfromFisherScientic,Inc.AschematicoftheexperimentalsetupisshowninFigure 7-1 .Thesystemconsistedoftheimpingingjetelectrochemicalcell,electrolytesolutionreservoir,centrifugalpump,piping,value,temperaturecontroller,pHmeterandperipher-als.Thepiping,connectionsandsolutionreservoirweremadeofpropylene.Acoolingcoilmadeofglasswassubmergedinthereservoirtomaintainthede-siredtemperatureofelectrolyteinthereservoir.ThecoolanttemperaturewascontrolledbyaFisherScienticIsotempRefrigeratedModel910.Thetempera-tureoftheelectrolytewascontrolledat250.1C.Temperaturecontrolisveryimportantinaqueouselectrochemistry,astransportpropertiesofvariousspeciesexhibitaverystrongdependenceonthetemperatureoftheelectrolyte.ThepHofthesolutionwasmonitoredbyaFisherScienticmodelAccumetpHMeter915.Theelectrolyteinthereservoirwasbubbledwithairtosaturateitwithoxygen.AirwasrstpassedthroughaCO2scrubbertoremovethecarbondioxidefromair.Thepuriedairwasthenbubbledforaboutthirtyminutesbeforestartofeachexperiment.Theowsystempumpedelectrolytesolution,viaacentrifugalpumppoweredbyaVARIACpowersupply.TheuidvelocityinthejetwascontrolledbyVARIACoutputanduseofabypassline.Theowinthebypasslinewascon-trolledbyathrottlevalve.Theuidowrateintheimpingingjetwasadjustedbetween0.2gallon/minuteto3.0gallon/minute.Theelectrochemicalcell,presentedschematicallyinFigure 7.2a ,consistedofasubmergedaxisymmetricjet,acounterelectrode,aworkingelectrodewithholder,aportforreferenceelectrode,andonewindowtomonitorthesampleinsitu.Thecounterelectrodewasinsertedintothecellthroughthetopplate.Itwasmadeofplatinumfoilwithdimensionsofca.40.0mm40.0mm.Theimpingingjetwascenteredoverthesample.ImportantcalldimensionsareshowninFigure 7.2b

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136 Figure7-1:Experimentalsetupusedforthestudyofoxygenreductionreaction.

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137 a bFigure7-2:Schematicdiagramofimpingingjetelectrochemicalcell.aLayoutofthecellwithitscomponent.bImportantcelldimensions.

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138 Table7.2:Propertiesofoxygensaturated0.1MNaClat25oC. Property 0.1MNaCl pH 6:991/ohm)]TJ/F20 7.97 Tf 6.586 0 Td[(1cm)]TJ/F20 7.97 Tf 6.586 0 Td[(1 0:011/cm2sec)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1:0110)]TJ/F20 7.97 Tf 6.586 0 Td[(2DO2/cm2sec)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1:9310)]TJ/F20 7.97 Tf 6.586 0 Td[(5CbO2/molcm)]TJ/F20 7.97 Tf 6.586 0 Td[(3 2:2510)]TJ/F20 7.97 Tf 6.586 0 Td[(7 ThepotentialandcurrentinthecellwerecontrolledandmeasuredbySola-tran1286potentiostat/galvenostat.AfrequencyresponseanalyzerSolatran1250FRAwasusedtoapplyaperturbationsignalandthenmeasurethecorrespond-ingimpedanceofthesystem.FRAwasconnectedinseriestothepotentiostat.BoththepotentiostatandFRAwereconnectedtoacomputerviaaGPIBIEEE488.2controllercard.ThecontrolcommandsweresenttothepotentiostatandFRA,anddatawascollectedbyasoftwaredevelopedin-houseutilizingLab-view,109agraphicalinterfacingsoftware.AsaturatedcalomelelectrodeSCEwasusedasareferenceelectrode.Allpotentialsarereportedwithrespecttothesatu-ratedcalomelelectrode.Polarizationexperimentswerecarriedoutonbothdiskandhemisphericalelec-trode.Theelectrodepotentialwasvariedfromacathodicpotentialtoamoreposi-tivepotential.Beforethestartofapolarizationrun,theelectrodewaspolarizedat-1.2Vforthreeminutes.Thisprocessprovidescathodiccleaningoftheelectrodesurface,thuseliminatinganyoxidelayerformedattheopencircuitpotential.Impedanceexperimentswereconductedinpotentiostaticmode.Aperturba-tionsignalof10mVwasappliedontopofthebiaspotential.Thefrequencyrangewaschosentobe65KHzto0.5Hzwithtenlogarthimcallyspacedpointsperdecadeforthediskelectrodeand65KHzto1.0Hzforthehemisphericalelectrode.Alargeautointegration%closureerroroptionoffrequencyresponseanalyzerwasused,andthechannelforintegrationwasthatcorrespondingtocurrent.

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139 Figure7-3:Imageofthehemisphericalelectrodeduringthepolarizationmeasure-mentofoxygenreductionreaction. 7.3PolarizationMeasurementsPolarizationcurvesfordiskelectrodeandhemisphericalelectrodewerecol-lectedatthreedifferentjetvelocities.Theelectrodepotentialwasscannedinan-odicdirectionstartingwithinitialpotentialof-1.2Vtonalvalueof0.2V.Itwasvariedinstepsof2mV.Theresultingcurrentwasmeasuredafteradelayoftwoseconds.PolarizationcurvesfordiskelectrodearepresentedinFigure 7-4 .Polar-izationcurvesforthehemisphericalelectrodearepresentedinFigure 7-5 .Thetransportlimiteddiffusioncurrentwasobservedbetweenpotentialrangeof-0.8Vto-1.15V.ItisobservedinFigures 7-4 and 7-4 thatthediffusion-limitedcurrentvariesslightlywithpotential.Thisisattributedtohydrogenevolutionreaction.95Amiddlepointofthediffusion-limitedrangewaschosenasmass-transfer-limitedcurrentfordifferentexperimentalconditions.Valuesofthediffusion-limitedcur-rentatthediskelectrodeareplottedasafunctionofsquarerootofaverageuidvelocityinthejet.ThisispresentedinFigure 7-6 .Alinearttodatapointsyieldsastraightlinepassingthroughoriginwithaslopeof0:666640:01973.Asimilarplotdepictingmasstransferlimitedcurrentasafunctionofsquarerootoftheav-eragejetvelocityforhemisphericalelectrodeispresentedinFigure 7-7 .Alinear

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140 Figure7-4:Polarizationcurvefortheoxygenreductionreactioncollectedatthediskelectrode.Thesolidlinecorrespondstoaverageuidjetvelocityof1.99m/s,dashlinecorrespondsto2.99m/s,anddottedlinecorrespondsto3.98m/s. Figure7-5:Polarizationcurvefortheoxygenreductionreactioncollectedatthehemisphericalelectrode.Thesolidlinecorrespondstoaverageuidjetvelocityof1.99m/s,dashlinecorrespondsto2.99m/s,anddottedlinecorrespondsto3.98m/s.

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141 Figure7-6:Diffusionlimitedcurrentforoxygenreductionin0.1MNaClasafunc-tionofsquarerootofthejetvelocityforNi270diskelectrode.Thedashedlineisalinearttothedatapoints. tyieldsaslopeof1:405360:04216.Novisualchangeattheelectrodesurfacewasobservedduringthecourseofexperiments.Theelectrodesurfacewasmon-itoredwithavideocamera,andimageswerecollectedeverythirtyseconds.ArepresentativeimageisillustratedinFigure 7-3 .APourbaixdiagram,110ofnickelinsodiumchlorideispresentedinFigure 7-8 .ThesecalculationswereperformedusingCorrosionAnalyzer1.3softwaredevel-opedbyOLISystems,Inc.TheverticaldashedlineinthisgurecorrespondstoapHofoxygensaturated0.1Msodiumchlorideelectrolyte.ThespeciesandreactionsconsideredingeneratingFigure 7-8 arelistedinTable 7.3 .Forthesecal-culations,theactivityofnickelionswasassumedtobe110)]TJ/F20 7.97 Tf 6.587 0 Td[(6M.ThisvaluewasalsousedbyPourbaix.110Thepourbaixdiagramindicatesthatnickelshouldremainimmuneatcathodicpotentialsinthesodiumchlorideelectrolytesolution.Thevalueofthehydrodynamicconstantforeachexperimentalconditionwascalculatedfromthemass-transferlimitedcurrentforbothdiskandhemispherical

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142 Figure7-7:Diffusionlimitedcurrentforoxygenreductionin0.1MNaClasafunctionofsquarerootofthejetvelocityforNi270hemisphericalelectrode.Thedashedlineisalinearttothedatapoints. Table7.3:SpeciesconsideredincalculationofthePourbaixdiagrampresentedasFigure 7-8 AqueousPhase SolidPhase VaporPhase Water Nickel WaterChlorideion-1 NickelIIchloridedihydrate HydrogenHydrogen NickelIIchloridehexahydrate HydrogenchlorideHydrogenchloride NickelIIchloridetetrahydrate NitrogenHydrogenion+1 NickelIIhydroxide OxygenHydroxideion-1 NickelIIoxide Nickelion+2 NickelIIIhydroxide Nickelion+3 NickelIVoxide NickelIIhydroxide Sodiumchloride NickelIImonochlorideion+1 Sodiumhydroxide NickelIImonohydroxideion+1 Sodiumhydroxidemonohydrate NickelIItrihydroxideion-1 Trinickeltetraoxide Nitrogen Oxygen Sodiumion+1

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143 Figure7-8:Thepotential-pHdiagramofnickelinoxygensaturatedsodiumchlo-ridesolution.Thepotentialisreportedwithrespecttostandardhydrogenelec-trodeSHE.TheverticaldashlinecorrespondstopHof0.1Msodiumchloridesolution.ThisdiagramwasgeneratedbycomputersoftwareCorrosionAnalyzer1.3Revision1.3.33.OLISystems,Inc.Theactivityofnickelionswasassumedtobe1:010)]TJ/F20 7.97 Tf 6.587 0 Td[(6M.

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144 Table7.4:Computedvaluesofhydrodynamicconstantaforthediskelectrode. JetValocity,vJ Diffusionlimitedcurrent,ilim HydrodynamicsConstant,am/s amp /s 1:99 0:95710)]TJ/F20 7.97 Tf 6.587 0 Td[(3 699:622:99 1:14010)]TJ/F20 7.97 Tf 6.587 0 Td[(3 992:773:98 1:33110)]TJ/F20 7.97 Tf 6.587 0 Td[(3 1351:27 Table7.5:Calculatedvaluesofhydrodynamicsconstantaforthehemisphericalelectrode. JetValocity,vJ Diffusionlimitedcurrent,ilim HydrodynamicsConstant,am/s amp /s 1.99 1:9110)]TJ/F20 7.97 Tf 6.586 0 Td[(3 4109:102.99 2:3810)]TJ/F20 7.97 Tf 6.586 0 Td[(3 6380:123.98 2:9010)]TJ/F20 7.97 Tf 6.586 0 Td[(3 9472:75 electrodes.Themass-transfer-limitedcurrentisgivenbyIlim=0:85r20nFCbO2p aSc)]TJ/F20 7.97 Tf 6.587 0 Td[(2=3-4forthediskelectrode,111andIlim=0:352r20nFCbO2p aSc)]TJ/F20 7.97 Tf 6.586 0 Td[(2=3-5forhemisphericalelectrode.Inderivingequation 7-5 ,itwasassumedthatcur-rentintheseparatedpartoftheboundarylayerissameasatthepointofboundary-layerseparation.Thepointofseparationoccuratanangleof54.8asderivedinChapter 2 .ThecalculatedvaluesofhydrodynamicconstantsforthediskelectrodearepresentedinTable 7.4 ,andthevaluesforthehemisphericalelectrodearelistedinTable 7.5 .ItisevidentfromTables 7.4 and 7.5 thatthehydrodynamicsconstantsaresixtimeshigherforthehemisphericalelectrodethanforthediskelectrode.7.4ImpedanceMeasurementsImpedancemeasurementswereperformedforthissystematdifferentjetveloc-ityandbiaspotential.Foreachjetvelocity,twopotentialpointswerechosenfrom

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145 Table7.6:Experimentalconditionsforimpedancescanofoxygenreductionatdiskandhemisphericalelectrode JetVelocity Currentlevel BiasPotential BiasPotentialm/s i=ilim DiskElectrode HemisphericalElectrode 1.99 1/4 -0.535V -0.525V1.99 1/2 -0.650V -0.665V 2.99 1/4 -0.540V -0.570V2.99 1/2 -0.635V -0.665V 3.98 1/4 -0.535V -0.570V3.98 1/2 -0.665V -0.730V polarizationcurvespresentedinprevioussection.Thetwopotentialpointscor-respondedtohalfandquarterofmass-transfer-limitedcurrent,respectively.Theexperimentalconditionforallimpedancemeasurementsatthediskandhemi-sphericalelectrodearelistedinTable 7.6 .Complexplaneplotsofrstimpedancespectrumatthediskelectrodefordif-ferentjetvelocitiesandbiaspotentialsarepresentedinFigure 7-9 .Thersttwelvedatapointsweredeletedfromeachspectrum.Thesedatapointsshowedaclearpresenceofinstrumentartifactsathighfrequency.AsseeninFigure 7-9 ,theimpedancevaluechangesasjetvelocityisincreased.Impedanceswerehigheratlowervaluesofbiaspotential.NyquistplotsoftherstimpedancescanatthehemisphericalelectrodefordifferentjetvelocitiesandbiaspotentialarepresentedinFigure 7-10 .Inthiscase,thersttwelvedatapointswerealsodeletedfromeachspectrumathigh-frequencyend.Toshowtemporalvariationofimpedance,complexplane,andrealandimaginaryasafunctionoffrequencyofimpedancespectraobtainedatjetvelocityof1.99m/sandbiaspotentialof-0.54Varepre-sentedinFigure 7-11 .7.5MeasurementModelAnalysisMeasurementmodelanalysiswasperformedtoestimatethestochasticcontri-butionoferrorinimpedance.TheconsistencyofimpedancewiththeKramers-

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146 Figure7-9:Firstimpedancescancollectedduringthestudyofoxygenreductionatthediskelectrodeundersubmergedjetimpingement.Theimpedancespectrumwerecollectedfordifferentjetvelocitiesandbiaspotential. Figure7-10:Firstimpedancescancollectedduringthestudyofoxygenreductionatthehemisphericalelectrodeundersubmergedjetimpingement.Theimpedancespectrumwerecollectedfordifferentjetvelocitiesandbiaspotential.

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147 a bFigure7-11:Collectedimpedancespectrumforjetvelocityof2.99m/sandbiaspotentialof-0.540V.aComplexplaneplot;Realandimaginaryimpedancearenormalizedwithsurfacearea;bRealandimaginaryimpedanceasafunctionoffrequency.

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148 Kronigconsistencyrelationswascheckedtodeterminetheinconsistentportionofimpedancespectraateachexperimentalcondition.7.5.1DeterminationofStochasticErrorStructureThestochasticerrorstructureoftheimpedancemeasurementswereobtainedusingmeasurementmodelapproach.TheVoigt-element-basedmeasurementmodelwasselected,andmodulusweightingwasusedinregressionofthemeasurementmodel.AmaximumpossiblenumberofstatisticallysignicantVoigtelementswereregressedtotheimpedancespectraateachexperimentalcondition.Thecal-culationprocedurewasasdescribedinChapter 6 .Thevarianceofthestochasticerrorwasestimatedforalldatasetsateachexperimentalcondition.Theresultsofthecalculationaredescribedbelow.ForimpedancedatapresentedinFigure 7-9 byopensymbols,fourVoigtel-ementmeasurementmodelwereregressedtoeachdataset.Thesedatacorre-spondtolowervalueofthebiaspotentialquarterofmass-transferlimitedcur-rent.Someimpedancespectrawerenotusedforthecalculation,asitwasfoundthatthenoiselevelinthedatawashigher,andregressionwasabletoresolvefewerVoigtelements.TheestimatederrorstructureispresentedinFigure 7.12a .Athighervalueofbiaspotential,whichcorrespondtohalfofdiffusion-limitedcurrent,onlythreestatisticallysignicantVoigtelementscouldberesolvedfromregressionforeachdataset.TheresultingerrorstructureforthiscaseispresentedinFigure 7.12b .Aparalleltreatmentwasappliedtoimpedancedatasetcollectedonthehemi-sphericalelectrode.AfourVoigtelementmeasurementmodelwasregressedtodatasetscollectedatdifferentjetvelocitiesandbiaspotentialscorrespondingtoaquarterofmass-transfer-limitedcurrent.StandarddeviationsofstochasticerrorforrealandimaginarypartsarepresentedinFigure 7.13a .Asimilarprocedure

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149 a bFigure7-12:StandardDeviationsofstochasticerrorsfortheimpedancedatacol-lectedondiskelectrode.Arepresentativerstscanoftheanalyzeddataispre-sentedinFigure 7-9 .Theresultsarepresentedfordifferentjetvelocitiesandap-pliedbiaspotentials.aValuesofbiaspotentialswasselectedtoprovidetheav-eragecurrentlevelataboutquarterofmass-transfer-limitedcurrent;bValuesofbiaspotentialswasselectedtoprovidetheaveragecurrentlevelatabouthalfofmass-transfer-limitedcurrent.

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150 a bFigure7-13:StandardDeviationsofstochasticerrorsfortheimpedancedatacol-lectedondiskelectrode.Arepresentativerstscanoftheanalyzeddataispre-sentedFigure 7-10 .Theresultsarepresentedfordifferentjetvelocitiesandap-pliedbiaspotentials.aValuesofbiaspotentialswasselectedtoprovidetheav-eragecurrentlevelataboutquarterofmass-transfer-limitedcurrent;bValuesofbiaspotentialswasselectedtoprovidetheaveragecurrentlevelatabouthalfofmass-transfer-limitedcurrent.

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151 Table7.7:Modelparametersoferrorstructurefordifferentexperimentalcondi-tionsondiskelectrode. JetVelocity,BiasPotential m/s,V 1.99,-0.535 1:03510)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:1462.99,-0.540 0:30110)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:1933.98,-0.535 0:51810)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:052 1.99,-0.650 1:09910)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:1732.99,-0.635 1:11510)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:2743.98,-0.665 1:32210)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:283 Table7.8:Modelparametersoferrorstructurefordifferentexperimentalcondi-tionsonhemisphericalelectrode. JetVelocity,BiasPotential m/s,V 1.99,-0.525 0:63710)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:1122.99,-0.570 1:65610)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:0633.98,-0.575 0:63710)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:112 1.99,-0.665 2:25310)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:0692.99,-0.665 1:85310)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:0793.98,-0.730 0:65610)]TJ/F20 7.97 Tf 6.586 0 Td[(5 0:083 wasfollowedforimpedancespectraathigherbiaspotentials.OnlythreeVoigtelementscouldberesolvedwithregressionofdata.Theestimatedstandarddevi-ationofstochasticerrorsispresentedinFigure 7.13b .ThestandarddeviationsoferrorareonthesameorderinFigures 7.13a and 7.13b Ageneralizederror-structuremodelcouldnotbeobtainedforalltheexperimentalconditions.Differenterror-structuremodelsweredevelopedforeachexperimentalconditionaccordingtoequation 6-28 .ThetermRmwasas-signedatavalueof100.0asthisvalueofcurrentmeasuringresistorwasusedforallexperimentalconditions.Only,andcouldbeobtainedfromtheerror-structuremodelbecausethecondenceintervalforandincludedzero.TheresultsofthelinearmodelregressionarepresentedinTable 7.7 forthediskelec-trode.Theerror-structuremodelparametersatthehemisphericalelectrodeare

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152 giveninTable 7.8 .7.5.2Kramers-KronigConsistencyCheckKramers-Kronigtransformsrelatetherealpartoftheimpedancedatatotheimaginarypart.Theconsistencycheckensuresthatelectrochemicalsystemwaslinear,causal,stable,andstationaryduringtheexperiments.ErrorstructureweightingwasusedinregressionofimpedancespectrumtocheckforKramers-Kronigtransforms.Thisstrategyassignslessweighttonoisydataandmoreweighttogooddata.Theconsistencycheckswereperformedforstspectrumofeachdataset.TheprocedurewasfollowedasoutlinedinChapter 6 .Somerepresentativeresultsarepresentedhere.DiskElectrode.TheVoigtmeasurementmodelwasregressedtotheimag-inarypartofrstimpedancescancollectedat1.99meter/secjetvelocityand-0.535Vbiaspotentialquarterofmass-transferlimitedcurrent.ThisdatasetisrepresentedbyopencirclesinFigure 7-9 .Theregressionyieldedfourstatisticallysignicantlineshapes.VoigtelementparametersarelistedinthesecondcolumnofTable 7.9 .NormalizedresidualerrorsfortheimaginarypartarepresentedinFigure 7.14a wheredottedlinesrepresentthe2boundforthestochasticer-rorstructuredeterminedintheprevioussection.ThecorrespondingpredictionoftherealpartisgiveninFigure 7.14b wheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.Thevalueoftheso-lutionresistancewasxedat63.28inthecalculation.ThisdatasetwasfoundtobeconsistentwithKramers-Kronigrelation.Aparalleltreatmentwasperformedforrstimpedancescancollectedat1.99meter/secjetvelocityand-0.650Vbiaspotentialhalfofmass-transfer-limitedcurrent.ThedatasetisrepresentedbyopentrianglesinFigure 7-9 .Inthiscase,onlythreestatisticallysignicantline

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153 Table7.9:ModelparametersforthetofaVoigtmeasurementmodeltoimagi-narypartofrstimpedancescansatdiskelectrode.Thejetvelocityforthissetofexperimentswasat1.99meter/sec. Variable Spectrum#1 Spectrum#1 -0.530V -0.650V 1/10)]TJ/F20 7.97 Tf 6.587 0 Td[(4s 4:701:48 )]TJ/F22 11.955 Tf -181.878 -14.446 Td[(R1/ 14:561:86 )]TJET1 0 0 1 208.136 595.859 cmq[]0 d0 J0.398 w0 0.199 m235.379 0.199 lSQ1 0 0 1 -208.136 -595.859 cmBT/F22 11.955 Tf 214.114 585.747 Td[(2/10)]TJ/F20 7.97 Tf 6.587 0 Td[(3s 4:730:71 1:140:26R2/ 42:877:81 14:511:64 3/10)]TJ/F20 7.97 Tf 6.587 0 Td[(2s 2:150:15 0:960:12R3/ 430:0438:86 63:087:28 4/10)]TJ/F20 7.97 Tf 6.587 0 Td[(2s 5:820:22 6:820:21R4/ 664:5243:54 830:6513:14 Re/ 63:28xed 75:81xed shapescouldberesolvedbyregression.TheVoigtelementparametersobtainedaregiveninthethirdcolumnofTable 7.9 .Normalizedresidualerrorsforimagi-narypartsarepresentedinFigure 7.15a .Again,thecorrespondingpredictionoftherealpartisgiveninFigure 7.15b .HemisphericalElectrode.Asimilartreatmentwasappliedtotheimpedancedatacollectedatthehemisphericalelectrode.Theresultsoftheconsistencytestarepresentedforajetuidvelocityof3.98meter/sec.Themeasurementmodelwasregressedtotheimaginarypartofrstimpedancescancollectedat-0.57Vbiaspo-tentialquarterofmass-transfer-limitedcurrent.ThisdatasetisrepresentedbyhalflledcirclesinFigure 7-10 .Inthiscase,fourstatisticallysignicantlineshapeswereobtainedbyregressiontotheimaginarypartofthespectrum.TheobtainedVoigtmodelparametersarelistedinthesecondcolumnofTable 7.10 .TheresultsoftheregressionarepresentedinFigure 7-16 .NormalizedresidualerrorsfortheimaginarypartareshowninFigure 7.16a .ThecorrespondingpredictionoftherealpartisgiveninFigure 7.16b .Thevalueofsolutionresistanceforthiscalcula-tionwasxedat39.52.Aparalleltreatmentwasappliedtotherstimpedancescancollectedat-0.73Vbiaspotentialhalfofmass-transfer-limitedcurrent.The

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154 a bFigure7-14:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftheimpedancespectrumpresentedinFigure 7-9 byopencircles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.

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155 a bFigure7-15:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimagi-narypartoftheimpedancespectrumpresentedinFigure 7-9 byhalflledcircles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.

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156 a bFigure7-16:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftheimpedancespectrumpresentedinFigure 7-10 byopentraingles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.

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157 a bFigure7-17:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftherstimpedancespectrumpresentedinFigure 7-10 byinvertedhalflledtriangles.attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.

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158 Table7.10:ModelparametersforthetofaVoigtmeasurementmodeltoimagi-narypartofrstimpedancescansathemisphericalelectrode.Thejetvelocityforthissetofexperimentswasat3.98meter/sec. Variable Spectrum#1 Spectrum#1 -0.570V -0.730V 1/10)]TJ/F20 7.97 Tf 6.587 0 Td[(4s 4:991:60 5:881:78R1/ 9:961:28 9:160:84 2/10)]TJ/F20 7.97 Tf 6.587 0 Td[(3s 3:271:05 5:810:72R2/ 23:949:92 35:404:12 3/10)]TJ/F20 7.97 Tf 6.587 0 Td[(2s 1:210:26 )]TJ/F22 11.955 Tf -179.341 -14.446 Td[(R3/ 135:9135:76 )]TJET1 0 0 1 210.673 537.279 cmq[]0 d0 J0.398 w0 0.199 m230.304 0.199 lSQ1 0 0 1 -210.673 -537.279 cmBT/F22 11.955 Tf 216.651 527.167 Td[(4/10)]TJ/F20 7.97 Tf 6.587 0 Td[(2s 3:050:22 2:750:08R4/ 296:4343:12 225:224:36 Re/ 39:52xed 39:52xed datasetisrepresentedbyhalflledinvertedtrianglesinFigure 7-10 .AthreeVoigtelementmeasurementmodelwasregressedtoimaginarypartofthedatasetwitherrorstructureweighting.ThevaluesofregressedVoigtelementparametersarelistedinTable 7.10 .NormalizedresidualerrorsforimaginarypartsarepresentedinFigure 7.17a .Theregressionofmeasurementmodelprovidesttingerrorsoftheorderofstochasticerrorofthemeasurements.ThecorrespondingpredictionoftherealpartisgiveninFigure 7.17b .AsystematicdepartureoftherealpartofdatasetatlowfrequencyisevidentinFigure 7.17b .Atotalofsixdatapointslieoutsidethepredictionbandstartingatthefrequencyof4.1Hzto1Hz.Thisisinagreementwiththeobservationthatthemeasuringinstrument,SolartronFRA1250,recordedanerrorcode11282atthesedatapoint.Theerrorcode82isanindi-cationthattheautointegrationofmeasuredthecurrentwasnotsuccessfulintheallottedtime.7.6ProcessModelInterpretationofimpedanceinthecomplexplanediagramsFigures 7-9 and 7-10 suggeststhataconstantphaseelementbasedempiricalprocessmodelcould

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159 describethesystem.Thetransferfunctionforthisequivalentcircuitisgivenby:Z!=Rsol+Rct 1+j!1)]TJ/F23 7.97 Tf 6.586 0 Td[(-6whereRsolisthesolutionresistance,Rctisthechargetransferresistance,!isfre-quencyinradians,andisdistributionparameter.Thisequivalentcircuitprocessmodelwasdevelopedafterfollowingsphenom-enawereobservedinthecollectedimpedancespectrumofboththediskandthehemisphericalelectrode. 1. Complexplaneplotsdonotdistinctlyshowthemass-transferdiffusionimpedancecharacteristics,and 2. impedanceincomplexplaneshowthecharacteristicsofdepressedsemicir-cles.Undertheaforementionedobservationsaboutimpedance,aconstantphaseelementbasedprocessmodelcoulddescribetheimpedanceofthesystem.Itwasfurtherassumedthattheoxygenreductionoccurviafourelectronprocessdescribedbyequation 7-1 .ACPEelement113isknowntodescribethedistri-butionofthereactionrateattheelectrodesurface.ItishypothesizedthattheCPEparameterscanberelatedtothecurrentdistribution.Orazemandcoworkers117establishedarelationshipbetweenparametersofCPEelementstovoigtele-ments.Theauthorsconcludedthatlargerthevalueof,themoretimeconstantscanberesolvedfromimpedance.Hence,representsdegreeofnonuniformityinreactionrateattheelectrodesurface.Theprocessmodeldescribedinequation 7-6 wasregressedtotheKramers-Kronigconsistentpartofeachimpedancespectrum.Theerror-structureweightingwasusedinregression.Themodelsoferror-structureforeachtheexperimentalconditionsaregiveninTable 7.7 and 7.8 .AregressionresultispresentedinFigure

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160 Table7.11:EstimatedmodelparametersofaCPEequivalentcircuitmodeltoimpedancedatacollectedatthediskelectrode.Reportedparametersvaluesareaverageofsevenreplicatespectrumcollectedatanexperimentalcondition. ExperimentalConditions Rsol/ Rct/ /10)]TJ/F20 7.97 Tf 6.587 0 Td[(2s JetVelocitym/s,BiasPotential 1:99,)]TJ/F15 11.955 Tf 9.299 0 Td[(0:535V 68:0 1223:9 6:56 0:12152:99,)]TJ/F15 11.955 Tf 9.299 0 Td[(0:540V 73:1 1251:8 5:92 0:11643:98,)]TJ/F15 11.955 Tf 9.299 0 Td[(0:540V 66:5 994:1 5:21 0:1161 1:99,)]TJ/F15 11.955 Tf 9.299 0 Td[(0:650V 74:9 1064:3 10:04 0:15022:99,)]TJ/F15 11.955 Tf 9.299 0 Td[(0:635V 72:9 814:2 6:53 0:1631 Table7.12:EstimatedmodelparametersofaCPEequivalentcircuitmodeltoimpedancedatacollectedatthehemisphericalelectrode.Reportedparametersvaluesareaverageofsevenreplicatespectrumcollectedatanexperimentalcon-dition. ExperimentalConditions Rsol/ Rct/ /10)]TJ/F20 7.97 Tf 6.586 0 Td[(2s JetVelocitym/s,BiasPotential 1:99,)]TJ/F15 11.955 Tf 9.298 0 Td[(0:535V 41:2 754:8 4:71 0:13392:99,)]TJ/F15 11.955 Tf 9.298 0 Td[(0:540V 41:2 399:4 3:31 0:14383:98,)]TJ/F15 11.955 Tf 9.298 0 Td[(0:540V 41:5 475:2 3:16 0:1281 7-18 fortherstspectrumcollectedatajetvelocityof2.99m/sandbiaspotentialof-0.540V.AttoimpedanceindataispresentedinFigure 7.18a .Thecor-respondingvaluesofrealandimaginaryresidualerrorsarepresentedinFigure 7.18b .Forfewexperimentalconditions,theCPEprocessmodelcouldnotbere-gressedsuccessfully.Thiswasduetofactthattheconsistencycheckeliminatedamajorityofdatapointandameaningfulregressionwasnotpossibleforreduceddateset.CPEparametersvalues,obtainedfromregressionofimpeancedatasetatthediskelectrode,aregiveninTable 7.11 .Thereportedvaluesofparametersaretheaverageofsevenreplicatespectracollectedateachexperimentalcondi-tion.TheresultsofregressionforimpedancedatasetathemisphericalelectrodearegiveninTable 7.12 .ThevaluesofforthediskelectrodeinTable 7.11 suggeststhatcurrentdis-

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161 a bFigure7-18:ACPEequivalentcircuitmodelttotheimpedancedatacollectatthejetvelocityof2.99m/s.Thebiaspotentialwassetat-0.540V.aColpmexplaneplotofthettothedata;bRealandimaginaryresidualerrorsasafunctionoffrequency.

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162 tributionismoreuniformatlowervalueofbiaspotentialascomparedtohighervalues.However,nosuchconclusioncanbedrawnabouthemisphericalelectrodebecausevaluescouldnotbeobtainedfortheimpedancedatacollectedathigherbiaspotential.Acomparisonforthediskandthehemispheresuggeststhatcur-rentdistributionismoreuniformatdiskelectrodeforsamelevelofbiaspotential.7.7SummaryThischapterhaspresentedresultsofelectrochemicalmeasurementsofoxy-genreductionreactionatbothdiskandhemisphericalelectrodes.Theanalysisofimpedancedatasuggestthatthecurrentdistributionifmoreuniformatdiskelectrodethanathemisphericalelectrode.Thisisquitecontrarytothecalcula-tionsresultsinChapter 5 .However,theresultsoftheanalysiscouldbedebatedbecauseprocessmodelwaslimitedbyitsassumptions.

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CHAPTER8ELECTROCHEMICALMEASUREMENTSOFFERRICYANIDEREDUCTIONATNICKELELECTRODEThischapterpresentsanexperimentalstudyofreductionofferricyanideatthenickelelectrodeintheelectrolytemadeofsodiumhydroxide,potassiumfer-rocyanide,andpotassiumferricyanide.ElectrochemicalImpedanceexperimentswereconductedonboththediskandhemisphericalelectrodegeometryundersubmergedjetimpingement.Polarizationandimpedancemeasurementswereconductedinordertoexplorethedifferencesassociatedwiththediskandhemi-sphericalelectrodegeometries.ImpedancedatawasgraphicallyanalyzedandelectoratesurfacewascharacterizedwithEnergyDispersiveSpectroscopy.Theresultsandanalysisoftheimpedanceexperimentsarepresentedinthischapter.8.1IntroductionReductionofferricyanideisaone-electronchargetransferreaction.Thesto-chiometryofthisredoxsystemcanberepresentedas:FeCN3)]TJ/F20 7.97 Tf -4.234 -7.892 Td[(6+e)]TJ/F25 11.955 Tf 10.406 -4.936 Td[(!FeCN4)]TJ/F20 7.97 Tf -4.235 -7.892 Td[(6-1whereferricyanideionFeCN3)]TJ/F20 7.97 Tf -4.234 -7.879 Td[(6isreducedtoferrocyanideionFeCN4)]TJ/F20 7.97 Tf -4.234 -7.879 Td[(6bycombiningwithoneelectron.Historically,thissystemhasbeenthoughttoberelativelysimpletounderstandduetofollowingreasons. 1. Theelectrotransferreactioniskineticallyfast.Thispropertyofthesystemcausesthekineticandmass-transfereffectstobedistinctlyobservedduringimpedanceexperiments.118 163

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164 Table8.1:Electrolytepropertiesusedinexperiments. SupportingElectrolyte 1MNaOH 0.1MNaOH T/C 25 25r=cOH)]TJET1 0 0 1 210.334 661.125 cmq[]0 d0 J0.478 w0 0.239 m48.243 0.239 lSQ1 0 0 1 -210.334 -661.125 cmBT/F20 7.97 Tf 210.334 654.252 Td[(cNa++cK+ 0:97 0:740pH 13:87 12:831/ohm)]TJ/F20 7.97 Tf 6.586 0 Td[(1cm)]TJ/F20 7.97 Tf 6.587 0 Td[(1 0:17418 0:02516/cm2sec)]TJ/F20 7.97 Tf 6.586 0 Td[(1 1:08910)]TJ/F20 7.97 Tf 6.587 0 Td[(2 0:91210)]TJ/F20 7.97 Tf 6.586 0 Td[(2 2. Theeffectofthesupportingelectrolyteonmigrationisquantied,andpres-enceofexcesssupportingelectrolyteconcentrationdiminishestheeffectofohmiccontribution.23Theobjectiveofthisstudyistounderstandthedifferenceinimpedancere-sponseatdiskandhemisphericalelectrodegeometry.8.2ExperimentalMethodTheexperimentalsetupwasthesameasdescribedinChapter 7 forstudyofoxygenreductionstudies.Theelectrolyteconsistedof17Mcmdeionizedwater,0.005MK3FeCN6potassiumferricyanide,and0.005MK4FeCN6potassiumferrocyanide.SodiumhydroxideNaOHwasusedasthesupportingelectrolyteatconcentrationsofeither1or0.1M.ThecorrespondingelectrolytepropertiesaregiveninTable 8.1 .Thevalueofr=cOH)]TJ/F22 11.955 Tf 6.752 -0.299 Td[(=cNa++cK+indicatestheextenttowhichmigrationinuencesmasstransfer.Forthereductionofferricyanide,migrationactstore-ducethemass-transfer-limitedcurrentdensity,buttheeffectissmall.54Themass-transfer-limitedcurrentdensityforr=0:952isroughlyequalto98percentofthediffusion-limitedcurrentdensity.Forr=0:995,theinuenceofmigrationissuppressedcompletely.119,120Thediskelectrodeelectrodeandhemisphericalelectrodeswerepolishedbe-foreuse,withthenalpolishingstageconsistingof0.05maluminaslurryona1200gridemerycloth.Forthehemisphericalelectrode,theemeryclothwasheld

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165 Table8.2:Calculatedvaluesofsolutionresistanceforprimarycurrentdistribution,RPsol,usingelectricalconductivitiesofelectrolytelistedintable 8.1 SupportingElectrolyte 1MNaOH 0.1MNaOH Rpsol=DiskElectrode 2:30 15:7Rpsol=HemisphericalElectrode 1:44 9:96 inaconcavecavitytoavoiddeformationoftheelectrodeshape.Theelectrodeswerecleanedfor15minutesinanultrasonicbathina1:1mixtureofdeionizedwaterandethylalcohol.Argonwasbubbledinelectrolytereservoirtoreducetheconcentrationofdissolvedoxygen.ElectrochemicalmeasurementswereconductedwithaSolartron1286potentio-statandaSolartron1250frequencyresponseanalyzer.Impedancemeasurementswereconductedunderpotentialmodulation.8.3ExperimentalResultsTheexperimentalresultsservedtoverifytheinuenceofboundarylayersep-aration.Theseresultsanddiscussionarepresentedinthefollowingsections.8.3.1Steady-StateMeasurementApolarizationcurveispresentedinFigure 8-1 forthediskelectrodein1MNaOHsupportingelectrolyte.Thepotentialoftheelectrodeisreportedwithre-specttothesaturatedcalomelelectrodeSCE.Theelectrodepotentialwasvariedinstepsof2mV,andcurrentmeasurementwasmadeafteratwosecondsini-tialdelay.ThepolarizationcurveinFigure 8-1 issimilartotheonereportedbyDurbha.118Amass-transfer-limitedplateauwasnotclearlydened,andthemasstransferlimitedcurrentwasassumedtobeatthecurrentatapotentialof-0.3VSCE.Theelectrodepotentialcorrespondingtoanaveragecurrentequaltoonequarterofthemass-transfer-limitedcurrentwasdeterminedtobe+0.195VSCE.

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166 Figure8-1:Polarizationcurveofnickeldiskelectrodeinthesolutionof1.0MNaOH,0.005MK3FeCN6andK4FeCN6.Theaverageuidvelocityinthejetwas1.99meter/second. 8.3.2ImpedanceMeasurementImpedancemeasurementswereconductedunderpotentiostaticmodulationwithapotentialselectedtoprovideanaveragecurrentsuchthatiavg=ilimavg=4.Impedancedatacollectedonthediskelectrodein1MNaOHsupportingelec-trolytearepresentedinFigure 8-2 .Thefrequencyrangeforimpedancemeasure-mentwas65kHzto0.1Hzwithfrequencieslogarithmicallyspacedattenpointsperdecade.Thersttwelvehigh-frequencydatapointswerefoundtobecor-ruptedbyinstrumentalartifactsandweredeletedfromeachmeasurement.Cor-respondingimpedancedataforthehemisphericalelectrodeareshowninFigure 8-3 .Theimpedancedatadiffersignicantlyfromthespectrareportedinthelit-eratureforreductionofferricyanideonnickelelectrodes.Forboththediskandhemisphericalgeometries,theimpedanceincreasedwithimmersiontime.Asec-ondarylowerfrequencyfeaturewasseenthatwasattributedtooxygenreduction.Evidently,theexperimentalowloopsystemcouldnotbepurgedadequatelyof

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167 Figure8-2:Impedancespectraobtainedforthereductionofferricyanideonanickeldiskelectrodeundersubmergedjetimpingement.Theaverageuidve-locityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelectrolyteforthissetofexperimentsconsistedof1.0MNaOH,0.005MK3FeCN6andK4FeCN6. Figure8-3:CollectedImpedancespectraforthereductionofferricyanideonanickelhemisphericalelectrodeundersubmergedjetimpingement.Theaverageuidvelocityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelectrolyteforthissetofexperimentsconsistedof1.0MNaOH,0.005MK3FeCN6andK4FeCN6.

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168 a bFigure8-4:Complex-planeplotsofimpedanceobtainedonthediskelectrode.at=60s;andbt=1;860s. dissolvedoxygen.Itisknownthattheferricyanidereductionisinuencedbybothpoisoningandsurfaceblocking,15but,inpreviousexperiments,thehigh-frequencyasymptoteyieldedaslopeof45forthecomplex-planeplotoftheimpedance.Inthepresentwork,theslopeofthecomplex-planeplotofimpedancedatapresentedinFigures 8-2 and 8-3 forthediskandhemisphericalelectrode,respec-tively,differathighfrequenciesfromtheexpectedvalueof45.AsshowninFigure 8.4a ,theslopeofthecomplex-planeplotinthehighfrequencyregionisinitiallyslightlylargerthan22.5.AsseeninFigure 8.4b ,afteraperiodofabout30minutes,theslopeapproachesvalueof22.5.Theseresultsareconsistentwiththeformationofaporouselectrode.121,122Thetemporalevolutionshownin

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169 a bFigure8-5:Complex-planeplotsofimpedanceobtainedonthehemisphericalelec-trode.at=60s;andbt=1;860s. Figure 8-4 suggeststhattheporouslmformsrapidlyandbehavesasaporouselectroderatherthanasapassivediffusionbarrier.Thebehavioronthehemisphericalelectrodedifferedfromthatonthedisk.AsseeninFigure 8.5a theslopeoftherstimpedancescanforhemisphericalelectrodepresentedisneartheexpectedvalueof45.After30minutes,asseeninFigure 8.5b ,theslopeofthehigh-frequencyregionliesbetween22.5and45.Evenafteranimmersionofover3hours,theslopeneverapproachedthevalueof22.5seenforthediskelectrode.Theresultsobtainedforthehemisphericalelectrodesuggestthattheelectrodesurfacemayhavebeenpartiallycoveredbyaporouselectrode;whereas,incontrast,theporouselectrodecoveredthediskcompletely.

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170 Table8.3:Modelofobtainederrorstructureforimpedancespectraondiskandhemisphericalelectrode. System DiskElectrode 4:1710)]TJ/F20 7.97 Tf 6.587 0 Td[(5 )]TJ/F15 11.955 Tf 9.299 0 Td[(3:4710)]TJ/F20 7.97 Tf 6.587 0 Td[(4HemisphericalElectrode 11:8810)]TJ/F20 7.97 Tf 6.586 0 Td[(5 )]TJ/F15 11.955 Tf 9.299 0 Td[(9:5610)]TJ/F20 7.97 Tf 6.587 0 Td[(4 8.4MeasurementModelAnalysisMeasurementmodelanalysiswasperformedtodeterminethestochasticcom-ponentoferrorinimpedancemeasurement.Kramers-Kronigconsistencycheckwasalsoperformedtodeterminetheuncorruptedpartofthedatasetfrombiaserrorsandinstrumentalartifacts.8.4.1DeterminationofErrorStructureTheVoigtmeasurementmodelwasusedtodeterminethestochasticnoiselevelintheimpedancedata.Eachimpedancespectrumcollectedatthediskelectrodewasregressedtosevenvoigtelementmeasurementmodel.Modulusweightingwasusedinregression.Thevarianceofstochasticerrorwascalculatedusingpro-ceduredescribedinchapter 6 .ResultsoftheanalysisarepresentedinFigure 8-6 .Asimilaranalysiswasperformedforimpedancemeasurementsatthehemispher-icalelectrode.Inthiscase,sixvoigtelementmeasurementmodelwasregressedtoeachspectrum.Theresultingstochasticerrorsfortherealandimaginarycom-ponentsofimpedanceisshowninFigure 8-7 .Amodelforerrorstructurewasdevelopedusingequation 6-28 .Auniversalmodelcouldnotbeobtained,asthemodeldidnotprovideagoodttobothdatasetpresentedinFigures 8-6 and 8-7 .Anindividualerrorstructuremodelwasdevelopedfromestimatedstochasticer-rorsatthediskandthehemisphericalelectrode.Onlyandcouldbeextractedfromdatabecausevaluesofandincludedzerointheircondenceintervals.SolidlinesinFigures 8-6 and 8-7 representtheerror-structuremodel.ThemodelparametersaretabulatedinTable 8.3

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171 Figure8-6:StandardDeviationsforthedatapresentedinFigure 8-2 .Thesolidlinerepresentsthettotheerrorstructure. Figure8-7:StandardDeviationsforthedatapresentedinFigure 8-3 .Thesolidlinerepresentsthettotheerrorstructure.

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172 8.4.2Kramers-KronigConsistencyCheckTheKramers-KronigconsistencycheckwereperformedfortherstimpedancescanpresentedinFigures 8-2 and 8-3 .Theprocedurewasfollowedasoutlinedinchapter 6 .Theerrorstructuremodelforstochasticcomponentoferrorswereusedasweightsintheregression.Avoigtmeasurementmodelwasregressedtotheimaginarypartofrstimpeda-ncescancollectedat1.99meter/secjetvelocityandquarter.0mVbiaspoten-tialofmasstransferlimitedatthediskelectrode.ThisimpedancespectrumisrepresentedbycirclesinFigure 8-2 .Inthiscase,sevenstatisticallysignicantlineshapeswereobtainedbyregressionofthemeasurementmodel.VoigtelementmodelparametersarelistedininsecondcolumnofTable 8.4 .TheresultsoftheregressionareshowninFigure 8-8 .NormalizedresidualerrorsforimaginarypartarepresentedinFigure 8.8a wheredottedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection.ThecorrespondingpredictionoftherealpartisgiveninFigure 8.8b wheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheestimatedparameters.Thevalueofsolutionresistancewasxedat1.7.TheimpedancespectrumwasfoundtobeconsistentwithKramers-Kronigtransforms.Aparalleltreatmentwasperformedfortherstimpedancescancollectedathemisphericalelectrode.Measurementmodelwasregressedtoimaginarypartoftherstimpedancescan.Inthiscase,onlysixstatisticallysignicantlineshapescouldbeobtained.VoigtmodelparametersaregiveninthethirdcolumnofTable 8.4 .TheresultsoftheregressionareshowninFigure 8-9 .NormalizedresidualerrorsforimaginarypartarepresentedinFigure 8.9a wheredottedlinesrepre-sentthe2boundforthestochasticerrorstructuredeterminedintheprevious

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173 Table8.4:ModelparametersforthetofaVoigtmeasurementmodeltoimaginarypartofrstimpedancescansatdiskandhemisphericalelectrode Variable Spectrum#1 Spectrum#1 DiskElectrode HemisphericalElectrode 1/10)]TJ/F20 7.97 Tf 6.586 0 Td[(5s 3:490:34 3:5851:148R1/ 2:1460:072 1:8920:236 2/10)]TJ/F20 7.97 Tf 6.586 0 Td[(4s 3:0960:292 3:4390:476R2/ 1:9680:238 0:9250:376 3/10)]TJ/F20 7.97 Tf 6.586 0 Td[(3s 1:0660:216 1:6710:342R3/ 2:5630:418 1:3430:118 4/10)]TJ/F20 7.97 Tf 6.586 0 Td[(3s 3:5190:78 8:6763:214R4/ 3:4850:388 3:3461:764 5/10)]TJ/F20 7.97 Tf 6.586 0 Td[(2s 1:7050:192 2:6130:516R5/ 13:751:51 8:8571:648 6/10)]TJ/F20 7.97 Tf 6.586 0 Td[(2s 5:9381:718 21:6117:86R6/ 7:5661:31 0:970:58 7/10)]TJ/F20 7.97 Tf 6.586 0 Td[(1s 2:8870:444 )]TJ/F22 11.955 Tf -225.542 -14.446 Td[(R7/ 7:061:12 )]TJET1 0 0 1 170.167 434.565 cmq[]0 d0 J0.398 w0 0.199 m311.318 0.199 lSQ1 0 0 1 -170.167 -434.565 cmBT/F22 11.955 Tf 176.144 424.453 Td[(Re/ 3:1936xed 1:446xed section.ThecorrespondingpredictionoftherealpartisgiveninFigure 8.9b wheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheesti-matedparameters.Thevalueofthesolutionresistanceforthiscalculationwasxedat2.34.Normalizedrealresidualerrorsliebetween95:4%condenceband,whichindicatesthatthespectrumisconsistentwithKramers-Kronigrelations.8.5SurfaceAnalysisforDiskElectrodeOpticalmicrographsofthediskelectrodearepresentedinFigure 8-10 .AsseeninFigure 8.10a ,thediskelectrodeiscompletelycoveredwithdeposit.Toshowthatcontrastbetweenmetalsurfaceanddeposits,thelmwasremovedfromonesideofthediskelectrodebyrubbingwithsandpaper.TheresultingimageispresentedinFigure 8.10b .Theopticalimagessupporttheconclusionthatadepositisformedonthediskelectrode.

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174 a bFigure8-8:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftherstimpedancespectrumpresentedinFigure 8-2 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheesti-matedparameters.

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175 a bFigure8-9:ResidualerrorsforthetofaVoigtmeasurementmodeltotheimag-inarypartoftherstimpedancespectrumpresentedinFigure 8-3 .attotheimaginarypart,wheredashedlinesrepresentthe2boundforthestochasticerrorstructuredeterminedintheprevioussection;bpredictionoftherealpartwheredashedlinesrepresentthe95:4%condenceintervalforthemodelobtainedbyMonteCarlosimulationusingthecalculatedcondenceintervalsfortheesti-matedparameters.

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176 a bFigure8-10:Topviewofthediskelectrodeafterimpedanceexperiments.aUndis-turbedimageofelectrode.bImageobtainedaftertherightsideofdiskelectrodewascleanedwithsandpapertohighlightthecontrastbetweenmetalsurfaceanddeposits. AscanningelectronmicroscopicimageofsurfacedepositsispresentedinFig-ure 8-11 .Theimage,takenwithaJSM-6400ScanningMicroscope,showsthatthedepositsareintheformofclustersofcrystals.Theelementalcompositionofsur-facedepositswasdeducedwithEnergyDispersiveSpectroscopyEDS,alsocar-riedoutwithJSM-6400.Theenergyoftheelectronbeamwasvariedbetween0.0and65.0kVolts.TheresultsofanalysisarepresentedinFigure 8-12 ,wherepeaksinthespectrumcorrespondstodifferentelements.TheEDSanalysisrevealedthepresenceofcarbon,oxygen,potassium,iron,nickel,andsodium.Similaranalysisconductedonthepolishedmetalrevealedthepresenceofnickel.Theratiosob-tainedforelementalspeciesareinconclusiveduetothepossibleinuenceofthesubstrateandevaporatedsalts,butthepresenceofironandnickelisconsistentwiththeresultsofthethermodynamicanalysis.8.6OpticalMicrographsoftheHemisphericalElectrodeAseriesofopticalimagesofthethehemisphericalelectrodeafterimpedanceexperimentsintheelectrolytesupportedby1.0MNaOHarepresentedinFig-ure 8-13 .Atopviewoftheelectrode,presentedinFigure 8.13a ,revealsthat

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177 Figure8-11:ScanningElectronspectroscopyofofadiskelectrodeafterimmersionintheelectrolytesupportedby1.0MNaOH. Figure8-12:EnergyDispersiveSpectroscopyEDSanalysisofadiskelectrodeafterimmersionintheelectrolytesupportedby1.0MNaOH.

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178 a b c d Figure8-13:Imagesofhemisphericalelectrodeafterimpedanceexperiments.

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179 theelectrodewasonlypartiallycoveredwithsurfacedeposits.Themorphologyofthedepositismuchsmootherthanthegranularmorphologyofthedepositonthediskelectrode.AsideviewoftheelectrodeispresentedinFigure 8.13b .Inthisimagealso,onecanseethatthehemisphericalelectrodewasonlypartiallycovered,conrmingtheimagepresentedinFigure 8.13a .AnenlargedimageoflowerportionoftheelectrodeispresentedinFigure 8.13c .Thisimageempha-sizestheabsenceofcoverageofthelowerpartoftheelectrode.Sometendrilsofdepositcanbeseen,but,incontrasttotheobservationsnearthepoleofthehemi-sphere,thesurfaceneartheinsulatingplanewasnotuniformlycovered.AlineofdemarcationbetweenthecoveredandthecleanareasoftheelectrodeisevidentinFigure 8.13d .Thepositionatwhichthesurfacechangesfromcompletecoveragetomostlyuncoveredislocatedatanangleof55fromthepole.Aftertheseimagesweretaken,thehemisphericalelectrodewasrinsedwithdeionizedwater.TheresultingimageispresentedinFigure 8-14 .InagreementwiththePourbaixdiagram,thesurfacelayerisunstableinwaterofneutralpH,andtheresultingelectrodesurfaceiscompletelycleanwithoutsurfacedeposits.Itcanbeinferredfromthistreatmentthatthedepositsformingtheporouslayerarewatersolubleandthattheelectrodereturnedtoitsoriginalmirror-likesurface.Theunevenformationofdepositsonthehemisphericalelectrodecannotbeat-tributedtoanonuniformcurrentdistributionbecausetheaveragecurrentdensitywassufcientlysmallastoallowauniformcurrentdistribution.Thedemarkationatanangleof55betweenzonesofevendepositionandmostlyuncoveredsurfacesuggeststhatthehigheruidvelocitiesandshearforcesintherecirculationzoneeitherinhibiteddepositionorremovedthedeposit.Theangleof55isquiteclosetothevalueof54.8predictedbysemi-analyticalhydrodynamicmodeltobethepointofboundarylayerseparation.123Theseobservationsprovideexperimental

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180 Figure8-14:Sideviewofthehemisphericalelectrodeafterwashingitwithdeion-izedwater. vericationoftheboundary-layercalculations.1238.7ThermodynamicAnalysisThehypothesisthattheimpedanceresponseisinuencedbyformationofaporouslayerissupportedbythermodynamiccalculationsperformedusingCor-rosionAnalyzer1.3softwaredevelopedbyOLISystems,Inc.124,125ThespeciesandreactionsconsideredingeneratingFigure 8-15 arelistedinTable 8.5 .Forthesecal-culations,theactivityofnickelionswasassumedtobe110)]TJ/F20 7.97 Tf 6.586 0 Td[(6M.ThisvaluewasalsousedbyPourbaix.110Useofasmallervaluewillgenerallyreducetheregionsofstabilityforsolidlms.Activitycoefcientcorrectionswereusedbasedonthepubliclibraryusedinthesoftware.Impedancespectrawerecollectedatabiaspotentialof0.195VSCE,whichcorrespondsto0.4Voltswithrespecttothestandardhydrogenreferenceelectrode.Fortheelectrolytewith1.0MNaOH,0.005MK3FeCN6andK4FeCN6,thePour-baixdiagramsuggeststhatdi-ironnickeltetraoxideNiFe2O4wouldbestableun-dertheconditionsoftheexperiment.AtlowervaluesofpH,nosolidphasewillexistinequilibriumwithnickel.Calculationsperformedassumingtheabsenceofdissolvedoxygenshowedthatthedi-ironnickeltetraoxideNiFe2O4wasnot

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181 stableandthatnooxidelayershouldexistonthenickel.Thislatterresultcon-formstotheexperimentalconditionsreportedintheliterature.TotesttheconclusionthattheNiFe2O4solidphaseshouldbeunstableatlowervaluesofpH,experimentswereconductedwithareduced0.1MconcentrationofNaOH.Again,somedatapointsatthehighfrequencyendwerefoundtobecorruptedbyinstrumentartifactsandweredeletedfromthespectra.Animpeda-ncespectrum,obtainedaftervehoursofimmersion,ispresentedinFigure 8-16 .Evenafteranimmersiontimeofvehoursintheelectrolyte,thehigh-frequencyportionoftheimpedanceshowedaslopeof45asexpectedforaplanarelectrode.Visualexaminationoftheelectrodeafterexperimentsdidnotshowanychangeinsurfacecharacteristics.Thus,observationthroughimpedancespectroscopyofaporouselectrodestructureforanelectrolytesupportedwith1.0MNaOHisinagreementwiththermodynamicpredictionsofaNiFe2O4layer;andtheabsenceofporouselectrodebehaviorforanelectrolytesupportedwith0.1MNaOHisalsoinagreementwiththermodynamicpredictions.8.8DiscussionTheboundarylayerseparationatthestationaryhemisphericalelectrodewaspredictedbythesemi-analyticalhydrodynamicmodelandcomputationaluidmechanicalsimulationofthehydrodynamicequations.Theexperimentalevi-denceofthephenomenawasfoundinstudyofferricyanidereductionatthenickelhemisphericalelectrodeintheelectrolyteconsistingof1MNaOHwithdissolvedoxygen.ThepointofseparationfromthesethreeobservationsarelistedinTable 8.6 .Theformationofporouslayerfortheferricyanidereductionwasunexpected.Thepresenceofthelayerwasevidentinimpedancemeasurements,whichwasconrmedbythemicrographsofbothelectrodesurfacesaftertheexperiments.

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182 Figure8-15:Thepotential-pHdiagramfornickelinwatercontainingsodiumhy-droxide,potassiumferricyanide,potassiumferrocyanide,anddissolvedoxygen.ThepotentialisreportedwithrespecttostandardhydrogenelectrodeSHE.TheverticaldashedlinesrepresentthepHofelectrolytesolutionusedinthepresentstudy.Thelineontheleftcorrespondstoasolutioncontaining0.1MNaOH,0.005MK3FeCN6andK4FeCN6,andthelineontherightcorrespondsto1.0MNaOH,0.005MK3FeCN6andK4FeCN6.ThisdiagramwasgeneratedusingCorrosio-nAnalyzer1.3Revision1.3.33byOLISystems,Inc.Theactivityofnickelionswasassumedtobe110)]TJ/F20 7.97 Tf 6.587 0 Td[(6M.

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183 Table8.5:SpeciesconsideredincalculationofthePourbaixdiagrampresentedasFigure 8-15 AqueousPhase SolidPhase VaporPhase Chlorideion-1 0.947-Ironoxide HydrocyanicacidCyanideion-1 Diironnickeltetraoxide HydrogenDihydrogenferrocyanideion-2 Iron HydrogenchlorideDiironIIIdihydroxideion+4 IronIIIchloride NitrogenHydrocyanicacid IronIIchloride OxygenHydrogen IronIIchloridedihydrate WaterHydrogenchloride IronIIchloridehexahydrate HydrogenferrocyanideIIion-3 IronIIchloridetetrahydrate Hydrogenion+1 IronIIhydroxide Hydroxideion-1 IronIIoxide IronIIIchloride IronIIIchloride2.5hydrate Ironion+2 IronIIIchloridedihydrate Ironion+3 IronIIIchloridehexahydrate IronIIchloride IronIIIhexacyanoferrateII IronIIhexacyanideion-4 IronIIIhydroxide IronIIhydroxide IronIIIoxide IronIImonochlorideion+1 IronIIIoxidehydroxide IronIImonohydroxideion+1 Nickel IronIItetrahydroxideion-2 NickelIIchloridedihydrate IronIItrihydroxideion-1 NickelIIchloridehexahydrate IronIIIdichlorideion+1 NickelIIchloridetetrahydrate IronIIIdihydroxideion+1 NickelIIhydroxide IronIIIhexacyanideion-3 NickelIIoxide IronIIIhydroxide NickelIItetracyanonickel IronIIImonochlorideion+2 NickelIIIhydroxide IronIIImonohydroxideion+2 NickelIVoxide IronIIItetrachlorideion-1 Potassiumchloride IronIIItetrahydroxideion-1 Potassiumcyanide IronVItetraoxideion-2 PotassiumferricyanideIII Nickelion+2 PotassiumferrocyanideII Nickelion+3 PotassiumferrocyanideIItrihydrate NickelIIhydroxide Potassiumhydroxide NickelIImonochlorideion+1 Potassiumhydroxidedihydrate NickelIImonohydroxideion+1 Potassiumhydroxidemonohydrate NickelIItetracyanideion-2 Sodiumchloride NickelIItrihydroxideion-1 Sodiumcyanide Nitrogen Sodiumcyanidedihydrate Oxygen Sodiumhydroxide Potassiumchloride Sodiumhydroxidemonohydrate PotassiumferricyanideIIIion-2 SodiumironIIIdioxide PotassiumferrocyanideIIion-3 Triirontetraoxide Potassiumion+1 Trinickeltetraoxide Sodiumion+1 Water

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184 Table8.6:Theboundarylayerpointofseparationatthestationaryhemisphericalelectrode. Phenomena SeparationPointindegrees boundary-layerhydrodynamicmodel 54:8Computationaluidmechanicsimulation 62:0Reductionofferricyanide 552 Figure8-16:CollectedImpedancespectrumforthereductionofferricyanideonanickeldiskelectrodeundersubmergedjetimpingement.Theaverageuidve-locityinthejetwassetat1.99meter/secondandabiaspotentialof+0.195Vwasappliedtotheelectrode.Theelectrolyteforthisexperimentconsistedof0.1MNaOH,0.005MK3FeCN6andK4FeCN6.Therepresentedimpedancespectrumwascollectedafter5hoursofimmersionoftheelectrodeintheelectrolyte. Fromthemicrographsofhemispheres,theangleofseparationwasfoundtobeabout55.0degreeswithvariationof2degrees.Theenergydispersivespec-troscopyanalysisindicatedofcompositionoftheporouslayer,whichwasdeter-minedtobedi-ironnickeltetraoxideNiFe2O4fromtheequilibriumthermody-namicanalysisintermsofPourbaixdiagram.Moreworkneedtobedonetovalidatethechemicalcompositionoftheporouslayer.Astudyofthesystemintherotatingdiskandhemisphericalelectrodeisneededtofurtherunderstandtheresponseofimpedance.8.9ConclusionsThenumericalsimulationsandtheobservationofnonuniformdepositiononthehemisphericalelectrodeconrmthepresenceofboundary-layerseparation

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185 onthestationaryhemisphericalelectrodeunderasubmergedimpingingjet.Thestationaryhemisphericalelectrodeisthereforeinappropriateforkineticstudiesofelectrochemicalreactionsinwhichdepositionisexpected.Adepositwasformedonthenickelwhen1.0MNaOHwasusedasthesupport-ingelectrolyteandwhendissolvedoxygenwaspresent.Thislmwasobservedinelectronandopticalmicrographs.APourbaixanalysisindicatedthatdi-ironnickeltetraoxideNiFe2O4isthermodynamicallystableundertheseconditions.EDSanalysisoftheelectrodesrevealedNi,Fe,andOwerepresentonthenickelelectrodeafterthelmwasdeposited,butthatonlyNiwasseenontheelectrodebeforetheexperiment.Impedancemeasurementsshowthatthelayerbehaves,notasaninertbarriertomasstransfer,butasaporouselectrode.AsatestofthePourbaixanalysis,experimentswereconductedatalowerpHatwhichNiFe2O4waspredictedtobeunstable.Nolmwasobservedinopticalmicrographs,andtheimpedanceresponsewasthatofacleansurface.

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CHAPTER9CONCLUSIONSThequestforanstationaryelectrodesystemwithuniformcurrentdistribu-tionwasexploredinthiswork.Tothistask,stationaryhemisphericalelectrodeundersubmergedjetimpingementwassuggestedasacandidateelectrodegeom-etry.Thehydrodynamicmodeldevelopedusingboundary-layertheorypredictedaseparationofboundarylayer.Theangleofseparationwaspredictedtooccurat54.8.ThecomputationaluiddynamicsCFDmodel,developedatVrijeUni-versiteitBrussel,Belgium,predictedtheseparationpointat62.0.Theconvective-diffusionsolutionformass-transfer-limited-currentshowedanonuniformdistri-butionattheelectrodesurface.TheCFDsolutionofconvective-diffusionpre-dictedaminimuminmass-transfer-limitedcurrentatthepointofboundarylayerseparation.TheCFDsolutiondisplayedanenhancementinmass-transferintheseparatedpartofboundarylayer.Thecalculationsofcurrentandpotentialdistributionbelowthelimitingcon-ditionsshowedthatthecurrentdistributionshouldbecomeuniformatabout25%oftotalmass-transferlimitedcurrent.Thesecalculationswereperformedundertheassumptionthatcurrentdistributionisuniformbeyondthepointofsepara-tion.Thecurrentandpotentialdistributioncalculationsforrotatinghemispheri-calelectrodewerealsoperformed.Thesesimulationsaccountedforcorrectioninmass-transferduetoniteScnumber.Theresultingdistributionofcurrentde-pictedtheeffectofScnumbercorrectionatthedifferentlevelsofaveragecurrent.Asystematicstudywasundertakentoevaluatethemeasurementmodelap-proachforassessingtheerrorstructureofimpedancemeasurements.Transfer 186

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187 functionandVoigt-elementbasedmeasurementmodelwereappliedtoestimatestochasticerrorsfortheimpedancemeasurementscollectedattherotatingdiskelectrodeofferricyanidereduction.Theestimatederrorstructurewasfoundtobeindependentofchoiceofmeasurementmodel.Furthermore,thesamedatasetwasalsoanalyzedforKramers-Kronigconsistencycheckusingthetwomea-surementmodels.Thecondenceintervalsofparametersfortwomeasurementmodelwerefoundtobesignicantlydifferent.Voigt-elementmeasurementmodelyieldedatightercondenceintervalformodelparameters.Asaresult,Kramers-KronigconsistencycheckwasmoresensitiveforVoigt-elementbasedmeasure-mentmodel.Anexperimentalstudyofoxygenreductionwasexploredatthenickeldiskandhemisphericalelectrodes.Repeatedelectrochemicalimpedancemeasurementswerecollected.Aconstant-phase-elementbasedequivalentcircuitmodelwasregressedtotheKramers-Kronigconsistentimpedancedata.Theregressedparametersshowedthatthecurrentdistributionismoreuniformatthediskelectrodethanatthehemi-sphericalelectrodeforthesamelevelofaveragecurrent.Impedancespectracollectedatthediskelectrodeforferricyanidereductionre-vealedthebehaviorofaporouselectrode.Theinterpretationofimpedancedataatthehemisphericalelectrodeshowedabehaviorofpartiallycoveredsurface.Thisfeatureofelectrodesurfacewascorroboratedwithopticalmicrographsoftheelectrodeafterexperiments,equilibriumthermodynamicanalysis,andenergydispersivespectroscopy.Itissurmisedthatthevortexintheseparatedpartoftheboundarylayerpartiallyremovedthedepositsformedatthetheelectrodesurface.Baseduponthiswork,Itcanbestatedthatthestationaryhemisphericalelec-trodeundersubmergedjetimpingementisnotasuitablecandidateforelectroan-alyticalexperiments.

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CHAPTER10SUGGESTEDFUTURERESEARCHTheconclusionofthisworkhasledtothefactthatboundarylayersepara-tioninhibitstheuseofstationaryhemisphericalelectrodeforfutureconvective-diffusionimpedancemodeldevelopment.Aninterestingstudywouldbetode-velopasemi-analyticalmodelforuidowintheseparatedpartoftheboundarylayer.Themodelthencanbeincorporatedintothesolutionofconvectivediffusionandcurrentdistributioncalculations.Thepresentworksuggeststhattherotatinghemisphericalelectrodeisthebestcandidateforhighcurrentelectrochemicalimpedanceexperiments.Thehydrody-namicmodeldevelopedforrotatinghemisphericalelectrodeinthisworkshouldbeusedtodevelopanaccuratemodelforconvective-diffusionimpedance.Therotatinghemisphericalelectrodecanbeusedinconjunctionwithastationarydiskelectrodeunderjetimpingement,whereelectrodesurfacecanbemonitoredinsitu.Theobservationsfromstationaryelectrodecanbecorrelatedwiththemodelforimpedanceattherotatinghemisphericalelectrode. 188

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APPENDIXAHYDRODYNAMICEQUATIONSINSERIESEXPANSIONThisappendixpresentsthehydrodynamicgoverningequationsforvariableH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1introducedinsection 2.4 ofChapter 2 .A.1OrdinaryDifferentialEquationsforH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1ThegoverningequationsforH3andF3aregivenbyF1F3)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F1 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F3 @=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(8 3+1 2@2F3 @2A-1and2F3)]TJ/F15 11.955 Tf 13.15 8.088 Td[(1 6F1=2@H3 @A-2whereequations A-1 and A-2 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH5andF5aregivenby1 46F1F5+3F23)]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F3 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H5@F1 @=32 15+1 2@2F5 @2 A-3 and3F5)]TJ/F15 11.955 Tf 13.15 8.088 Td[(1 6F3)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F1=2@H5 @A-4whereequations A-3 and A-4 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH7andF7aregivenby[2F1F7+2F3F5])]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F5 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H5@F3 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F1 @=)]TJ/F15 11.955 Tf 10.494 8.088 Td[(256 315+1 2@2F7 @2 A-5 189

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190 and4F7)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F5)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F3)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F1=2@H7 @A-6whereequations A-5 and A-6 representsthemomentumandcontinuityequa-tions,respectively,forH5andF5.ThegoverningequationsforH9andF9aregivenby1 410F1F9+10F3F7+5F25)]TJ/F22 11.955 Tf 11.956 0 Td[(H1@F9 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H3@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F5 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H7@F3 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H9@F1 @=48 9!+1 2@2F9 @2 A-7 and5F9)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 6F7)]TJ/F15 11.955 Tf 16.077 8.087 Td[(1 90F5)]TJ/F15 11.955 Tf 19.004 8.087 Td[(1 945F3)]TJ/F15 11.955 Tf 21.93 8.087 Td[(1 9450F1=2@H9 @A-8whereequations A-7 and A-8 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH11andF11aregivenby[3F1F11+3F3F9+3F5F7])]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F11 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H3@F9 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F3 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H11@F1 @=)]TJ/F15 11.955 Tf 10.563 8.088 Td[(410 11!+1 2@2F11 @2 A-9 and6F11)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 6F9)]TJ/F15 11.955 Tf 16.077 8.087 Td[(1 90F7)]TJ/F15 11.955 Tf 19.004 8.087 Td[(1 945F5)]TJ/F15 11.955 Tf 19.273 8.087 Td[(1 9450F3)]TJ/F15 11.955 Tf 24.857 8.087 Td[(1 93555F1=2@H11 @ A-10 whereequations A-9 and A-10 representsthemomentumandcontinuityequa-tions,respectively.

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191 ThegoverningequationsforH13andF13aregivenby1 414F1F13+14F3F11+14F5F9+7F27)]TJ/F22 11.955 Tf 9.299 0 Td[(H1@F13 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H3@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F9 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F7 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H9@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H11@F3 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H13@F1 @=412 13!+1 2@2F13 @2 A-11 and7F13)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F11)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F9)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F7)]TJ/F15 11.955 Tf 19.274 8.088 Td[(1 9450F5)]TJ/F15 11.955 Tf 24.856 8.088 Td[(1 93555F3)]TJ/F15 11.955 Tf 30.709 8.088 Td[(691 638512875F1=2@H13 @ A-12 whereequations A-11 and A-12 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH15andF15aregivenby[4F1F15+4F3F13+4F5F11+4F7F9])]TJ/F22 11.955 Tf 9.298 0 Td[(H1@F15 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F13 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F9 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H9@F7 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H11@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H13@F3 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H15@F1 @=)]TJ/F15 11.955 Tf 10.564 8.088 Td[(414 15!+1 2@2F15 @2 A-13 and8F15)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F13)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F11)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F9)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F7)]TJ/F15 11.955 Tf 22.2 8.088 Td[(1 93555F5)]TJ/F15 11.955 Tf 30.709 8.088 Td[(691 638512875F3)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F1=2@H15 @ A-14 whereequations A-13 and A-14 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH17andF17aregivenby1 4[18F1F17+18F3F15+18F5F13+18F7F11+9F29)]TJ/F22 11.955 Tf 11.956 0 Td[(H1@F17 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F15 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H5@F13 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H7@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F9 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H11@F7 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H13@F5 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H15@F3 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F1 @=416 17!+1 2@2F17 @2 A-15

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192 and9F17)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F15)]TJ/F15 11.955 Tf 16.078 8.088 Td[(1 90F13)]TJ/F15 11.955 Tf 19.003 8.088 Td[(1 945F11)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F9)]TJ/F15 11.955 Tf 24.856 8.088 Td[(1 93555F7)]TJ/F15 11.955 Tf 28.053 8.088 Td[(691 638512875F5)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F3)]TJ/F15 11.955 Tf 36.563 8.088 Td[(3617 325641566250F1=2@H17 @ A-16 whereequations A-15 and A-16 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH19andF19aregivenby[5F1F19+5F3F17+5F5F15+5F7F13+5F9F11])]TJ/F22 11.955 Tf 11.956 0 Td[(H1@F19 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F17 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H5@F15 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H7@F13 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H11@F9 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H13@F7 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H15@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F3 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H19@F1 @=)]TJ/F15 11.955 Tf 10.563 8.088 Td[(418 19!+1 2@2F19 @2 A-17 and10F19)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F17)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F15)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F13)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F11)]TJ/F15 11.955 Tf 24.857 8.088 Td[(1 93555F9)]TJ/F15 11.955 Tf 28.053 8.088 Td[(691 638512875F7)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F5)]TJ/F15 11.955 Tf 36.563 8.088 Td[(3617 325641566250F3)]TJ/F15 11.955 Tf 36.833 8.088 Td[(43867 38979295480125F1=2@H19 @ A-18 whereequations A-17 and A-18 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH21andF21aregivenby1 4[22F1F21+22F3F19+22F5F17+22F7F15+22F9F13+11F211)]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F21 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H3@F19 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H5@F17 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F15 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F13 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H11@F11 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H13@F9 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H15@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F5 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H19@F3 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H21@F1 @=420 21!+1 2@2F21 @2 A-19

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193 and11F21)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F19)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F17)]TJ/F15 11.955 Tf 19.003 8.088 Td[(1 945F15)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F13)]TJ/F15 11.955 Tf 24.857 8.088 Td[(1 93555F11)]TJ/F15 11.955 Tf 28.053 8.088 Td[(691 638512875F9)]TJ/F15 11.955 Tf 33.637 8.088 Td[(2 18243225F7)]TJ/F15 11.955 Tf 36.563 8.088 Td[(3617 325641566250F5)]TJ/F15 11.955 Tf 36.833 8.088 Td[(43867 38979295480125F3)]TJ/F15 11.955 Tf 42.415 8.088 Td[(174611 1531329465290625F1=2@H21 @ A-20 whereequations A-19 and A-20 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH23andF23aregivenby[6F1F23+6F3F21+6F5F19+6F7F17+6F9F15+6F11F13])]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F23 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H3@F21 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H5@F19 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H7@F17 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F15 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H11@F13 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H13@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H15@F9 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H19@F5 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H21@F3 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H23@F1 @=422 23!+1 2@2F23 @2 A-21 and12F23)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F21)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F19)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F17)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F15)]TJ/F15 11.955 Tf 22.2 8.088 Td[(1 93555F13%)]TJ/F15 11.955 Tf 30.709 8.088 Td[(691 638512875F11)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F9)]TJ/F15 11.955 Tf 33.906 8.088 Td[(3617 325641566250F7)]TJ/F15 11.955 Tf 39.489 8.088 Td[(43867 38979295480125F5)]TJ/F15 11.955 Tf 42.416 8.088 Td[(174611 1531329465290625F3)]TJ/F15 11.955 Tf 42.685 8.088 Td[(155366 13447856940643125F1=2@H23 @ A-22 whereequations A-21 and A-22 representsthemomentumandcontinuityequa-

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194 tions,respectively.ThegoverningequationsforH25andF25aregivenby1 4[26F1F25+26F3F23+26F5F21+26F7F19+26F9F17+26F11F15+13F213)]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F25 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H3@F23 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F21 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H7@F19 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F17 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H11@F15 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H13@F13 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H15@F11 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F9 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H19@F7 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H21@F5 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H23@F3 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H25@F1 @=424 25!+1 2@2F25 @2 A-23 and13F25)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F23)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F21)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 945F19)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F17)]TJ/F15 11.955 Tf 22.2 8.088 Td[(1 93555F15)]TJ/F15 11.955 Tf 30.71 8.088 Td[(691 638512875F13)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F11)]TJ/F15 11.955 Tf 33.906 8.088 Td[(3617 325641566250F9)]TJ/F15 11.955 Tf 39.489 8.088 Td[(43867 38979295480125F7)]TJ/F15 11.955 Tf 39.759 8.088 Td[(174611 1531329465290625F5)]TJ/F15 11.955 Tf 45.342 8.088 Td[(155366 13447856940643125F3)]TJ/F15 11.955 Tf 45.612 8.088 Td[(236364091 201919571963756521875F1=2@H25 @ A-24 whereequations A-23 and A-24 representsthemomentumandcontinuityequa-tions,respectively.ThegoverningequationsforH27andF27aregivenby[7F1F27+7F3F25+7F5F23+7F7F21+7F9F19+7F11F17+7F13F15])]TJ/F22 11.955 Tf 11.955 0 Td[(H1@F27 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H3@F25 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H5@F23 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H7@F21 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H9@F19 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H11@F17 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H13@F15 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H15@F13 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H17@F11 @)]TJ/F22 11.955 Tf 9.299 0 Td[(H19@F9 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H21@F7 @)]TJ/F22 11.955 Tf 11.956 0 Td[(H23@F8 @)]TJ/F22 11.955 Tf 11.955 0 Td[(H25@F3 @)]TJ/F22 11.955 Tf 9.298 0 Td[(H27@F1 @=426 27!+1 2@2F27 @2 A-25

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195 and14F27)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 6F25)]TJ/F15 11.955 Tf 16.077 8.088 Td[(1 90F23)]TJ/F15 11.955 Tf 19.003 8.088 Td[(1 945F21)]TJ/F15 11.955 Tf 21.93 8.088 Td[(1 9450F19)]TJ/F15 11.955 Tf 22.2 8.088 Td[(1 93555F17)]TJ/F15 11.955 Tf 30.709 8.088 Td[(691 638512875F15)]TJ/F15 11.955 Tf 33.636 8.088 Td[(2 18243225F13)]TJ/F15 11.955 Tf 33.906 8.088 Td[(3617 325641566250F11)]TJ/F15 11.955 Tf 39.489 8.088 Td[(43867 38979295480125F9)]TJ/F15 11.955 Tf 42.416 8.088 Td[(174611 1531329465290625F7)]TJ/F15 11.955 Tf 42.685 8.088 Td[(155366 13447856940643125F5)]TJ/F15 11.955 Tf 48.269 8.088 Td[(236364091 201919571963756521875F3)]TJ/F15 11.955 Tf 48.539 8.088 Td[(1315862 11094481976030578125F1=2@H27 @ A-26 whereequations A-25 and A-26 representsthemomentumandcontinuityequa-tions,respectively.A.2SolutionsofH2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1TheobtainedsolutionprolesofH2i)]TJ/F20 7.97 Tf 6.586 0 Td[(1andF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1arepresentedinthissection.H1andF1asafunctionofarepresentedinFigure A.1a ,andH3andF3areshowninFigure A.1b .ValuesofH3andF3haveoppositesignscomparedtoH1andF1.ProlesofH5andF5aredrawninFigure A.1c .Similarly,H7andF7aregiveninFigure A.1d .AsseeninFigure A-1 ,themagnitudeofH5,F5,H7,andF7issmallercomparetothatofH1andF1.ThisisattributedtotheforcingterminthedifferentialequationofH5,F5,H7,andF7seeequations A-3 A-4 A-5 ,and A-6 andtheirfareldboundaryconditionat=1.TheobtainedsolutionsofH9andF9asafunctionofispresentedinFig-ure A.2a .ProlesofH11andF11aredisplayedinFigure A.2b .H13andF13aredrawninFigure A.2c ,andH15andF15arepresentedinFigure A.2d .F9andF11shownoscillationsclosetotheelectrodesurfacewhereasoscillationsinF13andF15aredampeddown.ThemagnitudesandthesignsofH9,F9,H11,F11,H13,F13,H15,andF15aresameinFigure A-2

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196 a b c dFigureA-1:Calculatedprolesofdimensionlessradialandcolatitudefunctionsintheexpansionof 2-24 and 2-25 .aH1andF1asafunction,bH3andF3asafunction,cH5andF5asafunction,anddH7andF7asafunction.

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197 a b c dFigureA-2:Calculatedprolesofdimensionlessradialandcolatitudefunctionsintheexpansionof 2-24 and 2-25 .aH9andF9asafunctionof,bH11andF11asafunctionof,cH13andF13asafunctionof,anddH15andF15asafunctionof.

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198 a b c dFigureA-3:Calculatedprolesofdimensionlessradialandcolatitudeintheex-pansionof 2-24 and 2-25 .aH17andF17asafunctionof,bH19andF19asafunctionof,cH21andF21asafunctionof,anddH23andF23asafunctionof.

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199 a bFigureA-4:Calculatedprolesofdimensionlessradialandcolatitudeintheex-pansionof 2-24 and 2-25 .aH25andF25asafunctionof,bH27andF27asafunctionof ThecalculatedsolutionsofH17andF17asafunctionofispresentedinFigure A.3a .Similarly,H19andF19aregiveninFigure A.3b .H21andF21aredrawninFigure A.3c ,andH23andF23arepresentedinFigure A.3d .ThecalculatedsolutionsofH25andF25asafunctionofispresentedinFigure A.4a .ProlesofH27andF27areplottedinFigure A.4b .A.3ExtrapolationofFiniteDifferenceValuesforF02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1ValuesofF02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1at=0wereobtainedbyextrapolatingthemtozeromeshsize.Thecalculationmethodhasbeengivenindetailinsection 2.4.1 ofChapter 2 .F02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1asfunctionofH2for1i4aregiveninFigure A-5 .F02i)]TJ/F20 7.97 Tf 6.586 0 Td[(1asfunctionofH2for5i8areprovidedinFigure A-5 .F02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1asfunctionofH2for9i12aredisplayedinFigure A-5 .F02i)]TJ/F20 7.97 Tf 6.587 0 Td[(1asfunctionofH2fori=13,and14areshowninFigure A-5 .Intheaforementionedgures,astraightlinewasregressedtothedataset.Thettothedatabyasolidlineisalsogiveningures.

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200 a b c dFigureA-5:FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1at=0fordifferentgridspacing.aF01asafunctionofH2,bF03asafunctionofH2,cF05asafunctionofH2,anddF07asafunctionofH2.

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201 a b c dFigureA-6:FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1at=0fordifferentgridspacing.aF09asafunctionofH2,bF011asafunctionofH2,cF013asafunctionofH2,anddF015asafunctionofH2.

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202 a b c dFigureA-7:FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1at=0fordifferentgridspacing.aF017asafunctionofH2,bF019asafunctionofH2,cF0210asafunctionofH2,anddF023asafunctionofH2.

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203 a bFigureA-8:FirstderivativeofdimensionlesscolatitudevelocitycoefcientsF2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1at=0fordifferentgridspacing.aF025asafunctionofH2,bF027asafunctionofH2

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APPENDIXBSOLUTIONOFCONVECTIVE-DIFFUSIONEQUATIONFORINFINITESCHMIDTNUMBERAsolutionofequationset 3-7 mentionedinChapter 3 wereobtainedanalyti-cally.Theresultingsolutionsof1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zfor2i14aregivenbybelow.1;3Z=0:5456995862847621Zexp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-1 1;5Z=)]TJ/F15 11.955 Tf 5.479 -9.684 Td[(0:019549161933433Z+0:22980950047037Z4exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-2 1;7Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:046385239894569Z)]TJ/F15 11.955 Tf 11.955 0 Td[(0:081892175730629Z4+0:064519511997433Z7exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-3 1;9Z=)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(0:037197466305511Z+0:044578750069712Z4)]TJ/F15 11.955 Tf 9.298 0 Td[(0:055197684466942Z7+0:013585493919529Z10exp)]TJ/F22 11.955 Tf 10.494 8.087 Td[(Z3H001 3 B-4 1;11Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:032471175096286Z+9:53121020077975910)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z4+0:034390061141628Z7)]TJ/F15 11.955 Tf 11.955 0 Td[(0:01898559227903Z10+2:28849399908543810)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z13exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-5 204

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205 1;13Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:029495946388488Z+0:013705692210355Z4)]TJ/F15 11.955 Tf 9.299 0 Td[(8:15540495528495310)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+0:015905953680082Z10)]TJ/F15 11.955 Tf 9.298 0 Td[(4:48741775016131110)]TJ/F20 7.97 Tf 6.586 0 Td[(3Z13+3:21249808488844410)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z16exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-6 1;15Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:027617257051674Z+0:012206800552896Z4+5:60169327633569610)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7)]TJ/F15 11.955 Tf 11.955 0 Td[(7:91486617210218810)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z10+4:8900076501783710)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z13+8:14837181092425210)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z16+3:86535322838121210)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z19exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-7 1;17Z=)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(0:026472305023423Z+0:011739595754222Z4+2:02751142373247610)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+3:71561009660302210)]TJ/F20 7.97 Tf 6.586 0 Td[(3Z10)]TJ/F15 11.955 Tf 9.299 0 Td[(3:46029910511240810)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z13+1:10691905160830610)]TJ/F20 7.97 Tf 6.586 0 Td[(3Z16)]TJ/F15 11.955 Tf 9.298 0 Td[(1:20587482661878210)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z19+4:0695234011606910)]TJ/F20 7.97 Tf 6.586 0 Td[(6Z22exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-8 1;19Z=)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(0:025853481275731Z+0:011448664255586Z4+2:62681508853612610)]TJ/F20 7.97 Tf 6.586 0 Td[(3Z7)]TJ/F15 11.955 Tf 11.955 0 Td[(6:71839170085972310)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z10+1:9396899835117810)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z13)]TJ/F15 11.955 Tf 11.955 0 Td[(1:00116540485177510)]TJ/F20 7.97 Tf 6.586 0 Td[(3Z16+1:97350869068478710)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z19)]TJ/F15 11.955 Tf 11.955 0 Td[(1:50899495695756910)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z22+3:80842485286888410)]TJ/F20 7.97 Tf 6.586 0 Td[(7Z25exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-9

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206 1;21Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:02563560273734Z+0:011340988284241Z4+2:49774669830168510)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+6:36761372629233110)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z10)]TJ/F15 11.955 Tf 9.299 0 Td[(7:73532590961440910)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z13+6:914041226502410)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z16)]TJ/F15 11.955 Tf 9.299 0 Td[(2:1669221979534810)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z19+2:89815227764637810)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z22)]TJ/F15 11.955 Tf 9.298 0 Td[(1:63760067172553210)]TJ/F20 7.97 Tf 6.586 0 Td[(6Z25+3:20767042898225510)]TJ/F20 7.97 Tf 6.586 0 Td[(8Z28exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-10 1;23Z=)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(0:02574081291381Z+0:011377417297367Z4+2:51635144390892610)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+3:14535079830940210)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z10+3:27458142187333810)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z13)]TJ/F15 11.955 Tf 11.955 0 Td[(3:70867386951635110)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z16+1:7992548272323710)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z19)]TJ/F15 11.955 Tf 11.955 0 Td[(3:74208402114799310)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z22+3:6167400729100310)]TJ/F20 7.97 Tf 6.587 0 Td[(6Z25)]TJ/F15 11.955 Tf 11.955 0 Td[(1:56997497299369810)]TJ/F20 7.97 Tf 6.586 0 Td[(7Z28+2:45607375954592710)]TJ/F20 7.97 Tf 6.586 0 Td[(9Z31exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-11 1;25Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:026120175464161Z+0:011536033857682Z4+2:54799097880054210)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+3:85320431785129510)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z10)]TJ/F15 11.955 Tf 9.299 0 Td[(4:30415566047665110)]TJ/F20 7.97 Tf 6.587 0 Td[(5Z13+1:71773713784855610)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z16)]TJ/F15 11.955 Tf 9.299 0 Td[(1:18374169689625710)]TJ/F20 7.97 Tf 6.587 0 Td[(4Z19+3:64282241570646810)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z22)]TJ/F15 11.955 Tf 9.299 0 Td[(5:37063166120477210)]TJ/F20 7.97 Tf 6.587 0 Td[(6Z25+3:92393066905919710)]TJ/F20 7.97 Tf 6.586 0 Td[(7Z28)]TJ/F15 11.955 Tf 9.298 0 Td[(1:34860131407899110)]TJ/F20 7.97 Tf 6.586 0 Td[(8Z31+1:72386998874351910)]TJ/F20 7.97 Tf 6.586 0 Td[(10Z34exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-12

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207 1;27Z=)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(0:0267433955015Z+0:011803072108809Z4+2:60520549238241910)]TJ/F20 7.97 Tf 6.587 0 Td[(3Z7+3:81919203056887910)]TJ/F20 7.97 Tf 6.586 0 Td[(4Z10+6:36145736144478610)]TJ/F20 7.97 Tf 6.587 0 Td[(5Z13)]TJ/F15 11.955 Tf 11.955 0 Td[(5:99978182100372510)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z16+6:50900863230861910)]TJ/F20 7.97 Tf 6.587 0 Td[(5Z19)]TJ/F15 11.955 Tf 11.955 0 Td[(2:82648719245025910)]TJ/F20 7.97 Tf 6.586 0 Td[(5Z22+6:00566339890483510)]TJ/F20 7.97 Tf 6.587 0 Td[(6Z25)]TJ/F15 11.955 Tf 11.955 0 Td[(6:59009941867131710)]TJ/F20 7.97 Tf 6.586 0 Td[(7Z28+3:76562232724568910)]TJ/F20 7.97 Tf 6.587 0 Td[(8Z31)]TJ/F15 11.955 Tf 11.955 0 Td[(1:0496338027710)]TJ/F20 7.97 Tf 6.586 0 Td[(9Z34+1:11687740892852510)]TJ/F20 7.97 Tf 6.586 0 Td[(11Z37exp)]TJ/F22 11.955 Tf 10.494 8.088 Td[(Z3H001 3 B-13

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APPENDIXCBOUNDARY-LAYERPROGRAMLISTINGTheprogramlistingpresentsalloftheFORTRANcodetosolvegoverningequationsoftheboundarylayeruidmodelforstationaryhemisphericalelectrodeundersubmergedjetimpingement.Theprogramwasdevelopedwith'CompaqVisualFortran,Version6.1'withdoubleprecisionaccuracy.Themainprogram'BOUNDARYLAYER'calledthesubroutinecontaininggoverningequationsandboundaryconditions.ThegoverningequationsfortheboundarylayerhydrodynamicmodelwereprogrammedinsubroutinesINNERH#F#,where#variesfrom1to27.Thebound-aryconditionsforthegoverningequationsattheelectrodesurfacewereprogrammedinsubroutinesBC1H#F#,andfareldboundaryconditionsareprogrammedinsubroutinesBC2H#F#.TheboundaryvalueproblemwasnumericallysolvedbysubroutinesBANDandMATINV,whichweredevelopedbyNewman.TheprogramwasiterateduntilallrelativevaluesforH#i,1,F#i,1werewithinaspeciedtolerancelimit.C.1ProgramListingC.1.1MainProgram PROGRAM BOUNDARYLAYER IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION ALLOCATABLE ::C:,:,H1:,:,F1:,: DOUBLE PRECISION ALLOCATABLE ::H3:,:,F3:,: DOUBLE PRECISION ALLOCATABLE ::H5:,:,F5:,: DOUBLE PRECISION ALLOCATABLE ::H7:,:,F7:,: DOUBLE PRECISION ALLOCATABLE ::H9:,:,F9:,: DOUBLE PRECISION ALLOCATABLE ::H11:,:,F11:,: DOUBLE PRECISION ALLOCATABLE ::H13:,:,F13:,: DOUBLE PRECISION ALLOCATABLE ::H15:,:,F15:,: 208

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209 DOUBLE PRECISION ALLOCATABLE ::H17:,:,F17:,: DOUBLE PRECISION ALLOCATABLE ::H19:,:,F19:,: DOUBLE PRECISION ALLOCATABLE ::H21:,:,F21:,: DOUBLE PRECISION ALLOCATABLE ::H23:,:,F23:,: DOUBLE PRECISION ALLOCATABLE ::H25:,:,F25:,: DOUBLE PRECISION ALLOCATABLE ::H27:,:,F27:,: DOUBLE PRECISION ALLOCATABLE ::H29:,:,F29:,: DOUBLE PRECISION ALLOCATABLE ::DH:,:,DF:,: DOUBLE PRECISION maxvalG,Zeta DOUBLE PRECISION FP,FPP DOUBLE PRECISION H,H2 INTEGER NJL,NJLIST CHARACTER NAMEFP*15,NAME1*25,NAME2*25,NAME3*25,FNUM*6 CHARACTER NAME4*25,NAME5*25,NAME6*25,NAME7*25,NAME8*25,NAME9*25 CHARACTER NAME10*25,NAME11*25,NAME12*25,NAME13*25,NAME14*25 CHARACTER NAME15*25 NAMELIST /par/ERRSUB,Zeta,NMAX,N INCLUDE 'COT_TERM.f'NJLIST=9NJL=20001NJL=40001NJL=80001NJL=100001NJL=160001NJL=200001NJL=250001NJL=400001NJL=500001NJL=1000001 open FILE ='INPUT.DAT', STATUS ='UNKNOWN' READ ,par CLOSE OPEN unit =16, file ='FP.txt' CLOSE unit =16, status ='delete' OPEN unit =16, file ='FP.txt' OPEN unit =17, file ='FPP.txt' CLOSE unit =17, status ='delete' OPEN unit =17, file ='FPP.txt' DO ii=1,NJLISTNJ=NJLiiH=Zeta/NJ-1H2=H*H INCLUDE 'ALC.f' INCLUDE 'HF.f' INCLUDE 'CALFP_J1.f' INCLUDE 'DAC.f' END DO END C.1.2MainSubroutinesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere.

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210 ***************** SUBROUTINE SETUP ***************************** SUBROUTINE SETUPHFC,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25,H27,F27*,H29,F29 INCLUDE 'DEFINEVAR.f' DOUBLE PRECISION CONSTCONST=0.0d0F1,1=0.0d0H1,1=0.0d0F3,1=0.0d0H3,1=0.0d0F1,NJ=BCH1,NJ=0.0d0F3,NJ=BCH3,NJ=0.0d0F5,1=0.0d0H5,1=0.0d0F5,NJ=BCH5,NJ=0.0d0F7,1=0.0d0H7,1=0.0d0F7,NJ=BCH7,NJ=0.0d0F9,1=0.0d0H9,1=0.0d0F9,NJ=BCH9,NJ=0.0d0F11,1=0.0d0H11,1=0.0d0F11,NJ=BCH11,NJ=0.0d0H13,1=0.0d0F13,1=0.0d0H13,NJ=BCF13,NJ=0.0d0H15,1=0.0d0F15,1=0.0d0H15,NJ=BCF15,NJ=0.0d0H17,1=0.0d0F17,1=0.0d0H17,NJ=BCF17,NJ=0.0d0H19,1=0.0d0F19,1=0.0d0H19,NJ=BCF19,NJ=0.0d0H21,1=0.0d0F21,1=0.0d0H21,NJ=BCF21,NJ=0.0d0H23,1=0.0d0F23,1=0.0d0H23,NJ=BCF23,NJ=0.0d0

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211 H25,1=0.0d0F25,1=0.0d0H25,NJ=BCF25,NJ=0.0d0H27,1=0.0d0F27,1=0.0d0H27,NJ=BCF27,NJ=0.0d0H29,1=0.0d0F29,1=0.0d0H29,NJ=BCF29,NJ=0.0d0 DO 20II=2,NJ-1F1,II=CONSTH1,II=CONSTF3,II=CONSTH3,II=CONSTH5,II=CONSTF5,II=CONSTH7,II=CONSTF7,II=CONSTH9,II=CONSTF9,II=CONSTH11,II=CONSTF11,II=CONSTH13,II=CONSTF13,II=CONSTH15,II=CONSTF15,II=CONSTH17,II=CONSTF17,II=CONSTH19,II=CONSTF19,II=CONSTH21,II=CONSTF21,II=CONSTH23,II=CONSTF23,II=CONSTH25,II=CONSTF25,II=CONSTH27,II=CONSTF27,II=CONSTH29,II=CONSTF29,II=CONST20 CONTINUE RETURN END ***************** SUBROUTINE BC1H1F1 ***************************** SUBROUTINE BC1H1F1J,C,H1,F1 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0

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212 END DO END DO G=-H1,JB,1=1.0d0G=-F1,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H3F3 ***************************** SUBROUTINE BC1H3F3J,C,H3,F3 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H3,JB,1=1.0d0G=-F3,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H5F5 ***************************** SUBROUTINE BC1H5F5J,C,H5,F5 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H5,JB,1=1.0d0G=-F5,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H7F7 ***************************** SUBROUTINE BC1H7F7J,C,H7,F7 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H7,JB,1=1.0d0

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213 G=-F7,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H5F5 ***************************** SUBROUTINE BC1H9F9J,C,H9,F9 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H9,JB,1=1.0d0G=-F9,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H11F11 ***************************% SUBROUTINE BC1H11F11J,C,H11,F11 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H11,JB,1=1.0d0G=-F11,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H13F13 ************************** SUBROUTINE BC1H13F13J,C,H13,F13 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H13,JB,1=1.0d0G=-F13,JB,2=1.0d0 RETURN END

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214 ***************** SUBROUTINE BC1H15F15 ************************** SUBROUTINE BC1H15F15J,C,H15,F15 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H15,JB,1=1.0d0G=-F15,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H17F17 ************************* SUBROUTINE BC1H17F17J,C,H17,F17 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H17,JB,1=1.0d0G=-F17,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H19F19 ************************* SUBROUTINE BC1H19F19J,C,H19,F19 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H19,JB,1=1.0d0G=-F19,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H21F21 ************************* SUBROUTINE BC1H21F21J,C,H21,F21 INCLUDE 'DEFINEVAR.f' DO i=1,N

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215 Gi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H21,JB,1=1.0d0G=-F21,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H23F23 ************************** SUBROUTINE BC1H23F23J,C,H23,F23 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H23,JB,1=1.0d0G=-F23,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H25F25 ************************* SUBROUTINE BC1H25F25J,C,H25,F25 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO G=-H25,JB,1=1.0d0G=-F25,JB,2=1.0d0 RETURN END ***************** SUBROUTINE BC1H27F27 ************************ SUBROUTINE BC1H27F27J,C,H27,F27 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0

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216 Di,k=0.0d0 END DO END DO G=-H27,JB,1=1.0d0G=-F27,JB,2=1.0d0 RETURN END ***************** SUBROUTINE INNERH1F1 ************************* SUBROUTINE INNERH1F1J,C,H1,F1 INCLUDE 'DEFINEVAR.f'CContinuityequationforH1andF1 G=2.0d0*H1,J-H1,J-1/H-F1,J+F1,J-1/2.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0A,2=1.0d0/2.0d0B,2=1.0d0/2.0d0D,2=0.0d0CMomentumequationforH1andF1 G=F1,J+1-2.0d0*F1,J+F1,J-1/2.0d0/H2*+FT*+H1,J*F1,J+1-F1,J-1/H/2.0d0*-F1,J*F1,J/4.0d0A,1=0.0d0B,1=-F1,J+1-F1,J-1/H/2d0D,1=0.0d0A,2=-1.0d0/2.0d0/H2+H1,J/2.0d0/HB,2=1.0d0/H2+F1,J/2.0d0D,2=-1.0d0/2.0d0/H2-H1,J/2.0d0/H RETURN END ***************** SUBROUTINE INNERH3F3 ************************* SUBROUTINE INNERH3F3J,C,H1,F1,H3,F3 INCLUDE 'DEFINEVAR.f'CContinuityequationforH3andF3G=2.0d0*H3,J-H3,J-1/H-F3,J+F3,J-1*+t2*F1,J+F1,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0A,2=1.0d0B,2=1.0d0D,2=0.0d0CMomentumequationforH3andF3 G=F3,J+1-2.0d0*F3,J+F3,J-1/2.0d0/H2*+FT*+H1,J*F3,J+1-F3,J-1/H/2.0d0*+H3,J*F1,J+1-F1,J-1/H/2.0d0*-F1,J*F3,JA,1=0.0d0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0d0A,2=-1.0d0/2.0d0/H2+H1,J/2.0d0/H

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217 B,2=1.0d0/H2+F1,JD,2=-1.0d0/2.0d0/H2-H1,J/2.0d0/H RETURN END ***************** SUBROUTINE INNERH5F5 ************************* SUBROUTINE INNERH5F5J,C,H1,F1,H3,F3,H5,F5 INCLUDE 'DEFINEVAR.f'CContinuityequationforH5andF5 G=2.0d0*H5,J-H5,J-1/H-1.5d0*F5,J+F5,J-1*+t3*F1,J+F1,J-1/4.0d0*+t2*F3,J+F3,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=1.5d0B,2=1.5d0D,2=0.0d0CMomentumequationforH5andF5 G=F5,J+1-2.0d0*F5,J+F5,J-1/2.0d0/H2*+FT*+H1,J*F5,J+1-F5,J-1/H/2.0d0*+H3,J*F3,J+1-F3,J-1/H/2.0d0*+H5,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F1,J*F5,J+3.0d0*F3,J*F3,J/4.0d0A,1=0.0d0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0d0A,2=-1.0d0/2.0d0/H2+H1,J/2.0d0/HB,2=1.0d0/H2+6.0d0*F1,J/4.0d0D,2=-1.0d0/2.0d0/H2-H1,J/2.0d0/H RETURN END ***************** SUBROUTINE INNERH7F7 ************************* SUBROUTINE INNERH7F7J,C,H1,F1,H3,F3,H5,F5,H7,F7 INCLUDE 'DEFINEVAR.f'CContinuityequationH7andF7 G=2.0*H7,J-H7,J-1/H-2.0d0*F7,J+F7,J-1*+t4*F1,J+F1,J-1/4.0d0*+t3*F3,J+F3,J-1/4.0d0*+t2*F5,J+F5,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=2.0d0B,2=2.0d0D,2=0.0d0CMomentumequationF7,H7 G=F7,J+1-2.0d0*F7,J+F7,J-1/2.0d0/H2*+FT*+H1,J*F7,J+1-F7,J-1/H/2.0d0*+H3,J*F5,J+1-F5,J-1/H/2.0d0*+H5,J*F3,J+1-F3,J-1/H/2.0d0*+H7,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F1,J*F7,J+8.0*F3,J*F5,J/4.0d0A,1=0.0

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218 B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+8.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH9F9 ************************* SUBROUTINE INNERH9F9J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9 INCLUDE 'DEFINEVAR.f'CContinuityequationforH9,F9 G=2.0*H9,J-H9,J-1/H-2.5d0*F9,J+F9,J-1*+t5*F1,J+F1,J-1/4.0d0*+t4*F3,J+F3,J-1/4.0d0*+t3*F5,J+F5,J-1/4.0d0*+t2*F7,J+F7,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=2.5d0B,2=2.5d0D,2=0.0d0CMomentumequationforH9,F9 G=F9,J+1-2.0d0*F9,J+F9,J-1/2.0d0/H2*+FT*+H1,J*F9,J+1-F9,J-1/H/2.0d0*+H3,J*F7,J+1-F7,J-1/H/2.0d0*+H5,J*F5,J+1-F5,J-1/H/2.0d0*+H7,J*F3,J+1-F3,J-1/H/2.0d0*+H9,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F1,J*F9,J+5.0d0*F5,J*F5,J*+10.0*F3,J*F7,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+10.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH11F11 *********************** SUBROUTINE INNERH11F11J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11CContinuityequationforH11,F11 G=2.0*H11,J-H11,J-1/H-3.0d0*F11,J+F11,J-1*+t6*F1,J+F1,J-1/4.0d0*+t5*F3,J+F3,J-1/4.0d0*+t4*F5,J+F5,J-1/4.0d0*+t3*F7,J+F7,J-1/4.0d0*+t2*F9,J+F9,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=3.0d0B,2=3.0d0D,2=0.0d0

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219 CMomentumequationforH11,F11 G=F11,J+1-2.0d0*F11,J+F11,J-1/2.0d0/H2*+FT*+H1,J*F11,J+1-F11,J-1/H/2.0d0*+H3,J*F9,J+1-F9,J-1/H/2.0d0*+H5,J*F7,J+1-F7,J-1/H/2.0d0*+H7,J*F5,J+1-F5,J-1/H/2.0d0*+H9,J*F3,J+1-F3,J-1/H/2.0d0*+H11,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F5,J*F7,J+12.0d0*F1,J*F11,J*+12.0*F3,J*F9,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+12.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH13F13 *********************** SUBROUTINE INNERH13F13J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13 INCLUDE 'DEFINEVAR.f'CContinuityequationforH13,F13 G=2.0*H13,J-H13,J-1/H-3.5d0*F13,J+F13,J-1*+t7*F1,J+F1,J-1/4.0d0*+t6*F3,J+F3,J-1/4.0d0*+t5*F5,J+F5,J-1/4.0d0*+t4*F7,J+F7,J-1/4.0d0*+t3*F9,J+F9,J-1/4.0d0*+t2*F11,J+F11,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=3.5d0B,2=3.5d0D,2=0.0d0CMomentumequationforH13,F13 G=F13,J+1-2.0d0*F13,J+F13,J-1/2.0d0/H2*+FT*+H1,J*F13,J+1-F13,J-1/H/2.0d0*+H3,J*F11,J+1-F11,J-1/H/2.0d0*+H5,J*F9,J+1-F9,J-1/H/2.0d0*+H7,J*F7,J+1-F7,J-1/H/2.0d0*+H9,J*F5,J+1-F5,J-1/H/2.0d0*+H11,J*F3,J+1-F3,J-1/H/2.0d0*+H13,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F3,J*F11,J+14.0d0*F5,J*F9,J*+14.0d0*F1,J*F13,J+7.0d0*F7,J*F7,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+14.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H

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220 RETURN END ***************** SUBROUTINE INNERH15F15 *********************** SUBROUTINE INNERH15F15J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO CContinuityequationforH15,F15 G=2.0*H15,J-H15,J-1/H-4.0d0*F15,J+F15,J-1*+t8*F1,J+F1,J-1/4.0d0*+t7*F3,J+F3,J-1/4.0d0*+t6*F5,J+F5,J-1/4.0d0*+t5*F7,J+F7,J-1/4.0d0*+t4*F9,J+F9,J-1/4.0d0*+t3*F11,J+F11,J-1/4.0d0*+t2*F13,J+F13,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=4.0d0B,2=4.0d0D,2=0.0d0CMomentumequationforH15,F15 G=F15,J+1-2.0d0*F15,J+F15,J-1/2.0d0/H2*+FT*+H1,J*F15,J+1-F15,J-1/H/2.0d0*+H3,J*F13,J+1-F13,J-1/H/2.0d0*+H5,J*F11,J+1-F11,J-1/H/2.0d0*+H7,J*F9,J+1-F9,J-1/H/2.0d0*+H9,J*F7,J+1-F7,J-1/H/2.0d0*+H11,J*F5,J+1-F5,J-1/H/2.0d0*+H13,J*F3,J+1-F3,J-1/H/2.0d0*+H15,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F3,J*F13,J+16.0d0*F7,J*F9,J*+16.0d0*F1,J*F15,J+16.0d0*F5,J*F11,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+4.0d0*F1,JD,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH17F17 ************************ SUBROUTINE INNERH17F17J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17 INCLUDE 'DEFINEVAR.f'CContinuityequationforH17,F17

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221 G=2.0*H17,J-H17,J-1/H-4.5d0*F17,J+F17,J-1*+t9*F1,J+F1,J-1/4.0d0*+t8*F3,J+F3,J-1/4.0d0*+t7*F5,J+F5,J-1/4.0d0*+t6*F7,J+F7,J-1/4.0d0*+t5*F9,J+F9,J-1/4.0d0*+t4*F11,J+F11,J-1/4.0d0*+t3*F13,J+F13,J-1/4.0d0*+t2*F15,J+F15,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=4.5d0B,2=4.5d0D,2=0.0d0CMomentumequationforH17,F17 G=F17,J+1-2.0d0*F17,J+F17,J-1/2.0d0/H2*+FT*+H1,J*F17,J+1-F17,J-1/H/2.0d0*+H3,J*F15,J+1-F15,J-1/H/2.0d0*+H5,J*F13,J+1-F13,J-1/H/2.0d0*+H7,J*F11,J+1-F11,J-1/H/2.0d0*+H9,J*F9,J+1-F9,J-1/H/2.0d0*+H11,J*F7,J+1-F7,J-1/H/2.0d0*+H13,J*F5,J+1-F5,J-1/H/2.0d0*+H15,J*F3,J+1-F3,J-1/H/2.0d0*+H17,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F3,J*F15,J+9.0d0*F9,J*F9,J*+18.0d0*F7,J*F11,J*+18.0*F1,J*F17,J+18.0d0*F5,J*F13,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+18.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH19F19 *********************** SUBROUTINE INNERH19F19J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19 INCLUDE 'DEFINEVAR.f'CContinuityequationforH19,F19 G=2.0*H19,J-H19,J-1/H-5.0d0*F19,J+F19,J-1*+t10*F1,J+F1,J-1/4.0d0*+t9*F3,J+F3,J-1/4.0d0*+t8*F5,J+F5,J-1/4.0d0*+t7*F7,J+F7,J-1/4.0d0*+t6*F9,J+F9,J-1/4.0d0*+t5*F11,J+F11,J-1/4.0d0*+t4*F13,J+F13,J-1/4.0d0*+t3*F15,J+F15,J-1/4.0d0*+t2*F17,J+F17,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/H

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222 D,1=0.0d0A,2=5.0d0B,2=5.0d0D,2=0.0d0CMomentumequationforH19,F19 G=F19,J+1-2.0d0*F19,J+F19,J-1/2.0d0/H2*+FT*+H1,J*F19,J+1-F19,J-1/H/2.0d0*+H3,J*F17,J+1-F17,J-1/H/2.0d0*+H5,J*F15,J+1-F15,J-1/H/2.0d0*+H7,J*F13,J+1-F13,J-1/H/2.0d0*+H9,J*F11,J+1-F11,J-1/H/2.0d0*+H11,J*F9,J+1-F9,J-1/H/2.0d0*+H13,J*F7,J+1-F7,J-1/H/2.0d0*+H15,J*F5,J+1-F5,J-1/H/2.0d0*+H17,J*F3,J+1-F3,J-1/H/2.0d0*+H19,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F3,J*F17,J+20.0d0*F9,J*F11,J*+20.0d0*F7,J*F13,J*+20.0*F1,J*F19,J+20.0d0*F5,J*F15,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+20.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH21F21 ************************ SUBROUTINE INNERH21F21J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21 INCLUDE 'DEFINEVAR.f'CContinuityequationforH21,F21 G=2.0d0*H21,J-H21,J-1/H-5.5d0*F21,J+F21,J-1*+t11*F1,J+F1,J-1/4.0d0*+t10*F3,J+F3,J-1/4.0d0*+t9*F5,J+F5,J-1/4.0d0*+t8*F7,J+F7,J-1/4.0d0*+t7*F9,J+F9,J-1/4.0d0*+t6*F11,J+F11,J-1/4.0d0*+t5*F13,J+F13,J-1/4.0d0*+t4*F15,J+F15,J-1/4.0d0*+t3*F17,J+F17,J-1/4.0d0*+t2*F19,J+F19,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=5.5d0B,2=5.5d0D,2=0.0d0CMomentumequationforH21,F21 G=F21,J+1-2.0d0*F21,J+F21,J-1/2.0d0/H2*+FT*+H1,J*F21,J+1-F21,J-1/H/2.0d0*+H3,J*F19,J+1-F19,J-1/H/2.0d0

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223 *+H5,J*F17,J+1-F17,J-1/H/2.0d0*+H7,J*F15,J+1-F15,J-1/H/2.0d0*+H9,J*F13,J+1-F13,J-1/H/2.0d0*+H11,J*F11,J+1-F11,J-1/H/2.0d0*+H13,J*F9,J+1-F9,J-1/H/2.0d0*+H15,J*F7,J+1-F7,J-1/H/2.0d0*+H17,J*F5,J+1-F5,J-1/H/2.0d0*+H19,J*F3,J+1-F3,J-1/H/2.0d0*+H21,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F1,J*F21,J+11.0d0*F11,J*F11,J*+22.0d0*F7,J*F15,J+22.0d0*F5,J*F17,J*+22.0d0*F3,J*F19,J+22.0d0*F9,J*F13,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+22.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH23F23 *********************** SUBROUTINE INNERH23F23J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25,H27,F27*,H29,F29 INCLUDE 'DEFINEVAR.f'CContinuityequationforH23,F23 G=2.0d0*H23,J-H23,J-1/H-6.0d0*F23,J+F23,J-1*+t12*F1,J+F1,J-1/4.0d0*+t11*F3,J+F3,J-1/4.0d0*+t10*F5,J+F5,J-1/4.0d0*+t9*F7,J+F7,J-1/4.0d0*+t8*F9,J+F9,J-1/4.0d0*+t7*F11,J+F11,J-1/4.0d0*+t6*F13,J+F13,J-1/4.0d0*+t5*F15,J+F15,J-1/4.0d0*+t4*F17,J+F17,J-1/4.0d0*+t3*F19,J+F19,J-1/4.0d0*+t2*F21,J+F21,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=6.0d0B,2=6.0d0D,2=0.0d0CMomentumequationforH23,F23 G=F23,J+1-2.0d0*F23,J+F23,J-1/2.0d0/H2*+FT*+H1,J*F23,J+1-F23,J-1/H/2.0d0*+H3,J*F21,J+1-F21,J-1/H/2.0d0*+H5,J*F19,J+1-F19,J-1/H/2.0d0*+H7,J*F17,J+1-F17,J-1/H/2.0d0*+H9,J*F15,J+1-F15,J-1/H/2.0d0*+H11,J*F13,J+1-F13,J-1/H/2.0d0*+H13,J*F11,J+1-F11,J-1/H/2.0d0*+H15,J*F9,J+1-F9,J-1/H/2.0d0

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224 *+H17,J*F7,J+1-F7,J-1/H/2.0d0*+H19,J*F5,J+1-F5,J-1/H/2.0d0*+H21,J*F3,J+1-F3,J-1/H/2.0d0*+H23,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F1,J*F23,J+24.0d0*F7,J*F17,J*+24.0d0*F5,J*F19,J+24.0d0*F3,J*F21,J*+24.0d0*F11,J*F13,J+24.0d0*F9,J*F15,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+24.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNER ***************************** SUBROUTINE INNERH25F25J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25CContinuityequationforH25,F25 G=2.0*H25,J-H25,J-1/H-6.5d0*F25,J+F25,J-1*+t13*F1,J+F1,J-1/4.0d0*+t12*F3,J+F3,J-1/4.0d0*+t11*F5,J+F5,J-1/4.0d0*+t10*F7,J+F7,J-1/4.0d0*+t9*F9,J+F9,J-1/4.0d0*+t8*F11,J+F11,J-1/4.0d0*+t7*F13,J+F13,J-1/4.0d0*+t6*F15,J+F15,J-1/4.0d0*+t5*F17,J+F17,J-1/4.0d0*+t4*F19,J+F19,J-1/4.0d0*+t3*F21,J+F21,J-1/4.0d0*+t2*F23,J+F23,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=5.5d0B,2=5.5d0D,2=0.0d0CMomentumequationforH25,F25 G=F25,J+1-2.0d0*F25,J+F25,J-1/2.0d0/H2*+FT*+H1,J*F25,J+1-F25,J-1/H/2.0d0*+H3,J*F23,J+1-F23,J-1/H/2.0d0*+H5,J*F21,J+1-F21,J-1/H/2.0d0*+H7,J*F19,J+1-F19,J-1/H/2.0d0*+H9,J*F17,J+1-F17,J-1/H/2.0d0*+H11,J*F15,J+1-F15,J-1/H/2.0d0*+H13,J*F13,J+1-F13,J-1/H/2.0d0*+H15,J*F11,J+1-F11,J-1/H/2.0d0*+H17,J*F9,J+1-F9,J-1/H/2.0d0*+H19,J*F7,J+1-F7,J-1/H/2.0d0*+H21,J*F5,J+1-F5,J-1/H/2.0d0*+H23,J*F3,J+1-F3,J-1/H/2.0d0*+H25,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F13,J*F13,J+26.0d0*F11,J*F15,J

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225 *+26.0d0*F7,J*F19,J+26.0d0*F5,J*F21,J*+26.0d0*F3,J*F23,J+26.0d0*F1,J*F25,J*+26.0d0*F9,J*F17,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+26.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ***************** SUBROUTINE INNERH27F27 *********************** SUBROUTINE INNERH27F27J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25,H27,F27 INCLUDE 'DEFINEVAR.f'CContinuityequationforH27,F27 G=2.0*H27,J-H27,J-1/H-7.0d0*F27,J+F27,J-1*+t14*F1,J+F1,J-1/4.0d0*+t13*F3,J+F3,J-1/4.0d0*+t12*F5,J+F5,J-1/4.0d0*+t11*F7,J+F7,J-1/4.0d0*+t10*F9,J+F9,J-1/4.0d0*+t9*F11,J+F11,J-1/4.0d0*+t8*F13,J+F13,J-1/4.0d0*+t7*F15,J+F15,J-1/4.0d0*+t6*F17,J+F17,J-1/4.0d0*+t5*F19,J+F19,J-1/4.0d0*+t4*F21,J+F21,J-1/4.0d0*+t3*F23,J+F23,J-1/4.0d0*+t2*F25,J+F25,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=7.0d0B,2=7.0d0D,2=0.0d0CMomentumequationforH27,F27 G=F27,J+1-2.0d0*F27,J+F27,J-1/2.0d0/H2*+FT*+H1,J*F27,J+1-F27,J-1/H/2.0d0*+H3,J*F25,J+1-F25,J-1/H/2.0d0*+H5,J*F23,J+1-F23,J-1/H/2.0d0*+H7,J*F21,J+1-F21,J-1/H/2.0d0*+H9,J*F19,J+1-F19,J-1/H/2.0d0*+H11,J*F17,J+1-F17,J-1/H/2.0d0*+H13,J*F15,J+1-F15,J-1/H/2.0d0*+H15,J*F13,J+1-F13,J-1/H/2.0d0*+H17,J*F11,J+1-F11,J-1/H/2.0d0*+H19,J*F9,J+1-F9,J-1/H/2.0d0*+H21,J*F7,J+1-F7,J-1/H/2.0d0*+H23,J*F5,J+1-F5,J-1/H/2.0d0*+H25,J*F3,J+1-F3,J-1/H/2.0d0*+H27,J*F1,J+1-F1,J-1/H/2.0d0*-.0d0*F5,J*F23,J+28.0d0*F3,J*F25,J*+28.0d0*F7,J*F21,J+28.0d0*F1,J*F27,J

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226 *+28.0d0*F13,J*F15,J+28.0d0*F11,J*F17,J*+28.0*F9,J*F19,J/4.0d0A,1=0.0B,1=-F1,J+1-F1,J-1/H/2.0d0D,1=0.0A,2=-1.0/2.0/H2+H1,J/2.0/HB,2=1.0/H2+28.0d0*F1,J/4.0d0D,2=-1.0/2.0/H2-H1,J/2.0/H RETURN END ********************** SUBROUTINE BC2H1F1 ********************** SUBROUTINE BC2H1F1J,C,H1,F1 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH1 G=2.0d0*H1,J-H1,J-1/H-F1,J+F1,J-1/2.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0A,2=1.0d0/2.0d0B,2=1.0d0/2.0d0D,2=0.0d0CBoundaryConditiononF1 G=BC-F1,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H3F3 ********************** SUBROUTINE BC2H3F3J,C,H1,F1,H3,F3 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH3 G=2.0d0*H3,J-H3,J-1/H-F3,J+F3,J-1*+t2*F1,J+F1,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0A,2=1.0d0B,2=1.0d0D,2=0.0d0CBoundaryConditiononF3 G=BC-F3,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H5F5 ********************** SUBROUTINE BC2H5F5J,C,H1,F1,H3,F3,H5,F5 INCLUDE 'DEFINEVAR.f'G=2.0d0*H5,J-H5,J-1/H-1.5d0*F5,J+F5,J-1*+t3*F1,J+F1,J-1/4.0d0*+t2*F3,J+F3,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=1.5d0B,2=1.5d0D,2=0.0d0

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227 CBoundaryConditiononF5 G=BC-F5,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H7F7 ********************** SUBROUTINE BC2H7F7J,C,H1,F1,H3,F3,H5,F5,H7,F7 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH7 G=2.0*H7,J-H7,J-1/H-2.0d0*F7,J+F7,J-1*+t4*F1,J+F1,J-1/4.0d0*+t3*F3,J+F3,J-1/4.0d0*+t2*F5,J+F5,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=2.0d0B,2=2.0d0D,2=0.0d0CBoundaryConditiononF7 G=BC-F7,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H9F9 *********************** SUBROUTINE BC2H9F9J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH9 G=2.0*H9,J-H9,J-1/H-2.5d0*F9,J+F9,J-1*+t5*F1,J+F1,J-1/4.0d0*+t4*F3,J+F3,J-1/4.0d0*+t3*F5,J+F5,J-1/4.0d0*+t2*F7,J+F7,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=2.5d0B,2=2.5d0D,2=0.0d0CBoundaryConditiononF9 G=BC-F9,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H11F11 ******************** SUBROUTINE BC2H11F11J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH11 G=2.0*H11,J-H11,J-1/H-3.0d0*F11,J+F11,J-1*+t6*F1,J+F1,J-1/4.0d0*+t5*F3,J+F3,J-1/4.0d0*+t4*F5,J+F5,J-1/4.0d0*+t3*F7,J+F7,J-1/4.0d0*+t2*F9,J+F9,J-1/4.0d0A,1=2.0d0/H

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228 B,1=-2.0d0/HD,1=0.0d0A,2=3.0d0B,2=3.0d0D,2=0.0d0CBoundaryConditiononF11 G=BC-F11,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BCH13F13 ********************** SUBROUTINE BC2H13F13J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH13 G=2.0*H13,J-H13,J-1/H-3.5d0*F13,J+F13,J-1*+t7*F1,J+F1,J-1/4.0d0*+t6*F3,J+F3,J-1/4.0d0*+t5*F5,J+F5,J-1/4.0d0*+t4*F7,J+F7,J-1/4.0d0*+t3*F9,J+F9,J-1/4.0d0*+t2*F11,J+F11,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=3.5d0B,2=3.5d0D,2=0.0d0CBoundaryConditiononF13 G=BC-F13,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H15F15 ******************** SUBROUTINE BC2H15F15J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH15 G=2.0*H15,J-H15,J-1/H-4.0d0*F15,J+F15,J-1*+t8*F1,J+F1,J-1/4.0d0*+t7*F3,J+F3,J-1/4.0d0*+t6*F5,J+F5,J-1/4.0d0*+t5*F7,J+F7,J-1/4.0d0*+t4*F9,J+F9,J-1/4.0d0*+t3*F11,J+F11,J-1/4.0d0*+t2*F13,J+F13,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=4.0d0B,2=4.0d0D,2=0.0d0CBoundaryConditiononF15 G=BC-F15,JB,2=1.0d0

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229 RETURN END ***************** SUBROUTINE BC2H17F17 ************************* SUBROUTINE BC2H17F17J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17 INCLUDE 'DEFINEVAR.f' DO i=1,NGi=0.0d0 DO k=1,NAi,k=0.0d0Bi,k=0.0d0Di,k=0.0d0 END DO END DO CBoundaryConditiononH17 G=2.0*H17,J-H17,J-1/H-4.5d0*F17,J+F17,J-1*+t9*F1,J+F1,J-1/4.0d0*+t8*F3,J+F3,J-1/4.0d0*+t7*F5,J+F5,J-1/4.0d0*+t6*F7,J+F7,J-1/4.0d0*+t5*F9,J+F9,J-1/4.0d0*+t4*F11,J+F11,J-1/4.0d0*+t3*F13,J+F13,J-1/4.0d0*+t2*F15,J+F15,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=4.5d0B,2=4.5d0D,2=0.0d0CBoundaryConditiononF17 G=BC-F17,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H19F19 ********************* SUBROUTINE BC2H19F19J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH19 G=2.0*H19,J-H19,J-1/H-5.0d0*F19,J+F19,J-1*+t10*F1,J+F1,J-1/4.0d0*+t9*F3,J+F3,J-1/4.0d0*+t8*F5,J+F5,J-1/4.0d0*+t7*F7,J+F7,J-1/4.0d0*+t6*F9,J+F9,J-1/4.0d0*+t5*F11,J+F11,J-1/4.0d0*+t4*F13,J+F13,J-1/4.0d0*+t3*F15,J+F15,J-1/4.0d0*+t2*F17,J+F17,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=5.0d0B,2=5.0d0

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230 D,2=0.0d0CBoundaryConditiononF19 G=BC-F19,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H21F21 ********************* SUBROUTINE BC2H21F21J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21CBoundaryConditiononH21 G=2.0d0*H21,J-H21,J-1/H-5.5d0*F21,J+F21,J-1*+t11*F1,J+F1,J-1/4.0d0*+t10*F3,J+F3,J-1/4.0d0*+t9*F5,J+F5,J-1/4.0d0*+t8*F7,J+F7,J-1/4.0d0*+t7*F9,J+F9,J-1/4.0d0*+t6*F11,J+F11,J-1/4.0d0*+t5*F13,J+F13,J-1/4.0d0*+t4*F15,J+F15,J-1/4.0d0*+t3*F17,J+F17,J-1/4.0d0*+t2*F19,J+F19,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=5.5d0B,2=5.5d0D,2=0.0d0CBoundaryConditiononF21 G=BC-F21,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H23F23 ********************* SUBROUTINE BC2H23F23J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH23 G=2.0*H23,J-H23,J-1/H-6.0d0*F23,J+F23,J-1*+t12*F1,J+F1,J-1/4.0d0*+t11*F3,J+F3,J-1/4.0d0*+t10*F5,J+F5,J-1/4.0d0*+t9*F7,J+F7,J-1/4.0d0*+t8*F9,J+F9,J-1/4.0d0*+t7*F11,J+F11,J-1/4.0d0*+t6*F13,J+F13,J-1/4.0d0*+t5*F15,J+F15,J-1/4.0d0*+t4*F17,J+F17,J-1/4.0d0*+t3*F19,J+F19,J-1/4.0d0*+t2*F21,J+F21,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=6.0d0B,2=6.0d0D,2=0.0d0

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231 CBoundaryConditiononF23 G=BC-F23,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H25F25 ********************* SUBROUTINE BC2H25F25J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25CBoundaryConditiononH25 G=2.0*H25,J-H25,J-1/H-6.5d0*F25,J+F25,J-1*+t13*F1,J+F1,J-1/4.0d0*+t12*F3,J+F3,J-1/4.0d0*+t11*F5,J+F5,J-1/4.0d0*+t10*F7,J+F7,J-1/4.0d0*+t9*F9,J+F9,J-1/4.0d0*+t8*F11,J+F11,J-1/4.0d0*+t7*F13,J+F13,J-1/4.0d0*+t6*F15,J+F15,J-1/4.0d0*+t5*F17,J+F17,J-1/4.0d0*+t4*F19,J+F19,J-1/4.0d0*+t3*F21,J+F21,J-1/4.0d0*+t2*F23,J+F23,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/HD,1=0.0d0A,2=5.5d0B,2=5.5d0D,2=0.0d0CBoundaryConditiononF25 G=BC-F25,JB,2=1.0d0 RETURN END ********************** SUBROUTINE BC2H27F27 ******************** SUBROUTINE BC2H27F27J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21*,H23,F23,H25,F25,H27,F27 INCLUDE 'DEFINEVAR.f'CBoundaryConditiononH27 G=2.0*H27,J-H27,J-1/H-7.0d0*F27,J+F27,J-1*+t14*F1,J+F1,J-1/4.0d0*+t13*F3,J+F3,J-1/4.0d0*+t12*F5,J+F5,J-1/4.0d0*+t11*F7,J+F7,J-1/4.0d0*+t10*F9,J+F9,J-1/4.0d0*+t9*F11,J+F11,J-1/4.0d0*+t8*F13,J+F13,J-1/4.0d0*+t7*F15,J+F15,J-1/4.0d0*+t6*F17,J+F17,J-1/4.0d0*+t5*F19,J+F19,J-1/4.0d0*+t4*F21,J+F21,J-1/4.0d0*+t3*F23,J+F23,J-1/4.0d0*+t2*F25,J+F25,J-1/4.0d0A,1=2.0d0/HB,1=-2.0d0/H

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232 D,1=0.0d0A,2=7.0d0B,2=7.0d0D,2=0.0d0CBoundaryConditiononF27 G=BC-F27,JB,2=1.0d0 RETURN END ************ SUBROUTINE GETMAXG ********************* SUBROUTINE GETMAXG_DHDFN,NJ,NJP,DH,DF,maxval INTEGER N,NJ,NJP,NN DOUBLE PRECISION DH,NJ,DF,NJ,maxvalNJP=1maxval=0.0d0NN=N/2 DO 20j=1,NN DO 10i=2,NJ-1 IF dabsDHj,i.GT.maxval THEN maxval=dabsDHj,iNJP=i END IF 10 CONTINUE 20 CONTINUE DO 30j=1,NN DO 40i=2,NJ-1 IF dabsDFj,i.GT.maxval THEN maxval=dabsDFj,iNJP=i END IF 40 CONTINUE 30 CONTINUE RETURN END ************ factrl for ********************** FUNCTION factrln INTEGER n REAL *8factrl INTEGER j,ntop DOUBLE PRECISION aSAVEntop,aa=1.0d0 DO 11j=1,naj+1=j*aj11 CONTINUE factrl=an+1 return END ************************************************

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233 C.1.3IncludeFilesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere. ***************** Include File ALC f *********************** ALLOCATE C,NJ,H1,NJ,F1,NJ ALLOCATE H3,NJ,F3,NJ ALLOCATE H5,NJ,F5,NJ ALLOCATE H7,NJ,F7,NJ ALLOCATE H9,NJ,F9,NJ ALLOCATE H11,NJ,F11,NJ ALLOCATE H13,NJ,F13,NJ ALLOCATE H15,NJ,F15,NJ ALLOCATE H17,NJ,F17,NJ ALLOCATE H19,NJ,F19,NJ ALLOCATE H21,NJ,F21,NJ ALLOCATE H23,NJ,F23,NJ ALLOCATE H25,NJ,F25,NJ ALLOCATE H27,NJ,F27,NJ ALLOCATE DH,NJ,DF,NJ ***************** Include File COTTERM f *********************** CDefiningCottermt2=1.0d0/3.0d0t3=1.0d0/45.0d0t4=2.0d0/945.0d0t5=1.0d0/4725.0d0t6=2.0d0/93555.0d0t7=1382.0d0/638512875.0d0t8=4.0d0/18243225.0d0t9=3617.0d0/162820783125.0d0t10=87734.0d0/38979295480125.0d0t11=349222.0d0/1531329465290625.0d0t12=310732.0d0/13447856940643125.0d0t13=472728182.0d0/201919571963756521875.0d0t14=2631724.0d0/11094481976030578125.0d0t15=13571120588.0d0/564653660170076273671875.0d0CDefininingforcingtermFT=1.0d0FT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrlFT=-.0d0**-1/factrlFT=.0d0**-1/factrl Definining Boundary Condition on $F_ {2 i -1} infty $

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234 BC=2.0d0BC=-.0d0**3/factrlBC=.0d0**5/factrlBC=-.0d0**7/factrlBC=.0d0**9/factrlBC=-.0d0**11/factrlBC=.0d0**13/factrlBC=-.0d0**15/factrlBC=.0d0**17/factrlBC=-.0d0**19/factrlBC=.0d0**21/factrlBC=-.0d0**23/factrlBC=.0d0**25/factrlBC=-.0d0**27/factrlBC=.0d0**29/factrl ********** INCLUDE FILE HF f for main program ************ NAME15:10='VelHF_ALL_' CALL fnumberNJ,FNUMNAME15:16=FNUMNAME15:20='.txt' OPEN unit =15, file =NAME15 CLOSE unit =15, status ='delete' OPEN unit =15, file =NAME15CSetupinitialvelocityprofileforHandF CALL SETUPHFC,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25,H27,F27*,H29,F29CINITILIZEALLCOEFFICIENTMATRIX DO I=1,N DO K=1,NXI,K=0.0d0YI,K=0.0d0 END DO END DO JJ=1101J=0351J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H1F1J,C,H1,F1CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH1F1J,C,H1,F1 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H1F1J,C,H1,F1

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235 END IF CALL BANDJ,C IF J.NE.NJ GOTO 351 DO I=1,NJH1,I=H1,I+C,IF1,I=F1,I+C,I END DO DO i=1,NJ IF dabsH1,i.GT.0.0d0 THEN DH,i=dabsC,i/H1,i ELSE DH,i=1.0d0 ENDIF IF dabsF1,i.GT.0.0d0 THEN DF,i=dabsC,i/F1,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 101 ********** For H3 and F3 ************ JJ=1102J=0352J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H3F3J,C,H1,F1,H3,F3 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH3F3J,C,H1,F1,H3,F3 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H3F3J,C,H1,F1,H3,F3 END IF CALL BANDJ,C IF J.NE.NJ GOTO 352 DO I=1,NJH3,I=H3,I+C,IF3,I=F3,I+C,I END DO DO i=1,NJ IF dabsH3,i.GT.0.0d0 THEN DH,i=dabsC,i/H3,i ELSE

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236 DH,i=1.0d0 ENDIF IF dabsF3,i.GT.0.0d0 THEN DF,i=dabsC,i/F3,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 102 ********** For H5 and F5 ************ JJ=1103J=0353J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H5F5J,C,H1,F1,H3,F3,H5,F5 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH5F5J,C,H1,F1,H3,F3,H5,F5 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H5F5J,C,H1,F1,H3,F3,H5,F5 END IF CALL BANDJ,C IF J.NE.NJ GOTO 353 DO I=1,NJH5,I=H5,I+C,IF5,I=F5,I+C,I END DO DO i=1,NJ IF dabsH5,i.GT.0.0d0 THEN DH,i=dabsC,i/H5,i ELSE DH,i=1.0d0 ENDIF IF dabsF5,i.GT.0.0d0 THEN DF,i=dabsC,i/F5,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 103

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237 ********** For H7 and F7 ************ JJ=1104J=0354J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H7F7J,C,H1,F1,H3,F3,H5,F5,H7,F7 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH7F7J,C,H1,F1,H3,F3,H5,F5,H7,F7 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H7F7J,C,H1,F1,H3,F3,H5,F5,H7,F7 END IF CALL BANDJ,C IF J.NE.NJ GOTO 354 DO I=1,NJH7,I=H7,I+C,IF7,I=F7,I+C,I END DO DO i=1,NJ IF dabsH7,i.GT.0.0d0 THEN DH,i=dabsC,i/H7,i ELSE DH,i=1.0d0 ENDIF IF dabsF7,i.GT.0.0d0 THEN DF,i=dabsC,i/F7,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 104 ********** For H9 and F9 ************ JJ=1105J=0355J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0

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238 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H9F9J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH9F9J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H9F9J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9 END IF CALL BANDJ,C IF J.NE.NJ GOTO 355 DO I=1,NJH9,I=H9,I+C,IF9,I=F9,I+C,I END DO DO i=1,NJ IF dabsH9,i.GT.0.0d0 THEN DH,i=dabsC,i/H9,i ELSE DH,i=1.0d0 ENDIF IF dabsF9,i.GT.0.0d0 THEN DF,i=dabsC,i/F9,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 105 ********** For H11 and F11 ************ JJ=1106J=0356J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H11F11J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11*,F11CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH11F11J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN

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239 CALL BC2H11F11J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11 END IF CALL BANDJ,C IF J.NE.NJ GOTO 356 DO I=1,NJH11,I=H11,I+C,IF11,I=F11,I+C,I END DO DO i=1,NJ IF dabsH11,i.GT.0.0d0 THEN DH,i=dabsC,i/H11,i ELSE DH,i=1.0d0 ENDIF IF dabsF11,i.GT.0.0d0 THEN DF,i=dabsC,i/F11,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 106 ********** For H13 and F13 ************ JJ=1107J=0357J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H13F13J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11*,F11,H13,F13 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH13F13J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H13F13J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21*,H23,F23,H25,F25,H27,F27,H29,F29 END IF CALL BANDJ,C IF J.NE.NJ GOTO 357 DO I=1,NJH13,I=H13,I+C,IF13,I=F13,I+C,I

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240 END DO DO i=1,NJ IF dabsH13,i.GT.0.0d0 THEN DH,i=dabsC,i/H13,i ELSE DH,i=1.0d0 ENDIF IF dabsF13,i.GT.0.0d0 THEN DF,i=dabsC,i/F13,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 107 ********** Solving for H15 and F15 ************ JJ=1108J=0358J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H15F15J,C,H15,F15 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH15F15J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H15F15J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15 END IF CALL BANDJ,C IF J.NE.NJ GOTO 358 DO I=1,NJH15,I=H15,I+C,IF15,I=F15,I+C,I END DO DO i=1,NJ IF dabsH15,i.GT.0.0d0 THEN DH,i=dabsC,i/H15,i ELSE DH,i=1.0d0 ENDIF IF dabsF15,i.GT.0.0d0 THEN DF,i=dabsC,i/F15,i

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241 ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 108 ********** For H17 and F17 ************ JJ=1109J=0359J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H17F17J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11*,F11,H13,F13,H15,F15,H17,F17CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH17F17J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H17F17J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17 END IF CALL BANDJ,C IF J.NE.NJ GOTO 359 DO I=1,NJH17,I=H17,I+C,IF17,I=F17,I+C,I END DO DO i=1,NJ IF dabsH17,i.GT.0.0d0 THEN DH,i=dabsC,i/H17,i ELSE DH,i=1.0d0 ENDIF IF dabsF17,i.GT.0.0d0 THEN DF,i=dabsC,i/F17,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 109 ********** For H19 and F19 ************

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242 JJ=1110J=0360J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H19F19J,C,H19,F19CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH19F19J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H19F19J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19 END IF CALL BANDJ,C IF J.NE.NJ GOTO 360 DO I=1,NJH19,I=H19,I+C,IF19,I=F19,I+C,I END DO DO i=1,NJ IF dabsH19,i.GT.0.0d0 THEN DH,i=dabsC,i/H19,i ELSE DH,i=1.0d0 ENDIF IF dabsF19,i.GT.0.0d0 THEN DF,i=dabsC,i/F19,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 110 ********** For H21 and F21 ************ JJ=1111J=0361J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0

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243 DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H21F21J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11*,F11,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH21F21J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H21F21J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21 END IF CALL BANDJ,C IF J.NE.NJ GOTO 361 DO I=1,NJH21,I=H21,I+C,IF21,I=F21,I+C,I END DO DO i=1,NJ IF dabsH21,i.GT.0.0d0 THEN DH,i=dabsC,i/H21,i ELSE DH,i=1.0d0 ENDIF IF dabsF21,i.GT.0.0d0 THEN DF,i=dabsC,i/F21,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 111 ********** For H23 and F23 ************ JJ=1112J=0362J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO C Call BC1forJ=1 IF J.EQ.1 CALL BC1H23F23J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11*,F11,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23CEquationforinteriorregion IF J.GT.1.AND.J.LT.NJ THEN

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244 CALL INNERH23F23J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23 END IF CEquationforsecondboundarycondition IF J.EQ.NJ THEN CALL BC2H23F23J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23 END IF CALL BANDJ,C IF J.NE.NJ GOTO 362 DO I=1,NJH23,I=H23,I+C,IF23,I=F23,I+C,I END DO DO i=1,NJ IF dabsH23,i.GT.0.0d0 THEN DH,i=dabsC,i/H23,i ELSE DH,i=1.0d0 ENDIF IF dabsF23,i.GT.0.0d0 THEN DF,i=dabsC,i/F23,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 112 ********** For H25 and F25 ************ JJ=1113J=0363J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H25F25J,C,H25,F25 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH25F25J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H25F25J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21*,H23,F23,H25,F25 END IF

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245 CALL BANDJ,C IF J.NE.NJ GOTO 363 DO I=1,NJH25,I=H25,I+C,IF25,I=F25,I+C,I END DO DO i=1,NJ IF dabsH25,i.GT.0.0d0 THEN DH,i=dabsC,i/H25,i ELSE DH,i=1.0d0 ENDIF IF dabsF25,i.GT.0.0d0 THEN DF,i=dabsC,i/F25,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 113 ********** For H27 and F27 ************ JJ=1114J=0364J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1H27F27J,C,H27,F27 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERH27F27J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21,H23,F23,H25,F25,H27,F27 END IF Equation for second boundary condition IF J.EQ.NJ THEN CALL BC2H27F27J,C,H1,F1,H3,F3,H5,F5,H7,F7,H9,F9,H11,F11*,H13,F13,H15,F15,H17,F17,H19,F19,H21,F21*,H23,F23,H25,F25,H27,F27 END IF CALL BANDJ,C IF J.NE.NJ GOTO 364 DO I=1,NJH27,I=H27,I+C,IF27,I=F27,I+C,I END DO DO i=1,NJ IF dabsH27,i.GT.0.0d0 THEN

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246 DH,i=dabsC,i/H27,i ELSE DH,i=1.0d0 ENDIF IF dabsF27,i.GT.0.0d0 THEN DF,i=dabsC,i/F27,i ELSE DF,i=1.0d0 ENDIF END DO CALL GETMAXG_DHDFN,NJ,NJP,DH,DF,maxvalGJJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.NMAX GO TO 114 ****************** Include file COMMON f ***************** COMMON /BA/A,2,B,2,D,5,G,X,2,Y,2,N,NJ COMMON /BB/H,H2 COMMON /COTTERM/t2,t3,t4,t5,t6,t7,t8,t9,t10,t11,t12,t13,t14,t15 COMMON /FANDBCTERM/FT,BC ******************* Include File Cotterm f **************** FP=-F1,3+4.0d0*F1,2/2.0d0/HFP=-F3,3+4.0d0*F3,2/2.0d0/HFP=-F5,3+4.0d0*F5,2/2.0d0/HFP=-F7,3+4.0d0*F7,2/2.0d0/HFP=-F9,3+4.0d0*F9,2/2.0d0/HFP=-F11,3+4.0d0*F11,2/2.0d0/HFP=-F13,3+4.0d0*F13,2/2.0d0/HFP=-F15,3+4.0d0*F15,2/2.0d0/HFP=-F17,3+4.0d0*F17,2/2.0d0/HFP=-F19,3+4.0d0*F19,2/2.0d0/HFP=-F21,3+4.0d0*F21,2/2.0d0/HFP=-F23,3+4.0d0*F23,2/2.0d0/HFP=-F25,3+4.0d0*F25,2/2.0d0/HFP=-F27,3+4.0d0*F27,2/2.0d0/HFP=-F29,3+4.0d0*F29,2/2.0d0/H WRITE ,1410H,H2,FPij,ij=1,15FPP=-F1,4+4.0d0*F1,3-5.0d0*F1,2/H2FPP=-F3,4+4.0d0*F3,3-5.0d0*F3,2/H2FPP=-F5,4+4.0d0*F5,3-5.0d0*F5,2/H2FPP=-F7,4+4.0d0*F7,3-5.0d0*F7,2/H2FPP=-F9,4+4.0d0*F9,3-5.0d0*F9,2/H2FPP=-F11,4+4.0d0*F11,3-5.0d0*F11,2/H2FPP=-F13,4+4.0d0*F13,3-5.0d0*F13,2/H2FPP=-F15,4+4.0d0*F15,3-5.0d0*F15,2/H2FPP=-F17,4+4.0d0*F17,3-5.0d0*F17,2/H2FPP=-F19,4+4.0d0*F19,3-5.0d0*F19,2/H2FPP=-F21,4+4.0d0*F21,3-5.0d0*F21,2/H2FPP=-F23,4+4.0d0*F23,3-5.0d0*F23,2/H2FPP=-F25,4+4.0d0*F25,3-5.0d0*F25,2/H2FPP=-F27,4+4.0d0*F27,3-5.0d0*F27,2/H2FPP=-F29,4+4.0d0*F29,3-5.0d0*F29,2/H2 WRITE ,*H,H2,FPPij,ij=1,151410 FORMAT f57.52,2x

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APPENDIXDPROGRAMLISTINGFORCONVECTIVEDIFFUSIONCALCULATIONSTheprogramlistingpresentsalloftheFORTRANcodetocalculatethecon-vectivediffusionmodelforstationaryhemisphericalelectrodeundersubmergedjetimpingement.Theprogramwasdevelopedwith'CompaqVisualFortran,Ver-sion6.1'withdoubleprecisionaccuracy.Themainprogram'SOLUTIONOFCON-VECTIVE'calledthesubroutinecontaininggoverningequationsandboundaryconditions.Thegoverningequationsfortheconvectivediffusionmodelwereprogrammedinsubroutines'INNERPALL'and'INNERPALLSC'.Varible1;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereprogrammedin'INNERPALL',while2;2i)]TJ/F20 7.97 Tf 6.587 0 Td[(1wereprogrammedin'INNERPALLSC'.Theelec-trodesurfaceandfareldboundaryconditionsforthesubroutines'INNERPALL'wereprogrammedin'BC1PALL'and'BC2PALL',respectively.Similarly,Theelec-trodesurfaceandfareldboundaryconditionsforthesubroutines'INNERPALLSC'wereprogrammedin'BC1PALL'and'BC2PALLSC',respectively.TheboundaryvalueproblemwasnumericallysolvedbysubroutinesBANDandMATINV,whichweredevelopedbyNewman.Theprogramwasiterateduntilallrelativevaluesfor1;2i)]TJ/F15 11.955 Tf 10.213 0 Td[(1,;2i)]TJ/F15 11.955 Tf 10.213 0 Td[(1werewithinaspeciedtolerancelimit.D.1ProgramListingD.1.1MainProgram IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION ALLOCATABLE ::C:,:,P1: DOUBLE PRECISION ALLOCATABLE ::P3:,P5:,P7:,P9: 247

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248 DOUBLE PRECISION ALLOCATABLE ::P11:,P13:,P15:,P17: DOUBLE PRECISION ALLOCATABLE ::P19:,P21:,P23:,P25: DOUBLE PRECISION ALLOCATABLE ::GG:,P27: DOUBLE PRECISION ALLOCATABLE ::P1C:,P3C:,P5C:,P7C: DOUBLE PRECISION ALLOCATABLE ::P9C:,P11C:,P13C:,P15C: DOUBLE PRECISION ALLOCATABLE ::P17C:,P19C:,P21C:,P23C: DOUBLE PRECISION ALLOCATABLE ::P25C:,P27C:,DP:,: DOUBLE PRECISION maxvalG,Z,PP,PCP DOUBLE PRECISION FP,FPP,HPP,HPPP DOUBLE PRECISION H,H2 INTEGER NJL,NJLIST,ITMAX INCLUDE 'VELOCITYDATA.f'NJLIST=1NJL=2001NJL=100001NJL=120001NJL=140001NJL=160001NJL=20001NJL=80001ERREQN=0.0ERRSUB=1.0E-12Z=20.0d0N=14ITMAX=1000 OPEN unit =16, file ='PHP.txt' CLOSE unit =16, status ='delete' OPEN unit =16, file ='PHP.txt' OPEN unit =161, file ='PCHP.txt' CLOSE unit =161, status ='delete' OPEN unit =161, file ='PCHP.txt' DO ii=1,NJLISTNJ=NJLiiH=Z/NJ-1H2=H*H INCLUDE 'ALC.f' INCLUDE 'GETFNAME.f' INCLUDE 'PALL.f' INCLUDE 'CAL_PP.f' INCLUDE 'DLC.f' END DO END D.1.2MainSubroutinesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere.C***************** SUBROUTINE SETUP***************************** SUBROUTINE SETUPPALLP1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21,P23*,P25,P27,P1C,P3C,P5C,P7C,P9C,P11C,P13C,P15C*,P17C,P19C,P21C,P23C,P25C,P27C IMPLICIT DOUBLE PRECISION A-H,O-Z IMPLICIT INTEGER I-N

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249 DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P21*,P23*,P25*,P27* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DOUBLE PRECISION P21C*,P23C*,P25C*,P27C* COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ DO 20II=1,NJP1II=0.0d0P3II=0.0d0P5II=0.0d0P7II=0.0d0P9II=0.0d0P11II=0.0d0P13II=0.0d0P15II=0.0d0P17II=0.0d0P19II=0.00d0P21II=0.00d0P23II=0.00d0P25II=0.00d0P27II=0.00d0P1CII=0.0d0P3CII=0.0d0P5CII=0.0d0P7CII=0.0d0P9CII=0.0d0P11CII=0.0d0P13CII=0.0d0P15CII=0.0d0P17CII=0.0d0P19CII=0.00d0P21CII=0.00d0P23CII=0.00d0P25CII=0.00d0P27CII=0.00d020 CONTINUE P1=1.0d0 RETURN END ***************** SUBROUTINE BC1PALL ************************** SUBROUTINE BC1PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21,P23*,P25,P27 IMPLICIT DOUBLE PRECISION A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P21*,P23*,P25*,P27*G=1.0d0-P1JB,1=1.0d0G=0.0d0-P3JB,2=1.0d0G=0.0d0-P5JB,3=1.0d0G=0.0d0-P7J

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250 B,4=1.0d0G=0.0d0-P9JB,5=1.0d0G=0.0d0-P11JB,6=1.0d0G=0.0d0-P13JB,7=1.0d0G=0.0d0-P15JB,8=1.0d0G=0.0d0-P17JB,9=1.0d0G=0.0d0-P19JB,10=1.0d0G=0.0d0-P21JB,11=1.0d0G=0.0d0-P23JB,12=1.0d0G=0.0d0-P25JB,13=1.0d0G=0.0d0-P27JB,14=1.0d0 RETURN END ***************** SUBROUTINE INNER ***************************** SUBROUTINE INNERPALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21*,P23,P25,P27 IMPLICIT DOUBLE PRECISION A-H,O-Z IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P21*,P23*,P25*,P27* DOUBLE PRECISION ZZ=J-1*H DO i=1,N DO jj=1,NAi,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=P1J+1-2.0d0*P1J+P1J-1/H2*+Z*Z*HPP1*P1J+1-P1J-1/2.0d0/HA,1=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,1=2.0d0/H2D,1=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P3J+1-2.0d0*P3J+P3J-1/H2*+Z*Z*HPP1*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP3*P1J+1-P1J-1/2.0d0/H*-Z*FP1*P3JA,1=Z*Z*HPP3/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP3/2.0d0/HA,2=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,2=2.0d0/H2+Z*FP1

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251 D,2=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P5J+1-2.0d0*P5J+P5J-1/H2*+Z*Z*HPP1*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP3*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP5*P1J+1-P1J-1/2.0d0/H*-Z*FP3*P3J*-2.0d0*Z*FP1*P5JA,1=Z*Z*HPP5/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP5/2.0d0/HA,2=Z*Z*HPP3/2.0d0/HB,2=Z*FP3D,2=-Z*Z*HPP3/2.0d0/HA,3=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,3=2.0d0/H2+2.0d0*Z*FP1D,3=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P7J+1-2.0d0*P7J+P7J-1/H2*+Z*Z*HPP1*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP3*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP5*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP7*P1J+1-P1J-1/2.0d0/H*-Z*FP5*P3J-2.0d0*Z*FP3*P5J-3.0d0*Z*FP1*P7JA,1=Z*Z*HPP7/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP7/2.0d0/HA,2=Z*Z*HPP5/2.0d0/HB,2=Z*FP5D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP3/2.0d0/HB,3=2.0d0*Z*FP3D,3=-Z*Z*HPP3/2.0d0/HA,4=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,4=2.0d0/H2+3.0d0*Z*FP1D,4=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P9J+1-2.0d0*P9J+P9J-1/H2*+Z*Z*HPP1*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP3*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP5*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP7*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP9*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP7*P3J-2.0d0*Z*FP5*P5J-3.0d0*Z*FP3*P7J*-4.0d0*Z*FP1*P9JA,1=Z*Z*HPP9/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP9/2.0d0/HA,2=Z*Z*HPP7/2.0d0/HB,2=1.0d0*Z*FP7D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP5/2.0d0/HB,3=2.0d0*Z*FP5D,3=-Z*Z*HPP5/2.0d0/HA,4=Z*Z*HPP3/2.0d0/HB,4=3.0d0*Z*FP3D,4=-Z*Z*HPP3/2.0d0/HA,5=-1.0d0/H2+Z*Z*HPP1/2.0d0/H

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252 B,5=2.0d0/H2+4.0d0*Z*FP1D,5=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P11J+1-2.0d0*P11J+P11J-1/H2*+Z*Z*HPP1*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP3*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP5*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP7*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP9*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP11*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP9*P3J-2.0d0*Z*FP7*P5J-3.0d0*Z*FP5*P7J*-4.0d0*Z*FP3*P9J-5.0d0*Z*FP1*P11JA,1=Z*Z*HPP11/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP11/2.0d0/HA,2=Z*Z*HPP9/2.0d0/HB,2=1.0d0*Z*FP9D,2=-Z*Z*HPP9/2.0d0/HA,3=Z*Z*HPP7/2.0d0/HB,3=2.0d0*Z*FP7D,3=-Z*Z*HPP7/2.0d0/HA,4=Z*Z*HPP5/2.0d0/HB,4=3.0d0*Z*FP5D,4=-Z*Z*HPP5/2.0d0/HA,5=Z*Z*HPP3/2.0d0/HB,5=4.0d0*Z*FP3D,5=-Z*Z*HPP3/2.0d0/HA,6=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,6=2.0d0/H2+5.0d0*Z*FP1D,6=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P13J+1-2.0d0*P13J+P13J-1/H2*+Z*Z*HPP1*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP3*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP5*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP7*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP9*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP11*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP13*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP11*P3J-2.0d0*Z*FP9*P5J-3.0d0*Z*FP7*P7J*-4.0d0*Z*FP5*P9J-5.0d0*Z*FP3*P11J-6.0d0*Z*FP1*P13JA,1=Z*Z*HPP13/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP13/2.0d0/HA,2=Z*Z*HPP11/2.0d0/HB,2=1.0d0*Z*FP11D,2=-Z*Z*HPP11/2.0d0/HA,3=Z*Z*HPP9/2.0d0/HB,3=2.0d0*Z*FP9D,3=-Z*Z*HPP9/2.0d0/HA,4=Z*Z*HPP7/2.0d0/HB,4=3.0d0*Z*FP7D,4=-Z*Z*HPP7/2.0d0/HA,5=Z*Z*HPP5/2.0d0/HB,5=4.0d0*Z*FP5D,5=-Z*Z*HPP5/2.0d0/HA,6=Z*Z*HPP3/2.0d0/H

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253 B,6=5.0d0*Z*FP3D,6=-Z*Z*HPP3/2.0d0/HA,7=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,7=2.0d0/H2+6.0d0*Z*FP1D,7=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P15J+1-2.0d0*P15J+P15J-1/H2*+Z*Z*HPP1*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP3*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP5*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP7*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP9*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP11*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP13*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP15*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP13*P3J-2.0d0*Z*FP11*P5J-3.0d0*Z*FP9*P7J*-4.0d0*Z*FP7*P9J-5.0d0*Z*FP5*P11J-6.0d0*Z*FP3*P13J*-7.0d0*Z*FP1*P15JA,1=Z*Z*HPP15/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP15/2.0d0/HA,2=Z*Z*HPP13/2.0d0/HB,2=1.0d0*Z*FP13D,2=-Z*Z*HPP13/2.0d0/HA,3=Z*Z*HPP11/2.0d0/HB,3=2.0d0*Z*FP11D,3=-Z*Z*HPP11/2.0d0/HA,4=Z*Z*HPP9/2.0d0/HB,4=3.0d0*Z*FP9D,4=-Z*Z*HPP9/2.0d0/HA,5=Z*Z*HPP7/2.0d0/HB,5=4.0d0*Z*FP7D,5=-Z*Z*HPP7/2.0d0/HA,6=Z*Z*HPP5/2.0d0/HB,6=5.0d0*Z*FP5D,6=-Z*Z*HPP5/2.0d0/HA,7=Z*Z*HPP3/2.0d0/HB,7=6.0d0*Z*FP3D,7=-Z*Z*HPP3/2.0d0/HA,8=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,8=2.0d0/H2+7.0d0*Z*FP1D,8=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P17J+1-2.0d0*P17J+P17J-1/H2*+Z*Z*HPP1*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP3*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP5*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP7*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP9*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP11*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP13*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP15*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP17*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP15*P3J-2.0d0*Z*FP13*P5J-3.0d0*Z*FP11*P7J*-4.0d0*Z*FP9*P9J-5.0d0*Z*FP7*P11J-6.0d0*Z*FP5*P13J*-7.0d0*Z*FP3*P15J-8.0d0*Z*FP1*P17JA,1=Z*Z*HPP17/2.0d0/H

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254 B,1=0.0d0D,1=-Z*Z*HPP17/2.0d0/HA,2=Z*Z*HPP15/2.0d0/HB,2=1.0d0*Z*FP15D,2=-Z*Z*HPP15/2.0d0/HA,3=Z*Z*HPP13/2.0d0/HB,3=2.0d0*Z*FP13D,3=-Z*Z*HPP13/2.0d0/HA,4=Z*Z*HPP11/2.0d0/HB,4=3.0d0*Z*FP11D,4=-Z*Z*HPP11/2.0d0/HA,5=Z*Z*HPP9/2.0d0/HB,5=4.0d0*Z*FP9D,5=-Z*Z*HPP9/2.0d0/HA,6=Z*Z*HPP7/2.0d0/HB,6=5.0d0*Z*FP7D,6=-Z*Z*HPP7/2.0d0/HA,7=Z*Z*HPP5/2.0d0/HB,7=6.0d0*Z*FP5D,7=-Z*Z*HPP5/2.0d0/HA,8=Z*Z*HPP3/2.0d0/HB,8=7.0d0*Z*FP3D,8=-Z*Z*HPP3/2.0d0/HA,9=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,9=2.0d0/H2+8.0d0*Z*FP1D,9=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P19J+1-2.0d0*P19J+P19J-1/H2*+Z*Z*HPP1*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP3*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP5*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP7*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP9*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP11*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP13*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP15*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP17*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP19*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP17*P3J-2.0d0*Z*FP15*P5J-3.0d0*Z*FP13*P7J*-4.0d0*Z*FP11*P9J-5.0d0*Z*FP9*P11J-6.0d0*Z*FP7*P13J*-7.0d0*Z*FP5*P15J-8.0d0*Z*FP3*P17J-9.0d0*Z*FP1*P19JA,1=Z*Z*HPP19/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP19/2.0d0/HA,2=Z*Z*HPP17/2.0d0/HB,2=1.0d0*Z*FP17D,2=-Z*Z*HPP17/2.0d0/HA,3=Z*Z*HPP15/2.0d0/HB,3=2.0d0*Z*FP15D,3=-Z*Z*HPP15/2.0d0/HA,4=Z*Z*HPP13/2.0d0/HB,4=3.0d0*Z*FP13D,4=-Z*Z*HPP13/2.0d0/HA,5=Z*Z*HPP11/2.0d0/HB,5=4.0d0*Z*FP11D,5=-Z*Z*HPP11/2.0d0/H

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255 A,6=Z*Z*HPP9/2.0d0/HB,6=5.0d0*Z*FP9D,6=-Z*Z*HPP9/2.0d0/HA,7=Z*Z*HPP7/2.0d0/HB,7=6.0d0*Z*FP7D,7=-Z*Z*HPP7/2.0d0/HA,8=Z*Z*HPP5/2.0d0/HB,8=7.0d0*Z*FP5D,8=-Z*Z*HPP5/2.0d0/HA,9=Z*Z*HPP3/2.0d0/HB,9=8.0d0*Z*FP3D,9=-Z*Z*HPP3/2.0d0/HA,10=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,10=2.0d0/H2+9.0d0*Z*FP1D,10=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P21J+1-2.0d0*P21J+P21J-1/H2*+Z*Z*HPP1*P21J+1-P21J-1/2.0d0/H*+Z*Z*HPP3*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP5*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP7*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP9*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP11*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP13*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP15*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP17*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP19*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP21*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP19*P3J-2.0d0*Z*FP17*P5J-3.0d0*Z*FP15*P7J*-4.0d0*Z*FP13*P9J-5.0d0*Z*FP11*P11J-6.0d0*Z*FP9*P13J*-7.0d0*Z*FP7*P15J-8.0d0*Z*FP5*P17J-9.0d0*Z*FP3*P19J*-10.0d0*Z*FP1*P21JA,1=Z*Z*HPP21/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP21/2.0d0/HA,2=Z*Z*HPP19/2.0d0/HB,2=1.0d0*Z*FP19D,2=-Z*Z*HPP19/2.0d0/HA,3=Z*Z*HPP17/2.0d0/HB,3=2.0d0*Z*FP17D,3=-Z*Z*HPP17/2.0d0/HA,4=Z*Z*HPP15/2.0d0/HB,4=3.0d0*Z*FP15D,4=-Z*Z*HPP15/2.0d0/HA,5=Z*Z*HPP13/2.0d0/HB,5=4.0d0*Z*FP13D,5=-Z*Z*HPP13/2.0d0/HA,6=Z*Z*HPP11/2.0d0/HB,6=5.0d0*Z*FP11D,6=-Z*Z*HPP11/2.0d0/HA,7=Z*Z*HPP9/2.0d0/HB,7=6.0d0*Z*FP9D,7=-Z*Z*HPP9/2.0d0/HA,8=Z*Z*HPP7/2.0d0/HB,8=7.0d0*Z*FP7D,8=-Z*Z*HPP7/2.0d0/H

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256 A,9=Z*Z*HPP5/2.0d0/HB,9=8.0d0*Z*FP5D,9=-Z*Z*HPP5/2.0d0/HA,10=Z*Z*HPP3/2.0d0/HB,10=9.0d0*Z*FP3D,10=-Z*Z*HPP3/2.0d0/HA,11=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,11=2.0d0/H2+10.0d0*Z*FP1D,11=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P23J+1-2.0d0*P23J+P23J-1/H2*+Z*Z*HPP1*P23J+1-P23J-1/2.0d0/H*+Z*Z*HPP3*P21J+1-P21J-1/2.0d0/H*+Z*Z*HPP5*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP7*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP9*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP11*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP13*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP15*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP17*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP19*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP21*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP23*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP21*P3J-2.0d0*Z*FP19*P5J-3.0d0*Z*FP17*P7J*-4.0d0*Z*FP15*P9J-5.0d0*Z*FP13*P11J-6.0d0*Z*FP11*P13J*-7.0d0*Z*FP9*P15J-8.0d0*Z*FP7*P17J-9.0d0*Z*FP5*P19J*-10.0d0*Z*FP3*P21J-11.0d0*Z*FP1*P23JA,1=Z*Z*HPP23/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP23/2.0d0/HA,2=Z*Z*HPP21/2.0d0/HB,2=1.0d0*Z*FP21D,2=-Z*Z*HPP21/2.0d0/HA,3=Z*Z*HPP19/2.0d0/HB,3=2.0d0*Z*FP19D,3=-Z*Z*HPP19/2.0d0/HA,4=Z*Z*HPP17/2.0d0/HB,4=3.0d0*Z*FP17D,4=-Z*Z*HPP17/2.0d0/HA,5=Z*Z*HPP15/2.0d0/HB,5=4.0d0*Z*FP15D,5=-Z*Z*HPP15/2.0d0/HA,6=Z*Z*HPP13/2.0d0/HB,6=5.0d0*Z*FP13D,6=-Z*Z*HPP13/2.0d0/HA,7=Z*Z*HPP11/2.0d0/HB,7=6.0d0*Z*FP11D,7=-Z*Z*HPP11/2.0d0/HA,8=Z*Z*HPP9/2.0d0/HB,8=7.0d0*Z*FP9D,8=-Z*Z*HPP9/2.0d0/HA,9=Z*Z*HPP7/2.0d0/HB,9=8.0d0*Z*FP7D,9=-Z*Z*HPP7/2.0d0/HA,10=Z*Z*HPP5/2.0d0/HB,10=9.0d0*Z*FP5

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257 D,10=-Z*Z*HPP5/2.0d0/HA,11=Z*Z*HPP3/2.0d0/HB,11=10.0d0*Z*FP3D,11=-Z*Z*HPP3/2.0d0/HA,12=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,12=2.0d0/H2+11.0d0*Z*FP1D,12=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P25J+1-2.0d0*P25J+P25J-1/H2*+Z*Z*HPP1*P25J+1-P25J-1/2.0d0/H*+Z*Z*HPP3*P23J+1-P23J-1/2.0d0/H*+Z*Z*HPP5*P21J+1-P21J-1/2.0d0/H*+Z*Z*HPP7*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP9*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP11*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP13*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP15*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP17*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP19*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP21*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP23*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP25*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP23*P3J-2.0d0*Z*FP21*P5J-3.0d0*Z*FP19*P7J*-4.0d0*Z*FP17*P9J-5.0d0*Z*FP15*P11J-6.0d0*Z*FP13*P13J*-7.0d0*Z*FP11*P15J-8.0d0*Z*FP9*P17J-9.0d0*Z*FP7*P19J*-10.0d0*Z*FP5*P21J-11.0d0*Z*FP3*P23J-12.0d0*Z*FP1*P25JA,1=Z*Z*HPP25/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP25/2.0d0/HA,2=Z*Z*HPP23/2.0d0/HB,2=1.0d0*Z*FP23D,2=-Z*Z*HPP23/2.0d0/HA,3=Z*Z*HPP21/2.0d0/HB,3=2.0d0*Z*FP21D,3=-Z*Z*HPP21/2.0d0/HA,4=Z*Z*HPP21/2.0d0/HB,4=3.0d0*Z*FP21D,4=-Z*Z*HPP21/2.0d0/HA,5=Z*Z*HPP17/2.0d0/HB,5=4.0d0*Z*FP17D,5=-Z*Z*HPP17/2.0d0/HA,6=Z*Z*HPP15/2.0d0/HB,6=5.0d0*Z*FP15D,6=-Z*Z*HPP15/2.0d0/HA,7=Z*Z*HPP13/2.0d0/HB,7=6.0d0*Z*FP13D,7=-Z*Z*HPP13/2.0d0/HA,8=Z*Z*HPP11/2.0d0/HB,8=7.0d0*Z*FP11D,8=-Z*Z*HPP11/2.0d0/HA,9=Z*Z*HPP9/2.0d0/HB,9=8.0d0*Z*FP9D,9=-Z*Z*HPP9/2.0d0/HA,10=Z*Z*HPP7/2.0d0/HB,10=9.0d0*Z*FP7D,10=-Z*Z*HPP7/2.0d0/H

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258 A,11=Z*Z*HPP5/2.0d0/HB,11=10.0d0*Z*FP5D,11=-Z*Z*HPP5/2.0d0/HA,12=Z*Z*HPP3/2.0d0/HB,12=11.0d0*Z*FP3D,12=-Z*Z*HPP3/2.0d0/HA,13=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,13=2.0d0/H2+12.0d0*Z*FP1D,13=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P27J+1-2.0d0*P27J+P27J-1/H2*+Z*Z*HPP1*P27J+1-P27J-1/2.0d0/H*+Z*Z*HPP3*P25J+1-P25J-1/2.0d0/H*+Z*Z*HPP5*P23J+1-P23J-1/2.0d0/H*+Z*Z*HPP7*P21J+1-P21J-1/2.0d0/H*+Z*Z*HPP9*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP11*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP13*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP15*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP17*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP19*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP21*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP23*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP25*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP27*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP25*P3J-2.0d0*Z*FP23*P5J-3.0d0*Z*FP21*P7J*-4.0d0*Z*FP19*P9J-5.0d0*Z*FP17*P11J-6.0d0*Z*FP15*P13J*-7.0d0*Z*FP13*P15J-8.0d0*Z*FP11*P17J-9.0d0*Z*FP9*P19J*-10.0d0*Z*FP7*P21J-11.0d0*Z*FP5*P23J-12.0d0*Z*FP3*P25J*-13.0d0*Z*FP1*P27JA,1=Z*Z*HPP27/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP27/2.0d0/HA,2=Z*Z*HPP25/2.0d0/HB,2=1.0d0*Z*FP25D,2=-Z*Z*HPP25/2.0d0/HA,3=Z*Z*HPP23/2.0d0/HB,3=2.0d0*Z*FP23D,3=-Z*Z*HPP23/2.0d0/HA,4=Z*Z*HPP21/2.0d0/HB,4=3.0d0*Z*FP21D,4=-Z*Z*HPP21/2.0d0/HA,5=Z*Z*HPP19/2.0d0/HB,5=4.0d0*Z*FP19D,5=-Z*Z*HPP19/2.0d0/HA,6=Z*Z*HPP17/2.0d0/HB,6=5.0d0*Z*FP17D,6=-Z*Z*HPP17/2.0d0/HA,7=Z*Z*HPP15/2.0d0/HB,7=6.0d0*Z*FP15D,7=-Z*Z*HPP15/2.0d0/HA,8=Z*Z*HPP13/2.0d0/HB,8=7.0d0*Z*FP13D,8=-Z*Z*HPP13/2.0d0/HA,9=Z*Z*HPP11/2.0d0/HB,9=8.0d0*Z*FP11

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259 D,9=-Z*Z*HPP11/2.0d0/HA,10=Z*Z*HPP9/2.0d0/HB,10=9.0d0*Z*FP9D,10=-Z*Z*HPP9/2.0d0/HA,11=Z*Z*HPP7/2.0d0/HB,11=10.0d0*Z*FP7D,11=-Z*Z*HPP7/2.0d0/HA,12=Z*Z*HPP5/2.0d0/HB,12=11.0d0*Z*FP5D,12=-Z*Z*HPP5/2.0d0/HA,13=Z*Z*HPP3/2.0d0/HB,13=12.0d0*Z*FP3D,13=Z*Z*HPP3/2.0d0/HA,14=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,14=2.0d0/H2+13.0d0*Z*FP1D,14=-1.0d0/H2-Z*Z*HPP1/2.0d0/H RETURN END ********************** SUBROUTINE BC2PALL ********************** SUBROUTINE BC2PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21*,P23,P25,P27 IMPLICIT DOUBLE PRECISION A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P21*,P23*,P25*,P27*G=0.0d0-P1JB,1=1.0d0G=0.0d0-P3JB,2=1.0d0G=0.0d0-P5JB,3=1.0d0G=0.0d0-P7JB,4=1.0d0G=0.0d0-P9JB,5=1.0d0G=0.0d0-P11JB,6=1.0d0G=0.0d0-P13JB,7=1.0d0G=0.0d0-P15JB,8=1.0d0G=0.0d0-P17JB,9=1.0d0G=0.0d0-P19JB,10=1.0d0G=0.0d0-P21JB,11=1.0d0G=0.0d0-P23JB,12=1.0d0G=0.0d0-P25JB,13=1.0d0G=0.0d0-P27JB,14=1.0d0

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260 RETURN END *************** SUBROUTINE BC1PALLSC ************************ SUBROUTINE BC1PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,P13C,*P15C,P17C,P19C,P21C,P23C,P25C,P27C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ COMMON /BB/H,H2 DOUBLE PRECISION C,* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DOUBLE PRECISION P21C*,P23C*,P25C*,P27C* DO i=1,N DO jj=1,NAi,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=0.0d0-P1CJB,1=1.0d0G=0.0d0-P3CJB,2=1.0d0G=0.0d0-P5CJB,3=1.0d0G=0.0d0-P7CJB,4=1.0d0G=0.0d0-P9CJB,5=1.0d0G=0.0d0-P11CJB,6=1.0d0G=0.0d0-P13CJB,7=1.0d0G=0.0d0-P15CJB,8=1.0d0G=0.0d0-P17CJB,9=1.0d0G=0.0d0-P19CJB,10=1.0d0G=0.0d0-P21CJB,11=1.0d0G=0.0d0-P23CJB,12=1.0d0G=0.0d0-P25CJB,13=1.0d0G=0.0d0-P27CJB,14=1.0d0 RETURN END *************** SUBROUTINE INNERPALLSC ********************** SUBROUTINE INNERPALLSCJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*P21,P23,P25,P27,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C,P21C,P23C,P25C,P27C IMPLICIT REAL *8A-H,O-Z

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261 IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P21*,P23*,P25*,P27* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DOUBLE PRECISION P21C*,P23C*,P25C*,P27C* DOUBLE PRECISION ZZ=J-1*H DO i=1,N DO jj=1,NAi,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=P1CJ+1-2.0d0*P1CJ+P1CJ-1/H2*+Z*Z*HPP1*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P1J+1-P1J-1/2.0d0/H/3.0d0A,1=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,1=2.0d0/H2D,1=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P3CJ+1-2.0d0*P3CJ+P3CJ-1/H2*+Z*Z*HPP1*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP3*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P1J+1-P1J-1/2.0d0/H/3.0d0*-Z*FP1*P3CJ-0.50d0*Z*Z*FPP1*P3JA,1=Z*Z*HPP3/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP3/2.0d0/HA,2=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,2=2.0d0/H2+Z*FP1D,2=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P5CJ+1-2.0d0*P5CJ+P5CJ-1/H2*+Z*Z*HPP1*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP3*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP5*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP3*P3CJ-2.0d0*Z*FP1*P5CJ*-0.50d0*Z*Z*FPP3*P3J-1.00d0*Z*Z*FPP1*P5JA,1=Z*Z*HPP5/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP5/2.0d0/HA,2=Z*Z*HPP3/2.0d0/HB,2=Z*FP3D,2=-Z*Z*HPP3/2.0d0/HA,3=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,3=2.0d0/H2+2.0d0*Z*FP1D,3=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P7CJ+1-2.0d0*P7CJ+P7CJ-1/H2*+Z*Z*HPP1*P7CJ+1-P7CJ-1/2.0d0/H

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262 *+Z*Z*HPP3*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP5*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP7*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP5*P3CJ-2.0d0*Z*FP3*P5CJ-3.0d0*Z*FP1*P7CJ*-0.50d0*Z*Z*FPP5*P3J-1.00d0*Z*Z*FPP3*P5J-1.50d0*Z*Z*FPP1*P7JA,1=Z*Z*HPP7/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP7/2.0d0/HA,2=Z*Z*HPP5/2.0d0/HB,2=Z*FP5D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP3/2.0d0/HB,3=2.0d0*Z*FP3D,3=-Z*Z*HPP3/2.0d0/HA,4=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,4=2.0d0/H2+3.0d0*Z*FP1D,4=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P9CJ+1-2.0d0*P9CJ+P9CJ-1/H2*+Z*Z*HPP1*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP3*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP5*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP7*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP9*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP7*P3CJ-2.0d0*Z*FP5*P5CJ-3.0d0*Z*FP3*P7CJ*-4.0d0*Z*FP1*P9CJ-0.50d0*Z*Z*FPP7*P3J-1.00d0*Z*Z*FPP5*P5J*-1.50d0*Z*Z*FPP3*P7J-2.00d0*Z*Z*FPP1*P9JA,1=Z*Z*HPP9/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP9/2.0d0/HA,2=Z*Z*HPP7/2.0d0/HB,2=1.0d0*Z*FP7D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP5/2.0d0/HB,3=2.0d0*Z*FP5D,3=-Z*Z*HPP5/2.0d0/HA,4=Z*Z*HPP3/2.0d0/HB,4=3.0d0*Z*FP3D,4=-Z*Z*HPP3/2.0d0/HA,5=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,5=2.0d0/H2+4.0d0*Z*FP1D,5=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P11CJ+1-2.0d0*P11CJ+P11CJ-1/H2*+Z*Z*HPP1*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP3*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP5*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP7*P5CJ+1-P5CJ-1/2.0d0/H

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263 *+Z*Z*HPP9*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP11*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP9*P3CJ-2.0d0*Z*FP7*P5CJ-3.0d0*Z*FP5*P7CJ*-4.0d0*Z*FP3*P9CJ-5.0d0*Z*FP1*P11CJ-0.50d0*Z*Z*FPP9*P3J*-1.00d0*Z*Z*FPP7*P5J-1.50d0*Z*Z*FPP5*P7J*-2.00d0*Z*Z*FPP3*P9J-2.50d0*Z*Z*FPP1*P11JA,1=Z*Z*HPP11/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP11/2.0d0/HA,2=Z*Z*HPP9/2.0d0/HB,2=1.0d0*Z*FP9D,2=-Z*Z*HPP9/2.0d0/HA,3=Z*Z*HPP7/2.0d0/HB,3=2.0d0*Z*FP7D,3=-Z*Z*HPP7/2.0d0/HA,4=Z*Z*HPP5/2.0d0/HB,4=3.0d0*Z*FP5D,4=-Z*Z*HPP5/2.0d0/HA,5=Z*Z*HPP3/2.0d0/HB,5=4.0d0*Z*FP3D,5=-Z*Z*HPP3/2.0d0/HA,6=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,6=2.0d0/H2+5.0d0*Z*FP1D,6=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P13CJ+1-2.0d0*P13CJ+P13CJ-1/H2*+Z*Z*HPP1*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP3*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP5*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP7*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP9*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP11*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP13*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP11*P3CJ-2.0d0*Z*FP9*P5CJ-3.0d0*Z*FP7*P7CJ*-4.0d0*Z*FP5*P9CJ-5.0d0*Z*FP3*P11CJ-6.0d0*Z*FP1*P13CJ*-0.50d0*Z*Z*FPP11*P3J-1.00d0*Z*Z*FPP9*P5J-1.50d0*Z*Z*FPP7*P7J*-2.00d0*Z*Z*FPP5*P9J-2.50d0*Z*Z*FPP3*P11J-3.00d0*Z*Z*FPP1*P13JA,1=Z*Z*HPP13/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP13/2.0d0/HA,2=Z*Z*HPP11/2.0d0/HB,2=1.0d0*Z*FP11D,2=-Z*Z*HPP11/2.0d0/H

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264 A,3=Z*Z*HPP9/2.0d0/HB,3=2.0d0*Z*FP9D,3=-Z*Z*HPP9/2.0d0/HA,4=Z*Z*HPP7/2.0d0/HB,4=3.0d0*Z*FP7D,4=-Z*Z*HPP7/2.0d0/HA,5=Z*Z*HPP5/2.0d0/HB,5=4.0d0*Z*FP5D,5=-Z*Z*HPP5/2.0d0/HA,6=Z*Z*HPP3/2.0d0/HB,6=5.0d0*Z*FP3D,6=-Z*Z*HPP3/2.0d0/HA,7=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,7=2.0d0/H2+6.0d0*Z*FP1D,7=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P15CJ+1-2.0d0*P15CJ+P15CJ-1/H2*+Z*Z*HPP1*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP3*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP5*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP7*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP9*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP11*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP13*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP15*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP13*P3CJ-2.0d0*Z*FP11*P5CJ*-3.0d0*Z*FP9*P7CJ*-4.0d0*Z*FP7*P9CJ-5.0d0*Z*FP5*P11CJ*-6.0d0*Z*FP3*P13CJ*-7.0d0*Z*FP1*P15CJ-0.50d0*Z*Z*FPP13*P3J-1.0d0*Z*Z*FPP11*P5J*-1.50d0*Z*Z*FPP9*P7J-2.0d0*Z*Z*FPP7*P9J-2.50d0*Z*Z*FPP5*P11J*-3.0d0*Z*Z*FPP3*P13J-3.50d0*Z*Z*FPP1*P15JA,1=Z*Z*HPP15/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP15/2.0d0/HA,2=Z*Z*HPP13/2.0d0/HB,2=1.0d0*Z*FP13D,2=-Z*Z*HPP13/2.0d0/HA,3=Z*Z*HPP11/2.0d0/HB,3=2.0d0*Z*FP11D,3=-Z*Z*HPP11/2.0d0/HA,4=Z*Z*HPP9/2.0d0/HB,4=3.0d0*Z*FP9D,4=-Z*Z*HPP9/2.0d0/HA,5=Z*Z*HPP7/2.0d0/HB,5=4.0d0*Z*FP7D,5=-Z*Z*HPP7/2.0d0/HA,6=Z*Z*HPP5/2.0d0/HB,6=5.0d0*Z*FP5D,6=-Z*Z*HPP5/2.0d0/H

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265 A,7=Z*Z*HPP3/2.0d0/HB,7=6.0d0*Z*FP3D,7=-Z*Z*HPP3/2.0d0/HA,8=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,8=2.0d0/H2+7.0d0*Z*FP1D,8=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P17CJ+1-2.0d0*P17CJ+P17CJ-1/H2*+Z*Z*HPP1*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP3*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP5*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP7*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP9*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP11*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP13*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP15*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP17*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP15*P3CJ-2.0d0*Z*FP13*P5CJ-3.0d0*Z*FP11*P7CJ*-4.0d0*Z*FP9*P9CJ-5.0d0*Z*FP7*P11CJ-6.0d0*Z*FP5*P13CJ*-7.0d0*Z*FP3*P15CJ-8.0d0*Z*FP1*P17CJ-0.50d0*Z*Z*FPP15*P3J*-1.0d0*Z*Z*FPP13*P5J-1.50d0*Z*Z*FPP11*P7J-2.0d0*Z*Z*FPP9*P9J*-2.50d0*Z*Z*FPP7*P11J-3.0d0*Z*Z*FPP5*P13J*-3.50d0*Z*Z*FPP3*P15J-4.00d0*Z*Z*FPP1*P17JA,1=Z*Z*HPP17/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP17/2.0d0/HA,2=Z*Z*HPP15/2.0d0/HB,2=1.0d0*Z*FP15D,2=-Z*Z*HPP15/2.0d0/HA,3=Z*Z*HPP13/2.0d0/HB,3=2.0d0*Z*FP13D,3=-Z*Z*HPP13/2.0d0/HA,4=Z*Z*HPP11/2.0d0/HB,4=3.0d0*Z*FP11D,4=-Z*Z*HPP11/2.0d0/HA,5=Z*Z*HPP9/2.0d0/HB,5=4.0d0*Z*FP9D,5=-Z*Z*HPP9/2.0d0/HA,6=Z*Z*HPP7/2.0d0/HB,6=5.0d0*Z*FP7D,6=-Z*Z*HPP7/2.0d0/HA,7=Z*Z*HPP5/2.0d0/HB,7=6.0d0*Z*FP5D,7=-Z*Z*HPP5/2.0d0/HA,8=Z*Z*HPP3/2.0d0/HB,8=7.0d0*Z*FP3D,8=-Z*Z*HPP3/2.0d0/H

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266 A,9=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,9=2.0d0/H2+8.0d0*Z*FP1D,9=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P19CJ+1-2.0d0*P19CJ+P19CJ-1/H2*+Z*Z*HPP1*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP3*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP5*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP7*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP9*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP11*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP13*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP15*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP17*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP19*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP17*P3CJ-2.0d0*Z*FP15*P5CJ-3.0d0*Z*FP13*P7CJ*-4.0d0*Z*FP11*P9CJ-5.0d0*Z*FP9*P11CJ-6.0d0*Z*FP7*P13CJ*-7.0d0*Z*FP5*P15CJ-8.0d0*Z*FP3*P17CJ-9.0d0*Z*FP1*P19CJ*-0.50d0*Z*Z*FPP17*P3J-1.00d0*Z*Z*FPP15*P5J*-1.50d0*Z*Z*FPP13*P7J-2.00d0*Z*Z*FPP11*P9J*-2.50d0*Z*Z*FPP9*P11J-3.00d0*Z*Z*FPP7*P13J*-3.50d0*Z*Z*FPP5*P15J-4.00d0*Z*Z*FPP3*P17J*-4.50d0*Z*Z*FPP1*P19JA,1=Z*Z*HPP19/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP19/2.0d0/HA,2=Z*Z*HPP17/2.0d0/HB,2=1.0d0*Z*FP17D,2=-Z*Z*HPP17/2.0d0/HA,3=Z*Z*HPP15/2.0d0/HB,3=2.0d0*Z*FP15D,3=-Z*Z*HPP15/2.0d0/HA,4=Z*Z*HPP13/2.0d0/HB,4=3.0d0*Z*FP13D,4=-Z*Z*HPP13/2.0d0/HA,5=Z*Z*HPP11/2.0d0/HB,5=4.0d0*Z*FP11D,5=-Z*Z*HPP11/2.0d0/HA,6=Z*Z*HPP9/2.0d0/HB,6=5.0d0*Z*FP9D,6=-Z*Z*HPP9/2.0d0/HA,7=Z*Z*HPP7/2.0d0/HB,7=6.0d0*Z*FP7D,7=-Z*Z*HPP7/2.0d0/HA,8=Z*Z*HPP5/2.0d0/HB,8=7.0d0*Z*FP5

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267 D,8=-Z*Z*HPP5/2.0d0/HA,9=Z*Z*HPP3/2.0d0/HB,9=8.0d0*Z*FP3D,9=-Z*Z*HPP3/2.0d0/HA,10=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,10=2.0d0/H2+9.0d0*Z*FP1D,10=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P21CJ+1-2.0d0*P21CJ+P21CJ-1/H2*+Z*Z*HPP1*P21CJ+1-P21CJ-1/2.0d0/H*+Z*Z*HPP3*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP5*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP7*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP9*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP11*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP13*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP15*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP17*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP19*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP21*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P21J+1-P21J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP21*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP19*P3CJ-2.0d0*Z*FP17*P5CJ-3.0d0*Z*FP15*P7CJ*-4.0d0*Z*FP13*P9CJ-5.0d0*Z*FP11*P11CJ-6.0d0*Z*FP9*P13CJ*-7.0d0*Z*FP7*P15CJ-8.0d0*Z*FP5*P17CJ-9.0d0*Z*FP3*P19CJ*-10.0d0*Z*FP1*P21CJ-0.50d0*Z*Z*FPP19*P3J*-1.00d0*Z*Z*FPP17*P5J-1.50d0*Z*Z*FPP15*P7J*-2.00d0*Z*Z*FPP13*P9J-2.50d0*Z*Z*FPP11*P11J*-3.00d0*Z*Z*FPP9*P13J-3.50d0*Z*Z*FPP7*P15J*-4.00d0*Z*Z*FPP5*P17J-4.50d0*Z*Z*FPP3*P19J*-5.00d0*Z*Z*FPP1*P21JA,1=Z*Z*HPP21/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP21/2.0d0/HA,2=Z*Z*HPP19/2.0d0/HB,2=1.0d0*Z*FP19D,2=-Z*Z*HPP19/2.0d0/HA,3=Z*Z*HPP17/2.0d0/HB,3=2.0d0*Z*FP17D,3=-Z*Z*HPP17/2.0d0/HA,4=Z*Z*HPP15/2.0d0/HB,4=3.0d0*Z*FP15D,4=-Z*Z*HPP15/2.0d0/HA,5=Z*Z*HPP13/2.0d0/HB,5=4.0d0*Z*FP13D,5=-Z*Z*HPP13/2.0d0/HA,6=Z*Z*HPP11/2.0d0/H

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268 B,6=5.0d0*Z*FP11D,6=-Z*Z*HPP11/2.0d0/HA,7=Z*Z*HPP9/2.0d0/HB,7=6.0d0*Z*FP9D,7=-Z*Z*HPP9/2.0d0/HA,8=Z*Z*HPP7/2.0d0/HB,8=7.0d0*Z*FP7D,8=-Z*Z*HPP7/2.0d0/HA,9=Z*Z*HPP5/2.0d0/HB,9=8.0d0*Z*FP5D,9=-Z*Z*HPP5/2.0d0/HA,10=Z*Z*HPP3/2.0d0/HB,10=9.0d0*Z*FP3D,10=-Z*Z*HPP3/2.0d0/HA,11=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,11=2.0d0/H2+10.0d0*Z*FP1D,11=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P23CJ+1-2.0d0*P23CJ+P23CJ-1/H2*+Z*Z*HPP1*P23CJ+1-P23CJ-1/2.0d0/H*+Z*Z*HPP3*P21CJ+1-P21CJ-1/2.0d0/H*+Z*Z*HPP5*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP7*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP9*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP11*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP13*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP15*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP17*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP19*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP21*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP23*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P23J+1-P23J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P21J+1-P21J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP21*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP23*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP21*P3CJ-2.0d0*Z*FP19*P5CJ-3.0d0*Z*FP17*P7CJ*-4.0d0*Z*FP15*P9CJ-5.0d0*Z*FP13*P11CJ-6.0d0*Z*FP11*P13CJ*-7.0d0*Z*FP9*P15CJ-8.0d0*Z*FP7*P17CJ-9.0d0*Z*FP5*P19CJ*-10.0d0*Z*FP3*P21CJ-11.0d0*Z*FP1*P23CJ*-0.50d0*Z*Z*FPP21*P3J-1.0d0*Z*Z*FPP19*P5J*-1.50d0*Z*Z*FPP17*P7J-2.00d0*Z*Z*FPP15*P9J*-2.50d0*Z*Z*FPP13*P11J-3.0d0*Z*Z*FPP11*P13J*-3.50d0*Z*Z*FPP9*P15J-4.0d0*Z*Z*FPP7*P17J*-4.50d0*Z*Z*FPP5*P19J-5.00d0*Z*Z*FPP3*P21J*-5.50d0*Z*Z*FPP1*P23JA,1=Z*Z*HPP23/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP23/2.0d0/H

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269 A,2=Z*Z*HPP21/2.0d0/HB,2=1.0d0*Z*FP21D,2=-Z*Z*HPP21/2.0d0/HA,3=Z*Z*HPP19/2.0d0/HB,3=2.0d0*Z*FP19D,3=-Z*Z*HPP19/2.0d0/HA,4=Z*Z*HPP17/2.0d0/HB,4=3.0d0*Z*FP17D,4=-Z*Z*HPP17/2.0d0/HA,5=Z*Z*HPP15/2.0d0/HB,5=4.0d0*Z*FP15D,5=-Z*Z*HPP15/2.0d0/HA,6=Z*Z*HPP13/2.0d0/HB,6=5.0d0*Z*FP13D,6=-Z*Z*HPP13/2.0d0/HA,7=Z*Z*HPP11/2.0d0/HB,7=6.0d0*Z*FP11D,7=-Z*Z*HPP11/2.0d0/HA,8=Z*Z*HPP9/2.0d0/HB,8=7.0d0*Z*FP9D,8=-Z*Z*HPP9/2.0d0/HA,9=Z*Z*HPP7/2.0d0/HB,9=8.0d0*Z*FP7D,9=-Z*Z*HPP7/2.0d0/HA,10=Z*Z*HPP5/2.0d0/HB,10=9.0d0*Z*FP5D,10=-Z*Z*HPP5/2.0d0/HA,11=Z*Z*HPP3/2.0d0/HB,11=10.0d0*Z*FP3D,11=-Z*Z*HPP3/2.0d0/HA,12=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,12=2.0d0/H2+11.0d0*Z*FP1D,12=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P25CJ+1-2.0d0*P25CJ+P25CJ-1/H2*+Z*Z*HPP1*P25CJ+1-P25CJ-1/2.0d0/H*+Z*Z*HPP3*P23CJ+1-P23CJ-1/2.0d0/H*+Z*Z*HPP5*P21CJ+1-P21CJ-1/2.0d0/H*+Z*Z*HPP7*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP9*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP11*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP13*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP15*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP17*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP19*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP21*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP23*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP25*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P25J+1-P25J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P23J+1-P23J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P21J+1-P21J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P11J+1-P11J-1/2.0d0/H/3.0d0

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270 *+Z*Z*Z*HPPP17*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP21*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP23*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP25*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP23*P3CJ-2.0d0*Z*FP21*P5CJ-3.0d0*Z*FP19*P7CJ*-4.0d0*Z*FP17*P9CJ-5.0d0*Z*FP15*P11CJ-6.0d0*Z*FP13*P13CJ*-7.0d0*Z*FP11*P15CJ-8.0d0*Z*FP9*P17CJ-9.0d0*Z*FP7*P19CJ*-10.0d0*Z*FP5*P21CJ-11.0d0*Z*FP3*P23CJ-12.0d0*Z*FP1*P25CJ*-0.50d0*Z*Z*FPP23*P3J-1.00d0*Z*Z*FPP21*P5J*-1.50d0*Z*Z*FPP19*P7J-2.00d0*Z*Z*FPP17*P9J*-2.50d0*Z*Z*FPP15*P11J-3.00d0*Z*Z*FPP13*P13J*-3.50d0*Z*Z*FPP11*P15J-4.00d0*Z*Z*FPP9*P17J*-4.50d0*Z*Z*FPP7*P19J-5.00d0*Z*Z*FPP5*P21J*-5.50d0*Z*Z*FPP3*P23J-6.00d0*Z*Z*FPP1*P25JA,1=Z*Z*HPP25/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP25/2.0d0/HA,2=Z*Z*HPP23/2.0d0/HB,2=1.0d0*Z*FP23D,2=-Z*Z*HPP23/2.0d0/HA,3=Z*Z*HPP21/2.0d0/HB,3=2.0d0*Z*FP21D,3=-Z*Z*HPP21/2.0d0/HA,4=Z*Z*HPP19/2.0d0/HB,4=3.0d0*Z*FP19D,4=-Z*Z*HPP19/2.0d0/HA,5=Z*Z*HPP17/2.0d0/HB,5=4.0d0*Z*FP17D,5=-Z*Z*HPP17/2.0d0/HA,6=Z*Z*HPP15/2.0d0/HB,6=5.0d0*Z*FP15D,6=-Z*Z*HPP15/2.0d0/HA,7=Z*Z*HPP13/2.0d0/HB,7=6.0d0*Z*FP13D,7=-Z*Z*HPP13/2.0d0/HA,8=Z*Z*HPP11/2.0d0/HB,8=7.0d0*Z*FP11D,8=-Z*Z*HPP11/2.0d0/HA,9=Z*Z*HPP9/2.0d0/HB,9=8.0d0*Z*FP9D,9=-Z*Z*HPP9/2.0d0/HA,10=Z*Z*HPP7/2.0d0/HB,10=9.0d0*Z*FP7D,10=-Z*Z*HPP7/2.0d0/HA,11=Z*Z*HPP5/2.0d0/HB,11=10.0d0*Z*FP5D,11=-Z*Z*HPP5/2.0d0/HA,12=Z*Z*HPP3/2.0d0/HB,12=11.0d0*Z*FP3D,12=-Z*Z*HPP3/2.0d0/HA,13=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,13=2.0d0/H2+12.0d0*Z*FP1D,13=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P27CJ+1-2.0d0*P27CJ+P27CJ-1/H2

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271 *+Z*Z*HPP1*P27CJ+1-P27CJ-1/2.0d0/H*+Z*Z*HPP3*P25CJ+1-P25CJ-1/2.0d0/H*+Z*Z*HPP5*P23CJ+1-P23CJ-1/2.0d0/H*+Z*Z*HPP7*P21CJ+1-P21CJ-1/2.0d0/H*+Z*Z*HPP9*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP11*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP13*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP15*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP17*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP19*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP21*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP23*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP25*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP27*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P27J+1-P27J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P25J+1-P25J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P23J+1-P23J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P21J+1-P21J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP21*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP23*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP25*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP27*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP25*P3CJ-2.0d0*Z*FP23*P5CJ-3.0d0*Z*FP21*P7CJ*-4.0d0*Z*FP19*P9CJ-5.0d0*Z*FP17*P11CJ-6.0d0*Z*FP15*P13CJ*-7.0d0*Z*FP13*P15CJ-8.0d0*Z*FP11*P17CJ-9.0d0*Z*FP9*P19CJ*-10.0d0*Z*FP7*P21CJ-11.0d0*Z*FP5*P23CJ-12.0d0*Z*FP3*P25CJ*-13.0d0*Z*FP1*P27CJ*-0.50d0*Z*Z*FPP25*P3J-1.00d0*Z*Z*FPP23*P5J*-1.50d0*Z*Z*FPP21*P7J-2.00d0*Z*Z*FPP19*P9J*-2.50d0*Z*Z*FPP17*P11J-3.00d0*Z*Z*FPP15*P13J*-3.50d0*Z*Z*FPP13*P15J-4.00d0*Z*Z*FPP11*P17J*-4.50d0*Z*Z*FPP9*P19J-5.00d0*Z*Z*FPP7*P21J*-5.50d0*Z*Z*FPP5*P23J-6.00d0*Z*Z*FPP3*P25J*-6.50d0*Z*Z*FPP1*P25JA,1=Z*Z*HPP27/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP27/2.0d0/HA,2=Z*Z*HPP25/2.0d0/HB,2=1.0d0*Z*FP25D,2=-Z*Z*HPP25/2.0d0/HA,3=Z*Z*HPP23/2.0d0/HB,3=2.0d0*Z*FP23D,3=-Z*Z*HPP23/2.0d0/HA,4=Z*Z*HPP21/2.0d0/HB,4=3.0d0*Z*FP21D,4=-Z*Z*HPP21/2.0d0/HA,5=Z*Z*HPP19/2.0d0/HB,5=4.0d0*Z*FP19D,5=-Z*Z*HPP19/2.0d0/H

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272 A,6=Z*Z*HPP17/2.0d0/HB,6=5.0d0*Z*FP17D,6=-Z*Z*HPP17/2.0d0/HA,7=Z*Z*HPP15/2.0d0/HB,7=6.0d0*Z*FP15D,7=-Z*Z*HPP15/2.0d0/HA,8=Z*Z*HPP13/2.0d0/HB,8=7.0d0*Z*FP13D,8=-Z*Z*HPP13/2.0d0/HA,9=Z*Z*HPP11/2.0d0/HB,9=8.0d0*Z*FP11D,9=-Z*Z*HPP11/2.0d0/HA,10=Z*Z*HPP9/2.0d0/HB,10=9.0d0*Z*FP9D,10=-Z*Z*HPP9/2.0d0/HA,11=Z*Z*HPP7/2.0d0/HB,11=10.0d0*Z*FP7D,11=-Z*Z*HPP7/2.0d0/HA,12=Z*Z*HPP5/2.0d0/HB,12=11.0d0*Z*FP5D,12=-Z*Z*HPP5/2.0d0/HA,13=Z*Z*HPP3/2.0d0/HB,13=12.0d0*Z*FP3D,13=-Z*Z*HPP3/2.0d0/HA,14=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,14=2.0d0/H2+13.0d0*Z*FP1D,14=-1.0d0/H2-Z*Z*HPP1/2.0d0/H RETURN END ******************* SUBROUTINE BC2PALLSC ********************** SUBROUTINE BC2PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C,P21C,P23C,P25C,P27C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ DOUBLE PRECISION C,* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DOUBLE PRECISION P21C*,P23C*,P25C*,P27C* DO i=1,N DO jj=1,NAi,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=0.0d0-P1CJB,1=1.0d0G=0.0d0-P3CJB,2=1.0d0G=0.0d0-P5CJB,3=1.0d0G=0.0d0-P7CJB,4=1.0d0G=0.0d0-P9CJ

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273 B,5=1.0d0G=0.0d0-P11CJB,6=1.0d0G=0.0d0-P13CJB,7=1.0d0G=0.0d0-P15CJB,8=1.0d0G=0.0d0-P17CJB,9=1.0d0G=0.0d0-P19CJB,10=1.0d0G=0.0d0-P21CJB,11=1.0d0G=0.0d0-P23CJB,12=1.0d0G=0.0d0-P25CJB,13=1.0d0G=0.0d0-P27CJB,14=1.0d0 RETURN END *************** SUBROUTINE GETMAXG_CONC *********************** SUBROUTINE GETMAXG_CONCNJ,NJP,DP,maxval INTEGER NJ,NJP DOUBLE PRECISION DP,*,maxvalNJP=1maxval=0.0d0 DO 20j=1,14 DO 10i=2,NJ-1 IF dabsDPj,i.GT.maxval THEN maxval=dabsDPj,iNJP=i END IF 10 CONTINUE 20 CONTINUE RETURN END D.1.3IncludeFilesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere. ********** INCLUDE FILE PALL f for main program ************ maxvalG=1.0d0 DO I=1,N DO K=1,NXI,K=0.0d0YI,K=0.0d0 END DO END DO JJ=1110J=0

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274 360J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1PALL for J =1 IF J.EQ.1 CALL BC1PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21*,P23,P25,P27 Equation for interior region IF J.GT.1.AND.J.LT.NJ THEN CALL INNERPALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21*,P23,P25,P27 END IF Equation for second boundary condition C IF J.EQ.NJ THEN CALL BC2PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P21*,P23,P25,P27 END IF CALL BANDJ,C IF J.NE.NJ GOTO 360 DO I=1,NJP1I=P1I+C,IP3I=P3I+C,IP5I=P5I+C,IP7I=P7I+C,IP9I=P9I+C,IP11I=P11I+C,IP13I=P13I+C,IP15I=P15I+C,IP17I=P17I+C,IP19I=P19I+C,IP21I=P21I+C,IP23I=P23I+C,IP25I=P25I+C,IP27I=P27I+C,I END DO INCLUDE 'CHECKCONVERGE.f'JJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.ITMAX GO TO 110maxvalG=1.0d0 DO I=1,N DO K=1,NXI,K=0.0d0YI,K=0.0d0 END DO END DO JJ=1111J=0361J=J+1SUMH=SUMH+H

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275 DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1PALLSC for J =1 C IF J.EQ.1 CALL BC1PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,P13C,*P15C,P17C,P19C,P21C,P23C,P25C,P27C Equation for interior region C IF J.GT.1.AND.J.LT.NJ THEN CALL INNERPALLSCJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*P21,P23,P25,P27,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C,P21C,P23C,P25C,P27C END IF Equation for second boundary condition C IF J.EQ.NJ THEN CALL BC2PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C,P21C,P23C,P25C,P27C END IF CALL BANDJ,C IF J.NE.NJ GOTO 361 DO I=1,NJP1CI=P1CI+C,IP3CI=P3CI+C,IP5CI=P5CI+C,IP7CI=P7CI+C,IP9CI=P9CI+C,IP11CI=P11CI+C,IP13CI=P13CI+C,IP15CI=P15CI+C,IP17CI=P17CI+C,IP19CI=P19CI+C,IP21CI=P21CI+C,IP23CI=P23CI+C,IP25CI=P25CI+C,IP27CI=P27CI+C,I END DO INCLUDE 'CHECKCONVERGE1.f'JJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.ITMAX GO TO 111 ****** INCLUDE FILE VELOCITYDATA f for main program ********** OPEN unit =2, file ='HFP.txt' DO i=1,14 READ ,*ii,FPi,FPPi,HPPi,HPPPi ENDDO HPP1=HPPHPP3=HPPHPP5=HPPHPP7=HPPHPP9=HPPHPP11=HPPHPP13=HPP

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276 HPP15=HPPHPP17=HPPHPP19=HPPHPP21=HPPHPP23=HPPHPP25=HPPHPP27=HPPFP1=FPFP3=FPFP5=FPFP7=FPFP9=FPFP11=FPFP13=FPFP15=FPFP17=FPFP19=FPFP21=FPFP23=FPFP25=FPFP27=FPHPPP1=HPPPHPPP3=HPPPHPPP5=HPPPHPPP7=HPPPHPPP9=HPPPHPPP11=HPPPHPPP13=HPPPHPPP15=HPPPHPPP17=HPPPHPPP19=HPPPHPPP21=HPPPHPPP23=HPPPHPPP25=HPPPHPPP27=HPPPFPP1=FPPFPP3=FPPFPP5=FPPFPP7=FPPFPP9=FPPFPP11=FPPFPP13=FPPFPP15=FPPFPP17=FPPFPP19=FPPFPP21=FPPFPP23=FPPFPP25=FPPFPP27=FPP CLOSE ****** INCLUDE FILE CAL_PP f for main program ********** PP=-P1+4.0d0*P1-3.0d0*P1/2.0d0/HPP=-P3+4.0d0*P3/2.0d0/HPP=-P5+4.0d0*P5/2.0d0/HPP=-P7+4.0d0*P7/2.0d0/H

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277 PP=-P9+4.0d0*P9/2.0d0/HPP=-P11+4.0d0*P11/2.0d0/HPP=-P13+4.0d0*P13/2.0d0/HPP=-P15+4.0d0*P15/2.0d0/HPP=-P17+4.0d0*P17/2.0d0/HPP=-P19+4.0d0*P19/2.0d0/HPP=-P21+4.0d0*P21/2.0d0/HPP=-P23+4.0d0*P23/2.0d0/HPP=-P25+4.0d0*P25/2.0d0/HPP=-P27+4.0d0*P27/2.0d0/H WRITE ,*H,H2,PP,PP,PP,PP,PP,PP6,PP,PP*,PP,PP,PP,PP,PP,PP14PCP=-P1C+4.0d0*P1C/2.0d0/HPCP=-P3C+4.0d0*P3C/2.0d0/HPCP=-P5C+4.0d0*P5C/2.0d0/HPCP=-P7C+4.0d0*P7C/2.0d0/HPCP=-P9C+4.0d0*P9C/2.0d0/HPCP=-P11C+4.0d0*P11C/2.0d0/HPCP=-P13C+4.0d0*P13C/2.0d0/HPCP=-P15C+4.0d0*P15C/2.0d0/HPCP=-P17C+4.0d0*P17C/2.0d0/HPCP=-P19C+4.0d0*P19C/2.0d0/HPCP=-P21C+4.0d0*P21C/2.0d0/HPCP=-P23C+4.0d0*P23C/2.0d0/HPCP=-P25C+4.0d0*P25C/2.0d0/HPCP=-P27C+4.0d0*P27C/2.0d0/H WRITE ,*H,H2,PCP,PCP,PCP,PCP,PCP,PCP,PCP,*PCP,PCP,PCP,PCP,PCP,PCP,PCP ****** INCLUDE FILE ALC f for main program ********** ALLOCATE C,NJ,P1NJ ALLOCATE P3NJ,P5NJ,P7NJ,P9NJ ALLOCATE P11NJ,P13NJ,P15NJ,P17NJ ALLOCATE P19NJ,P21NJ,P23NJ,P25NJ ALLOCATE GGNJ,P27NJ ALLOCATE P1CNJ,P3CNJ,P5CNJ,P7CNJ ALLOCATE P9CNJ,P11CNJ,P13CNJ,P15CNJ ALLOCATE P17CNJ,P19CNJ,P21CNJ,P23CNJ ALLOCATE P25CNJ,P27CNJ,DP,NJ ****** INCLUDE FILE DLC f for main program ********** DEALLOCATE C,P1 DEALLOCATE P3,P5,P7,P9 DEALLOCATE P11,P13,P15,P17 DEALLOCATE P19,P21,P23,P25 DEALLOCATE GG,P27 DEALLOCATE P1C,P3C,P5C,P7C DEALLOCATE P9C,P11C,P13C,P15C DEALLOCATE P17C,P19C,P21C,P23C DEALLOCATE P25C,P27C,DP ****** INCLUDE FILE COMMON f for main program ********** COMMON /BA/A,14,B,14,D,29,G,X,14,Y,14,N,NJ COMMON /BB/H,H2 COMMON /VELOCITY/HPP1,HPP3,HPP5,HPP7,HPP9,HPP11,HPP13,*HPP15,HPP17,HPP19,HPP21,HPP23,HPP25,HPP27,*HPPP1,HPPP3,HPPP5,HPPP7,HPPP9,HPPP11,HPPP13,*HPPP15,HPPP17,HPPP19,HPPP21,HPPP23,HPPP25,HPPP27,

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278 *FP1,FP3,FP5,FP7,FP9,FP11,FP13,FP15,FP17,FP19,FP21,*FP23,FP25,FP27,FPP1,FPP3,FPP5,FPP7,FPP9,FPP11,FPP13,*FPP15,FPP17,FPP19,FPP21,FPP23,FPP25,FPP27 ****** INCLUDE FILE CHECHCONVERGE f for main program ********** DO i=1,NJ IF dabsP1i.GT.0.0d0 THEN DP,i=dabsC,i/P1i ELSE DP,i=1.0d0 ENDIF IF dabsP3i.GT.0.0d0 THEN DP,i=dabsC,i/P3i ELSE DP,i=1.0d0 ENDIF IF dabsP5i.GT.0.0d0 THEN DP,i=dabsC,i/P5i ELSE DP,i=1.0d0 ENDIF IF dabsP7i.GT.0.0d0 THEN DP,i=dabsC,i/P7i ELSE DP,i=1.0d0 ENDIF IF dabsP9i.GT.0.0d0 THEN DP,i=dabsC,i/P9i ELSE DP,i=1.0d0 ENDIF IF dabsP11i.GT.0.0d0 THEN DP,i=dabsC,i/P11i ELSE DP,i=1.0d0 ENDIF IF dabsP13i.GT.0.0d0 THEN DP,i=dabsC,i/P13i ELSE DP,i=1.0d0 ENDIF IF dabsP15i.GT.0.0d0 THEN DP,i=dabsC,i/P15i ELSE DP,i=1.0d0 ENDIF IF dabsP17i.GT.0.0d0 THEN DP,i=dabsC,i/P17i ELSE DP,i=1.0d0 ENDIF IF dabsP19i.GT.0.0d0 THEN DP,i=dabsC,i/P19i ELSE DP,i=1.0d0 ENDIF

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279 IF dabsP21i.GT.0.0d0 THEN DP,i=dabsC,i/P21i ELSE DP,i=1.0d0 ENDIF IF dabsP23i.GT.0.0d0 THEN DP,i=dabsC,i/P23i ELSE DP,i=1.0d0 ENDIF IF dabsP25i.GT.0.0d0 THEN DP,i=dabsC,i/P25i ELSE DP,i=1.0d0 ENDIF IF dabsP27i.GT.0.0d0 THEN DP,i=dabsC,i/P27i ELSE DP,i=1.0d0 ENDIF END DO CALL GETMAXG_CONCNJ,NJP,DP,maxvalG ****** INCLUDE FILE CHECHCONVERGE1 f for main program ********** DO i=1,NJ IF dabsP1Ci.GT.0.0d0 THEN DP,i=dabsC,i/P1Ci ELSE DP,i=1.0d0 ENDIF IF dabsP3Ci.GT.0.0d0 THEN DP,i=dabsC,i/P3Ci ELSE DP,i=1.0d0 ENDIF IF dabsP5Ci.GT.0.0d0 THEN DP,i=dabsC,i/P5Ci ELSE DP,i=1.0d0 ENDIF IF dabsP7Ci.GT.0.0d0 THEN DP,i=dabsC,i/P7Ci ELSE DP,i=1.0d0 ENDIF IF dabsP9Ci.GT.0.0d0 THEN DP,i=dabsC,i/P9Ci ELSE DP,i=1.0d0 ENDIF IF dabsP11Ci.GT.0.0d0 THEN DP,i=dabsC,i/P11Ci ELSE DP,i=1.0d0 ENDIF IF dabsP13Ci.GT.0.0d0 THEN

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280 DP,i=dabsC,i/P13Ci ELSE DP,i=1.0d0 ENDIF IF dabsP15Ci.GT.0.0d0 THEN DP,i=dabsC,i/P15Ci ELSE DP,i=1.0d0 ENDIF IF dabsP17Ci.GT.0.0d0 THEN DP,i=dabsC,i/P17Ci ELSE DP,i=1.0d0 ENDIF IF dabsP19Ci.GT.0.0d0 THEN DP,i=dabsC,i/P19Ci ELSE DP,i=1.0d0 ENDIF IF dabsP21Ci.GT.0.0d0 THEN DP,i=dabsC,i/P21Ci ELSE DP,i=1.0d0 ENDIF IF dabsP23Ci.GT.0.0d0 THEN DP,i=dabsC,i/P23Ci ELSE DP,i=1.0d0 ENDIF IF dabsP25Ci.GT.0.0d0 THEN DP,i=dabsC,i/P25Ci ELSE DP,i=1.0d0 ENDIF IF dabsP27Ci.GT.0.0d0 THEN DP,i=dabsC,i/P27Ci ELSE DP,i=1.0d0 ENDIF END DO CALL GETMAXG_CONCNJ,NJP,DP,maxvalG

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APPENDIXEPROGRAMLISTINGFORCALCULATINGTHECURRENTDISTRIBUTIONATTHESTATIONARYHEMISPHERICALELECTRODEUNDERSUBMERGEDJETIMPINGEMENTTheprogramlistingpresentsalloftheFORTRANcodetosolvegoverningequationsforcalculatingthecurrentdistributionatthestationaryhemisphericalelectrodeundersubmergedjetimpingement.Theprogramwasdevelopedwith'CompaqVisualFortran,Version6.1'withdoubleprecisionaccuracy.Themainprogram'CURRDISTIJHSE'calledthesubroutinecontaininggoverningequations.Thegoverningequationswereprogrammedinsubroutines'THETA'and'CSOLPONT'.Thesubroutine'THETA'calculatedthecurrentattheelectrodesurfaceduetoelec-trodekineticsandmass-transfer.Thesubroutine'CSOLPONT'calculatedthepo-tentialdistributionalongtheelectrodesurface.Thevalueofelectrodepotentialwascalculatedatgridpointineachiteration.Theprogramwasterminatedonceelectrodepotentialbecameuniformfortheentireelectrodesurface.E.1ProgramListingE.1.1MainProgram IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N DOUBLE PRECISION ALLOCATABLE ::SOLPONT: DOUBLE PRECISION ALLOCATABLE ::CI:,THC:,ZCI: DOUBLE PRECISION ALLOCATABLE ::CONC:,CURT1:,CURT2:,ILIM: DOUBLE PRECISION ALLOCATABLE ::AA:,BB: DOUBLE PRECISION ALLOCATABLE ::EACT:,CUR:,ETEST:,ESOLPONT: DOUBLE PRECISION ALLOCATABLE ::BM:,RESV: DOUBLE PRECISION ALLOCATABLE ::PPN: DOUBLE PRECISION ALLOCATABLE ::XGAUSS:,WGAUSS: DOUBLE PRECISION maxvalG,Z,LP,PP,ILIM_AVG,IAVG DOUBLE PRECISION FP,FPP,HPP,HPPP DOUBLE PRECISION H,H2,percent,percent1,percent2 INTEGER SUR_CONC,SUR_DER,N_STEP,IPP,JMID,JMAX 281

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282 DOUBLE PRECISION SC,SD,ESOLPONTP,TH INTEGER NJL,NJLIST,NGAUSS,LMAX,NMAX DOUBLE PRECISION DCI,XL,XU,T,ZZ DOUBLE PRECISION CURR_NN,EJJ,alpha,beta,gamma,pi2 DOUBLE PRECISION tplus,ETAC0,ETAS0,CONC0,CUR0,TAF,res DOUBLE PRECISION dpi2,BOLD,DB,DAMP,DAMP1,PHI,V,XACC,CUR_AVG CHARACTER filename*13 CHARACTER filename1*13 REAL *8GRADT NAMELIST /par/CURR_NN,EJJ,alpha,beta,gamma,tplus,LMAX,NMAX,IPMAX,&NGAUSS,DAMP,DAMP1,JMID,JMAX,XACC,CONC0,filename,&filename1 EXTERNAL LP,GRADT Read input parameters open FILE ='INPUT.DAT', STATUS ='UNKNOWN' READ ,par CLOSE pi2=acos-1.0d0/2.0d0 ALLOCATE CILMAX,THCLMAX,ZCILMAX ALLOCATE CONCLMAX,CURT1LMAX,CURT2LMAX,ILIMLMAX ALLOCATE AALMAX,BBLMAX ALLOCATE EACTLMAX,CURLMAX,ETESTLMAX,ESOLPONTLMAX ALLOCATE BMNMAX,RESVIPMAX-1 ALLOCATE XGAUSSNGAUSS,WGAUSSNGAUSSBM=0.0d0JCOUNT=0 CALL gauleg.0d0,pi2,XGAUSS,WGAUSS,NGAUSS CALL GETCITHCLMAX,CI,THC,ZCI,AA,BB,CIMAX,TH CALL CALINIPONTEJJ,CURR_NN,alpha,beta,gamma,tplus,CONC0,ETAC0*,ETAS0,CUR0,TAF DO i=1,LMAXCONCi=CONC0EACTi=ETAS0+ETAC0 END DO CUR=CUR012BOLD=BMJCOUNT=JCOUNT+1 CALL THETALMAX,CIMAX,EJJ,CURR_NN,alpha,beta,gamma,tplus,TAF,*CONC,EACT,CUR,ZCI,AA,BB INCLUDE 'POLYFIT.f' CALL CSOLPONTLMAX,CUR,THC,NMAX,BM,ESOLPONT,IP,PPN,*WGAUSS,XGAUSS,NGAUSS,TH,ESOLPONTP DEALLOCATE PPNV=ESOLPONT+EACT IF JCOUNT.LT.JMID THEN DO 18i=2,LMAXPHI=VPHI=PHI-ESOLPONTiEACTi=EACTi+DAMP*PHI-EACTi18 CONTINUE END IF IF JCOUNT.GE.JMID THEN DO 19i=2,LMAXPHI=VPHI=PHI-ESOLPONTi

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283 EACTi=EACTi+DAMP1*PHI-EACTi19 CONTINUE END IF DB=dabsBOLD-BM/dabsBM IF DABSDB.GE.XACC.AND.JCOUNT.LE.JMAX GO TO 12 PRINT *, CALCULATION COMPLETE PRINT *, BOLD = ,BOLD INCLUDE 'WRITEOUTPUT.f' DEALLOCATE BM,RESV DEALLOCATE EACT,CUR,ETEST,ESOLPONT DEALLOCATE AA,BB DEALLOCATE CI,THC,ZCI DEALLOCATE CONC,CURT1,CURT2,ILIM DEALLOCATE XGAUSS,WGAUSS END E.1.2MainSubroutinesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutinesarelistedhere. ***************** SUBROUTINE GETCITHC *************************** SUBROUTINE GETCITHCLMAX,CI,THC,ZCI,A,B,CIMAX,TH INTEGER LMAX,NSTEP DOUBLE PRECISION CILMAX,THCLMAX,ZCILMAX,TH DOUBLE PRECISION ALMAX,BLMAX,EX,EX1,CIMAX,pi2 DOUBLE PRECISION ALLOCATABLE ::CI_FF:,THC_FF: ALLOCATE CI_FF,THC_FFpi2=dacos-1.0d0/2.0d0 DO i=1,400THi=dfloati-1*pi2/400.0d0 END DO TH=pi2 OPEN unit =10, file ='CITH.txt' DO i=1,10001 READ ,*CI_FFi,THC_FFi END DO NSTEP=-1/LMAX-1CIMAX=CI_FF DO i=1,LMAXj=i-1*NSTEP+1CIi=CI_FFjTHCi=THC_FFj END DO DO i=1,LMAXZCIi=SQRTGRADTTHCi END DO ZCILMAX=0.12EX=2.0d0/3.0d0EX1=1.0d0/3.0d0 DO L=1,LMAXAA=DFLOATLAL=2.0*AA**EX-AA+1.0d0**EX-AA-1.0d0**EXBL=AA**EX-AA-1.0d0**EX

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284 END DO DEALLOCATE CI_FF,THC_FF RETURN END ********** SUBROUTINE CALINIPONT ************************** SUBROUTINE CALINIPONTEJJ,CURR_NN,ALPHA,BETA,GAMMA,TPLUS,*C0,ETAC,ETAS,CUR0,TAF DOUBLE PRECISION EJJ,C1,C2,EXCH,CURR_NN DOUBLE PRECISION ALPHA,BETA,GAMMA,TPLUS,C0 DOUBLE PRECISION ETAC,ETAS,TAF,CUR0,xacc,DETAS DOUBLE PRECISION F,FPC1=1.5788643711074880d0C2=1.1198465217221860d0*CURR_NNxacc=1.0e-14EXCH=1/EJJTAF=1.0d0 IF EXCH-4007,7,66TAF=0.0d0EXCH=1.0d07ETAC=DLOGC0+TPLUS*-C0CUR0=-.0d0-C0*C1*C2*EXCH/C0**GAMMAETAS=DLOGTAF-CUR0/BETA IF CUR08,10,88 DO 9J=1,100F=TAF*DEXPALPHA*ETAS-DEXP-BETA*ETASFP=TAF*ALPHA*DEXPALPHA*ETAS+BETA*DEXP-BETA*ETASDETAS=F-CUR0/FP IF DABSDETAS.LT.xacc GO TO 10ETAS=ETAS-DETAS9 CONTINUE 10CUR0=-.0d0-C0*C1*C2 RETURN END ***************** SUBROUTINE INNERP1 *************************** SUBROUTINE THETALMAX,CIMAX,EJJ,CURR_NN,al,be,ga,tp,TAF,*CONC,EACT,CUR,ZC,A,B INTEGER LMAX DOUBLE PRECISION EJJ,CURR_NN,al,be,ga,tp DOUBLE PRECISION CONCLMAX,EACTLMAX,CURLMAX DOUBLE PRECISION ALPHA,BETA,GAMMA,TPLUS,EXCH DOUBLE PRECISION ZCLMAX,ALMAX,BLMAX DOUBLE PRECISION C1,C2,C3,EX,EX1,AA,PN,DTH DOUBLE PRECISION CIMAX,DCI,Z,TAF,S,DZ,ETA,XACC DOUBLE PRECISION DX3,X3,DS INTEGER N,i,J,K,NZ,NJC2=1.1198465217221860d0C1=C2*.0d0-CONCEX=2.0d0/3.0d0EX1=1.0d0/3.0d0DCI=CIMAX/DFLOATLMAX-1ALPHA=alBETA=beGAMMA=gaTPLUS=tpEXCH=1.0d0/EJJ

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285 DZ=CIMAX/DFLOATLMAX-1S=CONCPN=CURR_NNXACC=1.0e-15S=CONC DO 60NZ=2,LMAXZ=NZ-1*DZSUM=0.0d0 IF NZ.LE.2 GO TO 42 DO 40J=3,NZK=NZ-J+1SUM=SUM+CONCJ-1*AK40 CONTINUE 42ETA=EACTNZNJ=NZ-1 DO 56N=1,100X1=EJJ*TAF*S**GAMMA-ALPHA**DEXPALPHA*ETA*DEXPALPHA*TPLUS*S-1.0d0X2=EJJ*TAF*S**GAMMA+BETA**DEXP-BETA*ETA*DEXPBETA*TPLUS*.0d0-SDX1=X1*GAMMA-ALPHA/S+ALPHA*TPLUSDX2=X2*GAMMA+BETA/S-BETA*TPLUSC3=1.50d0*C2*ZCNZ/DZ**EX1X3=-C3*PN*CONC*BNJ+SUM-S-C1*PN*ZCNZ/Z**EX1DX3=C3*PNDS=X1-X2-X3/DX1-DX2-DX3CURNZ=X3 IF DABSDS.LE.XACC GO TO 10S=S-DS56 CONTINUE 10CONCNZ=S60 CONTINUE RETURN END ************ SUBROUTINE CSOLPONT ******************************** SUBROUTINE CSOLPONTLMAX,CUR,THC,NMAX,BM,ESOLPONT,IP,PPN,*W,X,NGAUSS,TH,ESOLPONTP INTEGER LMAX,NMAX,N,i,j,IP,NGAUSS DOUBLE PRECISION CURLMAX,ESOLPONTLMAX,BMNMAX,THCLMAX DOUBLE PRECISION LP,PPNIP,BMMNMAX,ESOLPONT1LMAX DOUBLE PRECISION WNGAUSS,XNGAUSS,ss DO N=1,NMAXBMN=0.0d0 CALL qgauss*N-1,ss,IP,PPN,W,X,NGAUSSBMN=ss*dfloat*N-3/dfloat*N-1 END DO DO 20i=1,LMAXESOLPONTi=0.0d0 DO N=1,NMAXESOLPONTi=ESOLPONTi+BMN*LP*N-1,dcosTHCi END DO 20 CONTINUE RETURN END **************** SUBROUTINE POLYFITCURTHETA **********************

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286 SUBROUTINE POLYFITCURTHETALMAX,CUR,THC,IP,PPN,res INTEGER LMAX,IP DOUBLE PRECISION CURLMAX,THCLMAX,PPNIP,DET,sig,res DOUBLE PRECISION ALLOCATABLE ::Y:,:,X:,:,XX:,:,XXINV:,: DOUBLE PRECISION ALLOCATABLE ::PC:,:,CURC:,: ALLOCATE YLMAX,1,XLMAX,IP,XXIP,IP,XXINVIP,IP ALLOCATE PCIP,1,CURCLMAX,1res=0.0d0 DO 10i=1,LMAXsig=1.0d0Yi,1=CURi/sigTH=THCi DO 20j=1,IPXi,j=TH**dfloat*j-2/sig20 CONTINUE 10 CONTINUE XX=MATMULTRANSPOSEX,X CALL CALINVERSEXX,IP,XXINV,DETPC=MATMULXXINV,MATMULTRANSPOSEX,YCURC=MATMULX,PC DO 40j=1,IPPPNj=PCj,140 CONTINUE DEALLOCATE PC,CURC DEALLOCATE Y,X,XX,XXINV RETURN END *************** SUBROUTINE qgauss ****************************** SUBROUTINE qgaussNC,ss,IP,PP,W,X,NGAUSS DOUBLE PRECISION a,b,ss,func INTEGER j,NC,IP,NGAUSS DOUBLE PRECISION dx,xm,xr DOUBLE PRECISION WNGAUSS,XNGAUSS,PPIPss=0.0d0 DO 11j=1,NGAUSSdx=Xj CALL get_funcNC,dx,func,IP,PPss=ss+Wj*func11 CONTINUE RETURN END ********************* SUBROUTINE get_func ******************* SUBROUTINE get_funcN,x,func,IP,PP INTEGER N,IP DOUBLE PRECISION x,func,cur,LP,xstar DOUBLE PRECISION PPIP DOUBLE PRECISION PPNIP,1,XX,IP,curr,1XX,1=1.0d0PPN,1=PPxstar=0.9562841591392380d0 DO i=2,IP IF x.le.xstar THEN XX,i=x**dfloat*i-2 ELSE XX,i=xstar**dfloat*i-2

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287 END IF PPNi,1=PPi END DO curr=MATMULXX,PPNcur=curr,1func=cur*dsinx*LPN,dcosx RETURN END **************** FUNCTION LP *********************************** REAL *8 FUNCTION LP*8N,X DOUBLE PRECISION P1,P2,P,X INTEGER NM1,NU,NP1=1.0P2=X IF N-11,2,31LP=P1 RETURN 2LP=P2 RETURN 3NM1=N-1 DO 4NU=1,NM1,1P=X*dfloat*NU+1*P2-dfloatNU*P1/dfloatNU+1P1=P2P2=P4 CONTINUE LP=P RETURN END ********************** SUBROUTINE gauleg ********************** SUBROUTINE gaulegx1,x2,x,w,n INTEGER n DOUBLE PRECISION x1,x2,x*,w* DOUBLE PRECISION EPS PARAMETER EPS=3.d-14 INTEGER i,j,m DOUBLE PRECISION p1,p2,p3,pp,xl,xm,z,z1m=n+1/2xm=0.5d0*x2+x1xl=0.5d0*x2-x1 do 12i=1,mz=cos.141592654d0*i-.25d0/n+.5d01 continue p1=1.d0p2=0.d0 do 11j=1,np3=p2p2=p1p1=.d0*j-1.d0*z*p2-j-1.d0*p3/j11 continue pp=n*z*p1-p2/z*z-1.d0z1=zz=z1-p1/pp if absz-z1.gt.EPS goto 1xi=xm-xl*zxn+1-i=xm+xl*z

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288 wi=2.d0*xl/.d0-z*z*pp*ppwn+1-i=wi12 continue RETURN END *************** SUBROUTINE CALILIMAVG *************************** SUBROUTINE CALILIMIAVGLMAX,CIMAX,CURR_NN,*CONC,CUR,THC,ZC,ILIM,ILIM_AVG,CUR_AVG INTEGER LMAX DOUBLE PRECISION CURR_NN DOUBLE PRECISION CONCLMAX,THCLMAX,CURLMAX DOUBLE PRECISION ZCLMAX,ILIMLMAX,ILIM_AVG,ILIMAVGLMAX DOUBLE PRECISION C1,C2,C3,EX,EX1,AA,PN,DTH,CURAVGLMAX DOUBLE PRECISION CIMAX,Z,TAF,S,DZ,pi2,BETA,CUR_AVG INTEGER N,i,J,K,NZ,NJpi2=acos-1.0d0BETA=1.0C1=1.5788643711074880d0C2=1.1198465217221860d0*CURR_NNEX=2.0d0/3.0d0EX1=1.0d0/3.0d0DZ=CIMAX/DFLOATLMAX-1ILIM=-C1*C2 DO 60NZ=2,LMAXZ=NZ-1*DZILIMNZ=-C2*ZCNZ/Z**EX160 CONTINUE DO i=1,LMAXILIMAVGi=dsinTHCi*ILIMiCURAVGi=dsinTHCi*CURi END DO ILIM_AVG=0.0d0CUR_AVG=0.0d0 DO i=2,LMAXDTH=THCi-THCi-1ILIM_AVG=ILIMAVGi+ILIMAVGi-1*DTH*0.50d0+ILIM_AVGCUR_AVG=CURAVGi+CURAVGi-1*DTH*0.50d0+CUR_AVG END DO DTH=pi2-THCLMAXILIM_AVG=ILIM_AVG+BETA*ILIMLMAX*dcosTHCLMAXCUR_AVG=CUR_AVG+BETA*CURLMAX*dcosTHCLMAX RETURN END E.1.3IncludeFilesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere.C***************** INCLUDE FILE WRITEOUTPUT.f****************** CALL CALILIMIAVGLMAX,CIMAX,CURR_NN,*CONC,CUR,THC,ZCI,ILIM,ILIM_AVG,CUR_AVG900 FORMAT 'THETACONCCURRENTILIMCURRENT/AVG

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289 *CURRENT/ILIMVAGILIM/ILIMAVGIAVG/ILIMAVG*EACTSEACTCESOLETOT'901 FORMAT 'THETAESOL' OPEN unit =10, file =filename OPEN unit =11, file =filename1 WRITE ,900 WRITE ,9011410 FORMAT e13.5,2x1411 FORMAT e13.5,2xCURAVG=-BOLD/*1.50d0/CURR_NN/CIMAX**.0d0/3.0d0/1.1198465217221860d0CURAVG=-BOLDpi2=dacos-1.0d0/2.0d0 DO i=1,LMAXETAC0=DLOGCONCi+TPLUS*-CONCi WRITE ,1410THCi*90.0d0/pi2,CONCi,CURi,ILIMi,*CURi/CUR_AVG,CURi/ILIM_AVG,ILIMi/ILIM_AVG*,CUR_AVG/ILIM_AVG,EACTi-ETAC0,ETAC0,ESOLPONTi*,ESOLPONTi+EACTi END DO DO i=1,401 WRITE ,1411THi*90.0d0/pi2,ESOLPONTPi END DO CLOSE CLOSE E.1.4InputFileThefollowingvariableswerereadinthemainprogrambythefollowinginputle.. ************************ INPUT DAT ************************** CURR_NN=20.0d0EJJ=5.0d0alpha=0.50d0beta=0.50d0gamma=0.50d0tplus=0.50d0LMAX=201NMAX=51IPMAX=20NGAUSS=400DAMP=0.02DAMP1=0.0200JMID=1000JMAX=10000XACC=1.0e-8CONC0=0.50d0filename='N20J5NCP5.txt'filename1='N20J5NPP5.txt'

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APPENDIXFPROGRAMLISTINGFORCALCULATINGTHECURRENTDISTRIBUTIONATTHEROTATINGHEMISPHERICALELECTRODETheprogramlistingpresentsalloftheFORTRANcodetosolvegoverningequationsforcalculationofcurrentdistributionattherotatinghemisphericalelec-trodewithniteSchmidtnumber.Theprogramwasdevelopedwith'CompaqVisualFortran,Version6.1'withdoubleprecisionaccuracy.Themainprogram'CURRDISTRHEWITHSCCORR'calledthesubroutinecontaininggoverningequa-tions.Theinputparametersforthemainprogramwerereadfromthele'INPUT.DAT'.Aninitialsurfaceconcentrationdistributionwasestablished.Theconvective-diffusiongoverningequationforthecurrentdistributionweresolvedbycallingtheincludele'PALL.f'and'CALCONCCURR.f'.Thesurfacepotentialalongtheelectrodesurfacewasestimatedbysubroutine'CSOLPONT'.Theelectrodepoten-tialwascalculatedandtotaloverpotentialwasupdatedateachnode.Assumingthecurrentalongtheelectrodesurfacefrompreviousiteration,anewsurfacecon-centrationdistributionwasestimatedinthesubroutine'THETA'Apolynomialwasregressedtotheobtainedconcentrationdistribution.Thecoefcientsofthepolynomialwereusedtocalculatethecurrentdistributionduetomass-transferagainusingsubroutine'CALCONCCURR.f'.Theprocedurewasiterateduntilquantitiesconverged.F.1ProgramListingF.1.1MainProgram IMPLICIT REAL *8A-H,O-Z 290

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291 IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION ALLOCATABLE ::C:,:,GG:,DP:,: DOUBLE PRECISION ALLOCATABLE ::P1:,P3:,P5:,P7:,P9: DOUBLE PRECISION ALLOCATABLE ::P11:,P13:,P15:,P17: DOUBLE PRECISION ALLOCATABLE ::P19: DOUBLE PRECISION ALLOCATABLE ::P1C:,P3C:,P5C:,P7C:,P9C: DOUBLE PRECISION ALLOCATABLE ::P11C:,P13C:,P15C:,P17C: DOUBLE PRECISION ALLOCATABLE ::P19C: DOUBLE PRECISION ALLOCATABLE ::THC:,CONC: DOUBLE PRECISION ALLOCATABLE ::EACT:,CUR:,ETEST:,ESOLPONT: DOUBLE PRECISION ALLOCATABLE ::BM:,RESV: DOUBLE PRECISION ALLOCATABLE ::XGAUSS:,WGAUSS: DOUBLE PRECISION ALLOCATABLE ::CI:,ZCI:,AA:,BB: DOUBLE PRECISION ALLOCATABLE ::CUR1:,CUR2: DOUBLE PRECISION maxvalG,Z,LP,PP,PPC,CUR_NN,TH DOUBLE PRECISION FP,FPP,HPP,HPPP DOUBLE PRECISION H,H2,percent,percent1,percent2 INTEGER SUR_CONC,SUR_DER,N_STEP,IPP,NJP,NT DOUBLE PRECISION SC,SD,SCOLD,DSCMAX INTEGER NJL,NJLIST,NGAUSS,LMAX,NMAX,IP DOUBLE PRECISION DCI,XL,XU,T,ZZ,SCN,SCNTERM DOUBLE PRECISION CURR_NN,EJJ,alpha,beta,gamma,pi2 DOUBLE PRECISION tplus,ETAC0,ETAS0,CONC0,CUR0,TAF,res CHARACTER filename*15 DOUBLE PRECISION dpi2,BOLD,DB,DAMP,PHI,V,XACC1,XACC2,CIMAX,ERRSUB NAMELIST /par/CURR_NN,EJJ,alpha,beta,gamma,tplus,LMAX,NMAX,IPMAX,NT&,NGAUSS,DAMP,XACC1,XACC2,CONC0,Z,NJ,ERRSUB,filename,SCN EXTERNAL LP Reading the parameters open FILE ='INPUT.DAT', STATUS ='UNKNOWN' READ ,par CLOSE pi2=acos-1.0d0/2.0d0IP=10 ALLOCATE THCLMAX,CONCLMAX ALLOCATE EACTLMAX,CURLMAX,ETESTLMAX,ESOLPONTLMAX ALLOCATE BMNMAX,RESVIPMAX-1 ALLOCATE XGAUSSNGAUSS,WGAUSSNGAUSS ALLOCATE CILMAX,ZCILMAX,AALMAX,BBLMAX ALLOCATE CUR1LMAX,CUR2LMAX INCLUDE 'VELOCITYDATA.f'SUR_CONC=1SUR_DER=0 DO i=1,10SCi=0.0d0SDi=0.0d0SCOLDi=0.0d0 END DO SC=1.0d0-CONC0 INCLUDE 'ALC.f' CALL SETUPPALLP1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*P1C,P3C,P5C,P7C,P9C,P11C,P13C,P15C,P17C,P19C CALL GETCITHCLMAX,THC,pi2 CALL gauleg.0d0,pi2,XGAUSS,WGAUSS,NGAUSS

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292 BM=0.0d0JCOUNT=012BOLD=BMN=10H=Z/NJ-1H2=H*H INCLUDE 'PALL.f' INCLUDE 'CALCONCCURR.f'CUR0=CUR CALL CALINIPONTLMAX,EJJ,CURR_NN,alpha,beta,gamma,tplus,CONC,ETAC0*,ETAS0,CUR,EACT,TAF CALL CSOLPONTLMAX,CUR,THC,NMAX,BM,ESOLPONT,IP,PP,PPC,SCN,*WGAUSS,XGAUSS,NGAUSSV=ESOLPONT+EACT DO 18i=2,LMAXPHI=VPHI=PHI-ESOLPONTiEACTi=EACTi+DAMP*PHI-EACTi18 CONTINUE CALL THETALMAX,EJJ,CURR_NN,alpha,beta,gamma,tplus,TAF,*CONC,EACT,CUR,THC CALL POLYFITCONCTHETA2LMAX,CONC,THC,NT,IP,SC,res CALL FINDDSCMAXIP,SC,SCOLD,DSCMAXSC=1.0d0-CONC0JCOUNT=JCOUNT+1DB=dabsBOLD-BM/dabsBM IF DABSDB.GE.XACC1.OR.DSCMAX.GE.XACC2 GO TO 12 INCLUDE 'WRITEOUTPUT.f' INCLUDE 'DLC.f' DEALLOCATE CUR1,CUR2 DEALLOCATE CI,ZCI,AA,BB DEALLOCATE XGAUSS,WGAUSS DEALLOCATE BM,RESV DEALLOCATE EACT,CUR,ETEST,ESOLPONT DEALLOCATE THC,CONC END F.1.2MainSubroutinesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere. ***************** SUBROUTINE SETUPPALL ************************* SUBROUTINE SETUPPALLP1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*P1C,P3C,P5C,P7C,P9C,P11C,P13C,P15C,P17C,P19C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,6,B,6,D,13,G,X,6,Y,6,N,NJ DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19*,P1C*,P3C*,P5C*,P7C* DOUBLE PRECISION P9C*,P11C*,P13C*,P15C*,P17C*,P19C* DO 20II=1,NJP1II=0.0d0P3II=0.0d0

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293 P5II=0.0d0P7II=0.0d0P9II=0.0d0P11II=0.0d0P13II=0.0d0P15II=0.0d0P17II=0.0d0P19II=0.00d0P1CII=0.0d0P3CII=0.0d0P5CII=0.0d0P7CII=0.0d0P9CII=0.0d0P11CII=0.0d0P13CII=0.0d0P15CII=0.0d0P17CII=0.0d0P19CII=0.00d020 CONTINUE RETURN END ***************** SUBROUTINE BC1PALL ************************** SUBROUTINE BC1PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*SUR_CONC,SUR_DER,SC,SD IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,10,B,10,D,21,G,X,10,Y,10,N,NJ COMMON /BB/H,H2 DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19* INTEGER SUR_CONC,SUR_DER DOUBLE PRECISION SC*,SD* DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO IF SUR_CONC.EQ.1 THEN G=SC-P1JB,1=1.0d0G=SC-P3JB,2=1.0d0G=SC-P5JB,3=1.0d0G=SC-P7JB,4=1.0d0G=SC-P9JB,5=1.0d0G=SC-P11JB,6=1.0d0G=SC-P13JB,7=1.0d0

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294 G=SC-P15JB,8=1.0d0G=SC-P17JB,9=1.0d0G=SC-P19JB,10=1.0d0 END IF RETURN END ***************** SUBROUTINE INNERPALL ************************ SUBROUTINE INNERPALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19 IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19* DOUBLE PRECISION ZZ=J-1*H DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=P1J+1-2.0d0*P1J+P1J-1/H2*+Z*Z*HPP1*P1J+1-P1J-1/2.0d0/HA,1=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,1=2.0d0/H2D,1=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P3J+1-2.0d0*P3J+P3J-1/H2*+Z*Z*HPP1*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP3*P1J+1-P1J-1/2.0d0/H-Z*FP1*P3JA,1=Z*Z*HPP3/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP3/2.0d0/HA,2=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,2=2.0d0/H2+Z*FP1D,2=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P5J+1-2.0d0*P5J+P5J-1/H2*+Z*Z*HPP1*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP3*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP5*P1J+1-P1J-1/2.0d0/H*-Z*FP3*P3J-2.0d0*Z*FP1*P5JA,1=Z*Z*HPP5/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP5/2.0d0/HA,2=Z*Z*HPP3/2.0d0/HB,2=Z*FP3D,2=-Z*Z*HPP3/2.0d0/HA,3=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,3=2.0d0/H2+2.0d0*Z*FP1D,3=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P7J+1-2.0d0*P7J+P7J-1/H2

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295 *+Z*Z*HPP1*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP3*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP5*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP7*P1J+1-P1J-1/2.0d0/H*-Z*FP5*P3J-2.0d0*Z*FP3*P5J-3.0d0*Z*FP1*P7JA,1=Z*Z*HPP7/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP7/2.0d0/HA,2=Z*Z*HPP5/2.0d0/HB,2=Z*FP5D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP3/2.0d0/HB,3=2.0d0*Z*FP3D,3=-Z*Z*HPP3/2.0d0/HA,4=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,4=2.0d0/H2+3.0d0*Z*FP1D,4=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P9J+1-2.0d0*P9J+P9J-1/H2*+Z*Z*HPP1*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP3*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP5*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP7*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP9*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP7*P3J-2.0d0*Z*FP5*P5J-3.0d0*Z*FP3*P7J*-4.0d0*Z*FP1*P9JA,1=Z*Z*HPP9/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP9/2.0d0/HA,2=Z*Z*HPP7/2.0d0/HB,2=1.0d0*Z*FP7D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP5/2.0d0/HB,3=2.0d0*Z*FP5D,3=-Z*Z*HPP5/2.0d0/HA,4=Z*Z*HPP3/2.0d0/HB,4=3.0d0*Z*FP3D,4=-Z*Z*HPP3/2.0d0/HA,5=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,5=2.0d0/H2+4.0d0*Z*FP1D,5=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P11J+1-2.0d0*P11J+P11J-1/H2*+Z*Z*HPP1*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP3*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP5*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP7*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP9*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP11*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP9*P3J-2.0d0*Z*FP7*P5J-3.0d0*Z*FP5*P7J*-4.0d0*Z*FP3*P9J-5.0d0*Z*FP1*P11JA,1=Z*Z*HPP11/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP11/2.0d0/HA,2=Z*Z*HPP9/2.0d0/HB,2=1.0d0*Z*FP9D,2=-Z*Z*HPP9/2.0d0/H

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296 A,3=Z*Z*HPP7/2.0d0/HB,3=2.0d0*Z*FP7D,3=-Z*Z*HPP7/2.0d0/HA,4=Z*Z*HPP5/2.0d0/HB,4=3.0d0*Z*FP5D,4=-Z*Z*HPP5/2.0d0/HA,5=Z*Z*HPP3/2.0d0/HB,5=4.0d0*Z*FP3D,5=-Z*Z*HPP3/2.0d0/HA,6=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,6=2.0d0/H2+5.0d0*Z*FP1D,6=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P13J+1-2.0d0*P13J+P13J-1/H2*+Z*Z*HPP1*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP3*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP5*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP7*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP9*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP11*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP13*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP11*P3J-2.0d0*Z*FP9*P5J-3.0d0*Z*FP7*P7J*-4.0d0*Z*FP5*P9J-5.0d0*Z*FP3*P11J-6.0d0*Z*FP1*P13JA,1=Z*Z*HPP13/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP13/2.0d0/HA,2=Z*Z*HPP11/2.0d0/HB,2=1.0d0*Z*FP11D,2=-Z*Z*HPP11/2.0d0/HA,3=Z*Z*HPP9/2.0d0/HB,3=2.0d0*Z*FP9D,3=-Z*Z*HPP9/2.0d0/HA,4=Z*Z*HPP7/2.0d0/HB,4=3.0d0*Z*FP7D,4=-Z*Z*HPP7/2.0d0/HA,5=Z*Z*HPP5/2.0d0/HB,5=4.0d0*Z*FP5D,5=-Z*Z*HPP5/2.0d0/HA,6=Z*Z*HPP3/2.0d0/HB,6=5.0d0*Z*FP3D,6=-Z*Z*HPP3/2.0d0/HA,7=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,7=2.0d0/H2+6.0d0*Z*FP1D,7=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P15J+1-2.0d0*P15J+P15J-1/H2*+Z*Z*HPP1*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP3*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP5*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP7*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP9*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP11*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP13*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP15*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP13*P3J-2.0d0*Z*FP11*P5J-3.0d0*Z*FP9*P7J*-4.0d0*Z*FP7*P9J-5.0d0*Z*FP5*P11J-6.0d0*Z*FP3*P13J*-7.0d0*Z*FP1*P15J

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297 A,1=Z*Z*HPP15/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP15/2.0d0/HA,2=Z*Z*HPP13/2.0d0/HB,2=1.0d0*Z*FP13D,2=-Z*Z*HPP13/2.0d0/HA,3=Z*Z*HPP11/2.0d0/HB,3=2.0d0*Z*FP11D,3=-Z*Z*HPP11/2.0d0/HA,4=Z*Z*HPP9/2.0d0/HB,4=3.0d0*Z*FP9D,4=-Z*Z*HPP9/2.0d0/HA,5=Z*Z*HPP7/2.0d0/HB,5=4.0d0*Z*FP7D,5=-Z*Z*HPP7/2.0d0/HA,6=Z*Z*HPP5/2.0d0/HB,6=5.0d0*Z*FP5D,6=-Z*Z*HPP5/2.0d0/HA,7=Z*Z*HPP3/2.0d0/HB,7=6.0d0*Z*FP3D,7=-Z*Z*HPP3/2.0d0/HA,8=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,8=2.0d0/H2+7.0d0*Z*FP1D,8=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P17J+1-2.0d0*P17J+P17J-1/H2*+Z*Z*HPP1*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP3*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP5*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP7*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP9*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP11*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP13*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP15*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP17*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP15*P3J-2.0d0*Z*FP13*P5J-3.0d0*Z*FP11*P7J*-4.0d0*Z*FP9*P9J-5.0d0*Z*FP7*P11J-6.0d0*Z*FP5*P13J*-7.0d0*Z*FP3*P15J-8.0d0*Z*FP1*P17JA,1=Z*Z*HPP17/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP17/2.0d0/HA,2=Z*Z*HPP15/2.0d0/HB,2=1.0d0*Z*FP15D,2=-Z*Z*HPP15/2.0d0/HA,3=Z*Z*HPP13/2.0d0/HB,3=2.0d0*Z*FP13D,3=-Z*Z*HPP13/2.0d0/HA,4=Z*Z*HPP11/2.0d0/HB,4=3.0d0*Z*FP11D,4=-Z*Z*HPP11/2.0d0/HA,5=Z*Z*HPP9/2.0d0/HB,5=4.0d0*Z*FP9D,5=-Z*Z*HPP9/2.0d0/HA,6=Z*Z*HPP7/2.0d0/HB,6=5.0d0*Z*FP7D,6=-Z*Z*HPP7/2.0d0/H

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298 A,7=Z*Z*HPP5/2.0d0/HB,7=6.0d0*Z*FP5D,7=-Z*Z*HPP5/2.0d0/HA,8=Z*Z*HPP3/2.0d0/HB,8=7.0d0*Z*FP3D,8=-Z*Z*HPP3/2.0d0/HA,9=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,9=2.0d0/H2+8.0d0*Z*FP1D,9=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P19J+1-2.0d0*P19J+P19J-1/H2*+Z*Z*HPP1*P19J+1-P19J-1/2.0d0/H*+Z*Z*HPP3*P17J+1-P17J-1/2.0d0/H*+Z*Z*HPP5*P15J+1-P15J-1/2.0d0/H*+Z*Z*HPP7*P13J+1-P13J-1/2.0d0/H*+Z*Z*HPP9*P11J+1-P11J-1/2.0d0/H*+Z*Z*HPP11*P9J+1-P9J-1/2.0d0/H*+Z*Z*HPP13*P7J+1-P7J-1/2.0d0/H*+Z*Z*HPP15*P5J+1-P5J-1/2.0d0/H*+Z*Z*HPP17*P3J+1-P3J-1/2.0d0/H*+Z*Z*HPP19*P1J+1-P1J-1/2.0d0/H*-1.0d0*Z*FP17*P3J-2.0d0*Z*FP15*P5J-3.0d0*Z*FP13*P7J*-4.0d0*Z*FP11*P9J-5.0d0*Z*FP9*P11J-6.0d0*Z*FP7*P13J*-7.0d0*Z*FP5*P15J-8.0d0*Z*FP3*P17J-9.0d0*Z*FP1*P19JA,1=Z*Z*HPP19/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP19/2.0d0/HA,2=Z*Z*HPP17/2.0d0/HB,2=1.0d0*Z*FP17D,2=-Z*Z*HPP17/2.0d0/HA,3=Z*Z*HPP15/2.0d0/HB,3=2.0d0*Z*FP15D,3=-Z*Z*HPP15/2.0d0/HA,4=Z*Z*HPP13/2.0d0/HB,4=3.0d0*Z*FP13D,4=-Z*Z*HPP13/2.0d0/HA,5=Z*Z*HPP11/2.0d0/HB,5=4.0d0*Z*FP11D,5=-Z*Z*HPP11/2.0d0/HA,6=Z*Z*HPP9/2.0d0/HB,6=5.0d0*Z*FP9D,6=-Z*Z*HPP9/2.0d0/HA,7=Z*Z*HPP7/2.0d0/HB,7=6.0d0*Z*FP7D,7=-Z*Z*HPP7/2.0d0/HA,8=Z*Z*HPP5/2.0d0/HB,8=7.0d0*Z*FP5D,8=-Z*Z*HPP5/2.0d0/HA,9=Z*Z*HPP3/2.0d0/HB,9=8.0d0*Z*FP3D,9=-Z*Z*HPP3/2.0d0/HA,10=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,10=2.0d0/H2+9.0d0*Z*FP1D,10=-1.0d0/H2-Z*Z*HPP1/2.0d0/H RETURN END

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299 ********************** SUBROUTINE BC2 ************************** SUBROUTINE BC2ALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19 IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,10,B,10,D,21,G,X,10,Y,10,N,NJ DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19* DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=0.0d0-P1JB,1=1.0d0G=0.0d0-P3JB,2=1.0d0G=0.0d0-P5JB,3=1.0d0G=0.0d0-P7JB,4=1.0d0G=0.0d0-P9JB,5=1.0d0G=0.0d0-P11JB,6=1.0d0G=0.0d0-P13JB,7=1.0d0G=0.0d0-P15JB,8=1.0d0G=0.0d0-P17JB,9=1.0d0G=0.0d0-P19JB,10=1.0d0 RETURN END ***************** SUBROUTINE BC1PALLSC *********************** SUBROUTINE BC1PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,P13C,*P15C,P17C,P19C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,10,B,10,D,21,G,X,10,Y,10,N,NJ COMMON /BB/H,H2 DOUBLE PRECISION C,* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=0.0d0-P1CJ

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300 B,1=1.0d0G=0.0d0-P3CJB,2=1.0d0G=0.0d0-P5CJB,3=1.0d0G=0.0d0-P7CJB,4=1.0d0G=0.0d0-P9CJB,5=1.0d0G=0.0d0-P11CJB,6=1.0d0G=0.0d0-P13CJB,7=1.0d0G=0.0d0-P15CJB,8=1.0d0G=0.0d0-P17CJB,9=1.0d0G=0.0d0-P19CJB,10=1.0d0 RETURN END ***************** SUBROUTINE INNERPALLSC ********************** SUBROUTINE INNERPALLSCJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P1C,*P3C,P5C,P7C,P9C,P11C,P13C,P15C,P17C,P19C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N INCLUDE COMMON .f' DOUBLE PRECISION C,* DOUBLE PRECISION P1*,P3*,P5*,P7*,P9*,P11*,P13* DOUBLE PRECISION P15*,P17*,P19* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DOUBLE PRECISION ZZ=J-1*H DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=P1CJ+1-2.0d0*P1CJ+P1CJ-1/H2*+Z*Z*HPP1*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P1J+1-P1J-1/2.0d0/H/3.0d0A,1=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,1=2.0d0/H2D,1=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P3CJ+1-2.0d0*P3CJ+P3CJ-1/H2*+Z*Z*HPP1*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP3*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P1J+1-P1J-1/2.0d0/H/3.0d0*-Z*FP1*P3CJ-0.50d0*Z*Z*FPP1*P3JA,1=Z*Z*HPP3/2.0d0/HB,1=0.0d0

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301 D,1=-Z*Z*HPP3/2.0d0/HA,2=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,2=2.0d0/H2+Z*FP1D,2=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P5CJ+1-2.0d0*P5CJ+P5CJ-1/H2*+Z*Z*HPP1*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP3*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP5*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP3*P3CJ-2.0d0*Z*FP1*P5CJ*-0.50d0*Z*Z*FPP3*P3J-1.00d0*Z*Z*FPP1*P5JA,1=Z*Z*HPP5/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP5/2.0d0/HA,2=Z*Z*HPP3/2.0d0/HB,2=Z*FP3D,2=-Z*Z*HPP3/2.0d0/HA,3=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,3=2.0d0/H2+2.0d0*Z*FP1D,3=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P7CJ+1-2.0d0*P7CJ+P7CJ-1/H2*+Z*Z*HPP1*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP3*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP5*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP7*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP5*P3CJ-2.0d0*Z*FP3*P5CJ-3.0d0*Z*FP1*P7CJ*-0.50d0*Z*Z*FPP5*P3J-1.00d0*Z*Z*FPP3*P5J-1.50d0*Z*Z*FPP1*P7JA,1=Z*Z*HPP7/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP7/2.0d0/HA,2=Z*Z*HPP5/2.0d0/HB,2=Z*FP5D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP3/2.0d0/HB,3=2.0d0*Z*FP3D,3=-Z*Z*HPP3/2.0d0/HA,4=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,4=2.0d0/H2+3.0d0*Z*FP1D,4=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P9CJ+1-2.0d0*P9CJ+P9CJ-1/H2*+Z*Z*HPP1*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP3*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP5*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP7*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP9*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P3J+1-P3J-1/2.0d0/H/3.0d0

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302 *+Z*Z*Z*HPPP9*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP7*P3CJ-2.0d0*Z*FP5*P5CJ-3.0d0*Z*FP3*P7CJ*-4.0d0*Z*FP1*P9CJ-0.50d0*Z*Z*FPP7*P3J*-1.00d0*Z*Z*FPP5*P5J-1.50d0*Z*Z*FPP3*P7J*-2.00d0*Z*Z*FPP1*P9JA,1=Z*Z*HPP9/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP9/2.0d0/HA,2=Z*Z*HPP7/2.0d0/HB,2=1.0d0*Z*FP7D,2=-Z*Z*HPP7/2.0d0/HA,3=Z*Z*HPP5/2.0d0/HB,3=2.0d0*Z*FP5D,3=-Z*Z*HPP5/2.0d0/HA,4=Z*Z*HPP3/2.0d0/HB,4=3.0d0*Z*FP3D,4=-Z*Z*HPP3/2.0d0/HA,5=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,5=2.0d0/H2+4.0d0*Z*FP1D,5=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P11CJ+1-2.0d0*P11CJ+P11CJ-1/H2*+Z*Z*HPP1*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP3*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP5*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP7*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP9*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP11*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP9*P3CJ-2.0d0*Z*FP7*P5CJ-3.0d0*Z*FP5*P7CJ*-4.0d0*Z*FP3*P9CJ-5.0d0*Z*FP1*P11CJ*-0.50d0*Z*Z*FPP9*P3J-1.00d0*Z*Z*FPP7*P5J-1.50d0*Z*Z*FPP5*P7J*-2.00d0*Z*Z*FPP3*P9J-2.50d0*Z*Z*FPP1*P11JA,1=Z*Z*HPP11/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP11/2.0d0/HA,2=Z*Z*HPP9/2.0d0/HB,2=1.0d0*Z*FP9D,2=-Z*Z*HPP9/2.0d0/HA,3=Z*Z*HPP7/2.0d0/HB,3=2.0d0*Z*FP7D,3=-Z*Z*HPP7/2.0d0/HA,4=Z*Z*HPP5/2.0d0/HB,4=3.0d0*Z*FP5D,4=-Z*Z*HPP5/2.0d0/HA,5=Z*Z*HPP3/2.0d0/HB,5=4.0d0*Z*FP3D,5=-Z*Z*HPP3/2.0d0/HA,6=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,6=2.0d0/H2+5.0d0*Z*FP1D,6=-1.0d0/H2-Z*Z*HPP1/2.0d0/H

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303 G=P13CJ+1-2.0d0*P13CJ+P13CJ-1/H2*+Z*Z*HPP1*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP3*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP5*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP7*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP9*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP11*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP13*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP11*P3CJ-2.0d0*Z*FP9*P5CJ-3.0d0*Z*FP7*P7CJ*-4.0d0*Z*FP5*P9CJ-5.0d0*Z*FP3*P11CJ-6.0d0*Z*FP1*P13CJ*-0.50d0*Z*Z*FPP11*P3J-1.0d0*Z*Z*FPP9*P5J-1.50d0*Z*Z*FPP7*P7J*-2.0d0*Z*Z*FPP5*P9J-2.50d0*Z*Z*FPP3*P11J-3.0d0*Z*Z*FPP1*P13JA,1=Z*Z*HPP13/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP13/2.0d0/HA,2=Z*Z*HPP11/2.0d0/HB,2=1.0d0*Z*FP11D,2=-Z*Z*HPP11/2.0d0/HA,3=Z*Z*HPP9/2.0d0/HB,3=2.0d0*Z*FP9D,3=-Z*Z*HPP9/2.0d0/HA,4=Z*Z*HPP7/2.0d0/HB,4=3.0d0*Z*FP7D,4=-Z*Z*HPP7/2.0d0/HA,5=Z*Z*HPP5/2.0d0/HB,5=4.0d0*Z*FP5D,5=-Z*Z*HPP5/2.0d0/HA,6=Z*Z*HPP3/2.0d0/HB,6=5.0d0*Z*FP3D,6=-Z*Z*HPP3/2.0d0/HA,7=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,7=2.0d0/H2+6.0d0*Z*FP1D,7=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P15CJ+1-2.0d0*P15CJ+P15CJ-1/H2*+Z*Z*HPP1*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP3*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP5*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP7*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP9*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP11*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP13*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP15*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P5J+1-P5J-1/2.0d0/H/3.0d0

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304 *+Z*Z*Z*HPPP13*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP13*P3CJ-2.0d0*Z*FP11*P5CJ-3.0d0*Z*FP9*P7CJ*-4.0d0*Z*FP7*P9CJ-5.0d0*Z*FP5*P11CJ-6.0d0*Z*FP3*P13CJ*-7.0d0*Z*FP1*P15CJ*-0.50d0*Z*Z*FPP13*P3J-1.0d0*Z*Z*FPP11*P5J-1.50d0*Z*Z*FPP9*P7J*-2.0d0*Z*Z*FPP7*P9J-2.50d0*Z*Z*FPP5*P11J-3.0d0*Z*Z*FPP3*P13J*-3.50d0*Z*Z*FPP1*P15JA,1=Z*Z*HPP15/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP15/2.0d0/HA,2=Z*Z*HPP13/2.0d0/HB,2=1.0d0*Z*FP13D,2=-Z*Z*HPP13/2.0d0/HA,3=Z*Z*HPP11/2.0d0/HB,3=2.0d0*Z*FP11D,3=-Z*Z*HPP11/2.0d0/HA,4=Z*Z*HPP9/2.0d0/HB,4=3.0d0*Z*FP9D,4=-Z*Z*HPP9/2.0d0/HA,5=Z*Z*HPP7/2.0d0/HB,5=4.0d0*Z*FP7D,5=-Z*Z*HPP7/2.0d0/HA,6=Z*Z*HPP5/2.0d0/HB,6=5.0d0*Z*FP5D,6=-Z*Z*HPP5/2.0d0/HA,7=Z*Z*HPP3/2.0d0/HB,7=6.0d0*Z*FP3D,7=-Z*Z*HPP3/2.0d0/HA,8=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,8=2.0d0/H2+7.0d0*Z*FP1D,8=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P17CJ+1-2.0d0*P17CJ+P17CJ-1/H2*+Z*Z*HPP1*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP3*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP5*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP7*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP9*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP11*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP13*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP15*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP17*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP15*P3CJ-2.0d0*Z*FP13*P5CJ-3.0d0*Z*FP11*P7CJ*-4.0d0*Z*FP9*P9CJ-5.0d0*Z*FP7*P11CJ-6.0d0*Z*FP5*P13CJ*-7.0d0*Z*FP3*P15CJ-8.0d0*Z*FP1*P17CJ*-0.50d0*Z*Z*FPP15*P3J-1.0d0*Z*Z*FPP13*P5J-1.50d0*Z*Z*FPP11*P7J

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305 *-2.0d0*Z*Z*FPP9*P9J-2.50d0*Z*Z*FPP7*P11J-3.0d0*Z*Z*FPP5*P13J*-3.50d0*Z*Z*FPP3*P15J-4.0d0*Z*Z*FPP1*P17JA,1=Z*Z*HPP17/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP17/2.0d0/HA,2=Z*Z*HPP15/2.0d0/HB,2=1.0d0*Z*FP15D,2=-Z*Z*HPP15/2.0d0/HA,3=Z*Z*HPP13/2.0d0/HB,3=2.0d0*Z*FP13D,3=-Z*Z*HPP13/2.0d0/HA,4=Z*Z*HPP11/2.0d0/HB,4=3.0d0*Z*FP11D,4=-Z*Z*HPP11/2.0d0/HA,5=Z*Z*HPP9/2.0d0/HB,5=4.0d0*Z*FP9D,5=-Z*Z*HPP9/2.0d0/HA,6=Z*Z*HPP7/2.0d0/HB,6=5.0d0*Z*FP7D,6=-Z*Z*HPP7/2.0d0/HA,7=Z*Z*HPP5/2.0d0/HB,7=6.0d0*Z*FP5D,7=-Z*Z*HPP5/2.0d0/HA,8=Z*Z*HPP3/2.0d0/HB,8=7.0d0*Z*FP3D,8=-Z*Z*HPP3/2.0d0/HA,9=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,9=2.0d0/H2+8.0d0*Z*FP1D,9=-1.0d0/H2-Z*Z*HPP1/2.0d0/HG=P19CJ+1-2.0d0*P19CJ+P19CJ-1/H2*+Z*Z*HPP1*P19CJ+1-P19CJ-1/2.0d0/H*+Z*Z*HPP3*P17CJ+1-P17CJ-1/2.0d0/H*+Z*Z*HPP5*P15CJ+1-P15CJ-1/2.0d0/H*+Z*Z*HPP7*P13CJ+1-P13CJ-1/2.0d0/H*+Z*Z*HPP9*P11CJ+1-P11CJ-1/2.0d0/H*+Z*Z*HPP11*P9CJ+1-P9CJ-1/2.0d0/H*+Z*Z*HPP13*P7CJ+1-P7CJ-1/2.0d0/H*+Z*Z*HPP15*P5CJ+1-P5CJ-1/2.0d0/H*+Z*Z*HPP17*P3CJ+1-P3CJ-1/2.0d0/H*+Z*Z*HPP19*P1CJ+1-P1CJ-1/2.0d0/H*+Z*Z*Z*HPPP1*P19J+1-P19J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP3*P17J+1-P17J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP5*P15J+1-P15J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP7*P13J+1-P13J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP9*P11J+1-P11J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP11*P9J+1-P9J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP13*P7J+1-P7J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP15*P5J+1-P5J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP17*P3J+1-P3J-1/2.0d0/H/3.0d0*+Z*Z*Z*HPPP19*P1J+1-P1J-1/2.0d0/H/3.0d0*-1.0d0*Z*FP17*P3CJ-2.0d0*Z*FP15*P5CJ-3.0d0*Z*FP13*P7CJ*-4.0d0*Z*FP11*P9CJ-5.0d0*Z*FP9*P11CJ-6.0d0*Z*FP7*P13CJ*-7.0d0*Z*FP5*P15CJ-8.0d0*Z*FP3*P17CJ-9.0d0*Z*FP1*P19CJ*-0.50d0*Z*Z*FPP17*P3J-1.0d0*Z*Z*FPP15*P5J-1.50d0*Z*Z*FPP13*P7J*-2.0d0*Z*Z*FPP11*P9J-2.50d0*Z*Z*FPP9*P11J-3.0d0*Z*Z*FPP7*P13J

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306 *-3.50d0*Z*Z*FPP5*P15J-4.0d0*Z*Z*FPP3*P17J-4.50d0*Z*Z*FPP1*P19JA,1=Z*Z*HPP19/2.0d0/HB,1=0.0d0D,1=-Z*Z*HPP19/2.0d0/HA,2=Z*Z*HPP17/2.0d0/HB,2=1.0d0*Z*FP17D,2=-Z*Z*HPP17/2.0d0/HA,3=Z*Z*HPP15/2.0d0/HB,3=2.0d0*Z*FP15D,3=-Z*Z*HPP15/2.0d0/HA,4=Z*Z*HPP13/2.0d0/HB,4=3.0d0*Z*FP13D,4=-Z*Z*HPP13/2.0d0/HA,5=Z*Z*HPP11/2.0d0/HB,5=4.0d0*Z*FP11D,5=-Z*Z*HPP11/2.0d0/HA,6=Z*Z*HPP9/2.0d0/HB,6=5.0d0*Z*FP9D,6=-Z*Z*HPP9/2.0d0/HA,7=Z*Z*HPP7/2.0d0/HB,7=6.0d0*Z*FP7D,7=-Z*Z*HPP7/2.0d0/HA,8=Z*Z*HPP5/2.0d0/HB,8=7.0d0*Z*FP5D,8=-Z*Z*HPP5/2.0d0/HA,9=Z*Z*HPP3/2.0d0/HB,9=8.0d0*Z*FP3D,9=-Z*Z*HPP3/2.0d0/HA,10=-1.0d0/H2+Z*Z*HPP1/2.0d0/HB,10=2.0d0/H2+9.0d0*Z*FP1D,10=-1.0d0/H2-Z*Z*HPP1/2.0d0/H RETURN END ********************** SUBROUTINE BC2 ************************** SUBROUTINE BC2ALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C IMPLICIT REAL *8A-H,O-Z IMPLICIT INTEGER I-N COMMON /BA/A,10,B,10,D,21,G,X,10,Y,10,N,NJ DOUBLE PRECISION C,* DOUBLE PRECISION P1C*,P3C*,P5C*,P7C*,P9C*,P11C* DOUBLE PRECISION P13C*,P15C*,P17C*,P19C* DO i=1,10 DO jj=1,10Ai,jj=0.0d0Bi,jj=0.0d0Di,jj=0.0d0 END DO END DO G=0.0d0-P1CJB,1=1.0d0G=0.0d0-P3CJB,2=1.0d0G=0.0d0-P5CJB,3=1.0d0

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307 G=0.0d0-P7CJB,4=1.0d0G=0.0d0-P9CJB,5=1.0d0G=0.0d0-P11CJB,6=1.0d0G=0.0d0-P13CJB,7=1.0d0G=0.0d0-P15CJB,8=1.0d0G=0.0d0-P17CJB,9=1.0d0G=0.0d0-P19CJB,10=1.0d0 RETURN END ********************* SUBROUTINE THETA ************************** SUBROUTINE THETALMAX,EJJ,CURR_NN,al,be,ga,tp,TAF,*CONC,EACT,CUR,THC INTEGER LMAX DOUBLE PRECISION EJJ,CURR_NN,al,be,ga,tp DOUBLE PRECISION CONCLMAX,EACTLMAX,CURLMAX,THCLMAX DOUBLE PRECISION ALPHA,BETA,GAMMA,TPLUS,EXCH DOUBLE PRECISION CUR0,DX1,DX2 DOUBLE PRECISION TAF,S,ETA,XACC,DS INTEGER N,NZ,NJALPHA=alBETA=beGAMMA=gaTPLUS=tpEXCH=1.0d0/EJJXACC=1.0e-15S=CONC DO 60NZ=2,LMAXETA=EACTNZCUR0=CURNZ DO 56N=1,100X1=EJJ*TAF*S**GAMMA-ALPHA**DEXPALPHA*ETA*DEXPALPHA*TPLUS*S-1.0d0X2=EJJ*TAF*S**GAMMA+BETA**DEXP-BETA*ETA*DEXPBETA*TPLUS*.0d0-SDX1=X1*GAMMA-ALPHA/S+ALPHA*TPLUSDX2=X2*GAMMA+BETA/S-BETA*TPLUSDS=X1-X2-CUR0/DX1-DX2 IF DABSDS.LE.XACC GO TO 10S=S-DS56 CONTINUE 10CONCNZ=S60 CONTINUE RETURN END

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308 F.1.3IncludeFilesInthissection,alltheincludeleswhicharecalledinthemainprogramaswellasinvarioussubroutimesarelistedhere. *************** INCLUDE FILE VELOCITYDATA f **************** OPEN unit =2, file ='HFP_Benard.txt' DO i=1,10 READ ,*ii,FPi,FPPi,HPPi,HPPPi ENDDO HPP1=HPPHPP3=HPPHPP5=HPPHPP7=HPPHPP9=HPPHPP11=HPPHPP13=HPPHPP15=HPPHPP17=HPPHPP19=HPPFP1=FPFP3=FPFP5=FPFP7=FPFP9=FPFP11=FPFP13=FPFP15=FPFP17=FPFP19=FPHPPP1=HPPPHPPP3=HPPPHPPP5=HPPPHPPP7=HPPPHPPP9=HPPPHPPP11=HPPPHPPP13=HPPPHPPP15=HPPPHPPP17=HPPPHPPP19=HPPPFPP1=FPPFPP3=FPPFPP5=FPPFPP7=FPPFPP9=FPPFPP11=FPPFPP13=FPPFPP15=FPPFPP17=FPPFPP19=FPP CLOSE ********************* INCLUDE FILE ALC f ****************** ALLOCATE C,NJ,GGNJ,DP,NJ ALLOCATE P1NJ,P3NJ,P5NJ,P7NJ,P9NJ

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309 ALLOCATE P11NJ,P13NJ,P15NJ,P17NJ ALLOCATE P19NJ ALLOCATE P1CNJ,P3CNJ,P5CNJ,P7CNJ,P9CNJ ALLOCATE P11CNJ,P13CNJ,P15CNJ,P17CNJ ALLOCATE P19CNJNJLIST=1NJL=50001ERREQN=0.0N=10 ********************* INCLUDE FILE PALL f ****************** maxvalG=1.0d0 INITILIZE ALL COEFFICIENT MATRIX C DO I=1,N DO K=1,NXI,K=0.0d0YI,K=0.0d0 END DO END DO JJ=1110J=0360J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 IF J.EQ.1 CALL BC1PALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,*SUR_CONC,SUR_DER,SC,SD Equation for interior region C IF J.GT.1.AND.J.LT.NJ THEN CALL INNERPALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19 END IF Equation for second boundary condition C IF J.EQ.NJ THEN CALL BC2ALLJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19 END IF CALL BANDJ,C IF J.NE.NJ GOTO 360 DO I=1,NJP1I=P1I+C,IP3I=P3I+C,IP5I=P5I+C,IP7I=P7I+C,IP9I=P9I+C,IP11I=P11I+C,IP13I=P13I+C,IP15I=P15I+C,IP17I=P17I+C,IP19I=P19I+C,I END DO

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310 INCLUDE 'CHECKCONVERGE.f'JJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.25 GO TO 110maxvalG=1.0d0 INITILIZE ALL COEFFICIENT MATRIX C DO I=1,N DO K=1,NXI,K=0.0d0YI,K=0.0d0 END DO END DO JJ=1 IF SCN.GT.0.0d0 THEN 111J=0361J=J+1SUMH=SUMH+H DO I=1,NGI=0.0d0 DO K=1,NAI,K=0.0d0BI,K=0.0d0DI,K=0.0d0 ENDDO ENDDO Call BC1 for J =1 C IF J.EQ.1 CALL BC1PALLSCJ,C,P1C,P3C,P5C,P7C,P9C,P11C,*P13C,P15C,P17C,P19C Equation for interior region C IF J.GT.1.AND.J.LT.NJ THEN CALL INNERPALLSCJ,C,P1,P3,P5,P7,P9,P11,P13,P15,P17,P19,P1C,*P3C,P5C,P7C,P9C,P11C,P13C,P15C,P17C,P19C END IF Equation for second boundary condition C IF J.EQ.NJ THEN CALL BC2ALLSCJ,C,P1C,P3C,P5,P7,P9,P11,P13,P15,P17,P19 END IF CALL BANDJ,C IF J.NE.NJ GOTO 361 DO I=1,NJP1CI=P1CI+C,IP3CI=P3CI+C,IP5CI=P5CI+C,IP7CI=P7CI+C,IP9CI=P9CI+C,IP11CI=P11CI+C,IP13CI=P13CI+C,IP15CI=P15CI+C,IP17CI=P17CI+C,IP19CI=P19CI+C,I END DO INCLUDE 'CHECKCONVERGE1.f'JJ=JJ+1 IF maxvalG.GT.ERRSUB.AND.JJ.LT.25 GO TO 111

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311 F.1.4InputFileThefollowingvariableswerereadinthemainprogrambythefollowinginputle.. ************************ INPUT DAT **************************% CURR_NN=125.0d0EJJ=5.0d0alpha=0.50d0beta=0.50d0gamma=0.50d0tplus=0.50d0LMAX=501NMAX=51IPMAX=20NT=6NGAUSS=400DAMP=0.05XACC1=1.0e-6XACC2=1.0e-5CONC0=0.50d0Z=10.0d0NJ=10001ERRSUB=1.0e-7filename='NSCY125C0P5.txt'SCN=1000.0d0

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BIOGRAPHICALSKETCHPavanKumarShuklagrewupinKanpur,India.HeattendedtheIndianInsti-tuteofTechnology,Kanpur,forhisundergraduateandmaster'sdegreeinchem-icalEngineering.HenishedhisundergraduatedegreeinSpring1994andhismastrer'sdegreeinSpring1996.HecametotheU.S.inJuly1996toworkonaNASAproject.HestartedhisPh.D.programinthefallof1998attheUniversityofFloridaunderthedirectionofProfessorMarkE.Orazem.Hegraduatedinthesummerof2004afterspendingsixyearsbeingeducatedinchemicalandelectro-chemicalengineering. 321


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STATIONARY HEMISPHERICAL ELECTRODE UNDER SUBMERGED JET
IMPINGEMENT AND VALIDATION OF MEASUREMENT MODEL CONCEPT
FOR IMPEDANCE SPECTROSCOPY












By

PAVAN KUMAR SHUKLA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2004














ACKNOWLEDGMENTS

I would like to sincerely thank my advisor, Professor Mark E. Orazem, for his

technical guidance throughout my graduate research. I will always be grateful for

his patience and understanding. I would like to pay special thanks to Dr. Oscar

Crisalle for his expertise and guidance in the parameter estimation theory and pro-

gramming. His time and effort were invaluable to me, especially in understanding

transfer function methodology.

I would like to express my gratitude to Dr. Gert Nelissan for his hard work in

numerical simulations. His simulation results buttressed my work alongside my

experiments.

I gratefully acknowledge OLI Systems, Inc. (108 American Way, Morris Plains,

NJ USA, www.olisystems.com) for use of their CorrosionAnalyzer 1.3 software.

I would like to express my appreciation to the members of my committee, Dr.

Jason Butler, Dr. Anuj Chauhan and Dr. Darryl Butt, for their contributions in my

dissertation defense.

I would like to thank the previous and present members of the electrochemi-

cal engineering group, Mike Membrino, Chen Chen Qui, Kerry Allahar, Nellian

Perez-Garcia, and Vicky Huang. I was fortunate to be a member of this group.

I would like to express my heartfelt thanks and gratitude to my family mem-

bers and friends who have always encouraged and facilitated my academic pur-

suits. I would like to mention my high school physics teacher Mr. Gyanendra

Sharma, whose love for physics inspired me to be curious about science.















TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ................... .......... ii


LIST OF TABLES ................... ............... vii


LIST OF FIGURES ................... ............ x


ABSTRACT ................... .................. xxi


CHAPTER

1 INTRODUCTION ................... ........... 1

1.1 History of Electrode Systems ....... ........... .... 3
1.2 Measurement Model Concept ................ ... 4
1.3 Scope and Structure of the Thesis ....... ........ 6

2 HYDRODYNAMIC MODELS FOR A STATIONARY ELECTRODE UN-
DER SUBMERGED JET IMPINGEMENT ................. 9

2.1 Schematic Illustration of the System ................. 9
2.2 Governing Equations ................... ........ 10
2.3 Potential Flow Calculation ................. ..... 12
2.4 Boundary Layer Flow Calculation .................. 14
2.4.1 Solution Method ...... ........... ...... 17
2.4.2 Results ................... .......... 18
2.5 Boundary Layer Separation ...... ........... .... 21
2.6 Numerical Simulation ....... .......... ....... 22
2.6.1 Governing Equations. ....... ........... 23
2.6.2 Numerical Method ................. ..... 25
2.6.3 Simulation Results ...... ........... ..... 25
2.7 Summary ......................... .............. 26

3 CONVECTIVE-DIFFUSION MODELS FOR A STATIONARY HEMISPHER-
ICAL ELECTRODE UNDER SUBMERGED JET IMPINGEMENT .... 29

3.1 Governing Equations ................... ........ 29
3.2 Solution Method and Results .................. ... 32
3.3 Mass Transfer Limited Current .................. .. 33









3.4 Numerical Simulations ...... ......... ....... 36
3.5 Conclusion ................ ............. 39

4 HYDRODYNAMIC AND MASS-TRANSFER MODELS FOR A ROTAT-
ING HEMISPHERICAL ELECTRODE ..................... 41

4.1 Schematic Illustration of the System ................. 41
4.2 Hydrodynamic Model ................. ....... 42
4.2.1 Governing Equations. ....... ........... .. 43
4.2.2 Results ............... ........... 46
4.2.3 Fluid Flow at the Corner ...... ........... 47
4.3 Mass Transfer ............. ............ 48
4.4 Summary ............. .............. 53

5 CURRENT AND POTENTIAL DISTRIBUTION AT AXISYMMETRIC ELEC-
TRODES .......................................... 54

5.1 Introduction ............... ................. 54
5.2 Development of Mathematical Model ................ 55
5.2.1 Hydrodynamics ....... ........... ...... 56
5.2.2 Mass Transfer ...... ......... ........ 57
5.2.3 Electrode Kinetics ....... .......... .... 59
5.2.4 Concentration Overpotential ................. 59
5.2.5 Solution Potential in Outer Region .......... 60
5.2.6 Electrode Potential ......... .......... ... 62
5.3 Dimensionless Quantities ......... .......... ... 62
5.4 Calculation Procedure ...... ......... ....... 66
5.4.1 Disk electrode ...... ......... ........ 66
5.4.2 Hemispherical electrode ..... .......... .. 68
5.5 Current Distribution at Disk Electrode .............. 70
5.5.1 Primary Distribution ..... .......... 70
5.5.2 Secondary Current Distribution ............. 71
5.5.3 Tertiary Current Distribution ............... 72
5.6 Current Distribution at Hemispherical Electrode .......... 76
5.6.1 Primary Distribution ..... .......... 76
5.6.2 Secondary Distribution ..... .......... .. 78
5.6.3 Tertiary Distribution ..... .......... .. 78
5.7 Current Distribution on the Rotating Hemispherical Electrode .. 88
5.7.1 Governing Equations. ....... ........... .. 89
5.7.2 Numerical Procedure . .. .......... ... 89
5.7.3 Results ............... ........... 91
5.8 Summary ............. .............. 92

6 VALIDATION OF THE MEASUREMENT MODEL CONCEPT .... 96

6.1 Introduction ............. ............. 96
6.2 Definition of Errors .............. ... ........ 99
6.3 Equivalence of Measurement Models .............. 99
6.4 Kramers-Kronig Relations ................... ... 101









6.5 Complex Nonlinear Least-square Regression . . ..... 102
6.5.1 Solution Method . . . . ... . 106
6.5.2 Convergence Criterion . . . . . 107
6.5.3 Weighting Strategy . . . . . 108
6.5.4 Computer Program Implementation . . ..... 110
6.5.5 Confidence Interval . . . ..... .. 110
6.6 Method ....... . . ............... 111
6.7 Results ..... ............... ............... 112
6.7.1 Evaluation of Stochastic Errors . . . 112
6.7.2 Evaluation of Bias Errors . . . . 119
6.8 Conclusions . . . . . . . . 130

7 ELECTROCHEMICAL MEASUREMENTS OF OXYGEN REDUCTION AT
NICKEL ELECTRODE . . . . . . . 131

7.1 Reaction Mechanism of Oxygen Reduction . . ..... 132
7.2 Experimental ................... ............ 134
7.3 Polarization Measurements .................. 139
7.4 Impedance Measurements ................... ... 144
7.5 Measurement Model Analysis . ............. .. 145
7.5.1 Determination of Stochastic Error Structure ......... 148
7.5.2 Kramers-Kronig Consistency Check ............. 152
7.6 Process Model ................... ............ 158
7.7 Summary ................. ............ 162

8 ELECTROCHEMICAL MEASUREMENTS OF FERRICYANIDE REDUC-
TION AT NICKEL ELECTRODE ...... . . . .. 163

8.1 Introduction . . . . . . . . 163
8.2 Experimental Method ................... ........ 164
8.3 Experimental Results ................... ........ 165
8.3.1 Steady-State Measurement .................. 165
8.3.2 Impedance Measurement .................. 166
8.4 Measurement Model Analysis . ... .......... .. 170
8.4.1 Determination of Error Structure ............ 170
8.4.2 Kramers-Kronig Consistency Check ............. 172
8.5 Surface Analysis for Disk Electrode ................. 173
8.6 Optical Micrographs of the Hemispherical Electrode ........ 176
8.7 Thermodynamic Analysis ......... .......... ... 180
8.8 Discussion ................ ............. 181
8.9 Conclusions ...... .......... ............ 184

9 CONCLUSIONS . . . . . . . 186


10 SUGGESTED FUTURE RESEARCH . . . . .... 188









APPENDIX

A HYDRODYNAMIC EQUATIONS IN SERIES EXPANSION ....... 189

A.1 Ordinary Differential Equations for H2,-1(0) and F2,-1(0) ...... 189
A.2 Solutions of H2,-1(0) and F2,-1(0) ..... ............ 195
A.3 Extrapolation of Finite Difference Values for Fi_, (0) ....... 199

B SOLUTION OF CONVECTIVE-DIFFUSION EQUATION FOR INFINITE
SCHMIDT NUMBER ....... ......... ............ 204


C BOUNDARY-LAYER PROGRAM LISTING ............... 208

C.1 Program Listing ...... ............ ........ 208
C.1.1 Main Program ...... ..... ..... ....... 208
C.1.2 Main Subroutines ......... .......... .... 209
C.1.3 Include Files .............. .... ......... 233

D PROGRAM LISTING FOR CONVECTIVE DIFFUSION CALCULATIONS247

D.1 Program Listing ...... ............ ........ 247
D.1.1 Main Program ...... ..... ..... ....... 247
D.1.2 Main Subroutines ......... .......... .... 248
D.1.3 Include Files .............. .... ......... 273

E PROGRAM LISTING FOR CALCULATING THE CURRENT DISTRIBU-
TION AT THE STATIONARY HEMISPHERICAL ELECTRODE UNDER
SUBMERGED JET IMPINGEMENT ................ 281

E.1 Program Listing ...... ............ ........ 281
E.1.1 Main Program ...... ..... ..... ....... 281
E.1.2 Main Subroutines ......... .......... .... 283
E.1.3 Include Files .............. .... ......... 288
E.1.4 Input File ............ ............ 289

F PROGRAM LISTING FOR CALCULATING THE CURRENT DISTRIBU-
TION AT THE ROTATING HEMISPHERICAL ELECTRODE ...... 290

F.1 Program Listing .................... ........... 290
F.1.1 Main Program ...... ..... ..... ....... 290
F.1.2 Main Subroutines ......... .......... .... 292
F.1.3 Include Files .............. .... ......... 308
F.1.4 Input File ............ ............ 311

REFERENCES ................... .. ............. 312


BIOGRAPHICAL SKETCH ....... ....... ........... 321















LIST OF TABLES


Table page

2.1 Series expansion coefficients Fi,_,(0) and H2i-1(0) in the equations
(2-34) and (2-35) for H(O, ) and F(0, ) at 0. . . 21

2.2 Series expansion coefficients F -_,(0) and H2i- (0) in the equations
(2-34) and (2-35) for H(O, ) and F(O, ) at = 0. . . 22

3.1 Calculated values for coefficients '1,2i-1(0) and 2-,2i-1(0) used in
equation (3-14) for mass-transfer-limited current distribution. 33

3.2 Physical properties of the electrolyte used in the numerical solution
of equation (3-21). . . . . . . ... 38

4.1 ^i,_,(0) and H2'-1(0) coefficients in the series expansion of equa-
tions (4-16) and (4-15) for H(0, ) and F(0, ) at ( = 0. The third
column in the table lists the values reported by Barcia et al. and
the fourth column lists the values calculated using the continuity
equation. ...... . . . .............. 47

4.2 Calculated values for coefficients used in equation (4-29) for calcu-
lating mass-transfer-limited current distribution. . . 50

5.1 Calculated values for uniformity parameter Tdisk (see equation (5-
55)), iavg/ilim, and ir=o/iavg for the current distributions presented in
Figures 5-3 and 5-4. The values of J and N was 5 and 125, respec-
tively. ..... . . .... ............... 76

5.2 Values of uniformity parameter Ths (see equation (5-60)) for for sta-
tionary The calculated values are for hemispherical electrode under
jet impingement. the current distributions presented in Figure 5-7.
Parameter J was fixed at 5. . . . .... . 86

5.3 Values of iavg/(iiim),avg for stationary hemispherical electrode under
jet impingement. The calculated values are for the current distribu-
tions presented in Figure 5-7. Parameter J was fixed at 5. . 87









5.4 Values of Ths (see equation (5-60)) for stationary hemispherical elec-
trode under jet impingement. The calculated values are for the cur-
rent distributions presented in Figure 5-10. Parameter N was fixed
at 20. . . . . . . . . . 87

5.5 Values of iavg/(iiim),av for stationary hemispherical electrode under
jet impingement. The calculated values are for the current distribu-
tions presented in Figure 5-10. Parameter N was fixed at 20. . 87

5.6 Values of iavg/(iiim)avg and Ths for the rotating hemispherical elec-
trode. The calculated values are for the current distributions pre-
sented in Figure 5.14(a). Parameters N and J was fixed at 125 and
5, respectively. . . . . . . ....... 94

6.1 Model parameters for the fit of a Voigt measurement model to impeda-
nce scans #1, #5, and #25 presented in Figure 6-3. . . 128

6.2 Model parameters for the fit of a Transfer function measurement
model to the impedance scan #1,#5, and #25 presented in Figure 6-3. 129

7.1 Chemical composition of Nickel 270 .. . . . 131

7.2 Properties of oxygen saturated 0.1 M NaCI at 250C. . . 138

7.3 Species considered in calculation of the Pourbaix diagram presented
as Figure 7-8. ...... . . . ........ 142

7.4 Computed values of hydrodynamic constant a for the disk electrode. 144

7.5 Calculated values of hydrodynamics constant a for the hemispheri-
cal electrode. . . . . . . . . 144

7.6 Experimental conditions for impedance scan of oxygen reduction at
disk and hemispherical electrode . . . . 145

7.7 Model parameters of error structure for different experimental con-
ditions on disk electrode. . . . . ... 151

7.8 Model parameters of error structure for different experimental con-
ditions on hemispherical electrode. .. . . . 151

7.9 Model parameters for the fit of a Voigt measurement model to imag-
inary part of first impedance scans at disk electrode. The jet velocity
for this set of experiments was at 1.99 meter/sec. . . 153









7.10 Model parameters for the fit of a Voigt measurement model to imag-
inary part of first impedance scans at hemispherical electrode. The
jet velocity for this set of experiments was at 3.98 meter/sec. . 158

7.11 Estimated model parameters of a CPE equivalent circuit model to
impedance data collected at the disk electrode. Reported parame-
ters values are average of seven replicate spectrum collected at an
experimental condition. . . . . . . 160

7.12 Estimated model parameters of a CPE equivalent circuit model to
impedance data collected at the hemispherical electrode. Reported
parameters values are average of seven replicate spectrum collected
at an experimental condition. . . . . .. 160

8.1 Electrolyte properties used in experiments. . . ..... 164

8.2 Calculated values of solution resistance for primary current distri-
bution, RP1, using electrical conductivities of electrolyte listed in
table 8.1 ....... . . . . ....... 165

8.3 Model of obtained error structure for impedance spectra on disk
and hemispherical electrode. . . . .. .. 170

8.4 Model parameters for the fit of a Voigt measurement model to imag-
inary part of first impedance scans at disk and hemispherical electrode 173

8.5 Species considered in calculation of the Pourbaix diagram presented
as Figure 8-15. ............... . . ....... . 183

8.6 The boundary layer point of separation at the stationary hemispher-
ical electrode. . . . . . . ...... 184















LIST OF FIGURES


Figure page

2-1 Schematic illustration of a stationary hemispherical submerged im-
pinging jet electrode system. . . . .... . 10

2-2 Computed flow trajectories corresponding to the potential flow so-
lution, given as equation (2-12), for the hemispherical electrode sub-
jected to a submerged impinging jet system with -_ce r0 2 as a pa-
rameter.. ... . . ............... 14

2-3 Distribution of the dimensionless pressure gradient given as equa-
tion (2-15). ...... . . .... ............. 15

2-4 Schematic diagram of grid for calculation domain H is the spacing
between adjacent nodes. . . . . . 18

2-5 Dimensionless radial and colatitude functions H1 () and Fi (0) as a
function of (see equations (2-24) and (2-25)). . . . 19

2-6 Calculated dimensionless surface shear stress as a function of an-
gle 0. Solid lines represent the result for the stationary hemisphere
under submerged jet impingement and dashed lines represent the
result for the rotating hemisphere. . . . . 23

2-7 Schematic representation of the simulated flow geometry. The di-
mensions are given in units of m. The arrow represents the general
direction of flow, and the cylindrical electrode is located at the origin. 24

2-8 Fluid streamlines in the vicinity of the electrode for an inlet Reynolds
number of 1,100. The color map indicate the pressure distribution.
The radial dimension is given in units of m. . . . 26

2-9 Fluid streamlines in the vicinity of the electrode for an inlet Reynolds
number of 11,000. The color map indicate the pressure distribution.
The radial dimension is given in units of m. Figure 2.9(b) provides
an enlarged image of the recirculation shown in Figure 2.9(a). 27









3-1 Calculated mass-transfer limited current density for a hemispher-
ical electrode subjected to a submerged impinging jet. Solid lines
represent results for the stationary electrode, and the dashed lines
represent results for the rotating hemispherical electrode.a) Contri-
bution to equation (3-17) for an infinite Schmidt number; b) Contri-
bution to equation (3-17) providing correction for a finite Schmidt
number........ . . . . ......... 35

3-2 Reactant concentration distribution as a function of distance from
electrode surface, obtained through numerical simulation of equa-
tion (3-21). The blue line corresponds to zero concentration on the
electrode surface, whereas red corresponds to the bulk reactant con-
centration. The radial dimension is given in units of cm. These sim-
ulations were performed for Re 11300 in the nozzle. . . 38

3-3 Calculated mass-transfer-limited current density for different Reynolds
number at the inlet of the nozzle. The vertical dash line at 620 is the
point of boundary layer separation. The physical properties of the
electrolyte used in the simulations are listed in Table 3.2. . 39

4-1 Schematics illustration of Rotating Hemispherical Electrode. . 41

4-2 Shear Stress Distribution at the electrode surface. Solid line repre-
sent results of Barcia et al. the dashed line represent the results of
Chin, and the dotted line represent the result of Manohar. . 46

4-3 A two dimensional depiction of boundary layer at the intersection
of electrode and insulating plane. . . . . 48

4-4 Calculated mass-transfer limited current density for a rotating hemi-
spherical electrode. a) Contribution to equation (4-32) for an infinite
Schmidt number; b) Contribution to equation (4-32) providing cor-
rection for a finite Schmidt number. . . . . 52

4-5 Relative error in mass-transfer-limited current given by expressions
(4-36) and (4-37) as a function of Schmidt number. . . 53

5-1 Schematics illustration of an axisymmetric body in a curvilinear co-
ordinate system. The horizontal dash line represents the axis of
symmetry, and the fluid field is assumed to be symmetric around
this axis. ..... . . .... .............. 56

5-2 Primary current distribution at the disk electrode. The value of local
current approaches to infinity as r/ro 1 . . ..... 72









5-3 Calculated current, concentration, and solution potential distribu-
tion at the disk electrode. The simulations were done for J = 5,
N = 125, and C(0) = 0.5 to 0.9 in incremental steps of 0.1. a) i/i*im
as a function of r/ro. b) Dimensionless concentration distribution
as a function of r/ro. c) Dimensionless solution potential at the elec-
trode surface as a function of r/ro. . . . . 74

5-4 Calculated current, concentration, and solution potential distribu-
tion at the disk electrode. The simulations were done for J = 5,
N = 125, and C(0) 0.4, 0.3, 0.2, 0.1, 0.05. a) i/iim as a function of
r/ro. b) Dimensionless concentration distribution as a function of
r/ro. c) Dimensionless solution potential at the electrode surface as
a function of r/ro. ...... . . . .... ...... 75

5-5 1 ir=o/iavg as a function Tdisk for different values of C(O). . 77

5-6 i/iavg as a function r/ro for different values of C(O). . . 77

5-7 Calculated current distribution as a function 0 at the stationary hemi-
spherical electrode under submerged jet impingement. The simula-
tion were done for different values of pole concentrations C(O), and
parameters J and N. The vertical dash line represent the point of
boundary layer separation. (a) N 125 and J = 5, (b) N = 50 and
J = 5, (c) N = 20 and J = 5, and (d) N = 5 and J = 5. . . 80

5-8 Calculated concentration profile corresponding to the current distri-
bution presented in Figure 5-7 as a function 0 at the stationary hemi-
spherical electrode under submerged jet impingement. The simula-
tion were done for different values of pole concentrations C(O), and
parameters J and N. The vertical dash line represent the point of
boundary layer separation. (a) N 125 and J = 5, (b) N = 50 and
J = 5, (c) N = 20 and J = 5, and (d) N = 5 and J = 5. . . 81

5-9 Calculated values of solution potential corresponding to the cur-
rent distribution presented in Figure 5-7 as a function of 0 at the
stationary hemispherical electrode under submerged jet impinge-
ment. The simulation were done for different values of pole con-
centrations C(O), and parameters J and N. The vertical dash line
represent the point of boundary layer separation. (a) N 125 and
J = 5, (b) N = 50 and J = 5, (c) N = 20 and J = 5, and (d) N = 5
and J = 5. ..... . . .... ............. 82









5-10 Calculated current distribution as a function of 0 at the stationary
hemispherical electrode under submerged jet impingement. The
simulation were done for N = 20, and different values of pole con-
centrations C(0) and parameters J. The vertical dash line represent
the point of boundary layer separation. (a) N = 20 and J 100, (b)
N 20 and J = 10, (c) N 20 and J = and (d) N 20 and J = 0.1. 83

5-11 Calculated concentration profile corresponding to the current distri-
bution presented in Figure 5-10 as a function of 0 at the stationary
hemispherical electrode under submerged jet impingement. The
simulation were carried out for N = 20, and different values of
pole concentrations C(0) and parameters J. The vertical dash line
represent the point of boundary layer separation. (a) N = 20 and
J = 100, (b) N = 20 and J = 10, (c) N = 20 and J = 1, and (d)
N = 20 and J = 0.1....... . . . ... ...... 84

5-12 Calculated values of solution potential corresponding to the current
distribution presented in Figure 5-10 as a function of 0 at the sta-
tionary hemispherical electrode under submerged jet impingement.
The simulation were carried out for N = 20, and different values of
pole concentrations C(0) and parameters J. The vertical dash line
represent the point of boundary layer separation. (a) N = 20 and
J = 100, (b) N = 20 and J = 10, (c) N = 20 and J = 1, and (d)
N = 20 and J = 0.1....... . . . ... ...... 85

5-13 Current distribution calculations presented in the paper by Nisancioglu
and Newman.(a) Figure 6 of the paper by Nisanciiglu et al. (b) Fig-
ure 2 of the paper by Nisancioglu et al. .. . . 88

5-14 Calculated values of current distribution, concentration distribu-
tion, surface overpotential, and concentration overpotential as a func-
tion of 0 at the rotating hemispherical electrode. The lines in black
color corresponds to the calculations for infinite Schmidt number,
and lines in blue color corresponds to calculated results with Sc =
1000.0. These calculations were performed for J = 5 and N = 125.
(a) Current distribution as a function of 0, (b) Dimensionless Con-
centration distribution as a function of 0, (c) Dimensionless surface
overpotential as a function of 0, (d) Dimensionless concentration
overpotential as a function of 0. . . . . 93

5-15 Calculated dimensionless Solution potential along the electrode sur-
face as a function of 0. The results corresponds to the current distri-
butions given in Figure 5.14(a). (a) Dimensionless solution potential
without Schmidt number correction, (b) Dimensionless solution po-
tential with Schmidt number correction. . . . 94

6-1 A schematic representation of a Voigt element measurement model. 97









6-2 Impedance spectra obtained for the reduction of ferricyanide on a
platinum rotating disk electrode. . . . . 111

6-3 Current measurements before and after the impedance scans shown
in Figure 6-2. The data sets singled out for error analysis are high-
lighted. ....... . . . . ....... 112

6-4 Relative departures from the mean value for the first four spectra
given in Figure 6-2: a) real part and b) imaginary part of the impeda-
nce. ....... . . . . ....... 115

6-5 Residual errors for the fit of a transfer-function measurement model,
equation (6-3), to the impedance data presented in Figure 6-2: a) real
part and b) imaginary part of the impedance. . .... 116

6-6 Residual errors for the fit of a Voigt measurement model, equation
(6-1), to the impedance data presented in Figure 6-2: a) real part and
b) imaginary part of the impedance. .. . . . 117

6-7 Standard Deviations for the data presented in Figure 6-2, obtained
from the residual errors presented in Figures 6-5 and 6-6. The dashed
line represents the results obtained for the Kramers-Kronig -consistent
data in set 2 and3. . . . . ... . 118

6-8 Relative departures from the mean value for the second four spec-
tra given in Figure 6-2: a) real part and b) imaginary part of the
impedance. . ......... . ........... 120

6-9 Residual errors for the fit of a transfer-function measurement model,
equation (6-3), to the impedance data presented in Figure 6-2: a) real
part and b) imaginary part of the impedance. . .... 121

6-10 Residual errors for the fit of a Voigt measurement model, equation
(6-1), to the impedance data presented in Figure 6-2: a) real part and
b) imaginary part of the impedance. .. . . . 122

6-11 Standard Deviations for the data presented in Figure 6-2: a) results
obtained from the residual errors presented in Figures 6-9 and 6-10,
and b) results obtained from the residual errors for Data set 3. The
dashed line represents the results obtained for the Kramers-Kronig
-consistent data in set 2 and 3. . . . . . 123









6-12 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the first impedance spectrum presented in Fig-
ure 6-2. a) fit to the imaginary part, where dashed lines represent
the 2o- bound for the stochastic error structure determined in the
previous section; b) prediction of the real part where dashed lines
represent the 95.4% confidence interval for the model obtained by
Monte Carlo simulation using the calculated confidence intervals
for the estimated parameters. . . . . .. 125

6-13 Residual errors for the fit of a transfer-function measurement model
to the imaginary part of the first impedance spectrum presented in
Figure 6-2. a) fit to the imaginary part, where dashed lines represent
the 2, bound for the stochastic error structure determined in the
previous section; b) prediction of the real part where dashed lines
represent the 95.4% confidence interval for the model obtained by
Monte Carlo simulation using the calculated confidence intervals
for the estimated parameters. . . . . .. 127

7-1 Experimental setup used for the study of oxygen reduction reaction. 136

7-2 Schematic diagram of impinging jet electrochemical cell. a) Layout
of the cell with its component. b) Important cell dimensions. . 137

7-3 Image of the hemispherical electrode during the polarization mea-
surement of oxygen reduction reaction. . . . 139

7-4 Polarization curve for the oxygen reduction reaction collected at the
disk electrode. The solid line corresponds to average fluid jet veloc-
ity of 1.99 m/s, dash line corresponds to 2.99 m/s, and dotted line
corresponds to 3.98 m/s. . . . . ..... . 140

7-5 Polarization curve for the oxygen reduction reaction collected at the
hemispherical electrode. The solid line corresponds to average fluid
jet velocity of 1.99 m/s, dash line corresponds to 2.99 m/s, and dot-
ted line corresponds to 3.98 m/s. . . . . 140

7-6 Diffusion limited current for oxygen reduction in 0.1 M NaCI as a
function of square root of the jet velocity for Ni 270 disk electrode.
The dashed line is a linear fit to the data points. . .... 141

7-7 Diffusion limited current for oxygen reduction in 0.1 M NaCI as a
function of square root of the jet velocity for Ni 270 hemispherical
electrode. The dashed line is a linear fit to the data points. . 142









7-8 The potential-pH diagram of nickel in oxygen saturated sodium
chloride solution. The potential is reported with respect to stan-
dard hydrogen electrode(SHE). The vertical dash line corresponds
to pH of 0.1 M sodium chloride solution. This diagram was gener-
ated by computer software CorrosionAnalyzer 1.3 Revision 1.3.33.
OLI Systems, Inc. The activity of nickel ions was assumed to be
1.0 x 10-6M. . . ...... ............ 143

7-9 First impedance scan collected during the study of oxygen reduc-
tion at the disk electrode under submerged jet impingement. The
impedance spectrum were collected for different jet velocities and
bias potential. . . . . . . ... 146

7-10 First impedance scan collected during the study of oxygen reduc-
tion at the hemispherical electrode under submerged jet impinge-
ment. The impedance spectrum were collected for different jet ve-
locities and bias potential. . . . . ... . 146

7-11 Collected impedance spectrum for jet velocity of 2.99m/s and bias
potential of -0.540 V. a) Complex plane plot; Real and imaginary
impedance are normalized with surface area; b) Real and imaginary
impedance as a function of frequency. . . . 147

7-12 Standard Deviations of stochastic errors for the impedance data col-
lected on disk electrode. A representative first scan of the analyzed
data is presented in Figure 7-9. The results are presented for differ-
ent jet velocities and applied bias potentials. a) Values of bias po-
tentials was selected to provide the average current level at about
quarter of mass-transfer-limited current; b) Values of bias poten-
tials was selected to provide the average current level at about half
of mass-transfer-limited current. . . . . 149

7-13 Standard Deviations of stochastic errors for the impedance data col-
lected on disk electrode. A representative first scan of the analyzed
data is presented Figure 7-10. The results are presented for different
jet velocities and applied bias potentials. a) Values of bias potentials
was selected to provide the average current level at about quarter
of mass-transfer-limited current; b) Values of bias potentials was
selected to provide the average current level at about half of mass-
transfer-limited current. . . . . .. . 150









7-14 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the impedance spectrum presented in Figure 7-9
by open circles. a) fit to the imaginary part, where dashed lines rep-
resent the 2o- bound for the stochastic error structure determined
in the previous section; b) prediction of the real part where dashed
lines represent the 95.4% confidence interval for the model obtained
by Monte Carlo simulation using the calculated confidence inter-
vals for the estimated parameters. . . . . 154

7-15 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the impedance spectrum presented in Figure 7-
9 by half filled circles. a) fit to the imaginary part, where dashed
lines represent the 2a bound for the stochastic error structure de-
termined in the previous section; b) prediction of the real part where
dashed lines represent the 95.4% confidence interval for the model
obtained by Monte Carlo simulation using the calculated confidence
intervals for the estimated parameters. . . . 155

7-16 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the impedance spectrum presented in Figure 7-
10 by open traingles. a) fit to the imaginary part, where dashed
lines represent the 2a bound for the stochastic error structure de-
termined in the previous section; b) prediction of the real part where
dashed lines represent the 95.4% confidence interval for the model
obtained by Monte Carlo simulation using the calculated confidence
intervals for the estimated parameters. . . . 156

7-17 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the first impedance spectrum presented in Fig-
ure 7-10 by inverted half filled triangles. a) fit to the imaginary part,
where dashed lines represent the 2a bound for the stochastic er-
ror structure determined in the previous section; b) prediction of
the real part where dashed lines represent the 95.4% confidence in-
terval for the model obtained by Monte Carlo simulation using the
calculated confidence intervals for the estimated parameters. . 157

7-18 A CPE equivalent circuit model fit to the impedance data collect at
the jet velocity of 2.99 m/s. The bias potential was set at -0.540 V.
a) Colpmex plane plot of the fit to the data; b) Real and imaginary
residual errors as a function of frequency. . . .. 161

8-1 Polarization curve of nickel disk electrode in the solution of 1.0 M
NaOH, 0.005 M K3Fe(CN)6 and K4Fe(CN)6. The average fluid veloc-
ity in the jet was 1.99 meter/second. .. . . . 166


xv11









8-2 Impedance spectra obtained for the reduction of ferricyanide on a
nickel disk electrode under submerged jet impingement. The av-
erage fluid velocity in the jet was set at 1.99 meter/second and a
bias potential of +0.195 V was applied to the electrode. The elec-
trolyte for this set of experiments consisted of 1.0 M NaOH, 0.005 M
K3Fe(CN)6 and K4Fe(CN)6. ..... . . . . .. 167

8-3 Collected Impedance spectra for the reduction of ferricyanide on a
nickel hemispherical electrode under submerged jet impingement.
The average fluid velocity in the jet was set at 1.99 meter/second
and a bias potential of +0.195 V was applied to the electrode. The
electrolyte for this set of experiments consisted of 1.0 M NaOH, 0.005
M K3Fe(CN)6 and K4Fe(CN)6........ . ..... . 167

8-4 Complex-plane plots of impedance obtained on the disk electrode.
a) t 60s; andb) t = 1,860s.. ..... . . ....... .168

8-5 Complex-plane plots of impedance obtained on the hemispherical
electrode. a) t= 60 s; and b) t 1, 860 s. . . ... 169

8-6 Standard Deviations for the data presented in Figure 8-2. The solid
line represents the fit to the error structure. . . .... 171

8-7 Standard Deviations for the data presented in Figure 8-3. The solid
line represents the fit to the error structure. . . .... 171

8-8 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the first impedance spectrum presented in Fig-
ure 8-2. a) fit to the imaginary part, where dashed lines represent
the 2o- bound for the stochastic error structure determined in the
previous section; b) prediction of the real part where dashed lines
represent the 95.4% confidence interval for the model obtained by
Monte Carlo simulation using the calculated confidence intervals
for the estimated parameters. . . . . .. 174

8-9 Residual errors for the fit of a Voigt measurement model to the
imaginary part of the first impedance spectrum presented in Fig-
ure 8-3. a) fit to the imaginary part, where dashed lines represent
the 2o- bound for the stochastic error structure determined in the
previous section; b) prediction of the real part where dashed lines
represent the 95.4% confidence interval for the model obtained by
Monte Carlo simulation using the calculated confidence intervals
for the estimated parameters. . . . . .. 175


xv111









8-10 Top view of the disk electrode after impedance experiments. a)
Undisturbed image of electrode. b) Image obtained after the right
side of disk electrode was cleaned with sand paper to highlight the
contrast between metal surface and deposits. . . . 176

8-11 Scanning Electron spectroscopy of of a disk electrode after immer-
sion in the electrolyte supported by 1.0 M NaOH. . . 177

8-12 Energy Dispersive Spectroscopy (EDS) analysis of a disk electrode
after immersion in the electrolyte supported by 1.0 M NaOH. 177

8-13 Images of hemispherical electrode after impedance experiments. .. 178

8-14 Side view of the hemispherical electrode after washing it with deion-
ized water. . . . . ....... ....... 180

8-15 The potential-pH diagram for nickel in water containing sodium
hydroxide, potassium ferricyanide, potassium ferrocyanide, and dis-
solved oxygen. The potential is reported with respect to standard
hydrogen electrode(SHE). The vertical dashed lines represent the
pH of electrolyte solution used in the present study. The line on
the left corresponds to a solution containing 0.1 M NaOH, 0.005 M
K3Fe(CN)6 and K4Fe(CN)6, and the line on the right corresponds
to 1.0 M NaOH, 0.005 M K3Fe(CN)6 and K4Fe(CN)6. This diagram
was generated using CorrosionAnalyzer 1.3 Revision 1.3.33 by OLI
Systems, Inc. The activity of nickel ions was assumed to be 1 x 10-6 M.182

8-16 Collected Impedance spectrum for the reduction of ferricyanide on
a nickel disk electrode under submerged jet impingement. The av-
erage fluid velocity in the jet was set at 1.99 meter/second and a bias
potential of +0.195 V was applied to the electrode. The electrolyte
for this experiment consisted of 0.1 M NaOH, 0.005 M K3Fe(CN)6
and K4Fe(CN)6. The represented impedance spectrum was collected
after 5 hours of immersion of the electrode in the electrolyte. . 184

A-1 Calculated profiles of dimensionless radial and colatitude functions
in the expansion of (2-24) and (2-25). (a) H,1() and Fi() as a func-
tion (b) H3() and F3( ) as a function (c) H5() and F5() as a
function and (d) H7() and F7() as a function . . 196

A-2 Calculated profiles of dimensionless radial and colatitude functions
in the expansion of (2-24) and (2-25). (a) H9() and F9g() as a func-
tion of (b) H11 () and F1 () as a function of (c) H13 () and F13 ()
as a function of and (d) H15() and F15() as a function of . 197









A-3 Calculated profiles of dimensionless radial and colatitude in the ex-
pansion of (2-24) and (2-25). (a) H17() and F17() as a function of
(b) H19() and F1g( ) as a function of (c) H21(0) and F21( ) as a
function of and (d) H23(0) and F23(0) as a function of . 198

A-4 Calculated profiles of dimensionless radial and colatitude in the ex-
pansion of (2-24) and (2-25). (a) H25(0) and F25(0) as a function of ,
(b) H27(0) and F27( ) as a function of . . . . 199

A-5 First derivative of dimensionless colatitude velocity coefficients F2i-, ()
at -0 for different grid spacing. (a) F,(0) as a function of H2 (b)
F3(0) as a function of H2, (c) F5(0) as a function of H2, and (d) F(0O)
as a function of H2. . . . . ...... . 200

A-6 First derivative of dimensionless colatitude velocity coefficients F2i- ()
at -0 for different grid spacing. (a) F9(0) as a function of H2 (b)
F,,(0) as a function of H2, (c) F3(O) as a function of H2, and (d)
F,5(O) as a function of H2. . . . . . . 201

A-7 First derivative of dimensionless colatitude velocity coefficients F2i- ()
at -0 for different grid spacing. (a) F7,(0) as a function of H2 (b)
F,9(O) as a function of H2, (c) F'01) as a function of H2, and (d)
F23(0) as a function of H2 . . . . . . 202

A-8 First derivative of dimensionless colatitude velocity coefficients F2i- ()
at -0 for different grid spacing. (a) F2,(0) as a function of H2 (b)
F,7(0) as a function of H2 ....... . . ..... 203














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


STATIONARY HEMISPHERICAL ELECTRODE UNDER SUBMERGED JET
IMPINGEMENT AND VALIDATION OF MEASUREMENT MODEL CONCEPT
FOR IMPEDANCE SPECTROSCOPY

By

Pavan Kumar Shukla

August 2004

Chair: Mark E. Orazem
Major Department: Chemical Engineering


Interpretation of electrochemical impedance measurements requires an ade-

quate understanding of electrode surface phenomena, current distribution, and

stochastic error structure. Most importantly, nonuniform current distributions ob-

fuscate impedance analysis using regression. Traditional electrode systems such

as the rotating disk electrode have a nonuniform current distribution; therefore,

use of the rotating disk electrode is not suitable for impedance studies at high cur-

rent levels. In this work, a stationary hemispherical electrode under submerged

jet impingement is suggested to be an alternative. Primary and secondary current

distributions on stationary hemispherical electrode system are uniform, increasing

the likelihood of uniform tertiary current distribution. Moreover, electrochemical

processes can be monitored in situ on a stationary hemispherical electrode. In the

present work, a hydrodynamic model was devised using the boundary layer the-

ory and compared to the computational fluid dynamic model developed at Vrije

Universiteit Brussel, Belgium. Both models predicted a separation of boundary









layer at the stationary hemispherical electrode under submerged jet impingement.

The hydrodynamic model results were used to obtain a solution of convective-

diffusion at the mass-transfer limiting condition. Calculations for steady-state

current and potential distribution below the mass-transfer limited current were

performed to obtain the conditions for uniform current.

Reduction of oxygen and ferricyanide were studied on both the disk and the

hemispherical electrode under jet impingement. The objective was to understand

the differences in impedance response of the disk and the hemispherical elec-

trodes. Repeated impedances measurements were conducted on both electrode

systems. The impedance analysis of ferricyanide reduction showed the evidence

of boundary layer separation at the hemispherical electrode.

A systematic study was undertaken to evaluate the measurement model ap-

proach for assessing the error structure of electrochemical impedance measure-

ments. The remaining question was whether the error structure obtained with this

model was a property of the measurement or depended on the arbitrary selection

of a measurement model. Transfer function and Voigt-element based models were

used to assess the error structure of impedance measurements. In spite of differ-

ences in the fitting errors and numbers of parameters needed for the regression,

the values for the frequency-dependent stochastic errors were found to be inde-

pendent of the measurement model used. These results confirm the measurement

model approach for error analysis. The confidence intervals for the parameter es-

timates differed for the two models. The Voigt-element based model was found

to provide the tightest confidence intervals and was more suited for evaluation of

consistency with the Kramers-Kronig relations.


xx11














CHAPTER 1
INTRODUCTION

Accurate determination of physical properties for electrochemical systems re-

mains a challenge. Electrochemical impedance Spectroscopy (EIS) provides a frame-

work whereby different physical properties of the system can be estimated simul-

taneously even for complex systems. Interpretation of electrochemical measure-

ments is facilitated when experiments are conducted under well-defined and eas-

ily characterized flow conditions. Experimental systems such as the rotating disk

electrode (RDE)1 and the stationary disk electrode under a submerged imping-

ing jet2'3 have been employed extensively in electrochemical investigations. The

rotating and impinging jet disk electrode geometries are attractive because an ac-

curate solution is available for convective diffusion, and the current distribution

is uniform at the mass-transfer-limited conditions.

Experimental investigations of electrochemical reaction mechanisms, however,

are not generally conducted under mass-transfer limitations. The current and po-

tential distribution on a disk electrode below the mass transfer limited current is

not uniform,4-6 and it has been shown that neglect of the nonuniform current dis-

tribution introduces error in estimation of kinetic parameters from steady state

(DC) measurements.7-9

Even the most complete expressions available for convective diffusion impeda-

nce on a rotating disk electrode10'11 or on a disk electrode under a submerged

impinging jet" assume that the system may be treated as having a uniform cur-
rent distribution. Numerical calculations presented by Appel and Newman13 and

Durbha et al.14 illustrated the influence of a non-uniform current distribution






2

on the impedance response. Orazem et al.15 suggested that the discrepancy be-

tween experimental measurements and a detailed mathematical model could be

attributed partially to the influence of the non-uniform current distribution below

the mass-transfer-limited value. This claim was discussed further by Orazem and

Tribollet."1 Matos et al.16 have demonstrated experimentally that the impedance

response on a disk electrode was significantly different than that on a rotating

hemisphere electrode (RHE), for which the primary current and potential distri-

butions are uniform.

Current mathematical models for the impedance of a disk electrode with nonuni-

form current distributionl1314 are too complex for regression analysis. The pre-

ferred approach for experimental investigation of electrode kinetics is to use ge-

ometries for which mass-transfer is well-defined and current distribution is uni-

form at the experimental condition. The rotating hemispherical electrode, intro-

duced by Chin,17 has a uniform primary current distribution and would therefore

be a suitable configuration for experiments conducted under conditions such that

the current distribution is not influenced by the non-uniform accessibility to mass

transfer. Nisanqibglu and Newman"1 demonstrated that current distribution in the

RHE is uniform. This condition of uniformity is achieved when the total current

is smaller than 68 percent of the average mass-transfer-limited value. A refined

mathematical model for the convective diffusion impedance of a RHE, developed

by Barcia et al.,19 provided an excellent match to experimental impedance mea-

surements conducted under these conditions.

Systems that employ a stationary electrode facilitate use of in situ observation

or surface-analysis techniques. Orazem et al.,12 for example, used in situ video

microscopy to obtain images of a disk electrode under a submerged impinging jet.

These images were then used to interpret impedance measurements in terms of









viscoelastic properties of corrosion product films.20 Experiments using scanning

ellipsometry on a disk electrode under a submerged impinging jet were employed

to distinguish between the influence of convective diffusion and hydrodynamic

shear.21 Flow channel experiments have been employed by Alkire and Cangellari22

to illustrate the role of current distribution on formation of salt films.

To date, no experimental system exists in the literature exhibiting a uniform

primary current distribution, a stationary electrode amenable to in situ observa-

tion, and well defined flow characteristics allowing control of convective diffu-

sion. The objective of the present work was to develop the hydrodynamic, convec-

tive diffusion and current distribution calculations for a stationary hemispherical

electrode subjected to a submerged impinging jet. The use of a stationary elec-

trode was intended to facilitate in situ observation of electrode processes, and the

hemispherical electrode geometry was intended to ensure that the primary and

secondary current distributions would be uniform.23 The present work provides a

foundation for the design of electrode systems and for development of models for

the impedance response.

1.1 History of Electrode Systems

Geometries such as disks, spheres, and cylinders have been widely explored in

fluid mechanics, heat, and mass-transfer studies. The idea to employ a disk geom-

etry as an electrochemical experimental tool was envisioned after Levich24 treated

the convective diffusion problem at a rotating disk electrode. Levich showed that

the surface of a rotating disk has uniform mass-transfer for the limiting conditions.

For a long time it was assumed that the current below the mass-transfer limited

value is also uniform for the rotating disk electrode.

Rotating cylinders were suggested to be an alternative to the disk electrode for

mass-transfer research as reviewed by Eisenberg et al.25 The disk electrode has









been widely used in numerous studies because of its simple design and ease of

operation. The surface of the disk electrode can be easily polished and reused

without losing its geometric features. Riddiford26 provided a detailed account of

the evolution of the disk electrode and its use in electroanalytical studies. New-

man's27 treatment of mass-transfer coupled with potential distribution, and elec-

trode kinetics for the rotating disk electrode showed that the current distribution

at the electrode surface is highly nonuniform even at current levels slightly below

the mass-transfer limited value.

Chin17 proposed the rotating spherical electrode to be an alternative to the disk

electrode for high-rate deposition and dissolution studies. Matlosz and cowork-

ers28 proposed a hybrid electrode geometry with central disk and a surrounding

hemispherical electrode. The resulting electrode was subsequently called a disk-

hemispherical electrode. The geometric features of the system allowed the pri-

mary current distribution to be finite at the edge of the electrode. Madore et al.29

suggested a cylindrical hull electrode. They calculated the primary current dis-

tribution for the system with different cell parameters. Dinan et al.30 proposed a

recessed rotating disk electrode. This geometry provided a uniform current dis-

tribution; however, uniform accessibility to mass-transfer was lost due to its geo-

metric feature.

1.2 Measurement Model Concept

Measurement model concept was first introduced by Agarwal et al.31 3The

methodology was devised for the following reasons.

1. To estimate stochastic error structure of electrochemical impedance spec-

troscopy data, and

2. to check for consistency of impedance data with Kramers-Kronig relations.









The proposed measurement model consisted of Voigt elements and a solution re-

sistance connected in series with each other. Agarwal et al.31 showed the applica-

bility of measurement model to various impedance data. The Voigt measurement

model with sufficient parameters was able to fit the impedance data within the

noise level. Later, Agarwal et al.32 devised a method to filter the replication errors

of impedance data in order to distinguish between stochastic errors and deter-

ministic errors. In the subsequent paper, Agarwal et al.33 showed the applicability

of Voigt measurement model to assess the consistency of impedance data with

Kramers-Kronig relations.

Pauwels et al.34 have recently proposed a transfer function based measurement

model. In light of Pauwels's model, the measurement model concept was reeval-

uated for estimation of stochastic errors. The purpose of this work was to answer

the question whether the error structure obtained with Voigt model was a prop-

erty of the measurement or depended on the arbitrary selection of a measurement

model. Both measurement models were applied to estimate stochastic errors in the

impedance measurements collected at the rotating disk electrode of ferricyanide

reduction. Furthermore, the same data set was also analyzed for Kramers-Kronig

consistency check using the two measurement models. The estimated error struc-

ture was found to be independent of choice of measurement model even though

transfer function model required fewer parameters to fit the impedance data. The

confidence intervals for the parameter estimates differed for the two models. The

Voigt-element based model was found to provide the tightest confidence inter-

vals. As a result, Kramers-Kronig consistency check was more sensitive for Voigt

element based measurement model.









1.3 Scope and Structure of the Thesis

The structure of the thesis can be divided into three parts. The first part of the

thesis is presented in Chapters 2, 3, 4, and 5. This part deals with the hydrody-

namic models, convective diffusion models, and current distributions calculations

for submerged stationary hemispherical electrode under jet impingement and ro-

tating hemispherical electrode. The second part of the thesis presents a study of

measurement model concepts for three different measurement models in Chap-

ter 6. The third part deals with experimental investigation of two electrochemical

systems in Chapters 7 and 8. A reader can go through the first and third part of

this thesis exclusively without losing the continuity. The second part can be read

independently.

Chapter 2 provides a rigorous treatment of fluid mechanics for stationary elec-

trode under jet impingement. Two hydrodynamic models were developed for

the system. The first model was developed using boundary layer theory, and

the governing equations were solved by a series method. The model predicted

the separation of boundary layer at an angle of 54.80 from the pole. However,

this model is valid up to the point of boundary layer separation, after which the

fluid mechanics becomes undefined in the region beyond separation. A com-

putational fluid dynamic model (CFD), developed by Dr. Gert Nelissen at Vrije

Universiteit Brussel, Belgium, was used to identify the fluid mechanics over the

entire electrode surface. The CFD model predicted vortex formation in the sep-

arated part of the boundary layer. The angle of separation was predicted to be

620 by the CFD model. A solution of convective-diffusion equation is provided in

Chapter 3. A solution of convective-diffusion was developed with series method

which predicted the mass-transfer-limited current until the boundary layer sepa-

ration point. A complementary CFD model of convective-diffusion, developed by









Dr. Gert Nelissen at Vrije Universiteit Brussel, Belgium, solved the governing

equation over the entire surface.

Chapter 4 presents a review of the hydrodynamics and the mass-transfer for

a rotating hemispherical electrode. The governing equations were solved using

the series solution. The objective of this chapter was to provide a correction to the

solution given by Barcia et al.19

A generalized mathematical model to obtain the current and potential distri-

bution at axisymmetric electrodes was developed in Chapter 5. The model was

then applied to calculate the distribution at the submerged stationary disk and

hemispherical electrode under jet impingement. A numerical calculation proce-

dure was developed to solve the governing equations. A modified mathematical

model was also developed to obtain the current and potential distribution at the

rotating hemispherical electrode. This model accounted for correction in the mass-

transfer to the electrode due to a finite value of the Schmidt number. An algorithm

was developed to solve the governing equations.

Chapter 6 reviews the measurement model concept for estimation of stochastic

errors in impedance spectroscopy data. The chapter presents the three different

models. Impedance data collected at the rotating disk electrodes for ferricyanide

reduction were analyzed for stochastic errors. The data were also analyzed for

consistency with Kramers-Kronig relations.

Chapter 7 presents experimental study of oxygen reduction at the nickel elec-

trode. Electrochemical measurements were performed at the disk and hemispher-

ical electrodes. Repeated impedance spectrum were collected at different experi-

mental conditions.

Chapter 8 provides an experimental study of ferricyanide reduction at the

disk and hemispherical electrode under submerged jet impingement. Impedance






8

measurements were carried out at electrodes made of nickel. Analysis of impeda-

nce at the stationary hemispherical electrode provided an evidence of boundary

layer separation.

Conclusions from this work are provided in Chapter 9, and suggestions for

future research is in Chapter 10.














CHAPTER 2
HYDRODYNAMIC MODELS FOR A STATIONARY ELECTRODE UNDER

SUBMERGED JET IMPINGEMENT

This chapter presents a detailed description of the hydrodynamics of a station-

ary submerged hemispherical electrode under submerged jet impingement. The

electrode is amenable to in situ observation and has a uniform primary and sec-

ondary current distribution below the mass-transfer-limited value. The present

work is intended to provide a foundation for the design of stationary hemispher-

ical electrode systems and for development of steady state mass transfer and ter-

tiary current distribution calculations for the system.

2.1 Schematic Illustration of the System

A schematic illustration of a stationary hemispherical electrode under a sub-

merged impinging jet is presented in Figure 2-1, where a hemispherical electrode

protrudes out of a planar insulating surface and a nozzle is placed above the hemi-

sphere. The center of nozzle is axisymmetric with the hemisphere. The whole

system is submerged in an aqueous electrolyte assumed to have uniform fluid

properties. The dimensions of the nozzle are sufficiently large and its placement

is sufficiently apart from the electrode such that flow field of the fluid coming

out nozzle can be described as being a potential flow with uniform axial veloc-

ity. A detailed description of flow field generated from the nozzle can be found

in the paper by Scholtz and Trass.35 The walls of the enclosure were assumed to

be sufficiently distant that they do not influence the flow patterns near the elec-

trode surface. A spherical polar coordinate system is employed to describe the

system, where r represent the radial outward direction, 0 represents the colatitude

9










nozzle







hemispherical
Al / O electrode
insulating plane
Figure 2-1: Schematic illustration of a stationary hemispherical submerged im-
pinging jet electrode system.

direction, and 0 is along the body of the revolution. The corresponding fluid field

velocity components are Vr, veo, and v6, respectively.

2.2 Governing Equations

The steady-state fluid flow around hemisphere under submerged jet impinge-

ment can be treated by dividing the flow field into two regions: the outer or po-

tential flow region, where inertial forces dominate, and the inner or boundary

layer region, where viscous and inertial forces are of the same magnitude. The

fluid flow in the boundary layer region is described by Navier-Stokes and mass-

conservation equations, and the flow in the potential flow region is described by

mass-conservation only.

Howarth36 first derived the governing equations for fluid flow around a rotat-

ing sphere in the spherical coordinate system. These equations are valid for high

Reynolds numbers, which corresponds to a high rotation speed of the sphere. The

equations for fluid flow in the boundary layer can be modified by setting the 0

component of fluid velocity equal to zero. This is a valid assumption, because

the fluid flow around this hemisphere is axisymmetric. Under assumption of

constant fluid properties, the equations governing a thin boundary layer on an









axisymmetric body of rotation37'36 are conservation of momentum in the colati-

tude direction


9ve 9v 1 Ops 12,,,
-- +v r1 --- +v 7,1 (2-1)
ro 90 Or pro 90 9r2

and conservation of mass


S+ -+ cot(O) = 0 (2-2)
ro 80 Or ro

The underlying assumptions in deriving the above equations (2-1) and (2-2)

are:

1. The fluid flow in the boundary layer is laminar, and the gradients of all quan-

tities are large in the direction normal to the surface: however, their tangen-

tial gradients are relatively small.

2. The momentum flow in the r-direction is much smaller than the 0-direction.

From this assumption, it can be concluded that the pressure gradient in the

r-direction vanishes.

3. The thickness of the momentum boundary layer, 60, is much smaller than the

radius of the hemisphere.

The governing equation for the potential flow region is

1 21 a
r1a (r2) + in()a (sin (0) ',) =0 (2-3)
r Or sin (0) 60

which is the continuity equation in the outer flow region. Equations (2-1), (2-2),

and (2-3) complete the description of fluid flow around the stationary hemisphere.

The objective is to determine the fluid flow field within the boundary layer. The

solution procedure progressed in two stages. First, following the usual bound-

ary layer development for forced flow,37 a solution was obtained for the potential









flow region. The potential flow solution provided the pressure distribution over

the electrode surface and the far-field boundary conditions needed for solution of

the boundary layer equations. Second, the boundary layer equation were solved

using series expansion discussed by Barcia et al.19 for the rotating hemispherical

electrode.

2.3 Potential Flow Calculation

The velocity potential 0 satisfies Laplace's equation, which can be written in

spherical polar coordinates as


a r (2 + a(sin (0) = 0 (2-4)
Or Or ) sin (0) 90 90

where the radial component of the fluid velocity is given by


v, a (2-5)
Or

the angular or colatitude component of the fluid velocity is given by

18a
S'go (2-6)
r 60

and r and 0 are the radial and angular components, respectively.

The no-penetration boundary conditions can be expressed as


S0 (2-7)
Sr,e=7v/2

for the insulating plane and as

S = 0 (2-8)
r=ro,0

for the electrode surface. A symmetry condition for the centerline can be expressed

as

S- (2-9)
0r,0=








Under the assumption that the flow can be considered to be of uniform velocity

towards the insulating plane and the presence of hemisphere does not effect the

fluid field far away from the electrode, the velocity potential should approach an

asymptotic behavior and can be expressed as


r-oo,o c r (3cos2(0) 1) (2-10)

where c6 is a hydrodynamic constant. Equation (2-10) was previously applied in

the development of the potential flow solution for a submerged jet impingement

onto a flat disk.38 Thus, use of equation (2-10) constitutes a statement that, far from

the electrode, the influence of the shape of the hemispherical electrode should

diminish.

The solution of equation (2-4) subjected to boundary conditions (2-7) to (2-10)

is given by

2 -C) (3cos3COS2() 1)(2-11)

with the corresponding stream function b given by


S-C r2 r) sin2 ) oS(0) (2-12)

Computed values for flow trajectories, given as 1 ro02, are presented in Figure

2-2 as a function of dimensionless position scaled by the hemisphere radius ro.

The boundary layer calculation presented in the subsequent section employs the

pressure gradient obtained from Bernoulli's equation


p + tpo2 = constant (2-13)

where the velocity is given by the potential flow solution. Thus, given that v,|r =

0 and, from equations (2-11) and (2-6), that

5cro
2= sin(20) (2-14)










-100 \ \ \100-
/ 50 50 \
4 10 5 1 0 1 5 10

0 3

2

1

0
-5 -4 -3 -2 -1 0 1 2 3 4 5
r Ir

Figure 2-2: Computed flow trajectories corresponding to the potential flow solu-
tion, given as equation (2-12), for the hemispherical electrode subjected to a sub-
merged impinging jet system with -_cj r0 2 as a parameter.

the pressure gradient at the electrode surface can be expressed as

1 p 25
= sin(40) (2-15)
pC2r 2 0 4
The dimensionless pressure gradient along the electrode surface is given in Fig-

ure 2-3 as a function of colatitude angle 0. The dimensionless pressure gradient

changes sign at a position of 0 = r/4. As -shown in the subsequent section, the

reversal of the pressure driving force for flow induces separation of the velocity

boundary layer.

2.4 Boundary Layer Flow Calculation

The solution technique employed to solve equations (2-1) and (2-2) closely fol-

lows closely the development presented by Barcia et al.19 for the rotating hemi-

spherical electrode.

Equations (2-1) and (2-2) can be conveniently written in dimensionless form by

introducing dimensionless variable (, H(O, ), F(O, ). The dimensionless momen-

tum and continuity equations are:

1 ( F(0, H) ( F(0, ) sin(4) 1 2 F,
4F(, ) 4H( 2 + 2(2-16)
4 80 8( 4 2 8(2













S 4


I 0


-4


-8
0 15 30 45 60 75 90
o / degree

Figure 2-3: Distribution of the dimensionless pressure gradient given as equation
(2-15).

and
1 F(O, E) aH(O, E) 1
-2 9F(0, O) cot(0)= -O (2-17)
2 80 2 2

respectively, where the pressure gradient was introduced from equation (2-15), (

is the dimensionless radial position given in terms of the hydrodynamic constant

a as

r ro) (2-18)

H(O, ) is the dimensionless radial velocity, such that


v= -2 avH(0, ) (2-19)


and F(O, ) is the dimensionless colatitude velocity, such that


a, rF(0,) (2-20)
2

The no-slip boundary conditions at the electrode surface for the colatitude and

radial velocity components can be expressed as


(2-21)


F(O, ) =o- =o0









and

H(, |) =o- =0 (2-22)

respectively. The condition that the flow must approach the potential flow solution

far from the surface is expressed by


F(0, \) lco = sin(20) (2-23)


Comparison between equations (2-23) and (2-14) reveals that a = 5c6, which pro-

vides that the boundary layer equations corresponding to a jet impinging upon a

planar surface are recovered for 0 = 0.

Following Howarth,36 H(0, ) and F(0, ) can be expanded in terms of 0 and (

as

H(O,) = o2i- 2H2,- ) (2-24)
i= 1
and

F(O,) -= 2i-1F2i-1,() (2-25)
i= 1
respectively. The sin(40) term arising in equation (2-16) from the colatitude pres-

sure gradient can be expanded as

43 45 A7 f_ n+l14 92n-l
sin(40) = 40 43 + )o 7 + -(4 +Oj (2-26)
3! 5! 7! (2n 1)!

and the cot(0) term appearing in equation (2-17) can be expanded as

1 0 03 205
cot(O) 1 0 (2-27)
8 3 45 945

The number of terms in the series n can be arbitrarily selected to achieve a desired

level of accuracy. In the present work, the number of terms in the expansions (2-

24) to (2-27) was limited to n = 14 because terms of higher-order in the expansion

(2-26) for the colatitude pressure gradient were negligibly small as compared to

the largest terms in equation (2-16). Introduction of equations (2-24)-(2-27) into









equations (2-16) and (2-17), and collecting the terms of given orders of 0 yields a

series of 28 coupled ordinary differential equations for H2i-1 () and F2ji-,(). For

example, the equation for H1 () and Fi () were obtained by collecting the terms

of order of 0 in the momentum balance. It is represented as

1 dF1 1 d2 (2-28)
Ff2() H 1() =1 + (2-28)
4 d< 2 d<2

Similarly, collecting the terms of order 0' in the continuity equation yields

1 dH1(()
F1( ) = (2-29)
2 d<

The higher order terms of 0 in the momentum and the continuity equations, listed

in Appendix A, give the governing equations for H2i-1 () and F2i- (0.

The no-slip boundary condition at the electrode surface for v, and ',. is related

to H2i-1(_) and F2i- 1() by the following


H2-i() F2i-i() = 0.0 at = 0 (2-30)

and the far-field boundary condition for ',, yields

(_)-122i-1
F2i-1(00) 1)- (2-31)

Thus, the governing ordinary differential equations for H2i-1 () and F2i-1 () with

boundary conditions (2-30) and (2-31) describe the fluid flow within the boundary

layer. The solution procedure is described in the next section.

2.4.1 Solution Method

The above set of ordinary differential equations were solved using the BAND

algorithm introduced by Newman.39 The boundary condition (2-31) for the colat-

itude velocity, i.e., F2i_( = oo) was applied at ( = 40.0. The calculation domain

was divided into a grid of N nodes. The nodes were spaced at distance of H with

each other. The momentum equations were discritized at node J as presented in









J-2 J-1 J J+1 J+2


H-

Figure 2-4: Schematic diagram of grid for calculation domain H is the spacing
between adjacent nodes.

Figure 2-4. The corresponding continuity equations were discritized at half point

between node J and J 1. The discretization procedure ensures that the order of

resulting equations at each node is of H2. The disctitized form of continuity and

momentum equations for Hi() and Fi(0) (equations (2-28) and (2-29)) are given

as

(Fi(J + 1) 2Fi(J) + Fi(J 1)) H1(J) (Fi(J + 1) Fi(J 1)) Fi(J)2
2H2 2H 4
(2-32)

and
G(2) (H(J) Hi(J 1)) (Fi(J) + Fi(J 1))
G(2) = 2 (2-33)
H 2

where G(1) and G(2) are the residuals for the momentum and the continuity equa-

tion at node J. The BAND algorithm solves equations (2-33) and (2-32) with

boundary conditions (2-30) and (2-31) such that the residuals G(1) and G(2) are

effectively equal to zero within the specified tolerance at each node. The same

procedure was followed for rest of the equations listed in Appendix A. A FOR-

TRAN code was used to solve the equations. Listing of the code with its main

program and subroutines is presented in Appendix C.

2.4.2 Results

The solutions obtained for Hi() and Fi(0) as a function of are presented

in Figure 2-5. The results are in agreement with those obtained by Homann40 for

disk electrode under submerged jet impingement. The obtained solution for H3(),

F3(), H5(), F5(), ., F27() are presented in Appendix A.2.










10
9 -H()
8- ----FF( )
7

LL.
5-
4
I 3

2 --- ----------- -----
1 -


0 1 2 3 4 5 6 7 8 9 10


Figure 2-5: Dimensionless radial and colatitude functions H1() and Fi(0) as a
function of (see equations (2-24) and (2-25)).

The velocity distribution near the electrode surface can be approximated by

Taylor's series expansions for F(O, ) and H(O, ) as
14 t 14
F(Y, ) [ 02i-1F2il(0) Y 02i-1F _1(0) ( (2-34)
i=F1 i=+2 (21

and


H(, ) = 02i-1H2i 2 +2i- 1H2 -() (2-35)

where the first and second terms of equation (2-34) represent the first and the

second order derivative of ',. with respect to respectively. Similarly, the first and

second terms of equation (2-35) represent the second and the third order derivative

of v, with respect to respectively. The first order derivative of v, with respect to

is zero because of no penetration condition. Velocity expansions (2-34) and (2-35)

provide a convenient way to represent the fluid flow field within the boundary

layer. Later, the coefficient of the velocity flow field are utilized in the solution of









convective-diffusion equation. This is a valid approach for convective-diffusion

processes with large Sc number, because the mass transfer boundary layer is much

thinner compare to the momentum boundary layer, and the fluid velocities ', and

v, can be approximated with a quadratic and third-degree polynomial in within

the mass transfer boundary layer.

The gradient expressions j,_,1(0) at the electrode surface were calculated from

the obtained solution using three point forward difference method.41 In order

to minimize the influence of finite-difference errors on evaluation of ',_ 1 and

H2-1,, the differential equations were approximated to the order of the square of

the mesh-size H, and the numerical values were obtained by extrapolation to zero

mesh size. Plots 2,_,1(0) vs H2 are given in Appendix A. The number of digits

given in Table 2.1 are consistent with the standard deviation obtained through the

regression procedure.

Two methods can be employed in obtaining H2ji-(0). The first method utilizes

the solution of ordinary differential equations and difference schemes. The sec-

ond method estimates Hi_-1(0) by substituting the calculated values of F-i_,(0)

in the corresponding continuity equations. Most importantly, the second method

reduces errors in the evaluation of the expressions H2ji1(0) by ensuring that the

residuals for mass-balance are zero at the electrode surface.

The higher order expressions F -_,(0) were obtained by substituting the no-

slip condition in the momentum equation (2-16). Thus, the expression for second

derivative of F(0, ) at the electrode surface is given by

2F(O, sin4 (2-36)
a2 20 2

After substitution of the series expansion for F(O, ) (equation (2-25)) and further

expansion of sin 40 in terms of 0, values of F- _,(0) were obtained. The expressions

H2-1 (0) were obtained by double differentiating the the continuity equation (2-16)









Table 2.1: Series expansion coefficients F.2
and (2-35) for H(O, 0) and F(O, ) at ( 0.


S(0) and H2ji (0) in the equations (2-34)


S F2i-1(0) H2i-1(0)

1 2.6238754 1.3119377
2 -3.99262600 -4.21128229
3 1.71640917 2.89275550
4 -0.41810335 -0.95844785
5 0.014014149 0.062315385
6 -0.0325842274 -0.0973086579
7 -0.025404936 -0.086129459
8 -0.02354329 -0.09186753
9 -0.0221291 -0.0974599
10 -0.02122090 -0.10411441
11 -0.02068333 -0.11185299
12 -0.02043571 -0.12075969
13 -0.02042491 -0.13093150
14 -0.02061600 -0.14248419


and substituting the second derivative of F(O, ) at ( = 0. The substitution yields:


a3H(0, )
8(3 0 o


cos 40 cos 20 cos2 0
2 2


(2-37)


Again, after substitution of the series expansion for the series expansion for H(O, )

(see equation (2-24)) and further expansion of trigonometric functions in terms

of 0, the values of F2i-1(0) were obtained. The calculated values for coefficients

F2i-1(0) and H2i_-(0) for i=1,...,14 are given in Table 2.1. Similarly, values of

Fj _,(0) and H2_1 (0) are provided in Table 2.2.

2.5 Boundary Layer Separation

Boundary layer separation takes place at the location where the normal deriva-

tive of the colatitude velocity, i.e., has a value equal to zero. Thus, bound-

ary layer separation is observed at the value of 0 where the dimensionless shear

stress,


(2-38)


B(O) = 02i-1F2i-1(0)1
i=1









Table 2.2: Series expansion coefficients F-,1 (0) and H2 1 (0) in the equations (2-34)
and (2-35) for H(O, ) and F(O, 0) at 0.


F :- 1_(0) H2,-1(0)

1 -2 -1
2 16/3 11/2
3 -64/15 -41/6
4 512/315 161/45
5 -1024/2'. -641/630
6 8192/155925 2561/14175
7 -32768/6081075 -133/6075
8 262144/638512875 81922/42567525
9 -262144/10854718875 -163841/1277025750
10 2097152/1856156927625 93623/13956067125
11 -8 ;- 111S/19489647700625 -2621441/9280784638125
12 6710tW. 1/49308808782358125 146654/14992036723125
13 -134217728/3698160658676859375 -5991863/21132346621010625
14 1073741824/1298054391195577640625 335544322/48076Ii1-'.,12799171875


has a value equal to zero. The value of 0 at which separation was calculated de-

pended slightly on the number of terms retained in the series expansion. The

point of separation reached a value of 54.8 degrees for n = 14. A plot of B(0) is

presented in Figure 2-6 as a function of 0, showing clearly the point of boundary

layer separation. The corresponding result obtained by Barcia et al.,19 and repro-

duced in the present work, for the rotating hemisphere is presented in Figure 2-6

to provide comparison. The fluid dynamics calculations of a rotating hemisphere

does not predict boundary layer separation, although a small region of is theo-

retically possible near the singularity where the electrode contacts the insulating

plane.

2.6 Numerical Simulation

In the present section, computational fluid mechanics calculations were per-

formed for the stationary hemispherical electrode under jet impingement. The

hydrodynamic model developed here involved simultaneous numerical solution









1.0 I I I

stationary hemisphere
0.8


0.6


0.4 -
rotating hemisphere

0.2 -



0 15 30 45 60 75 90

9 / degree

Figure 2-6: Calculated dimensionless surface shear stress as a function of angle 0.
Solid lines represent the result for the stationary hemisphere under submerged jet
impingement and dashed lines represent the result for the rotating hemisphere.

of the Navier-Stokes and the continuity equations. Numerical solution of the gov-

erning equations was developed by Dr. Gert Nelissen, Vrije Universiteit Brussel,

Belgium.

2.6.1 Governing Equations

As shown in Figure 2-7, a two-dimensional cylindrical coordinate system was

employed to describe the system. The r-coordinate corresponded to the horizontal

axis, and z-coordinate was assumed in the vertical direction. In this representa-

tion, the third dimension, i.e., 0-coordinate, was assumed to be around the vertical

z-coordinate. The flow was symmetric in the 0-coordinate, therefore the 0 velocity

component and the derivative of quantities in the 0 direction were substituted by

zero in the governing equations. The mathematical development and numerical

approach used in the present work are described by Nelissen et al.42 The flow was

assumed to be steady state, and the fluid was assumed to be incompressible. Thus,












0.1






0.05







0 0.05

Figure 2-7: Schematic representation of the simulated flow geometry. The dimen-
sions are given in units of m. The arrow represents the general direction of flow,
and the cylindrical electrode is located at the origin.

conservation of momentum in r, and z-coordinates could be expressed by

( v v, a9p a l(rVr) ) 2r
Vr ar v a +- + z (2-39)
Or Oz or Or r Or Oz2

and
( Ovz Ovz Op + ( avz \ a2vz
P vyr + +V- -a a+ L rr + 2 (2-40)
Or Oz Oz ror or Iz2

where p is the fluid density, p is the pressure, p is the molecular viscosity of the

fluid. For the incompressible fluid, conservation of mass is represented by

1a(rv,) av
+ 0 (2-41)
r Or dz

Under turbulent conditions, Reynolds averaged NavierStokes (RANS) equations

were used.42 The boundary conditions for equations (2-39), (2-40), and (2-41) were

that no-slip and no-penetration conditions applied at solid surfaces, and that a uni-

form fluid velocity profile was imposed at the inlet to the system









(corresponding to the nozzle). In addition, a reference point for the pressure was

located at the outlet of the system.

The boundary conditions for equations (2-39), (2-40), and (2-41) are:

1. The no slip boundary condition i.e., v, = v, = 0 at the electrode surface.

2. Imposed fluid velocity profile from the inlet. In this case, fluid field was

assumed to be emanating with a constant velocity across the nozzle.

3. A reference point for the pressure i.e., p = pref = 0 was assumed to be located

at the outlet of the system.

2.6.2 Numerical Method

The partial differential equations (2-39), (2-40), and (2-41) were solved with

residual distribution method.43 The discretization was done on the grids of tri-

angles in the geometric domain of interest. The Lax-Wendroff43 scheme was ap-

plied to the convection terms of the momentum balances. The viscous terms were

treated in a standard finite element manner. The numerical scheme provided a

second order accuracy. The resulting non-linear set of equations were solved by

using the Newton-Raphson method, with explicit calculation of the jacobian ma-

trices. An incomplete LU preconditions gmres was used to approximate the solu-

tion of the linear system. The grid in the boundary layer regime contained at least

ten elements in the direction normal to the electrode.

2.6.3 Simulation Results

The calculated results presented here correspond to two different inlet flow

rates. Fluid properties were assumed to be those of water at 25 'C. Simulations cor-

responding to an inlet Reynolds number of 1,100 are presented in Figure 2-8. The

flow in the inlet region is laminar. The false color images indicate that the pressure

near the electrode is large at the stagnation point (0 = 0), decreases in the region











0.007

0.006
0.005

0.004

0.003 -

0.002

0.001

0.000
0.000 0.002 0.004 0.006 0.008

Figure 2-8: Fluid streamlines in the vicinity of the electrode for an inlet Reynolds
number of 1,100. The color map indicate the pressure distribution. The radial
dimension is given in units of m.

of (0 = 7/4), and increases near the electrode-insulator interface (0 = r/2). The

adverse pressure gradient seen for angles larger that 0 = 7/4 induces a boundary

layer separation, just as predicted by potential flow calculations. The fluid field

shows a circulation zone starting at an angle of about 620 which is slightly larger

than the value of 54.80 obtained by the boundary-layer hydrodynamic model.

The flow field for an inlet Reynolds number of 11,000 is presented in Figure 2-

9. In this case, the inlet flow is turbulent. The results again show that a circulation

zone is formed at an angle near 620. Figure 2.9(b) provides an enlarged image of

the recirculation zone shown near the electrode-insulator interface in Figure 2.9(a).

2.7 Summary

This chapter has presented two hydrodynamic models for fluid flow around a

stationary submerged hemisphere under jet impingement. The first model was

based upon semi-analytical solution of the momentum and mass conservation

equation. The results of the calculation show a formation of boundary layer

separation for the system. The point of boundary layer separation was predicted

to occur at 54.80. The second model numerically solved the momentum and the












0.010






0.005






0.000

0.000 0.005 0.010


(a)



0.0015



0.0010



0.0005



0.0000 -

0.0025 0.0030 0.0035 0.0040 0.0045


(b)
Figure 2-9: Fluid streamlines in the vicinity of the electrode for an inlet Reynolds
number of 11,000. The color map indicate the pressure distribution. The radial
dimension is given in units of m. Figure 2.9(b) provides an enlarged image of the
recirculation shown in Figure 2.9(a).






28

continuity equations without invoking any approximations. This model also suc-

cessfully show the formation of vortex in separated part of the boundary layer,

and the point of boundary layer septation was predicted at 620. The results of the

hydrodynamic models are used in subsequent chapters.















CHAPTER 3
CONVECTIVE-DIFFUSION MODELS FOR A STATIONARY HEMISPHERICAL

ELECTRODE UNDER SUBMERGED JET IMPINGEMENT

This chapter provides a detailed description of convective-diffusion processes

taking place in the boundary layer of a hemispherical electrode under submerged

jet impingement. A solution for convective-diffusion equation of the system is

provided in this chapter. The obtained solution provide a framework for further

investigation of impinging jet hemispherical electrode.

3.1 Governing Equations

Under the assumptions that the Peclet number is large and that the concen-

tration of the reactant CR is small with respect to the supporting electrolyte, the

steady-state convective diffusion equation within the boundary layer can be writ-

ten as
OCR 'a.OCR c2CR
v, + = DR (3-1)
Or ro 60 or2

where v, and ', are the radial and colatidute velocity components of fluid flow

field with in the boundary layer. Their expressions have been obtained from the

fluid mechanics development presented in Chapter 2. The followings assumptions

were made in deriving equation (3-1):


1. the fluid flow in the boundary layer is laminar and normal derivative of all

quantities are much larger compare to the tangential derivatives,

2. the diffusion of reactant CR in the tangential direction direction is much smaller

compare to the normal direction at the electrode surface, and









3. the thickness of the mass transfer boundary layer, 6m, is infinitely small com-

pare to the radius of the hemisphere.


It is important to note that equation (3-1) is only valid in the unseparated part of

the momentum boundary layer at the electrode surface.

The objective here is the calculation of the mass-transfer-limited current distri-

bution at the electrode surface; thus, the boundary conditions for equation (3-1)

are given as

CR r=ro,o = co (3-2)

CR Ir=,o = c- (3-3)

and

CR r,o=0 = c- (3-4)

The concentration CR can be expanded as a series function of 0, and Sc-3 as

CR CR 2i-2 12i(-1) + Sc- 2-2 ) (3-5)
CO C 1 i 1

such that the first term of the expansion provides the solution under the assump-

tion that the Schmidt number Sc is infinitely large, and the second term provides

a correction for a finite value of Sc.

The characteristic dimensionless distance for mass transfer can be defined to

be

Z = Sc1/3 (3-6)

which accounts for the difference in scale between the convection and mass trans-

fer boundary layer thicknesses.

Fourteen coupled ordinary differential equations for #41,2i-1,1 and 42,2i-1 were

obtained through following steps:










1. Substitution of v, and ',. in terms of a Taylor series dimensionless velocities

F(O, ) and H(O, ) from equation (2-34) and (2-35) with n = 14 into equation

(3-1),


2. introduction of dimensionless concentration from equation and scaled dis-

tance Z from equation (3-5) and (3-6), respectively, and


3. collection of terms corresponding to 02i and 02iSc-1/3


The derived equations can be written in general form as following:


d2 41,2 -1(Z) H d 4-1,2i- (Z) 2F, j(O)Z41,2i-1(Z)


=z2 H d 1,-i-I-(Z)
n= 1
2 ddZ
n=l
n=i-1
+Z 2nF2(-) + (0)1,2n+1(Z) (3-7)
n=l

and


d2 2,2 i-2(Z) 1 (0)2d 2,2i-1(Z) 2F (0)Z41,2i- (Z)


z3 t"' d 2,2i-1(Z)
=-H2+ Z1H (0)
d 2 2(Z d Z


n=0
n=i-1
+Z2 Y ( ) +1(0)(&1,2(i-n)+1(Z)
n=O



n=i-1
+Z 2(1 n)F I(0)(2,2(i-n)+1(Z)
n=l
n-i
(3-8)


where equation sets (3-7) and (3-8) represent 14 ordinary differential equations for

#1,2i-1 and )2,2i-1, respectively.









The boundary conditions for )1,2i-1 and -2,2i-1 are


41,1 1.0 (3-9)

l1,2i-1 = 0.0

12,2i-1 = 0.0




at Z = 0, and


1i,2i-1 = 0.0 (3-10)

12,2i-1 = 0.0




at Z = oo. The analytic solutions of 41,2i- and a numerical solutions of 41,2i-1 are

provided in the next section.

3.2 Solution Method and Results

The equation set (3-7) corresponding to )1,2i-1 with boundary conditions (3-9)

and (3-10) were solved analytically. For i = 1, the solution corresponds to the disk

electrode under submerged jet impingement,44 and it is given as


41,1(Z) 1.0 0.8500069 exp (0)] dZ (3-11)
0
For i > 1, the general solution of the equation set (3-7) is given by


j=i1
ch,2i-I(Z) exp H (0) (AZ3 -2) (3-12)


where values of Aj were deduced by substitution of equation (3-12) into equation

(3-7). The complete expressions of 1,2i-I for 2 < i < 14 are given in Appendix B.

The equations (3-8) for 1 < i < 14 were solved numerically using tridiagonal

BAND algorithm described by Newman.39 The expressions involving 1,2i-I in









Table 3.1: Calculated values for coefficients 1',2i-1 (0) and )2,2i- (0) used in equa-
tion (3-14) for mass-transfer-limited current distribution.


i 1,2i-1 (0) 4)2,2i-1(0)

1 -0.8500077 0.0719099
2 0.5456994 -0.0850449
3 0.1954955 -0.0193952
4 0.4638516 -0.0203633
5 0.3719751 -0.0182829
6 0.3247121 -0.0172154
7 0.2949598 -0.0166429
8 0.2761729 -0.0164106
9 0.2647235 -0.0164366
10 0.2585353 -0.0166741
11 0.2563565 -0.0170954
12 0.2574086 -0.0176849
13 0.2612022 -0.0184347
14 0.2674345 -0.0194908


equations set (3-8) were substituted with their analytical expressions as provided

in Appendix B. The numerical simulation were performed for different values of

grid spacing H. The quantities of interest for mass-transfer-limited current are first

derivatives of )1,2i-I and )1,2i-I with respect to Z at the electrode surface. In order

to minimize the influence of finite-difference errors, the differential equations were

approximated to the order of the square of the mesh-size, and the numerical values

of 4)2,2i-1(0) were obtained by extrapolation to zero mesh size. Calculated values

for V1,2i- (0) and '2,2i-1(0) are provided in Table 3.1.

3.3 Mass Transfer Limited Current

The flux at the electrode surface is given by


D OCR
DRa


(3-13)


STro









which can be evaluated in the form of a mass-transfer-limited current density in

terms of the dimensionless variables introduced above as


inim(O) = nF (c co) /3 Re1/2 2-21 -1(0) + Sc-1/3 o2i-242i
ro i= 1 i= )
(3-14)

where the Reynolds number Re is defined to be


Re = a (3-15)


Equation (3-14) can be expressed in terms of a characteristic number


N* F (c co) D Sc1/3 Re1/2 (3-16)
ro

as

Nm) (0) + Sc- A(0) (3-17)

where T(0) is the mass-transfer-limited current density for an infinite Schmidt

number, and A(O) is the correction to account for the finite value of the Schmidt

number. Thus,

(0) 02i-2'2- (3-18)
i= 1
and

A(O) = 02i-24 -1(0) (3-19)
i= 1
The calculated values for T(0) and A(0) are presented in Figures 3.1(a) and 3.1(b),

respectively, as functions of colatitude angle 0.

The boundary-layer solution is valid only up to the point of boundary layer

separation. The result obtained for the stationary hemispherical electrode is in

stark contrast to that obtained by Barcia et al.19 for the rotating hemispherical

electrode, also shown in Figure 3-1, which does not show such a boundary layer

separation. For the stationary electrode in Figure 3.1(a), the region of circulation

is represented by a uniform extension of the value of T at the point of separation.



























0 15 30 45 60
0 / degrees


0.1



0



-0.1



-0.2


75 90


0 15 30 45 60 75 90


o / degrees

(b)
Figure 3-1: Calculated mass-transfer limited current density for a hemispherical
electrode subjected to a submerged impinging jet. Solid lines represent results
for the stationary electrode, and the dashed lines represent results for the ro-
tating hemispherical electrode.a) Contribution to equation (3-17) for an infinite
Schmidt number; b) Contribution to equation (3-17) providing correction for a fi-
nite Schmidt number.









Work is needed to provide a more correct estimation of the mass-transfer rate in

this region. A preliminary approach could be to assume a uniform value for a

mass-transfer coefficient kMT. Thus, within the region of circulation, 1iim can be

expressed as

ihim = nFkMT (co co) (3-20)

Integration of equations (3-14) and (3-20) over the electrode surface is required to

obtain a value for the total current which is accessible from experimental measure-

ment.

3.4 Numerical Simulations

The aforementioned model gives an expression of convective-diffusion pro-

cesses up to the point of boundary-layer separation. Numerical solutions, how-

ever, were used to obtain a solution of convective-diffusion over the entire elec-

trode surface. Numerical simulation of the governing equation was performed by

Dr. Gert Nelissen, Vrije Universiteit Brussel, Belgium. The convective-diffusion

model in the cylindrical coordinate system is given by


Ir + i'9 = DR tz r9 + 2CR (3-21)
r or Oz [Or r r ) OZ2

where v, and v, are the fluid velocity component calculated by the computational

fluid dynamic model in the previous chapter, CR is the concentration of reacting

species, and DR is the molecular diffusivity of the reacting species. Fluid flow was

assumed to be laminar in the mass transfer boundary layer. The transport of the

reactant due to electric field in equation (3-21) has been neglected by assuming

the presence of excess supporting electrolyte. The system is assumed to operate

under isothermal conditions. At the electrode surface, the boundary conditions

for equation (3-21) were:


CR = 0


(3-22)









and in the bulk

CR = Coo (3-23)

Equation (3-21) was solved with boundary conditions (3-22) and (3-23) using nu-

merical scheme presented in the previous chapter. To ensure positivity, the con-

vection term in the convective-diffusion equation were discretized using the N-

scheme.45' 42Standard finite element discretization was applied to the diffusion term.

In numerical simulations, the diameter of electrode and nozzle was fixed to be 1/4

inch, and nozzle was assumed to be placed at 5.0 cm from the electrode. The phys-

ical properties of the electrolyte used in the simulations are listed in Table 3.2. The

concentration distribution of reactant as a function of distance from the electrode

surface is shown in Figure 3-2. Directly on the electrode surface the reactant con-

centration is zero, which is depicted by the blue line. The area represented by the

red corresponds to the bulk reactant concentration. Above the electrode surface,

the concentration distribution of reactant exhibit a very sharp gradient, shown in

Figure 3-2 by marked change in color. However, at the point of separation (0=620),

the color variation becomes wider signaling a drop in concentration gradient. This

confirms that the mass-transfer is minimal at the point of boundary layer separa-

tion. The expression of mass-transfer-limited current is given by:

ium = nFD CR (3-24)
ar

The limiting current density vector at the electrode surface was calculated using

the simulation. The parameters used in simulations are given in Table 3.2. The

results of simulations are presented in Figure 3-3. Simulation were done for two

Reynolds number of fluid in the nozzle. At high Re, the current is higher. Current

reaches a minimum at the point of boundary layer separation for both Re numbers.

The current in the separated zone is higher than at the point of separation for both

cases.














4.185e+0





3.336e+0





2.486e+0





1.637e+0





7.878e-1


-6.169e-2 1
-1.424e-1 5.765e-1 1.295e+0 2.014e+0 2.733e+0 3.452e+0


100.0


90.0


80.0


70.0


60.0


50.0


40.0


30.0


20.0


10.0


0.0


Figure 3-2: Reactant concentration distribution as a function of distance from elec-
trode surface, obtained through numerical simulation of equation (3-21). The blue
line corresponds to zero concentration on the electrode surface, whereas red corre-
sponds to the bulk reactant concentration. The radial dimension is given in units
of cm. These simulations were performed for Re = 11300 in the nozzle.






Table 3.2: Physical properties of the electrolyte used in the numerical solution of
equation (3-21).


Property
v/m2 sec-1
DR/m2 sec-1
c /mol m-3


H-


Value
1.0 x 10-6
7.0 x 10-10
100.0









-14 -o- Re = 11,300
-A- Re = 1,130
-12

-10

E -8
< 0
-6 0 oQ-









S/ degree
-2 ~






Figure 3-3: Calculated mass-transfer-limited current density for different
Reynolds number at the inlet of the nozzle. The vertical dash line at 620 is the
point of boundary layer separation. The physical properties of the electrolyte used
in the simulations are listed in Table 3.2.
3.5 Conclusion

Steady state mass-transfer was obtained for the system. Two models were pre-

sented. The first model was based upon semi-analytical series expansion method.

The models used the fluid velocity fields obtained in Chapter 2. The second

model made use of a numerical scheme to solve the convective-diffusion equa-

tion. Both model predicted a finite value of mass-transfer-limited current at the

point of boundary layer separation. The second model was able to calculate the

mass-transfer in the separated region of the boundary. The mass-transfer is in-

creased in the separated region with a minimum at the point of separation. This

result is analogous to the heat-transfer in sphere, where heat-transfer is enhanced

in the separated part of the boundary layer.46 An implication of this calculation

is that the current can be assumed to be constant in the region of boundary layer







40

separation. This approximation will be used in the calculation of the current dis-

tribution for the system in Chapter 5.














CHAPTER 4
HYDRODYNAMIC AND MASS-TRANSFER MODELS FOR A ROTATING

HEMISPHERICAL ELECTRODE

The hemispherical electrode under jet impingement represents a modification

to the rotating hemispherical electrode described in this work. The present work

provides a review of previous hydrodynamic and the steady-state mass-transfer

models of a rotating hemispherical electrode system. New calculation results are

being presented alongside results of Bercia et al.19 and subsequently compared.

This study provides a correction to the published work.19

4.1 Schematic Illustration of the System

The rotating hemispherical electrode was first suggested by Chin17 as an alter-

native to rotaing disk electrode for study of electrochemical systems. The advan-

tage of the hemisphere over rotating disk is that the current distribution remains

uniform even at large fraction of mass transfer limited current.18 This uniformity

of current can be exploited to study electrochemical processes at higher current

density. A schematic of the system is presented in Figure 4-1.

In Figure 4-1, a hemispherical electrode is fixed in a insulating material with a

no gap at the point of intersection with the insulating plane. The system is rotated




Planar surface
H of insulating
Hemispherical r material
Electrode c^


Figure 4-1: Schematics illustration of Rotating Hemispherical Electrode.
41









in the electrolytic solution in the azimuthal direction. The spherical polar coordi-

nate system is used to describe the system, where r is the radially outward normal

direction with origin at the center of hemisphere, 0 is colatitude direction, and 0

is the azimuthal direction; and Vr, vo, and v6 are the components of fluid velocity

field in r, 0, and 0 direction, respectively.

4.2 Hydrodynamic Model

The hydrodynamic model of the rotating hemisphere has been addressed in

literature by several workers.19,17,47 It has been based on the hydrodynamic model

of a rotating sphere in a quiescent fluid. Howarth36 first introduced the problem of

a rotating sphere in 1951. He provided a solution of the model equations using a

polynomial series expansion for vr, vo, and v6 in terms of 0. His solution was lim-

ited to the first two terms in the expansion. He also suggested that the fluid flow

near the equator can not be described by boundary layer equations. Nigam48 sug-

gested a different form of series expansion than that of Howarth.36 He proposed

that the velocity expansions in terms of trigonometric functions of 0 for velocity

components, and provided a solution for first three terms of vr, vo, and v6. His

calculations suggested that the boundary layer remains intact at the equator; thus,

boundary layer equations adequately describe the flow near equator. Stewartson49

stated that the fluid flow at equator is outward along the equatorial plane; there-

fore, boundary layer will break down near the equator. He suggested that the

thickness of the region, where boundary layer assumptions fail, is within O(v1/2)

distance of the equator. Banks50 improved the solution by solving for coefficients

in the series expansion up to four terms. His series expansions were based upon

Howarth's36 model. Manohar,51 on the other hand, solved the boundary layer

equations using a finite difference technique. He reported a better convergence of

solution than that of Howarth's36 series method.









Chin17,47 treated the hydrodynamics of the rotating hemispherical electrode

(RHE), as described in Figure 4-1, like that of rotating spherical electrode. He used

Howarth's method of series expansion for velocity components and limited it to

four terms. More recently, Barcia and coworkers19 extended the series expansion

up to ten terms. Inclusion of additional terms was needed to obtain the accuracy

needed for impedance calculations.

The next section revisits the hydrodynamic model of the RHE. This was moti-

vated by the observation that the results provided by Barcia et al.19 do not satisfy

the continuity equation at the electrode surface. The results presented in this chap-

ter provide a correction to the results of Barcia and coworkers.19

4.2.1 Governing Equations

The governing equations,36 which describe the fluid motion within the bound-

ary layer of RHE, are written as

l + + cot(O) = 0 (4-1)
ro 60 Or ro

O,*9ve 9v e V2 (1)2,
0+ v ~ cot (0) v (4-2)
ro d0 Or ro Or2

S+ V, + cot (0) v (4-3)
ro d0 Or ro dr2
where equation (4-1) is the continuity equation, and equations (4-2) and (4-3) are

the momentum balances in 0 and 0 directions, respectively. The underlying as-

sumption is that derivatives of quantities with respect to 0 vanish and there is no

imposed pressure gradient. The boundary conditions for equations (4-1)-(4-3) are


vr = i, 0 = 0 v6 = wro sin(O) at r = ro (4-4)


and

O 0 at r oo (4-5)









where ro is the radius of the electrode, and w is its rotation speed. Dimensionless

variables for r, v,, ve, and v6 can be given by


V = (r ro) (4-6)

vr- w)H(0, ) (4-7)

,,=(row) F(, ) (4-8)

and

v = (row) G(6, ) (4-9)

where dimesionless quantities H(0, ),F(0, ), and G(0, ) are defined as polyno-

mial series expansion with respect to 0 such that


H(0,) = 02i-1H2, ) (4-10)
i= 1

F(0,) = 02i-1F2i-() (4-11)
i= 1

G(, )= 2i-1G2i-1) 1(4-12)
i=1
where n is the number of terms included in the expansion. Substitution of equa-

tions (4-6) to (4-12) into equations (4-1), (4-2), and (4-3) yields 3n ordinary differ-

ential equations for H2 i-1(), F2,-1i(), and G2i-l(). The boundary conditions for

the derived equations are


H2i-1(0) = F2,_1(O) 0 G2i-(0) ( 1! (4-13)

at ( = 0 and

F2i-1(00) 0, G2i-1(00) 0 (4-14)

The Taylor series expansion of H(O, () and F(0, () close to the electrode surface is

given by:


H(0,) = 02i-1H2i-1(0) 2 + t 2i-+1H2 i-(0) 3 (4-15)
i=1 i=1









and

F(0,^ )= 0i-2i-1F i(0) + 02i-1-(0) ( (4-16)
Fi= 1 (i= 1
and the dimensionless shear stress at the electrode surface is given by:


B(0) = 02i-1 F2i (0) (4-17)
i= 1

The dimensionless shear stress derived by Barcia et al.,19 Chin,47 and Manohar51'52

are plotted in Figure 4-2. Barcia and coworkers19 solution is presented by a solid

line, whereas those of Chin47 and Manohar51'52 are presented by dashed and dot-

ted lines, respectively. Newman52 obtained an expression for dimensionless shear

stress by fitting a polynomial in 0 to the results of Manohar.51 It is rewritten as


B(O) = 0.510230 0.180881903 0.04408 sin3(0) (4-18)


Newman52 considered Manohar's51 solution to be more accurate than that of Chin47

for large angle 0. As seen in Figure 4-2, the B(O) presented by dotted line has the

lowest value of the at large angles.

The objective of the hydrodynamics calculations is to estimate the values of

F2-,_(0) and H2i-1(0). These coefficients are used directly in the steady-state mass
transfer, current distribution, and convective diffusion impedance calculations.

The values of F -_,(0) and H21 -(0O)5 are obtained by substituting boundary con-

ditions. They are given by
_-l22i-2
Fi-i(0) ( (4-19)

S 2 if i = 1
H2i1 3 (0)22 if >1 (4-20)
3 (2i-2)!
The values of F- _,(0) and H2-1 (0) are determined from above equation, whereas,

coefficient expressions i1_,(0) and H2i-1(0) are obtained from the numerical solu-

tion of governing equations describing the boundary-layer hydrodynamic model.









0.35 .. .. .

0.30

0.25

0.20

M 0.15

0.10

0.05

0.00 ** ** *
0 30 60 90

9 / degrees
Figure 4-2: Shear Stress Distribution at the electrode surface. Solid line represent
results of Barcia et al. the dashed line represent the results of Chin, and the dotted
line represent the result of Manohar.

4.2.2 Results

Barcia et al.19 solved the ordinary differential equations for hydrodynamics

with n = 10 using BAND39,23 algorithm. They showed that their calculations were

in agreement with that of Chin17,47 for n = 4. However, it was found that the

substitution of j,_,j(0) in the continuity equations for i > 4 does not yield the

reported values19 of Hi-j (0). For example, the value of jH9'(0) is reported as

0.30978 x 10-3, whereas continuity equation yields a value of -0.20763 x 10-3. The

error in the values of coefficients will have a larger implication in estimation of

mass-transfer-limited current and convective-diffusion impedance. It is assumed

in this work that the values of coefficients F,_, (0) reported Barcia et al.19 are suf-

ficiently accurate. The corresponding values of H2ji1 (0) are calculated from con-

tinuity equations. They are tabulated in Table 4.1. In comparing the values of

'H2Hi- (0) from Table 4.1, it is observed that the values differ for n > 5 than that of









Table 4.1: F 1(0) and H2'-1 (0) coefficients in the series expansion of equations (4-
16) and (4-15) for H(0, ) and F(0, 0) at ( 0. The third column in the table lists the
values reported by Barcia et al. and the fourth column lists the values calculated
using the continuity equation.


i 1-(0) Hi-1 (0) reported by Barcia et al.,19 jHi-1(0)

1 0.51023 -0.51023 -0.51023
2 -0.22128 0.52761 0.52760
3 0.20711E-1 -0.93344E-1 -0.93344E-1
4 -0.18905E-2 0.90951E-2 0.90951E-2
5 -0.11499E-4 0.30978E-3 -0.20763E-3
6 -0.41534E-4 0.21299E-3 0.23024E-3
7 -0.11468E-4 0.70451E-4 0.71604E-4
8 -0.18727E-5 0.12325E-4 0.12434E-4
9 -0.89351E-6 0.75296E-5 0.75405E-5
10 -0.21900E-6 0.20009E-5 0.20020E-5

Barcia's et al.19

4.2.3 Fluid Flow at the Corner

The fluid motion of at the corner i.e., at the intersection of hemispherical elec-

trode and insulating plane, is discussed here. Existing literature has overlooked

this detail. The problem has been ignored in the literature, however, there may be

some compelling implication of this issue. For example, Stewartson49 suggested

that boundary layer equations are not valid within O(v1/2) distance of the equator.

This would mean that for aqueous electrolytes the distance would be O(0.1cm),

which represents a considerable portion of a 0(0.635cm) diameter electrode. On

other hand, Nisanqiiglu18 suggested that the thickness of the region where bound-

ary layer fails is of the order of 0 (e) where Reynolds number Re is defined as

Re (4-21)

Thus, the thickness of the region can be reduced by high rotation speed of the

electrode. It is important to note that Nisanqiiglu's18 as well as Stewartson's49

statements are for rotating spherical electrodes. Fluid motion in RHE will be more




















e = 9001 v

r

Figure 4-3: A two dimensional depiction of boundary layer at the intersection of
electrode and insulating plane.
complicated than that that of rotating spherical electrode. A schematic represen-

tation of the fluid motion in the boundary layer near the corner is shown in Fig-

ure 4-3. As fluid moves along the boundary layer on the spherical surface of the

electrode, it would encounter a 90.00 change in direction as it approaches the flat

insulating plane where fluid will move along the plane in the outward r direction.

The bulk of aforementioned paragraph describes a rotating hemispherical elec-

trode. However, for ease of operation, insulating plane can also rotate with the

electrode. As a result, the fluid flow will be further accelerated along the insulat-

ing plane due to its rotation, causing the boundary layer at the corner to shrink.

This problem requires further investigation, and it is suggested as future work.

4.3 Mass Transfer

The steady state convective-diffusion equation governing the mass transfer in

the boundary layer can be written as

OCR '.. cR 82CR
R, + = Dar2 (4-22)
Or roR0 r2

where CR is the reacting species, v, and ,. are the radial and colatidute velocity

components of fluid flow field with in the boundary layer, and DR is the diffusivity









of the reacting species. The objective is to calculate the steady-state mass-transfer-

limited current distribution at the electrode surface, utilizing the fluid velocity

coefficients reported in this work. The boundary conditions for equation (4-22)

are given as

CRr=ro,o = co (4-23)

CR r=,o = Coo (4-24)

and

CR r,o=0 = c- (4-25)

As stated in Chapter 3, CR can be expanded as a functions of 0, and and Sc-3 as:

CR -1 c fi-2 1,iZ-1(2) + Sc--2 (4-26)
CO C 1 i 1

such that the first term of the expansion provides the solution under the assump-

tion that the Schmidt number(Sc) is infinitely large and the second term provides

a correction for a finite value of Sc.

The characteristic dimensionless distance for mass transfer can be defined as


Z = Sc1/3 (4-27)


which accounts for the difference in scale between the convection and mass trans-

fer boundary layer thicknesses.

Ten coupled ordinary differential equations for 1,2i-I and #42,2i-1 obtained by

following steps:


1. Substitution of v, and ',. with dimensionless velocities H(O, ) and F(O, ()

given by equations (4-15) and (4-16), respectively with n = 10.

2. Representing CR with the dimensionless concentrations 041,2i-1 and 42,2i-1

given in equation (4-26).









Table 4.2: Calculated values for coefficients used in equation (4-29) for calculating
mass-transfer-limited current distribution.


i 41,2i-1(0) )'2,2i-1(0)

1 -0.62045 0.18490
2 0.12831 -0.43440E-1
3 0.34750E-2 -0.15360E-2
4 0.14694E-2 -0.55010E-3
5 0.35468E-3 -0.19855E-3
6 0.10783E-3 -0.56713E-4
7 0.34006E-4 -0.17257E-4
8 0.99642E-5 -0.60689E-5
9 0.32183E-5 -0.19643E-5
10 0.10249E-5 -0.66980E-6


3. Substitution of r with Z as given by equation (4-27).


4. The above three substitutions were made in equation (4-22). The terms cor-

responding to 02i and 02iSc-3 were collected to get the governing equations.

The obtained equations were solved using Newman's BAND54 algorithm. The ob-

tained results in the form of first derivatives of )1,2i-I(Z) and #42,2i-1(Z) at Z = 0

are tabulated in Table 4.2. The value of V1,2i-1(0) and '2,2i-1(0) are not in agree-

ment with those calculated by Barcia et al.19 for i > 4.

The concentration flux at the electrode surface is given by

aCR
NR = DR (4-28)
Or r=ro

which can be evaluated in the form of a mass-transfer-limited current density in

terms of the dimensionless variables introduced above as

nF (c co)DR n \
iiim(O) c -o) /3Re 1/2 2i-2 -1(0) + Sc-1/3 o2i-2_'2 i
ro i=1 i= 1
(4-29)

where the Reynolds number Re is defined to be


Re = r (4-30)









Equation (4-29) can be expressed in terms of a characteristic number


N* F (c co) D Sc/3Re 1/2 (4-31)
ro

as

N(= T(0) + Sc-'A(0) (4-32)

where '(0) is the mass-transfer-limited current density for an infinite Schmidt

number, and A(O) is the correction to account for the finite value of the Schmidt

number. Both T(0) and A(0) are given by


(0) 02-2 12i-1(0) (4-33)
i= 1

and

A(O) 02i-24',2i-1(0) (4-34)
i= 1
The calculated values for T(0) and A(0) are presented in Figures 4.4(a) and 4.4(b),

respectively, as functions of 0. The average of mass Transfer limited current is

given by:
2 2
= J (0) sin(O)d0 + Sc- J A(0) sin(O0)dO (435)
o o
Upon integrating T(0) sin(O) and A(O) sin(O) with respect 0, one obtains:


Iim 0.45710 ( 0.27896Sc-') (4-36)


whereas Barcia et al.,19 reported Ilim as


Sim= 0.45636 (i 0.28002Sc- (4-37)


For Sc = 1000, the relative error between the two expressions would be about

0.I-'.. A graph of relative error as a function of Sc number is presented in

Figure 4-5. The relative error is minimum for infinite Sc number, and increase

as value of Sc number is decreased.


























0.2

0.1 -

0.0 ...
0 15 30 45 60 75 90

o / degrees


(a)

-0.3 ...




-0.2 -J


-0.1 0





0 15 30 45 60 75 90

6 / degrees


(b)
Figure 4-4: Calculated mass-transfer limited current density for a rotating hemi-
spherical electrode. a) Contribution to equation (4-32) for an infinite Schmidt num-
ber; b) Contribution to equation (4-32) providing correction for a finite Schmidt
number.










0.35 1 1,1, 1


i 0.30
0
0
LUJ
( 0.25


Q 0.20


0.15 1
100 101 102 103 104
Schmidt Number
Figure 4-5: Relative error in mass-transfer-limited current given by expressions
(4-36) and (4-37) as a function of Schmidt number.

4.4 Summary

Following Barcia and coworker,19 the hydrodynamic and convective-diffusion

models for a rotating hemispherical system was presented in this chapter. The

second order radial velocity coefficients Hj _,(0) were calculated such that resid-

uals of the continuity equation at the electrode surface is equal to zero. The cal-

culated velocity coefficients were then utilized to obtain an expression of mass-

transfer limited current. The relative error in the mass-transfer-limited current

presented in this chapter and to the one presented by Barcia and coworkers19 is

about 0.18% for a value of Sc = 1000. An accurate development of convective-

diffusion impedance can be achieved using the hydrodynamic model presented

here.















CHAPTER 5
CURRENT AND POTENTIAL DISTRIBUTION AT AXISYMMETRIC

ELECTRODES

In Chapters 2 and 3, hydrodynamic and convective diffusion models were de-

veloped for a stationary submerged hemispherical electrode under jet impinge-

ment. This chapter discusses the current and potential distribution of a stationary

hemispherical electrode. The current distribution for a disk electrode was also

calculated and compared to the hemispherical electrode results. The shear stress,

obtained by the semi-analytical hydrodynamic model in Chapter 2, was used in

the current distribution calculations for the stationary hemispherical electrode.

A generalized axisymmetric model for current distribution is developed in this

chapter. The model can describe the current distribution for the disk and the hemi-

spherical geometries. Calculations for a rotating hemisphere are also performed

in this chapter. These calculations accounted for corrections in mass-transfer due

to finite Schmidt number. The shear stress obtained in Chapter 4 was used in case

of rotating hemispherical electrode system.

5.1 Introduction

Current distribution plays an essential role in electrochemical fabrication tech-

nologies and in interpretation of electrochemical processes.55 Significant effort

has been made on development of new rotating electrode designs to obtain the

ideal balance between uniform current distributions, well-defined mass transfer,

and ease of surface characterization.56 The need to couple surface characteriza-

tion with electrochemical measurements on uniform current distributions was ad-

dressed by Matlosz and coworkerset al.,28 who used a removable disk electrode









inserted in a rotating hemispherical electrode. The flat disk was suitable for ex-

situ surface analysis, and numerical simulations were used to identify conditions

under which the influence of the flat surface on the current distribution could be

neglected. Dinan et al.30 proposed a recessed rotating disk electrode that would

provide a uniform current distribution by compromising the uniform accessibility

of the rotating disk.

The advantage of using a disk electrode under jet impingement is that, so

long as the disk lies within the stagnation region of flow, an accurate solution

is available for convective diffusion, and the current distribution is uniform under

mass-transfer-limited conditions. The current and potential distribution on a disk

electrode below the mass-transfer-limited current is not uniform,5 and it has been

shown that neglect of the nonuniform current distribution introduces error in es-

timation of kinetic parameters from steady-state measurements.7-9 Similar errors

are observed when impedance measurements are interpreted under the assump-

tion of a uniform current distribution.13,14

Stationary electrodes are attractive because they can easily be adapted to use of

in-situ observation or surface analysis techniques. The objective of this study was

to understand the influence of fluid mechanics, convective-diffusion, and elec-

tric field below the mass-transfer-limited current for submerged stationary disk

and hemispherical electrodes under jet impingement. The current distribution at

the rotating hemispherical electrode was also explored. The calculations were ac-

counted for the effect of finite Schmidt number on mass-transfer.

5.2 Development of Mathematical Model

A two-dimensional mathematical model describing current and potential dis-

tribution was developed for axisymmetric bodies in a curvilinear coordinate sys-

tem. Examples of axisymmetric electrochemical systems are the disk and






















Figure 5-1: Schematics illustration of an axisymmetric body in a curvilinear coor-
dinate system. The horizontal dash line represents the axis of symmetry, and the
fluid field is assumed to be symmetric around this axis.

hemispherical electrodes with symmetric fluid flow field. A schematic representa-

tion of an axisymmetric system in curvilinear coordinates is illustrated in

Figure 5-1. The geometry of the axisymmetric bodies can be described by x, y,

and R(x) where x is an arc length measured along a meridian section from point

of stagnation, y is perpendicular to surface, and R(x) is the radius of the section

of the body perpendicular to its axis of symmetry. In this development, the fluid

flow field is assumed to be symmetric with respect to the y axis at x = 0.

5.2.1 Hydrodynamics

The velocity components of the fluid field in the boundary layer are given by

vx and vy. vx is assumed to be known and is represented as


vz = P(x>y (5-1)

where P(x) is the tangential shear stress, obtained by solving the Navier-Stokes

equations along with the continuity equation. The continuity equation for incom-

pressible fluid within the boundary layer of axisymmetric flow can be represented

by
1 9(R(x)v) (v5
t(+ = (5-2)
R(x) Ox 6y









Substitution of v, from equation (5-1) into equation (5-2) yields the following ex-

pression for vy.

1 2 1 do3(x)R(x) (53)
2 R(x) dx

Hence, vx and v, can be expressed in terms of tangential shear stress in curvilinear

coordinates as presented by equations (5-1) and (5-3), respectively.

5.2.2 Mass Transfer

The convective diffusion equation in the curvilinear coordinate system within

the boundary layer can be written as:

aCR aCR a2CR
DvR + )OCR = DR (5-4)
ax y 9x2

where CR is the concentration of reacting species, and DR is the diffusivity of the

reacting species. Equation (5-4) is valid under the following assumptions.

1. The Peclet number is large and diffusion in the y direction can be neglected.

2. Concentration of the reactant CR is small with respect to the supporting elec-

trolyte. As a result, the migration of CR due to the electric field is negligible.

The objective here is to solve equation (5-4) along with the following boundary

conditions:

CR = cR(x) at y = 0 (5-5)

and

CR = Co at y oo (5-6)

where c. is the bulk concentration of CR. However, it is convenient to solve equa-

tion (5-4) using the Lighthill transformation57 and then transforming the solution

for boundary conditions (5-5) and (5-6). The boundary conditions for Lighthill

transformations are given as


CR = Co at y = 0


(5-7)








where co is the uniform surface concentration of the reactant along the electrode

surface,

CR = Co at y oo (5-8)

and

CR = Coo at x = 0 (5-9)

The solution of equation (5-4) using the Lighthill's transformation can be repre-

sented by

N(x) D1/3 (5-10)
(4) 9DRJ R(x) VR(x)(x)dx)

where N(x) is flux of the reactant to the electrode. The corresponding mass trans-

fer limited current is given by

nFD (co co) VR(x)O(x)

9DR f R(x) R(x)O(x)dx

where F is the Faraday's constant, n is the number of electrons taking part in the

reduction of reactant, tR is the transference number of the species, and SR is the

stoichiometric coefficient of the reactant.

Duhamel's theorem58 was used to transform equation (5-11) for nonuniform

surface concentration given by equation (5-5) and bulk concentration condition

given by equation (5-6). The resulting integral equation can be expressed as


I(X) -nFDR R(x) 3(x) [ dc(x) dxo (512)
i(x (= (5-12)
(t t )sRF (4) J dx xx 1/3
( I0 9DR f R(x) R(x)3(x)dx


+ (CR(X 0) Co) 1

(9DR f R(x) R(x)3(x)dx









Equation (5-12) provides a solution of the mass transfer to the electrode under

nonuniform surface concentration. This approach has several advantages, which

will be elucidated later.

5.2.3 Electrode Kinetics

The current generated due to electrode-reactant charge transfer can be de-

scribed empirically by the Butler-Volmer equation. This equation relates the sur-

face overpotential Tr, to the current by


t(o) C io exp zFrs(x) exp c zFr, (x) (5-13)

where io is the exchange current density for the bulk concentration of the reactant,

aa and ac, are the anodic and cathodic charge transfer coefficients, respectively, R
is the gas constant, T is the temperature of the system, and Z = -z+z_/(z+ z_)

for a binary salt, and Z = -n for a reactant with excess supporting electrolyte.

The term ( CR) provides a correction to the exchange current density for sur-

face concentration of the reactant, where the constant 7 depends on the kinetic

mechanism of the reaction.

5.2.4 Concentration Overpotential

The concentration gradient across the mass-transfer boundary layer leads to a

concentration overpotential. A general form of concentration overpotential can be

expressed as

C RT ( )CR(X) + tR ( -CR(X) (5-14)
ZF C ) C )\
where rlc(x) is the concentration overpotential as a function of x. In the develop-

ment of equation (5-14), the concentration variation is assumed to be linear within

the diffusion layer. Furthermore, the conductivity variations of electrolyte are neg-

ligible.









5.2.5 Solution Potential in Outer Region

If there are no concentration variation and electrolyte is electrically neutral, the

potential of the solutions in the diffuse part of the solution can be described by

Laplace's equation

V24 0 (5-15)

and the current flowing in the electrolyte solution is given by


i = KV@ (5-16)


where K is the electrical conductivity of the electrolyte. Under steady state, the

current calculated from equation (5-16) would balance the current from electrode

kinetics and mass-transport. A solution of equation (5-14) for the disk and hemi-

spherical geometries is discussed in the following sections.

Disk Electrode A disk electrode of radius ro embedded in infinitely large insu-

lating plane is considered here. The potential far from the disk can be assumed to

be equal to zero, i.e.,

S= 0 at z2 + r2 00 (5-17)

where z and r are the axial and cylindrical coordinates, respectively. The current

on the insulating plane is equal to zero, hence, equation (5-16) yields


0 at z = 0, r > ro (5-18)


The solution of Laplace's equation satisfying above boundary conditions can be

expressed in the rotational elliptical coordinate system. The coordinate system is

defined as

z ro(rI, r ro ( + (2) 2)

where ( and TI are the coordinate axes for the rotational elliptical system. The local

potential of the solution potential in rotational elliptical coordinate system can be









expressed by
RT
4)= ZF BnP2n () M1 ( (5-19)
n=O
In the absence of concentration variation, the potential 4 is related to current den-

sity i according to equation (5-16); hence, the current at the electrode can also be

given by

80 a 8i ,RT
az rK F BP2 (TI) M2. (0) (5-20)
0z '_ ror 0( c=( ror]ZF

The coefficient B, are calculated by applying the orthogonality property of Leg-

endre polynomials. Thus,
1
roZF
B, = iZF ()P2. (T) d (5-21)
I- 1 'O' RT
0
An explicit expression for local solution potential as a function of position is ob-

tained by substitution of B, into equation (5-19).

Hemispherical Electrode A hemispherical electrode of radius ro embedded in

a infinitely insulating plane is considered here. Spherical polar coordinates ad-

equately describe the system. The potential far away from the electrode can be

assumed to be equal to zero. Thus, the potential boundary condition at r oo is

given by

S) 0 (5-22)

The current at the insulating plane is equal to zero; hence


-0 at 0-= (5-23)
00 2

The above condition will also be valid at 0 = 0 to satisfy the condition of symmetry.

The solution of Laplace's equation (5-15), subjected to boundary

conditions (5-22) and (5-23), can be expressed as

RT TO 2n+1
=- B, P2n (OS 0) () (5-24)
n=o0









Application of equation (5-16) relates the current at the electrode to the potential

according to

04 KroZF 2n (COS ) ( ro 2n+1
i(O) -K B), 1) 1 (5-25)
r RT Zrn Ic
r=ro n=o

and the coefficient B, are obtained by applying the orthogonality property of Leg-

endre polynomials. Thus, the expression for B, is given by


B= (4 + ) i(0)P2cos ) sin Od (5-26)
RT (2n + 1)
0

An explicit expression for local solution potential as a function of position is ob-

tained by substitution of B, in equation (5-24).

5.2.6 Electrode Potential

The electrode potential V with respect to a reference electrode can be parti-

tioned as

V ,ref = s + reC + c o (5-27)

where Gref is the potential of a reference electrode, %o is the solution potential

near the electrode surface, r, and rc are the surface and concentration potentials,

respectively. Under the assumption that the thickness of the diffuse layer is negli-

gible, %o can be assumed to be the electrolyte solution potential along the electrode

surface.

5.3 Dimensionless Quantities

The disk electrode can be easily described in cylindrical coordinates. The curvi-

linear coordinate relates to the cylindrical coordinate system through the follow-

ing equations:


x = r, y = z, R(x) = r









where r and z are the cylindrical coordinate axes. The dimensionless shear stress

P(x) is given by
3/2
3(x) -= (r) cah r (5-28)

where Ch is a constant, obtained by solving the Navier-Stokes and the Continuity

equations for the disk electrode, and ah is the hydrodynamic constant. For the

rotating disk electrode, ah is replaced by disk rotation speed, U. The value of

Ch is 0.51023 for a rotating disk electrode, and 0.36023 for a disk electrode under

submerged jet impingement.

After substitution of above quantities in equation (5-12), an expression for cur-

rent at the disk electrode is given as

.-nFD (3c)l/3 1/3 h dcr) dr
idisk (r) nF(3ch) 1)/ a r- d-(r) d (5-29)
(1 tR)sR F ( 9DD v dr = (r3 X3)13

+ (coo CR(0))

A dimensionless current ijisk, along with a dimensionless concentration and pa-

rameter N are given by the following

S oZF (5-30)
KRT

C(x) = CR( (5-31)
coo
nZF^2 Er2 a 1/3
N nZF=Dc,, roah V (5-32)
RTK(1 t4) v 9DR)
where N is measure of the mass-transfer resistance to the ohmic resistance.

After substitution of above quantities into equation (5-29), the dimensionless

current at the disk electrode can be written as

idisk(r) rN( Id _d/3 + (C(0)- 1) (5-33)
(4)dr r=x (r3 +3) (C

Equation (5-33) provides a convenient method to calculate the at the disk electrode

surface by specifying parameter N and surface concentration distribution.








For the hemispherical electrode system, the curvilinear coordinates x, y, and

R(x) are related to the spherical-polar coordinates by

x = rO, y = r, R(x) = ro sin 0

and the corresponding shear stress P(x) is expressed as
3/2
3(x) = 3(0) = hoB(0) (5-34)

An analytical expression of B(O) for stationary hemispherical electrode under jet

impingement has been obtained in Chapter 2. Similarly, B(0) for the rotating

hemispherical electrode has been derived in Chapter 4. After substitution of above

coordinate relationships and dimensionless quantities i*, C, and N into equation

(5-12), an expression for dimensionless current is given by:


N sin OB(0) fdC dOo
'hemisphere() F () Jdx 0 1/3 (5-35)
0 f sin 0e sineB(e)dO)


+ (C(0)- 1)

f sin 0 /sin OB(0)d)

Equation (5-35) provides a convenient way to calculate the current distribution for

a given surface concentration distribution at the hemispherical electrode. Equa-

tions (5-33) and (5-35) can also used to calculate surface concentration distribution

for a given current distribution at the electrode surface.

The dimensionless quantities J, E,, Ec, and E are defined as:

ioroZF ZFR, ZFTI
J Es E, E = E, + E,
RTK' RT RT '

where J represents the ratio of ohmic resistance of electrolyte to kinetic resistance,

E, is the dimensionless surface overpotential, Ec is the dimensionless









concentration overpotential, and E is the total dimensionless overpotential. With

the introduction of above mentioned dimensionless quantities, equation (5-14) for

concentration overpotential and equation (5-13) for electrode kinetics can be as

rewritten as


E,(x) = E(x) E,(x) = log(C(x)) + tR(1 C(x)) (5-36)


and

i*(x) = JC7 [exp (aaE,) exp (-aE,)] (5-37)

respectively. Equations (5-36) and (5-37) can be combined to eliminate the surface

overpotential E,. The resulting equation can be expressed as following


i*(x) = J [C(-a ) exp (aaE aaR (1 C)) C(7+ao) exp (-aE + atn (1 C))]

(5-38)

Equation (5-38) provides a convenient way to calculate current distribution. Its

usefulness will be elucidated in the subsequent section.

The dimensionless solution potential is defined as


) ZF (5-39)
RT

Equation (5-19) can be recast using the dimensionless solution potential as
00
ZBP2()3[ (() (5-40)
n=O

where coefficients B. are defined in terms of dimensionless current distribution

diisk as
1

Bn = '(0)- jisk T)P2n()d (5-41)
0
Similarly, dimensionless solution potential for hemispherical electrode can be ex-

pressed as

B P2(cos) O 2n+1 (5-42)
n=o









where coefficients B, are defined in terms of dimensionless current distribution

'hemisphere as

4n +1
2n + 1 J lhemisphere(O)P2n(COS 8) sin OdO (5-43)
0
and the dimensionless electrode potential V* can be expressed by


V* = E, + E,+ E (5-44)


The objective is to obtain the current profile below the mass-transfer-limited cur-

rent. This involves a simultaneous solution of mass-transport, electrode kinetics,

and calculation of electric field in the electrolyte. The dimensionless form of gov-

erning equations describing mass-transport, electrode kinetics, and solution po-

tential provide a convenient way to calculate the current distribution. The dimen-

sionless parameter J and N are the only variables present in the dimensionless

form of the governing equations. Therefore, by altering J and N, a current distri-

bution profile can be obtained for the disk and the hemispherical electrode.

5.4 Calculation Procedure

The Algorithm for calculating the current and potential distribution is outlined.

The procedure presented is applicable for submerged electrode systems under jet

impingement. The calculation procedure for disk electrode is presented first. Since

the hydrodynamic model predicts the separation of boundary layer at the hemi-

spherical electrode, a modified algorithm is presented in the subsequent subsec-

tion. The mathematical model was programmed using FORTRAN with double

precision accuracy. The program listing is given in Appendix E.

5.4.1 Disk electrode

The following procedure calculated the current distribution at the disk elec-

trode.









1. Values of J and N were assigned, and C was assumed to have a value be-

tween 0.0 and 1.0 at the center of the disk. Alternatively, values of electrode

potential or current level can also be chosen. This adds an extra step in the
calculation procedure, which iterates on the C(O) at the center of the elec-

trode.

2. The r/ro domain was discretized in irregularly spaced grid as outlined by

Acrivos et al.59

3. The value of current at r = 0 was calculated by

S() 1.57 137117488(1- C(0)) N
disk(o) = -( (5-45)
I ( )
This expression was derived by taking the limit of equation (5-33) at r = 0.

4. The values of surface overpotential E, and concentration overpotential Ec

were calculated at r/ro = 0 using equations (5-36) and (5-37), respectively.

5. As an initial guess, the values of E,, Ec, and i'isk along each point at the

electrode surface were assumed to be the same as at r/ro = 0. The values

C(r/ro) were obtained from equation (5-35) using the method devised by

Acrivos et al.59

6. The current distribution was calculated using equation (5-38) in the dis-

cretized domain.

7. The coefficients B, for solution potential were calculated from

equation (5-43). The number of terms in the summation were limited to 51.

Additional terms in equation (5-43) did not improve the calculated solution

potential.

8. The solution potential adjacent to the electrode surface was obtained by

equation (5-42).









9. The value of the electrode potential at r/ro = 0 was obtained using

equation (5-44).

10. A new overpotential distribution E was calculated using



E(r/ro) = E(r/ro) + A(V*(0) 4*(0) E(r/ro)) (5-46)

where A can have a value between 0 and 1. In this procedure, a value of 0.05

was chosen.

11. The relative percentage difference of coefficient Bo was used as termination

criterion. The Bo represent the average dimensionless current at the elec-

trode surface. If (Bo,new Bo,oid)/Bo,old was found to be less than 1.0-6, cal-

culation was terminated; otherwise, the calculation procedure was repeated

starting from step 6 to 10.

12. The calculated current, potential, and concentration distribution were writ-

ten to the file.


5.4.2 Hemispherical electrode

The algorithm for hemispherical electrode required modification due to bound-

ary layer separation. At the point of separation, the value of B(0) is zero. The

method devised by Acrivos et al.59 predicts that the current will also be zero at the

point of separation. However, the numerical difficulty can be avoided by termi-

nating the calculations just before the point of separation. The current distribution

calculations were performed up to 54.4', just before the point of boundary layer

separation predicted to occur at 54.80 by the boundary layer theory. The following

modified procedure was used to calculate the current distribution at the hemi-

spherical electrode.









1. Values of J and N were assigned, and C was assumed to have a value be-

tween 0.0 and 1.0 at 0 = 0. Values of electrode potential or current level can

also be chosen. This adds an extra step in the calculation procedure, which

iterates on the C(0) at 0 = 0.

2. The 0 domain was discretized from 0O to 54.40 in an irregularly spaced grid

as outlined by Acrivos et al.59

3. The value of current at 0 = 0 was calculated by

1.57-. 137117488 (1 C(0)) N
ihemisphere(0) (4 (5-47)

This expression was derived by taking the limit of equation (5-35) at 0 = 0.

4. The values of surface overpotential E, and concentration overpotential Ec

were calculated at 0 = 0 using equations (5-36) and (5-37), respectively.

5. As an initial guess, the values of E,, Ec, i-emisphere at each node was assumed

to be same as at 0 = 0. The concentration distribution was obtained using

equation (5-35) with the method devised by Acrivos et al.59

6. The current distribution was obtained using equation (5-38) in the discretized

domain. The current beyond the point of boundary layer separation was as-

sumed to be uniformly constant, and was assigned the obtained value at

0 = 54.40.

7. The coefficients B, for solution potential were calculated using

equation (5-43). The number of terms in the summation were limited to 51.

8. The solution potential adjacent to the electrode surface was obtained using

equation (5-42).









9. The value of the electrode potential was obtained from equation (5-44) at

0 = 0.

10. A new overpotential distribution E(O) was calculated using


E(0) = E(0) + A(V*(0) V*(0) E(0)) (5-48)


where A can have a value between 0 and 1. In this procedure, a value of 0.02

was selected.


11. The relative percentage difference of coefficient Bo was used as termination

criterion. The Bo represent the average dimensionless current at the elec-

trode surface. If (Bo,ne Boood)/Bo,old was found to be less than 1.0-6, calcu-

lations was terminated, otherwise, the calculation procedure was repeated

starting from step 6 to 10.

5.5 Current Distribution at Disk Electrode

The primary, secondary, and tertiary current distribution at the disk electrode

are presented in this section. The simulation results for current distribution below

the mass-transfer-limited current are discussed below.

5.5.1 Primary Distribution

If concentrations are uniform and the electrode reactions are fast, then E, and

E, can be set equal to zero in the governing equations. As a result, the solution

potential adjacent to the electrode will be equal to the electrode metal potential and

will have a uniform value. This condition is satisfied by equation (5-40) for n = 0,

and the resulting distribution is the primary current and potential distribution.

The solution potential 4p is given by


#P = 1l 2tan- ( (5-49)









where V4) is the solution potential at the electrode surface. The superscript P

stands for primary distribution. The current distribution at the disk surface was

evaluated from equation (5-20)

9^ 2n^o
S= (5-50)
z=0 T 02 r 2

the total current is

I = 27 irdr = 4-ro< (5-51)
0
and the resistance is
## 1
RP = (5-52)
IP 4Kro

For convenience, equation (5-50) is recast in terms of average current as

i 0.5
(5-53)
iavg 1 () 2


where the average current is defined to be

I 4K4)p
a o2 (5-54)
tavg 9-Vr2 Trro

A graph of i/iav, as a function of r/ro is presented in Figure 5-2. The current is

fairly well behaved near the center of the electrode, but it approaches infinity at

the edge of the electrode. As a result, the primary current distribution is highly

non-uniform for the disk electrode.

5.5.2 Secondary Current Distribution

A secondary current distribution is a outcome of the balance between electrode

kinetic rate and Ohm's law. For this case, the Laplace's equation for solution po-

tential is solved with the Butler-Volmer equation for electrode kinetics, which acts

as a boundary condition for the current at the electrode surface. The current given

by Ohm's law is equated to the current generated at the electrode surface due to

charge-transfer reactions. The secondary current distribution at the disk electrode









2.0



1.5



1.0



0.5



0.0 I I I
0.0 0.2 0.4 0.6 0.8 1.0
r/ro
Figure 5-2: Primary current distribution at the disk electrode. The value of local
current approaches to infinity as r/ro -+ 1

has been discussed in detail by Newman.23 The kinetics factor limits the value

of current at the electrode edge. The final value of current distribution depends

on the parameter J. Newman has shown that the current distribution becomes

uniform at J = 0.1.

5.5.3 Tertiary Current Distribution

The current distribution below the mass-transfer-limited value was obtained

at the disk electrode using the numerical algorithm presented in the previous

section. The fluid mechanics coefficient Ch was used for the disk electrode un-

der jet impingement. Numerical simulations were carried out for J = 5 and

N = 125. The values of C(0) were selected from 0.05 to 0.9. The values of aa,

ac, 7, and tR were kept fixed at 0.5. A plot of i/ium distribution is presented in

Figure 5.3(a). The nonuniform behavior of the current can be seen as r/ro ap-

proaches 1.0. The current has a maximum value at r/ro = 1 for all C(0) except

at 0.5. For this condition, the maximum occurs before the periphery of the elec-

trode. This observation is consistent with results reported by Durbha.6 It can









be explained as following: As unreacted reactant material moves along the electro

surface in the mass-transfer boundary layer, the concentration gradient, i.e., aC/az

builds up. This causes aC/az to be higher before the periphery. Therefore, a max-

imum in current is seen before r/ro = 1. The dimensionless concentration of the

reactant at the electrode surface as a function of r/ro is presented in Figure 5.3(b).

The dimensionless solution potential Vo is plotted in Figure 5.3(c). The solution

potential distribution becomes uniform for C(O) =0.9.

A parallel set of current distribution calculations for C(O) = 0.4, 0.3, 0.2, 0.1, and

0.05 are presented in Figure 5.4(a). The corresponding concentration distribution

is shown in Figure 5.4(b). The dimensionless solution potential along the electrode

surface is presented in Figure 5.4(c).

The following expression was used to quantify the uniformity of current dis-

tribution
ro )2
f -- 1 rdr
Tdisk ro (5-55)
f rdr
0
where Tdisk represents the uniformity parameter for current distribution at the

disk electrode. The quantity Tdisk was calculated for the current distribution de-

scribed in Figures 5-3 and 5-4. The obtained values are listed in Table 5.1. The

value of Tdisk is minimum for C(O) equal to 0.05, and maximum for 0.7. The val-

ues of iavg/ium and ir=o/iiim are also given in Table 5.1. A graph of 1 ir=o/izim

as a function of Tdisk for different value of C(O) is presented in Figure 5-5. This

plot shows a monotonically increasing relationship between 1 ir=o/ijim and Tdisk

up to C(O) = 0.7. The ratio i/iavg as a function of r/ro for different values of C(O)

is presented in Figure 5-6. The current distribution is most uniform for C(O) =

0.05 as seen in Figure 5-6, and the average current is about 98.9% of mass-transfer-

limited current. The Tdisk is 0.021 for C(O) = 0.05. This value of Tdisk is chosen













































0.2 0.4 0.6
r/ro


0.6



0.4


.8 1. 0.0
0.8 1.0 0.


0


0.2 0.4 0.6
r/ro


-140

-120

-100

-80

-60


-40


I I C(0)=0 5
c(0)=0 6
S--c(0)=07
---C(0)=0 8
----- c(0)=0 9


-----------------------




------------~----~--- --------




0 0.2 0.4 0.6 0.8 1.0
r/ro


(c)

Figure 5-3: Calculated current, concentration, and solution potential distribution

at the disk electrode. The simulations were done for J = 5, N = 125, and C(O) =

0.5 to 0.9 in incremental steps of 0.1. a) i/ium as a function of r/ro. b) Dimen-

sionless concentration distribution as a function of r/ro. c) Dimensionless solution

potential at the electrode surface as a function of r/ro.


Si I I

-------------- ------



- - -



-- C(0)=0 5
- C(0)=06 6
S- .C(0)=0 7
(----(0)=08 \-
----C(0)=0 9


0.0 L
0.0


0.8 1.0


S I I


C(0)=05
C(0)=0 6
-- C(0)=0 6
-- C(0)=08
-*-.C(0)=0 9








-.-------.- -- -- -- -- -
.. .. .. .. .. .. .. .













































0.2 0.4 0.6 0.8
r/r-


I .u


c(0)=0 4
0.8 - c(0)=0 3
c(0)=0 2
--- c(0)=o 1
0.6 --- c(0)=0 05


0.4


0.2 ...


0r--- -- - -.
0.0 -
0.0 0.2 0.4 0.6 0.8 1.0
r Iro


-140


-130


-120


e -110


-100 I


-90 I


-8 0 1
0.0 0.2 0.4 0.6 0.8 1.0
r Iro



(c)

Figure 5-4: Calculated current, concentration, and solution potential distribution

at the disk electrode. The simulations were done for J = 5, N = 125, and C(O) =

0.4, 0.3, 0.2, 0.1, 0.05. a) i/ium as a function of r/ro. b) Dimensionless concentra-

tion distribution as a function of r/ro. c) Dimensionless solution potential at the

electrode surface as a function of r/ro.


1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0
0.


.. ..--

-


- c(0)=0 4
- c(0)=0 3
. *c(0)=0 2
--- c(0)=0 1
-----c(0)= 05

L I I I


0


c(0)=0 4
C(0)=0 3
c(0)=0 2
---- C(0)=0 1



.0
:- ----------- ^ ^< f\
'51


. .


-









Table 5.1: Calculated values for uniformity parameter Tdisk (see equation (5-55)),
iavg//irm, and ir=o/iavg for the current distributions presented in Figures 5-3 and
5-4. The values of J and N was 5 and 125, respectively.


C(O) Tdisk lavg/ lim tr=0 /avg
0.00 0.000 1.000 1.000
0.05 0.021 0.989 0.960
0.10 0.044 0.975 0.922
0.20 0.097 0.937 0.854
0.30 0.161 0.879 0.796
0.40 0.235 0.797 0.752
0.50 0.312 0.690 0.725
0.60 0.376 0.563 0.711
0.70 0.392 0.422 0.710
0.80 0.282 0.266 0.757
0.90 0.238 0.128 0.782


as condition of uniformity. Therefore, current distributions with Tdisk <0.021 are

uniform.


5.6 Current Distribution at Hemispherical Electrode


The primary, secondary, and tertiary current distribution at the stationary hemi-

spherical electrode under submerged jet impingement are discussed in this sec-

tion.

5.6.1 Primary Distribution

If no concentration variations exist in the system and reaction kinetics is not

a limiting factor, the solution potential is a result of equation (5-24) equation for


n = 0. It is given by


( = roo


(5-56)


and the current is given by


K0r
"'r=rn


(5-57)


The total current to the hemisphere is expressed as


I = 27rKro( (


Kdo
0


(5-58)





































0.0 0.1 0.2 0.3

cfisk


0.4 0.5 0.6


Figure 5-5: 1


ir=o/iav, as a function Tdisk for different values of C(O).


4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0
0.


0


0.2 0.4 0.6 0.8 1.0
r/ro
0


Figure 5-6: i/iav, as a function r/ro for different values of C(O).


0.4 |-


0.2 k-


o c(0)=0.0
A c(0)=0.05
v C(O)=0.1
0 C(0)=0.2
< C(0)=0.3
S > C(0)=0.4
o c(O)=0.5
A> c(O)=0.6
0 c(O)=0.7
< e C(O)=0.8
K A c(0)=0.9 .


0 A

I I I *


-0.1 1
-0.


1


I I


c()=0.05
-----C(O)=0.1
--------( )=0.2
--- (0)=0.3
--------- c(0)=0.4
---------- )=0.5
................... c(O )= 0 .6
----------- )=0. 7
-- --------- c0)=0.7



- _- -- '


i/
/-



,.^----'.
4 ^':





___________


p3
,
o
jl
,
I


-


..









Consequently, the resistance to the flow of current is


RS J 0 o (5-59)
I 27Kro

The RPo1 provide a value of solution resistance to the current flow for the primary

current distribution.

5.6.2 Secondary Distribution

When the rate of mass-transfer to the electrode is infinite, and slow electrode

kinetics is taken into account, the resulting current calculation is called as sec-

ondary distribution. In the absence of mass-transfer resistance, the secondary

current distribution in hemispherical geometry would yield a uniform current

throughout the electrode surface. This can be easily deduced by solving the Butler-

Volmer equation for the electrode kinetics with the Laplace's equation for the so-

lution potential simultaneously.

5.6.3 Tertiary Distribution

The current distribution, dimensionless surface concentration, and solution po-

tential below the mass-transfer limited conditions were obtained for the stationary

hemispherical electrode under submerged jet impingement. Several simulations

were carried out for various values J, N, and C(O). The parameter C(O) was var-

ied between 0.5 and 0.9 in incremental steps of 0.1 for each value of J and N. The

values of aa, ac, 7, and tR were kept fixed at 0.5.

The calculated current distributions for four values of N (125, 50, 20, and 5),

and C(O) = 0.5, 0.6, 0.7, 0.8, 0.9, and for a fixed value J = 5 are given in Figure

5.7(a). Comparison of Figures 5.7(a) to 5.7(d) show that the distribution of current

becomes fairly uniform for the pole concentration of C(O) = 0.9. Simulation results

in Figure 5.7(a) for C(O) = 0.5, 0.6, 0.7, and 0.8 display a maximum in current. The

maximum is observed at the 0 values between the pole and the point of boundary