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Simplified methodology for designing parabolic trough solar power plants

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Title:
Simplified methodology for designing parabolic trough solar power plants
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Vasquez Padilla, Ricardo
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University of South Florida
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Tampa, Fla
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Subjects / Keywords:
Air Cooled Condensers
Levelized Cost Of Electricity
Regenerative Rankine Cycle
Solar Radiation
Solar Shading
Dissertations, Academic -- Engineering Energy Alternative Energy -- Doctoral -- USF   ( lcsh )
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ABSTRACT: The performance of parabolic trough based solar power plants over the last 25 years has proven that this technology is an excellent alternative for the commercial power industry. Compared to conventional power plants, parabolic trough solar power plants produce significantly lower levels of carbon dioxide, although additional research is required to bring the cost of concentrator solar plants to a competitive level. The cost reduction is focused on three areas: thermodynamic efficiency improvements by research and development, scaling up of the unit size, and mass production of the equipment. The optimum design, performance simulation and cost analysis of the parabolic trough solar plants are essential for the successful implementation of this technology. A detailed solar power plant simulation and analysis of its components is needed for the design of parabolic trough solar systems which is the subject of this research. Preliminary analysis was carried out by complex models of the solar field components. These components were then integrated into the system whose performance is simulated to emulate real operating conditions. Sensitivity analysis was conducted to get the optimum conditions and minimum levelized cost of electricity (LCOE). A simplified methodology was then developed based on correlations obtained from the detailed component simulations. A comprehensive numerical simulation of a parabolic trough solar power plant was developed, focusing primarily on obtaining a preliminary optimum design through the simplified methodology developed in this research. The proposed methodology is used to obtain optimum parameters and conditions such as: solar field size, operating conditions, parasitic losses, initial investment and LCOE. The methodology is also used to evaluate different scenarios and conditions of operation. The new methodology was implemented for a 50 MWe parabolic trough solar power plant for two cities: Tampa and Daggett. The results obtained for the proposed methodology were compared to another physical model (System Advisor Model, SAM) and a good agreement was achieved, thus showing that this methodology is suitable for any location.
Thesis:
Disseration (Ph.D.)--University of South Florida, 2011.
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Includes bibliographical references.
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by Ricardo Vasquez Padilla.
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Includes vita.

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SimpliedMethodologyforDesigningParabolicTroughSolarPowerPlants by RicardoVasquezPadilla Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofChemicalandBiomedicalEngineering CollegeofEngineering UniversityofSouthFlorida Co-MajorProfessor:D.YogiGoswami,Ph.D. Co-MajorProfessor:EliasStefanakos,Ph.D. MuhammadM.Rahman,Ph.D. JohnT.Wolan,Ph.D. YunchengYou,Ph.D. DateofApproval: April4,2011 Keywords:Solarradiation,Solarshading,RegenerativeRankinecycle,Levelizedcostof electricity,Aircooledcondensers Copyright2011,RicardoVasquezPadilla

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Dedication DedicatedtoJehovahandmytwoloves:JenniandNoah

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Acknowledgements Iexpressmymostsinceregratitudetomyadvisors,Dr.YogiGoswamiandDrStefanakosfortheirguidance,patience,understandingandencouragementthroughoutthis work.Ialsoliketothankthemembersofmycommittee,Drs.MuhammadRahman,John WolanandYunchengYou.SpecialthankstoMs.GinnyCosmides,Ms.BarbaraGraham andMr.CharlesGarretsonwhowiththeiradviceandkindnessweregreathelpduringmy stayintheCleanEnergyResearchCenterCERC. IwouldliketothankmyfriendsSesha,Saeb,June,Chennan,Antonio,Jamie,Ko, YangandGokmen,intheCERC,fortheirhelpandsupport,andtheUniversidaddelNorte fortheeconomicsupportinmydoctoralstudies.Iwouldalsoliketothankmyfriends: Viviana,Homero,Henry,PaulaLezama,Sophia,Cesar,PaulaAlgarin,Pedro,Ceciliaand Mrs.Ena.IcannotforgetmyfriendRollandwhoisnolongerwithusbutwillliveforever inmymemories.SpecialacknowledgegoestoMs.AnaRiveraandAlondrafortheir spiritualsupportandfriendship.Iwouldliketogivespecialthankstomymother,Nubia, myfather,Armando,andmyauntSara,becausetheyhavealwaysbelievedinme.Thanks tomyparentsinlawSantiagoandMariadeLourdes,withouttheirhelpthisdissertation wouldnothavebeenwritten.Finally,IwouldliketoacknowledgeJehovahforgiving mestrengthinmomentsofweaknessandmywifeJenniferandmysonNoahfortheir unwaveringsupportandforbeingmyinspiration.

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TableofContents ListofTablesiv ListofFiguresviii ListofSymbolsxv Abstractxix Chapter1Introduction1 1.1LiteratureReview5 Chapter2SolarRadiation9 2.1SolarAngles10 2.2HourlySolarRadiationModels20 2.3SingleAxisTracking28 2.3.1HorizontalTrackingAxis30 2.4Results31 2.5SolarShading36 2.6Conclusions48 Chapter3HeatTransferAnalysisofParabolicTroughSolarReceiver50 3.1Introduction50 3.2SolarReceiverModel53 3.2.1HeatTransferfromAbsorbertoHeatTransferFluid56 3.2.1.1CircularPipe57 3.2.1.2ConcentricAnnulus60 3.2.2HeatTransferfromAbsorbertoGlassEnvelope62 3.2.2.1VacuuminAnnulusP < 1Torr63 3.2.2.2PressureinAnnulusP > 1Torr65 3.2.2.3RadiationHeatTransferfromReceivertoEnvelope67 3.2.2.4HeatConductionThroughSupportBrackets76 3.2.3HeatTransferfromGlassEnvelopetotheAmbient82 3.2.3.1HeatConvection84 3.2.3.2RadiationHeatTransferSkyandCollectorSurface85 i

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3.2.4SolarEnergyAbsorption90 3.3NumericalSolution97 3.4ModelValidation100 3.5ResultsandDiscussion103 3.6Non-LinearRegressionHeatLossModel111 3.7Conclusions112 Chapter4PowerBlock114 4.1Introduction114 4.2ReheaterandSuperheater118 4.3Boiler122 4.4Preheater125 4.5ClosedFeedwater126 4.6OpenFeedwaterDeaerator129 4.7Turbine130 4.8Pump134 4.9Condenser137 4.9.1CoolingTower138 4.9.1.1DesignProcedure139 4.9.2CoolingTowerPerformanceatOffDesignConditions144 4.9.3AirCooledCondensers147 4.9.3.1DesignoftheAirCooledCondensers152 4.10NetElectricWork166 4.11Results168 4.12LinearRegressionModel174 4.13Conclusions180 Chapter5SolarFieldPipingandThermalLosses181 5.1SolarFieldLayout181 5.1.1HFieldLayout181 5.1.2IFieldLayout183 5.2PressureDropintheSolarField184 5.3ThermalLosses192 5.4ExpansionTank195 5.5Conclusions201 Chapter6IntegrationofSystemComponents202 6.1TransientAnalysis203 6.2EconomicAnalysis209 6.3Results212 6.3.1ResultsforAirCooledCondensers218 Chapter7ConclusionsandRecommendations223 ii

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ListofReferences225 Appendices239 AppendixA:ThermophysicalPropertiesofGases240 AppendixB:DataofParabolicTroughCollectors246 AppendixC:ThermophysicalPropertiesofHeatTransferFluidHTF247 AppendixD:PipeGeometry255 AbouttheAuthorEndPage iii

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ListofTables Table1.1CharacteristicsofConcentratingSolarPowerCSPsystems2 Table2.1Monthlyaveragesolardeclinationangle, d s ,andsunearthdistance correctionfactor, R 26 Table2.2Locationsusedforthedesignofparabolictroughsolarplants31 Table2.3Inputparametersforsolarshadingsimulation46 Table3.1Nusseltnumberforconcentricannulusunderlaminarow61 Table3.2Nusseltnumberforconcentricannulusunderlaminarowfordevelopingtemperatureanddevelopedvelocityprole61 Table3.3Thermalconductivity,densityandspecicheatfor304L,316Land 321Hstainlesssteel,temperatureinC63 Table3.4Moleculardiameterofdifferentgases65 Table3.5ConstantsforuseinEquation.63forlonghorizontalsquarecylindersinanisothermalenvironment[55]81 Table3.6ConstantsforuseinEquation.65forlonghorizontalsquarecylinders[84]subjectedtoacrossowofair82 Table3.7Polynomialcoefcientsforthermalconductivityandvolumetricheat capacity84 Table3.8ConstantsforEquation.73foracylinderincrossow[53]85 Table3.9Effectiveopticalefciencyterms91 Table3.10Incidentanglemodierfordifferentsolarcollectors92 Table3.11RadiativepropertiesofdifferentheatcollectionelementsHCE96 Table3.12Coatingemittanceofdifferentsolarreceivers96 iv

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Table3.13SpecicationsforaSEGSLS-2parabolictroughsolarcollectortest101 Table3.14Coefcientsobtainedbypolynomialregressionofthermalproperties ofSyltherm800102 Table3.15ComparisonofrootmeansquareerrorRMSEbetweentheproposed heattransfermodelandothernumericalmodelsforthecermetcoatingcase107 Table3.16ComparisonofrootmeansquareerrorRMSEbetweentheproposed heattransfermodelandothernumericalmodelsfortheblackchrome coatingcase108 Table3.17ComparisonofrootmeansquareerrorRMSEfordifferentheatloss convectionfactors109 Table3.18Specicationsusedfortheheatlossmodel111 Table3.19Heatlosscorrelationcoefcients112 Table4.1Typicalhigh-ntubedata151 Table4.2Typicalvaluesofoverallheattransfercoefcientinaircooledheat exchangers154 Table4.3AircooledcondenserparametersforTampa163 Table4.4AircooledcondenserparametersforDaggett164 Table4.5Heatexchangerparameterscalculatedfortheaircooledcondenser166 Table4.6Cycleparametersassumedforthesimulation169 Table4.7Cycleparametersobtainedatnominalconditions169 Table4.8Inputsparametersforthepowerblocksimulation178 Table4.9CoefcientsusedfortheproposedlinearcorrelationgivenbyEquation4.159,HTF:VP-1179 Table4.10CoefcientsusedfortheproposedlinearcorrelationgivenbyEquation4.159,HTF:Hitec179 Table5.1Maximumallowablestressksifordifferentmaterials185 Table5.2 K valuesfordifferentpipettingsusedinthesolareld187 v

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Table5.3FittingsusedintheHeatCollectionElementHCEloop188 Table5.4Headerlengthandttingsusedinthesolareldpipinglayout189 Table5.5Pipingttingandlengthusedforatypicalpowerblockunit190 Table5.6MinimumandmaximumallowableworkingtemperatureandpressurefordifferentHTFs190 Table5.7Thermalconductivity,inkW/mK,ofpipeinsulationmaterials193 Table5.8Coefcientsforcalculationoftheoptimumeconomicthickness194 Table6.1Costs,taxesanddiscountrateassumedfortheeconomicanalysis211 Table6.2Parameterusedforthehourlysimulation212 Table6.3ResultsobtainedforTampa213 Table6.4ResultsobtainedforDaggett214 Table6.5EffectofthecondensertypeontheannualperformanceofthePTC solarpowerplant221 TableA.1ThermophysicalcoefcientsofairEquationsA.2-A.4241 TableA.2ThermophysicalcoefcientsofhydrogenEquationA.2242 TableA.3ThermophysicalcoefcientsofhydrogenEquationsA.3-A.4243 TableA.4ThermophysicalcoefcientsofargonEquationsA.2-A.4244 TableA.5ThermophysicalcoefcientsofnitrogenEquationsA.2-A.4245 TableB.1Geometricalandopticaldataforparabolictroughcollectors246 TableC.1CoefcientsforuseinEquationC.1247 TableC.2CoefcientsforuseinEquationC.2248 TableC.3CoefcientsforuseinEquationC.3249 TableC.4CoefcientsforuseinEquationC.4251 TableC.5CoefcientsforuseinEquationsC.5-C.7252 TableC.6CoefcientsforuseinEquationC.8254 vi

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TableD.1Wallthickness,inmm,fordifferentnominalpipesizes PipeSchedule A-G255 TableD.2Wallthickness,inmm,fordifferentnominalpipesizes PipeSchedule H-M256 TableD.3Insidediameter,inmm,fordifferentnominalpipesizes PipeSchedule A-G257 TableD.4Insidediameter,inmm,fordifferentnominalpipesizes PipeSchedule H-M258 vii

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ListofFigures Figure1.1Currentandprojectedworldenergyusebyfueltype1 Figure1.2SchematicofaPTCsolarpowerplant3 Figure1.3PartsofaSolarCollectorAssemblySCA4 Figure1.4SchematicofLS-3solarcollectorloop4 Figure2.1Motionoftheearthaboutthesun10 Figure2.2Variationofthedeclinationangle, d s ,throughouttheyear11 Figure2.3EarthsurfacecoordinatesystemforobserveratQshowingthesolar azimuthangle a s ,thesolaraltitudeangle a andthesolarzenith angle z foracentralsunrayalongthedirectionvector S 0 12 Figure2.4Fundamentalsunangles:hourangle h ,latitude L anddeclination d s 13 Figure2.5Equationoftime EOT 13 Figure2.6Earthcentercoordinatesystemforthesunraydirectionvector S denedintermsofhourangle h ,anddeclinationangle d s 15 Figure2.7Earthsurfacecoordinatesaftertranslationfromtheearthcenter C to theobserverat Q 16 Figure2.8Geometricviewofthesun'spathasseenbyanobserveratQ19 Figure2.9Extraterrestrialsolarradiationspectruminvacuumbelow280nm, inairabove280nm;alsoshownareequivalentblackbodyand atmosphere-attenuatedspectraSMARTS2,U.S.Standard AtmosphereUSSA,ruralaerosolmodel,Z=48.19Airmass1.520 Figure2.10Variationof D o = D 2 throughouttheyear21 Figure2.11Attenuationofsolarradiationasitpassesthroughtheatmosphere22 viii

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Figure2.12Solarradiationonahorizontalsurface24 Figure2.13Variationof r d ,diffuseconversionfactor,withthesunsethourangle fordifferenttimesoftheday25 Figure2.14TrackingmodeforPTCs28 Figure2.15Asingle-axistrackingaperture28 Figure2.16Single-axistrackingsystemcoordinate29 Figure2.17Rotationof u b ,and r from z w ,and n coordinatesaboutthe z axis30 Figure2.18Comparisonofdifferenthourlyradiationmodels32 Figure2.19EffectoftrackingaxisanddataradiationsourceonthemonthlyaveragebeamradiationforTampa33 Figure2.20EffectoftrackingaxisanddataradiationsourceonthemonthlyaveragebeamradiationforDaggett33 Figure2.21Comparisonoftheannualtotalbeamradiationfordifferenttracking axisandsolarradiationdatasource34 Figure2.22SolardirectbeamradiationmapforUSA34 Figure2.23SolarbeamradiationmapNorth-SouthaxistrackingforUSA35 Figure2.24SolarradiationmapforFlorida35 Figure2.25Solarshadingproblemwithonlyoneconcentratingcollector36 Figure2.26Geometryusedtocalculatetheshadowofanobject37 Figure2.27Simpliedgeometryusedforoneconcentratingcollector37 Figure2.28Geometryusedinasolarshadingproblemwithonlyoneconcentratingcollector38 Figure2.29Solarshadingproblemwithtwoconcentratingcollectors40 Figure2.30Geometryusedtocalculate q and s 0 40 Figure2.31Interceptionofthesolarshadingwiththeprojectedareaofthe parabolictrough41 Figure2.32Differentcongurationsforthesolarshadingarea43 ix

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Figure2.33Shadingareaforconguration1-aand2-a'44 Figure2.34Shadingareaforconguration1-b45 Figure2.35Shadingareaforconguration2-b'46 Figure2.36Logicowforcalculationofsolarshading47 Figure2.37ComparisonbetweentheproposedshadingmodelandthemodeldevelopedbyStuetzle[8]48 Figure3.1PartsofaheatcollectionelementHCEandcontrolvolumeusedfor theheattransferanalysis54 Figure3.2Heattransferandthermalresistancemodelinacrosssectionatthe heatcollectionelementHCE55 Figure3.3Controlvolumeoftheheattransferuid57 Figure3.4Controlvolumeusedfortheabsorberanalysis62 Figure3.5Annulusgeometry68 Figure3.6Surfacesonacoaxialcylinder70 Figure3.7Nodepositionforcoaxialcylinders71 Figure3.8Viewfactorsforneighboringsurfacesonshellinteriorofcoaxial cylinders, R = 1 : 572 Figure3.9Viewfactorsforneighboringsurfacesonshellinteriorofcoaxial cylinders, R = 2 : 073 Figure3.10Supportbracket77 Figure3.11Comparisonoftheheatlossesthroughsupportbracketsfordifferent connectiontablengths,andthemodelusedbyForristall[6]79 Figure3.12Controlvolumeofglassenvelope83 Figure3.13Zoneanalysisoftheradiationheatlossfromthereceivertothe ambient86 Figure3.14Skyviewfactors, F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky and F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c 88 Figure3.15Incidentanglemodierfordifferentsolarcollectors93 x

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Figure3.16Collectorgeometricalendlosses93 Figure3.17Parabolageometryforarimangleof j m 94 Figure3.18Endlossfactorfordifferentcollectorsandassumptions95 Figure3.19Effectoftemperatureontheemissivityofborosilicateglassfortwo thicknesses.35and12.7mm97 Figure3.20Gridindependentanalysisfordifferentcollectorsegments,case:air intheannulus102 Figure3.21Comparisonofcollectorefciencycalculatedfromtheproposed modelwithexperimentaldata[39]andothersolarreceivermodels [6,41]104 Figure3.22Comparisonofthermallossescalculatedfromtheproposedmodel withexperimentaldata[39]andothersolarreceivermodels[6,41], on-suncase105 Figure3.23Comparisonofthermallossescalculatedfromtheproposedmodel withexperimentaldata[39]andothersolarreceivermodels[6,41], off-suncase106 Figure3.24Comparisonoftheoreticalandexperimental[39]collectorefciency andthermallossesobtainedfordifferentheatconvectionlossfactors110 Figure3.25Comparisonofheatlossesobtainedfromthenon-linearcorrelation Equation3.123andtheproposedmodel113 Figure4.1RegenerativeRankinecycleconguration115 Figure4.2Steamgenerationprocess119 Figure4.3Reheaterandsuperheaterheatexchanger119 Figure4.4Boilerheatexchanger123 Figure4.5Preheaterheatexchanger126 Figure4.6Closedfeedwaterheater128 Figure4.7Openfeedwaterheater130 Figure4.8Enthalpy-entropydiagramofsteamexpansioninamulti-stageturbinestages131 xi

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Figure4.9Highpressureturbinestages132 Figure4.10Effectofthrottleowratioontheturbineefciency135 Figure4.11Pump135 Figure4.12Effectofthrottleowratioonthepumpefciency136 Figure4.13Schematicofthecondenser137 Figure4.14Coolingtowerprocessheat139 Figure4.15Schematicofacoolingtower140 Figure4.16CongurationofanAframeaircooledcondenser148 Figure4.17Congurationofforcedandinduceddraftaircooledheatexchanger149 Figure4.18Cumulativefrequencydistributionofthedrybulbtemperature153 Figure4.19Aircooledcondenserlayout153 Figure4.20Effectofturbineworkonthegeneratorefciency167 Figure4.21Temperature-entropydiagramoftheregenerativeRankinecycle168 Figure4.22Effectofthepowerplantsizeonthenormalizedelectricoutput W e = W e ; nom andthenormalizedcondenserheattransferrate Q c = Q c ; nom HTF:VP-1170 Figure4.23Comparisonofthenormalizedelectricoutput W e = W e ; nom obtained bytheproposedpowerblockmodelandthemodeldevelopedby Patnode[13]171 Figure4.24Effectofthenormalizedsteammassowrate m steam = m steam ; nom ,condenserpressureandHTFinlettemperature T HTF ; a onthenormalized networkoutput W net = W net ; nom ,HTF:VP-1172 Figure4.25Effectofthenormalizedsteammassowrate m steam = m steam ; nom and condenserpressureonthenormalizedturbineextractionmassow rates m 15 = m 10 and m 18 = m 10 T HTF ; a = 390 C,HTF:VP-1173 Figure4.26Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenserpressureonthenormalizednetworkoutput W net = W net ; nom T HTF ; a = 390 C175 xii

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Figure4.27Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenserpressureonthenormalizedcondenserheattransferrate Q c = Q c ; nom T HTF ; a = 390 C176 Figure4.28Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenserpressureonthereturnHTFtemperature, P c = 0 : 08bar177 Figure4.29Comparisonofthedimensionlessnetworkoutputandcondenserheat transferrateobtainedfromthelinearcorrelationwiththeproposed powerblockmodel,HTF:VP-1andHitec180 Figure5.1Hsolareldlayout182 Figure5.2Isolareldlayout183 Figure5.3Thermallossesfromaverticaltank197 Figure6.1LogicowforthepreliminarydesignofthePTCsolarplants204 Figure6.2NodeanalysisofthesolarcollectorassemblySCA205 Figure6.3Thermalcapacitanceanalysisofthepipeheader206 Figure6.4ThermalinertiadistributionforIlayout207 Figure6.5ThermalinertiadistributionforHlayout207 Figure6.6LogicowusedforthedynamicsimulationofthePTCsolarpower plant208 Figure6.7FrequencydistributionofDirectNormalIrradianceDNI215 Figure6.8Monthlyaveragedistributionofthenetpoweroutputcalculatedat minimumLCOE R 216 Figure6.9ComparisonoftheLCOE R andannualnetpoweroutputbetweenthe proposedmodelandSystemAdvisorModelSAM[15],location: Tampa216 Figure6.10ComparisonoftheLCOE R andannualnetpoweroutputbetweenthe proposedmodelandSystemAdvisorModelSAM[15],location: Daggett217 Figure6.11EffectofthenumberofloopsontheLCOE R andtheutilization factor218 xiii

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Figure6.12Effectofthecondensingmethodonthepowercycleperformance219 Figure6.13Monthlyaveragedistributionofthecondenserpressure220 Figure6.14Effectofthecondensertypeonthemonthlynetpoweroutput222 Figure6.15Annualnetpoweroutputforcoolingtowerandaircooledcondenser222 FigureC.1DensityfordifferentHTFs247 FigureC.2SpecicheatatconstantpressurefordifferentHTFs248 FigureC.3SpecicenthalpyfordifferentHTFs250 FigureC.4ThermalconductivityfordifferentHTFs251 FigureC.5AbsoluteviscosityfordifferentHTFs253 FigureC.6VaporpressurefordifferentHTFs254 xiv

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ListofSymbols AArea[m 2 ] a s Azimuthangle[ ] CSpecicheat[kJ/kgK] DDiameter[m] dDistancebetweenrowcollectors[m] E Rateofenergytransferbyheat,workandmass[kW] FGeometricfactorforconcentriccylinders,Viewfactor fFrictionfactor,Focallength[m] gGravity[m/s 2 ] HCEHeatcollectionelement hEnthalpy[kJ/kg],Convectiveheattransfercoefcient[kW/m 2 K] HTFHeattransferuid iAngleofincidence[ ] KIncidentanglemodier kThermalconductivity[kW/mK] KnKnudsennumber L b Lengthbetweensupportbrackets[m] LAxiallengthofcylinder[m],Projectedlength[m],Object'slength[m] m Massowrate[kg/s] xv

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NuNusseltnumber PPressure[Torr],Annulusgaspressure[Torr],Perimeter[m] PrPrandtlnumber PTCParabolictroughcollector Q Rateofheattransfer[kW] q Radiationheatux[kW/m 2 ] R 2 Coefcientofdetermination RaRayleighnumber ReReynoldsnumber RGasconstant[kJ/kgK] rRadius[m],Radialfocaldistance[m] tTime[s] TTemperature[C] VVelocity[m/s] wCollectorwidth[m] xAxialdistance[m],Shadingdistance[m],Cartesiancoordinate[m] yCartesiancoordinate[m] zAxiallength[m] GreekSymbols a Altitudeangle[ ],Accommodationcoefcient,Absorptance b Volumetricthermalexpansioncoefcient[K )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 ] D Incrementalvalue d Moleculardiameterofannulusgas[cm] e Piperoughness[m],Emissivity xvi

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h o Pickopticalefciency g Intercepfactor l Meanfreepath[cm] m Absoluteviscosity[kg/ms] n Kinematicviscosity[m 2 /s] r Density[kg/m 3 ],Reectivity,Trackingangle[] f Relativehumidity y Collectorgeometricalendlosses s StefanBoltzmannconstant[5 : 67 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 W = m 2 K 4 ] t Transmittance q Ratioofspecicheats,Excesstemperatureofn[C] e Skyemissivity j Shadingfactor x Variableusedinradiationanalysis Subscripts a-absAbsorber-absorbed aAbsorber,Intersectionpoint,Absorbercoating bBaseofn, cCollectorsurface,Crossarea clClean condConduction convConvection ctConnectiontab,Coolingtower cv,aControlvolumeforabsorberanalysis xvii

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cv,eControlvolumeforenvelopeanalysis DDiameter e-absEnvelope-absorbed eGlassenvelope esEnvelope-sky fFluid gGas i,aInner,absorber i-jFromsurfaceitosurfacej L c Characteristiclength LLengthofconnectiontab mPartiallydeveloped,Mean SHShading sky-cFromskytocollectorsurface wWall windWind Fullydeveloped,Ambientconditions Superscripts Average 0 Perunitoflength,Perunitlengthofapertureareawidth xviii

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Abstract Theperformanceofparabolictroughbasedsolarpowerplantsoverthelast25yearshas proventhatthistechnologyisanexcellentalternativeforthecommercialpowerindustry. Comparedtoconventionalpowerplants,parabolictroughsolarpowerplantsproducesignicantlylowerlevelsofcarbondioxide,althoughadditionalresearchisrequiredtobring thecostofconcentratorsolarplantstoacompetitivelevel.Thecostreductionisfocusedon threeareas:thermodynamicefciencyimprovementsbyresearchanddevelopment,scaling upoftheunitsize,andmassproductionoftheequipment.Theoptimumdesign,performancesimulationandcostanalysisoftheparabolictroughsolarplantsareessentialforthe successfulimplementationofthistechnology.Adetailedsolarpowerplantsimulationand analysisofitscomponentsisneededforthedesignofparabolictroughsolarsystemswhich isthesubjectofthisresearch. Preliminaryanalysiswascarriedoutbycomplexmodelsofthesolareldcomponents. Thesecomponentswerethenintegratedintothesystemwhoseperformanceissimulatedto emulaterealoperatingconditions.Sensitivityanalysiswasconductedtogettheoptimum conditionsandminimumlevelizedcostofelectricityLCOE.Asimpliedmethodology wasthendevelopedbasedoncorrelationsobtainedfromthedetailedcomponentsimulations. xix

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Acomprehensivenumericalsimulationofaparabolictroughsolarpowerplantwas developed,focusingprimarilyonobtainingapreliminaryoptimumdesignthroughthesimpliedmethodologydevelopedinthisresearch.Theproposedmethodologyisusedto obtainoptimumparametersandconditionssuchas:solareldsize,operatingconditions, parasiticlosses,initialinvestmentandLCOE.Themethodologyisalsousedtoevaluate differentscenariosandconditionsofoperation. Thenewmethodologywasimplementedfora50MWeparabolictroughsolarpower plantfortwocities:TampaandDaggett.TheresultsobtainedfortheproposedmethodologywerecomparedtoanotherphysicalmodelSystemAdvisorModel,SAMandagood agreementwasachieved,thusshowingthatthismethodologyissuitableforanylocation. xx

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Chapter1 Introduction Currentworldenergyconsumptionshowsthatapproximately84.7%oftheworldenergyissuppliedbyfossilfuels,andonly9.9%byrenewableenergysources[1].Figure 1.1showsthattheworldenergyconsumptionisprojectedtoincreaseby50%from2005 to2030.ForthespeciccaseofU.S.,in2009only8.2%oftheenergyconsumedwas producedbyrenewablesources,andthemajorityoftherenewableenergywascoming frombiomassandhydroelectricplants[2].Differentfactorssuchas:risingfossilfuel prices,energysecurity,andgreenhousegasemissions,haveencouragedtheworldtoshift Figure1.1Currentandprojectedworldenergyusebyfueltype.Adaptedfrom[1] 1

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theenergypolicytowardsrenewablesources.Renewableenergysourcesareattractivefor environmentalreasons,especiallyincountrieswherereducinggreenhousegasemissionsis ofparticularconcern. Powerplantswithsolarconcentratorsareoneofthemainrenewableenergyalternativesfortheproductionofelectricity.Currently,fourtechnologiesareproposed:Parabolic TroughCollectorPTC,LinearFresnelReectorSystemLF,PowerTowerorCentral ReceiverSystemCRS,andDish/EnginesystemDE.Table1.1summarizesthecharacteristicsofeachsolartechnology. Table1.1CharacteristicsofConcentratingSolarPowerCSPsystems.Adaptedfrom[3] System Peak Efciency% Annual Efciency% AnnualCapacity Factor% Trough/linearFresnel 2110d 24d 14p Powertower 2314p 25p Dish/engine 2918p 25p ddemonstrated,pProjected,basedonpilot-scaletesting ThisdissertationwillfocusonlyonthePTC,whichisconsideredasoneofthemost matureapplicationsofsolarenergyinthiseld[4].Inordertoreducethecosts,PTC solarplantsshouldbedesignedforlargesizesorbepartofahybridsystemwhichincludes aregularfossilbackup.PTCsarecomposedofparabolictrough-shapedmirrors,which reecttheincidentradiationfromthesunonthesolarreceivertube.Thereceivertube islocatedatthefocusofaparabolainwhosesidesthemirrorsarelocated.Asshown inFigure1.2,thecirculatingheattransferuidHTFpassesthroughthereceiverandis heatedupbytheradiantenergyabsorbed.Then,theHTFiscollectedtobesenttothe 2

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Figure1.2SchematicofaPTCsolarpowerplant powerblock,whereitpassesthroughaseriesofheatexchangersandproducessuperheated steamathightemperatures.Thesuperheatedsteamowsthroughasteamturbinewhere rotationalmechanicalworkisthenconvertedintoelectricity. AsolareldconsistsofhundredsofSolarCollectorAssemblies,whichareindependentlytrackingassembliesofparabolictroughsolarcollectors.EachSCAhasthefollowingcomponents:metallicsupportstructure,mirrors,solarreceiver,andcollectorbalance ofthesystem.Figure1.3showsdifferentpartsofaSolarCollectorAssemblySCAfora LS3solarcollector.Inordertoreachtheoperationalconditions,thesolarcollectorassembliesSCAarearrangedinaseriescongurationnormallyknownasaloop.Thelength oftheloopdependsonthePTCperformance,butasshowninFigure1.4,itusuallyhas aUshapetominimizethepressuredropthroughthepipeheader.UsuallythePTCsare 3

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Figure1.3PartsofaSolarCollectorAssemblySCA.Adaptedfrom[3] orientedNorth-Southandtrackingthesunfromeasttowest,butthisalsodependsonthe landconstraints. TheeconomicfeasibilityofPTCsolarplantsisbasedonndingtheoptimumsizefora givenelectricoutput.Thepreliminaryanalysisisperformedbytheintegrationofthecomplexmodelsandcomponents,whichareintegratedtosimulaterealoperatingconditions. Thisdissertationproposesthedevelopmentofasimplemethodologyfortheinitialdesign ofparabolictroughsolarsystemsbasedonphysicalmodels.Themethodologyisfocused inobtainingapreliminaryoptimumdesignthroughasimpliedmethodologybasedon correlationsobtainedfromdetailedcomponentsimulations.Thismethodologyisexpected Figure1.4SchematicofLS-3solarcollectorloop.Adaptedfrom[3] 4

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tobegreathelpforengineersforthedesignandperformanceanalysisofparabolictrough solarsystems. 1.1LiteratureReview OneoftherstsolarpowerplantsimulationswasdevelopedbyLippke[5].Atypical 30MWeSEGSplantwasstudiedusingadetailedthermodynamicmodel.Inthismodel, correlationsfortheperformanceofparabolictroughsolarcollectorswerederivedbased onmeasureddataunderdifferentconditions;thesecorrelationswereusedtocalculatethe energyobtainedfromthesolareld.Themodelwasusedtocalculatethegrossandnet electricityoutputunderdifferentoperatingconditions. Adetailedheattransferanalysisandmodelingofaparabolictroughsolarreceiverwas carriedoutbyForristall[6].Oneandtwodimensionalenergybalanceswereusedforshort andlongreceiversrespectively.Thismodelwasusedtodeterminethethermalperformanceofparabolictroughcollectorsunderdifferentoperatingconditions.Jonesetal.[7] accomplishedacomprehensivemodelofthe30MWeSEGSVIparabolictroughplantin TRNSYS.Thismodelincludedsolarandpowercycleperformancewithoutfossilbackup. ThemodelwascreatedtoaccuratelypredicttheSEGSVIplantbehaviorandtoexamine transienteffectssuchasstartup,shutdown,andsystemresponse.Likewise,Stuetzle[8,9] investigatedthethermalperformanceofSEGSVIparabolictroughplant.Theanalysisconsistedofadynamicmodelforthecollectorandasteady-statemodelforthepowerplant. Thissimulationexaminedthelinearmodelpredictivecontrolstrategyformaintainingthe solareldataconstantoutlettemperature,althoughmaximizingofthegrosselectricity producedwasnotpursued.Quaschning[10]presentedamethodologyforeconomicopti5

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mizationofsolarplantdesignasafunctionofthesolarirradiance.Thismodelisableto ndtheoptimalsolareldsizeforanyspecicprojectlocation. Acomplexmodelforaparabolictrough,includingacomprehensiveeconomicanalysis,wasdevelopedbyNRELNationalRenewableEnergyLaboratory[11].Theparabolic troughsolartechnologywasmodeledusingthemethodologydevelopedbyStineandHarrigan[12].ThemodeliscapableofmodelingaRankine-cycleparabolictroughplant, withorwithoutthermalstorage,andwithorwithoutfossil-fuelbackup.Anothersimulation[13]forthesolareldbasedonSEGSVIwasmodeledinTRNSYS.Inthissimulation,theRankinepowercyclewasseparatelymodeled;thesteady-statepowercycleperformancewasregressedintermsoftheheattransferuidtemperature,heattransferuid massowrate,andcondensingpressure,andimplementedinTRNSYS.Boththesolar eldandpowercyclemodelswerevalidatedwithmeasuredtemperatureandowratedata fromtheSEGSVIplant.AthermaleconomicalmodelcalledSolarAdvisorModelSAM wasdevelopedbyNRELandSandiaNationalLaboratory[14,15].Thismodelcalculates costs,nances,andperformanceofconcentratingsolarpowerandalsoallowsexamining theimpactofvariationinthephysicalparametersonthecostsandoverallperformance. Amorerecentmethodologyfortheeconomicoptimizationofthesolarareaineither parabolictroughorcomplexsolarplantswascarriedoutbyMontesetal.[16].Thermalperformancefordifferentsolarpowerplantswasanalyzedatnominalandloadconditions.Onceannualelectricenergygenerationisknown,levelizedcostofenergyLCOE forthesolarplantcanbecalculated,yieldingaminimumLCOEvalueforacertainoptimumsolararea.Similarly,ananalyticmodelforasolarthermalelectricgeneratingsystem 6

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withparabolictroughcollectorswasdonebyRolimetal.[17].Threeeldsofdifferent collectorswereconsidered,thersteldwithevacuatedabsorbers,thesecondwithnonevacuatedabsorbersandthethirdwithbareabsorbers.MittelmanandEpstein[18]proposedanewpowerblockbyusingabottomingKalinacycle.Thiscyclehastheadvantage thatelectricitycanbeproducedatlowsolarinsolation-400W/m 2 Asitwasshownbefore,theparabolictroughsolarpowerplantsimulationistheresult ofacombinationofcomplexthermalandcostmodels.Ingeneraltheoptimizationsof thesesystemsrequireanextensivecomputationaleffortandsoftwareresources.Asimpliedconceptualandnumericalmethodologyfordesigningofparabolictroughsolarenergy systemswasproposedbyStineandHarrigan[19].Thismethodproposedadesignchart calledstoragesizinggraph,whichobtainstheoptimumcollectorareaforcertainlocation. Thestoragesizinggraphsareanexcellentinitialtoolforpreliminarydesignofsolarpower plants,buttheyincludeasimpliedthermalmodelsandcostanalysis. Basedonthemotivationofthisdissertation,thechaptersfollowthefollowingoutline. InChapter2,aheattransferanalysisofthePTCsolarreceiverisperformed.Thenew modelincludesnewconvectivecorrelationsandacomprehensiveradiationanalysis.Attingequationfortheheatlossesisobtainedfortwosolarcollectors:LS2andLS3.Chapter 3presentsthedesignoftypicalpowerblockforPTCsolarplants.Thepowerblockisthen simulatedunderpartialloadconditionsandattingequationisobtainedtocalculatethe netcyclepoweroutput,condenserheattransferrateandheattransferuidHTFreturn temperature.Chapter4showsthedesignofthesolarpipinglayoutandthecalculationof thepumpingpowerrequirementsforcirculatingtheHTFinthesolareld.Determination 7

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ofthermallossesinpipesandexpansionvesselarealsoincludedinthischapter.Chapter5 presentstheintegrationofallpreviouscomponentsandsystemsanddescribesthemethodologyfortheinitialandoptimumdesignofPTCsolarplants.Twodistinctsitesareused fortheapplicationoftheproposedmethodology,andevaluationofdifferentcondensersystemsisalsocarriedout.Finally,chapter6showstheconclusionsandrecommendationsfor furtherresearch. 8

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Chapter2 SolarRadiation Detailedinformationaboutsolarradiationavailabilityatanylocationisessentialfor thedesignandeconomicevaluationofparabolictroughsolarpowerplants.Longterm measureddataofsolarradiationareavailableforalargenumberoflocationsintheUnited Statesandotherpartsoftheworld.Forthoselocationswherelongtermmeasureddataare notavailabledifferentphysicsandsatellitemodelscanbeusedtoestimatethesolarenergy availability. Solarenergyisintheformofelectromagneticradiationwiththewavelengthsranging fromabout0 : 3 m m )]TJ/F51 8.9664 Tf 6.967 0 Td [(6 mtoover3 m m,whichcorrespondtoultravioletlessthan0 : 4 m m,visible : 4 m mand0 : 7 m m,andinfraredover0 : 7 m m;mostofthisenergyis concentratedinthevisibleandthenear-infraredwavelengthrange.Theincidentsolar radiation,sometimescalledinsolation,ismeasuredasirradiance,ortheenergyperunit timeperunitareakW/m 2 .Theaverageamountofsolarradiationfallingonasurface normaltotheraysofthesunoutsidetheatmosphereoftheearth,extraterrestrialinsolation, atmeanearth-sundistance D o iscalledthesolarconstant, I o .Recently,newmeasurements havefoundthevalueofsolarconstanttobe1366.1W/m 2 [20]. 9

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Figure2.1Motionoftheearthaboutthesun.Adaptedfrom[21] 2.1SolarAngles Thevariationinseasonalsolarradiationavailabilityatthesurfaceoftheearthcanbe understoodfromthegeometryoftherelativemovementoftheeartharoundthesun.The distancebetweentheearthandthesunchangesthroughouttheyear,theminimumbeing 1.471 10 11 matwintersolsticeDecember21andthemaximumbeing1 : 521 10 11 m atsummersolsticeJune21.Theyearroundaverageearthsundistanceis1 : 496 10 11 m.Theamountofsolarradiationinterceptedbytheearth,thereforevariesthroughoutthe year,themaximumbeingonDecember21andtheminimumonJune21Figure2.1. Theaxisoftheearth'sdailyrotationarounditselfisatanangleof23.45totheaxisof itseclipticorbitalplanearoundthesun.Thistiltisthemajorcauseoftheseasonalvariation ofthesolarradiationavailableatanylocationontheearth.Theanglebetweentheearth-sun linethroughtheircenterandtheplanethroughtheequatoriscalledthesolardeclination angle, d s Figure2.2.Thedeclinationanglevariesbetween-23.45onDecember21to +23.45onJune21.Thesolardeclinationisgivenby: 10

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Figure2.2Variationofthedeclinationangle, d s ,throughouttheyear d s = 23 : 45 sin 360 284 + n 365 .1 where n isthedaynumberwithJanuary1being n = 1.Ingeneral,thedeclinationis assumedtoremainconstantduringaspecicday.Theanalysisofthesunmotionisbased onthePtolemaictheory,whichassumesthattheearthisxedandthesunrotatesaround theearth.ThePtolemaicviewdescribestherelativesunmotionwithacoordinatesystem xedtotheearthwithitsoriginatthelocationofinterest. Thepositionofthesuncanbedescribedatanytimebytwoangles,thealtitudeand azimuthangleFigure2.3.Thesolaraltitudeangle, a ,istheanglebetweenalinecollinear withthesunraysandthehorizontalplane[21].Thesolarazimuthangle, a s ,istheangle betweenaduesouthlineandthehorizontalprojectionofthelinejoiningthesitetothe sun[21].Thesignconventionusedforazimuthangleispositivewestofsouthandnegative eastofsouth.Thesolarzenithangle,zistheanglebetweenthesitetosunlineandthe verticalatthesitelocation: z = 90 )]TJ/F54 11.9552 Tf 10.949 0 Td [(a .2 11

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Figure2.3EarthsurfacecoordinatesystemforobserveratQshowingthesolarazimuth angle a s ,thesolaraltitudeangle a andthesolarzenithangle z foracentralsunray alongdirectionvector S 0 .Alsoshownunitvectors i 0 j 0 k 0 alongtheirrespectiveaxes. Adaptedfrom[12] Thesolaraltitudeangleandazimuthanglesarenotfundamental;hence,theymustbe relatedtothefundamentalanglesFigure2.4:hourangle h s ,latitude L anddeclination angle d s .Thesolarhourangleisbasedonthenominaltimeof24hoursrequiredforthe suntomove360aroundtheearthor15perhour[21],itisdenedas: h = 15 t s )]TJ/F51 11.9552 Tf 10.949 0 Td [(12 .3 where t s isthesolartimeinhours.Thesolartimeiscalculatedfromthelocaltimebythe followingexpression[21]: t s = t + EOT + l st )]TJ/F59 11.9552 Tf 10.95 0 Td [(l local 4min = degree.4 where l st isthestandardtimemeridian,and l local isthelocaltimemeridian. EOT isthe equationoftimeFigure2.5,whichaccountsforthevariationoftherotationalspeedof 12

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Figure2.4Fundamentalsunangles:hourangle h ,latitude L anddeclination d s .Adapted from[12] theearth.Anapproximationforcalculatingtheequationoftime, EOT ,inminutesisgiven byWoolf[22]andisaccuratetowithinabout30secondsduringdaylighthours: EOT = 0 : 258cos x )]TJ/F51 11.9552 Tf 10.949 0 Td [(7 : 416sin x )]TJ/F51 11.9552 Tf 10.949 0 Td [(3 : 648cos2 x )]TJ/F51 11.9552 Tf 10.95 0 Td [(9 : 228sin2 x .5 x = 360 n )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 365 : 242 .6 Figure2.5Equationoftime EOT 13

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Thelatitudeangle L Figure2.4istheanglebetweenthelinefromthecenterofthe earthtositeandtheequatorialplane.Thelatitudeisconsideredpositivenorthoftheequator andnegativesouthoftheequator.Expressionforsolaraltitudeandsolarazimuthmaybe denedintermsoflatitude L ,hourangle h ,anddeclinationangle d s .Asitisshown inFigure2.3,theunitdirectionvector S 0 pointingtowardthesunfromtheobserver Q is denedas: S 0 = S 0 i i 0 + S 0 j j 0 + S 0 k k 0 .7 where i 0 j 0 k 0 areunitvectorsalong n w ,and z axesrespectively.Thedirectioncosines correspondingto n w ,and z axesare S 0 i S 0 j ,and S 0 k ,respectively.Theycanbewrittenin termsofthesolaraltitudeandazimuthas: S 0 i = )]TJ/F51 11.9552 Tf 10.618 0 Td [(cos a cos a s S 0 j = cos a sin a s .8 S 0 k = sin a Similarly,adirectionvectorpointingtothesuncanbedescribedatthecenterofthe earthasshowninFigure2.6.Usinganewsetofcoordinates,thedirectionvector S pointing tothesunmaybedescribedintermsofdirectioncosines S i S j ,and S k as: S = S i i + S j j + S k k .9 S i = )]TJ/F51 11.9552 Tf 10.618 0 Td [(cos d s cos h S j = cos d s sin h .10 S k = sin d s 14

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Figure2.6Earthcentercoordinatesystemforthesunraydirectionvector S denedinterms ofhourangle h ,anddeclinationangle d s .Adaptedfrom[12] Thesetwosetsofcoordinatesareinterrelatedbyarotationabout e axisFigure2.7 throughthecomplementofthelatitudeangle )]TJ/F59 11.9552 Tf 11.2 0 Td [(L andthetranslationalongtheearth radius QC .Thetranslationalongtheearth'sradiusisnegligiblesincethisisabout1 = 23525 ofthedistancefromtheearthtothesun.Notethattherotationaboutthe e axisisinthe negativesensebasedontheright-handrule.Inmatrixrotationthistakestheform: S 0 i S 0 j S 0 k = sin L 0cos L 010 )]TJ/F51 11.9552 Tf 10.617 0 Td [(cos L 0sin L S i S j S k .11 15

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Figure2.7Earthsurfacecoordinatesaftertranslationfromtheearthcenter C totheobserver at Q .Adaptedfrom[12] Solving,itisobtainedthat: S 0 i = S i sin L + S k cos L S 0 j = S j .12 S 0 k = )]TJ/F59 11.9552 Tf 9.289 0 Td [(S i cos L + S k sin L Afterreplacingthecorrespondingdirectioncosines,thefollowingsetofequationsareobtained: )]TJ/F51 11.9552 Tf 10.949 0 Td [(cos a cos a s = )]TJ/F51 11.9552 Tf 10.617 0 Td [(cos d s sin L cos h + sin d s cos L .13a cos a sin a s = cos d s sin h .13b sin a = cos d s cos L cos h + sin d s sin L .13c Equation.13cisanexpressionforthesolaraltitudeangleintermsoftheobserver's locationLatitudeangle,thehourangletimeandthesun'sdeclinationdate.Solving forthesolaraltitudeangle: a = sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 sin d s sin L + cos d s cos L cos h .14 16

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Twoexpressionsforcalculatingthesolarazimuthangle a s fromeitherEquation .13aor.13bwereobtained.Thesolarazimuthanglecanbeinanyofthefourtrigonometricquadrantsdependingonthelocation,timeofday,andtheseason.BothEquations .13aor.13brequireatesttoknowtheproperquadrant.ForEquation.13athe appropriateprocedureforcalculating a s is: a s = cos )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 cos d s sin L cos h )]TJ/F51 11.9552 Tf 10.95 0 Td [(sin d s cos L cos a .15 if h < 0then a s = )]TJ 10.617 0.048 Td [(j a s j ForEquation.13b,theprocedureisasfollows: a s = sin )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 cos d s sin h cos a .16 For L > d s : ifcos h < tan d s tan L then j a s j > 90 and a s = 8 > > > < > > > : )]TJ/F51 11.9552 Tf 9.289 0 Td [(180 + j a s j h < 0 180 )]TJ 10.949 0.048 Td [(j a s j h 0 .17 ifcos h tan d s tan L then j a s j 90 and if h < 0then a s = )]TJ 10.617 0.048 Td [(j a s j 17

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For L d s thesunremainsnorthoftheeast-westline j a s j > 90andthevalueof a s isgiven byEquation.17.Thepreviousprocedureisderivedfromthepathofthesunacrossthe sky,whichcanbeviewedasadiscdisplacedfromtheobserver.Thisgeometricperspective ofthesun'spathishelpfultovisualizethesunmovementsandobtainexpressionsfor testingthesunangles[12]. ThesunmaybeviewedastravelingaboutadiscofradiusRataconstantrateof15 degreesperhour.AsshowninFigure2.8a,thecenterofthisdiscappearsatdifferent seasonallocationsalongthepolaraxis,whichpassesthroughtheobserveratQandis inclinedtothehorizonbythelatitudeangle.Thecenterofthedisciscoincidentwiththe observerQattheequinoxesandisdisplacedfromtheobserverbyadistanceof R tan d s at othertimesoftheyear.Theextremesofthistravelareatthesolstices.Figure2.8bisa sideviewofthesun'sdisclookingfromtheeast.Inthesummer,thesunpathdiscofradius R hasitscenterYdisplacedabovetheobserverQ.PointXisdenedbyaperpendicular fromQ.Inthe n-z plane,theprojectionofthepositionSontothelinecontainingXandY willbewherethehourangle, h ,is.Theappropriatetestforthesunbeinginthenorthern skyisthen: R cos h < XY .18 ThedistanceXYis: XY = R tan d s tan L .19 then 18

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aSolardisc bSideviewofthesun'sdisc Figure2.8Geometricviewofthesun'spathasseenbyanobserveratQ.Adaptedfrom [12,21] 19

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cos h < tan d s tan L .20 Sunriseandsunsettimescanbeestimatedbyndingthehourangleat a = 0,then: h sr or h ss = cos )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F51 11.9552 Tf 11.946 0 Td [(tan L tan d s .21 2.2HourlySolarRadiationModels Theaverageamountofsolarradiationfallingonasurfacenormaltotheraysofthesun outsidetheatmosphereoftheearthextraterrestrialatmeanearth-sundistanceiscalled thesolarconstant, I o .Inthissimulationthesolarconstanthasavalueof1366.1W/m 2 as calculatedbyGueymard[20].Figure2.9showstheextraterrestrialsolarradiationspecFigure2.9Extraterrestrialsolarradiationspectruminvacuumbelow280nm,inair above280nm;alsoshownareequivalentblackbodyandatmosphere-attenuatedspectra SMARTS2,U.S.StandardAtmosphereUSSA,ruralaerosolmodel,Z=48.19Airmass 1.5.Adaptedfrom[20,23] 20

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trum,withthesolarconstantof1366.1W/m 2 ,withtheequivalentblackbodynormalized curveandtheatmosphereattenuatedspectrumforairmassof1.5.Theseasonalvariationof extraterrestrialsolarradiationatthesurfaceoftheearthiswellunderstoodfromtherelative movementoftheeartharoundthesun.Theextraterrestrialradiationvariesbytheinverse distancesquarefromtheearthtothesunas: I = I o D o = D 2 .22 where D isthedistancebetweenthesunandtheearth,and D o istheyearlymeanearth-sun distance.496 10 11 m.Thefactor D o = D 2 iscalculatedas[21]: D o = D 2 = 1 : 00011 + 0 : 034221cos x + 0 : 00128sin x + 0 : 000719cos 2 x + 0 : 000077sin 2 x .23 x = 360 n )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 = 365 Figure2.10showsthevariationofthefactor D o = D 2 throughouttheyear.Asextraterrestrialsolarradiation, I ,passesthroughtheatmosphere,apartofitisreectedbackinto Figure2.10Variationof D o = D 2 throughouttheyear 21

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thespace,apartisabsorbedbytheairandwatervapor,andsomegetsscatteredbythe moleculesofair,watervapor,aerosolsanddustparticlesFigure2.11[21].Thepartof solarradiationthatreachesthesurfaceoftheearthwithessentiallynochangeindirection iscalleddirectorbeamradiation.Thescattereddiffuseradiationreachingthesurfacefrom theskyiscalledskydiffuseradiation. Figure2.11Attenuationofsolarradiationasitpassesthroughtheatmosphere.Adapted from[21] Determinationofthehourlysolarradiationreceivedduringtheaveragedayofeach monthisprimordialforcalculatingthesolarcollectorperformancethroughouttheday. Thelongtermmodelsprovidethemeanhourlydistributionofglobalradiationoverthe averagedayofeachmonth.Giventhelongtermaveragedailytotalanddiffuseirradiation onahorizontalsurface, H h and D h respectively,itispossibletondthelongtermhourly solarradiation: I h and I d ; h .Valuesof H h and D h canbeobtainedfromeitherground measurementdataorsatellitedata[24].Satellitedataprovideinformationaboutsolar 22

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radiationandmeteorologicalconditionsinlocationswheregroundmeasurementdataare notavailable. Forsolarradiationcalculation,DailyintegrationapproachModelDI[25]wasused asthehourlyradiationmodel.Gueymard[25]developedtheDailyintegrationapproach topredictthemonthly-averagehourlyglobalirradiationbyusingalargedatasetof135 stationswithdiversegeographiclocations.58Nto67.68Sandclimates.Gueymard comparedhisproposedmodelwithprevioushourlyradiationmodels,Collares-Pereiraand RablModelCP&R[26]andCollares-PereiraandRablModelmodiedbyGueymard[27], andconcludedthatthedailyintegrationmodelisthemostaccuratecomparedtotheother models. Totalinstantaneoussolarradiationonahorizontalsurface, I h ,isthesumofthebeamor directradiation, I b ; h ,andtheskydiffuseradiation I d ; h : I h = I b ; h + I d ; h .24 ReferringtoFigure2.12, I h maybeexpressedas: I h = I b ; N cos z + I d ; h .25 or I h = I b ; N sin a + I d ; h .26 Thehourlybeamradiation, I b ; h isobtainedfromEquation.24: I b ; h = I h )]TJ/F59 11.9552 Tf 10.95 0 Td [(I d ; h .27 23

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Figure2.12Solarradiationonahorizontalsurface.Adaptedfrom[21] Introducingthehourlytodailyratios r d and r t as: r d = I d ; h D h .28 r t = I h H h .29 Then,thebeamradiationis: I b ; h = r t H h )]TJ/F59 11.9552 Tf 10.949 0 Td [(r d D h .30 LiuandJordan[28]showedthat r d iswellexpressedbyFigure2.13: r d = p T cos h )]TJ/F51 11.9552 Tf 10.949 0 Td [(cos h ss sin h ss )]TJ/F59 11.9552 Tf 10.949 0 Td [(h ss cos h ss .31 Thedaily-averageextraterrestrialirradiationonahorizontalsurface, H o ,maybecalculated asafunctionofthesolarconstant, E sc ,as: H o = 24 p h ss RE sc sin h o .32 24

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Figure2.13Variationof r d ,diffuseconversionfactor,withthesunsethouranglefordifferenttimesoftheday.Adaptedfrom[26,28] where h ss isthesunsethourangle + inradiansdenedbyEquation2.21.Duringpolar daysandnights,thevalueof h ss isgivenby: h ss = 8 > > > < > > > : 0if )]TJ/F51 11.9552 Tf 10.949 0 Td [(tan L tan d s > 1 p if )]TJ/F51 11.9552 Tf 10.949 0 Td [(tan L tan d s < )]TJ/F51 11.9552 Tf 9.289 0 Td [(1 .33 where h o isthedailyaveragesolarelevationoutsideoftheatmosphere,denedby: h o = qA h ss = h ss .34 and q = cos L cos d s .35 A h ss = sin h ss )]TJ/F59 11.9552 Tf 10.949 0 Td [(h ss cos h ss .36 25

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Thesolargeometryneedbedeterminedforonlyanaveragedayofeachmonth.The monthlyaveragedeclination, d s ,andthemonthlyaveragesunearthcorrectionfactor, R = D o = D 2 ,areshowninTable2.1. Table2.1Monthlyaveragesolardeclinationangle, d s ,andsunearthdistancecorrection factor, R .Adaptedfrom[25,29] Month d s R 1-20.711.032 2-12.811.025 3-1.801.011 49.770.994 518.830.978 623.070.969 721.160.967 813.650.975 92.890.99 10-8.721.007 11-18.371.022 12-22.991.031 Thedailyaverageclearnessindex, K t ,isgivenby[30]: K t = H h H o .37 Thedaylengthinhoursisobtainedas: S o = 24 p h ss .38 26

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Theratioofthehorizontalhourlyradiationtothetotalhorizontaldailyradiation, r t ,isgiven by: r t = r d 1 + q a 2 = a 1 A h ss r d 24 = p 1 + q a 2 = a 1 B h ss = A h ss .39 where a 2 = a 1 representstheatmosphericextinctioneffect. a 1 and a 2 wereobtainedfrom amultipleleast-squarest: a 1 = 0 : 41341 K t + 0 : 61197 K 2 t )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 a : 01886 K t S o + 0 : 00759 S o .40 a 2 = Max )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(0 : 054 ; 0 : 28116 + 2 : 2475 K t )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 : 7611 K 2 t )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 : 84535sin h o + 1 : 681sin 3 h o .41 and B h ss isdenedas: B h ss = w s )]TJ/F51 11.9552 Tf 5.475 -9.689 Td [(0 : 5 + cos 2 h ss )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 : 75sin 2 h ss .42 Foraparabolictroughsolarcollector,onlythebeamradiationontheaperturearea, I b ; c isneeded[21].Thebeamradiation, I b ; c ,iscalculatedas: I b ; c = )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(r t H h )]TJ/F59 11.9552 Tf 10.949 0 Td [(r d D h cos i sin a .43 where i istheangleofincidence,whichdependsonthetrackingmodeandtheposition ofthesun.InordertooptimizethePTCperformance,twodifferenttrackingmodeare usedFigure2.14:North-SouthEast-WestaxistrackingandEast-WestNorth-South axistracking.Thenextsectionshowshowtocalculatecos i forsingleaxistracking. 27

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aEast-WestNorth-Southaxistracking bNorth-SouthEast-Westaxistracking Figure2.14TrackingmodeforPTCs 2.3SingleAxisTracking Asitwasmentionedbefore,PTCsaredesignedtooperatewithtrackingaboutonlyone axis.Atrackingdrivesystemrotatesthecollectoraboutanaxisofrotationuntilthesun centralrayandtheaperturenormalareaarecoplanar.Figure2.15showshowtherotation ofacollectorapertureaboutatrackingaxis r .Thetrackingangle, r ,bringsthecentralray unitvector S intotheplaneformedbytheaperturenormalandthetrackingaxis.Towrite expressionsfor i and r intermsofthecollectororientationandsolarangles,itisnecessary totransformtheinitialcoordinates, x 0 ; y 0 ; z 0 Figure2.3toanewcoordinatesystemthat hasthetrackingaxisasoneofitsthreeorthogonalaxes.Theothertwoaxesareoriented Figure2.15Asingle-axistrackingaperture.Adaptedfrom[12] 28

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Figure2.16Single-axistrackingsystemcoordinate.Adaptedfrom[12] suchasthatoneaxisisparalleltothesurfaceoftheearth.Figure2.16showsthenew coordinatesystem,where r isthetrackingaxis, b istheaxisthatalwaysremainsparallelto theearthsurfaceand u isthethirdorthogonalaxis.Notethattheaperturenormal N rotates inthe u )]TJ/F59 11.9552 Tf 11.29 0 Td [(b plane.Both i and r canbedenedintermsofthedirectioncosinesofthe centralrayunitvector S alongthe u b ,and r axes.Thetrackingangleisdenedby 1 : tan r = )]TJ/F59 11.9552 Tf 10.484 8.093 Td [(S u S b .44 andthecosineofincidenceangle, i ,isgivenby: cos i = q S 2 b + S 2 u .45 or cos i = q 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(S 2 r .46 1 Thesignminusisbasedontherighthandrule 29

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2.3.1HorizontalTrackingAxis Todescribethiscategory,wemustrotatethe u b ,and r coordinatesbyanangle g from the z w ,and n coordinatesthatwerepreviouslyusedtodescribethesunrayunitvector Figure2.16.Sincetheaxistrackingremainsparalleltotheearthsurface,therotation takesplaceaboutthe z axisasshowninFigure2.17.Thedirectioncosinesof S new coordinatesystemarecalculatedas 2 : S r S b S u = cos g )]TJ/F51 11.9552 Tf 10.618 0 Td [(sin g 0 sin g cos g 0 001 S 0 i S 0 j S 0 k .47 Figure2.17Rotationof u b ,and r from z w ,and n coordinatesaboutthe z axis.Adapted from[12] SubstitutingintoEquation.44,itisobtainedthatthetrackingangleis: tan r = sin g )]TJ/F59 11.9552 Tf 10.949 0 Td [(a s tan a .48 2 Notethatthisrotationisinthenegativedirectionbasedontheright-handrule 30

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Theangleofincidenceforasingleaxis,horizontaltrackingcollectoris: cos q i = q 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(cos 2 a cos 2 g )]TJ/F59 11.9552 Tf 10.949 0 Td [(a s .49 Whenthetrackingaxisisorientedinthenorth-southdirection g = 0,theequationsabove reduceto: tan r = )]TJ/F51 11.9552 Tf 10.485 8.093 Td [(sin a s tan a .50 and cos i = p 1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(cos 2 a cos 2 a s .51 Whenthetrackingaxisisorientedintheeast-westdirection g = 90,theEquations.48 and.49become: tan r = cos a s tan a .52 and cos i = q 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(cos 2 a sin 2 a s .53 2.4Results Forthedesignofparabolictroughsolarplants,twolocationswereselected:Tampaand Daggett.ThelatitudeandlongitudeoftheselocationsarespeciedinTable2.2. Table2.2Locationsusedforthedesignofparabolictroughsolarplants LocationLatitudeLongitude Tampa,Fl27.97 -82.53 Daggett,CA34.85 -116.80 31

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Theresultsobtainedforsolarradiationcalculations,usingtwodifferentradiationdata sources:TMY3[32]andSurfacemeteorologyandSolarEnergyNASA-SSE[24],are showninFigures2.18and2.20.Figure2.18showsthecomparisonofthethreehourlysolar radiationmodels:CP&R,CP&RGandDImodel.Thethreemodelshowminordifferences forthemonthlyaveragebeamradiation.ThisagreeswiththeresultsobtainedbyGueymard [25],whofoundthatDImodelshowedbetterperformanceathighlatitudesthantheother modelsbutatlowlatitudesthedifferencesweresmall.Figures2.19and2.20showthe effectofthetrackingaxisonthebeamradiation.Asitwasexpected,forbothlocations, theNorth-Southaxistrackingpresentsbetterperformanceduringmostoftheyear.Forthat reasonmostofthePTCsareorientedwiththeirtrackingaxisintheNorth-Southdirection. Figure2.21showstheannualtotalradiationobtainedforeachtrackingaxis,againthenew resultsaremorefavorablefortheNorth-Southaxistracking,thereforethisconguration willbeadoptedforthecalculationofsolarradiationinthisdissertation. aTampa bDaggett Figure2.18Comparisonofdifferenthourlyradiationmodels.Radiationdatasource: NASA-SSE[24],hourlyradiationmodel:CP&R[31],CP&RG[27]andDImodel[25] 32

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aNorth-SouthAxisTracking bEast-WestAxisTracking Figure2.19Effectoftrackingaxisanddataradiationsourceonthemonthlyaveragebeam radiationforTampa.Radiationdatasource:NASA-SSE[24]andTMY3[32],hourly radiationmodel:DImodel[25] aNorth-SouthAxisTracking bEast-WestAxisTracking Figure2.20Effectoftrackingaxisanddataradiationsourceonthemonthlyaveragebeam radiationforDaggett.Radiationdatasource:NASA-SSE[24]andTMY3[32],hourly radiationmodel:DImodel[25] TheDImodel,alongwithradiationdatafromNASA-SSEwereusedtomapthedirect andbeamradiation.ThemapswereplottedusingtheMatplotlibBasemapToolkit[33] inPython2.6[34].GiventheresolutionoftheradiationdataobtainedfromNASA-SSE 33

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Figure2.21Comparisonoftheannualtotalbeamradiationfordifferenttrackingaxisand solarradiationdatasource.Radiationdatasource:NASA-SSE[24]andTMY3[32],location:TampaandDaggett,hourlyradiationmodel:DImodel[25] Figure2.22SolardirectbeamradiationmapforUSA.Radiationdatasource:NASA-SSE [24],hourlyradiationmodel:DImodel[25] 34

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Figure2.23SolarbeamradiationmapNorth-SouthaxistrackingforUSA.Radiationdata source:NASA-SSE[24],hourlyradiationmodel:DImodel[25] aDirectBeamRadiation bBeamRadiationforNorth-SouthAxis Tracking Figure2.24SolarradiationmapforFlorida.Radiationdatasource:NASA-SSE[24], hourlyradiationmodel:DImodel[25] 35

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1 ,itwasnecessarytouseabilinearinterpolation[33]forobtainingdataathigher resolution.Figures2.22and2.23showthemapsobtainedfortheNorth-SouthaxistrackingbeamradiationforUSA,whileFigure2.24showstheresultsforFlorida.Theresults showedthatingeneralFloridahasacceptablelevelsofbeamradiation.LowerlevelsofradiationincreasethecostoftheConcentratedSolarPowerCSPplantsandthereforetheir economicviability. 2.5SolarShading ThesolarshadingonPTCParabolictroughcollectorwasinitiallysolvedusingonly oneconcentratingcollectorFigure2.25. Figure2.25Solarshadingproblemwithonlyoneconcentratingcollector Forasimpleobject,itsshadowisdenedbyFigure2.26: L SH = L tan a .54 where L isthelengthoftheobject,and L SH isthelengthoftheobject'sshadow.ForaPTC, theshadingwascalculatedwiththeprojectedarearectangularprojectedsectionandeach 36

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Figure2.26Geometryusedtocalculatetheshadowofanobject shadowprojectionwasassumedtobeparallel.Figure2.27showsthegeometryusedto solvetheproblem.Usingthepreviousexpression,itisobtained: L S 1 = d 1 )]TJ/F59 11.9552 Tf 12.145 8.093 Td [(a 2 sin z tan a .55 Figure2.27Simpliedgeometryusedforoneconcentratingcollector 37

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L S 2 = d 1 + a 2 sin z tan a .56 with d 1 = f a + f b cos z AsseeninFigure2.27,theshadingareaisgivenby: A SH = b a a cos z + sin z sin a s tan a .57 Itisalsoimportanttoobtainanexpressionfor s Figure2.28: K = a cos 2 z + sin 2 z tan 2 a + sin2 z tan a sin a s 1 = 2 .58 Figure2.28Geometryusedinasolarshadingproblemwithonlyoneconcentratingcollector Usingsinelaw: sin s = a K sin z tan a cos a s .59 38

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Thenextexpressionfor s isobtained: s = sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 2 6 6 6 6 4 sin z cos a s tan a cos 2 z + sin 2 z tan 2 a + sin2 z tan a sin a s 1 = 2 3 7 7 7 7 5 .60 Aftersolvingfortheshadowofoneconcentratingcollector,theproblemwithtwoor moreconcentratingcollectorwasthensolved.Figure2.29showsthegeometryusedfor thisproblem.Twovariables, b 0 and x s ,werefound.ReferringtoFigure2.29,inorderto calculate x s ,itisnecessarytoobtain q rst.Theangle q Figure2.30aisgivenby: tan q = tan z cos s .61 x s isobtainedfromthefollowingexpression: x s = a sin z )]TJ/F75 11.9552 Tf 10.95 16.863 Td [( d )]TJ/F59 11.9552 Tf 10.949 0 Td [(a cos z cos s tan a cos q tan a + sin q .62 Thesecondcollectorisnotshadedwhen: a sin z d )]TJ/F59 11.9552 Tf 10.949 0 Td [(a cos z cos s tan a .63 or tan a tan a cr .64 a cr isdenedas: tan a cr = sin z cos s d = a )]TJ/F51 11.9552 Tf 10.949 0 Td [(cos z 39

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Figure2.29Solarshadingproblemwithtwoconcentratingcollectors a b Figure2.30Geometryusedtocalculate q and s 0 40

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Because0 a 90 o ,thelastexpressioncanberewrittenas: a a cr .65 Inordertocalculatetheshadingarea,itisnecessarytocalculate s 0 Figure2.30b: tan s 0 = tan s cos z .66 Now,itisnecessarytodenesomeconditionsbasedonpoints P 1 and P 2 Figure2.31. Point P 1 isdenedby: P 1 = x 1 ; y 1 .67 x 1 = L s 2 sin a s + a 2 cos z .68 y 1 = L s 2 cos a s .69 Figure2.31Interceptionofthesolarshadingwiththeprojectedareaoftheparabolictrough Theequationofastraightlineisgivenby: y )]TJ/F59 11.9552 Tf 10.949 0 Td [(y 1 = tan s x )]TJ/F59 11.9552 Tf 10.949 0 Td [(x 1 .70 41

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Point P 2 isdenedasfollows: P 2 = x 2 ; y 2 .71 x 2 = d )]TJ/F59 11.9552 Tf 12.145 8.094 Td [(a 2 cos z .72 y 2 = y 1 + m x 2 )]TJ/F59 11.9552 Tf 10.949 0 Td [(x 1 .73 c = y 2 .74 where x 2 )]TJ/F59 11.9552 Tf 10.949 0 Td [(x 1 isgivenby: x 2 )]TJ/F59 11.9552 Tf 10.95 0 Td [(x 1 = d )]TJ/F59 11.9552 Tf 10.95 0 Td [(a cos z )]TJ/F75 11.9552 Tf 10.949 13.276 Td [( d 1 + a 2 sin z sin a s tan a then y 2 = d 1 + a 2 sin z cos a s tan a + tan s d )]TJ/F59 11.9552 Tf 10.95 0 Td [(a cos z )]TJ/F75 11.9552 Tf 10.949 13.276 Td [( d 1 + a 2 sin z sin a s tan a .75 Thereisnoshadowwhen: y 2 b whichisequivalentto: y 0 2 R with y 0 2 = y 2 a and R = b a 42

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Figure2.32Differentcongurationsforthesolarshadingarea Whenthereisshading,asseeninFigure2.32,differentcongurationscanbeobserved. Forcase: x 0 s cos s 0 < 1.76 and 8 > > > < > > > : x 0 s sin s 0 + c 0 > RCase a x 0 s sin s 0 + c 0 < RCase b .77 x 0 s = x s a Forcase: x 0 s cos s 0 > 1.78 and 8 > > > < > > > : x 0 s sin s 0 + c 0 > RCase a x 0 s sin s 0 + c 0 < RCase b .79 43

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Forcases1-aand2-a',theshadingareaFigure2.33isgivenby: A sh = b )]TJ/F59 11.9552 Tf 10.949 0 Td [(c 2 2tan s 0 .80 Figure2.33Shadingareaforconguration1-aand2-a' TheexpressionforcFigure2.31waspreviouslyobtainedas: c = d 1 + a 2 sin z cos a s tan a + tan s d )]TJ/F59 11.9552 Tf 10.95 0 Td [(a cos z )]TJ/F75 11.9552 Tf 10.95 13.276 Td [( d 1 + a 2 sin z sin a s tan a .81 Thefractionofshadingareais: SH = A sh ab = b )]TJ/F59 11.9552 Tf 10.95 0 Td [(c 2 2 ab tan s 0 .82 Manipulatingthelastexpression: SH = R )]TJ/F59 11.9552 Tf 10.95 0 Td [(c 0 2 2 R tan s 0 .83 with R = b a and c 0 = c a Forcase1-b,theshadingareaFigure2.34isgivenby: 44

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A sh = b )]TJ/F59 11.9552 Tf 10.949 0 Td [(c )]TJ/F59 11.9552 Tf 12.144 8.093 Td [(x s 2 sin s 0 x s cos s 0 .84 Figure2.34Shadingareaforconguration1-b Thefractionofshadingareais: SH = A sh ab = R )]TJ/F59 11.9552 Tf 10.95 0 Td [(c 0 )]TJ/F59 11.9552 Tf 12.145 8.094 Td [(x 0 s 2 sin s 0 x 0 s R cos s 0 .85 x 0 s = x s a Forcase2-b',theshadingareaFigure2.35iscalculatedas: A sh = b )]TJ/F59 11.9552 Tf 10.949 0 Td [(c )]TJ/F59 11.9552 Tf 12.145 8.093 Td [(a 2 tan s 0 a .86 Inthiscase x s isgivenbythenextexpression: x s cos s 0 = a .87 Thefractionofshadingareais: SH = A sh ab = 1 R R )]TJ/F59 11.9552 Tf 10.95 0 Td [(c 0 )]TJ/F51 11.9552 Tf 12.145 8.094 Td [(1 2 tan s 0 .88 45

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Figure2.35Shadingareaforconguration2-b' Theshadingfactorisdenedby: j SH = 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(SH .89 TheprocedureforsolarshadingcalculationsisshowninFigure2.36.Inordertovalidatetheshadingmodel,itwascomparedwiththemodeldevelopedbyStuetzle[8]which assumesthattheshadowcoverstheentirelengthofthePTC.Table2.3showstheinput parametersusedforthesimulationofthesolarshading. Table2.3Inputparametersforsolarshadingsimulation DimensionValueReference a 5m[4] b 400m[13] d 15m[13] FocalLength f 1.49m[4] f a a = 2[12] f b a 2 = 16 f [12] ThesimulationwascarriedoutbyJune21andDecember21atSEGSVIlocation latitude37.21 Nandlongitude117.022 W,theresultsareshowninFigure2.37.The resultsobtainedshowthatthemodeldevelopedbyStuetzleismoreconservative,although 46

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Figure2.36Logicowforcalculationofsolarshading 47

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therearesmalldifferencesbetweenthemodelsowingtothedimensionsofthePTC.For smallersystems,likesmallcollectorsforheatingandelectricityproduction,theproposed modelseemstobemoreappropriate. a b Figure2.37Comparisonbetweentheproposedshadingmodelandthemodeldevelopedby Stuetzle[8].InputparametersaregiveninTable2.3 2.6Conclusions DImodelwasselectedasthehourlyradiationmodel.Theinputsforthismodelcan bemeasureddataorsatellitedata,whichgivescertainexibilitytothoselocationswhere measuredradiationdataarenotavailable.TheresultsshowthatNorth-Southaxistracking isthebesttrackingcongurationforthelocationsanalyzed:TampaandDaggett. Thesolarradiationmapsobtainedcanbeanaidtondpreliminarylocationsaroundthe worldwhereoptimumlevelsofradiationarepresent,althoughitisimportanttoemphasize thatsolarradiationisonlyoneofthecriteriatondoptimumlocationsforConcentrated SolarPowerplantsCSP. 48

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Asolarshadingmodelwasdeveloped;theresultsshowedthattheproposedmodelis suitableforanyPTCdimensions. 49

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Chapter3 HeatTransferAnalysisofParabolicTroughSolarReceiver 3.1Introduction Parabolictroughpowerplantscurrentlyrepresentthemostmaturetechnologyforsolar thermalpowerproduction.AparabolictroughsolarcollectorPTCtakestheradiantenergyfromthesunandconvertsittousefulthermalenergyintheheattransferuidHTF thatcirculatesthroughthesolareld.Oncethegeometryandthermalpropertiesaredened,thethermalperformanceandenergygainedbytheHTFcanbecalculatedunder differentcongurationsandmeteorologicalconditions.PTCsaretypicallyoperatedatup totemperaturesof400 CandsyntheticoiliscommonlyusedasHTF.Theheattransfer analysisofthesecollectorsisimportantforthecalculationofthermallossesandsizingof thesolarpowerplantduringpreliminarydesignandalsopermitstoevaluatetheeffects ofcollectordegradation,andHTFowratecontrolstrategiesontheoverallplantperformance[13].GiventheimportanceoftheheattransferanalysisinPTCs,sincethe1970s numerousmodelshavebeenproposed. Edenburn[35]predictedtheefciencyofaPTCbyusingananalyticalheattransfermodelforevacuatedandnonevacuatedcases.Theresultsshowedgoodagreement withmeasureddataobtainedfromSandiaNationalLaboratoriesSNLcollectortestfacility[36].Ratzeletal.[37]carriedoutbothanalyticalandnumericalstudyoftheheat 50

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conductionandconvectivelossesinanannularreceiverfordifferentgeometries.Three techniqueswereproposedtoreducetheconductionheatloss:evacuation,oversizingthe annularreceiverwhilekeepingtheRayleighnumberbelow1000overtherangeofoperationanduseofgaseswithlowthermalconductivity.Clark[38]analyzedtheeffectsof designandmanufacturingparametersthatinuencedthethermalandeconomicalperformanceofparabolictroughreceivers.Dudleyetal.[39]developedananalyticalmodelof SEGSLS-2parabolicsolarcollector.Thethermallossmodelfortheheatcollectionelementwasonedimensionalsteadystatemodelbasedonthermalresistanceanalysis.This modelwasvalidatedwithexperimentaldatacollectedbySNLfordifferentreceiverannulusconditions:vacuumintact,lostvacuumairinannulus,andbrokenannuluscoverbare tube.Theresultsshowedareasonableagreementbetweenthetheoreticalandexperimentalheatlosses.ThomasandThomas[40]developedasetofcurve-ttingequationsbased onanumericalheattransfermodelfortheheatlossesinthereceiverofaPTCfordifferent geometries,radiativepropertiesandmeteorologicalconditions.Adetailedheattransfer modelforthesolarreceiverwasdevelopedbyForristall[6].Onedimensionalenergybalanceforseveralsegmentswasusedforshortandlongreceiversrespectively.Thismodel wasusedtodeterminethethermalperformanceofparabolictroughcollectorsunderdifferentoperatingconditions.Stuetzle[8]proposedanunsteadystateanalysisofsolarcollector receivertocalculatethecollectoreldoutlettemperature.Themodelwassolvedbydiscretizingthepartialdifferentialequationsobtainedbyenergybalance.Theresultsobtained showedthattheoverallmatchbetweenthecalculatedandmeasuredoutlettemperatures wasgood.Garca-ValladaresandVelsquez[41]developedadetailednumericalmodelfor 51

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asinglepasssolarreceiverandvalidatedit.Thentheyextendedthemodeltoadouble passreceiver.Theirresultsshowedthattheproposedcongurationenhancesthethermal efciencyofthesolarcollectorcomparedwiththesinglepass.Recently,threedimensional heattransferanalysisofPTCswasperformedbycombiningtheMonteCarloRayTrace MethodMCRTandCFDanalysis[42].Theresultsindicatedthattheangesupport bracketandbellowundernon-vacuumconditionsbringahighconductiveheatloss. Inthemiddle90s,CohenandKerney[45]proposedtheuseofdirectsteamgeneration DSGcollectorasafuturedevelopmentoftheSolarElectricGeneratingStationSEGS inordertoeliminatethecostlysyntheticoil,intermediateheattransportpipingloopand theheatexchangerbetweenthesolareldandthepowerblock.Heidemannetal.[46] formulatedatwodimensionalheattransfermodelforcalculatingtheabsorberwalltemperatureofaDSGcollectorundersteadyandunsteadyconditions.Thenumericalsolution showedthatasuddendropofirradiationinducesaveryhightemperaturegradientinside theabsorbertubeinashortperiodoftime.Odehetal.[47]carriedoutamodelforheat lossofaDSGcollectorintermsofwalltemperatureratherthanworkinguidtemperature.TheresultswerecomparedwiththeSandiatestdataandtheresultsshowedthatthe modelunderestimatesthemeasuredloss.Thisunderestimationwasattributedtoomission ofheatlossfromthereceivertubevacuumbellows.Odehetal.[48]studiedthethermal performanceofparabolictroughsolarcollectorusedasdirectsteamgeneratorfordifferent solarradiationlevelsandgeometriccongurations.Thisheattransfermodelshowedbetter agreementwiththeinfocustestresultsthanthepolynomialcurvetequationobtained byDudleyetal.[39].Thethermallossescalculatedforwaterwerebasedonthereceiver 52

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walltemperatureandtheresultsshowedthatthermallossescalculatedforsteamastheheat transferuidwerelowerthanthoseobtainedforsyntheticoil. InthischapteradetailedonedimensionalheattransferanalysisofaPTCispresented. Theexistingmodelspublishedpreviouslyassumethatthereisnothermalinteractionbetweentheneighboringsurfacesabsorber-envelope,andenvelope-envelopeforthermal radiationlosses.Althoughthisassumptionssimpliestheanalysis,itunderestimatesthe radiationlossesathighabsorbertemperatures.Toaccountforthethermalinteractionbetweenadjacentsurfaces,acomprehensiveradiativeanalysiswasimplementedinourstudy fortheheatlossesintheabsorberandtheglassenvelope.Areviewofthecorrelationsfor convectiveheattransferlosseswasperformedaswellandnewcorrelationswereusedin thismodel.Thereceiverandenvelopeweredividedintoseveralsegments,andmassand energybalancewerecarriedoutineachcontrolvolume.Thepartialdifferentialequations werediscretizedbyusingthenitedifferencemethodandthesetofnonlinearalgebraic equationsweresolvedsimultaneously.Inordertovalidatetheproposednumericalmodel, itwascomparedwiththeexperimentaldataobtainedfromSandiaNationalLaboratories SNL,Dudleyetal.[39],andotherheattransfermodels[6,41]aswell. 3.2SolarReceiverModel TheheatcollectionelementHCEconsistsofanabsorbersurroundedbyaglassenvelopeFigure3.1.Theabsorberistypicallystainlesssteeltubewithaselectiveabsorber surfacewhichprovidestherequiredopticalandradiativeproperties.Selectivesurfaces combineahighabsorptanceforsolarradiationwithlowemittanceforthetemperature rangeinwhichthesurfaceemitsradiation.Thiscombinationofsurfacecharacteristicsis 53

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Figure3.1PartsofaheatcollectionelementHCEandcontrolvolumeusedfortheheat transferanalysis.Adaptedfrom[49] possiblebecause98percentoftheenergyinincomingsolarradiationiscontainedwithin wavelengthsbelow3 m m[21].Theglassenvelopeisanantireectiveevacuatedglasstube whichprotectstheabsorberfromdegradationandreducestheheatlosses.Thevacuum enclosureisusedprimarilytoreduceheatlossesathighoperatingtemperaturesandtoprotectthesolar-selectiveabsorbersurfacefromoxidation.TheHCEusesconventionalglass tometalsealsandmetalbellowsateitherendtoachievethenecessaryvacuumenclosure andforthermalexpansiondifferencebetweenthesteeltubingandtheglassenvelope[4]. ThebellowsalsoallowtheabsorbertoextendbeyondtheglassenvelopesothattheHeat 54

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CollectionElementscanformacontinuousreceiver.Thespacebetweenbellowsprovides aplacetoattachtheHCEsupportbrackets[6].Chemicalgettersareplacedintheannulus toabsorbhydrogen,whichcomesfromtheHTFanddecreasesthePTCperformance. Theheattransfermodelisbasedonanenergybalancebetweentheheattransferuid andthesurroundings.Figure3.2showstheheattransferinacrosssectionatthesolar collectorandthethermalresistancemodelusedintheheattransferanalysis. aHeattransfer bThermalcircuit Figure3.2HeattransferandthermalresistancemodelinacrosssectionattheheatcollectionelementHCE.Adaptedfrom[6,39] 55

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Thesolarenergyreectedbythemirrorsisabsorbedbytheglassenvelope Q e )]TJ/F59 8.9664 Tf 6.967 0 Td [(abs and theabsorbersurface Q a )]TJ/F59 8.9664 Tf 6.967 0 Td [(abs .Apartoftheenergyabsorbedintheabsorberistransferredto theHTFbyforcedconvection Q a )]TJ/F59 8.9664 Tf 8.311 0 Td [(f ; conv ,remainingenergyistransferredbacktotheglass envelopebyradiation Q a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad andnaturalconvection Q a )]TJ/F59 8.9664 Tf 6.966 0 Td [(e ; conv andlostthroughthesupportbracketsbyconduction Q cond ; bracket aswell.Theheatlosscomingfromtheabsorber radiationandnaturalconvectionpassesthroughtheglassenvelopebyconductionand alongwiththeenergyabsorbedbytheglassenvelope Q e )]TJ/F59 8.9664 Tf 6.967 0 Td [(abs islosttotheenvironmentby convection Q e )]TJ/F59 8.9664 Tf 6.967 0 Td [(sa ; conv andtoskybyradiation Q e )]TJ/F59 8.9664 Tf 6.966 0 Td [(s ; rad [6].Inthisheattransfermodelthe absorbedradiationwasassumedasaheatuxterm.Inmostofthesurfacesthethickness ofthesurfacelayeroverwhichabsorptionistakingplaceisverysmallcomparedwiththe overalldimensions,thereforetheerrorassociatedwiththisassumptionisverylow[50,51]. Inordertoobtainthepartialdifferentialequationsthatgoverntheheattransferphenomena,anenergybalancewasappliedoverasectionofthesolarreceiverFigure3.1.The equationsobtainedforeachcomponentareshowedbelow. 3.2.1HeatTransferfromAbsorbertoHeatTransferFluid ReferringtothecontrolvolumeinFigure3.3,auniformtemperatureandheatux distributionaroundthereceiverisassumed.Afterapplyingtheenergybalanceonthis controlvolume,assumingunsteadystateandincompressibleuidwith[52], h C p T ,the followingpartialdifferentialequationPDEisobtained: A i ; a r f C p ; f T f t = )]TJ/F51 11.9552 Tf 12.212 0 Td [( m f z C p ; f T f + V 2 f 2 + Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv .1 56

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Figure3.3Controlvolumeoftheheattransferuid V f = m f r f A i ; a .2 where A i ; a istheinternalcrosssectionalareaoftheabsorber, A i ; a = p = 4 D 2 i ; a ,and Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv istheheattransferbyconvectionfromtheabsorbertoHTFperunitlength.Theconvective heattransferisgivenby: Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv = p Nu f k f )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(T a )]TJ/F59 11.9552 Tf 10.949 0 Td [(T f .3 where Nu f istheNusseltnumber, k f isthethermalconductivityofHTF,and T a isthe absorberwalltemperature.Twocasesareconsideredinthischapter:circularpipeand concentricannulus. 3.2.1.1CircularPipe Therecommendedcorrelation[53]forfullydevelopedturbulentow Re D > 2300 convectiveheattransferincircularductsisgivenbyGnielinskicorrelation[54]: Nu f = )]TJ/F59 11.9552 Tf 4.877 -9.69 Td [(C f = 2 Re D )]TJ/F51 11.9552 Tf 10.95 0 Td [(1000 Pr 1 + 12 : 7 )]TJ/F59 11.9552 Tf 4.877 -9.689 Td [(C f = 2 1 2 )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(Pr 2 = 3 )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 Pr Pr w 0 : 11 .4 57

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Re D = V f D i ; a n f where C f isthefrictioncoefcientFanningfrictionfactor, Re D istheReynoldsnumber basedontheabsorberinnerpipediameter, Pr isthePrandtlnumber,and n f isthekinematic viscosityofHTF.Thiscorrelationisvalidfor2300 Re D 5 10 6 and0 : 5 Pr 2000. Thethermalpropertiesshouldbeevaluatedatthebulkmeanheattransferuidtemperature, except Pr w whichisevaluatedattheabsorberwalltemperature.Thefrictioncoefcientis calculatedfromtheFilonenkocorrelation[53]forisothermalowsinsmoothtubes: C f = 1 : 58 LnRe D )]TJ/F51 11.9552 Tf 10.95 0 Td [(3 : 28 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 10 4 Re D 10 7 .5 TheconvectionheattransfercoefcientforroughtubescanbecalculatedapproximatelybyusingthefrictioncoefcientdeterminedfromtheColebrookandWhiteequation[53]: 1 p C f = 3 : 48 )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 : 7372 Ln 2 e D i ; a + 9 : 35 Re D p C f 5 Re e 70.6 with Re e = V f e n f .7 Thebestexplicitcorrelationforpracticalfrictioncoefcientcomputationsinarough circularductisgivenbyChen[53]: 1 p C f = 3 : 48 )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 : 7372 Ln 2 e D i ; a )]TJ/F51 11.9552 Tf 12.145 8.093 Td [(16 : 2426 Re D LnA 2 .8 A 2 = 2 e = D i ; a 1 : 1098 6 : 0983 + 7 : 149 Re D 0 : 8981 58

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for4000 Re D 10 8 and2 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 2 e = D i ; a 0 : 1.Forthetransitionow 2100 Re D 4000theformuladevelopedbyBhattiandShahcanbeusedtocalculate thefrictioncoefcient[55]: C f = 0 : 0054 + 2 : 3 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 Re )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 = 2 D .9 Sinceentrylengthsforturbulentowaretypicallyshort10 z = D 60,itisoften reasonabletoassumethattheaverageNusseltnumberfortheentiretubeisequaltothe valueassociatedwiththethermallyfullydevelopedturbulentow[56].However,forshort tubesthemeanNusseltnumberforthermallydevelopingowcanbecalculatedusingAlArabi'scorrelation[55]: Nu f ; m Nu f ; = 1 + C z = D .10 where Nu standsforthefullydevelopedNusseltnumberand: C = z = D 0 : 1 Pr 1 = 6 0 : 68 + 3000 Re 0 : 81 D .11 Forlaminarow, Re D 2300,theNusseltnumberonwallswithuniformtemperature isgivenby: Nu f = 3 : 66.12 Foracirculartubesubjectedtoconstantsurfacetemperature,theaverageNusseltnumberforthethermalentranceregioncanbedeterminedfromHausen'scorrelation[55]: Nu f = 3 : 66 + 0 : 0668 z 1 = 3 )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(0 : 04 + z 2 = 3 .13 59

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where z = z = D = Re D Pr .Thethermalpropertiesshouldbeevaluatedatthebulkmean uidtemperature.Thefullydevelopedconditionsarereachedfor[57]: z = D Re D Pr 0 : 05 Thefrictioncoefcientforfullydevelopedlaminarowinacircularductisgivenby: C f = 16 Re D .14 3.2.1.2ConcentricAnnulus TheNusseltnumberinthiscasecanbedeterminedfromasuitableturbulentowcorrelation[58,59]Equation.4byusingahydraulicdiameterof D h = D i ; a )]TJ/F59 11.9552 Tf 11.593 0 Td [(D plug PetukhovandRoizen[60]recommendtoincludethefollowingcorrectionfactortoimprovetheaccuracyofNusseltnumberobtainedfromGnielinskicorrelation: F = 1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(0 : 14 r 0 : 6 .15 ThefrictioncoefcientiscalculatedfromFilonenkocorrelation[53]forisothermal owsinsmoothtubes: C f = 1 : 58 LnRe D l )]TJ/F51 11.9552 Tf 10.95 0 Td [(3 : 28 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 .16 Inordertogetmoreaccuratefrictioncoefcients.Jonesetal.[61]recommendtousethe laminarequivalentdiameterforconcentricannularductsratherthanthehydraulicdiameter. Thelaminarequivalentdiameterisdenedas: D l D h = 1 + r 2 + )]TJ/F51 11.9552 Tf 5.476 -9.69 Td [(1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(r 2 = ln r 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(r 2 .17 60

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ForshortsegmentsthemeanNusseltnumberforthermallydevelopingowcanbecalculatedusingAl-Arabi'scorrelationEquation.10.Forlaminarow,Table3.1shows theNusseltnumberforfullydevelopedow. Table3.1Nusseltnumberforconcentricannulusunderlaminarow.Adaptedfrom[62] r Nu oo 0.003.6568 0.023.9934 0.054.0565 0.104.1135 0.254.2321 0.504.4293 1.004.8608 Forthecasewhentheuidisthermallydevelopingforafullydevelopedlaminarprole, Table3.2showsthevaluesobtainedoftheNusseltnumberfordifferentvaluesof r and x ,whicharedenedas: r = D i ; a D plug z h = z D h Re D h Pr Table3.2Nusseltnumberforconcentricannulusunderlaminarowfordevelopingtemperatureanddevelopedvelocityprole.Adaptedfrom[62] x z h 0.020.050.100.250.51.0 0.0105.2175.2875.3595.5185.7626.260 0.0154.7324.7964.8625.0055.2325.705 0.0254.2984.3594.4194.5484.7575.207 0.0504.0314.0934.1504.2694.4684.902 0.1003.9944.0574.1144.2334.4304.861 0.1503.9934.0574.1144.2324.4294.861 0.2503.9934.0574.1144.2324.4294.861 0.5003.9934.0574.1144.2324.4294.861 1.0003.9934.0574.1144.2324.4294.861 3.9934.0574.1144.2324.4294.861 61

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3.2.2HeatTransferfromAbsorbertoGlassEnvelope Twoheattransfermechanismsoccurbetweentheabsorberandtheglassenvelope:convectionheattransferandthermalradiation.Convectionheattransferdependsontheannuluspressure;experimentalworkhasshownthatheattransferlossesareindependentof theannulusvacuumpressureforpressuresabove1Torr[37].Atpressuresbelow1Torr, molecularconductionistheheattransfermechanismwhileforpressureabove1Torr,naturalconvectiontakesplace.AfterapplyingtheenergybalanceFigure3.4,thenextPDE isobtained: A a r a C p ; a T a t = A a z k a T a z + Q 0 a ; abs )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv )]TJ/F51 11.9552 Tf 12.81 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 cond ; bracket .18 where A a isthecrosssectionalareaoftheabsorber, A a = p = 4 D 2 o ; a )]TJ/F59 11.9552 Tf 10.95 0 Td [(D 2 i ; a Q 0 a ; abs is thesolarabsorptionintheabsorberperreceiverlength, Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv istheheattransferby convectionfromabsorbertoHTFperunitlength, Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv istheheattransferbyconvectionfromabsorbertoenvelopeperunitlength, Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad istheheattransferbyradiation Figure3.4Controlvolumeusedfortheabsorberanalysis 62

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fromabsorbertoglassenvelopeperunitlength,and Q 0 cond ; bracket istheheatconduction throughsupportbracketsperunitlength.Stainlesssteelisnormallyusedastheabsorber tubematerial.Table3.3presentsthepropertiesofthreestainlesssteelcommonlyusedas theabsorbertubematerial. Table3.3Thermalconductivity,densityandspecicheatfor304L,316Land321Hstainlesssteel,temperatureinC.Datatakenfrom[63] Material k W/mK r kg/m 3 C p kJ/kgK 304L0 : 0130 T + 14 : 97328027.170.5024 316L0 : 0130 T + 14 : 97328027.170.5024 321H0 : 0151 T + 14 : 58378027.170.5024 3.2.2.1VacuuminAnnulusP < 1Torr Heatconductioningasesatvariouspressuresoccursinfourdistinctregimes;these regimesaredeterminedbyKnudsennumberK n ,whichistheratioofthemolecularmean free-path, l ,toacharacteristicdimensionofthesystem L c .Atverylowpressures K n > 10 collisionsbetweenmoleculesarerelativelyrare;thisisknownasthefreemoleculeregime. Atnormalpressures K n < 0 : 01thegasmaybeassumedasacontinuum.Betweenthese extremesliethetransition : 1 < K n < 10andthetemperaturejump : 01 < K n < 0 : 1. TheKnudsennumberrangesareapproximatedsincetheyaresomewhatgeometrydependentandalsodependontheaccommodationcoefcients.Thegoalofevacuatingacollector istomaketheheatlossesbyconductionandconvectioninsignicant,thereforethevacuum intheHCEmustbeinthefreemoleculeregimeornearfreemoleculeconditions.The 63

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pressureassociatedwiththisregimeisapproximately0.0001Torr.013Pa[4,64].The heattransfercoefcientfortheannularspaceisgivenby[65]: h a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e = k g D o ; a 2 Ln D i ; e = D o ; a + b l D i ; e = D o + 1 .19 Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv = h a )]TJ/F59 8.9664 Tf 6.966 0 Td [(e p D o ; a T a )]TJ/F59 11.9552 Tf 10.949 0 Td [(T e .20 where k g isthemeanconductivityofthegasintheannularplaceevaluatedat T a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e = T a + T e = 2, D o ; a isthediameterofthereceivertube, D i ; e istheinnerdiameter oftheglassenvelope, T a isthetemperatureofthereceivertube, T e isthetemperatureofthe glassenvelope,and l isthemeanfreepathm.Thecoefcient b isdenedby[65]: b = 2 )]TJ/F54 11.9552 Tf 10.95 0 Td [(a a 9 g )]TJ/F51 11.9552 Tf 10.949 0 Td [(5 2 g + 1 .21 where a istheaccommodationcoefcient,and g istheratioofspecicheats.Themean freepathmiscalculatedbyusingthenextexpression[65]: l = 2 : 331 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(20 T a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e P d 2 .22 where T a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e istheaverageannulusgastemperatureK )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 P istheannulusgaspressure TorrormmHg,and d isthemoleculardiameterofannulusgascm.Themolecular diametersfordifferentgasesareshowninTable3.4. Experimentalstudieshavereportedthevaluesforthethermalaccommodationcoefcient, a ,from0.01tonearlyunity[68].Thisvaluedependsoneitherthegassurface arrangementorthelevelofcontaminantgaslayersadsorbedonthesurface.Qualitative theoreticalargumentspredictthatthermalaccommodationtendstoincreasewiththein64

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Table3.4Moleculardiameterofdifferentgases Gas 1 d 10 8 cm Air 3.66 2 Hydrogen 2.97 3 Argon 3.42 3 1 Themoleculardiameterswereobtained frommeasuredgasviscosity 2 [66] 3 [67] creasinggasmolecularweightandwithroughnessforagivensurface[69].Sincetheexact natureofthethermalaccommodationcoefcientisstillanactiveproblem;almostallthe evidenceindicatesthatformostgas-solidinteractionsavalueof a = 1couldbeassumed intheabsenceofwelldocumentedinformation[70]. 3.2.2.2PressureinAnnulusP > 1Torr Theconductionlayermodelhasshowntoaccuratelypredicttheheattransferforhorizontalcylinders[55].Kakaetal.[53]recommendacorrelationgivenbyKuehnand Goldstein[73].Thiscorrelationusesaniterativemethodtoobtainthemeanbulktemperatureandisbasedonextensiveexperimentalandnumericalheattransferresults.This numericalmodelassumesthattheconductionlayersdonotoverlap.Theconvectiveheat transferisgivenby: Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv = p k g Nu a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e T a )]TJ/F59 11.9552 Tf 10.95 0 Td [(T e .23 65

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Thecorrelationfortheconvectionpartiswrittenas[73]: Nu D i conv = hD i k g = 2 ln 1 + 2 = Nu i 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 = Nu o .24 with Nu i = h 0 : 518 Ra 1 = 4 D i f 2 Pr i 15 + 0 : 1 Ra 1 = 3 D i 15 1 = 15 .25 Ra D i = g b T a )]TJ/F51 11.9552 Tf 13.639 2.379 Td [( T b D 3 i n 2 Pr .26 where g istheEarth'sgravity, b isthevolumetricthermalexpansioncoefcient,and n is thekinematicviscosity.Foranidealgas: b = 1 T a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e Nu o = 0 B B @ 8 > < > : 1 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(e )]TJ/F51 6.9738 Tf 8.162 3.533 Td [(1 4 5 3 + h 0 : 587 f 3 Pr Ra 1 = 4 D o i 5 3 # 3 5 9 > = > ; 15 + 0 : 1 Ra 1 = 3 D o 15 1 C C A 1 15 .27 Ra D o = g b T b )]TJ/F59 11.9552 Tf 10.949 0 Td [(T e D 3 o n 2 Pr .28 with f 2 Pr = 1 + 0 : 559 Pr 3 = 5 # )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 = 12 f 3 Pr = 1 + 0 : 6 Pr 0 : 7 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 + )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(0 : 4 + 2 : 6 Pr 0 : 7 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 # )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 = 5 66

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Fortheconductionpart,whichprevailsastheRayleighnumberapproacheszero,the heattransferbyconductionisgivenby: Nu D i cond = 2 cosh D 2 i + D 2 o = 2 D i D o .29 TheoverallNusseltnumbervalidforanyRayleighnumberis: Nu a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e = h Nu D i conv 15 + Nu D i cod 15 i 1 = 15 .30 Fluidpropertiesareevaluatedattheaveragetemperatureof T = T a + T e = 2, D i = D o ; a and D o = D i ; e .Theaveragebulktemperatureisobtainedfrom: T b )]TJ/F59 11.9552 Tf 10.95 0 Td [(T e T a )]TJ/F51 11.9552 Tf 13.639 2.379 Td [( T b = Nu D i conv Nu D o conv .31 and Nu D i conv = 2 ln [ 1 + 2 = Nu i ] .32 Nu D o conv = 2 )]TJ/F51 11.9552 Tf 11.945 0 Td [(ln [ 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 = Nu o ] .33 3.2.2.3RadiationHeatTransferfromReceivertoEnvelope Thermalradiationanalysisforonesurfaceimpliesthatallsurfacesthatcanexchange radiativeenergywiththesurfacemustbeconsideredsimultaneously.Howmuchenergy twosurfacesexchangedependsontheirsize,separationdistance,andorientation[51].In ordertocarryouttheradiativeheattransferanalysissomeviewfactorsmustbecalculated. 67

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Figure3.5Annulusgeometry Theviewfactorsforashortannulus[51,74]Figure3.5aregivenby: F 2 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 = 1 R )]TJ/F51 11.9552 Tf 17.238 8.094 Td [(1 p R cos )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 B A )]TJ/F51 11.9552 Tf 16.151 8.094 Td [(1 2 L q A + 2 2 )]TJ/F67 11.9552 Tf 10.949 0 Td [( 2 R 2 cos )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 B RA + Bsin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 1 R )]TJ/F54 11.9552 Tf 12.145 8.093 Td [(p A 2 .34 F 1 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = D 0 D i F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 .35 F 1 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s = 1 2 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(F 1 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 .36 F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = 1 )]TJ/F51 11.9552 Tf 12.808 8.094 Td [(1 R + 2 p R tan )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 2 p R 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 L )]TJ/F59 11.9552 Tf 20.207 8.093 Td [(L 2 p R p 4 R 2 + L 2 L sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 4 )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(R 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 + )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(L 2 = R 2 )]TJ/F59 11.9552 Tf 13.608 -9.69 Td [(R 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 L 2 + 4 R 2 )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 # )]TJ/F59 11.9552 Tf 10.618 0 Td [(sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 R 2 + p 2 p 4 R 2 + L 2 L )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 .37 68

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F 2 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s = 1 2 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 .38 with R = R o = R i L = l = R i A = L 2 + R 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 B = L 2 )]TJ/F59 11.9552 Tf 10.95 0 Td [(R 2 + 1 TheviewfactorsforneighboringsurfacesFigure3.6onshellinteriorcoaxialcylinder areasfollows F 21 0 [75], F 22 0 [76]: F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 = 1 R d Z c 0 B a 3 = 2 L )]TJ/F54 11.9552 Tf 10.949 0 Td [(d tan )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 L )]TJ/F54 11.9552 Tf 10.949 0 Td [(d a 1 = 2 + L + d tan )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 L + d a 1 = 2 )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 Ltan )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 L a 1 = 2 d q .39 with R = R o = R i L = l = R i a = R 2 + 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 RCos q B = R p R )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 2 R 2 + 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(R 1 + Cos q )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 Sin 2 q 2 R 2 + 1 )]TJ/F59 11.9552 Tf 9.289 0 Td [(R 1 + Cos q )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 R 2 Sin 2 q 2 69

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Figure3.6Surfacesonacoaxialcylinder c = cos )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R i = R o d = d = R i F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 = 1 + N 2 F 22 d 2 + N )]TJ/F67 11.9552 Tf 10.949 0 Td [( 1 + N F 22 d 1 + N + N 2 F 22 d N .40 with N = d = d d = d = R i F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 x = F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 L = x F 2 0 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s = N + 2 F 2 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s d 2 + N )]TJ/F67 11.9552 Tf 10.95 0 Td [( N + 1 F 2 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s d 1 + N .41 with N = d = d 70

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d = d = R i F 2 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s x = F 2 s L = x F 1 0 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s = N + 2 F 1 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s d 2 + N )]TJ/F67 11.9552 Tf 10.95 0 Td [( N + 1 F 1 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s d 1 + N .42 with N = d = d d = d = R i F 1 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s x = F 1 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s L = x Theviewfactors F 1 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 0 F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 ,and F 2 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 0 F 1 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s werecalculatedfordifferentpointsFigure3.7.TheresultsareshowninFigure3.8andFigure3.9.Theresultsshowthatview factorsforclosestnodearehighbutasthenodaldistanceincreasestheviewfactorsapproachzero. aIntermediateposition bSideposition Figure3.7Nodepositionforcoaxialcylinders 71

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a F 1 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 b F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 c F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 d F 1 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s Figure3.8Viewfactorsforneighboringsurfacesonshellinteriorofcoaxialcylinders, R = 1 : 5 72

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a F 1 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 b F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 c F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 d F 1 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s Figure3.9Viewfactorsforneighboringsurfacesonshellinteriorofcoaxialcylinders, R = 2 : 0 73

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Theradiationheattransfercalculationissimpliedbyassumingtheglassenvelope isopaquetoinfraredradiationandassuminggraysurfaces a = e .Makinganenergy balancetocalculatetheradiativeheattransferratebetweensurfaces,thenextexpressionis obtained[50]: q i = R i )]TJ/F59 8.9664 Tf 15.446 12.774 Td [(N j = 1 R j F i )]TJ/F59 8.9664 Tf 8.312 0 Td [(j i = 1 ; 2 ; 3 ;::: N .43 where R i istheradiosity,whichisthetotalheatuxleavingthesurfacei.Radiosityis denedas[50]: R i = e i s T 4 i + r i N j = 1 R j F i )]TJ/F59 8.9664 Tf 8.312 0 Td [(j i = 1 ; 2 ; 3 ;::: N .44 where e i istheemissivityofsurfacei, F i )]TJ/F59 8.9664 Tf 8.311 0 Td [(j istheviewfactorbetweensurfaceiandj, q i is thenetradiativeheatuxonthesurfacei,and s istheStefanBoltzmannconstantequalto 5 : 67 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 W = m 2 K 4 Fortheparticularcaseofthethermalradiationbetweenthereceiverandtheenvelope, foursurfacesareincludedFigure3.5.Forsimplication,itisassumedthattheright sideandleftsideareadiabaticsurfaces,whichmeansthattheymayreectallincoming butdonotemitanyradiantheat.Theserequirementsaresatisedif r r = r l = 1 : 0and e r = e l = 0[77].Theradiativeheattransferrateandradiosityforeachsurfaceareshown below.Fortheexternalsurfaceofreceiver,surfaceaandnodei: R a i = e a i s T 4 a i + r a i N j = 1 F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j R e j + F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(l R l # .45 q a i = R a i )]TJ/F75 11.9552 Tf 10.949 20.449 Td [(" N j = 1 F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F a i )]TJ/F59 8.9664 Tf 6.966 0 Td [(e j R e j + F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(l R l # .46 74

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F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j = 0 F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j = F 1 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 0 F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r = F 1 0 )]TJ/F59 8.9664 Tf 6.966 0 Td [(s F a i )]TJ/F59 8.9664 Tf 6.966 0 Td [(l = F 1 0 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s Fortheinternalsurfaceoftheenvelope,surfaceeandnodei: R e i = e e i s T 4 e i + r e i N j = 1 F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F e i )]TJ/F59 8.9664 Tf 6.966 0 Td [(e j R e j + F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F e i )]TJ/F59 8.9664 Tf 6.966 0 Td [(l R l # .47 q e i = R e i )]TJ/F75 11.9552 Tf 10.949 20.449 Td [(" N j = 1 F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F e i )]TJ/F59 8.9664 Tf 6.966 0 Td [(e j R e j + F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F e i )]TJ/F59 8.9664 Tf 6.966 0 Td [(l R l # .48 F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j = F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j = F 2 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r = F 2 0 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s F e i )]TJ/F59 8.9664 Tf 6.966 0 Td [(l = F 2 0 )]TJ/F59 8.9664 Tf 6.967 0 Td [(s Fortherightside,surfacer: R r = N j = 1 F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F r )]TJ/F59 8.9664 Tf 6.966 0 Td [(e j R e j + F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(l R l # .49 F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j = F s )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 0 F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j = F s )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 0 F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(r = 0 F r )]TJ/F59 8.9664 Tf 6.967 0 Td [(l = F s )]TJ/F59 8.9664 Tf 6.967 0 Td [(s Fortheleftside,surfacel: R l = N j = 1 F l )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j + N j = 1 F l )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j R e j + F l )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r # .50 F l )]TJ/F59 8.9664 Tf 6.966 0 Td [(a j = F s )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 0 F l )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j = F s )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 0 F l )]TJ/F59 8.9664 Tf 6.966 0 Td [(r = F s )]TJ/F59 8.9664 Tf 6.967 0 Td [(s F l )]TJ/F59 8.9664 Tf 6.966 0 Td [(l = 0 Equations.45and.47canbewritteninacompactwaybyintroducingKronecker's deltafunction[51].TheKronecker'sdeltafunctionisdenedas[51]: d ij = 8 > > > < > > > : 1 i = j 0 i 6 = j .51 75

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Thesimpliedequationsareasfollows: N j = 1 d ij e a i R a j )]TJ/F54 11.9552 Tf 12.145 8.094 Td [(r a i e a i F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j R e j )]TJ/F54 11.9552 Tf 12.145 8.094 Td [(r a i e a i F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F a i )]TJ/F59 8.9664 Tf 6.967 0 Td [(l R l = s T 4 a i .52 N j = 1 R e j e e i )]TJ/F54 11.9552 Tf 5.476 -9.69 Td [(d ij )]TJ/F54 11.9552 Tf 10.95 0 Td [(r e i F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(e j )]TJ/F54 11.9552 Tf 12.145 8.094 Td [(r e i e e i F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(a j R a j )]TJ/F54 11.9552 Tf 12.145 8.094 Td [(r e i e e i F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(r R r + F e i )]TJ/F59 8.9664 Tf 6.967 0 Td [(l R l = s T 4 e i .53 Theheattransferlossbyradiationperunitlengthontheexternalabsorbersurfaceis: Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad = q a i p D o ; a .54 Theheattransfergainedbyradiationperunitlengthontheinternalenvelopesurfaceis givenby: Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad 0 = q e i p D i ; e .55 3.2.2.4HeatConductionThroughSupportBrackets Thesolarreceiverisplacedatthecollectorfocallinebysupportbracketssupportedon thecollectorstructureandlocatedateachendofeverysolarreceiver.Forthisanalysis, thesupportbracketsaredividedintotwosegmentsconnectedinseries.Therstsegment isaconnectiontabrectangularcrosssectionthatconnectsthebaseofthesolarreceiver andthemetallicsupportandthesecondsegmentisametallicsupportsquaretubethat connectstheconnectiontabtothecollectorstructureFigure3.10. 76

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Figure3.10Supportbracket.Adaptedfrom[6] Initially,aheatconductionanalysisforanwithaprescribedtemperatureatthetipis studied.Thetemperaturedistributioninthenisgivenby[78]: q q b = sinh m L )]TJ/F59 11.9552 Tf 10.949 0 Td [(x + q L = q b sinh mx sinh mL .56 with q = T )]TJ/F59 11.9552 Tf 10.95 0 Td [(T m 2 = h b P k b A c where q b istheexcesstemperatureatthebaseofthen, q L istheexcesstemperatureatthe tipofthen, P isthenperimeter,and A c isthecrosssectionalareaofthen.Forthis conguration,Itwillbeassumedthattheconvectiveheattransfercoefcientisuniform 77

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overthelengthofthen.Usingthisapproximation,thetemperatureattheintersection pointisobtainedas: q a q b = 1 cosh m 1 L ct + z 2 z 1 sinh m 1 L ct .57 with z i = p h b k b P i A c ; i Thebasetemperatureisassumedtobetheabsorbertemperatureatthepointwherethe bracketislocated. q b = T a )]TJ/F59 11.9552 Tf 10.949 0 Td [(T Theheatlossesbyconductionaregivenby[78]: Q cond )]TJ/F59 8.9664 Tf 6.967 0 Td [(bracket = 2 z 1 q b cosh m 1 L ct )]TJ/F54 11.9552 Tf 12.145 8.094 Td [(q a q b sinh m 1 L ct .58 and Q 0 cond ; bracket = Q cond ; bracket = L b .59 where L b isthelengthbetweenthesupportbrackets.Basedonthegeometricinformation givenintheheattransfermodeldevelopedbyForristall[6],theconnectiontabhasalength intherangeof25 : 4 L ct 50 : 8mm.AcomparisonoftheresultsobtainedfromEquation.58,forconnectiontabswithdifferentlengths,andtheconductionmodelusedby Forristall[6]isshowninFigure3.11. 78

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Figure3.11Comparisonoftheheatlossesthroughsupportbracketsfordifferentconnection tablengths,andthemodelusedbyForristall[6] TheresultsshowthatthemodelusedbyForristallismoreconservative,whichmeans higherheatlossesfromsupportbrackets.Forthecurrentconductionanalysis,aconnection tabwithalengthof25.4mmwillbeassumed,thisassumptionisbasedonthemaximum heattransferlossesobtainedfromEquation.58.Forsimplication,theconvectiveheat transfercoefcientiscalculatedforthesquaretubebecausethispartofthesupportbracket hasmorethan99%oftheareaexposedtotheenvironment.Theconvectivecoefcientis obtainedfrom: h b = Nu L k b L .60 79

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Thethermalpropertiesofairareevaluatedatthelmtemperature[78] T f = T a + T = 2 andtheaveragesurfacetemperatureofthenis: T fin T a + m 2 L )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 T m 2 L .61 Thethermalconductivity, k b kW/m C ,forthesupportbracket,plaincarbonsteel,at certainlmtemperatureCwasobtainedbyttingthedata[56]toastraightline R 2 = 0 : 998: k b = )]TJ/F51 11.9552 Tf 9.289 0 Td [(0 : 0419 T f + 73 : 2357.62 Twoconvectiveheattransfermodesareusedtocalculatetheheattransfercoefcient: naturalconvection V wind = 0andforcedconvection V wind > 0. Forlongcylindersinstillair,heatlossesbynaturalconvectionaregreaterinorclose tothehorizontal,thanininclined[79,80],becausewhentubesareinclinedtheabnormal owpatterndecreasestheheattransfer,althoughthisdifferenceisnegligibleforacylinderinverticalposition.Itisassumedthatthepreviousanalysiscanbeextrapolatedfornon circularcylinders;thisassumptionisparticularlytrueforlongbodiesofarbitrarycrosssection[81],inwhichthegeometrycongurationcanbeapproximatedasacylindricalshape. Forthisanalysis,theconvectiveheattransfercoefcientiscalculatedinthehorizontalpositionsincethispositionleadstothehighestvalues.Forasquarecrosssectionalareathe averageNusseltnumberisgivenby[55]: Nu L =[ Nu l m + Nu t m ] 1 = m .63 Nu T = G C L Ra 1 = 4 .63a 80

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Nu l = C 2 ln 1 + C 2 = Nu T .63b Nu t = C t Ra 1 = 3 .63c Theconstant C L isdenedasfollows[55]: C L = 0 : 671 h 1 + 0 : 492 = Pr 9 = 16 i 4 = 9 .64 Theothersconstants G C 2 C t and m areshowninTable3.5.Giventhegeometryofthe supportbracket,aninclinationangleof0wasselectedforcalculatingtheaverageNusselt number. Table3.5ConstantsforuseinEquation.63forlonghorizontalsquarecylindersinan isothermalenvironment[55].OriginaldatatakenfromClemensetal.[82]forair.Correlationisvalidatedfor10 3 Ra 10 8 Geometry q GC 2 C t Pr = 0 : 71 m Nu = hL = k Ra = g b D TL 3 na P = 4 L 0 15 30 45 0.735 0.720 0.786 0.797 1.3 1.3 1.3 1.3 0.087 0.102 0.106 0.108 4.5 4.5 4.5 4.5 TheNusseltnumberforasquarecylindersubjectedtoacrossowofairisdened by[83]: Nu L = CRe m L .65 81

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Theconstants C m recommendedbySparrowetal.[83]foruseinEquation.65are showninTable3.6.Basedonthepreviouscorrelation,itcanbedemonstratedthat[83]: h square h diamond = 0 : 6 U L n 0 : 07 .66 5000 U L n 42000 Itwasfoundthatsquareorientationhashigherconvectiveheattransfercoefcientsthan thediamondorientation,thereforeasquareorientationwasselectedfortheheattransfer analysisunderforcedconvection. Table3.6ConstantsforuseinEquation.65forlonghorizontalsquarecylinders[84] subjectedtoacrossowofair Geometry Re L C m Square 5000 )]TJ/F51 11.9552 Tf 12.277 0 Td [(600000.14 0.666 Diamond 6000 )]TJ/F51 11.9552 Tf 12.277 0 Td [(600000.27 0.59 3.2.3HeatTransferfromGlassEnvelopetotheAmbient Theheattransferfromtheglassenvelopetothesurroundingsisbyconvectionandradiation.Convectionheattransferhastwocases:withwindforcedconvectionandno windnaturalconvection.Theradiationheattransferisbasicallybetweentheglassenvelopeandeithertheskyorthecollectorsurface,butthemaximumradiationheatlossestake placewhenthesolarreceiverisassumedtobesurroundedonlybytheskysurface. 82

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TheenergybalanceonthecontrolvolumeFigure3.12leadstothenextPDE: A e r e C p ; e T e t = A e z k e T e z + Q 0 e ; abs + Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv + Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad 0 )]TJ/F51 11.9552 Tf 12.81 2.653 Td [( Q 0 e )]TJ/F59 8.9664 Tf 6.967 0 Td [(sa ; conv )]TJ/F51 11.9552 Tf 14.47 2.653 Td [( Q 0 e )]TJ/F59 8.9664 Tf 6.967 0 Td [(s ; rad .67 where Q 0 e ; abs isthesolarabsorptionintheenvelopeperreceiverlength, Q 0 e )]TJ/F59 8.9664 Tf 6.966 0 Td [(sa ; conv isthe heattransferbyconvectionfromtheglassenvelopetothesurroundingairperunitlength, and Q 0 e )]TJ/F59 8.9664 Tf 6.967 0 Td [(s ; rad 0 istheheattransferbyradiationfromtheglassenvelopetotheskyperunit length. Figure3.12Controlvolumeofglassenvelope Borosilicateglassisusedcommonlyastheglassenvelopematerial.Thethermalconductivityandthevolumetricheatcapacity r C p arecalculatedbyusingapolynomialt obtainedfrom[85]: k e = k e ; o i a i T 298 : 15 i .68 83

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r C p e = r C p e ; o i b i T 298 : 15 i .69 ThecoefcientsfortheseequationsarepresentedinTable3.7. Table3.7Polynomialcoefcientsforthermalconductivityandvolumetricheatcapacity. Datatakenfrom[85] CoefcientValueCoefcientValue k e ; o 1.15 r C p e ; o 1770 a 0 0.7688 b 0 0.8716 a 1 0.2158 b 1 0.1634 a 2 0.0157 b 2 -0.035 3.2.3.1HeatConvection Theheattransferbyconvectionperunitlengthfromtheglassenvelopetothesurroundingairiscalculatedas: Q 0 e )]TJ/F59 8.9664 Tf 6.967 0 Td [(sa ; conv = h e p D o ; e T e )]TJ/F59 11.9552 Tf 10.95 0 Td [(T .70 with h e = Nu e k e D o ; e .71 Fornowindconditions,theexpressiongivenbyChurchillandChi[53]isrecommended forhorizontalcylinderundernaturalconvection: Nu e = 2 6 6 6 4 0 : 60 + 0 : 387 8 > > < > > : Ra D h 1 + 0 : 559 = Pr 9 16 i 16 9 9 > > = > > ; 1 6 3 7 7 7 5 2 .72 84

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Forwindconditions,theaverageNusseltnumberrecommendedforacylinderincross ow[53]isgivenby: Nu e = cRe m D Pr n Pr Pr w p .73 TheconstantssuggestedforthisequationaretabulatedinTable3.8.Thevalueof p depends ontheheatuxdirection: p = 0 : 25foruidheatingand p = 0 : 2foruidcooling.Fluid propertiesareevaluatedattheaveragetemperatureof T = T o + T = 2,except Pr w which isvaluatedatthewalltemperature. Table3.8ConstantsforEquation.73foracylinderincrossow[53] cmn Re D 0.760.40.371 )]TJ/F51 11.9552 Tf 10.95 0 Td [(4 10 1 0.520.50.374 10 1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(10 3 0.260.60.3710 3 )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 10 5 0.0230.80.42 10 5 )]TJ/F51 11.9552 Tf 10.95 0 Td [(10 7 3.2.3.2RadiationHeatTransferSkyandCollectorSurface Inthisanalysisthesolarreceiverissurroundedbyeitherthecollectororthesky.In ordertosimplifythemodel,itisassumedthathalfofthereceiversurfaceissurrounded bythemirrorandtheotherhalfbytheskyFigure3.13.Theheatuxandradiosityare calculatedforeachsurface.Fortheexternalsurfaceoftheenvelopeatnodei: q es i = R es i )]TJ/F59 11.9552 Tf 10.949 0 Td [(F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky )]TJ/F59 11.9552 Tf 10.949 0 Td [(F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(rs R rs )]TJ/F59 11.9552 Tf 10.949 0 Td [(F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(ls R ls )]TJ/F59 11.9552 Tf 10.95 0 Td [(F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .74 Fortheskysurface: q sky = R sky )]TJ/F59 8.9664 Tf 14.662 12.774 Td [(N i = 1 F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(es i R es i )]TJ/F59 11.9552 Tf 10.95 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky )]TJ/F59 11.9552 Tf 10.949 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(rs R rs )]TJ/F59 11.9552 Tf 10.949 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.966 0 Td [(ls R ls )]TJ/F59 11.9552 Tf 10.95 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .75 85

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Figure3.13Zoneanalysisoftheradiationheatlossfromthereceivertotheambient Forthecollectorsurface: q c = R c )]TJ/F59 8.9664 Tf 14.661 12.774 Td [(N i = 1 F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(es i R es i )]TJ/F59 11.9552 Tf 10.949 0 Td [(F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky )]TJ/F59 11.9552 Tf 10.949 0 Td [(F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(rs R rs )]TJ/F59 11.9552 Tf 10.95 0 Td [(F c )]TJ/F59 8.9664 Tf 6.966 0 Td [(ls R ls )]TJ/F59 11.9552 Tf 10.95 0 Td [(F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .76 Theviewfactor F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(es j isgivenby: F sky )]TJ/F59 8.9664 Tf 6.966 0 Td [(es j = F es j )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky A es j A sky .77 where A es j istheareaofanalysisenvelope,and A sky istheskyarea.Simplifying: F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(es j = F es j )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky 2 D z L D o ; e D c .78 86

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where L isthelengthofthecollector.Since D z = L and D o ; e = D c andareverysmall, D z = L D o ; e = D c 0, F sky )]TJ/F59 8.9664 Tf 6.966 0 Td [(es j approacheszero F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(es j 0.Usingtheprevioussimplication,itcanbedeterminedthat: F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(rs 0.79 F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(ls 0.80 Theviewfactors F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky and F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c foraninnitelylongcylinderarecalculatedbythe followingexpressions[86]: F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky = 1 )]TJ/F51 11.9552 Tf 12.922 8.093 Td [(2 p h )]TJ/F51 11.9552 Tf 5.476 -9.69 Td [(1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(R 2 1 = 2 + R sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R i .81 F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c = 2 p h )]TJ/F51 11.9552 Tf 5.475 -9.689 Td [(1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(R 2 1 = 2 + R sin )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R i )]TJ/F59 11.9552 Tf 10.95 0 Td [(R .82 and R = D o ; e = D c Figure3.14showsthevariationof F sky )]TJ/F59 8.9664 Tf 6.966 0 Td [(sky and F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c withtheparameter R .Forthisparticularanalysis R issmall,andtherefore F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky and F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c canbeapproximatedby: F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky 1 )]TJ/F51 11.9552 Tf 12.922 8.094 Td [(2 p .83 F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c 2 p .84 87

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Figure3.14Skyviewfactors, F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky and F sky )]TJ/F59 8.9664 Tf 6.966 0 Td [(c Thevaluesoftheviewfactorsforthesolarcollector, F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky and F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(c ,arecalculatedby usingsymmetry. F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(c 1 )]TJ/F51 11.9552 Tf 12.922 8.093 Td [(2 p .85 F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky 2 p .86 Theskyisoftenconsideredasaglobalblackbody[30,87],denedintermsofanequivalentskytemperature[88],whichisnottruebutusefulforpracticalcalculationsofheat exchangebetweentheskyandanysurfaceatgroundlevel.Usingthelastapproximation, e sky = 1,equationsforskyandcollectorsurfacecanbesimpliedas: q sky = R sky )]TJ/F59 11.9552 Tf 10.95 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky )]TJ/F59 11.9552 Tf 10.949 0 Td [(F sky )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .87 q c = R c )]TJ/F59 11.9552 Tf 10.95 0 Td [(F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky )]TJ/F59 11.9552 Tf 10.95 0 Td [(F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .88 88

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R sky = s T 4 sky .89 R c = e c s T 4 c + 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(e c F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky R sky + F c )]TJ/F59 8.9664 Tf 6.967 0 Td [(c R c .90 AddingEquations.87and.88fortheskyandthecollectorsurface,thefollowing equationisobtained: q sky + q c = 0.91 SubstitutingEquation.91intoEquation.87,theheatuxfortheskyisobtained: q sky = s T 4 sky )]TJ/F59 11.9552 Tf 10.95 0 Td [(T 4 c p 2 + 1 e c )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 .92 TheheatuxfortheareaofanalysiscanbecalculatedbysubstitutingEquation.92 intoEquation.74.Afterseveralsimplicationsthefollowingexpressionisobtained: q es i e es i = s T 4 es i )]TJ/F54 11.9552 Tf 10.95 0 Td [(s T 4 sky F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky x c + 1 + F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(rs + F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(ls )]TJ/F54 11.9552 Tf 10.95 0 Td [(s T 4 c F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky [ 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(x c ] .93 with x c = 1 e c )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 p 2 + 1 e c )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 .94 F es i )]TJ/F59 8.9664 Tf 6.967 0 Td [(sky = 1 2 N j = 1 F 1 i )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 0 j .95 Inthisanalysis,itisassumedthatthecollectormirrortemperatureisapproximatelythe ambienttemperature[91].Forthecasewhenthecollectorsurfaceisnotincludedinthe analysismaximumheattransferlossesbyradiation[6],whichmeansthatglassenvelope 89

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isassumedtobetotallycoveredbytheskysurface,theheatuxfortheareaofanalysisis givenby: q es i = se es i T 4 es i )]TJ/F59 11.9552 Tf 10.95 0 Td [(T 4 sky .96 Thelastequationissimilartothatusedinotherheattransfermodels[8,13,21,41,92]. Theheattransferlossbyradiationonexternalglassenvelopesurfaceperunitlengthis: Q 0 e )]TJ/F59 8.9664 Tf 6.966 0 Td [(e ; rad = q es i p D o ; e .97 Severalrelationshavebeenproposedtorelate T sky ,forclearskies,toothermeasured meteorologicalvariables.Intheabsenceofmeteorologicaldatasuchas:relativehumidity, dewpointtemperature,etc,asimplerelationgivenbySwinbank[93]maybeused: T sky = 0 : 0553 T 1 : 5 .98 3.2.4SolarEnergyAbsorption Inordertocalculatetheheattransferlossesthroughthesolarreceiver,opticalefciencytermsandsolarradiationabsorptionaredetermined.Theenergyabsorbedinthe solarreceiverisaffectedbytheopticalpropertiesandimperfectionsofthesolarcollector ensemble.Theimperfectionsineitherthereectororshapeoftheconcentratorareaccountedbytheinterceptfactor, g ,whichisafractionofthedirectsolarradiationreected bymirrorsthatdoesnotreachtheglasscover[3].Thefactorsthataffecttheinterceptfactor are[6,13]: Heatcollectionelementshadowingbellows,shielding,supports, g 1 Twistingandtrackingerror, g 2 90

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Geometryaccuracyofthecollectormirrors, g 3 Cleanmirrorreectivity, r cl t 0 : 935[6] Mirrorclearness, g 4 Dirtonheatcollectionelement, g 5 Miscellaneousfactors, g 6 Thevaluesfor g i areshowninTable3.9.Theinterceptfactoristhendenedas: g = 6 i = 1 g i .99 Table3.9Effectiveopticalefciencyterms.Adaptedfrom[6,13,15] FactorandOpticalpropertiesValue LuzBlackChrome g 1 0.974 LuzCermet g 1 0.971 Twistingandtrackingerror g 2 0.994 Geometryaccuracyofthecollectormirrors g 3 0.980 Mirrorclearness g 4 0.950 DirtonHCE g 5 0.980 Miscellaneousfactor g 6 0.960 Foraconcentratingcollector,theeffectiveopticalefciencyisdenedaslongasthe directbeamradiationisnormaltothecollectoraperturearea.Whenthebeamradiation isnotnormal,afactorcalledincidentanglemodier, K i ,isincludedtoaccountforall opticalandgeometriclossesduetoanincidentanglegreaterthan0[3].Theincidentangle 91

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modier,dependsonthegeometryandtheopticalcharacteristicsofthesolarcollector.The incidentanglemodierisdenedas[94]: K = h o i h o i = 0 .100 Theincidentanglemodierfunctionisdenedby: K = max 0 ; K i .101 Theincidentanglemodierdependsonthegeometryandopticsoftheconcentrator.Table3.10showstheincidentanglemodierfunctionfordifferentsolarcollectors.These functionswereplottedandareshowninFigure3.15. Table3.10Incidentanglemodierfordifferentsolarcollectors SolarCollectorIncidentAngleModierfunction 1 K i LS-2[39]1 + 0 : 000884 i cos i )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 : 00005369 i 2 cos i LS-3[3] 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 : 2307 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 i )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 : 1 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(4 i 2 + 3 : 18596 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(6 i 3 )]TJ/F51 11.9552 Tf 10.95 0 Td [(4 : 85509 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 i 4 IST[95]1 + 0 : 0003178 i cos i )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 : 00003985 i 2 cos i EuroTrough[16,96]1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(5 : 25097 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 i cos i )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 : 859621 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 i 2 cos i 1 i incidentangleindegrees Anothergeometricfactoriscalledthecollectorgeometricalendlosses, y i .This factoraccountsforthefractionofareceiverlengthwhichisnotilluminatedbytherays 92

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Figure3.15Incidentanglemodierfordifferentsolarcollectors.LS-2[39],LS-3[3], IST[95]andEuroTrough[16,96] incidentontheaperture[94,97].AsitisshowninFigure3.16,thepartofthereceiverthat isnotilluminated z isasfollows: z = r tan i .102 Thedistance r isshowninFigure3.17andisdenedby[12]: r = f + x 2 4 f .103 Figure3.16Collectorgeometricalendlosses 93

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Figure3.17Parabolageometryforarimangleof j m .Adaptedfrom[30] Thefractionofthereceiverthatisilluminatedis: y i = 1 )]TJ/F59 11.9552 Tf 15.245 8.094 Td [(r L c tan i .104 Lippkeetal.[5]proposedtotake r = f .Thisassumptioniswidelyusedinothermodels [8,13]andleadstominimumendlossesforcertaingeometricconguration.Aprevious workdevelopedbyGaulandRabl[94]suggeststheuseofanaveragevalueof r .This valueisusedinthepresentworkandisgivenby: r = 1 w = 2 Z w = 2 0 f + x 2 4 f dx = f 1 + w 2 48 f 2 .105 Replacingthevalueof r ,then,thecollectorgeometricalendloss[94,97]is: y i = max 0 ; 1 )]TJ/F59 11.9552 Tf 16.07 8.094 Td [(f L c 1 + w 2 48 f 2 tan i .106 94

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Figure3.18comparestheendlossfactorforthemodelofLippkeetal.[5]andthemodel ofGaulandRabl[94]fortwocollectorgeometries.ThemodelofGaulandRablshows lowerendlossfactorthanthemodelofLippke,whichmeansthatahigherfractionofthe receiverisnotilluminated. Figure3.18Endlossfactorfordifferentcollectorsandassumptions.Lippke[5], GaulandRabl[94] Thepeakopticalefciencyoftheparabolictroughcollectoris[3]: h o = r cl g t e a a n .107 where t e istheenvelopetransmittance,and a a isthecoatingabsorptance.Duetorereectionsandsubsequenttransmissions,amodiedvalueof t e a a n as1 : 01 t e a a n is recommendedbyStuetzle[8].Table3.11showstheradiativepropertiesoftheheatcollectionelement,andtheenvelopetransmittance,itisassumedthatboththeenvelopetransmittanceandthecoatingabsorptanceareindependentofthetemperature.Inordertoaccount fortheeffectoftemperatureontheradiationheatlossesthroughthesolarreceiver,coat95

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ingemittancewasapproximatedbyapolynomialfunctionobtainedbyForristall[6].The polynomialcoefcientsareshowninTable3.12. Otherimportantpropertiesforsolarradiationlossesaretheglassenvelopeabsorptance andemissivity.Inthismodelsitisassumedthatbothoftheseradiativepropertiesare independentofthetemperature;theenvelopeabsorptancehasavalueof a e = 0 : 023[100], whiletheemissivityhasanaveragevalueof e e = 0 : 90[100].Figure3.19showsthevalues Table3.11RadiativepropertiesofdifferentheatcollectionelementsHCE.Adaptedfrom [6] SelectiveCoating Envelope Transmittance Coating Absorptance Coating Emittance 100C 400C LuzBlackChrome0.9350.940 0.110 0.27 LuzCermet0.9350.920 0.060 0.15 SolelUVACCermeta0.9650.960 0.070 0.13 SolelUVACCermetb0.9650.950 0.080 0.15 SolelUVACAvg0.9650.955 0.076 0.14 SchottPTR[98]0.960.95 0.1 Table3.12Coatingemittanceofdifferentsolarreceivers.Adaptedfrom[6] SelectiveCoatingCoatingEmittance 1 LuzBlackChrome5 : 333 10 )]TJ/F115 8.9664 Tf 6.967 0 Td [(4 T + 273 : 15 )]TJ/F51 11.9552 Tf 10.95 0 Td [(0 : 0856 LuzCermet3 : 27 10 )]TJ/F115 8.9664 Tf 6.967 0 Td [(4 T + 273 : 15 )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 : 065971 SolelUVACCermeta2 : 249 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(7 T 2 + 1 : 039 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 T + 0 : 05599 SolelUVACCermetb 1 : 565 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(7 T 2 + 1 : 376 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 T + 0 : 06966 SolelUVACAvg 1 : 907 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(7 T 2 + 1 : 208 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 T + 0 : 06282 SchottPTR 2 [99] 2 : 00 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(7 T 2 + 0 : 062 1 TemperatureinC 2 Atanabsorbertemperatureof400Ctheemittanceuncertaintywas0.005 96

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Figure3.19Effectoftemperatureontheemissivityofborosilicateglassfortwothicknesses .35and12.7mm.Valuestakenfrom[100] oftheemissivityforborosilicateglassatdifferentthicknesses.Theenergyabsorbedonthe solarreceiverandtheglassenvelopeisgivenby: Q a )]TJ/F59 8.9664 Tf 6.967 0 Td [(abs = h o K i j i SHI 0 bn .108 and Q e )]TJ/F59 8.9664 Tf 6.967 0 Td [(abs = r cl ga e K i j SH j i I 0 bn .109 where j SH istheshadingfactor,with j SH = 0forthecollectorapertureareatotallyshaded. 3.3NumericalSolution ThepartialdifferentialequationsPDEwerediscretizedforsteadystateconditionsby usingthenitedifferencemethodandtakingintoaccountthedependenceofthethermal 97

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propertiesonthetemperature[101].Fortheheattransferuid,thediscretizationbyusing backwarddifferencegavethefollowingalgebraicequation: g i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 T f ; i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F54 11.9552 Tf 10.95 0 Td [(b i T f ; i + t i = 0 i = 1 ; 2 ;::: N .110 with g i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 = m f D z C p ; f ; i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 b i = m f D z C p ; f ; i + p Nu f ; i k f ; i t i = m f D z V 2 f 2 i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F75 11.9552 Tf 10.95 20.45 Td [( V 2 f 2 i # + p Nu f ; i k f ; i T a ; i Boundarynodes: i = 0 ; T f ; 0 = T e .111 Fortheabsorber,thediscretizationwascarriedoutbyusingcentraldifference.The algebraicequationobtainedwasasfollows: a i T a ; i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(a + i T a : i + a 0 i T a ; i + 1 + l q ; i = 0 i = 1 ; 2 ;::: N .112 with a i = A a k a ; i D z 2 a 0 i = A a k a ; i + 1 D z 2 a + i = a i + a 0 i 98

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l q ; i = j Q 0 a ; j i j Q 0 a ; j = Q 0 a ; abs )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 8.312 0 Td [(f ; conv )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q 0 cond ; bracket Boundarynodes: i = 1 ; T a ; 0 = T a ; 1 .113 i = N ; T a ; N + 1 = T a ; N .114 Fortheenvelope,usingcentraldifference,itwasobtainedthat: G i T a ; i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F66 11.9552 Tf 10.949 0 Td [(G + i T a : i + G 0 i T a ; i + 1 + D q ; i = 0 i = 1 ; 2 ;::: N .115 with G i = A e k e ; i D z 2 G 0 i = A e k e ; i + 1 D z 2 G + i = G i + G 0 i D q ; i = j Q 0 e ; j i j Q 0 e ; j = Q e ; abs + Q a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; conv + Q a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad 0 )]TJ/F51 11.9552 Tf 14.47 2.654 Td [( Q e )]TJ/F59 8.9664 Tf 6.966 0 Td [(sa ; conv )]TJ/F51 11.9552 Tf 14.471 2.654 Td [( Q e )]TJ/F59 8.9664 Tf 6.967 0 Td [(s ; rad Boundarynodes: i = 1 ; T e ; 0 = T e ; 1 .116 i = N ; T e ; N + 1 = T e ; N .117 99

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ThenonlinearalgebraicequationswerewritteninPython2.6[34]andsolvedsimultaneouslybyusingawrapperaroundMINPACK'shybrdandhybrjalgorithms[102]. Thecollectorefciencyandthermallossesarecalculatedasfollows: h c = m )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(h o ; f )]TJ/F59 11.9552 Tf 10.95 0 Td [(h i ; f I b A ap .118 ThermalLoss = i Q 0 a )]TJ/F59 8.9664 Tf 6.966 0 Td [(e ; conv i + i Q 0 a )]TJ/F59 8.9664 Tf 6.967 0 Td [(e ; rad i + j Q 0 cond ; bracket j # D z I b A ap .119 3.4ModelValidation Inordertovalidatetheheattransfermodel,itwascomparedwithexperimentaldata obtainedfromSandiaNationalLaboratoriesSNL[39].Inaddition,tocorroboratethe improvementinthecorrelationsandradiationanalysisproposedinthischapter,thenumericalmodelwasalsocomparedwiththeothersolarreceiverheattransfermodels[6,41]. TheexperimentalresultswereforasolarcollectorassemblyLS-2moduleplacedatthe AZTRAKrotatingplatformattheSNL.Duetolimitationsintheexperimentalsetup,a 2inchdiameterowrestrictiondevicesolidplugwascenteredintheinsidediameterof theabsorbertube.Twodifferentselectivecoatingswereusedinthistest:blackchrome andcermet.Cermethasbetterradiativepropertieslowemissivityathightemperatures thanblackchrome,anddoesnotoxidizeifthevacuumislost[39].TheSandiatestwas performedforbothfullsunandnosunconditionanddifferentscenariosfortheannulus oftheheatcollectionelementHCE:vacuumintacttheevacuatedannuluspressurewas 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 Torr,lostvacuumannuluslledwithambientair,andglasscovercompletelyremovedbaretube.Thethreepreviousconditionsweretestedwiththecermetcoatingbut brokenannuluswasnotincludedfortheblackchromecoatingcase.Alltheconditionsand 100

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Table3.13SpecicationsforaSEGSLS-2parabolictroughsolarcollectortest.Datataken from[39] ModuleSize7.8 5m Rimangle70 Reectors12thermallysaggedpanels Secondsurfacesilvered Lowironglass e c = 0 : 86[103] ApertureArea39.2m 2 FocalLength1.84m ConcentrationRatio71 ReceiverHCEEvacuatedtubedesign,metalbellowsateachend Absorberdiameter:70mm Length:4mpermoduleatSandia Pyrexglassenvelope:115mmdiameter Selectivecoating:CermetandBlackchrome i = 0, K i = 1, j i = 1 ConditionsofHCEAnnulusVacuumintacttheevacuatedannuluspressurewas10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 Torr Lostvacuumannuluslledwithambientair Glasscovercompletelyremovedbaretube AtmosphericAirPressure0.83atm HeatTransferFluidSyltherm800 specicationsusedintheexperimentaltestaresummarizedinTable3.13.Asiliconeheat transferuidSyltherm800wasusedintheexperimentalsetup.Thepropertiesforthis workinguidwerettedtotheexperimentaldatafrom[104]andapolynomialregression wasobtainedforeachproperty.Theequationsusedforthisttingwere: C p ; s 800 = 3 n = 0 C p ; n T n .120 r s 800 = 3 n = 0 r n T n .121 101

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k s 800 = 3 n = 0 k n T n .122 ThecoefcientobtainedbythepolynomialregressionareshowninTable3.14. Table3.14Coefcientsobtainedbypolynomialregressionofthermalpropertiesof Syltherm800.Experimentaldatatakenfrom[104] Property Coefcients, n R 2 0123 C p ; n 1.5741.707E-030.0000.0000.99 r n 953.164-0.9164.211E-04-1.670E-060.99 k n 0.138-1.880E-040.0000.0000.99 Alltheresultsinthischapterwereobtainedbydividingthesolarreceiverinto12segments, D x = 2 = 3m.Thisnumberofsegmentswasselectedaccordingtoagrid-independent solutionanalysis.Figure3.20showstheresultsobtainedfordifferentcollectorsegments Figure3.20Gridindependentanalysisfordifferentcollectorsegments,case:airinthe annulus 102

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withcermetcoatingasselectivecoatingandairintheannulus.Theresultsshowthat12 elementsaresuitableforvalidationoftheproposedheattransferanalysis. 3.5ResultsandDiscussion TheresultsobtainedforthecollectorefciencyandthermallossesareshowninFigures 3.21-3.23.ForthecollectorefciencyFigure3.21,themodelfollowsthetrendsofthe experimentalvaluesandalltheresultsarealwaysinsidetheexperimentalerrorbars.Asit wasexpected,thehigherefcienciesareobtainedwhentheannulusisundervacuum,butin bothcasesairandvacuumathightemperaturesthecollectorefciencydropsgradually, whichismorenotablefortheblackchromecoatingasitisshowninFigure3.21b.This isexplainedbytheradiativepropertiesofblackchromecoatingathightemperatures.For thecaseofcermetcoating,Figure3.21a,themodeldevelopedbyGarcia-Valladaresand Velsquez[41]showssomediscrepanciesatlowtemperaturesinthecollectorefciency duetotheirassumptions:negligibleconductionattheendsofeachtroughandonlyradiationheatlossestakeplacebetweenthereceiverandtheglassenvelopeforthecaseof vacuumintheannularspacereferto[41]formoredetails.Thoseassumptionswerenot madeinthecurrentmodelorNRELmodel.TheproposedmodelandNRELmodel[6] seemtoobtainsimilarcollectorefcienciesvalues,butadetailedrootmeansquareerror calculationTable3.15and3.16showsthattheproposedmodelachievedanimprovement ascomparedwiththeNRELandGarcia-ValladaresandVelsquezmodels.Theonlycase withoutimprovementwasforblackchromecoatingwithairintheannulusRMSEof0.855 %and0.808%fortheproposedmodelandNRELmodelrespectively,wherethereisnot muchdifferencebetweentheproposedandtheNRELmodels. 103

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aCermetcoating bBlackchromecoating Figure3.21Comparisonofcollectorefciencycalculatedfromtheproposedmodelwith experimentaldata[39]andothersolarreceivermodels[6,41] 104

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aCermetcoating bBlackchromecoating Figure3.22Comparisonofthermallossescalculatedfromtheproposedmodelwithexperimentaldata[39]andothersolarreceivermodels[6,41],on-suncase 105

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aCermetcoating bBlackchromecoating Figure3.23Comparisonofthermallossescalculatedfromtheproposedmodelwithexperimentaldata[39]andothersolarreceivermodels[6,41],off-suncase 106

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Table3.15ComparisonofrootmeansquareerrorRMSEbetweentheproposedheattransfermodelandothernumericalmodelsforthecermetcoatingcase.Experimentaldatataken from[39],NRELmodel[6],Garca-ValladaresandVelzquezModel[41] Model RMSE h c % HeatLossesW/m 2 VacuumSun Currentmodel1.01210.255 NRELmodel1.38214.718 Garca-ValladaresandVelzquezModel1.433 VacuumOffSun Currentmodel2.414 NRELmodel6.004 Garca-ValladaresandVelzquezModel4.671 AirSun Currentmodel1.2258.959 NRELmodel1.56213.594 Garca-ValladaresandVelzquezModel2.292 AirOffSun Currentmodel2.651 NRELmodel4.416 Garca-ValladaresandVelzquezModel3.865 Figures3.22and3.23showthethermallossescalculatedforthedifferentmodelsand comparedwiththeexperimentalvalues.Asinthecaseofcollectorefciency,thethermal lossesalsoshowedagoodagreementwiththeexperimentalresultsandmostofthevaluesareinsidetheexperimentalerrorbars.TheRMSEanalysisshowsthattheproposed modelgavethermallossvaluesclosesttotheexperimentalresultsascomparedtothose obtainedfromtheNRELandGarcia-ValladaresandVelsquezmodels.Theonlyexcep107

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Table3.16ComparisonofrootmeansquareerrorRMSEbetweentheproposedheattransfermodelandothernumericalmodelsfortheblackchromecoatingcase.Experimentaldata takenfrom[39],NRELmodel[6] Model RMSE h c % HeatLossesW/m 2 VacuumSun Currentmodel0.9264.260 NRELmodel1.1919.650 VacuumOffSun Currentmodel1.978 NRELmodel7.230 AirSun Currentmodel0.85513.196 NRELmodel0.80811.225 AirOffSun Currentmodel4.714 NRELmodel2.667 tionwasagainfortheblackchromecoating,forwhichtheNRELmodelachievedbetter RMSE.225W/m 2 and2.667W/m 2 foron-sunandoff-suncaserespectively,airinthe annulusscenariothantheproposedmodel.196W/m 2 and4.714.W/m 2 forthecurrent modelrespectively.Inthiscaseourmodelshowshigherheatlossesattemperaturesabove ambientof350 Corhigher.Itisdifculttosaywhichmodelismoreaccurate,although wefeelourmodelismoreconservative. Thethirdscenario,withtheglassenveloperemovedbaretubeandthesurroundingair indirectcontactwiththeabsorbertube,leadstohighestthermallossesasexpected.This scenarioispossibleinthesolarpowerplantsduringregularoperationwhentheglasstube 108

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getsbroken.Theheattransferanalysisforthisscenarioquantieshowmuchthecollector efciencyisdegraded.Forthiscase,windspeedtakesanimportantroleinthethermal lossesandthereforeinthecollectorefciencyaswell.NaeeniandYaghoubi[105,106] analyzedthewindowandthermaleldaroundthereceivertube.Theyconcludedthatthe localNusseltdistributionaroundthereceivertubeisdifferentfromthecrossowcondition. Giventhatintheproposedmodel,theconvectiveheatlosseswerecalculatedforacylinder incrossow,itcanbeconcludedthatforforcedconvection V wind > 0 ,themodelwill overpredictthermallossesandunderpredictcollectorefciency.ThiscanbeseeninFigure 3.24 V = 1.Forristall[6]recommendedtoincludeinhismodelhalfoftheconvective losses V = 0 : 5toreducetheoverestimationofthermallossesbyconvection.Inthispaper, afterattinganalysis,itwasfoundthatareductionof41.8%inheatconvectionlosses V = 0 : 582leadstogoodresults.Figure3.24comparestheresultsobtainedfordifferent heatconvectionfactors.Inordertocomparethedifferentheatconvectionfactors,RMSE wascalculatedforeachfactor.TheresultsobtainedareshowninTable3.17;thelowest RMSEwasobtainedforthefactorproposedinthisstudy, V = 0 : 582. Table3.17ComparisonofrootmeansquareerrorRMSEfordifferentheatlossconvection factors V HeatLossW/m 2 h c % 1.0Regularmodel91.6213.12 0.5RecommendedbyForristall[6]32.755.78 0.582Proposedmodel27.803.92 109

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aCollectorefciency bThermalloss Figure3.24Comparisonoftheoreticalandexperimental[39]collectorefciencyandthermallossesobtainedfordifferentheatconvectionlossfactors 110

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3.6Non-LinearRegressionHeatLossModel Theproposedequationfortheheatlossesmodelisasfollow: Q HL W/m = y o + y 1 )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T f )]TJ/F59 11.9552 Tf 10.949 0 Td [(T + y 2 T 2 f + y 3 T 3 f + I bn K cos i y 4 + y 5 T 2 f + V n wind y 6 + y 7 )]TJ/F59 11.9552 Tf 5.476 -9.689 Td [(T f )]TJ/F59 11.9552 Tf 10.949 0 Td [(T .123 with T f = T f ; i + T f ; o 2 .124 ThisequationisbasedonthecorrelationproposedbyPrice[11],althoughslightmodicationswereintroducedintheoriginalequation.Theconditionsandparametersusedfor theheatlossescalculationarepresentedinTable3.18. Table3.18Specicationsusedfortheheatlossmodel SolarCollectorsLS-2andLS-3 InletTemperature C 50,100,150, ::: 400 BeamRadiationW/m 2 0,250,500,750,1000 AmbientTemperature C 15,30,45 ReceiverHCESelectivecoating:UniversalVacuumAirCollector UVAC[6] i = 0 ,15 ,30 ,45 ,60 ,75 ,90 SH = 1 ConditionsofHCEAnnulusVacuumintacttheevacuatedannuluspressurewas 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 Torr Lostvacuumannuluslledwithambientair AtmosphericAirPressure1atm HeatTransferFluidVP-1 MassFlorratekg/s8 111

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Table3.19Heatlosscorrelationcoefcients Coefcients LS-2LS-3 VacuumAirVacuumAir y o 1.868-2.7201.930-3.229 y 1 2.515 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 1.0282.498 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 1.033 y 2 -1.080 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -1.165 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -1.097 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -1.201 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 y 3 6.639 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(6 6.700 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(6 6.671 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(6 6.747 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(6 y 4 1.771 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 -3.541 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 8.556 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(5 -4.596 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 y 5 4.398 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 1.771 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 5.569 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(8 2.784 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 y 6 -8.087 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 -3.686-8.401 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 -3.503 y 7 3.543 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 1.605 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 3.555 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 1.603 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 n 3.598 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 3.536 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 3.537 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 3.533 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R 2 0.99980.99960.99980.9996 RMSE1.644.001.694.01 Thecorrelationcoefcientsfordifferentconditionsinthereceiverannulusareshownin Table3.19.Table3.19alsoshowsthecoefcientofdetermination, R 2 ,whichisameasure ofhowwelltheheatlossescanbepredictedbytheproposedcorrelation,andtherootmean squareerrorRMSE.TheresultsshowthattheproposedttingEquation3.123predicts theheatlossesforLS-2andLS-3solarcollectorsveryaccurate.Thisiscorroborated inFigure3.25byplottingtheheatlossesobtainedfromthenon-linearregressionandthe proposedmodel. 3.7Conclusions Acomprehensiveheattransfermodelforthermalanalysisofparabolictroughsolar receiverswasdeveloped.Theproposedmodelincludedadetailedradiativeheattransfer analysisandmoreaccurateheattransfercorrelations.Theresultsobtainedshowedgood agreementwithexperimentaldataandincomparisonwithotherheattransfermodels,the proposedmodelpresentedingenerallowerRMSEvaluesandbetterperformance,espe112

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aVacuum bAir Figure3.25Comparisonofheatlossesobtainedfromthenon-linearcorrelationEquation 3.123andtheproposedmodel ciallyfortheheattransferlosses.Forthecaseofbaretubeglassenvelopebroken,itwas foundthatafactorof0.418inconvectiveheatlossesleadstoimprovementintheperformanceoftheheattransfermodel.Basedontheresultsobtained,itisconcludedthatthis modelissuitableforthecalculationofheatlossesandcollectorefciencyunderdifferent ow,selectivecoatingandoperatingconditions. Anon-linearregressionmodelwasobtainedforLS-2andLS-3solarcollectors.This modelisveryusefulforpredictingtheheatlossesinasolarcollectorunderdifferentradiationandmeteorologicalconditionswithoutusingtheproposedcomprehensivemodel. 113

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Chapter4 PowerBlock 4.1Introduction ThepowerblockcommonlyusedinconcentratingsolarthermalpowerplantsisaRegenerativeRankineCycle[13,16].ThethermodynamiccycleisshowninFigure4.1.The heattransferuidHTFpassesthroughthreeheatexchangerssimpliedmodel:superheater,boilerandpreheater.Inthepreheater,whichisnormallyashellandtubetypeheat exchanger,compressedwatercomingfromclosedfeedwaterheaterCF-1isheatedup untilsaturatedliquidconditionisreached.Then,thesaturatedliquidowsthroughthe boilerwhereachangeofphasefromliquidtovaporoccurs.Theboilersteamgeneratoris ashell-and-tubeheatexchangerwiththeHTFenteringthetubesideandliquidfeedwater owingthroughtheshellside.Aftertheboiler,thesaturatedvaporgoestothesuperheater whereadditionalenergyisaddedtothesteam,bringingittoasuperheatedvaporcondition.Thisheatexchangerisashell-and-tubetypeaswell.Thesuperheatedsteamis expandedthroughthehighpressureturbine. Twoextractions,11and12,aretakenfromthehighpressureturbinetotheclosed feedwatersCF-1,CF-2.Closedfeedwaterheatersareshell-and-tube-typerecuperators [52],whichareusedtoincreasethefeedwatertemperaturethroughcondensationofthe extractedsteam. 114

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Figure4.1RegenerativeRankinecycleconguration 115

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Thethermalefciencyofthecycleisincreasedbyincludingclosedfeedwaters,but theoptimumnumberofheatersisbasedoneconomicaloptimization,andforthiscasean optimumofveheatersisrecommended[107,108].Theremainingsteamisreheated; thereheatisusedtoallowhigherboilerpressureswithoutlow-qualitysteamproblemsat theturbineexhaustpressureandthereforeanincreaseintheoverallthermalefciencyof thecycleisachieved.Thereheatedsteamisthenexpandedintothelowpressure turbine;fourextractions,-,arebledtotheclosedfeedwatersCF-3,CF-4,CF-5 andanopenfeedwaterOF.Openfeedwatersareadirectcontact-typeheatexchanger[52] inwhichstreamsatdifferenttemperaturesaremixedtoformastreamatanintermediate temperatureatsaturatedliquidcondition.Openfeedwatersarealsousedforremovingair andotherdissolvedgaseswhichcancausecorrosionproblemordecreasetheperformance ofthecycle.Theexhaustturbinestreamismixedwiththefeedwatercomingfrom thetrapintheclosedfeedwater5CF-5.Atrapisavalvethatpermitsonlyliquidtopass throughtoaregionoflowerpressure[52].Themixturegoestothecondenser,ashell-andtubeheatexchanger,whereachangeofphasefrommixturetosaturatedliquidtakesplaces. ThefeedwaterleavingthecondenserSaturatedliquidconditionispumpedtotheopen feedwaterpressurethoughthecondenserpump.Thefeedwaterispreheatedbytheclosed feedwatersCF3,CF4andCF5untilitreachestheopenfeedwater.Saturatedliquid attheexitoftheopenfeedwaterispumpedatboilerpressureandpreheatedbyclosed feedwatersCF1andCF2andnallyreturnstothepreheater. Initially,massandenergybalance,understeadystateconditions,wascarriedoutin eachcomponentofthecycleandmassowrate,temperatureandpressurewereobtained 116

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foreachstream.Thethermodynamiccyclewasthenmodeledforpartialloadconditions. Thesepartialloadconditionsarepresentthroughoutthedayduetotheintermittentenergy absorbedbythesolareld.Theseconditionsaffectthetemperatureandmassowrate oftheHTFenteringthepowerblock.Themassowrateisvariedforthecasesinwhich theHTFtemperatureisheldconstant.Forpartialloadconditionsinheatexchangersthe approximationdevelopedbyPatnode[13]wasused: UA UA ref = m o m o ; ref 0 : 8 .1 where UA and UA ref aretheproductoftheoverallheattransfercoefcient, U ,andthe heattransferarea, A ,forthecurrentandreferenceconditionsrespectively,and m o and m o ; ref arethemassowratesoftheouteruidintheheatexchangerforthecurrentand referenceconditionsrespectively.ThisapproachisbasedonColburnequationforuids withconstantspropertiesanditassumesthatthemassowratesoftheinnerandouter uidsremaininthesameproportionatpartialloadconditionsasatthereferenceload. m i m o = m i ; ref m o ; ref .2 ThethermodynamicpropertiesofwaterandsteamwereimplementedinPython2.6[34] byusingtheinternational-standardIAPWS-IF97steamtables[109].Theanalysisforeach componentofthecycleisdevelopedbelow.Foralltheheatexchangersitisassumedthat: Theheatexchangersoperateundersteadystateandsteadyowconditions Heattransferlossestotheambientarenegligible 117

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Thereisnoheatgenerationintheheatexchangers Pressuredropisnegligible 4.2ReheaterandSuperheater Superheaterandreheaterincreasethetemperatureofthesaturatedornearsaturated steaminordertoincreasethethermodynamicefciencyoftheRankinecycle[110].They areshell-and-tubeheatexchangerswhosemaindifferenceistheoperatingpressure.Inthe superheater,thermalenergyisaddedtothesteamcomingfromtheboilertobringitto superheatedconditions.Forthereheater,steamcomingfromthehighpressureturbineexit isreheatedtoavoidproblemswiththesteamqualityleavingthelowpressureturbineand increasetheoverallefciencyoftheRankinecycle.Thefollowingassumptionsaremade forthesuperheaterandreheaterheatexchangers: Thesteamenteringthesuperheaterstream9 g isatsaturatedvaporconditions Thetemperatureofthestreamleavingthesuperheaterisgivenby: T 10 = T a )]TJ/F66 11.9552 Tf 10.677 0 Td [(D T pinch Figure4.2 Thetemperatureofthestreamleavingfromthereheateris: T 14 = T a )]TJ/F66 11.9552 Tf 10.949 0 Td [(D T pinch Figure4.3showsaschematicofthereheaterandsuperheater.Theheattransferuid HTFcomingfromthesolareldheatsupthesteamleavingtheboilersuperheateror highpressureturbinereheater.Counterowarrangement,one-shellpassandone-tube passisassumedandtheeffectivenessNTUmethodisusedforalltheheatexchangers. 118

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Figure4.2Steamgenerationprocess Thismethodisbasedonadimensionlessparametercalledtheheattransfereffectiveness, denedas[111]: e = Q Q max = Actualheattransferrate Maximumposibleheattransferrate .3 Figure4.3Reheaterandsuperheaterheatexchanger 119

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Theactualheattransferratecanbedeterminedfromtheenergybalanceonthehot HTFandcoldsteamuidsasfollows: Q = C HTF T HTF i )]TJ/F59 11.9552 Tf 10.949 0 Td [(T HTF o .4 Q = C Steam T Steam o )]TJ/F59 11.9552 Tf 10.95 0 Td [(T Steam i .5 withtheheatcapacityratesdenedas: C HTF = m HTF h HTF i )]TJ/F59 11.9552 Tf 10.95 0 Td [(h HTF o T HTF i )]TJ/F59 11.9552 Tf 10.949 0 Td [(T HTF o .6 C Steam = m Steam h Steam o )]TJ/F59 11.9552 Tf 10.949 0 Td [(h Steam i T Steam o )]TJ/F59 11.9552 Tf 10.949 0 Td [(T Steam i .7 where m HTF isthemassowrateoftheheattransferuid,and m Steam isthemassowrate ofthesteam.Themassbalanceforthisfeedwaterheateris: m HTF o = m HTF i .8 m Steam o = m Steam i .9 Themaximumtemperaturedifferenceinaheatexchangeristhedifferencebetweenthe inlettemperaturesoftheheattransferuidhotandtheinletsteamcold. D T max = T HTF i )]TJ/F59 11.9552 Tf 10.949 0 Td [(T Steam ; i .10 Themaximumheattransferrateinaheatexchangerisreachedwhenthesteamisheated totheinlettemperatureofthehotuidortheHTFiscooledtotheinlettemperatureofthe steam.Thesetwolimitingconditionswillnotbereachedsimultaneouslyunlesstheheat capacityratesoftheHTFandsteamareidentical C HTF = C Steam [111].Theuidwith 120

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thesmallerheatcapacityratewillreachthelargertemperaturedifference,therefore,the maximumpossibleheattransferrateinaheatexchangerisgivenby: Q max = C min )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF i )]TJ/F59 11.9552 Tf 10.95 0 Td [(T Steam ; i .11 and C min = min C HTF ; C Steam .12 Then,oncetheeffectivenessoftheheatexchangerisknown,theactualrateofheat transferisasfollows: Q = e C min )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF i )]TJ/F59 11.9552 Tf 10.949 0 Td [(T Steam ; i .13 Theeffectivenessofaheatexchangerdependsonthegeometryoftheheatexchangeras wellastheowarrangement[111].Twoparameterstypicallyinvolvedintheeffectiveness relationofaheatexchangerare: e = f UA C min ; C min C max .14 where UA = C min ,alsoknownasthenumberoftransferunits NTU ,isameasureofthe heattransfersurfacearea A ,and C min = C max isthedimensionlesscapacityratio, c .Fora counterowarrangementone-shellpassandone-tubepasstheeffectivenessisasfollows [111]: e = 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(exp [ )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(c ] 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(c exp [ )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(c ] .15 121

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Aspecialcaseitispresentedwhen c = 1,forthiscasetheeffectivenessissimpliedas[56]: e = NTU 1 + NTU .16 Theeffectivenesscanbealsoexpressedintermsofthemaximumheattransferrate: e = C HTF T HTF i )]TJ/F59 11.9552 Tf 10.95 0 Td [(T HTF o C min )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF i )]TJ/F59 11.9552 Tf 10.95 0 Td [(T Steam ; i .17 or e = C Steam T Steam o )]TJ/F59 11.9552 Tf 10.949 0 Td [(T Steam i C min )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF i )]TJ/F59 11.9552 Tf 10.95 0 Td [(T Steam ; i .18 Atpartialloadconditionstheproductoftheoverallheattransfercoefcient, U ,andthe heattransferarea, A ,isgivenby[112]: UA 0 = UA m 0 HTF i m HTF i 0 : 8 .19 where UA 0 isthenewproductoftheoverallheattransfercoefcientandtheheattransfer areacalculatedatthenewmassowrate m 0 HTF i 4.3Boiler Theboilerisaheatexchangerinwhichthefeedwaterchangesphasefromliquidto vaporatconstanttemperatureandpressure[111].TheboilerFigure4.4isshell-andtubetypeheatexchangerwiththeHTFenteringthetubesideandliquidfeedwaterowing throughtheshellside.Thefollowingassumptionsaremadetotheanalysisoftheboiler heatexchanger: 122

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Thefeedwaterstreamenteringtheboilerstream9 f isatsaturatedliquidconditions ThetemperatureoftheHTFstreamleavingtheboilerisgivenby: T b 0 = T sat @ T = T 9 + D T pinch Figure4.2 Figure4.4Boilerheatexchanger Theactualheattransferrateattheboilerisasfollows: Q boiler = C HTF )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF a 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T HTF b 0 .20 or Q boiler = m Feedwater h Steam )]TJ/F59 11.9552 Tf 10.949 0 Td [(h Feedwater = m Feedwater h fg @ T = T 9 .21 Thefeedwatercomingfromthepreheaterabsorbsalargeamountofheatatconstant temperatureduringthephase-changeprocess.Theheatcapacityrateofthesteamduring thephase-changeprocessapproachesinnitysincethetemperaturechangeiszero[111]. 123

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TheminimumheatcapacityrateisobtainedfromtheHTFandthedimensionlesscapacity ratioisdenedas: c boiler = C min ; boiler C max ; boiler = 0.22 with C min denedas: C min ; boiler = C HTF = m HTF )]TJ/F59 11.9552 Tf 5.476 -9.689 Td [(h HTF a 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h HTF b 0 T HTF a 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T HTF b 0 .23 Themassbalancefortheboilerisasfollows: m HTF b 0 = m HTF a 0 .24 m Steam = m Feedwater .25 Themaximumheattransferrateisgivenby: Q boiler ; max = C min )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF a 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T sat @ T = T 9 .26 Theeffectivenessisasfollows[111]: e boiler = 1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(exp )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU boiler .27 or e boiler = C HTF )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T HTF a 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T HTF b 0 C min )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(T HTF a 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T sat @ T = T 9 .28 e boiler = m Feedwater h fg @ T = T 9 C min )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(T HTF a 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T sat @ T = T 9 .29 124

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Atpartialloadconditionstheproductoftheoverallheattransfercoefcient, U boiler andtheheattransferarea, A boiler ,isgivenby[112]: UA 0 boiler = UA boiler m 0 HTF a 0 m HTF a 0 0 : 8 .30 where UA 0 boiler isthenewproductoftheoverallheattransfercoefcientandtheheat transferareaoftheboilercalculatedatthenewmassowrate m 0 HTF a 0 .TheHTFreturn temperatureandmassowratearecalculatedas: m HTF ; rec = m HTF + m HTF ; superheater .31 h HTF ; rec = m HTF h HTF ; a + m HTF ; superheater h HTF ; c m HTF + m HTF ; superheater .32 T HTF ; rec = f h HTF ; rec .33 4.4Preheater ThepreheaterFigure4.5isashellandtubeexchangerwhosemaingoalistobring theenteringfeedwatertosaturatedliquidconditions.Inordertosimplifytheanalysis,the preheaterisassumedtobeofvariableareathisassumptionisalsoadoptedbyPatnode [13]atpartialloadconditionssothattheexitingstream9 f isalwaysatsaturatedliquid condition.Thisassumptionreducesthenumberofguessvariablestobesolvedatpartial loadconditionsanddoesnothavemucheffectontheperformanceofthecycle. 125

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Figure4.5Preheaterheatexchanger TheactualheattransferratesforthehotHTFandcoldfeedwateruidsisasfollows: Q preheater = m HTF )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(h HTF b 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h HTF b .34 Q preheater = m F )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(h 9 f )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 9 .35 where h 9 f = h f @ P = P 10 and m F isthemassowrateofthefeedwater.Themassbalancefor thepreheateris: m HTF b = m HTF b 0 .36 m Feedwater ; 9 = m Feedwater ; 9 f .37 4.5ClosedFeedwater Closedfeedwaterheatersareshellandtubeheatexchangers,inwhichthesteambled fromtheturbinecondensesontheshellsidewhereasthefeedwaterstreamisheatedonthe tubeside.Thecondensateisremovedbyusingatrapandisallowedtopasstoafeedwater 126

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heateroperatingatlowerpressureorintothecondenserstream20'[52].Normallyclosed feedwatersincludethreeheattransferzonesasfollows[113]: Desuperheatingzonewherethesteamiscooledtoitssaturationtemperature Condensingzonewherethesteamiscondensedtosaturatedliquidconditionatconstantpressureandtemperature Subcoolingzonewheretheliquidiscooledbelowitssaturationtemperature Itisassumedthattheclosedfeedwaterheatersarealwaysworkingatthecondensingzone andthecondensedwaterleavingtheclosedfeedwaterisatthesaturatedliquidcondition. Theterminaltemperaturedifference,whichisthetemperaturedifferencebetweenthesaturationtemperatureattheextractionpressureandthefeedwatertemperatureleavingthe closedfeedwaterheater,isassumedas2.8 Catreferenceconditions[114].Theenergy andmassbalancefortheclosedfeedwaterheater2CF-2,Figure4.6aredetailedbelow. Initiallythermodynamicpropertiesofthestream12'mixturearecalculatedas: h 12 0 = 1 m 12 0 [ h 24 0 m 24 0 + h 12 m 12 ] .38 m 12 0 = m 24 0 + m 12 .39 Thetemperatureofstream12'isgivenby: T 12 0 = T P 12 ; h 12 0 .40 127

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Figure4.6Closedfeedwaterheater Themassbalanceshowsthat m 8 = m 7 m 23 = m 12 0 ,and m 23 0 = m 23 .Forthestreamafter thetrap,stream23',theenthalpyis h 23 0 = h f @ P = P 12 andtheenergybalanceisasfollows: m 12 0 h 12 0 + m 7 h 7 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 8 h 8 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 23 h 23 = 0.41 Theactualheattransferrateattheclosedfeedwateristhencalculatedas: Q CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = C F ; 7 )]TJ/F51 8.9664 Tf 6.966 0 Td [(8 T 8 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T 7 .42 Q CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = m 12 0 h 12 0 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h 23 .43 Theanalysisfortheclosedfeedwaterissimilartotheboileranalysispreviouslyperformed. c CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = C min ; CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 C max ; CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = 0.44 with C min denedas: C min ; CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = C F ; 7 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 = m 7 h 8 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h 7 T 8 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T 7 .45 128

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Themaximumheattransferrateisgivenby: Q CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 ; max = C min T 12 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T 7 .46 Theeffectivenessisasfollows[111]: e CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(exp )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU CF )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 .47 or e CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = C F ; 7 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 T 8 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T 7 C min T 12 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T 7 .48 e CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 = m 12 0 h 12 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 23 C min T 12 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T 7 .49 Atpartialloadconditionstheproductoftheoverallheattransfercoefcient, U CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 andtheheattransferarea, A CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 ,isgivenby[13]: UA 0 CF )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 = UA CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 m 0 7 m 7 0 : 8 .50 where UA 0 CF )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 isthenewproductoftheoverallheattransfercoefcientandtheheat transferareaoftheclosedfeedwaterCF-2calculatedatthenewmassowrate m 0 7 4.6OpenFeedwaterDeaerator Openfeedwaterheatersaredirectcontact-typeheatexchangers[52],whicharemore efcientthanclosedfeedwatersFigure4.7.Inthisfeedwaterheater,streamsatdifferent temperaturesandsamepressurearemixedtoachievestreamexitingatthesaturationconditioncorrespondingtotheinletpressure.OpenfeedwaterheatersDeaeratorarealsoused 129

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forremovingtheairandotherdissolvedgaseswhichcancausecorrosionproblems.The massandenergybalanceareobtainedas: m 6 = m 15 + m 5 + m 23 0 .51 h 15 m 15 + h 5 m 5 + h 23 0 m 23 0 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 6 m 6 = 0.52 Figure4.7Openfeedwaterheater 4.7Turbine TheproposedregenerativeRankinecycleusestwoturbines:highpressureandlow pressureturbine;thehighpressureturbinehastwostageswhilethelowpressureturbine hasvestages.Reheatisappliedbetweenthelaststageofthehighpressureturbineand therststageofthelowpressureturbine.Thereheatpressurewasselectedbasedonthe optimizationanalysiscarriedbyHabibetal.[115]andDincerandAl-Muslin[116].They concludedthattheoptimumreheatpressureshouldbebetween19-20%oftheboilerpressure. 130

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Forbothturbinestheconstantefciencymethod[114]wasemployedtocalculatethe entropyattheexitingstreamofeachstage.Theoptimumpressureateachextractionwas determinedbydividingthetotalisentropicdropinequalpartsaccordingtothenumberof stagesFigure4.8.Thiscongurationleadstooptimumturbineperformance[108,113]. Figure4.8Enthalpy-entropydiagramofsteamexpansioninamulti-stageturbinestages TheprocedureformodelingthehighpressureturbineisshownbelowFigure4.9. Initially,thepropertiesattheinletstreamarecalculated.Theexitpressureisdetermined fromtheoptimumreheatpressureaspreviouslyexplained. P 13 = f b P .53 with0 : 19 f b 0 : 2.Thevalueof f b isselectedsothattheminimumsteamqualityatthe exitis x 19 0 : 9. Theisentropicenthalpydropiscalculatedas: D h s ; T = h 10 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 12 ; s .54 131

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Figure4.9Highpressureturbinestages Theisentropicenthalpydropineachstagesiscalculatedbydividingthetotalisentropic enthalpydropbythenumberofstages.Theisentropicenthalpyiscalculatedateachextractionas: h 11 ; s = h 10 )]TJ/F66 11.9552 Tf 7.465 1.886 Td [(D h s ; T = 2.55 h 12 ; s = h 13 ; s = h 11 )]TJ/F66 11.9552 Tf 7.465 1.886 Td [(D h s ; T = 2.56 Theoptimumextractionpressuresaregivenby: P 11 = P h 11 ; s ; s 10 .57 P 12 = P 13 = P h 12 ; s ; s 10 .58 132

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Theenthalpyateachexitingstreamiscalculatedbyusingtheconstantefciencymethod [114]. h 11 = h 10 )]TJ/F54 11.9552 Tf 10.95 0 Td [(h H ; T h 10 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h 11 ; s .59 h 12 = h 13 = h 10 )]TJ/F54 11.9552 Tf 10.949 0 Td [(h H ; T h 10 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h 12 ; s .60 Thesteamowtothesteamturbineiscontrolledbythesteamcontrolvalves.Thereare twomainmethodstovarythesteamowthoughthesteamturbine[117]: changingthepositionoftheturbinecontrolvalves changingthesteampressureupstreamfromtheturbinevalvesbykeepingthemina xedposition Inthexedboilerpressurecontroltheturbinegovernorvalveisusedtocontrolthepower outputwhileboilerpressureiskeptalmostconstant.Thismethodworksunderrapidload changesbutlargethermalstressesareinducedduetotheinletsteamtemperatureuctuation [118].Forslidingpressurecontrol,boilerpressureisusedtocontroltheoutputpowerwhile theturbinegovernorvalvesiskeptwideopen.Althoughthiscontrolstrategyproduces minimalthermalstress,theresponseofthesystemisslowandthethermalefciencyis lowercomparedtothexedboilerpressurecontrolmethod[16].Inthisanalysis,sliding pressurecontrolisassumedsinceithasbetterperformanceforlargeloadchanges[16], whichiscommonlythecaseinsolarpowerplantsowingtotheintermittencyofthesolar irradiation. 133

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Thecalculationofthetransientoperatingconditionsforeachstageoftheturbineis donestartingwiththelaststageanduptotherststage.Atpartialloadconditions,the pressuredropateachstageiscalculatedbyusingStodolaLaw 1 [107]: P 2 i )]TJ/F59 11.9552 Tf 10.95 0 Td [(P 2 o P 2 i ; ref )]TJ/F59 11.9552 Tf 10.949 0 Td [(P 2 o ; ref = m m ref 2 .61 where P i istheinletpressureofthestage, P o istheoutletpressureofthestage, P i ; ref is theinletpressureofthestageatreferenceconditions, P o ; ref istheoutletpressureofthe stageatreferenceconditions,and m = m ref isthethrottleowratio.Theturbineefciency isaffectedatpartialloadconditionsaswell,specicallythereisareductionintheturbine efciencygivenby: h t = 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(R h t ; ref .62 Thereductionfactor R Figure4.10isasfollows 2 [119]: R = 0 : 191 )]TJ/F51 11.9552 Tf 10.95 0 Td [(0 : 409 m m ref + 0 : 218 m m ref 2 .63 4.8Pump Apumpisadevicetomoveorcompressliquids[52].Theproposedthermodynamic cyclehastwopumps:condenserpumpandopenfeedwaterpump.Thecondenserpump increasesthepressurefromthecyclelowpressuretotheopenfeedwaterpressurewhilethe openfeedwaterpumpincreasesthepressurefromtheopenfeedwaterpressuretotheboiler 1 Nocorrectionfactorfortemperatureisincludedbecausetheslidingpressurekeepsthesteaminlet temperaturealmostconstant 2 Condensingturbinewithonegoverningstageand3600rpm 134

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Figure4.10Effectofthrottleowratioontheturbineefciency pressure.Theworkinputrequiredtocompressthefeedwateristakenintoconsideration forcalculationofthenetworkobtainedfromthethermodynamiccycle,sincethiselectric workissubtractedfromthenetelectricworkobtainedfromtheturbines. Thepumpsareassumedtoworkundersteadyconditionsandheatlossestothesurroundingsarenegligible.ThemassandenergybalanceareasfollowsFigure4.11: m Feedwater i = m Feedwater o .64 Figure4.11Pump 135

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Theexitingenthalpyisgivenby: h Feedwater o = h Feedwateri i + )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(h Feedwater o ; s )]TJ/F59 11.9552 Tf 10.949 0 Td [(h Feedwateri i h pump .65 with h Feedwater o ; s = h P Feedwater o ; s Feedwater i .Thepumpworkisthen: W pump = m Feedwater i h Feedwater o )]TJ/F59 11.9552 Tf 10.95 0 Td [(h Feedwateri i .66 Atpartialloadconditionstheefciencyofthepumpisaffectedbythevariationofmass owrate.Thepumpefciencyisdenedby[5]: h pump h pump ; ref = e mo + 2 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(e mo m Feedwater i m Feedwater i ; ref )]TJ/F67 11.9552 Tf 10.95 0 Td [( 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(e mo m Feedwater i m Feedwater i ; ref 2 .67 where h pump ; ref istheefciencyofthepumpatreferenceconditions.Forconstantspeed, Lippke[5]recommendstouseavalueof e mo = 0.Figure4.12showstheeffectofthrottle owratioonthepumpefciency. Figure4.12Effectofthrottleowratioonthepumpefciency 136

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4.9Condenser Thecondenserisatwophaseowheatexchanger.Theheatgeneratedbythesteam phasechangefromvaportoliquidisremovedbyacoolant.Theshellandtubecondenser hasthecoolantowingonthetubesideandthesteamontheshellside.Coolingwater iscommonlyusedasthecoolantinthesolarpowerplantsalthoughproblemswithwater shortageshavebroughtaircooledcondensersasanalternative[120].Themassandenergy balanceforthecondenserareasfollowsFigure4.13: Q c = m 25 h 25 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 1 .68 m 25 = m 1 .69 where h 1 = h sat @ P = P 25 .TheefciencyoftheRankinecycleisaffectedbythecondenser pressure;ataxedboilerpressurethecondenserpressureshouldbekeptaslowaspossible [52].Thecondensertemperaturedependsonthecoolingmethodemployedtoremovethe latentheattothesurroundings.Theminimumambienttemperatureisgivenbythewet bulbtemperature;thus,mostofthesolarpowerplantsusecoolingtowerwhichkeepsthe Figure4.13Schematicofthecondenser 137

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coolingwatertemperatureatvaluesnearthistemperaturebyevaporationprocess.Forsites withwatershortages,aircooledcondenserscanbeused.Aircooledcondensersarelimited tothedrybulbtemperatureoftheairwhichdecreasesthethermalefciencyofthepower block.Twotypesofcoolantswereconsideredinthisstudy:waterandair.Forbothcooling uids,preliminarydesigntomeetthethermalrequirementsofthecondenserandhourly simulationsofoperationswereperformed.Adetailedanalysisofthecoolingmethodsand theirintegrationwiththecondenserisshownbelow. 4.9.1CoolingTower Coolingtowersarewidelyusedinindustrialprocessestoremovewasteheat.Twotypes ofcoolingtowersexist:naturalandmechanicaldraft.Inthissimulationmechanicaldraft coolingtowerwithairandwaterincounterowisanalyzed. Acoolingtowercoolswaterbyacombinationofheatandmasstransfer.Thehotwater comingfromthecondenserissprayedinthetowerinordertoexposealargesurfaceof watertotheatmosphericair.Duetothelowhumidityoftheairenteringatthecooling tower,apartofthefallingwaterevaporates[52],takingitslatentfromtheremainingwater intheliquidstate.Theneteffectisatemperaturedropintheexitingwater.Thistemperature dropinthecoolingwaterisknownasthetemperaturerangeFigure4.14.Anotherfactoris thecoolingtowerapproachFigure4.14whichisthedifferencebetweenthecoldwater temperatureleavingthetowerandtheambientwetbulbtemperature[121].Aminimum valueof5 Fisrecommendedforthisfactorsincethesizeofthetowervarieswidelywith theapproachtemperature[122]. 138

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Figure4.14Coolingtowerprocessheat.Adaptedfrom[121] Thewetbulbtemperatureisanimportantvariableofdesignforcoolingtowerssince thecondenserpressureisadirectfunctionofthecoldwatertemperature.Thedesignwet bulbtemperatureiscommonlyselectedasthe5%ambientwetbulbtemperature,whichis thewetbulbtemperaturethathashistoricallybeenexceeded,ontheaverage,byonly5% ofthehoursinthewarmestsummermonths[123]. 4.9.1.1DesignProcedure Foragivensetofdesignconditions,thereisanoptimumdesignforthecoolingtower. Theoptimumdesignconditionsarerelatedtominimumconstructionandoperationcost. Theoptimumoutletairtemperatureisgivenby[124]: T air ; o = T cw 3 + T cw 2 2 .70 139

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Forthisapproximation,theairowratewillbewithin 10%oftheoptimumdesign. AsshowninFigure4.15,thecoolwaterleavingthetowerispumpedtothecondenser. Duringthisprocessthetemperatureriseisnegligible[125],thatis T cw 2 T cw 1 .Thewater inletandoutlettemperaturesaredenedby: T cw 3 = T sat @ P = P 1 )]TJ/F66 11.9552 Tf 10.95 0 Td [(D T pinch ; c .71 T cw 1 = T wb ; in + D T approach .72 where D T pinch ; c isthepinchpointtemperaturebetweenthesteamandthecoolingwaterand T wb ; in isthedesignwetbulbtemperature.Althoughthewatermassowrateisnotconstant, theevaporationrateissmallandiscommonlyneglected[124].Fromtheenergybalance themassowratioofthewater L w totheair G isgivenby: L w G = h air ; o )]TJ/F59 11.9552 Tf 10.95 0 Td [(h air ; i C p ; w T CW 3 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T CW 2 .73 Figure4.15Schematicofacoolingtower 140

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Theairleavingthetowerisassumedtobeatsaturatedconditions.Thetowercharacteristic ka V = L w forcounterowtowerisgivenbytheMerkelequation[121]: ka V L w = Z T cw 3 T cw 1 C p ; air dT h s )]TJ/F59 11.9552 Tf 10.949 0 Td [(h air .74 where k isthemasstransfercoefcientinkg/hr-m 2 a isthecontactareainm 2 /m 3 V is theactivecoolingvolumeinm 3 /m 2 L w isthewatermassowrateperunitoftowercross sectionalareainkg/hr-m 2 h s istheenthalpyofthesaturatedairatwatertemperaturein kJ/kg air h istheenthalpyoftheairstreaminkJ/kg air ,and C p ; air isthespecicheatofthe airinkJ/Kg-K. Theperformanceofacoolingtowerisaffectedbythewaterandairmassowrates. Optimumcapitalandfanpowercostsareobtainedat G lessthan8800kg/hr-m 2 ,whilea poorwaterdistributionisobtainedatgreaterthan15000kg/hr-m 2 .MohiuddinandKant [126]recommendthefollowingproceduretodeterminethewaterandairowratesfor mechanicaldraughtcoolingtowers: Calculatetheratioofthewatertotheairmassowrate L w = G = L w = G fromEquation .73 Determine G foravalueof L w of12000kg/hr-m 2 Iftheresultingvalueof G exceeds7800kg/hr-m 2 ,itisnecessarytodeterminethe valueof L w thatcorrespondsto G = 7800kg/hr-m 2 141

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Therequiredllheight, Z tower ,iscalculatedfromthetowercharacteristics ka V = L w as: Z tower m = V = ka V L w L w ka .75 Inordertocalculate Z tower ,Leeper[124]suggestsavalueof ka =1600kg/hr-m 3 andthe valueof L w iscalculatedfromtheprocedurerecommendedbyMohiuddinandKant[126]. Thepumppowerisdeterminedfromthefollowingequation: W CT ; pump kW = r w gL w H p 1000 h ct ; pump .76 where r w isthedensityofthewater, H p isthepumpheadinm,and h ct ; pump istheefciency ofthecoolingtowerpump.Agoodapproximationforthepumpheadis[124]: H p m = Z tower + 34.77 Fanpowerrequirementiscalculatedbasedonanempiricalcorrelationwhichassumes thatonehorsepowerhp 0 : 75kWisrequiredpereach226.5m 3 /minofairmovedby thefan[126].Theairowrateiscalculatedbasedonthepositionofthefan.Forforced drafttoweriscalculatedattheinletairstreamwhileforinduceddrafttoweriscalculatedat theexitairstream.Thedensitykg d ; air /m 3 oftheairwatervapormixtureisgivenby[127]: r air = P R air T air K 1 + 1 : 6078 w .78 where w isthehumidityratio,denedastheratioofthemassofthewatervaporinthe airtothemassofdryair, M air = M v istheratioofthemolecularweightoftheairtothe 142

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molecularweightofthewatervapor, M air = M v = 1 : 6078, R air isthegasconstantfordry air, R air = 0 : 287kJ/kg-K,and P isthetotalpressureinkPa.Theairowrateisgivenby: F )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(m 3 /min = G 60 r air ; o .79 where r air ; o iscalculatedattheoutletairstream.Thefanpoweristhencalculatedas: W ct ; fan kW = F )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(m 3 /min 226 : 5 0 : 75.80 The NTU c numberoftransferunitsandeffectiveness, e c ,arecalculatedasbelow.The heatremovedfromthecondenserisgivenby: Q c = L w C p ; w T CW 3 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T CW 2 .81 Theminimumheatcapacityrateisobtainedfromthecoolingwaterandthedimensionless capacityratiois c c = 0,with C min denedas: C min ; c = L w C p ; w .82 Themaximumheattransferrateisgivenby: Q c ; max = C min ; c T 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T CW 2 .83 Theeffectivenessisasfollows[111]: e c = 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(exp )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU c .84 or e c = m 25 h 25 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h 1 C min ; c T 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T CW 2 .85 143

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e c = L w C p ; w T CW 3 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T CW 2 C min ; c T 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T CW 2 .86 The NTU c iscalculatedas: NTU c = UA 0 c C min ; c .87 Atpartialloadconditionstheproductoftheoverallheattransfercoefcient, U c ,and theheattransferarea, A c ,isgivenby[13]: UA 0 c = UA c L 0 w L w 0 : 8 .88 where UA 0 c isthenewproductoftheoverallheattransfercoefcientandtheheattransfer areaofthecondensercalculatedatthenewmassowrate L 0 w 4.9.2CoolingTowerPerformanceatOffDesignConditions Thechangeinthewatertemperatureacrossthetower,fornegligiblewaterlossdueto evaporation,isgivenby[128]: dT w dV = G L w C pw dh air dV .89 Aboveequationcanberewrittenintermsofonlyairenthalpies[128]: dh sw dV = GC s L w C pw dh air dV .90 with C s = dh s dT T = T w .91 Equation.90canbesolvedanalyticallybymakingananalogywithheatexchangers [129].Theairsideeffectivenessforacoolingtowerisdenedastheratiooftheactualheat 144

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transfertothemaximumairsideheattransferthatwouldoccuriftheexitingairstream weresaturatedatthetemperatureoftheincomingwater h air ; o = h s ; wi e air = Q G h s ; cw 3 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h air ; i .92 Then,theactualheattransferis: Q = e air G h s ; cw 3 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h air ; i .93 Makingananalogywithacounterowheatexchanger, e air isdenedas[128]: e air = 1 )]TJ/F51 11.9552 Tf 10.95 0 Td [(exp [ NTU ct 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(m 0 ] 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(m 0 exp [ NTU ct 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(m 0 ] .94 with m 0 = GC s L w C pw .95 and NTU ct = kaV G .96 Theexitairenthalpyisdeterminedbyreplacing Q fromtheenergybalanceoftheair: h air ; o = h air ; i + e air h s ; wi )]TJ/F59 11.9552 Tf 10.949 0 Td [(h air ; i .97 Theoutletwatertemperatureandthewaterlossbyevaporationarecalculatedas[128]: T cw 2 = L w ; i C pw T cw 3 )]TJ/F59 11.9552 Tf 10.949 0 Td [(G h air ; o )]TJ/F59 11.9552 Tf 10.949 0 Td [(h air ; i L w ; o C pw .98 145

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Theaveragesaturationspecicheat, C s ,isestimatedastheaverageslopebetweenthe inletandoutletwaterconditions: C s = h s ; cw 3 )]TJ/F59 11.9552 Tf 10.95 0 Td [(h s ; cw 1 T cw 3 )]TJ/F59 11.9552 Tf 10.95 0 Td [(T cw 1 .99 Thewaterlossisobtainedfromtheoverallmassbalance: L wo = L wi )]TJ/F59 11.9552 Tf 10.95 0 Td [(G w air ; o )]TJ/F54 11.9552 Tf 10.949 0 Td [(w air ; i .100 where w air ; o iscalculatedbyusinganaveragedvalueovertheentiretower[128]: w air ; o = w air ; e + w air ; i )]TJ/F54 11.9552 Tf 10.949 0 Td [(w air ; e exp )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU ct .101 Theeffectivesaturationhumidityratio, w air ; e ,isassociatedwiththeeffectivesaturation enthalpy h s ; e : h s ; e = h air ; i + h air ; o )]TJ/F59 11.9552 Tf 10.95 0 Td [(h air ; i 1 )]TJ/F51 11.9552 Tf 10.949 0 Td [(exp )]TJ/F59 11.9552 Tf 9.289 0 Td [(NTU ct .102 Thecoolingtowerperformanceisaffectedbythechangesinoperationalparameters. Oneoftheparameterswhichchangesmostoftenistheambientwetbulbtemperature.A changeintheinletwetbulbtemperaturedoesnotaffectthetowercharacteristicsbutrather affectstheapproachtemperature.Whenairandwaterowrateareadjusted,the NTU ct of thecoolingtowerisaffected.Theeffectoftheowrateson NTU ct isasfollows: NTU ct = c L w G 1 + n .103 where c and n areconstantsforagivencoolingtower.Leeper[124]suggestsavalueof c = )]TJ/F51 11.9552 Tf 9.289 0 Td [(0 : 6andthevalueof n isobtainedfromthedesignconditions. 146

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4.9.3AirCooledCondensers Traditionalcoolingmethodsforcondensersinthermalpowerplantsarewaterintensive processes.Analternativetotheseprocessesistheuseofdrycoolingcondensers,which completelyeliminatestheneedformakeupcoolingwater.Drycoolingisattractivebecauseofthenecessitytoreducetheconsumptionforthelimitedwaterresourcesforan expandingworldpopulation.Ifdrycoolingisusedratherthanwatercooling,itispossible tosavewaterforftythousandinhabitantspereach100MWecapacity[130].Someofthe advantagesofaircooledcondensersare[130,131]: Eliminationofwaterconsumptionforcoolingwatermakeup Noicingorfoggingproblems Reducedmaintenancecostsnochemicaladditivesorperiodiccleaningisrequired Eliminationofthecoolingtowerplume Reductionofcondensationonthemirrorsclosesttothecoolingtowerplumethis condensationaffectstheopticalperformanceofPTC Drycooledplantsofferpotentialeconomicandcollateraladvantagesduetoplant sitingexibility Drycoolingsystemsemitonlywarmandcleanair,withoutadverseenvironmental effects 147

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Ontheotherhand,aircooledsystemshavealsosomedisadvantages: Heattransferbyforcedairisalesseffectiveprocessthanevaporativeheattransfer Largerheatexchangerareasarerequiredtoachievetheequivalenttheheatrejection Greaterfanpowerwillberequiredtoachievetheequivalenttheheatrejection Theperformanceoftheaircooledcondenserisstronglyinuencedbytheambient conditionsdrybulbtemperature Therearetwobasictypesofdrycoolingtowers:directandindirect.Theindirectsystem issimilartoawettowerexceptthatthecoolingelementconsistsofalargennedcooling coilthatdoesnotallowthecoolingmediawatertobeincontactwiththeair.Inthedirect aircooledcondenser,thesteamowsthroughabankofnnedtubes,andtheambientairis blownacrossthetubesbyoneormoreaxialfans[133].AnAframeFigure4.16isoften Figure4.16CongurationofanAframeaircooledcondenser.Adaptedfrom[132] 148

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usedwherethecondensingsteamowsthroughthetubeswhichareorientatedatanglesof 45 or60 withthehorizontal[133]. Foraircooledheatexchangersusinghorizontaltubes,twocongurationsarepossible: forcedandinduceddraft.IntheforceddraftcongurationFigure4.17athefanislocated belowthetubebundle,whilefortheinduceddraftitislocatedabove.Althoughtheforced aForced-draft bInduced-draft Figure4.17Congurationofforcedandinduceddraftaircooledheatexchanger.Adapted from[133] 149

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draftisthesimplest,someproblemswiththerecirculatinghotairarepresentbecauseofthe lowvelocityoftheairstreamleavingtheheatexchanger.Thisproblemcanbereducedby usinganinduceddraftcongurationFigure4.17binwhichtheairowismoreuniform andthevelocityofthestreamleavingthetubebundleishigherthanintheforceddraft operation.Recirculatinghotairdecreasestheperformanceoftheaircooledheatexchanger andmoreheattransferareaandairmassowratearerequired.Thisexplainswhyinduced draftaircooledunitsusuallydonotrequiresignicantlymorepowerthantheinduceddraft units[133]. Anaircooledheatexchangerconsistsofoneormorefanbays.Abayconsistsofoneor moretubebundlesplacedsidebysideinthebay,thefans,drivesystem,frameworkandthe supportstructure.Tubebundlesarerectangularinshapeandusually6-12ftwide.Axial owfanshavediametersof6-18ftandtheirsizeisgenerally50hpkWorless.The designguidelinesforaircooledheatexchangersareasfollows[133]: Tubingselectionisbasedonthetubesideuidtemperatureandcorrosionresistance. ItisrecommendedtochooseoneofthetubingcongurationspresentedinTable4.1 Inordertoobtainuniformdistributionofairowcrossthetubebundle,thefanarea shouldbeatleast40%ofthebundlefacearea.Aminimumoffourtuberowsis suggested. Fortwofanbays,theratioofthetubelengthtobundlewidthshouldbeintherange of3-3.5 150

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Table4.1Typicalhigh-ntubedata.Adaptedfrom[133,134] FinandTubeDimensions1 = 2in.by95 = 8in.by10 RoottubeODin1.0001.000 Finheightin0.5000.625 FinOD2.0002.250 AverageFinheightin Tensionwoundorembedded0.012-0.0140.012-0.014 Bimetallicorintegral0.015-0.0250.015-0.025 Finsperinch910 Tubelayoutangle3030 TubePitchin2.25 4 2.5 4 A Tot = L 3.805.58 A Tot = A o 14.5021.40 A Tot = A i 13BWG17.926.3 14BWG17.425.6 16BWG16.724.5 Externalsurfaceareperunitbundlefacearea, A = A face Threetuberows60.680.4 Fourtuberows80.8107.2 Fivetuberows101.0134.0 Sixtuberows121.2160.8 Theaircooledheatexchangershouldbedesigntooperateatthesummerconditions. Usuallythetemperatureselectedis3-5%drybulbtemperature.Thisdrybulbtemperatureistheairtemperaturethathashistoricallybeenexceededanaverageofonly 3-5%forthewholeyear Forinduceddraftoperation,theoutletairtemperatureislimitedtoabout104.5 C inordertopreventdamageinthefan.Forceddraftunitshouldbeconsideredifthe outletairtemperatureisgreaterthan177 C Theairvelocitybasedonthebundleareaandstandardairconditionsisusuallyinthe rangeof400-800ft/min 151

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4.9.3.1DesignoftheAirCooledCondensers Fortheaircooledcondenserdesignacrossowaircooledheatexchangerhorizontal tubepositionisassumed,asimpledesignascomparedtotheAframeaircooledcondensers.Preliminarydesignusingaircooledheatexchangersisagoodstartingpointto calculatethefanpowerrequirementsparasiticlossesandtheeffectofenvironmentalconditionsairtemperatureonthenetpoweroutputwithoutcompromisingtheaccuracyof theresults 3 Twocitieswereselectedforthedesignofaircooledcondenserunit;forbothunits5 = 8 in.by10tubecongurationwasselectedTable4.1.Thedrybulbdesigntemperature wascalculatedbasedon3-5%drybulbtemperaturesuggestedbythedesignguidelines. Figure4.18showstheaccumulativefrequencydistributionofhourlydrybulbtemperature correspondingtoTampaandDaggett. Alayoutwith10-twofanbaysisanalyzedhereFigure4.19.Forthisconguration, theloadisdividedinto10equalparts. AninitialtemperaturedifferenceITD,whichisthetemperaturedifferencebetween thesaturatedsteamvaportemperatureandthedrybulbdesigntemperature,isassumed. TypicallyITDisintherangeof14-33.3 C-60 F[120,136].InthisdesignITDis assumedtobe22 : 2 C F.Then,thecondensertemperatureisgivenby: T sat ; c = T air ; d + ITD .104 3 Thissimplieddesignprocedurewasalsoassumedby[135] 152

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aTampa,Fl bDaggett,CA Figure4.18Cumulativefrequencydistributionofthedrybulbtemperature.Datataken from[32] Theairtemperaturerise D T air canbeapproximatedby[134]: D T air F = U D + 1 5 : 55 )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T sat ; c )]TJ/F59 11.9552 Tf 10.949 0 Td [(T air ; d .105 where T sat ; c isthecondensertemperature, T air ; d isthedrybulbdesigntemperature,and U D istheoverallheattransfercoefcientsintheaircooledheatexchangersinW = m 2 K.Typical Figure4.19Aircooledcondenserlayout 153

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valuesof U D areshowninTable4.2.Forpuresteamtheoverallheattransfercoefcientin aircooledheatexchangersisintherangeof U D 35 : 78 )]TJ/F51 11.9552 Tf 12.61 0 Td [(53 : 38W = m 2 K. Table4.2Typicalvaluesofoverallheattransfercoefcientinaircooledheatexchangers. Adaptedfrom[133,134] Service U D W = m 2 KBtu = h ft 2 F Lighthydrocarbons25.55-28.39.5-5.0 LightGasoline25.55.5 Lightnaphtha21.58-26.69.8-4.7 Heavynaphtha18.74-23.85.3-4.2 Reactorefuent21.58-26.69.8-4.7 Ammonia28.39-33.50.0-5.9 Aminereactivator26.69-32.37.7-5.7 Freon1219.88-23.85.5-4.2 Puresteam-20Psig35.78-53.38.3-9.4 Steamwithnon-condensables18.74.3 Theairmassowrateisdeterminedfrom: m air = Q = n bay C p ; air D T air .106 where n bay isthenumberofbaysand m Tot ; air = m air n bay .TheLogarithmicMeanTemperatureDifferenceLMTDis: D T ln = )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T sat )]TJ/F59 11.9552 Tf 10.95 0 Td [(T air ; d )]TJ/F67 11.9552 Tf 10.949 0 Td [( T sat )]TJ/F59 11.9552 Tf 10.95 0 Td [(T air ; o ln T sat )]TJ/F59 11.9552 Tf 10.95 0 Td [(T air ; d T sat )]TJ/F59 11.9552 Tf 10.95 0 Td [(T air ; o = D T air ln ITD ITD )]TJ/F66 11.9552 Tf 10.949 0 Td [(D T air .107 Inordertodeterminetheheattransfersurfacearea,aLMTDcorrectionfactorisassumedtobe F 1 : 0.Withmorethanfourtubes,the F factorisnearlythesameasfor unmixed-unmixedcrossow. 154

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Theheattransferrateiscalculatedas: Q = AU D F D T ln 10.108 therefore A = Q = 10 U D F D T ln .109 Thenextstepistodeterminethenumberoftuberows,tubelengthandthenumberof tubes.Thebundlefacearearequiredforagivenfacevelocitystandardis[133]: A face = m air r std V face .110 where r std istheairdensityatstandardconditions, r std = 1 : 2013kg/m 3 .Thefacevelocity isusuallyintherangeof2 : 54 )]TJ/F51 11.9552 Tf 12.326 0 Td [(3 : 56m/s,afacevelocityof3.05m/scanbeassumed.The ratioofheattransfersurfaceareatobundlefaceareaisthencalculated.The )]TJ/F59 11.9552 Tf 5.476 -9.69 Td [(A = A face cal valueiscomparedtothevaluesshowninTable4.1andtheclosestvaluebasedonthe numberoftuberowsisselectedas A = A face d .Usingthisvalue,therequiredfaceareais recalculated: A face = A = A = A face d .111 Atubelength, L ,ofthreetimesthebundlewidth, W ,basedonthedesignguidelines,is assumed. A face = WL = 3 W 2 .112 155

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Thenumberoftubesiscalculatedusingthevalueof A Tot = L d obtainedfromTable 4.1: n t = A A Tot = L d L .113 Thenumberoftubesisselectedbasedontheclosestintegerdivisiblebythenumberof tuberowspreviouslycalculatedwith A = A face d .Thecorrespondingbundlewidthisgiven by: W actual = D tube ; pitch n t ; row + D x .114 where n t ; row = n t = n tuberows ,andthesideclearance D x isassumedtobe2in.Theactual bundlefaceareaandstandardfacevelocityare: A face ; actual = W actual L actual .115 V face ; std = m air r std A face ; actual .116 Forthiscasethetemperaturechangeofthecondensingsteamisnegligibleandtherefore F 1[137].Therequiredoverallcoefcientis: U req = Q AF D T ln .117 Theoverallheattransfercoefcient U 0 D iscalculatedbelow.Athighvaporvelocitythe gravitationaleffectscanbeneglected,andthecondensatecollectsasathinannularlm aroundtheinsideofthetubewalls,withnostratication.Mostcondensersoperateinthis owregime[55].ThemodelforthelocalNusseltnumberisgivenby: Nu = Nu l F x .118 156

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Krger[138]recommendstousethecorrelationobtainedbyShah[62]: Nu cx = 0 : 023 Re 0 : 8 c Pr 0 : 4 c 1 )]TJ/F59 11.9552 Tf 10.95 0 Td [(x x 0 : 8 + 3 : 8 x 0 : 76 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(x x 0 : 04 p 0 : 38 r # .119 p r = P = P c where P c isthecriticalpressure, x x isthelocalquality,and A ts isthetubecrosssectional area.ByintegratingEquation.119forthecasewhen x x ; i = 1and x x ; o = 0,themean condensationNusseltnumberis: Nu c = 0 : 023 Re 0 : 8 c Pr 0 : 4 c 0 : 55 + 2 : 09 p 0 : 38 r .120 Allthepropertiesarecalculatedatsaturatedliquidconditions.TheReynoldsnumber Re c isgivenby: Re c = mD i n p = n t A ts m c .121 If Re c isintherangeof350 Re c 6300,thecongurationissatisfactory.Onepass, n p = 1,isrecommendedforaircooledcondensers[132].Theconvectiveheattransfer coefcientiscalculatedas: h i = k c Nu c = D i .122 with D i = 0 : 021m : 81in.Fortheconvectiveheattransfercoefcientforairside, h o ,the maximumairvelocityinthetubebankis V max = V face = A face = A min .123 157

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where A min istheminimumareainthetubebank,and A face isthefacearea.Forequilateral triangularpitch,theminimumareaisgivenby: A min = P T )]TJ/F59 11.9552 Tf 10.949 0 Td [(D r L )]TJ/F51 11.9552 Tf 10.95 0 Td [(2 h f Lb t .124 where D r istherootdiameter, P T isthetubepitch, L isthetubelengthand2 h f Lb t isthe approximateareaoccupiedbythens.Then: V max = P T V face P T )]TJ/F59 11.9552 Tf 10.949 0 Td [(D r )]TJ/F51 11.9552 Tf 10.949 0 Td [(2 h f b t .125 TheNusseltnumberiscalculatedby[133]: Nu = 0 : 38 Re 0 : 6 Pr 1 = 3 A Tot = A o )]TJ/F51 8.9664 Tf 6.967 0 Td [(0 : 15 .126 Thisequationisvalidfor: 1800 Re 10 5 0 : 011m D r 0 : 051m 0 : 006m b 0 : 019m 0 : 0003m t 0 : 0005m 0 : 0274m P T 0 : 0986m 1 A Tol = A o 50 275 : 6 nspermeter 433 : 1 158

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Thevalueof A Tot = A o isobtainedfromTable4.1.TheReynoldsnumberiscalculated as: Re = D r V max r air m air .127 Theairdensityshouldbecorrectedfortheelevationoftheaircooledunitlocation.The correctedatmosphericpressureis: r r o = P P o = exp )]TJ/F59 11.9552 Tf 10.485 8.093 Td [(M g = g c z RT .128 where M isthemolecularweightofair = 29 : 87, R isthegasconstant,8314J/kgmol K, T istheabsolutetemperatureinK, z istheelevationabovemeansealevelinm, P o ; r o are thepressureanddensityofairattemperature T andsealevel, P o 1atm,and P ; r arethe pressureanddensityofairattemperature T andelevation z Theoveralldesignheattransfercoefcientforhighntubingisasfollows: U D = 1 h i + R D i A Tot A i + A Tot L ln D o = D i 2 p k tube + R con A Tot A con + 1 h w h o + R D o h w )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 .129 where R con isthecontactresistancebetweennandtubewall, A con isthecontactarea betweennandtubewall, R D i isthethermalresistanceoftheinnerfoulinglayer, R D oi isthe thermalresistanceoftheouterfoulinglayer, h w h o istheconvectiveresistanceofthens, k tube isthethermalconductivityofthewalltube. Equation.129isapplicabletoalltypesoftubingwiththeexceptionofbimetallic tubes.Forbimetallictubes,thethermalresistanceoftheoutertubemustalsobeaccounted. 159

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Intheabsenceofavailabledataforthecontactresistance,thistermisusuallyneglected, R cond = 0 : Theproceduretoobtaintheweightedefciencyofthenis: y = r 2 c )]TJ/F59 11.9552 Tf 10.949 0 Td [(r 1 [ 1 + 0 : 35ln r 2 c = r 1 ] .130 with r 1 = D r = 2 r 2 = r 1 + b r 2 c = r 2 + t = 2 Thenefciencyisgivenby: h f = tanh m y m y .131 m = 2 h o = k t 0 : 5 Theextendedandprimesurfaceareasperinchofthetubelengtharecalculatedas: A fins = 2 N f p )]TJ/F59 11.9552 Tf 5.476 -9.689 Td [(r 2 2 c )]TJ/F59 11.9552 Tf 10.949 0 Td [(r 2 1 .132 A prime = 2 p r 1 )]TJ/F59 11.9552 Tf 5.476 -9.689 Td [(L )]TJ/F59 11.9552 Tf 10.949 0 Td [(N f t .133 Theweightedefciencyofthennedsurfaceisgivenby: h w = A prime A prime + A fins + h f A fins A prime + A fins .134 160

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Thepressuredrop,inPa,forowacrossabankofhighnnedtubesisgivenby[133]: D P air = fN r G 2 r air .135 G = r air V max where f istheFanningfrictionfactor, N r isthenumberoftuberows.Thefrictionfactoris calculatedbythefollowingexpression: f = 1 + 2 e )]TJ/F67 8.9664 Tf 6.967 0 Td [( a = 4 1 + a #" 0 : 021 + 27 : 2 Re eff + 0 : 29 Re 0 : 2 eff # .136 with a = )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(P T )]TJ/F59 11.9552 Tf 10.949 0 Td [(D f = D r Re eff = Re l = b D f = D r + 2 b l = 1 h f )]TJ/F54 11.9552 Tf 10.95 0 Td [(t Thefansshouldcovertatleast40%ofthebundlearea.Foratwofanbay,thefan diameteris: D fan 0 : 8 p A face ; actual 1 = 2 .137 Thevolumetricairow,perfaninacfmis: n fan = 0 : 5 m air r air .138 161

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Thetotalpressurechange,inPa,inacrossthefanisgivenby: D P total ; fan = D P air + a fr r fr V 2 fr 2 .139 where a fr isthekineticenergycorrectionfactor, a fr 1,and V fr istheairvelocityinthe fanring, V fr = n fan = p D 2 fr = 4 .Thebrakepoweriscalculatedby: W fan kW = D P total ; fan n fan 1000 h fan .140 where h fan isthefanefciency.Thepowersuppliedbythemotoris: W motor = W fan h sr .141 where h sr isthespeedreducerefciency. Thefollowingassumptionswereusedforthepreliminarydesignoftheaircooledcondenser: Tubingtype:Gtubettingwithsteeltubesandaluminumns Tubesize:0.0254minOD,16BWG,393.7nspermeternsperinch,n height0.0016m.625in Tubelayoutistriangular withatubepitchof0.0635m.5in Drafttype:InducedDraft,thecondenseroperatesbelow177 C F Headers:Plugtype Tables4.3and4.4showthespecicationsandthedesignsummaryoftheaircooled condenserobtainedforTampaandDaggettrespectively. 162

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Table4.3AircooledcondenserparametersforTampa ParameterValueUnits Q total 10MW n bay 10 T air ; d 32 C ITD 22.2 C T air ; out 47.49 C T sat ; c 54.22 C m Tot ; air 641.6kg/s )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(A = A face cal 129.87 A = A face d 134 Tuberows5 n t 180 n p 2 W actual 2.33m L actual 7.31m A face ; actual 17.09m 2 V face ; std 3.12m/s h i 6301.94W = m 2 K h o 54.34W = m 2 K k aluminum 238.5W = mK h w 0.848 k tube 45.0W = mK R D i 8.80 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 )]TJ/F51 11.9552 Tf 5.475 -9.689 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R D o 0 )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R con 0 )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 U d 34.47W = m 2 K U req 34.45W = m 2 K D fan 2.13m D P air 133.90Pa D P total ; fan 170.50Pa h fan 0.7 h sr 0.95 W motor 7.53kW W motor ; bay fans15.07kW W Tot ; motor bay150.7kW 163

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Table4.4AircooledcondenserparametersforDaggett ParameterValueUnits Q total 10MW n bay 10 T air ; d 39.11 C ITD 22.2 C T air ; out 54.49 C T sat ; c 61.33 C m Tot ; air 646.3kg/s )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(A = A face cal 129.16 A = A face d 134 Tuberows5 n t 180 n p 2 W actual 2.33m L actual 7.31m A face ; actual 17.09m 2 V face ; std 315m/s h i 5877.18W = m 2 K h o 54.57W = m 2 K k aluminum 238.5W = mK h w 0.847 k tube 45.0W = mK R D i 8.80 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 )]TJ/F51 11.9552 Tf 5.475 -9.689 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R D o 0 )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 R con 0 )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(W = m 2 K )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 U d 34.22W = m 2 K U req 34.21W = m 2 K D fan 2.13m D P air 139.38Pa D P total ; fan 176.71Pa h fan 0.7 h sr 0.95 W motor 7.85kW W motor ; bay fans15.69kW W Tot ; motor bay156.9kW 164

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Theeffectivenessoftheaircooledcondenserunitsisgivenby: e ACC = Q m Tot ; air C P ; air )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(T sat ; c )]TJ/F59 11.9552 Tf 10.949 0 Td [(T air ; d .142 ThenumberoftransferunitsNTUiscalculatedbasedontherelationforheatexchangers withphasechange: NTU ACC = )]TJ/F51 11.9552 Tf 10.617 0 Td [(ln 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(e ACC .143 The UA ACC oftheaircooledcondenseris: UA ACC = NTU m Tot ; air C P ; air .144 Forthehourlysimulationsthefollowingassumptionsaremade[13]: Fluidpropertiesremainconstantduringtheoperationoftheaircooledcondenserunit Theairmassowratecalculatedinthepreliminarydesignisassumedtobeconstant The UA oftheaircooledcondenserunitdoesnotchangeduringtheoperationofthe powerblock,sincetheeffectoftheheattransfercoefcientofthecondensingsteam on UA isrelativelysmallcomparedtotheheattransfercoefcientontheairside Thenalcondenserheatexchangerissizedbyaddingidenticalaircooledcondenser units N units Thelastassumptionimpliesthat NTU andtheeffectivenessoftheaircooledcondenser unitremainconstant.Usingtheseassumptions,itisfoundthat: e = m steam ; unit h 25 )]TJ/F59 11.9552 Tf 10.949 0 Td [(h 1 m Tot ; air C p ; air T sat @ P = P 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T air .145 165

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Thesteammassowrateisdistributeduniformlyinthecondenserunitsas: m steam ; unit = m steam = N units .146 Thetotalparasiticlossesintheaircooledcondenserare: W Tot ; ACC = W Tot ; motor N units .147 TheheatexchangerparametersobtainedfromthepreliminarydesignareshowninTable 4.5. Table4.5Heatexchangerparameterscalculatedfortheaircooledcondenser Tampa,FL e 0.696 NTU 1.192 UA 770.35kW/K Daggett,CA e 0.691 NTU 1.175 UA 765.20kW/K 4.10NetElectricWork Thetotalturbineworkisgivenby: W HP ; t = m 10 h 10 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 11 h 11 )]TJ/F51 11.9552 Tf 13.872 0 Td [( m 12 h 12 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 13 h 13 .148 W LP ; t = m 14 h 14 )]TJ/F51 11.9552 Tf 13.872 0 Td [( m 15 h 15 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 16 h 16 )]TJ/F51 11.9552 Tf 13.872 0 Td [( m 17 h 17 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 18 h 18 )]TJ/F51 11.9552 Tf 13.872 0 Td [( m 19 h 19 .149 W t = W HP ; t + W LP ; t .150 Thepumpworkisfoundasfollows: W pump ; cond = m 2 h 2 )]TJ/F51 11.9552 Tf 13.873 0 Td [( m 1 h 1 .151 166

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Figure4.20Effectofturbineworkonthegeneratorefciency W pump ; OF = m 7 h 7 )]TJ/F51 11.9552 Tf 13.872 0 Td [( m 6 h 6 .152 Thegrosselectricpoweroutput, W e ,iscalculatedbymultiplyingthenetpowerofthecycle bythegeneratorefciency: W e = W t h generator .153 Thenthenetworkis: W net = W e )]TJ/F51 11.9552 Tf 14.374 2.379 Td [( W pump ; cond )]TJ/F51 11.9552 Tf 14.375 2.379 Td [( W pump ; OF .154 Atpartialloadconditions,thegeneratorefciencyvarieswiththeload W t = W t ; nom Figure4.20asfollows[13]: h generator = 0 : 90 + 0 : 258Load )]TJ/F51 11.9552 Tf 10.949 0 Td [(0 : 3Load 2 + 0 : 12Load 3 .155 Load = W t W t ; nom 167

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Figure4.21Temperature-entropydiagramoftheregenerativeRankinecycle 4.11Results ThethermodynamicpropertiesofwaterandsteamwereimplementedinPython2.6[34] byusingtheinternational-standardIAPWS-IF97steamtables[109].Theinputparameters usedinthesimulationareshowninTable4.6.Theresultsobtainedatnominalconditions arepresentedinTable4.7,andthetemperature-entropydiagramoftheregenerativecycle isshowninFigure4.21. ThenonlinearalgebraicequationsforheatandmassbalanceweresolvedsimultaneouslybyusingawrapperaroundMINPACK'shybrdandhybrjalgorithms[102,139].In ordertoobtainageneralsolutionforthepowerblockatpartialloadconditions,normalized variableswereused.Thenormalizedvariableswereobtainedbydividingtheoutputcycle parametersbytheirrespectivevaluesatnominalconditions: W net = W net ; nom .156 168

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Table4.6Cycleparametersassumedforthesimulation ParameterValueReference HeatTransferFluidHTF InletTemperature390 C[13] FluidVP-1,Hitec[4] RankineCycle GrossElectricPower, W e 50MW e [16] HighPressure90bar[16] Highpressureturbineefciency85.50%[16] Lowpressureturbineefciency89.50%[16] ReheatPressure0 : 19 P high [115,116] CondenserPumpEfciency75%[16] OpenFeedwaterPumpEfciency78%[16] TerminalTemperaturedifferenceClosedFeedwater2.8 C[122] CondenserPressure0.08bar D T pinch 10 C[122] Table4.7Cycleparametersobtainedatnominalconditions VariableValueUnits Cycle W HP ; t 15778.3kW W LP ; t 35242.1kW W pump ; cond 47.6kW W pump ; OF 639.6kW W net 48385.9kW Q boiler + Q superheater 128713.5kW h cycle 37.6% Q c 78380.3kW m steam 54.8kg/s HeatTransferFluid m HTF ; VP 1 491.3kg/s m HTF ; VP 1 ; superheater 50.8kg/s T HTF ; VP 1 ; ret 292.6 C m HTF ; Hitec 776.4kg/s m HTF ; Hitec ; superheater 75.7kg/s T HTF ; Hitec ; ret 293.2 C 169

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a W e = W e ; nom b Q c = Q c ; nom Figure4.22Effectofthepowerplantsizeonthenormalizedelectricoutput W e = W e ; nom and thenormalizedcondenserheattransferrate Q c = Q c ; nom ,HTF:VP-1 Q c = Q c ; nom .157 T HTF ; rec = T HTF ; rec ; nom .158 Figure4.22showstheresultsobtainedfor50MW e and80MW e .Theresultsdemonstratethatthenormalizedbehaviorisindependentofthepowerblocksize.Inorderto corroboratethelastconclusion,theproposedpowermodelMW e wascomparedwith othersimulationcarriedoutbyPatnode[13]forapowerblockof35MW e .Asseenin Figure4.23,twodifferentboilingpressures,90and100bar,wereusedasinputtotheproposedpowerblockmodel.90baristhenominalboilerpressure,while100baristheboiler pressureusedbyPatnode[13].Theresultsshowthattheboilingpressureaffectsslightly thenormalizedgrosselectricoutputbutinbothcasesthevaluesareclosetotheresultsobtainedbyPatnode.Itisconcludedthattheproposedmodelcanbeusedatdifferentpower size,anditisrecommendedtousethesameinputparametersasgiveninTable4.6,since 170

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Figure4.23Comparisonofthenormalizedelectricoutput W e = W e ; nom obtainedbytheproposedpowerblockmodelandthemodeldevelopedbyPatnode[13] anychangeintheinputparameterswillaffecttheaccuracyoftheresultspresentedbythis simulation. Figure4.24showstheresultsobtainedforthenormalizednetworkoutputwithVP-1 astheheattransferuid.Thecondenserpressureandthenormalizedsteammassowrate affectadverselythenormalizednetworkoutput.AsshowninFigure4.24,thenormalized steammassowratecanbedecreaseddowntoacertainvalue,belowwhichnofeasiblesolutionsarereached.Thislimitingvalueisduetotherestrictionsonthemassandenergybalanceattheopenfeedwaterheaterandclosedfeedwaterheater5CF-5.Figure4.25shows theeffectofnormalizedsteammassowrateandcondenserpressureonthenormalized turbineextractionmassowrates m 15 = m 10 and m 18 = m 10 at390 C.Asitwasmentioned before,inordertosatisfythemassandenergybalance,themassowrates m 15 and m 18 171

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a P c = 0 : 03bar b P c = 0 : 08bar c P c = 0 : 25bar d P c = 0 : 50bar Figure4.24Effectofthenormalizedsteammassowrate m steam = m steam ; nom ,condenserpressureandHTFinlettemperature T HTF ; a onthenormalizednetworkoutput W net = W net ; nom ,HTF:VP-1 172

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a m 15 = m 10 b m 18 = m 10 Figure4.25Effectofthenormalizedsteammassowrate m steam = m steam ; nom andcondenser pressureonthenormalizedturbineextractionmassowrates m 15 = m 10 and m 18 = m 10 T HTF ; a = 390 C,HTF:VP-1 173

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shoulddecreaseasthenormalizedsteammassowrate m steam = m steam ; ref decreases.This tendencyisowingtotheincreaseinenthalpyassociatedwiththeslidingpressurecontrol method[108].Figure4.26showsacomparisonofthenormalizednetworkoutputobtained fortwodifferentHTFsVP-1andHitec.Theresultsshowthatthedifferencebetweenthese twoHTFsisnegligible.Themaximumnormalizednetworkoutputisobtainedatalow condenserpressureandhighheattransferuidinlettemperature.Therefore,tooptimize thenetpoweroutputatagivencondenserpressure,theHTFinlettemperatureshouldbe keptatthenominalconditions 4 byadjustingtheHTFmassowrate.Figure4.27shows theresultsobtainedforthenormalizedcondenserheattransferrate Q c = Q c ; nom .Thetrend obtainedissimilartothenormalizednetworkoutput.Anotherimportantvariableinthe analysisisthereturnHTFtemperature,Figure4.28showsthenormalizedreturnHTFtemperatureforthecondenserpressureatnominalconditions.08bar.Theresultsshowthat thenormalizedreturnHTFtemperatureisindependentofthecondenserpressureandis affectedbytheHTFinlettemperatureandthenormalizedsteammassowrate. 4.12LinearRegressionModel Theproposedequationforthelinearregressionoftheparametersobtainedfromthe powerblockmodelingisasfollows: ln F = y 0 + y 1 ln m HTF + y 2 [ ln m HTF ] 2 + y 3 T HTF ; i + y 4 )]TJ/F59 11.9552 Tf 5.476 -9.689 Td [(T HTF ; i 2 + y 5 ln P cond + y 6 [ ln P cond ] 2 + y 7 ln m HTF ln P cond + y 8 ln m HTF T HTF ; i + y 8 ln T HTF ; i ln P cond .159 4 ThiscontrolstrategyisalsousedbyMontesetal.[16] 174

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aVP-1 bHitec Figure4.26Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenser pressureonthenormalizednetworkoutput W net = W net ; nom T HTF ; a = 390 C 175

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aVP-1 bHitec Figure4.27Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenser pressureonthenormalizedcondenserheattransferrate Q c = Q c ; nom T HTF ; a = 390 C 176

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aVP-1 bHitec Figure4.28Effectofnormalizedsteammassowrate m steam = m steam ; nom andcondenser pressureonthereturnHTFtemperature, P c = 0 : 08bar 177

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Thenormalizedinputparametersareasfollows: m HTF = m HTF m HTF ; nom T HTF ; i = T HTF ; i T HTF ; i ; nom P cond = P cond P cond ; nom and F = W net W net ; nom ; Q c Q c ; nom ; T HTF ; ret T HTF ; ret ; nom TheregressionmodelwaswritteninPython2.6byusingleastsqmodule,whichis awrapperaroundMINPACKlmdifandlmderalgorithms[102].Theinputparameters usedforthepowerblocksimulationaredescribedinTable4.8.Tables4.9and4.10show thecoefcientsobtainedfortheproposedlinearregression.Valuesofthecoefcientof determination, R 2 ,androotmeansquareerrorRMSEarealsoshowninTables4.9and 4.10. Theresultsshowthatthelinearregressionaccuratelyrepresentsthebehavioroftheparameters W net = W net ; nom ; Q c = Q c ; nom ; T ret ; HTF = T ret ; HTF ; nom ,obtainedfromthepowerblock model.ThisiscorroboratedbytheFigure4.29,whichshowsacomparisonoftheparametersobtainedfromthelinearregressionwiththepowerblockmodel. Table4.8Inputsparametersforthepowerblocksimulation InletHTFTemperature, T HTF ; i C290-390 NominalInletHTFTemperature, T HTF ; i ; nom C390 DimensionlessHTFmassowrate, m HTF = m HTF ; nom 0.3-1.0 CondenserPressure, P cond bar0.03-1bar NominalcondenserPressure, P cond ; nom bar0.08 PowerBlock50MW e 178

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Table4.9CoefcientsusedfortheproposedlinearcorrelationgivenbyEquation4.159, HTF:VP-1 Coefcients W net = W net ; nom Q c = Q c ; nom T ret ; HTF = T ret ; HTF ; nom y 0 -7.118-4.473-1.172 y 1 8.864 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 -5.732 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 9.454 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 y 2 -1.228 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 -8.135 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 -3.832 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(3 3 y 3 10.9576.5791.836 y 4 -3.839-2.111-6.643 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 y 5 -2.202 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 4.075 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 7.979 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 y 6 -1.477 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 2.196 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 5.431 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 y 7 1.567 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 2.934 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -4.768 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(6 y 8 6.326 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 6.154 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 3.602 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 y 9 1.532 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 8.269 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -8.791 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 R 2 0.99950.99990.9998 RMSE0.01010.00350.0011 Table4.10CoefcientsusedfortheproposedlinearcorrelationgivenbyEquation4.159, HTF:Hitec Coefcients W net = W net ; nom Q c = Q c ; nom T ret ; HTF = T ret ; HTF ; nom y 0 -7.123-4.485-1.157 y 1 1.798 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 -1.055 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 7.796 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 y 2 -1.210 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 -7.624 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 -5.264 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 y 3 11.1856.7541.847 y 4 -4.063-2.274-6.911 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 y 5 -2.056 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 3.859 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(2 8.678 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 y 6 -1.492 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 2.294 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 5.863 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(5 y 7 1.589 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 2.721 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(3 -2.678 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(7 y 8 6.842 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 6.532 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 4.800 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 y 9 1.385 10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 1.023 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(2 -9.883 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(4 R 2 0.99991.00000.9997 RMSE0.00450.00140.0011 179

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a W net = W net ; nom b Q c = Q c ; nom Figure4.29Comparisonofthedimensionlessnetworkoutputandcondenserheattransfer rateobtainedfromthelinearcorrelationwiththeproposedpowerblockmodel,HTF:VP-1 andHitec 4.13Conclusions AcomprehensivemodelforthesimulationofaregenerativeRankinecyclewasdeveloped.Theresultsobtainedshowedthattheoutputpowerfromthecyclesisaffected mainlyby:theinletheattransferuidHTFtemperature,massowrateoftheHTFand thecondenserpressure.Thecycleparameterswerenormalizedanditwasfoundthatthe performancewasindependentofthepowerblocksize. Alinearregressionwasproposedbyusingthenormalizedvariables,theresultsshowed thatthelinearequationrepresentsaccuratelythetrendgivenbytheresultsobtainedinthe simulation. 180

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Chapter5 SolarFieldPipingandThermalLosses 5.1SolarFieldLayout ThesolarpipingsystemcirculatestheheattransferuidHTFinaclosedloopto andfromthepowerblockandsolareld.Thissystemisdesignedtomaintainanequally distributedowthroughallthesolarcollectorloopsandthustoavoidhotspotsorcold spots[140].Thesolarpipingmodelinthischapterisbasedonthemodeldevelopedby NationalRenewableLaboratoryNREL[141].Inthismodeltwolayoutswereproposed: anHeldlayoutforcollectorsareasgreaterthan400000m 2 ,andanIlayoutforareasless than400000m 2 5.1.1HFieldLayout AsshowninFigure5.1,inthislayoutthesolareldisdividedinto4headerpairs,with thepowerblocklocatedatthecenterofthesolareld.Colduidisdistributedfromthe coldheadertoeachsolarloopandreturnstothehotheaderwhereitgoesbacktothepower block.Eachcollectorloopconsistsof6LS-3solarcollectorswhicharearrangedinseries toincreasethetemperatureoftheHTFtotheoperatingconditions.Inasolarcollector loop,thesolarcollectorsareconnectedbyballjoints,whichareusedtoallowindependent rotationoftheadjacentsolarcollectors[3]. 181

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Figure5.1Hsolareldlayout.Adaptedfrom[3] 182

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Theballjointsarepreferredoverexiblehosesduetotheiradvantageswhichinclude: lowercost,reducedpressuredrop,andreducedheatlosses[3].Inordertoreducetheland requirementsandpipingpower,thecollectorloophastworowssuchthatthecoldandhot headersarelocatedinthesameside. 5.1.2IFieldLayout Inthislayoutthesolareldisdividedintoa2headerpairs,asshowninFigure5.2, withthepowerblocklocatedatthecenterofthesolareld.ColdandhotheaderrunEastWestdirection.Eachcollectorloopconsistsof16LS-2solarcollectors.AsinHsolar eldlayout,theLS-2collectorsarearrangedinserieswithtworows.Thedimensionsand propertiesofLS-2andLS-3solarcollectorsareshowninAppendixB. Figure5.2Isolareldlayout.Adaptedfrom[3] 183

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Forbothlayouts,themaximumpressuredropisgivenbythelongestpathintheHTF circuit.Thispressuredropiscalculatedforthesolareldandthepowerblockheatexchangers,internalpiping,etc.. 5.2PressureDropintheSolarField Thepressuredropinthesolarelddependson:theHTF,massowrate,innerdiameter ofthepipesandttingselbow,cross,reduction,expansion,valves,etc..Initially,itis necessarytodeterminetheoptimumdiameterforeachsection.Thecalculationsarebased ontheoptimumowvelocityforminimizingofthepipingcostsgivenbyNREL[141], whichisintherangeof2-4m/s.Aowvelocityof2m/s,alsousedin[16],wasselectedfor allthepipingmodeling.Thepipeineachsectionmustmeettheowconditionsoptimum velocityandthepressurerequirements.Forthepressurerequirementstherequiredwall thicknessiscalculatedas[142]: t wall = P abs )]TJ/F59 11.9552 Tf 10.95 0 Td [(P atm D o 2 [ S y + 0 : 4 P abs )]TJ/F59 11.9552 Tf 10.95 0 Td [(P atm ] .1 where P abs istheabsolutepressureofHTFinpsi, P atm istheatmosphericpressureinpsi, D o istheoutsidediameterininches,and S y istheallowablestressinpsi.Asrecommended byNREL[141],threedifferentpipematerialsareused: ASTMA106,GradeB,SeamlessCarbonSteelpipe ASTMA335P9,IntermediateAlloySteel ASTMTP347,StainlessSteel 184

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TheallowablestressforeachmaterialisshowninTable5.1.Theinnerdiameterandwall thicknessfordifferentpipeschedulesandnominaldiametersareshowninAppendixD. Table5.1Maximumallowablestressksifordifferentmaterials.Adaptedfrom[143] Temperature F MaximunAllowableStressksi A106TP347A335 400 ::: 15.5 ::: 500 ::: 14.914.4 600 ::: 14.714.2 6501514.713.9 70014.414.713.7 75013.014.713.2 80010.814.712.8 850 ::: 14.712.1 900 ::: 14.711.4 950 ::: 14.610.6 1000 ::: 14.07.4 1050 ::: 12.15.0 1100 ::: 9.13.3 1150 ::: 6.12.2 1200 ::: 4.41.5 Oncethediameterisselected,theowfrictionlossiscalculatedusingtheDarcyWeisbachequation[111]: h i = 2 C f L i D i V 2 i g .2 where C f isthefrictioncoefcientFanningfrictionfactor, L i isthelengthofthepipe sectioninm,and V i istheowvelocityinm/s.Thefrictioncoefcientcanbecalculated fromthecorrelationgivenbyChen[53]: 1 p C f = 3 : 48 )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 : 7372 Ln 2 e D i )]TJ/F51 11.9552 Tf 12.145 8.094 Td [(16 : 2426 Re D LnA 2 .3 185

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A 2 = 2 e = D i 1 : 1098 6 : 0983 + 7 : 149 Re D 0 : 8981 where Re D istheReynoldsnumber, e isthepiperoughnessinm.Thiscorrelationisvalid for4000 Re D 10 8 and2 10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(8 e = a 0 : 1.Forallthepipematerialsitisassumed thatthepiperoughnesshasavalueof e = 0 : 046mm[144].Thepressuredropinasection isgivenby: D P L i = r i gh i .4 Thefrictionlossesthroughthepipettingsarecalculatedas: L fi = K D i 2 C f .5 where L fi isthepipelengthwhichgivesthesamepressuredropasthetting.Thepressure dropthroughthettingis: h fi = 2 C f L fi D i V 2 i g .6 D P L fi = r i gh fi .7 The K factorsaregiveninTable5.2.Thelocationandnumberofttingsusedwas setbythedescriptiongivenbyNREL[141].Table5.3and5.4summarizethetypesand locationsofthettingsusedineachloopaswellasthelengthofthecoldandhotheader. AsseeninTable5.3,expansionloopsareeveryotherlooptomaintainpipestresseswithin thelimitsallowed[140]. 186

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Table5.2 K valuesfordifferentpipettingsusedinthesolareld.Adaptedfrom[141] Fitting K GateValve0.19 GlobeValve10.00 CheckValve2.50 StandardElbow0.90 MediumElbow0.75 LongElbow0.60 Weldolet1.80 BallJoint4.73 FlexibleHose20.60 Thepressuredropthroughthesolareldiscalculatedforthelongestpathtraveledby theHTF.Thepressuredropisthencalculatedas: D P SF = D P H + D P C + i D P i ; loop ; H + i D P i ; loop ; C + D P in ; out + D P collector .8 where D P H isthepressuredropintheoutlethotheader, D P C isthepressuredropinthe inletcoldheader, D P i ; loop ; H isthepressuredropinthehotheaderpipeatthepositionofthe loopi, D P i ; loop ; C isthepressuredropinthecoldheaderpipeatthepositionoftheloopi, D P in ; out isthepressuredropintheinletandoutletofthesolarcollectorloop,and D P collector isthepressuredropinthesolarcollectorloop.Thepressuredropthroughthepowerblock shouldbeincludedaswell.Table5.5showsthetypicallengthandttingsusedinapower blockunit.Thedescriptionofeachlineisasfollows: Line1:Expansionvesselorthermalstoragetanktopumpsuctionheader Line2:Individualpumpsuctionline,fromsuctionheadertopumpinlet Line3:Individualpumpdischargeline,frompumpdischargetodischargeheader 187

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Table5.3FittingsusedintheHeatCollectionElementHCEloop.Adaptedfrom[141] AccessoriesandPipeInletandOutletHCEToLoop a ToLoopFromLoop a FromLoop PipeLengthm b L Row + 40 L Loop = L HCE n SCE 2 L Row + 192 L Row 2 L Row + 192 L Row StandardElbows2100000 MediumElbows000000 LongElbows004080 GateValves200000 GlobeValves000000 CheckValves000000 LoopWeldolets200000 LoopControlValves100000 BallJoints c 02 + n SCE 0000 a Everyotherloop b L Row :Rowspacing, L Row 12 )]TJ/F51 11.9552 Tf 10.95 0 Td [(15m[3] c n SCE :Numberofsolarcollectorassemblies,LS-2 n SCE = 16,LS-3 n SCE = 6[3] 188

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Table5.4Headerlengthandttingsusedinthesolareldpipinglayout.Adaptedfrom [143] ColdHeaderHotHeader Lengthm PositionofPowerBlockCongurationICongurationH Center5020 + 0 : 5 L Loop North-South20 + 0 : 5 L Row 50 + 0 : 5 L Loop East-West5020 + 0 : 5 L Loop Accessories Standardelbows00 Mediumelbows00 Longelbows88 Gatevalves11 Globevalves00 Checkvalves10 LoopWeldolets00 Loopcontrolvalves00 BallJoints00 Line4:Pumpdischargeheader Line5:Collectoreldoutletheadertoexpansionvesselorthermalstoragetank Line6:Steamgeneratorsupplyheader Line7:Intersteamgeneratorpiping Line8:Steamgeneratorexitheadertoexpansionvesselorthermalstorage Thetotalpressuredropinthepowerblockisgivenby: D P PB = 7 i = 1 D P Line ; i + D P HE .9 189

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Table5.5Pipingttingandlengthusedforatypicalpowerblockunit.Adaptedfrom[141] AccessoriesandPipe Line 12345678 PipeLengthm20661030202520 Expansions11000000 Contractions00100000 Standardelbows00000000 Mediumelbows00000000 Longelbows40466666 Gatevalves11100000 Globevalves00000000 Checkvalves00100000 where D P HE isthepressuredropthroughtheheatexchanger,whichisassumedtobeapproximately4bar.Thetotalpressuredropisgivenby: D P total = D P SF + D P PB .10 Thepressuredropisiterateduntiltheminimumallowablepressureisreachedattheexpansiontank.TheminimumallowablepressureforvariousHTFsisshowninTable5.6. Table5.6MinimumandmaximumallowableworkingtemperatureandpressurefordifferentHTFs.Adaptedfrom[4,141] Fluid T min C T max C P min bar VP-137.840015 DowthermQ37.83306 DowthermRP37.83506 SolarSalt2606216 Hitec1495386 HitecXL1505006 190

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TheHTFiscirculatedinaclosedloopbyusingavariablespeedpump[140].Thepump poweristhehighestparasiticloadinthesolarplantandthereforeavariablespeedpumpis usedtoreducethepowerrequirementswhenthesolareldrequiresowrateslessthanthe nominalrate.Thepumpingrequirementiscalculatedas: W p ; nom = r HTF ; T cold g D P total .11 Atdifferentspeeds,thepumplawisusedandthepumpworkisgivenby[145]: W p = W p ; nom m SF m SF ; nom 3 .12 where m SF isthecurrentmassowrate,and m SF ; nom isthemassowrateatnominal condition.Thepumpefciencyisaffectedbythesolareldmassowrateaswell: h p ; SF h p ; SF ; nom = a o + 2 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(a o m SF m SF ; nom )]TJ/F67 11.9552 Tf 10.949 0 Td [( 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(a o m SF m SF ; nom 2 .13 with a o = )]TJ/F51 11.9552 Tf 9.289 0 Td [(0 : 4asrecommendedbyLippke[5].Thenetelectricpowerrequiredbythe solareldpumpis: W net ; pump = W p = h p ; SF .14 Duringnight,thesolareldoperatesat20%ofthenominalowratetokeepthepipes andsolarcollectorswarm,andavoidthermalshockduetosuddentemperaturedropsor rises.Whenthetemperaturedropsbeyondtheminimumallowabletemperature,anexternal heatersystemsisactivatedtoavoidfreezingproblems. 191

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5.3ThermalLosses Thermallossestakeplaceinthesolareldpipingowingtothetemperaturedifference betweentheHTFandthesurroundingair.Thewaytomitigatethethermallossesisby addinginsulationtothepipes,butaddingtoomuchinsulationgeneratesexcessivecost withlittleextrabenets.Forthisreason,anoptimumthicknessiscalculatedatnominal conditionswhichminimizesthecostsandthermallosses.Inthischapter,themethodology developedbyBahadoriandVuthaluru[146]isused.Thismethodologydeterminestheoptimumeconomicthicknessofthethermalinsulationasafunctionofthesteelpipediameter andthesurfacetemperature. Asaconservativeassumptionforinsulatedpipes,thermalresistancesofthepipewalls andtheexteriorairlmareneglected[147].Thisassumptionsimpliesthecalculationof thermallosses.Thetotalthermallossesaregivenby: Q piping = 2 p 2 6 6 4 k ins ; H T H )]TJ/F59 11.9552 Tf 10.95 0 Td [(T a i L i ln D oi ; pipe + 2 d ins ; i D oi ; pipe + k ins ; C T C )]TJ/F59 11.9552 Tf 10.949 0 Td [(T a j L j ln D oj ; pipe + 2 d ins ; j D oj ; pipe 3 7 7 5 .15 where k ins ; H isthethermalconductivityofthepipeinsulationevaluatedat T H = T H + T a = 2, k ins ; C isthethermalconductivityofthepipeinsulationevaluatedat T C = T C + T a = 2.The aboveEquationcanberewrittenas: Q piping = 2 p k ins ; H T H )]TJ/F59 11.9552 Tf 10.949 0 Td [(T a z H + k ins ; C T C )]TJ/F59 11.9552 Tf 10.95 0 Td [(T a z C .16 192

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z H = i L i ln D oi ; pipe + 2 d ins ; i D oi ; pipe z C = j L j ln D oj ; pipe + 2 d ins ; j D oj ; pipe Theinsulationthicknessisafunctionofthesteelouterpipe: ln d ins = 3 n = 0 a n D )]TJ/F59 8.9664 Tf 6.966 0 Td [(n o .17 Thecoefcients a n areexpressedasfunctionsofthethermalconductivityasfollows: a n = A n + B n k ins + C n k 2 ins + D n k 3 ins .18 Thefollowinginterpolationisusedfortheothersurfacetemperatures: d ins = d ins ; T 1 + T )]TJ/F59 11.9552 Tf 10.95 0 Td [(T 1 d ins ; T 2 )]TJ/F54 11.9552 Tf 10.949 0 Td [(d ins ; T 1 T 2 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T 1 .19 where d 1 and d 2 arecalculatedat T 1 and T 2 respectively.Duetothelowthermalresistance inthewallpipe,thesurfacetemperatureisassumedtobetheHTFtemperature.Thethermal conductivityoftypicalinsulationmaterialusedinCSPplantsisshowninTable5.7for differenttemperatures.Thecoefcients, A n B n C n and D n areshowninTable5.8. Table5.7Thermalconductivity,inkW/mK,ofpipeinsulationmaterials.Datatakenfrom [56,148] Temperature CMineralWoolTemperature CCalciumSilicate 380.033636.90.055 930.043291.90.059 1490.0525146.90.063 2040.0624256.90.075 2600.0730371.90.089 3160.0853476.90.104 3710.0997 :::::: 193

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Table5.8Coefcientsforcalculationoftheoptimumeconomicthickness.Adaptedfrom[146] Coefcients T s =100 C T s =300 C T s =500 C T s =700 C A 0 -1.619063838-1.467341621-1.306428735-1.50683293 B 0 -6.0440641629E-2-9.4579004057E-33.7689223988E-21.6368010607E-1 C 0 1.2992412636E-3-9.0991682769E-4-5.7653162538E-3-1.9272634157E-2 D 0 -1.0480516067E-52.1036093111E-51.5920740612E-46.081932874E-4 A 1 -5.675424778E-25.6420129717E-32.3856855469E-22.1494907991E-1 B 1 1.1266206576E-3-8.1324216389E-3-1.7666292295E-2-7.236908659E-2 C 1 -4.5476251244E-53.6013086233E-41.2527015231E-36.2521396767E-3 D 1 5.4011484658E-7-5.0991959691E-6-2.9095395202E-5-1.7702985302E-4 A 2 1.287145175E-3-1.0914548287E-3-2.2442814135E-3-1.2571952312E-2 B 2 -3.1321987972E-53.1336503528E-47.879778678E-43.6872340957E-3 C 2 1.3585299744E-6-1.3704000608E-5-5.6411819564E-5-3.1830356934E-4 D 2 -1.6951529528E-81.9239332368E-71.3264010427E-69.0141402462E-6 A 3 -1.1238180847E-51.4969220576E-53.3668468005E-51.4981731667E-4 B 3 3.574027836E-7-3.3674152566E-6-9.9103300792E-6-4.2562432156E-5 C 3 -1.5140460085E-81.4658761606E-77.1509244179E-73.6706099161E-6 D 3 1.8649575376E-10-2.0517177043E-9-1.6973934089E-8-1.0392224103E-7 194

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5.4ExpansionTank TheexpansiontankprovidesspaceforexpansionoftheHTFduetothechangein volumewhentheHTFisheatedupinthesolareldtotheoperatingtemperature.The expansiontankisusuallyinstalledatthehighestpointinthesolareldnexttothepump. ThistankisusedtoventthemoisturethataccumulatesintheHTFandtocreatepositive headpressuretothepumpinletaswell. Theexpansiontankshouldbesizedsothatitis25%fullatambienttemperatureand 75%fullatthenormaloperatingtemperature[149].Usuallytheexpansiontankisblanketedwithnitrogentomaintainanon-reactiveatmosphereandkeepingapositivepressure whichpreventsairandmoisturefromenteringthetank[149].Theheattransferlosses fromthetanktothesurroundingairaresignicantandshouldbetakenintoaccountalong withthethermalpipinglossestocalculatethetotalthermallossesinthesolareld.The incrementintheHTFvolumeisgivenby: D V HTF = r T ref r T C )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 V c + 1 4 V exp ; tank N tank + r T ref r T H )]TJ/F51 11.9552 Tf 10.95 0 Td [(1 V H + V collectors + V in ; out + r T ref r T H )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 V H .20 V c = p 4 i D 2 innner ; i L i .21 V H = p 4 j D 2 innner ; j L j .22 V collector = p 4 n loops D 2 receiver L c .23 195

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where r T ref isthedensityoftheHTFcalculatedatreferencecondition,25 C, r T C isthe densityoftheHTFcalculatedatcoldtemperature, r T H isthedensityoftheHTFcalculated athottemperature, V c isthetotalvolumeofthecoldheader, V in ; out isthetotalvolumeof theinletandoutletpipeofthecollectorloop,and V c isthetotalvolumeofsolarreceiver. Forthesolareld,multipleexpansiontankswithavolumeof283m 3 [140]were selected.Thedimensionsofthistankareapproximately: D tank = 6mand H tank = 10m. Thenumberoftanksiscalculatedas: N tank = 8 > > > > < > > > > : D V HTF = 1 2 V exp ; tank ifmod D V HTF ; 1 2 V exp ; tank = 0 int D V HTF = 1 2 V exp ; tank + 1ifmod D V HTF ; 1 2 V exp ; tank 6 = 0 .24 Theprocedureforheatlossesintheexpansiontankwastakenfromthemethodology developedbyKumanaandKothari[150].Fortheheatlosscalculationintheexpansion tankitisassumedthat: Foulingfactorsarenegligible Radiationlossesarenegligible Temperatureinthegasandliquidsectionsisuniform Thermalequilibriumexistsbetweenthegasandtheliquiduid AsshowninFigure5.3,thefourindividuallossesinthetankneedtobecalculated. Theheatlossfromeachsurfaceisshownbelow.Fordrysidewall: Q d = U d A d T gas )]TJ/F59 11.9552 Tf 10.949 0 Td [(T .25 196

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Figure5.3Thermallossesfromaverticaltank.Adaptedfrom[150] U d = 1 h ; d + D I ln D I = D o ; t 2 k I + D I ln D o ; t = D i ; t 2 k tw + 1 h gas ; d D I D i ; t )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 .26 A d = p D I L )]TJ/F59 11.9552 Tf 10.949 0 Td [(L w .27 D I = D i ; t + 2 t tw + 2 t I .28 Forwetsidewall: Q w = U W A Ww )]TJ/F59 11.9552 Tf 5.475 -9.69 Td [(T Liquid )]TJ/F59 11.9552 Tf 10.949 0 Td [(T .29 U w = 1 h ; w + D I ln D I = D o ; t 2 k I + D I ln D o ; t = D i ; t 2 k tw + 1 h Liquid ; w D I D i ; t )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 .30 A w = p D I L w .31 Fortankbottom: Q b = U b A b )]TJ/F59 11.9552 Tf 5.475 -9.689 Td [(T Liquid )]TJ/F59 11.9552 Tf 10.949 0 Td [(T .32 197

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U b = 1 h ; b + t I k I + t tw k tw + 1 h Liquid ; b )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 .33 A b = p D 2 i ; t = 4.34 Fortankroof: Q r = U r A r T Gas )]TJ/F59 11.9552 Tf 10.949 0 Td [(T .35 U r = 1 h ; r + t I k I + t tw k tw + 1 h gas ; r )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 .36 A r = p D 2 i ; t = 4.37 Therefore,thetotalheatlossesare: Q tank ; losses = Q d + Q w + Q b + Q r .38 Theconvectiveheattransfercoefcientsaredetailedbelow. h ; d and h ; w arecalculatedfortwocases:windornowindcondition.Forthewindconditionacylinderincross owisused.TheaverageNusseltnumberrecommendedforacylinderincrossow[53] isgivenby: Nu D = cRe m D Pr n Pr Pr w p .39 h ; d = Nu D k air D I .40 h ; w = Nu D k air D I .41 TheconstantssuggestedforthisequationaretabulatedinTable3.8.Thevalueof p depends ontheheatuxdirection: p = 0 : 25foruidheatingand p = 0 : 2foruidcooling.Forno 198

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windcondition,naturalconvectionisusedandtheverticalcylindercanbeapproximated asaverticalplate,Kakaetal.[53]recommendtousethefollowingexpression: Nu L = 8 > < > : 0 : 825 + 0 : 387 Ra 1 = 6 L h 1 + 0 : 492 = Pr 9 = 16 i 8 = 27 9 > = > ; 2 )]TJ/F51 11.9552 Tf 5.476 -9.69 Td [(10 )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 Ra L 10 12 .42 for D L 35 Gr 1 = 4 L .43 with Gr L = g b T s )]TJ/F59 11.9552 Tf 10.949 0 Td [(T L 3 n 2 .44 where L isthehighofthecylinderinm, D isthediameterofthecylinderinm, n :Kinematicviscosityinm 2 /s.When D = L isnotlargeenough,Kakaetal.[53]recommendthe followingcorrelation: Nu L = 4 3 7 Gr L Pr 2 5 20 + 21 Pr 1 = 4 + 4 272 + 315 Pr 35 64 + 63 Pr L = D .45 h ; d = Nu L k air L )]TJ/F59 11.9552 Tf 10.949 0 Td [(L w .46 h ; w = Nu L k air L w .47 Thenowindconditionisalsousedforthecalculationof h gas ; d and h Liquid ; w .Thesame cases,windandnowindcondition,arepresentedforhorizontalplatestopandbottom. Fornowindconditionaatsurfaceapproximationundernaturalconvectionisused. 199

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Naturalconvection,uppersurfaceofhotplateorlowersurfaceofcoldplate h ; r h gas ; r [111]: Nu L D = 0 : 54 Ra 1 = 4 L )]TJ/F51 11.9552 Tf 5.475 -9.689 Td [(10 4 Ra L 10 7 .48 Nu L D = 0 : 15 Ra 1 = 4 L )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(10 7 Ra L 10 11 .49 Naturalconvection,lowersurfaceofhotplateoruppersurfaceofcoldplate h ; b h Liquid ; r [111]: Nu L D = 0 : 27 Ra 1 = 4 L )]TJ/F51 11.9552 Tf 5.475 -9.69 Td [(10 5 Ra L 10 10 .50 with L D = A s P = D i ; t 4 .51 Ra L D = g b T s )]TJ/F59 11.9552 Tf 10.949 0 Td [(T L 3 D na .52 h = Nu L D k L D .53 where A s and P aretheplatesurfaceareaandperimeter,respectively.Forwindcondition h ; b and h ; r ,thewindenhancementfactor z w iscalculatedas[150]: z w = h ; w j wind h ; w j no ; wind .54 and h ; b wind = z w )]TJ/F59 11.9552 Tf 6.671 -9.69 Td [(h ; b nowind .55 h ; r j wind = z w )]TJ/F59 11.9552 Tf 6.671 -9.69 Td [(h ; r j nowind .56 200

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Inordertocalculatetheheattransfercoefcients,itisnecessarytoobtainthewall temperatureswhichrequireaniterativeprocedure.Thetotalheatlossesarecalculatedas: Q total ; tank = Q tank ; losses ; full N tank ; full + Q tank ; losses ; partial + Q tank ; losses ; empty N tank ; empty .57 where Q tank ; losses ; full N tank ; full isthetotalheatlossesforthetankscompletelylled : 75 V exp ; tank Q tank ; losses ; partial isthetotalheatlossesforthetankpartiallylled V < 0 : 75 V exp ; tank ,and Q tank ; losses ; empty N tank ; empty isthetotalheatlossesforthetanks lledatnominalconditions : 25 V exp ; tank 5.5Conclusions Acomprehensivemodelforthepressuredropandpumpingpowerrequirementofa solareldwasperformed.Themodelcalculatesthediameterandpressuredropineach headersectionbasedonthepipestresscalculationandHTFowratedistribution. Aheatlossesmodelwasalsocarriedoutforthepipingsystemandtheexpansiontank. Themodelcalculatesthethermallossesofthesolareldpipingfortheoptimumeconomic thicknessofthethermalinsulationineachheadersection.Fortheexpansiontank,the thermallossesarecalculatedindividuallyforthegasandliquidsections. 201

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Chapter6 IntegrationofSystemComponents InthepreviouschaptersthedifferentsubsystemsofthePTCsolarpowerplantwere designedfornominalandpartialconditions.Inthenalstep,thesubsystemsmustbe integratedintotheplantandworktogether.Forexample,theenergycomingfromthesolar eldrunsthepowerblock,thepowerblockisaffectedbytheambientconditionsandatthe sametimethereturnHTFtemperaturefromthepowerblockaffectstheoutputfromthe solareld.Thiskindofconnectionbetweenthesubsystemsofthesolarpowerplantaffect itsperformanceundersteadyandtransientconditions.Initiallyapreliminarydesignunder steadystateconditionsisperformed,inthefollowingsteps: CalculationofDirectNormalRadiationDNI. CalculationoftheDNIcumulativefrequencydistributionCFD,atthisstepaDNI with95%ofthecumulativefrequencyisselectedasthereferenceradiationforthe preliminarydesign. Powerblockdesign: HTFselection. Conditionsofoperationofthepowerblocklowpressure,highpressure,turbine efciency,etc. 202

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TemperaturedropandmassowrateoftheHTF. Calculationoftheparasiticlossesinthecondensersystem. Selectionofthesolarcollector. Solarcollectorperformanceunderreferenceconditions:massowrateatnominal conditionsandtheheatgained. Selectionofsolareldlayoutanddeterminationofthenumberofloops. Calculationofthemaximumpressuredropthroughthesolareld. Calculationofpumpingpowerrequirementandheatlosses. Calculationoftheexpansiontankvolumeandheatlosses. Hourlysimulation. CalculationoftheoptimumsizebyLCOEandsolarenergyutilization. Figure6.1showsaowchartofthedifferentstepsdescribedabove. 6.1TransientAnalysis Althoughitiscommonlyassumedthatthesolarpowerplantoperatesundersteady stateconditions,thisassumptionsisnottrueformostoftheoperationtime.Inthiscase,a transientanalysisisnecessarytoincorporatethethermalinertiaofthesystem.Thetransient energybalanceofaSolarCollectorAssemblySCAisgivenby[151]Figure6.2: d dt m collector C HTF T = m HTF C HTF T in )]TJ/F59 11.9552 Tf 10.949 0 Td [(T + Q u .1 203

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Figure6.1LogicowforthepreliminarydesignofthePTCsolarplants 204

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Figure6.2NodeanalysisofthesolarcollectorassemblySCA Integratingthelastexpression,itisobtainedthat: Z t 1 + D t t 1 0 dT T )]TJ/F59 11.9552 Tf 10.95 0 Td [(T in )]TJ/F51 11.9552 Tf 14.471 2.654 Td [( Q u = m HTF C HTF = )]TJ/F51 11.9552 Tf 20.37 8.093 Td [( m HTF m collector Z t 1 + D t t 1 0 dt .2 T t 1 + D t 1 = T t 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T HTF ; in )]TJ/F51 11.9552 Tf 37.498 10.747 Td [( Q u m HTF C HTF exp )]TJ/F51 11.9552 Tf 20.369 8.093 Td [( m HTF m collector D t + T HTF ; in + Q u m HTF C HTF .3 Usinganodalnotation,thelastexpressioncanberewrittenas: T n + 1 i = T n i )]TJ/F59 11.9552 Tf 10.949 0 Td [(T n i )]TJ/F51 8.9664 Tf 6.967 0 Td [(1 )]TJ/F51 11.9552 Tf 36.894 10.848 Td [( Q u ; i m HTF C HTF ; i exp )]TJ/F51 11.9552 Tf 15.741 8.093 Td [( m HTF V SCA r i D t .4 + T n i )]TJ/F51 8.9664 Tf 6.966 0 Td [(1 + Q u ; i m HTF C HTF ; i .5 V SCA = p 4 D 2 abs L c where T n + 1 i isthetemperatureatnodeiinthenexttimestep. Thesametransientanalysisisemployedforthethepipeheader,afterapplyingthe energybalancetothesystemFigure6.3thenextordinarydifferentialequationODEis found: d dt m T C T T = m f C HTF T in )]TJ/F59 11.9552 Tf 10.949 0 Td [(T )]TJ/F51 11.9552 Tf 14.471 2.654 Td [( Q loss .6 205

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Figure6.3Thermalcapacitanceanalysisofthepipeheader Q loss = z T )]TJ/F59 11.9552 Tf 10.949 0 Td [(T a Integrating: Z t 1 + D t t 1 0 dT T )]TJ/F59 11.9552 Tf 10.949 0 Td [(T p = )]TJ/F51 11.9552 Tf 13.408 8.449 Td [( m f C HTF + z m T C T Z t 1 + D t t 1 0 dt .7 T p = m f C HTF T in + z T a m f C HTF + z T t 1 + D t 1 = T p + T t 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(T p exp )]TJ/F51 11.9552 Tf 13.408 8.449 Td [( m f C HTF + z m T C T D t .8 whichmayberewrittenas: T n + 1 = T p + T n )]TJ/F59 11.9552 Tf 10.949 0 Td [(T p exp )]TJ/F51 11.9552 Tf 13.408 8.449 Td [( m f C HTF + z m T C T D t .9 ThelastEquationcanbeusedforthethermalcapacitancecalculationofthehotheader, coldheaterandexpansiontank.Theterm m T C T includesthethermalcapacitanceofthe HTF,pipewallsandinsulation.Inthissimulation,thethermalcapacitanceofthepipe wallsandinsulationwasdistributeduniformlyoverthesolareld.Thesameassumption wasusedfortheHTFinthehotandcoldheaders.Figures6.4and6.5showthedistribution 206

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Figure6.4ThermalinertiadistributionforIlayout Figure6.5ThermalinertiadistributionforHlayout 207

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Figure6.6LogicowusedforthedynamicsimulationofthePTCsolarpowerplant 208

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ofthethermalinertiaassumedthroughthesolareldforIandHlayouts.Figure6.6shows thelogicowusedforthedynamicsimulationofthesystems,inthiscaseallthesystems shouldbesimultaneouslysolved. 6.2EconomicAnalysis ThelevelizedcostofenergyLCOEisthecostthat,ifassignedtoeveryunitofelectricityproducedbythesolarpowerplantovertheprojectlife,willequalthetotallife-cycle costTLCCwhendiscountedbacktothebaseyear[152].LCOEisusedtocompare thecostofelectricitygeneratedbyarenewableresourceinthiscasesolarenergywith theequivalentfossilfuelunitortooptimizethesolareldunderdifferentscenarios.The LCOEwithoutincentivesisgivenby[152,153]: LCOE = I + N n = 1 Cost annual ; n = 1 + d n 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(TR N n = 1 E annual 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(R d n = 1 + d n .10 where I istheinitialinvestment, Cost annual istheannualFuelandO&Mcosts, d isthe discountrate, TR isthetaxrate, R d isthesystemdegradationrate, E annual isthenetannual poweroutputinkWh,and N istheprojectlife.Thetotalinvestmentisthesumofdirect andindirectcosts.Thedirectcostsaregivenby[15,152]: DC =[ SI + SF + HTF system A SF + C storage + C FB + C PB ] 1 + F contingency .11 where SI isthesiteimprovementscost, SF isthesolareldcost, HTF system isthecostof theHTF, A SF isthetotalareaofthesolareld, C storage isthecostofthestorageenergy, 209

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C FB isthecostoffossilbackup, C PB isthecostofthepowerblock,and F contingency isthe contingencyfactor.Theindirectcostsareasfollows: IC = DC EPC + PLM + ST .12 where EPC istheengineer,procureandconstructcost, PLM istheproject,landandmanagementcost,and ST isthesaleTax.Theannualcostsarecalculatedas: Cost annual = FC + FCCP nom + VCG E annual MWh + C Fuel .13 where FC isthexedannualCost, FCC isthexedcostbycapacity, VCG isthevariable costpergeneration,and C Fuel isthefuelcost.Table6.1showsthevaluesassumedforthe economicanalysis.Thediscountrateisusedforthecalculationofthepresentvalueby takingintoaccountthetimevalue.Twodifferentanalysescanbeperformedbyaccounting fortheination:nominaldiscountrateincludeinationaryeffects,andrealdiscountrate excludeinationaryeffects.Discountratecanbeconvertedfromrealtonominalandvice versabyusingthefollowingformulas[152]: d n = 1 + d r 1 + e )]TJ/F51 11.9552 Tf 10.949 0 Td [(1 d r =[ 1 + d r = 1 + e ] )]TJ/F51 11.9552 Tf 10.949 0 Td [(1.14 where d n isthenominaldiscountrate, d r istherealdiscountrate,and e istheinationrate. Ontheotherhand,theeffectivetaxrate,orcombinedstateandfederaltaxrateiscalculated as[152]: TR = STR + FTR 1 )]TJ/F59 11.9552 Tf 10.949 0 Td [(STR .15 where STR isthestatetaxrate,and FTR isthefederaltaxrate. 210

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Table6.1Costs,taxesanddiscountrateassumedfortheeconomicalanalysis.Valuestaken from[15,140,152] DirectCostValueUnits SiteImprovements25$/m 2 SolarField295$/m 2 HTFSystem90$/m 2 Storage80$/kWht FossilBackup0$/kW PowerBlockWet-Cooled940$/kW PowerBlockDry-Cooled1160$/kW Contingency10%DC IndirectCost Engineer,ProcureandConstruct15%DC Project,LandandManagement3.5%DC SalesTax7.75%DC O&MCost FixedAnnualCost0$/yr FixedCostbyCapacity70$/kW-yr VariableCostperGeneration3$/MWh FuelCost0$/MWh Taxes,InterestandSystemDegradation RealDiscountRate8% InationRate2.5% FederalTaxRate35% StateTaxRate8% AnnualDegradationRate0.5% ProjectLife30years 211

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6.3Results Thesolarpowerplantwassimulatedeveryhourbyusingthemonthlyaveragevalues. Table6.2showstheparametersusedforthehourlysimulation.Thesolareldsizewas increasedbyaddingevencollectorloopstothesolarlayout,andaparametricanalysiswas thencarriedouttogettheoptimumsize. Table6.2Parameterusedforthehourlysimulation NominalPowerOutput50MW e HoursofThermalStorageTES:0 SolarRadiationData:TMY3 Location:Tampa,Daggett HTF:VP1 SolarCollector:LS-3 SolarReceiver:UVAC Annulusundervacuum,P=10 )]TJ/F51 8.9664 Tf 6.966 0 Td [(4 Torr Layout:H h SF ; pump [5]:60% m SF ; night [141]:20% m SF ; day ; min :20% Condenser:CoolingTower AirCooledCondenser Theresultsobtainedforthecoolingtowerusedasthecoolingsystemforthepower blockarepresentedinTables6.3and6.4.Asitwasexpected,thereisaminimumLCOE whichcorrespondstotheoptimumsolarplantsize.ForTampathisnumbercorresponds to136collectorloops,whileforDaggetttheoptimumLCOEcorrespondsto88collector loops.Thedifferenceintheoptimumnumberofloopsisrelatedtothesolarradiation distribution. 212

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Table6.3ResultsobtainedforTampa.ParametersusedforthesimulationaregiveninTable6.2 A SF m 2 n loops W net ; cycle kW W par kW C Factor % LCOE R /kWh LCOE N /kWhNetPowerGWh 246343.67249312.82344.414.337.345.959.0 273715.28049312.82532.216.335.143.266.8 301086.78849312.82678.418.233.441.174.7 328458.29649312.82863.420.332.039.482.6 355829.710449312.83020.622.131.138.489.7 383201.211249312.83202.723.830.437.696.4 410572.812049312.83367.125.529.937.0102.8 437944.312849312.83555.727.129.736.7108.8 465315.813649312.83760.628.529.636.7113.8 492687.314449312.83899.629.629.836.8118.2 520058.815249312.84112.130.830.037.1122.3 547430.416049312.84294.431.830.337.6125.7 213

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Table6.4ResultsobtainedforDaggett.ParametersusedforthesimulationaregiveninTable6.2 A SF m 2 n loops W net ; cycle kW W par kW C Factor % LCOE R /kWh LCOE N /kWhNetPowerGWh 191600.65649312.82481.318.824.830.377.1 218972.26449312.82762.622.322.627.790.9 246343.77249312.83053.125.621.326.1103.6 273715.28049312.83321.528.220.825.6113.4 301086.78849312.83619.330.220.725.5120.7 328458.29649312.83891.931.421.226.1124.9 355829.810449312.84197.132.721.626.7129.0 383201.311249312.84468.333.922.127.2133.1 410572.812049312.84766.834.922.628.0136.2 437944.312849312.85047.635.723.328.9138.3 465315.813649312.85360.036.224.229.9139.5 214

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aTampa bDaggett Figure6.7FrequencydistributionofDirectNormalIrradianceDNI AsseeninFigure6.7,thefrequencydistributionofDirectNormalIrradianceDNI forTampashowsthatitsDNIisbelow500W/m 2 foralmost27%oftheyearthewhile forDaggettitisonlyfor15.8%oftheyear.Thisimpliesthat,forTampa,inorderto increasetheannualoutputfromthesolareldanappreciableincreaseinthesolareldsize isrequiredtocompensateforthosemonthswhenthesolarradiationislow.Forthecase ofDaggett,higherradiationisobtainedduringthewholeyearandthereforethesolareld sizeissmallerascomparedwithTampa.ThisisevidencedbyFigure6.8,whichshowsthat forDaggettamoreuniformmonthlyaveragenetpoweroutputisobtainedascomparedto Tampa. Inordertovalidatetheresultsobtainedfromtheproposedmodel,itwascompared withamodeldevelopedbyNREL,theSolarAdvisorModelSAM[15].Figures6.9 and6.10showtheresultsobtained,levelizedcostofelectricityevaluatedatrealdiscount rateLCOE R ,andtheannualnetpoweroutput,forTampaandDaggettrespectively.The 215

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aTampa bDaggett Figure6.8Monthlyaveragedistributionofthenetpoweroutputcalculatedatminimum LCOE R .ParametersusedforthesimulationaregiveninTable6.2 resultsshowthattheproposedmodelfollowsthetrendgivenbySAM,buttherearesome discrepancies.Theassumptionsandmodelingofthephysicalphenomenaaredifferent inbothofthesemodels,henceitisexpectedthattheresultsobtainedwouldhavesome aLCOE R bAnnualNetPowerOutput Figure6.9ComparisonoftheLCOE R andannualnetpoweroutputbetweentheproposed modelandSystemAdvisorModelSAM[15],location:Tampa.Parametersusedforthe simulationaregiveninTable6.2 216

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aLCOE R bAnnualNetPowerOutput Figure6.10ComparisonoftheLCOE R andannualnetpoweroutputbetweentheproposed modelandSystemAdvisorModelSAM[15],location:Daggett.Parametersusedforthe simulationaregiveninTable6.2 differences.Asitwasexplainedinchapter3,theproposedmodelismoreconservative thanthemodelsdevelopedbyNREL,whichexplainswhythenetpoweroutputobtained fromthemodelislowerthantheresultsobtainedbySAMinbothcities.TheLCOE R calculatedfromtheproposedmodelisalsolowerthanthevaluesobtainedfromSAMdue tothedifferenceintheannualcostwhichisproportionaltotheannualnetpoweroutput. AnotherimportantparameterforthedesignofthePTCsolarpowerplantistheutilizationofsolarenergywhichisameasureofhowmuchofthecollectedsolarenergyisused bythepowerblock[12].Theutilizationisdenedas: Utilization = Collected-Dissipated Solarenergy Collected Solarenergy .16 217

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aTampa bDaggett Figure6.11EffectofthenumberofloopsontheLCOE R andtheutilizationfactor.ParametersusedforthesimulationaregiveninTable6.2 Autilizationvaluelessthanunitymeansthatthecollectedenergyismorethanwhat canbeusedtorunthepowerblock,andthereforeapartoftheenergycollectedbythesolar eldisdumped.Figure6.11showstheutilizationvalueasfunctionofthesolareldsize. Utilizationvaluesequaltounityareobtainedforsolareldsizesclosetothereference conditions,butinordertodecreasetheLCOE,itisnecessarytoincreasethenetpower outputbyincreasingthesolareldsizeandconsequentlytheutilizationfactordecreases duetomoreenergyhavingtobedumped.Thiscanbeavoidedbyincorporatingastorage systemtothesolarplantorbydefocusingthesolarcollectorsathighsolarirradiance. 6.3.1ResultsforAirCooledCondensers Afterperformingtheanalysisusingtheconvectionalwetcoolingmethodforcondensers,ananalysisoftheeffectofthealternativecondensingmethodaircooledcondenseronthesolarpowerplantperformancewascarriedout.Initiallythenumberofair cooledcondenserunitswasdetermined.TheresultsobtainedarepresentedinFigure6.12; 218

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aTampa bDaggett Figure6.12Effectofthecondensingmethodonthepowercycleperformance thebestperformancecorrespondstotheevaporativemethodsincetheminimumcondensingwatertemperatureislimitedbythewetbulbairtemperature.Forthecaseofaircooled condenser,thenetpoweroutputislowerthantheevaporativecaseowingtothehighfan powerrequirement. AsseeninFigure6.12,theperformanceofaircooledcondenserhasanoptimumnumberofcondenserunits,inwhichthereisabalancebetweenthecyclepoweroutputand parasiticlosses.Forbothcities,theoptimumnumberofaircooledcondenserunitsare15. ForthecaseofTampa,thecondenserpressurenearthedesignconditions.08barcan beachieved,butforDaggetthighercondenserpressuresareexpectedduetoitslocation. Figure6.13showsthemonthlyaveragedistributionofthecondenserpressureforthetwo coolingmethodsproposed.ForTampa,highrelativehumidityandrelativehighairtemperaturearepresentduringmostoftheyear,thereforeaircooledcondensergivesalittle improvementinthecondenserpressurebutitshighpowerrequirementsdecreasethenet 219

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CoolingTower AirCooledCondenser aTampa CoolingTower AirCooledCondenser bDaggett Figure6.13Monthlyaveragedistributionofthecondenserpressure.Parameterusedforthe simulationaregiveninTable6.2 220

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poweroutput.TheoppositeisseeninDaggett,wherelowrelativehumidityandconsequentlylowwetbulbtemperaturearepresent;forthislocationevaporativecoolingmethod providesacceptablecondensingpressurestothepowerblock.ForDaggett,theaircooled condensersgenerateadecreaseinthecycleperformanceduetothehighercondensingpressuresandhigherparasiticlossesduetothehighairtemperatures. TheevaluationoftheaircooledcondenserwasperformedforthesolareldsizecorrespondingtotheminimumLCOE.Table6.5showstheresultsobtainedforevaporativeand aircooledcondenser. Table6.5EffectofthecondensertypeontheannualperformanceofthePTCsolarpower plant.ParametersusedforthesimulationaregiveninTable6.2 W net ; cycle kW W par kW LCOE R /kWh LCOE N /kWhNetPowerGWh Tampa,CoolingTower 49312.83760.629.636.7113.8 Tampa,AirCooledCondenser 49312.85088.931.939.5110.7 Daggett,CoolingTower 49312.83619.320.725.5120.7 Daggett,AirCooledCondenser 49312.85363.322.828.1116.2 ThemonthlyperformanceofthePTCsolarplantisshowninFigure6.14.Asitwas mentionedbefore,thefanpowerrequirementsandthehighairtemperaturesdecreasethe netpoweroutput,thedifferenceisremarkableduringsummerdays. 221

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aTampa bDaggett Figure6.14Effectofthecondensertypeonthemonthlynetpoweroutput.Parameterused forthesimulationaregiveninTable6.2 Figure6.15showstheannualoutputobtainedforeachcoolingmethod;forTampathe reductioninthenetpoweroutputis2.8%whileforDaggettis3.7%.Thisreductionin netpoweroutputalsoaffectstheLCOE,theincreaseintheLCOEis7.7%and10.1%for TampaandDaggettrespectively. Figure6.15Annualnetpoweroutputforcoolingtowerandaircooledcondenser.Parameter usedforthesimulationaregiveninTable6.2 222

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Chapter7 ConclusionsandRecommendations Inthisdissertationacomprehensivemethodologyfordesigningparabolictroughsolar powerplantswithoutthermalstoragewasdeveloped.Themethodologyisbasedonthe individualdesignofdifferentcomponentsandsubsequentintegrationofthecomponents intothewholesystem. Thevalidationoftheresultsobtainedshowedthattheproposedmethodologyissuitable foranylocationandthatanoptimumcongurationcanbeachievedbysensitivityanalysis. Inthiscase,thelevelizedcostofelectricityLCOEisausefulparameterforobtainingthe optimumsizeofthesolareld.LCOEisoneofthemainparameterstoanalyze,butthe utilizationfactorisalsoimportanttoassurethatthesolarenergycollectedisalmosttotally usedbythepowerblock. Ontheotherhand,theanalysisofalternativecondensersshowedthatalthoughair cooledcondensersareanexcellentalternative,theparasiticlossesfanpowerrequirements andthehighercondensingpressures,especiallyinhotlocations,makethistechnologyless attractiveexceptincertainlocationswherethereiswateravailable. 223

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Thefollowingrecommendationsshouldbeconsideredforfurtherresearchinthisarea: Inordertosimulatethepowerblockatdifferentreferenceconditions,acorrection factorthatcanbeappliedtotheoriginalpowerblockttingequationshouldbedetermined. Alternativeanddifferentcombinationsbottomingcyclesofpowerblocksshouldbe studiedformaximizingthenetpoweroutput. Thedesignoftheaircooledandevaporativecondensershouldbeimprovedbyusing morecomprehensivemodels. AmoredetailedcostanalysisandLCOEcalculationshouldbeincluded.Thecost analysisproposedinthepresentworkissimpleanddidnotincludetaxincentives andcommercialloans. Thermalstoragemodelshouldbeincludedinfuturework.Theheattransfermodel usedforthethermalexpansiontankcanbeextendedfortheheatlossesinthethermal storagesystem. Morerealisticcontrolstrategiesshouldbeimplementedintheprogramtokeepthe collectoroutlettemperaturealmostconstantwithoutdumpingenergy. 224

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Appendices 239

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AppendixA:ThermophysicalPropertiesofGases Thespecicheats,absoluteviscosities,andthermalconductivitiesareonlyfunctionof thetemperature.Toobtainthedensityofagas,theperfectgaslawmaybeused[55]: P = r RT A.1 Specicheat kJ = kgK ,absoluteviscosity m Pa s ,andthermalconductivity W = mK aredenedby[55]: C p = N i = 0 A i T i A.2 m = N i = 0 B i T i A.3 k = N i = 0 C i T i A.4 where T isinK 240

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AppendixA:Continued Physicalpropertiesofair[55]: Molecularweight kg = mol :28.966 GasConstant R kJ = kgK :0.287040 CriticalTemperature T c K :132.6 CriticalPressure P c Mpa :3.77 TableA.1ThermophysicalcoefcientsofairEquationsA.2-A.4.Adaptedfrom[55] 1 250 T < 1050 K 2 250 T < 600 K 3 600 T < 1050 K iA i 1 B i 2 B i 3 C i 1 01 : 03409 )]TJ/F51 10.9091 Tf 8.476 0 Td [(9 : 8601 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 4 : 8856745 )]TJ/F51 10.9091 Tf 8.476 0 Td [(2 : 276501 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 1 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 2848870 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 9 : 080125 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 5 : 43232 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 1 : 259848 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 20 : 7816818 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(6 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 17635575 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 4261775 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 481523 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 4970786 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(9 1 : 2349703 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 7 : 9306 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 1 : 735506 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(10 40 : 1077024 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 )]TJ/F51 10.9091 Tf 8.477 0 Td [(5 : 7971299 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(11 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 10398 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 066657 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(13 50002 : 476630 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(17 60000 241

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AppendixA:Continued Physicalpropertiesofhydrogen[55]: Molecularweight kg = mol :2.016 GasConstant R kJ = kgK :4.124289 CriticalTemperature T c K :33.3 CriticalPressure P c Mpa :1.3 TableA.2ThermophysicalcoefcientsofhydrogenEquationA.2.Adaptedfrom[55] 1 250 T < 425 K 2 425 T < 490 K 3 490 T < 1050 K iA i 1 A i 2 A i 3 05 : 0066253014 : 494714 : 920082 11 : 01569422 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 0 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 996917584 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 2 )]TJ/F51 10.9091 Tf 8.476 0 Td [(6 : 02891517 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 02 : 540615 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 32 : 73758940 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 0 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 7588954 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(10 4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(8 : 47582750 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 00 51 : 43800374 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(11 00 6 )]TJ/F51 10.9091 Tf 8.476 0 Td [(9 : 80724030 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(15 00 242

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AppendixA:Continued TableA.3ThermophysicalcoefcientsofhydrogenEquationsA.3-A.4.Adaptedfrom [55] 1 250 T < 500 K 2 500 T < 1050 K iB i 1 B i 2 C i 1 C i 2 0 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 1356662 : 729412 : 01 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 0 : 108 16 : 84115878 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 2 : 3224377 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 3 : 23 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 2 : 21 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 2 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 928747 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(7 : 6287854 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(6 2 : 16 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 2 : 26 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 31 : 8996 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 2 : 92585 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 )]TJ/F51 10.9091 Tf 8.477 0 Td [(6 : 49 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(9 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 74 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(10 4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(5 : 23104 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 )]TJ/F51 10.9091 Tf 8.476 0 Td [(5 : 2889938 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(13 5 : 52 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 4 : 65 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(14 57 : 4490972 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 000 6 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 250937 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(15 000 243

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AppendixA:Continued Physicalpropertiesofargon[55]: Molecularweight kg = mol :39.948 GasConstant R kJ = kgK :0.208129 CriticalTemperature T c K :150.8 CriticalPressure P c Mpa :4.87 TableA.4ThermophysicalcoefcientsofargonEquationsA.2-A.4.Adaptedfrom [55] 1 200 T < 1600 K 2 200 T < 1000 K 3 200 T < 540 K 4 540 T < 1000 K iA i 1 B i 2 B i 3 C i 4 00 : 52034 )]TJ/F51 10.9091 Tf 8.476 0 Td [(5 : 2839462 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 1 : 225734 : 03764 107 : 60706705 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 5 : 9456964 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 7 : 3665688 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 20 )]TJ/F51 10.9091 Tf 8.476 0 Td [(6 : 4749393 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(8 1 : 897011 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 3867 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 305 : 41874502 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(11 )]TJ/F51 10.9091 Tf 8.476 0 Td [(8 : 171242 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 1 : 127158 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(8 40 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 22024235 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(14 1 : 2939183 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 585569 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 501 : 17962552 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(17 )]TJ/F51 10.9091 Tf 8.476 0 Td [(7 : 5027442 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(13 0 60 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 86231745 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(21 00 244

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AppendixA:Continued Physicalpropertiesofnitrogen[55]: Molecularweight kg = mol :28.013 GasConstant R kJ = kgK :0.296798 CriticalTemperature T c K :126.2 CriticalPressure P c Mpa :3.4 TableA.5ThermophysicalcoefcientsofnitrogenEquationsA.2-A.4.Adaptedfrom [55] 1 280 T < 590 K 2 590 T < 1080 K 3 250 T < 1050 K iA i 1 A i 2 B i 3 C i 3 01 : 088041 : 4055072 : 5465 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 523178 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 1 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 55968 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 189456 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 7 : 533653 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 1 : 1887996 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 27 : 290760 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 4 : 785289 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(6 )]TJ/F51 10.9091 Tf 8.477 0 Td [(6 : 5156624 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 209284 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(7 3 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 886155 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(10 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 54016 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(9 4 : 34945 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(8 1 : 1556780 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(10 402 : 0849125 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(12 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 562245 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(11 )]TJ/F51 10.9091 Tf 8.477 0 Td [(6 : 3653734 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(14 50 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 790303 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(16 2 : 24966 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(15 1 : 4716702 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(17 60000 245

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AppendixB:DataofParabolicTroughCollectors TableB.1Geometricalandopticaldataforparabolictroughcollectors.Adaptedfrom[4] Collector w 1 m f 2 m L e 3 m L c 4 m LS-12.550.946.350.2 LS-25.001.49849.0 LS-35.761.711299.0 IST 8 2.300.766.149.0 EuroTrough5.761.7112150.0 SkyTrough 9 6.001.7113.9115.0 CollectorMirroraream 2 D 5 m C 6 h o 7 % LS-11280.0461:171 LS-22350.0771:176 LS-35450.0782:180 IST 8 4240.0450:178 EuroTrough8170.0782:180 SkyTrough 9 7500.0875:177 1 Aperturewidth 2 Focallength 3 Lengthperelement 4 Lengthpercollector 5 Receiverdiameter 6 Geometricconcentration 7 Peakopticalefciency 8 IndustrialSolarTechnology 9 Takenfrom:[154] 246

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AppendixC:ThermophysicalPropertiesofHeatTransferFluidHTF Density kg = m 3 isdenedby: r = N i = 0 a i T i C.1 where T isinC TableC.1CoefcientsforuseinEquationC.1 HTF a 0 a 1 a 2 a 3 VP-1 1 [155]1083 : 22 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 9027 : 369 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 287 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 D-Q 1 [156]982 : 18 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 7764 : 827 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 0 D-RP 1 [157]1042 : 39 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 668 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 924 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 0 SolarSalt 1 [158]2090 : 18 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 64000 Hitec 1 [159,160]2081 : 44 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 72800 HitecXL 2 [15]2240 : 00 )]TJ/F51 10.9091 Tf 8.476 0 Td [(0 : 82600 1 R 2 = 0 : 99 2 R 2 isnotgiven FigureC.1DensityfordifferentHTFs 247

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AppendixC:Continued Specicheatatconstantpressure kJ = kgK isdenedby: C p = N i = 0 b i T i C.2 where T isinC TableC.2CoefcientsforuseinEquationC.2 HTF b 0 b 1 b 2 b 3 b 4 VP-1 1 [155]1 : 4713 : 497 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 817 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 8 : 400 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(9 0 D-Q 1 [156]1 : 5893 : 198 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(5 : 288 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 00 D-RP 1 [157]1 : 5612 : 975 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 000 SolarSalt 1 [158]1 : 0933 : 755 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 322 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 2 : 112 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(8 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 2 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(11 Hitec[159,160]1 : 5600000 HitecXL 2 [15]1 : 536 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 624 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 139 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 00 1 R 2 = 0 : 99 2 R 2 isnotgiven FigureC.2SpecicheatatconstantpressurefordifferentHTFs 248

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AppendixC:Continued Specicenthalpy kJ = kg isdenedby: h = N i = 0 c i T i C.3 where T isinC TableC.3CoefcientsforuseinEquationC.3 HTF c 0 c 1 c 2 c 3 VP-1 1 [155] )]TJ/F51 10.9091 Tf 8.476 0 Td [(18 : 9771 : 5131 : 2908 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 1 : 201 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 D-Q 1 [156]53 : 6711 : 5891 : 599 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 762 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 D-RP 1 [157] )]TJ/F51 10.9091 Tf 8.476 0 Td [(15 : 7591 : 5611 : 4875 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 0 SolarSalt 1 [158] )]TJ/F51 10.9091 Tf 8.477 0 Td [(354 : 8451 : 0921 : 877 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(3 )]TJ/F51 10.9091 Tf 8.477 0 Td [(4 : 409 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(6 Hitec[159,160] )]TJ/F51 10.9091 Tf 8.476 0 Td [(232 : 3601 : 56000 HitecXL 2 [15]01 : 536 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 312 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.477 0 Td [(3 : 796 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(8 c 4 c 5 VP-1 1 [155]00 D-Q 1 [156]00 D-RP 1 [157]00 SolarSalt 1 [158]5 : 282 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 )]TJ/F51 10.9091 Tf 8.476 0 Td [(2 : 4 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 Hitec[159,160]00 HitecXL 2 [15]00 1 R 2 = 0 : 99 2 R 2 isnotgiven 249

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AppendixC:Continued FigureC.3SpecicenthalpyfordifferentHTFs 250

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AppendixC:Continued Thermalconductivity W = mK isdenedby: k = N i = 0 d i T i C.4 where T isinC TableC.4CoefcientsforuseinEquationC.4 HTF d 0 d 1 d 2 d 3 d 4 VP-1 1 [155]0 : 138 )]TJ/F51 10.9091 Tf 8.476 0 Td [(8 : 738 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(5 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 720 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 00 D-Q 1 [156]0 : 124 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 239 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(6 : 320 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(8 00 D-RP 1 [157]0 : 133 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 296 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 000 SolarSalt 1 [158]0 : 4411 : 953 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 000 Hitec[159,160]0 : 2213 : 457 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 669 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 165 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 6 : 07 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(12 HitecXL[141]0 : 5190000 1 R 2 = 0 : 99 FigureC.4ThermalconductivityfordifferentHTFs 251

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AppendixC:Continued Absoluteviscosity cP isdenedby: ln m = N i = 0 e i T i C.5 ForHitecandHitecXLthenextequationisused: ln m = N i = 0 e i ln T i C.6 Forsolarsalt: m = N i = 0 e i T i C.7 where T isinC TableC.5CoefcientsforuseinEquationsC.5-C.7 HTF e 0 e 1 e 2 e 3 e 4 VP-1 1 [155]2 : 008 )]TJ/F51 10.9091 Tf 8.477 0 Td [(2 : 989 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 1 : 207 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(2 : 714 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(7 2 : 370 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(10 D-Q 1 [156]2 : 125 )]TJ/F51 10.9091 Tf 8.477 0 Td [(3 : 960 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 2 : 090 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(5 : 935 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(7 6 : 460 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(10 D-RP 1 [157]5 : 147 )]TJ/F51 10.9091 Tf 8.477 0 Td [(7 : 174 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(2 3 : 981 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 087 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(6 1 : 108 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(9 SolarSalt 1 [158]22 : 713 )]TJ/F51 10.9091 Tf 8.477 0 Td [(1 : 200 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 2 : 281 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(4 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 474 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(7 0 Hitec 1 [159,160]33 : 324 )]TJ/F51 10.9091 Tf 8.476 0 Td [(9 : 2706 : 364 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 00 HitecXL 2 [141]14 : 132 )]TJ/F51 10.9091 Tf 8.477 0 Td [(3 : 364000 1 R 2 = 0 : 99 2 R 2 isnotgiven 252

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AppendixC:Continued FigureC.5AbsoluteviscosityfordifferentHTFs 253

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AppendixC:Continued Vaporpressure kPa isdenedby: P v = N i = 0 f i T i C.8 where T isinC TableC.6CoefcientsforuseinEquationC.8 HTF f 0 f 1 f 2 f 3 f 4 VP-1 1 [155]0 : 789 )]TJ/F51 10.9091 Tf 8.476 0 Td [(1 : 379 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 3 : 783 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(3 : 387 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(5 1 : 056 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 D-Q 1 [156]24 : 738 )]TJ/F51 10.9091 Tf 8.476 0 Td [(7 : 325 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(1 8 : 525 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(3 )]TJ/F51 10.9091 Tf 8.476 0 Td [(4 : 684 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(5 1 : 080 10 )]TJ/F51 7.9701 Tf 6.193 0 Td [(7 D-RP 2 [157] )]TJ/F51 10.9091 Tf 8.476 0 Td [(354 : 5604 : 768 )]TJ/F51 10.9091 Tf 8.476 0 Td [(2 : 155 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(2 3 : 318 10 )]TJ/F51 7.9701 Tf 6.192 0 Td [(5 0 1 R 2 = 0 : 99 2 R 2 = 1 : 00 FigureC.6VaporpressurefordifferentHTFs 254

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AppendixD:PipeGeometry TableD.1Wallthickness,inmm,fordifferentnominalpipesizes PipeSchedule A-G. Adaptedfrom[141] NominalPipeSchedule PipeSize,inABCDEFG 2.52.113.055.167.019.5314.02 ::: 32.113.055.497.6211.1315.24 ::: 42.113.053.964.786.028.5611.13 62.773.404.787.1110.9714.2718.26 82.773.766.357.048.1810.3112.70 103.404.194.786.357.809.2712.70 123.964.576.358.389.5310.3112.70 144.786.357.929.5311.1312.7015.09 164.786.357.929.5312.7014.3521.44 184.786.357.929.5311.1312.7014.27 205.546.359.5312.7015.0920.6226.19 226.359.5312.7022.2328.5834.9341.28 246.359.5312.7014.2717.4824.6130.96 267.929.5312.70 :::::::::::: 287.929.5312.7015.88 ::::::::: 307.929.5312.7015.88 ::::::::: 327.929.5312.7015.8817.48 :::::: 347.929.5312.7015.8817.48 :::::: 367.929.5312.7015.8819.05 :::::: 429.5312.7015.8819.05 ::::::::: 489.5312.7019.0525.40 ::::::::: 549.5312.7019.0525.40 ::::::::: 609.5312.7019.0525.40 ::::::::: 669.5312.7019.0525.40 ::::::::: 729.5312.7019.0525.40 ::::::::: 255

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AppendixD:Continued TableD.2Wallthickness,inmm,fordifferentnominalpipesizes PipeSchedule H-M. Adaptedfrom[141] NominalPipeSchedule PipeSize,inHIJKLM 2.5 :::::::::::::::::: 3 :::::::::::::::::: 413.4917.12 :::::::::::: 621.95 ::::::::::::::: 815.0918.2620.6222.2323.01 ::: 1015.0918.2621.4425.4028.58 ::: 1214.2717.4821.4425.4028.5833.32 1419.0523.8327.7931.7535.71 ::: 1626.1930.9636.5340.49 :::::: 1819.0523.8329.3634.9339.6745.24 2032.5438.1044.4550.01 :::::: 2247.6353.98 :::::::::::: 2438.8946.0252.3759.54 :::::: 26 :::::::::::::::::: 28 :::::::::::::::::: 30 :::::::::::::::::: 32 :::::::::::::::::: 34 :::::::::::::::::: 36 :::::::::::::::::: 42 :::::::::::::::::: 48 :::::::::::::::::: 54 :::::::::::::::::: 60 :::::::::::::::::: 66 :::::::::::::::::: 72 :::::::::::::::::: 256

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AppendixD:Continued TableD.3Insidediameter,inmm,fordifferentnominalpipesizes PipeSchedule A-G. Adaptedfrom[141] NominalPipeSchedule PipeSize,inABCDEFG 2.568.866.962.759.054.045.0 ::: 384.782.877.973.766.658.4 ::: 4110.1108.2106.4104.7102.397.292.0 6162.7161.5158.7154.1146.3139.7131.7 8213.5211.6206.4205.0202.7198.5193.7 10266.2264.7263.5260.4257.5254.5247.7 12315.9314.7311.2307.1304.8303.2298.5 14346.0342.9339.8336.6333.3330.2325.4 16396.8393.7390.6387.4381.0377.7363.5 18447.6444.5441.4438.2434.9431.8428.7 20496.9495.3489.0482.6477.8466.8455.6 22546.1539.8533.4514.4501.7489.0476.3 24596.9590.6584.2581.1574.6560.4547.7 26644.6641.4635.0 :::::::::::: 28695.4692.2685.8679.5 ::::::::: 30746.2743.0736.6730.3 ::::::::: 32797.0793.8787.4781.1777.8 :::::: 34847.8844.6838.2831.9828.6 :::::: 36898.6895.4889.0882.7876.3 :::::: 421047.81041.41035.11028.7 ::::::::: 481200.21193.81181.11168.4 ::::::::: 541352.61346.21333.51320.8 ::::::::: 601505.01498.61485.91473.2 ::::::::: 661657.41651.01638.31625.6 ::::::::: 721809.81803.41790.71778.0 ::::::::: 257

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AppendixD:Continued TableD.4Insidediameter,inmm,fordifferentnominalpipesizes PipeSchedule H-M. Adaptedfrom[141] NominalPipeSchedule PipeSize,inHIJKLM 2.5 :::::::::::::::::: 3 :::::::::::::::::: 487.380.1 :::::::::::: 6124.4 ::::::::::::::: 8188.9182.5177.8174.6173.1 ::: 10242.9236.5230.2222.3215.9 ::: 12295.3288.9281.0273.1266.7257.2 14317.5307.9300.0292.1284.2 ::: 16354.0344.5333.3325.4 :::::: 18419.1409.5398.5387.4377.9366.7 20442.9431.8419.1408.0 :::::: 22463.6450.9 :::::::::::: 24531.8517.6504.9490.5 :::::: 26 :::::::::::::::::: 28 :::::::::::::::::: 30 :::::::::::::::::: 32 :::::::::::::::::: 34 :::::::::::::::::: 36 :::::::::::::::::: 42 :::::::::::::::::: 48 :::::::::::::::::: 54 :::::::::::::::::: 60 :::::::::::::::::: 66 :::::::::::::::::: 72 :::::::::::::::::: 258

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AbouttheAuthor RicardoVasquezPadillawasborninBarranquilla,ColombiainMarch1978.HepursuedhisbachelorandmasteratUniversidaddelNorteinMechanicalEngineering.After hegotmarriedwithhisbelovedJennifer,hebegandoctoralstudiesatUniversityofSouth FloridaundertheguidanceofDr.YogiGoswamiandDr.EliasStefanakosintheClean EnergyResearchCenterCERC.Initially,heworkedintheexperimentalandtheoreticalanalysisoftheGoswamicycleandthenonthedesignofparabolictroughPTCsolar powerplants.InAugust2010hisrstsonNoahwasborn.Upongraduation,theauthorwill returntoUniversidaddelNorte,therehewillworkasfacultymemberintheDepartment ofMechanicalEngineeringintheareaofrenewableenergy. 259


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Simplified methodology for designing parabolic trough solar power plants
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ABSTRACT: The performance of parabolic trough based solar power plants over the last 25 years has proven that this technology is an excellent alternative for the commercial power industry. Compared to conventional power plants, parabolic trough solar power plants produce significantly lower levels of carbon dioxide, although additional research is required to bring the cost of concentrator solar plants to a competitive level. The cost reduction is focused on three areas: thermodynamic efficiency improvements by research and development, scaling up of the unit size, and mass production of the equipment. The optimum design, performance simulation and cost analysis of the parabolic trough solar plants are essential for the successful implementation of this technology. A detailed solar power plant simulation and analysis of its components is needed for the design of parabolic trough solar systems which is the subject of this research. Preliminary analysis was carried out by complex models of the solar field components. These components were then integrated into the system whose performance is simulated to emulate real operating conditions. Sensitivity analysis was conducted to get the optimum conditions and minimum levelized cost of electricity (LCOE). A simplified methodology was then developed based on correlations obtained from the detailed component simulations. A comprehensive numerical simulation of a parabolic trough solar power plant was developed, focusing primarily on obtaining a preliminary optimum design through the simplified methodology developed in this research. The proposed methodology is used to obtain optimum parameters and conditions such as: solar field size, operating conditions, parasitic losses, initial investment and LCOE. The methodology is also used to evaluate different scenarios and conditions of operation. The new methodology was implemented for a 50 MWe parabolic trough solar power plant for two cities: Tampa and Daggett. The results obtained for the proposed methodology were compared to another physical model (System Advisor Model, SAM) and a good agreement was achieved, thus showing that this methodology is suitable for any location.
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