Computer Simulation of Solar Combisystem Based on the Second Law Analysis

Xiao-Nan Wen

A Thesis

The Department

Building, Civil and Environmental Engineering

Presented in Partial Fulfillment of the Requirements

for the Degree of Master of Applied Science (Building Engineering) at

Concordia University

Montréal, Québec, Canada

December 2013

CONCORDIA UNIVERSTIY

School of Graduate Studies

This is to certify that the thesis prepared

By: Xiao-Nan Wen

Entitled: Computer Simulation of Solar Combisystem Based on the Second Law Analysis and submitted in partial fulfillment of requirements for the degree of

Master of Applied Science (Building Engineering) complies with the regulations of the University and meets the accepted standards with respect to originality and quality.

Signed by the final examining committee:

Chair Dr. Fariborz Haghighat

Examiner Dr. Marius Paraschivoiu

Examiner Dr. Fariborz Haghighat

Examiner Dr. S. Samuel Li

Supervisor Dr. Radu G. Zmeureanu

Approved by Dr. Tarek Zayed, GPD Department of Building, Civil and Environment Engineering

Dr. Christopher W. Trueman, Interim Dean Faculty of Engineering and Computer Science

Date

Abstract

ComputerSimulationofSolarCombisystemBasedontheSecondLawAnalysis XiaoͲNanWen ThisthesispresentsthedevelopmentofaSecondLawmodelofasolar combisystemforresidentialapplications.Theworkpresentedinthethesisincludes:1)A methodologyforthedevelopmentofacombisystemcomputermodel;2)The mathematicalmodelsofasolarcombisystemcomponents;3)ThefeaturesofanEESͲ basedcomputermodelofasolarcombisystembeingdeveloped;4)Thesimulated

operationofasolarcombisystem;and5)Theanalysisoftheperformanceofasimulated solarcombisystembasedontheSecondLawofthermodynamics.

ThesolarcombisystemhadanannualsystemCoefficientofPerformance(COP) of3.98,anannualsystemexergeticCOPof0.17,anannualsystemenergyefficiencyof

42.0%,anannualsystemexergyefficiencyof40.7%,anannualoverallsolarenergy contributionof89.5%,anannualoverallsolarexergycontributionof88.8%,anannual inputsolarenergycontributionof82.7%,andanannualinputsolarexergycontribution of26.5%.Comparedtoanelectricheatingsystem,thesolarcombisystemhadanannual solarenergyfractionof1.3andanannualsolarexergyfractionof0.1.

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Acknowledgements

Iamgratefultomanypeoplewhoencouragedandinspiredmeduringthegraduatestudy.

Firstofall,Ithankmyteachers:Dr.AndreasK.Athienitis,Dr.TheodoreStathopoulos,Dr. RamaniRamakrishnan(RyersonUniversity,Toronto),Dr.HashemAkbari,Dr.RaduG. Zmeureanu,Dr.PaulP.Fazio,Dr.FariborzHaghighat,andDr.MohammedZaheeruddin.Their courselectureswereenlightening.

Ibenefitedgreatlyfrompeerdiscussionwithmyfellowgraduatestudentsontechnicaltopics. Theirvariousstrengthsdirectlyorindirectlycontributedtomystudy.Thanksto:Justin Tamasauskasonicestorage;Dr.AhmedElShenawyonresearchmethodology;JunLiuonsystem control;LaurentMagnierontransportphenomena;VéroniqueTremblayonuncertainty;Eric McDonaldonsubstanceproperty;NicholasZibinoncalibration;LiyaLuoonverification.

Thecompletionofmythesisprojectwouldnotbepossiblewithouttheresourcesprovidedby ourgroup.Thanksto:JasonNgChengHinforextractingdatafromTRNSYSandmodelsby MitchellLecknerandAlexandreHugo;JustinTamasauskasforsharingdata,andDr.Danielle MonfetforEnergyPlusreferences.IalsothankthesupportandfriendshipofMarieͲClaude Hamelin,AndreeaMihai,andMathieuLeCam.

IoweveryspecialthankstomythesissupervisorDr.RaduG.Zmeureanuforlayingthe foundationandtakingmeonthisjourneyofresearch.Ihavebeenfortunatetositinhisclass andjoininhisresearchstudentgroup.Duringthisthesisproject,hisconstantchallenges motivatedmetodeepenmyunderstandingofthestudiedtopic.Hetaughtmefocusand determination.Iamtrulygratefulforhisguidance.

IwouldalsoliketothankMs.JennyDrapeaumyGraduateProgramsAdvisorandMs.Olga SoaresmyDepartmentAdministratorfortheirsupport.

TheresearchwasfundedbythefinancialsupportoftheNaturalSciencesandEngineering ResearchCouncilofCanadaandtheFacultyofEngineeringandComputerScienceofConcordia University.

Atlast,Ithankmyparentswhohavebeensupportivewithopenminds,andmysisterandher husbandwhohavebeenwithmeeverystepoftheway.

IN MEMORY OF

MY GRANDMOTHER, SHU-KUI WANG

TableofContents

1. Introduction...... 1

1.1. Background...... 1

1.2. Researchobjectives...... 4

2. Literaturereview...... 5

2.1. ComputermodelingofHVACsystems...... 5

2.2. Solarcombisystem...... 6

2.3. Stratifiedwaterstoragetank...... 10

2.4. Secondlawanalysis...... 15

2.5. Conclusions...... 19

3. Methodology...... 21

3.1. QuasiͲequilibriumprocess...... 21

3.2. Inputandoutput...... 22

3.3. Controlscheme...... 24

3.4. Mathematicalmodel...... 24

3.5. Computermodel...... 25

3.6. Testingandmodifications...... 25

3.7. Simulationandresultsanalysis...... 26

4. Developmentofmathematicalmodel...... 28

4.1. Solarcombisystemconfiguration...... 28

4.2. Mathematicalmodelofsolarthermalcollector...... 30

4.2.1. Solarheatabsorptionandloss...... 31

4.2.2. Enteringtemperatureofheattransferfluid...... 33

4.2.3. Leavingtemperatureofheattransferfluid...... 33

4.2.4. Exergyanalysisofsolarthermalcollector...... 34

4.3. MathematicalmodelofwaterͲtoͲwaterheatpump...... 37

4.3.1. Electricpowerandheatingcapacity...... 37

4.3.2. Flowratesandenteringtemperaturesofheattransferfluids...... 40

4.3.3. Energybalanceandleavingtemperaturesofheattransferfluids...... 42

4.3.4. Icemassfractionatevaporatoroutlet...... 44

4.3.5. Isentropicefficiencyofcompressor...... 45

4.3.6. Heatoutput...... 47

4.3.7. Exergyanalysisofheatpump...... 48

4.4. Mathematicalmodeloficestoragetank...... 50

4.4.1. EnergybalanceoficeͲlayerandwaterͲlayer...... 51

4.4.2. Masscontentinicestoragetank...... 53

4.4.3. Heattransferbetweenlayers...... 54

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4.4.4. Heattransferwithsurroundings...... 55

4.4.5. Passingflow...... 58

4.4.6. Temperaturesinicestoragetank...... 58

4.4.7. InterchangeofenergyinsolidͲliquidequilibrium...... 59

4.4.8. Enteringconditionsofheattransferfluid...... 62

4.4.9. Exergyanalysisoficestoragetank...... 63

4.5. Mathematicalmodelofwaterstoragetank...... 66

4.5.1. TheenergybalanceofwaterͲnodes...... 68

4.5.2. Heattransferofimmersionheatexchangers...... 70

4.5.3. Heattransferbetweennodes...... 78

4.5.4. Heattransferwithsurroundings...... 79

4.5.5. Domestichotwatermakeup...... 81

4.5.6. Temperaturesinwaterstoragetank...... 82

4.5.7. Electricpowerofheater...... 83

4.5.8. Heattransferwithheatsourceandheatsink...... 83

4.5.9. Heatingloadofwaterstoragetank...... 84

4.5.10. Exergyanalysisofwaterstoragetank...... 85

4.6. Mathematicalmodelofradiantfloor...... 88

4.6.1. Heattransferflow...... 89

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4.6.2. Supplyconditions...... 89

4.6.3. Exergyanalysisofradiantfloor...... 90

4.7. Mathematicalmodelofcirculatingpump...... 91

4.7.1. Electricpower...... 92

4.7.2. Flowrates...... 94

4.7.3. Exergyanalysisofcirculatingpump...... 94

4.8. Mathematicalmodelofsystem...... 95

4.8.1. Heatingload...... 95

4.8.2. Energysupplyandloss...... 95

4.8.3. Heattransferwithenvironment...... 96

4.8.4. Heatstorage...... 96

4.8.5. Systemperformance...... 97

4.9. Mathematicalmodelofcontrolsystem...... 98

4.9.1. Controlledvariables...... 99

4.9.2. Setpoints...... 102

4.9.3. Controllogic...... 105

4.10. Integrationofmathematicalmodelsincompletecomputermodel...... 110

4.11. Modelvalidationandverification...... 115

5. Resultsanddiscussions...... 117

5.1. Energyperformanceofsolarcombisystem...... 118

5.1.1. Overviewofannualenergyflow...... 118

5.1.2. Combisystemenergyperformance...... 120

5.1.3. ElectricityconsumptionandcombisystemCOP...... 123

5.1.4. Majorflowsofenergy...... 124

5.2. CombisystemoperationinnonͲheatingseason...... 127

5.3. Combisystemoperationinheatingseason...... 132

5.4. Responseforspaceheating...... 143

5.5. Operationofheatpumpwithicestoragetank...... 144

5.6. Operationofsolarcollectorwiththermalstorage...... 147

5.7. Operationofwaterstoragetank...... 150

5.8. Heattransitviathermalstorage...... 152

5.9. Exergyperformanceofsolarcombisystem...... 154

6. Conclusions...... 157

6.1. Futurework...... 161

References...... 163

Appendices...... 177

ListofFigures

Fig.1.1EnergyefficiencyeffectonenergyendͲuseinresidentialsectorofCanada...... 2

Fig.4.1Schematicofsolarcombisystemconfiguration...... 29

Fig.4.2Schematicofsolarthermalcollectormodel...... 31

Fig.4.3Schematicofheatpumpmodel...... 37

Fig.4.4TͲSdiagramofsingleͲstagevapourcompressionrefrigerationcycle...... 47

Fig.4.5Schematicoficestoragetankmodel...... 50

Fig.4.6Schematicofwaterstoragetankmodel...... 67

Fig.4.7Schematicofradiantfloormodel...... 89

Fig.4.8Schematicofcirculatingpumpmodel...... 91

Fig.4.9Performancecurveofcasepump...... 93

Fig.4.10Thermodynamicsystemboundaryofsolarcombisystem...... 95

Fig.4.11FlowchartofoperationcontrolalgorithmfornonͲheatingseason...... 107

Fig.4.12Flowchartofoperationcontrolalgorithmforheatingseason...... 108

Fig.4.13Simplifiedversionofthemodelingprocess(Sargent2011)...... 115

Fig.5.1Flowchartofenergyflowsofsolarcombisystemannualoperation...... 119

Fig.5.2Energysupplyandheatingloadofsolarcombisystem...... 125

Fig.5.3Heattransferbetweenstoragetanksandindoorsurroundings...... 126

Fig.5.4Heatstorageofsolarcombisystem...... 127

Fig.5.5HeatingloadofsolarcombisystemfromJuly20toJuly22...... 128

Fig.5.6SolarheatcollectionchargingwaterstoragetankfromJuly20toJuly22...... 128

Fig.5.7SolarheatcollectionandstoragefromJuly20toJuly22...... 129

Fig.5.8HeatstorageofwaterstoragetankfromJuly20toJuly22...... 130

Fig.5.9NodetemperatureofwaterstoragetankfromJuly20toJuly22...... 131

Fig.5.10ElectricityconsumptionofsolarcombisystemfromJuly20toJuly22...... 132

Fig.5.11Flowchartofenergyflowsofsolarcombisystematspaceheatingpeak...... 133

Fig.5.12HeatingloadofsolarcombisystemfromJan.11toJan.13...... 135

Fig.5.13HeatoutputofheatpumpfromJan.11toJan.13...... 136

Fig.5.14SolarincidenceandsolarheatcollectionfromJan.11toJan.13...... 137

Fig.5.15HeatinputtowaterstoragetankfromJan.11toJan.13...... 138

Fig.5.16HeatstorageofwaterstoragetankfromJan.11toJan.13...... 138

Fig.5.17HeatinputandheatoutputoficestoragetankfromJan.11toJan.13...... 139

Fig.5.18HeatstorageoficestoragetankfromJan.11toJan.13...... 140

Fig.5.19NodetemperatureofwaterstoragetankfromJan.11toJan.13...... 141

Fig.5.20WaterͲlayertemperatureoficestoragetankfromJan.11toJan.13...... 142

Fig.5.21IcemassfractionoficestoragetankfromJan.11toJan.13...... 142

Fig.5.22ElectricityconsumptionofsolarcombisystemfromJan.11toJan.13...... 143

Fig.5.23HeatpumpCOPinheatingseason,dailymean...... 145

Fig.5.24Temperatureofwaterenteringheatpumpevaporator,dailymean...... 146

Fig.5.25Temperatureofwaterenteringheatpumpcondenser,dailymean...... 146

Fig.5.26Icemassfractionoficestoragetankinheatingseason,dailymean...... 147

Fig.5.27NodetemperatureofWaterstoragetank,monthlymean...... 151

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Fig.5.28Heattransitviawaterstoragetank,byweek...... 153

Fig.5.29Heattransitviaicestoragetank,byweek...... 153

Fig.1Solutionstoquadraticinapplicationsofsolarthermalefficiency...... 179

Fig.2Upperandlowerboundsofoutdoortemperatureofcasesolarthermalefficiency

...... 180

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ListofTables

Table4.1Coefficientsandbaselineconditionofcaseheatpump...... 39

Table4.2Operatingparametersofcaseheatpump(GeoSmart2012)...... 47

Table4.3NodeexchangecontributionsofimmersionheatexchangerHX1inheat chargingmode...... 74

Table4.4NodeexchangecontributionsofimmersionheatexchangerHX2inheat chargingmode...... 75

Table4.5NodeexchangercontributionsofimmersionheatexchangerHX1inheat

dischargingmode...... 76

Table4.6Coefficientsofcitymaintemperatureofcasestudy...... 82

Table4.7Controlledvariablesofdevices...... 99

Table4.8Settingsofsolarcombisystemcomponentmodels...... 103

Table4.9Controlledvariablesofsolarcombisystemcomponent...... 105

Table4.10VariablesinparametrictablesofEESͲbasedsolarcombisystemmodel.....113

Table5.1Capacitiesandcontrolsettingsofsolarcombisystemsimulatedcase...... 118

Table5.2Referencesystemenergyusevs.solarenergycollectionofsolarcombisystem

...... 121

Table5.3ResultsofelectricityconsumptionandsystemCOPofsolarcombisystem annualoperation...... 123

Table5.4Resultsofenergyflowsofsolarcombisystemannualoperation...... 124

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Table5.5Resultsofspaceheatingdemandandheatsupplytoradiantfloor...... 144

Table5.6Monthlyresultsofheatpumpoperation...... 145

Table5.7Monthlyresultsofsolarcollectoroperation...... 148

Table5.8Monthlyresultsofthermalefficiencyofsolarthermalcollector...... 149

Table5.9Monthlyresultsofwaterstoragetanknodetemperatures...... 150

Table5.10Monthlyresultsofheatflowsofwaterstoragetank...... 152

Table5.11Referencesystemexergyusevs.combisystemsolarexergycollectionofsolar combisystem...... 154

Table5.12Monthlyresultsofexergydestroyedbysolarcombisystemoperation...... 156

Table5.13Annualresultsofexergeticefficiencyofsolarcombisystemoperation...... 156

Nomenclature

Symbols Descriptions Units ܣ Area m2 ஼௢௟௟ ܽ଴ Coefficient,asinߟ solarcollectorthermalefficiency dimensionless ஼௢௟௟ 2 ܽଵ Coefficient,asinߟ solarcollectorthermalefficiency W/(m ͼԨ) ஼௢௟௟ 2 2 ܽଶ Coefficient,asinߟ solarcollectorthermalefficiency W/(m ͼԨ ) ሶ ு௉ ܾ଴ǡܾଵǡǥܾସ Coefficients,asinܹ heatpumpelectricpowerinput dimensionless ሶ ௉௨௠௣ ܿ଴ǡܿଵǡǥ଼ܿ Coefficients,asinܹ circulatingpumppowerinput dimensionless ܿ௣ Constantpressurespecificheat J/(kgͼԨ) ܦ Diameter m ሶ ஼௢௡ௗ ݀଴ǡ݀ଵǡǥ݀ସ Coefficients,asinܳ heatpumpheatingcapacity dimensionless ܧ Energy J ܧሶ Energychangerate W ெ௔௜௡ ݁ Coefficient,asinܶ௪ citymainwatertemperature Ԩ ܨ Flowfractionofcirculatingpump dimensionless ெ௔௜௡ ݂ଵǡ ݂ଶǡǥ݂ଵ଻ Coefficients,asinܶ௪ citymainwatertemperature Ԩ ܩ Insolation J ܩሶ Insolationrateontiltedsurface W ܩݎ Grashofnumber dimensionless ݃ Gravityacceleration m/s2 ெ௔௜௡ ݃ଵǡ݃ଶǡǥ݃ଵ଻ Coefficients,asinܶ௪ citymainwatertemperature Ԩ ܪ Height m ݄ Specificenthalpy J/kg ݇ Thermalconductivity W/(mͼԨ) ܮ Length;latentheatoffusion m;J/kg Mass kg ݉ሶ Massflowrate kg/s ܰ Dayoftheyear dimensionless ݊ Numbers dimensionless ݑ Nusseltnumber dimensionlessܰ ܲ Pressure Pa ܲݎ Prandtlnumber dimensionless ܳ Heat J ܳሶ Heattransferrate W ܴ Surfacefilmresistanceofheattransfer (m2ͼԨ)/W ܴܽ Rayleighnumber dimensionless ݑ݊ Numberofsimulationruns dimensionlessܴ ݏ Specificentropy J/(kgͼԨ)

xvi

ܶ Temperature Ԩ ܶܭ Thermodynamictemperature K ݐ Time s ܶ݅݉݁ Simulationtime s ܷ Thermaltransmittance W/(m2ͼԨ) ݑ Specificinternalenergy J/kg ܸሶ Volumeflowrate m3/s ܹ Work J ܹሶ Electricalpowerinputrate W ܺ Exergy J ܺሶ Exergytransferrate W

GreekLetters Descriptions Units ߙ Nodeexchangecontribution dimensionless ߚ Volumeexpansioncoefficient 1/Ԩ ߛ Massfractionoficeinstoragetank dimensionless ο DifferenceͲ οݐ Timestep s ߝ CoolingeffectoficeͲlayerinstoragetank dimensionless ߟ Energeticefficiency dimensionless

ߟூூ Exergeticefficiency dimensionless ߠ Open/closesignalofflowcontrolvalve dimensionless ߤ Viscosity kg/(mͼs) ߥ Specificvolume m3/kg ߦ Massfractionoficeatevaporatoroutlet dimensionless ߩ Density kg/m3 ߪ Energy/exergyratio dimensionless ߬ Exergeticqualityfactor dimensionless ߮ specificnonͲflowexergy J/kg ߯ Qualityofsaturatedrefrigerant dimensionless ߰ specificflowexergy J/kg ெ௔௜௡ ߱ Coefficients,asinܶ௪ citymainwatertemperature dimensionless

xvii

Superscripts Descriptions ݑݔ Auxiliaryheaterܣ ܥ݋݈݈ Solarcollector ܥ݋݈݈Ͷܫܶ Circulationloopconnectingsolarcollectortoicestoragetank ܥ݋݈݈Ͷܹܶ Circulationloopconnectingsolarcollectortowaterstoragetank ܥ݋݈݈ܲ Circulatingpumpofcollectorloop ܥ݋݉݌ Compressor ܥ݋݊݀ Condenser ܥ݋݊݀ܲ Circulatingpumpofcondenserloop ܦܪܹ Domestichotwater ݒܽ݌ Evaporatorܧ ݒܽ݌ܲ Circulatingpumpofevaporatorloopܧ ܪܲ Heatpump ܪܲͶܴܨ Circulationloopconnectingheatpumptoradiantfloor ܪܲͶܹܶ Circulationloopconnectingheatpumptowaterstoragetank ܫ IceͲlayerinicestoragetank ܫʹܹ FromiceͲlayertowaterͲlayerinicestoragetank ܫܶ Icestoragetank ܫܶܮ Icestoragetankheatingload ܫܶܵ Icestoragetankheatingsource ܯܽ݅݊ Citymain ܰ WaterͲnodeinwaterstoragetank ܰ݅ WaterͲnode݅ ݑ݉݌ Circulationpumpܲ ܴܨ Radiantfloor ܴܨܮ Radiantfloorheatingload ܴܨܵ Radiantfloorheatingsource ܵܪ Spaceheating ݑ݊ Sunܵ System ݏݕܵ ݒ Expansionvalve݈ܸܽ ܹ WaterͲlayerinicestoragetank ܹʹܫ FromwaterͲlayertoiceͲlayerinicestoragetank ܹܶ Waterstoragetank ܹܶܮ Waterstoragetankheatingload ܹܶܵ Waterstoragetankheatingsource ܹܶͶܴܨ Circulationloopconnectingwaterstoragetanktoradiantfloor ഥ (overbar) Quantitymean ሶ (overdot) Quantityperunittime

xviii

Subscripts Descriptions ܽ݅ݎ Air ݒ݃ Averageܽ ܿ Condensation ܾ Base ܾܽݏ݁ Baselinecondition ݐ Contact݊ܿ ܿ݋݉ Thermalcomfort ݀݁݉݀ Demands ݐ Destructionݏ݁݀ ݋ݓ݊ Downward݀ ܧ Energetic ݁ Vaporization ݈݁݁ Electricity ݐ݉ Estimatedvalueݏ݁ ݔݐ Externalsurface݁ ݂ Flowinstoragetanks ݄ݏ Horizontalsurface ܪ HighͲtemperaturereservoir ܪܺ Heatexchanger ݅ܽ Indoorair ݅ܿ݁ Ice ݅݊ Inlet ݑ Thermalinsulationݏ݊݅ ݐ Internalsurface݊݅ ݅ݏ݁݊ Isentropic ܮ LowͲtemperaturereservoir ݈݋ܽ݀ Heatingload ݈݋ݏݏ loss ݔ Maximumܽ݉ ݉݁ܽ݊ Mean ݉݅݊ Minimum ݋ܽ Outdoorair ݋ݑݐ Outlet Overall ݈݈ܽݎ݋ݒ݁ ݌݈ܽ݊݁ Solarcollectorpanel ݎ Refrigerant ݎ݂݁ Referencesystem ݐ Setpoint݁ݏ ݏ݄݈݈݁ Tankshell ݕ Iceslurryݎݎݑ݈ݏ

xix

ݏ݋݈ Solar ݐ݋ Storageݏ ݑ݌݌ Supplyݏ Superheating ݄ݎݑ݌ݏ ݑ݊݅ݐ Unitarea ݑ݌ Upward Variation ݎݒܽ Verticalsurface ݏݒ ݓ Water ݓ݂ Liquidwater (ݓ݅ Solidwater(ice ܺ Exergetic Ͳ Referencestate ͳǡ ʹǡ ǥ ͺ Stateofrefrigerant

Abbreviation Descriptions ASHRAE AmericanSocietyofHeating,RefrigeratingandAirͲConditioningEngineers BLAST BuildingLoadsAnalysisandSystem COP CoefficientofPerformance EBC,ECBCS EnergyConservationinBuildingsandCommunitySystems EES EngineeringEquationSolver HVAC Heating,Ventilating,andAirͲConditioning HVACSIM+ HVACSimulationPlus IEA InternationalEnergyAgency MATLAB MaTrixLaBoratory NIST NationalInstituteofStandardsandTechnology REFPROP ReferenceFluidThermodynamicandTransportPropertiesDatabase SC Solarcontribution SF Solarfraction SPARK SimulationProblemAnalysisResearchKernel TRNSYS TransientSystemSimulationTool VAV VariableAirVolume

1. Introduction

1.1. Background Computersimulationistheprominentpracticetoaidlookingintosomephysical process.Throughmodelingphysicalprocessofamechanicaldevice,wecanbetter understandthedevicebehavior.Mankind’sdeepdependencyofmechanicaldevice arousestheeternalpursueforeffectivenessimprovementofenergyuse.Strengthening energysecurity,nowadays,isnotthematterofonenation,butworldwide.Thepressing

challengesonexpandingenergyneedsandnonͲstoppingemissionsofgreenhousegas

urgeactions.ThelatestreportissuedbytheU.N.climatepanelassessedthatatleast95%

oftheclimatechangearehumanͲcaused(IPCCWGI2013).Promotingtheeffectiveness

ofenergyuseisonesolutiontoreducethenegativeimpactontheglobe.Fig.1.1plots theenergyefficiencyeffectofCanadianresidentialenergyendͲuseoverthepasttwo decadesupto2009.Withouttheenergyefficiencyeffect,theenergyendͲusewouldrise dramaticallyfromthepastlevelof1990.Themostineffectivenessofenergyuse occurredinthethermodynamicprocessescanbeidentifiedwiththeSecondLawof thermodynamics.Throughexploringthequalityofenergy,themethodofSecondLaw analysisfacilitatestheleverageofpotentialmeasuresforenergyconservationand efficiencyimprovement.

ZLWKRXWHQHUJ\HIILFLHQF\HIIHFW ZLWKHQHUJ\HIILFLHQF\HIIHFW

(QHUJ\3-

Fig.1.1EnergyefficiencyeffectonenergyendͲuseinresidentialsectorofCanada Datasource:NRCanͲOEE(2012) Weembracedthetechnologythatusesenergyefficiently.Wealsoembracedthe

technology(suchassolartechnology)thatpromotestheenvironmentalsustainability

becauseweagreeontakingtherenewableenergiesintoourenergyschemeforgood.

Metz(1977)reviewedtheearlierstudiesandreportedthemosteconomichighͲ temperaturesolarsystemsservesseveralpurposesinacomplementaryfashion.The

authoralsoforecastedthatthemediumͲtemperaturesolarsystemswillhaveagreat

potentialby2000.Comingtopresentdays,solarsystemshavebeenseenappliedwidely.

Particularly,theapplicationsofsolartechnologydopresentthecomplementaryfashion

ofservingmultiplepurposes.AndthelowͲtemperaturesolarthermalsystemshave

takenoverthemajorityofsolarthermalcollectorsproducedintothiscentury.

SolarcombisystemisoneclassoflowͲtemperaturesolarthermalsystemthat

usessolarthermalenergy,togetherwithelectricity,toworkonthedualheating missionsinaresidentialhouse:toheatthedomesticwaterandtoheattheindoorspace.

2 Theextensivepracticesandtheactivesignsinthemarket,especiallyinEurope

(Ellehauge2003,Papillonetal.2010),haveconfirmedthevitalityofsolarcombisystem.

Invariousconfigurations,solarcombisystemsshareatypicalfeaturecomponent,the waterstoragetank.Becausetheavailabilityofsolarenergyvarieswithtime(Laughlin

2011),thestorageschemeprovidesthesolarenergysomedegreeofcontinuity.

Theheatpumpisoneoftheenergyconservativesolutionsthatuseslessofthe primaryenergythantheconventionalheatingsystems(IEAHPC2012).Harris(1957) countedthecombinationadvantageof"solarheater"with"thermalpump".Anon(1977)

reportedtherevivalofheatpumpswhiletheenergycostwassoaring;inparticular,the

heatpumpswerecoupledwiththe"heatͲreclaim"andsolarsystems.Steppingintothe

currentdecade,thecombinationofsolartechnologieswithheatpumphasredrawn

researchattention(IEAͲSHCT44/A382011).UpgradedfromtheAmericanSocietyof

Heating,RefrigeratingandAirͲConditioningEngineers(ASHRAE)TechnicalGroup1

Exergyinsummerof2012,theASHRAETechnicalCommittee7.4ExergyAnalysisfor

SustainableBuildingsaddressedthattheapplicationofsolarcollectorandheatpumpis preferredandmoresustainable,versusfossilfuel(ASHRAETC7.42013).

Whattheoperatingconditionswouldbewiththesolarcollectorandheatpump combinedinacombisystem?Howistheoperationperformance?Insightsfromthe pointofviewoftheSecondLawofthermodynamicsareworthforaninvestigation.

3 1.2. Researchobjectives Thisresearchaimstousecomputersimulationtechniquetoexaminetheenergy useofasolarcombisystemcoupledwithstoragetanksandaheatpump.Ourresearch grouphasaccumulatedsubstantialexperienceinaspecializedareadocumentedbyWu

(2004),Zheng(2006),Wei(2006),Zmeureanu&Wu(2007),Zmeureanu&Zheng(2008),

Leckner(2008),Hugo(2008),Wei&Zmeureanu(2009),Hugoetal.(2010),Leckner&

Zmeureanu(2011),NgChengHin(2013),andNgChengHinetal.(2013).Thisresearchis

proposedasacontributiontotheemploymentoftheSecondLawanalysisforthe

investigationofHVACsystemsbymeansofcomputermodeling.

Theresearchobjectivesare:

Formulatemathematicalmodelsofasolarcombisystemcomponents;

DevelopasolarcombisystemmathematicalmodelbasedontheSecondLawof

thermodynamics;

Implementthesolarcombisystemcomputermodel;

Analyzetheperformanceofasimulatedsolarcombisystem.

2. Literaturereview

2.1. ComputermodelingofHVACsystems ThecomputerͲbasedsimulationofHeating,Ventilating,andAirͲConditioning

(HVAC)systemsisperformedonHVACsystemcomputermodels,whicharecomposed ofmodularelementsrepresentingtheHVACcomponentequipmentandthecontrol system(ASHRAE2009a).SuchmodulartypeofHVACmodelshavebeenincludedin simulationpackagesandsoftware,suchas,ASHRAEHVAC1Toolkit(ASHRAE1994),

ASHRAEHVAC2Toolkit(Brandemeuhl1993,Brandemuehl&Gabel1994),TRNSYS(Klein

2006),EnergyPlus(BuildingTechnologiesOffice2013,Crawleyetal.2001),etc.

ComputermodelingstrategiesofHVACandbuildingsystemshaveemergedwiththree typicalschemes(ASHRAE2009a):systemͲbased,componentͲbased,andequationͲbased.

InasystemͲbasedscheme,calculationsareperformedforthebuildingzone loads,secondaryHVACsystems,andcentralplanttogetherateachtimestep,e.g. eQuest(JJH/LBNL2013),orcalculationsareprocessedinsequence,fromthebuilding zoneloadstothesecondaryHVACsystems,thenfromthecentralplanttotheeconomic

aspect,e.g.EnergyPlus(BuildingTechnologiesOffice2013).

InacomponentͲbasedscheme,calculationsareperformedatthecomponent level,andacompletesystemisaflexibleintegrationofthecomponents.Simulation programswithacomponentͲbasedschemeare,forexample,HVACSimulationPlus

5 (HVACSIM+)(Parketal.1985)andTransientSystemSimulationTool(TRNSYS)(Klein

2006).

TheequationͲbasedschemeenablesausertowritethemathematical formulationsofeverysinglecomponentandspecifytheinputsandoutputs.The equationͲbasedschemehasbeenusedby,forinstance,theSimulationProblemAnalysis

ResearchKernel(SPARK)(Buhletal.1993,Sowell&Moshier1995)andtheEngineering

EquationSolver(EES)(FͲChartSoftware,1989ascitedinPanek&Johnson1994,Klein andAlvarado,1991ascitedinMyers1992,Staus1997).

2.2. Solarcombisystem Solarcombisystems,beingmostlyusedinresidentialapplications,canbe classifiedto:solarcombisystem,solarcombiͲplussystem,andhighsolarcombiͲplus system(Balarasetal.2010).

Atypicalinstallationofasolarcombisystemcombinestheproductionof domestichotwaterandthespaceheating(Papillonetal.2010).Theinitiallydeveloped solarcombisystemwasstudiedbytheInternationalEnergyAgency(IEA)underthetask

26SolarCombisystems(SHCT262012).Thetaskgaveoneoftheconclusionsthatthe

solarcombisystemhasamaindrawbackofoversizing.Sizedbasedonthehigher

thermaldemandsinwinter,thesolarcombisystemis,therefore,oversizedforthelower

6 thermaldemandsinsummer.Thetaskreportconcludedthatthecaseswithlong heatingseasonsaremorepromising.

Duetothedrawbackofoversizing,thedevelopmentofsolarcombiͲplussystem wasmotivatedtoconnectthesolarthermalcollectorwiththethermaldrivencooling device(IEESolarCombi+2010).Combingsolarspaceheatingandcoolingwiththe domestichotwaterproduction,asolarcombiͲplussystemmayavoidthedrawbackof oversizing.

TheoperationofahighsolarcombiͲplussystemissameasofasolarcombiͲplus system,however,thesolarfractionishigher(EUHighCombi2011).Forinstance,the improvementmaylieinanintegrationofsolarheatstorage.Thesystemperformance wasexpectedtoimproveduringthespringandtheautumnwhentheheatingloadsare moderate.

TheIEAprojectSolarCombisystems,inactivefrom1998through2002,assessed twentyͲoneconfigurationsofsolarcombisystems.ForoneͲfamilyhouseapplication,the

solarcombisystemshadsolarthermalcollectorsof10to30m2andawaterstoragetank

of0.3to3m3.Lund(2005)analyzedthesizingofsolarcombisystems,intermsofthe areaofsolarthermalcollectorandthevolumeofshortͲtermheatstorage.Theuseof heatoutputoflargerareaofsolarthermalcollectorwasbetterinthecaseofsupplying bothdomestichotwaterandspaceheatingthaninthecaseonlyproducingdomestic hotwater.Enlargingtheareaofsolarthermalcollectorincreasedthesolarfractionofa completesystem,butreducedtheheatoutputofcollectorwithinunitareaofcollector

7 panel.Thestudyconcludedthattherewereonlymarginaleffectsontotheheatoutput ofcollectorfromtheincreaseofthesizeofheatstorage.Furbo,Vejen,etal.(2005) reportedalargescaleinstallationofasolarcombisysteminDenmark.Withthesolar thermalcollectorof336m2andhotwaterstoragetankof10m3,thesystemsupplied

thedistrictheatingnetworkandthehotwateruse.Raffeneletal.(2009)studiedthe sizingofsolarcombisystems.Twoobjectivefunctions,theannualfractionalenergy savingsandthemeanannualefficiency,weredevelopedasfunctionoftheareaofsolar thermalcollectorandthevolumeofheatstorage.Thesizingdecisionwasmadeby tradingoffbetweenthetwofunctions.

Amongvariousconfigurationsofsolarcombisystem,thoseincludeaheatpump arealsobelongtoatypeofsolarandheatpumpsystem(SHCT44A382011).Intermsof couplingsolarthermalcollectorswithheatpump,thecombinationcanbegenerally

classifiedasparallelsystem,seriessystem,orhybridsystem.Inaseriessystem,solar heatistransferredtoaheatpump(throughathermalstoragetank)forheating

(Freemanetal.1979).Chandrashekaretal.(1982)estimatedtheconditionsofCanada bysimulation.Theyconcludedthathigherenergysavingscouldbeobtainedbythe combination,comparedtoaresistanceheatingsystemandaconventionalairͲtoͲair heatpumpsystem.Thestudyalsoexaminedthe20Ͳyearlifecyclecost.Buttheresults differedamongthesingle,multiplexdwellingsatdifferentlocations.

Theoperationcontrolofasolarheatpumpcombinationiscomplex.The operationcontrolshouldtrytomaximizethedeliveryofthenonͲpurchasedsolarenergy.

8 Fromthisperspective,givingprioritytousethesolarsourceseemseasiertoberealized inaparalleledsystem.However,Day&Karayiannis(1994)concludedthatperformance improvementofparallelsystemwasnotasexpected.Inthereviewconductedby

Freemanetal.(1979),aparallelsystemwasthemostpracticalconfiguration.Theseries

systemiscomplexinthecontrolofbypassingaheatpumpanddirectlyusingthestorage tanktosupply,whenthesolesolaroperationissufficient.Bong(1983)pointedout, bypasstemperaturesarenotsubjecttoaunifieddecision,butdependontheprevailing buildingload.Theoperationofaheatpumpissubjectetothecapacityofsolarheat collection,ortheareaofsolarcollectorhadtobelargeenoughtakingintoaccountthe heatneedsofoperatingaheatpump.

Theoperationofasolarheatpumpsysteminvolvesthebalanceofusingthe

solarsourceand/ortheheatpump.Kush(1980)indicatedthatthethermalefficiencyof

solarthermalcollectorismoreimportantthantheCOPofheatpump.Karagiorgasetal.

(2010)useda“changeͲovercurve”,asafunctionoftheoutdoortemperatureandthe

solarradiationoncollectors,todecidetheoperationswitchbetweenadirectmodeand anindirectmode.Inthedirectmode,distributedairwasheatedusingsolarair collectors;intheindirectmode,aheatpumpwasusedtoreheattheairbeforebeing distributed.Theswitchofoperationreflectedtheneedsforoperatingaheatpump

whentheheatcollectionofsolarthermalcollectorisinsufficient.

Theoperationperformanceofsolarheatpumpsysteminstallationswas reported.Kugleetal.(1984)reportedthesolarheatpumpoperationforasingleͲfamily

9 housecouldbefreefromusingelectricalheatingelements.Asolaronlymodehadthe systemCOPupto35.Stojanoviđ&Akander(2010)testedtheperformanceofafullͲscale solarͲassistedheatpumpsystemfortheresidentialheatinginSweden.Intheheating season,agroundͲcoupledheatpumpusedsolarheatbeingstoredduringthemonthsof

latespring,summerandearlyautumn.TheheatpumphadaseasonalCOPof2.85,and thesystemhadaseasonalCOPof2.09.Bakirci&Yuksel(2011)monitoredasolarsource heatpumpsysteminstalledinabuildinginTurkey.ThesystemconsistedofflatͲplate solarcollectorsandaheatpumpusingasensiblestoragetankastheheatsource.The averageCOPwas3.8fortheheatpumpand2.9forthesystem.MorenoͲRodríguezetal.

(2012)studiedadirectͲexpansionsolarassistedheatpumpandfoundthestrong

dependenceoftheheatpumpCOPwiththeoutdoorconditions,especiallythesolar radiation.ThesimulatedCOPrangedfrom1.85to3.1,whereasthecalculatedusing experimentresultsrangedfrom1.7to2.9.

2.3. Stratifiedwaterstoragetank Waterstoragetankisamostcommontypeofsensiblethermalenergystorage

(ASHRAE2009b).Variousaspectsofthermalenergystoragesystemswereaddressedin

(Dincer&Rosen2011).Thermalenergystorageimprovetheintegrationofsystem

(Baker2008)andcreatethepotentialofeconomicsaving(Ehyaeietal.2010).Thermal stratificationisaphenomenonfeaturedbyapositivetemperaturegradientalongthe verticaldirectioninfluids(Turner1973).Themaintenanceofstratificationinathermal

10 storagetankenhancestheoperationperformanceofasystem(Nelsonetal.1998,

Duffie&Beckman2006).Attentionswerepaidtostratifiedstoragetanksinstudies.

Variousindicatorsregardingstratificationhaveappearedtoaddresstheperformanceof thermalstoragetank(Panthalookaranetal.2007):thestabilityindexandtheentropy generationnumberfromthe70’s;thestratificationcoefficient,stratificationfactor,

stratificationparameters,figureofmerit,andtheoverallsecondlawefficiencyfromthe

80’s;theMIXnumberandtheequivalenttemperaturefromthe90’s.Theemergeof newindicatorscontinued,suchas,theequivalentlosttankheightbyBahnfleth&Song

(2005),theenergyresponsefactorandtheentropygenerationratiobyPanthalookaran etal.(2007).Somestudiestracedthevariationofthermoclinetocomprehendthe thermalstateinstoragetanks(Behschnitt1996,Nelsonetal.1998,Hasnain1998,Rosen etal.2004,Shinetal.2004).

Thermalstratificationcanbedamagedinvariousways,includingtheturbulence ofjetflow,buoyancyeffect,andtheheatconductionalongthewallsofstoragetank.

Thejetflowisreferredtotheflowchanneledbytankportsfromtheinletstotheoutlets.

Thejetflowdisturbswaterbodyinthestoragetankcausingmixing.Theturbulenceof

jetflow,i.e.jetmixing,wasdescribedinthestudiesofVisser&VanDijk(1991),Halleret

al.(2008),andHalleretal.(2009).Turner(1973)gaveanaccountofflowmotionsdriven orinfluencedbythebuoyancyforceinastratifiedfluid:Alongtheverticaldirectionin fluids,theexistenceofnegativetemperaturegradientorpositivedensitygradient wouldoverturnthelocalstabilitybymovingthermals,intendingtouniformthelocal densityandsmooththetemperaturegradient.Fan&Furbo(2012a)andFan&Furbo

11 (2012b)showedthemagnitudeofvelocityofadownwardflowinawaterstoragetank underastandbyoperation.Theflowwascausedbythelossofheatfromthestorage tanktothesurroundings,whichledthedownwardflowalongthetanksidewallsand theupwardflowaroundtheverticalaxialofstoragetank.Theheatretainedwithin storagetankwallswouldconductdownward(Oppeletal.1986)causingthe degradationofthermalstratification(Nelsonetal.1999a).Cruickshank&Harrison(2010) reportedamodelingstudywiththeheatconductiontakingplacenotonlyacrossthe commonboundaryofstratifiednodesbutalsothroughthetankwalls.

Reversestratificationexistsinoperatedwaterstoragetank.Rheeetal.(2010) conductedanexperimentalstudyonahotwaterstoragetankwithadoublechimney device.Thedevicehastwonozzlestubstoroutethebuoyancydrivenflowthrough.The

experimentshowedthereversestratificationtakingplaceduringtheheatingperiodof thestoragetank.Thissuggestedthatrelativelystablestratificationneedstimetobe developedinanoperatedstoragetank.Cruickshank&Harrison(2011)reportedthermal responsetothechargeofheatinthestudiedmultiͲtanks.StratificationinaseriesͲ connectedthermalstoragedisappearedwhiletheflowrateinaheatchargeͲloop increased.

Differentapproacheswereemployedinthecomputermodelingofstratified storagetankforthepurposeofeliminatingthereversestratification.Forinstance,a reversioneliminationalgorithmwasimplementedinthemodel4PORTbyVisser&Van

Dijk(1991).Themodelforcesstratifiednodeinhighertemperaturetostayathigher

12 positionineachtimestep.Withinatimestep,wheneveramaximumtemperaturedoes notbelongtoatopnode,averagedtemperaturewouldbereassignedtothenodes, fromthebottomonetotheonewiththemaximumtemperature.Theaverage calculationandthereassignmentoftemperaturewouldrepeattillthedistributionof

thetemperaturesdemonstratingthermalstratification,thesimulationofnexttimestep

thenstarts.Withthisalgorithm,astablystratifiedstoreisachievedalways.Allardetal.

(2011)comparedfiveelectricalwaterheatersmodelsimplementedintheTRNSYS software.ThestudyconcludedthatthemodelTYPE534representedthebesttradeͲoff becausetheusercanhavecoldwaterenteringdifferentnodes(underafractionalinlet mode)and/oruseseveralheatingelements(underavirtualheatingmode).

Boththebuoyancyeffectandtheturbulenceofjetflowcausewatermixingina storagetank.Thisshouldbereflectedincertainwaysinthemodelingofstratified storagetank.Whenstablystratifiednaturalbodiesoffluidaredisturbedinanyway,

internalwavesaregenerated(Turner1973).Visser&VanDijk(1991)realizedthattaking

intoaccountonlytheinterͲnodalheatconductionresultedunreasonabletemperature

distribution;theychosetoemployareversioneliminationalgorithmtopresent stratification.Ismailetal.(1997)studiedastratifiedhotthermalstorageinan

experimentalsetup.Theyconcludedthattheuseofsoleconductioninmodelingwas

unsatisfying.ModelingthemixingwasincludedinLoehrkeetal.(1978),Oppeletal.

(1986),AlͲNajem&ElͲRefaee(1997),andNelsonetal.(1999b),howeverthesestudies didnotreachanagreedsolutionyet.

13 Kleinbachetal.(1993)validatedaoneͲdimensionalmultiͲnodemodel.

Cruickshank&Harrison(2010)reportedaoneͲdimensionalstratifiedwaterstoragetank modelwasacceptableforacoldͲdowntest.Kleinbach(1990)concludedhismodel predictionsdifferedmoreamongthemodelswithsmall(ч4)numberofnodesthan

amongthemodelswithlarge(ш8)numberofnodes.Visser&VanDijk(1991)suggested

usealargenumberofnodes,like100orevenlarger,like500,torepresentthe contributionofactualturbulentmixingproperly.Duffie&Beckman(2006)suggested thattheuseofthreeorfournodesmaybesuitable.Thepositionandthedesignoftank portsaffecttheoperationofstratifiedwaterstoragetank(Jordan&Furbo2005,Duffie

&Beckman2006).Theenergyperformanceofstoragetankbenefitsfromacoldwater portatthetankbottomandahotwaterportatthetanktop(Spuretal.2006,Ievers&

Lin2009).Keepingadistancebetweentheinletandoutletportsavoidstheoverlapping ofmixingzones(Palaciosetal.2012).Furbo,Andersen,etal.(2005)reportedthe thermalperformanceofstoragetankbenefitsfromthemultiͲdischargingofheat.The studyconcludedthatincreasingthefrequencyofheatchargingͲdischargingwasworth todoformakingmoreuseofthethermalstorageandavoidingthermalstagnation.

Glembin&Rockendorf(2012)reportedastratifiedchargingdeviceledhigherutilization ofsolarenergyinathermalstorageofasinglefamilyhouse.

14 2.4. Secondlawanalysis Fromthethermodynamicprinciple,theexergyindicatestheextentoftheenergy partingfromthetheoreticalmaximum(Moran&Shapiro1999),andtheSecondLawof thermodynamicsneedstobeconsidered(Rosen&Dincer2001,Dincer&Rosen2005,

Dewulfetal.2008,Rosenetal.2008,andAoetal.2008).Theexergyanalysishasbeen

fallenfarbehindtheenergyanalysisinthestudiesregardingbuildingenergysystems.

Toríoetal.(2009)gaveacriticalreviewofexergyanalysisstudiesconductedforsystems thatusetherenewableenergy.TheprogrammeEnergyConservationinBuildingsand

CommunitySystems(EBC,formerlyECBCS)carriedouttheresearchprojectAnnex37

LowExergySystemsforHeatingandCooling,from1999to2003,topromotetheuseof lowvaluedandenvironmentallysustainableenergysources(ECBCSA372003).The

ECBCScarriedoutanotherresearchprojectfrom2006to2010,theAnnex49Low

ExergySystemsforHighPerformanceBuildingsandCommunities,topromotetheuseof

exergyanalysismethodasthebasispracticefordesignersanddecisionmakerstotake energy/exergyandcostefficientmeasures(ECBCSA492010).

ThestateͲofͲartbuildingenergyanalysisprogramsuseonlytheenergybalance toestimatetheenergyuseinbuildings(Rosenetal.2001,Zmeureanu&Wu2007).Itard

(2005)implementedexergycalculationsintheH.e.n.k.program(Itard2003)foranearly designtocounttheflowsofexergywithinabuilding.Basedonthespreadsheet applications,e.g.MicrosoftExcel(Microsoft2013)andOpenOfficeCalc(Apache

Openoffice2013),afewofexergyanalysistoolsforbuildingsandcommunitieswere developed.Forinstance,thePreͲDesignTool(Schmidt2004,Torío2010)forbuildings,

15 theCascadia(Church2008)forneighborhooddesign,andtheSoftwareforExergy

Performance(SEPE)(Molinari2011)forexergydemandsofcoolingandheatingin buildings.Thesetoolsweredevelopedfordesign,preͲdesign,orgeneralassessment, performingthesteadystateexergyanalysis.AquasiͲsteadystateexergyanalysisstudy

wasreportedbyWu(2004)andZmeureanu&Wu(2007).TheEESprogram

demonstratedtobeanidealenvironmentintheexergyanalysisstudyofresidential

heatingsystems.Wei(2006)usedtheEESprogramtosimulatetheVariableAirVolume

(VAV)systemsofanofficebuilding.Gaggioli(2010)recommendedtheadoptionofthe

EESoracounterpartforthermodynamicproblems.TheroutinesoftheMaTrix

LaBoratory(MATLAB)program(Mathworks2011)alreadycandirectlyusetheNIST

ReferenceFluidThermodynamicandTransportPropertiesDatabase(REFPROP)

(Lemmonetal.2013).Thisprovidedanalternativeprogrammingenvironmentto performmoredetailedexergyanalysis.

TheexergyanalysiswasemployedinHVACsystemstudiesbyRosenetal.(1999),

Badescu(2002),Rosen(2001a),Dincer(2002),Shah&Furbo(2003),Rosenetal.(2004),

Singhetal.(2012),andSoni&Gupta(2012).Exergyanalysisstudyofpowerplantswere documentedbyDurmayaz&Yavuz(2001),Rosen(2001b),Verkhivker&Kosoy(2001),

andAljundi(2009).Resultsofenergeticperformanceandexergeticperformancearenot unanimous,withgreatdiscrepancyundermostofthecases.Forinstance,Bjurstrom&

CarlssonB.(1985)yieldeddifferentresultsregardingthetransitiontemperatureinthe latentheatstorageoperationfromtheenergyandexergyanalyses.Petela(2005)found theexergyefficiencyofasolarcylindricalͲparaboliccookerwasaround1%,about10

16 timessmallerthantherespectiveenergyefficiency.Zmeureanu&Wu(2007)reported

theheatrecoveryequipmentincreasedtheenergyefficiencybutdecreasedtheexergy efficiencyofthesystem.

Theexergyanalysisisappreciatedforbringingextraperspectivetotheviewof energyanalysis.Altaetal.(2010)foundthattheincreaseofairflowrateofasolar heaterimprovedtheenergyefficiencybutreducedtheexergyefficiency.Ezanetal.

(2011)analyzedaniceͲonͲcoilthermalenergystorage.Usingtheenergyanalysis method,thestoragecapacityandthesystemefficiencyincreasedwithdecreasedinlet temperatureofheattransferfluidandincreasedlengthoftube.Usingtheexergy analysismethod,theexergyefficiencywasincreasedwithincreasinginlettemperature

ofheattransferfluidandincreasinglengthoftube.Koroneos&Tsarouhis(2012)studied

asolarsystemappliedtoaresidenceinNorthernGreece.Comparingasolarheating,a

solarcooling,andasolardomesticwaterheatingsystems,theoverallsystemexergy efficiencywasthehighestinthesolarcoolingsystem,thelowestinthesolarheating system.

Theindustryofsolarthermalcollectoremploysenergyperformancetoassess collectorproducts(FSEC2010andSRCC2013).Intheresearcharea,variousindicators,

fromtheenergeticaspect,exergeticaspect,oreconomicaspect,wereemployedto

evaluatetheperformanceofvarioustypesofsolarcollector.Chow(2010)reviewedthe

indicatorsfortheperformanceofphotovoltaic/thermalhybridsolartechnology, includingthethermalefficiency,electricalefficiency,totalefficiency,energysaving

17 efficiency,exergeticefficiency,costpaybacktime,energypaybacktime,andthe greenhousegaspaybacktime.Intermsofcalculatingtheavailablesolarexergy,the

Carnotefficiencyandtheexergy/energyratioofradiationwereoftenbeingused,for example,byAltaetal.(2010),Akpinar&KoçyiŒit(2010),andAkyuzetal.(2012).

Studiesfocusingontheexergeticperformanceofsolarcollectorwere documentedsincethelate80’s.Bejanetal.(1981)analyzedtheirreversibility associatedwiththecollectionͲdeliveryprocessofsolarcollector.Suzuki(1988) presentedtheexpressionofexergybalanceofsolarthermalcollector.Thecapacityof

exergygainofevacuatedtubularcollectorwasnearlyequaltothatofflatplatecollector.

Thestudyalsoconcludedthatthelossduringtheabsorptionofsolarheattookthe largestproportionamongtheoverallexergyloss,whichincludedtheopticalloss, leakageloss,andtheheatconductionlossaswell.Petela(2005)concludedthatthe escapingofinsolationandtheheatlosstotheenvironmentdecreasedtheefficiencyof solarcollectiondevicefromtheexergeticviewpoint.InAkyuzetal.(2012),theexergy efficiencyofsolarphotovoltaicpanelwascalculatedwiththeexergylossduetowind speedbeingtakenintoaccount.

Themaximumexergyembodiedintheradiativeenergy,especiallythesolar

radiation,wasconcernedinstudies,sincethe60’stothe80’s.Petela(1964)derivedthe exergy/energyratioofradiationfortheestimationofthetheoreticalmaximumexergy embodiedintheradiativeheat.Parrott(1978)addressedthetheoreticalmaximum

exergyinsolarradiation.Edgerton(1980)questionedtheconvertibilityofsolarradiation

18 tothermodynamicwork.Jeter(1981)theoreticallyinvestigatedthemaximum conversionefficiencyofdirectsolarenergyutilization,whichwasastheCarnot efficiency.Petela(2010)andAgudelo&Cortés(2010)werethelatesttoreviewand summarizethestudiesregardingtheexergyanalysisofthermalradiationprocess.

2.5. Conclusions Basedontheliteraturereviewconductedforthisresearch,fourmainissuesgot theattention:(i)ThecomputermodelingofHVACsystemsarepopularamongthe applicationsinasystemͲbasedschemeortheapplicationsinacomponentͲbased scheme.AnequationͲbasedmodelingschemeissuperioratallowingspecificationsof

inputsandoutputs.(ii)Solarcombisystemshaveproofedthevitality,however,exergy analysisstudiesonsolarcombisystemsarerare;(iii)Themodelingofwaterstoragetank

isanindispensableaspectinthemodelingofasolarcombisystem.Researchinthe modelingofstratifiedstoragetankshouldbeconducted;and(iv)Includingaheatpump asacomponentofasolarcombisystemmakesachallengingsystemconfigurationfor investigation.

Toaddresstheseissues,amodeldevelopmentstudyshouldbedone,tousethe

EESprogramasthemodelingplatformdevelopingaSecondͲLawmodelofasolar combisystempresentingthesolarheatpumpcombination.Usingthecomponent

approach,thesolarcombisystemmodelbeingdevelopedisobtainedbyintegratingthe

modelcomponents,whicharetobedevelopedatfirst.Withthedevelopedsolar

19 combisystemcomputermodel,thesystemoperationcanbesimulatedandperformed withtheSecondͲLawanalyses.

3. Methodology Thischapterpresentsthemethodologyoftheresearchpresentedinthisthesis.

Thisresearchinvestigatedasolarcombisysteminaspecifiedconfigurationusing computersimulation.Thestepsfollowedare:

Choosingthemodelingapproach;

Determiningmodelinputsandoutputsandperformingcasedatacollection;

Establishingthecontrolschemeanddefiningthecontrolledvariables;

Developingthesolarcombisystemmathematicalmodel;

ImplementingthesolarcombisystemcomputermodelintheEESprogram;

Assigningthecasestudydataandcontrolsettings;

Modelvalidationandverification;

Simulatingthesolarcombisystemmodelandperformingresultsanalysis;

Drawingtheconclusionsbasedonthesimulatedsolarcombisystemoperation

andperformanceanalysis.

Regardingthestepslistedabove,thefollowedsectionswillprovidemoredetails.

3.1. QuasiͲequilibriumprocess ManyactualprocessesareapproximatedbythequasiͲequilibriumprocesses

withnegligibleerror(Cengel&Boles2002).InsuchaquasiͲsteadyapproach,the

21 combisystemmathematicalmodelwasformulated.Thethermodynamicbalancewas

writtenforrateofenergyinW.Thetimeincrementof15minuteswasused,whichis alsousedbyothermodelingprograms,forinstance,thesimulationprogramEnergyPlus

(BuildingTechnologiesOffice2013)usesasdefaultofsimulationtimestep.Inthe

developmentofIBLAST,aresearchversionofsimulationprogramBuildingLoads

AnalysisandSystem(BLAST),thevaluehasbeenidentifiedgivingstableresultswithout

largelyincreaseincomputationtime(BLASTSupportOffice1991).

Themodelingofthesolarcombisystemisinadynamicmodelingbecauseofthe

thermalenergystoragetanksbeingthecombisystemcomponentsinconnectionwith

othercomponentequipment.Themodelingofthermalenergystoragetankfollowsa

dynamicapproach,wherethecomputationresultsoftimeͲlaggedvariablesfromthe

previoustimeareprocessedastheinitialvaluesforcalculatingthetimeͲlaggedvariables

atthecurrenttime.

3.2. Inputandoutput Theinputandoutputofthesolarcombisystemcomputermodelwereidentified atthebeginningstage,beforestartingthemathematicalmodeldevelopment.Data collectionispartoftheinputsidentificationbecauseacomputermodelingofHVAC systemonlyrepresentsthesystemoperationscenariobycases,whichisthesametrue forthedevelopmentofacomputermodel.Thecollecteddatawillbeinputtothemodel duringthedevelopmentfortrialrun.

22 Thesolarcombisystemoutputswereidentifiedforthepurposeofoperation investigation,includingthetemperatureofworkingfluid,heattransferratebetween components,auxiliaryenergyuse,andperformanceindices.Themodelinputsidentified canbegroupedtoinputparametersandinputvariables.Theinputparametersare constantsincludingtheequipmentdimensionsandtheoperationconstraints.The operationconstraintsare,suchas,theoperationtemperaturelimits.Theinputvariables havevaluesvaryingtime,includingtheheatingrequirementsandtheboundary conditions.Theheatingrequirementsarethespaceheatingdemandandthedomestic hotwaterconsumption.Theboundaryconditionsarereferredtotheconditionsofthe environmentofthermodynamicsystem,includingtheinsolation,outdoortemperature, indoortemperature,andcitymaintemperature.

Theinputequipmentdimensionscanbefromthedocumentspublishedbythe manufacturers,whichcouldbetheinstallation,operationandmaintenancemanuals, specificationcatalogues,ratingcertifications,etc.However,tofixcertainparameters onemightneedtospecifytheexactproductmodeltosetthecapacity.Theinput variablescanbefromotherstudiesorapplications.Forinstance,thestudiedcaseused datafromotherstudiesfortheheatingdemands,indoortemperature,andcitymain temperature;usedweatherdatafromTRNSYSsimulationprogramfortheclimate

conditions.

23 3.3. Controlscheme Thecontrolschemepresentsthegeneralthoughtsabouthowtooperatethe controldevices(e.g.,valves,thermostats,etc.)toactuatethecomponentinorderto manipulatethesolarcombisystemoperation.Bearingacontrolschemeinmindwasone ofthepremisestoformulatethemathematicalmodel.

3.4. Mathematicalmodel Themathematicalmodelofsolarcombisystemwasdevelopedusingthe componentapproach.Inthisapproach,thewholecombisystemmodeliscomposedof componentmodels,includingthesolarthermalcollectormodel,heatpumpmodel,ice storagetankmodel,waterstoragetankmodel,circulatingpumpmodel,systemmodel

(todescribetheoperatingconditionsandtoevaluatetheoperationperformanceatthe

systemlevel),andcontrolsystemmodel.Bydevelopingrelationsamongthecomponent models,thewholecombisystemmodelwasformed.Thecomponentmathematical modelswereformulatedmainlyinequations,someinconditions.Theequationswereto describethethermodynamicprocessthateachofthecomponentsundergoes.The conditionswereforthedecisionalgorithmsofthecontrolsystemmodel.

Thepresentedmodelsinthisthesis,somepartsofcertaincomponentmodels arenewapproachesornewalgorithmsbeingdevelopedduringthisresearch.Theother partswereadoptedfromexistingmodelsdocumentedbyotherresearchers,withor

withoutmodification.

3.5. Computermodel Thesolarcombisystemcomputermodelwasobtainedbyimplementingthe

developedcomponentmathematicalmodelsintheEESprogram.Essentially,the programcansolvecouplednonͲlinearalgebraicanddifferentialequations.Theprogram hasthebuiltͲinfunctionscalculatingthethermodynamicandtransportpropertiesof substances.Theprogramcancheckunitconsistency,whichmayappeartobeaminor feature,butisespeciallyusefulforcomplexmodels;thesolarcombisystembeing

studiedinthisresearchisoneofthose.

Atthisstep,theinterfaceformodelinputswascreatedintheEESprogram.The equationsandconditionswereimplementedasprocedures.Mainprocedurewassetup

tocompilethecomputationdirectives,functions,andproceduressofortheexecutionin sequence.Duringimplementingtheprocedures,effortsweremadetopreparethe componentmodelsreadyforfuturereuseindifferentsystemconfigurations.

3.6. Testingandmodifications Thecomputermodelneededtestsandverificationbeforerunningthesimulation forfinalresults.Thetestingwasperformedbyrunningthemodelandobservingthe modelrespondsandmodeloutputstoensureiftheobservedareasbeingexpected

(Kennedy&O’Hagan2001).Thetestswereperformedforvarioussimulationtimespans,

25 forday,andforseason(i.e.,heatingseasonandnonͲheatingseason),beforeperforming onacompletesimulationyear.

Detailedoutputsfromthetestsimulationswereexamined.Theerrordetection atthisstepwastimeconsuming.Themodelwasexaminedwithenergybalance,aswell aswithcertainvariables,suchastemperatures,efficiencies,exergydestructions.For

instance,theefficiency,exergydestruction,anddimensionlessvariablestypicallyhave

valueswithincertainrange.Theiroutputscanexposeproblemsorerrorshidinginthe model.

3.7. Simulationandresultsanalysis Thedevelopedcomputermodelwasruntosimulatethesolarcombisystem annualoperation.Toeliminatetheimpactmadebytheinitialvalues,thesimulationof threecontinuousyearswasperformed,andonlytheresultsforthethirdyearwereused fortheanalysis.Theexportedsimulationresultswerevastsetsofdatarequiringdata treatment.TheprogramMATLABwasusedinthisstudyasapostͲprocessorforthis purpose.

Atthestepofdatatreatment,animportantaspectistodistinguishtheon/off statusofsolarthermalcollector,heatpump,andradiantfloor,becauseavariablethat belongstothemodelofeitheroneofthemisastepfunction.Whentheoperationisin

offstatus,thevariabledoesnothaveavalue.However,adummyvalueof0was

26 assignedoutofthecomputationneeds.Whenanalyzingthedataexported,thedummy valuemustbefilteredoutfirstduetothevaluedoesnotrepresentanyphysical meaning.Avariablethatbelongstothewaterstoragetankmodelortheicestorage tankmodelisnotastepfunction.Thecontrolledvariables(Section4.9.1)andthe variablesregardingtheheatingdemandswereusedinfilteringthedummyvalues.Any ofthemservedasaclassifyingindextorecognizedummyvaluesforcertainstep function.

4. Developmentofmathematicalmodel

4.1. Solarcombisystemconfiguration Thesolarcombisystemconsistsofsolarthermalcollector,awaterstoragetank, aradiantfloor,aheatpump,anicestoragetank,andthreecirculatingpumps.Thesolar combisystemconfigurationisillustratedinFig.4.1.Threecirculationloops(ܮͳǡ ܮʹǡ ܮ͵) makeconnectionsamongthesolarthermalcollector,waterstoragetank,icestorage

tank,heatpump,andradiantfloor.Thecirculatingpumpsdrivewater,theheattransfer

fluid,circulatingalongtheloopstotransferheatamongtheconnectedequipment.The solarthermalcollectortrapssolarheatinputtothecombisystem,andtheheatisstored inthewaterstoragetankandicestoragetank.Byusingsolarthermalenergyand

electricity,thecombisystemcoverstheneedsfordomestichotwaterandresidential spaceheating.Coldwaterfromthecitymainisheatedinthewaterstoragetank.While preparingdomestichotwater,thewaterstoragetankalsosuppliesheatbeingusedfor spaceheating.Thespaceheatingisprovidedbytheradiantfloorusingheatsupplied fromthewaterstoragetank,orbytheheatpump.Theheatpumpextractsheatfrom theicestoragetanktosupply,inagivensequence,theradiantfloor,andthewater storagetankaswell.

Fig.4.1Schematicofsolarcombisystemconfiguration Themodelingisbasedonthecombisystemcharacteristicsandresidentialdesign requirements.Thecombisystemcharacteristicsincludetheinsolationdensity,outdoor airdrybulbtemperature,andmainwatertemperature.Theresidentialdesign requirementsincludethesensibleheatingload,designindoortemperature,and domestichotwaterconsumption.Assumptionsbeingmadeforthemodelinginclude,

Thepressureofwaterisassumedtobeconstantatthestandardatmosphere

pressure;

Heattransfertotheheattransferwaterduetoelectricalpowerinputto

circulatingpumpsisneglected;

Heattransferthroughthecirculationlooppipesisneglected;

29 Heatstorageinpipewalls,deviceshellandattachedinsulations,floormaterial

andsurfaceattachmentisneglected;

Tankshellisassumedtobeuniformlyinsulated;neglectingpipeentranceon

tankshellaswellasjunctionsofthermalorcontrolinstruments;

Themagnitudeofimmersionheatexchangers,mountedinthewaterstorage

tank,isneglected.Thus,thevolumeofimmersionheatexchangerdoesnot

reducetheinsidevolumeofwaterstoragetank;

Theturbulenceofjetflow(duetowaterdrawnͲoff)inthewaterstoragetankis

neglected;

Thermaltransmittanceofthepipewalls,deviceshell,andinsulationmaterialis

constant.Filmresistanceontheexteriorsurfaceofstoragetanksisconstant.

Thischapterpresentsthemathematicalmodelsofthesolarcombisystem components,andtheintegrationofallmodelstocomprisethecombisystemcomputer model.

4.2. Mathematicalmodelofsolarthermalcollector Thesolarthermalcollector(Fig.4.2)absorbssolarradiationthatistransferredto

theheatsinkthroughthecirculationloopusingwaterastheworkingfluid.The mathematicalmodelcalculatestheheattransferredbytheworkingfluidandthefluid leavingtemperature.

Fig.4.2Schematicofsolarthermalcollectormodel

4.2.1. Solarheatabsorptionandloss

Theincidentsolarradiationonthesolarthermalcollectoriscalculatedas:

ሶ ஼௢௟௟ ሶ ஼௢௟௟ ஼௢௟௟ ܳ௦௢௟ ൌܩܣ௣௔௡௘௟݊ ǡ (4.1)

ሶ ஼௢௟௟ where,ܳ௦௢௟ istheinsolationrateupontiltedsolarthermalcollectorwhenthesolar

thermalcollectorisoperated;ܩሶisthepowerdensityofinsolationupontiltedcollector

஼௢௟௟ ஼௢௟௟ panelsurface;ܣ௣௔௡௘௟isthegrossareaofsinglesolarthermalcollectorpanel;݊ isthe

numberofsolarthermalcollectorpanel.

Theabsorbedsolarenergyiscalculatedasfunctionofthethermalefficiencyof

solarthermalcollector,ߟ஼௢௟௟,andtheincidentinsolationas:

ሶ ஼௢௟௟ ஼௢௟௟ ሶ ஼௢௟௟ ܳ ൌߟ ܳ௦௢௟ ǡ (4.2)

where,

஼௢௟௟ ஼௢௟௟ ଶ ܶ െܶ௢௔ ൫ܶ െܶ௢௔൯ ߟ஼௢௟௟ ൌܽ ൅ܽ ௜௡ǡ௪ ൅ܽ ௜௡ǡ௪ ǡ (4.3) ଴ ଵ ܩሶ ଶ ܩሶ

31 ஼௢௟௟ ܳሶ isthesolarheatabsorptionrateofsolarthermalcollector;ܽ଴ ǥܽଶarethe

஼௢௟௟ coefficientsofsolarcollectorthermalefficiency;ܶ௜௡ǡ௪ isthetemperatureofwater

enteringthesolarthermalcollector;ܶ௢௔istheoutdoorairtemperature.Forthe

calculationsofsolarcollectorthermalefficiency,twosituationsareexcluded:(1)ߟ஼௢௟௟<

0,and(2)ߟ஼௢௟௟>1.ConstraintsrequiredtoapplyEqn.(4.3)arepresentedinAppendixA.

ሶ ஼௢௟௟ Thesolarenergylossrateatthesolarthermalcollector,ܧ௟௢௦௦ ,iscalculatedas:

ሶ ஼௢௟௟ ஼௢௟௟ ሶ ஼௢௟௟ ܧ௟௢௦௦ ൌ൫ͳെߟ ൯ܳ௦௢௟ Ǥ (4.4)

஼௢௟௟ ஼௢௟௟ Thesurfaceareaܣ௣௔௡௘௟,thenumber݊ ,andthecoefficientsܽ଴ǡܽଵǡܽଶ,are

inputparameters.TheCertificationandRatingsheetofselectedsolarthermalcollector

productprovidessuchinformation.Forinstance,thestudiedcasesimulatedtheEmpire

ECͲ24glazedflatplatesolarcollector(SunEarth2012)of14panelsinparallel.Each

2 2 panelhastheareaof2.298m .Thecoefficientsareܽ଴=0.728,ܽଵ=Ͳ2.98830W/(m ͼԨ),

2 2 ܽଶ=Ͳ0.01930W/(m ͼԨ ).Thecoefficientܽ଴haspositivevalue;thecoefficientsܽଵǡܽଶ

havenegativevaluestotakeintoaccountthedecreaseofefficiencyduetoheatloss.

Theinsolationܩሶandthetemperatureܶ௢௔areinputvariables.Forinstance,the

casestudywasperformedforMontreal.Theinsolationdensityonthetiltedsurfaceof

60°duesouth,andthelocaloutdoordrybulbtemperaturewereused.Theseweather

informationwereextractedfromtheweatherdatafileTMY2ofTRNSYS16simulation

environment(Klein2006)at15minutestimestep.

32 4.2.2. Enteringtemperatureofheattransferfluid

Theenteringtemperatureofheattransferfluidisdeterminedbytheoperation

ofotherdeviceconnectedtothesolarthermalcollector.Theenteringtemperatureis

theinputofthesolarthermalcollectormodel.

Thewaterstoragetankandicestoragetankaretheheatsinksofsolarthermal

collectorinthesolarcombisystem.Operatingthesolarthermalcollectoragainstoneof

thestoragetanksatatime,thetemperatureofwaterinthestoragetankistakenasthe

temperatureofwaterenteringthesolarthermalcollector,

ேଵ ݐܽ݊݇ ݁݃ܽݎݐ݋ݏ ݎݐ ݓܽݐ݁ݏ݊݅ܽ݃ܽ ஼௢௟௟ ܶ௪ ǡ ܶ௜௡ǡ௪ ൌ൜ ௐ ǡ (4.5) ݐܽ݊݇ ݁݃ܽݎݐ݋ݏ ݁ܿ݅ ݐݏ݊݅ܽ݃ܽ ௪ ǡܶ

ேଵ where,ܶ௪ isthetemperatureofthewaterͲnodeܰͳinthewaterstoragetank(Section

ௐ 4.5.1);ܶ௪ isthetemperatureofthewaterͲlayerintheicestoragetank(Section4.4.1).

Theheattransferrateofsolarthermalcollectorܳሶ ஼௢௟௟isnotedasܳሶ ஼௢௟௟ସௐ்for

thetransfertothewaterstoragetankandܳሶ ஼௢௟௟ସூ்forthetransfertotheicestorage

tank.

4.2.3. Leavingtemperatureofheattransferfluid

஼௢௟௟ Thetemperatureofwaterleavingthesolarthermalcollector,ܶ௢௨௧ǡ௪,is

calculatedas:

ሶ ஼௢௟௟ ஼௢௟௟ ஼௢௟௟ ܳ ܶ௢௨௧ǡ௪ ൌܶ ൅ ǡ (4.6) ௜௡ǡ௪ ݉ሶ ஼௢௟௟ܥ ஼௢௟௟ ௪ ௣௪

33 and

஼௢௟௟ ஼௢௟௟ ஼௢௟௟ ݉ሶ ௪ ൌߩ௪ ܸ௪ሶ ǡ (4.7)

ሶ ஼௢௟௟ ሶ ஼௢௟௟ ஼௢௟௟ ஼௢௟௟ ܸ௪ ൌܸ௨௡௜௧ ܣ௣௔௡௘௟݊ Ǥ (4.8)

஼௢௟௟ ሶ ஼௢௟௟ ሶ ஼௢௟௟ where,݉ሶ ௪ isthemassflowrateofwater;ܸ௪ isthevolumeflowrateofwater,ܸ௨௡௜௧

isthevolumeflowrateperunitcollectorarea.Thewaterproperties,constantpressure

specificheatܥ ஼௢௟௟anddensityߩ஼௢௟௟,areestimatedattheaveragetemperatureof ௣௪ ௪

஼௢௟௟ waterthroughthesolarthermalcollector,ܶ௔௩௚ǡ௪,as:

஼௢௟௟ ஼௢௟௟ ൫ܶ ൅ܶ௢௨௧ǡ௪൯ ܶ஼௢௟௟ ൌ ௜௡ǡ௪ Ǥ (4.9) ௔௩௚ǡ௪ ʹ ሶ ஼௢௟௟ ሶ ஼௢௟௟ Ͳ6 Thevolumeflowrateܸ௨௡௜௧ istheinputparameter.Forinstanceܸ௨௡௜௧ =20.1×10

m3/(sͼm2)wasusedinthiscasestudyreferringtoCertificationandRatingsheetof

EmpireECͲ24(SunEarth2012).

4.2.4. Exergyanalysisofsolarthermalcollector

Thissectionpresentstheexergyanalysisofthesolarthermalcollectoroperation

byregardingthesolarthermalcollectorasthethermodynamicopensystem.Theexergy

balanceequationisestablishedconsideringtheabsorptionofsolarradiation,theflowof

water,andtheexergydestroyedduringtheoperation.

34 Specificexergyoffluids

Thevelocityofsubstanceandtheelevationofsubstanceareneglectedto calculatethespecificexergyofthefluid.Thespecificexergyoftheflowmass߰,e.g. liquidwater,iscalculatedas:

߰ൌ݄െܶܭ଴ݏǡ (4.10)

andthespecificexergyofthenonͲflowmass߮,e.g.solidwater(ice),iscalculatedas:

(ǡ (4.11ݏ଴ܭ߮ൌݑ൅ܲ଴ݒെܶ

isthespecificentropy;ݑisthespecificinternalݏ;where,݄isthespecificenthalpy

Ͳistheܭܶ;energy;ݒisthespecificvolume;ܲͲisthepressureatthereferencestate

absolutetemperatureatthereferencestate.Eqn.(4.10)and(4.11)arereferredto

Cengel&Boles(2002).

Solarexergyabsorption

ሶ Thesolarexergyinputrate,ܺ௦௢௟,duetotheabsorbedsolarenengyiscalculated as:

ሶ ሶ ஼௢௟௟ ܺ௦௢௟ ൌܳ หߪ௦௢௟หǡ (4.12)

where,ߪ௦௢௟istheexergy/energyratioofradiationwithrespecttotheSun’s temperature(Petela1964),

ସ Ͷ ܶܭ଴ ͳ ܶܭ଴ ߪ௦௢௟ ൌͳെ ൅ ቆ ቇ ǡ (4.13) ͵ ܶܭ௦௨௡ ͵ ܶܭ௦௨௡

ܶܭ௦௨௡istheabsolutetemperatureoftheSun,5700K(Petela2010).

35 Thesolarexergyabsorptioniscalculatedwiththeassociatedsolarenergy

absorption.Thesolarenergyabsorptionisthenetoftheincidentsolarenergyonthe

solarthermalcollectorpanelsurfaceandthelossoftheincidentoccurredatthesame

timeinvariousways.

Exergyofwaterflowthroughsolarthermalcollector

Theexergytransferratebywaterflowthroughthesolarthermalcollector,ܺሶ ஼௢௟௟,

iscalculatedas:

ሶ ஼௢௟௟ ஼௢௟௟ ஼௢௟௟ ஼௢௟௟ ܺ ൌ݉ሶ ௪ ൫߰௢௨௧ǡ௪ െ߰௜௡ǡ௪൯ǡ (4.14)

஼௢௟௟ ஼௢௟௟ where,߰௢௨௧ǡ௪isthespecificexergyofwaterleavingthesolarthermalcollector;߰௜௡ǡ௪is

thespecificexergyofwaterenteringthesolarthermalcollector.

Theexergytransferrateofsolarthermalcollectorܺሶ ஼௢௟௟isnotedasܺሶ ஼௢௟௟ସௐ்for

thetransfertothewaterstoragetank,andܺሶ ஼௢௟௟ସூ்forthetransfertotheicestorage

tank.

Exergydestructionandexergeticefficiency

ሶ ஼௢௟௟ Theexergydestructionratebythesolarthermalcollectoroperation,ܺௗ௘௦௧,is

obtainedfromtheexergybalanceequation:

ሶ ஼௢௟௟ ሶ ሶ ஼௢௟௟ ܺௗ௘௦௧ ൌܺ௦௢௟ െܺ ǡ (4.15)

஼௢௟௟ Theexergeticefficiencyofthesolarthermalcollectoroperation,ߟூூ ,iscalculatedas:

ܺሶ ஼௢௟௟ ߟ஼௢௟௟ ൌͳെ ௗ௘௦௧Ǥ (4.16) ூூ ሶ ܺ௦௢௟

4.3. MathematicalmodelofwaterͲtoͲwaterheatpump Theheatpump(Fig.4.3)isoperatedunderthevaporcompressionrefrigeration cycleoftherefrigerant.Theheatpumpevaporatorisconnectedtotheheatsourceby onecirculationloop.Theheatpumpcondenserisconnectedtotheheatsinkbyanother circulationloop.Byoperatingtheheatpump,heatistransferredfromtheheatsource totheheatsink.Themathematicalmodelcalculatestheelectricpowerinputtothe waterͲtoͲwaterheatpump,theheattransferredbyheattransferfluids,andthefluid

leavingtemperatures.

Fig.4.3Schematicofheatpumpmodel

4.3.1. Electricpowerandheatingcapacity

Theelectricpower,ܹሶ ு௉,andtheheatoutputrateatthecondenser,ܳሶ ஼௢௡ௗ,are

estimatedforgiveninlettemperatureandvolumeflowrateatbothofcondenserand evaporatorusingaregressionmodeldevelopedbyTang(2005):

஼௢௡ௗ ா௩௔௣ ஼௢௡ௗ ா௩௔௣ ܶܭ ܶܭ ܸ௪ሶ ܸሶ ܹሶ ு௉ ൌ൭ܾ ൅ܾ ௜௡ǡ௪ ൅ܾ ௜௡ǡ௪ ൅ܾ ൅ܾ ௪ ൱ܹሶ ு௉ ǡ ଴ ଵ ʹͺ͵Ǥͳͷ ଶ ʹͺ͵Ǥͳͷ ଷ ܸሶ ஼௢௡ௗ ସ ሶ ா௩௔௣ ௕௔௦௘ ௕௔௦௘ ܸ௕௔௦௘ (4.17)

஼௢௡ௗ ா௩௔௣ ஼௢௡ௗ ா௩௔௣ ܶܭ ܶܭ ܸ௪ሶ ܸሶ ܳሶ ஼௢௡ௗ ൌ൭݀ ൅݀ ௜௡ǡ௪ ൅݀ ௜௡ǡ௪ ൅݀ ൅݀ ௪ ൱ܳሶ ஼௢௡ௗǡ ଴ ଵ ʹͺ͵Ǥͳͷ ଶ ʹͺ͵Ǥͳͷ ଷ ܸሶ ஼௢௡ௗ ସ ሶ ா௩௔௣ ௕௔௦௘ ௕௔௦௘ ܸ௕௔௦௘ (4.18)

஼௢௡ௗ ா௩௔௣ where,ܶܭ௜௡ǡ௪ istheabsolutetemperatureofwaterenteringthecondenser;ܶܭ௜௡ǡ௪ is

஼௢௡ௗ theabsolutetemperatureofwaterenteringtheevaporator;ܸ௪ሶ isthevolumeflow

ሶ ா௩௔௣ rateofwaterthroughthecondenser;ܸ௪ isthevolumeflowrateofwaterthrough theevaporator.Thesubscript ௕௔௦௘denotesthephysicalquantityatthebaseline condition.ThesoͲcalledreferenceconditioninTang(2005)iscalledbaselineconditionin thisthesistoavoidtheconfusionwiththereferencestateofthethermodynamic analysis.

ሶ ு௉ ሶ ஼௢௡ௗ Thecoefficientsܾ଴ǡܾଵǡǥܾସand݀଴ǡ݀ଵǡǥ݀ସ,andthequantitiesܹ௕௔௦௘ǡܳ௕௔௦௘ ǡ

ሶ ஼௢௡ௗ ሶ ா௩௔௣ ܸ௕௔௦௘ andܸ௕௔௦௘ arespecificvaluesassignedbytheuserforaspecifiedheatpump.The tencoefficientsandfourquantitiesareidentifiedasconstantsusingregression

஼௢௡ௗ ா௩௔௣ technique.Theabsolutetemperaturesܶܭ௜௡ǡ௪ andܶܭ௜௡ǡ௪ ,andthevolumeflowrates

ሶ ஼௢௡ௗ ሶ ா௩௔௣ ܸ௪ andܸ௪ aredependentvariables.Table4.1showsthecoefficientsandthe

baselineconditionofthecaseheatpumpusedinthisstudypreparedforaheatpump

GeoSmartHS050(GeoSmart2012)usingStatGraphicssoftware(StatPoint2012).

38 Table4.1Coefficientsandbaselineconditionofcaseheatpump

Coefficientsof ܾ଴ ܾଵ ܾଶ ܾଷ ܾସ electricpowerinput, dimensionless Ͳ6.32321 0.498219 6.52788 0.0211943Ͳ0.115108

Coefficientsof ݀଴ ݀ଵ ݀ଶ ݀ଷ ݀ସ heatoutputrate, dimensionless Ͳ2.36262 4.11641Ͳ0.949693 0.0853777Ͳ0.0135187 ா௩௔௣ ܹሶ ு௉ ܳሶ ஼௢௡ௗ ܸሶ ஼௢௡ௗ ܸሶ Baselinecondition ௕௔௦௘ ௕௔௦௘ ௕௔௦௘ ௕௔௦௘ 3390W 21072W 9.46×10Ͳ3m3/s 9.46×10Ͳ3m3/s Eqns.(4.19)and(4.20)areusedtocalculatethepowerܹሶ ு௉andheatoutput rateܳሶ ஼௢௡ௗofthemodeledcaseheatpump.

ா௩௔௣ ܶܭ஼௢௡ௗ ܶܭ ܹሶ ு௉ ൌ ൭െ͸Ǥ͵ʹ͵ʹͳ ൅ ͲǤͶͻͺʹͳͻ ௜௡ǡ௪ ൅ ͸Ǥͷʹ͹ͺͺ ௜௡ǡ௪ ʹͺ͵Ǥͳͷ ʹͺ͵Ǥͳͷ (4.19) ஼௢௡ௗ ா௩௔௣ ܸ௪ሶ ܸሶ ൅ ͲǤͲʹͳͳͻͶ͵ െ ͲǤͳͳͷͳͲͺ ௪ ൱ ൈ ͵͵ͻͲǡ ͲǤͲͲͲͻͶ͸͵ͷ͵ ͲǤͲͲͲͻͶ͸͵ͷ͵

ா௩௔௣ ܶܭ஼௢௡ௗ ܶܭ ܳሶ ஼௢௡ௗ ൌ ൭െʹǤ͵͸ʹ͸ʹ ൅ ͶǤͳͳ͸Ͷͳ ௜௡ǡ௪ െ ͲǤͻͶͻ͸ͻ͵ ௜௡ǡ௪ ʹͺ͵Ǥͳͷ ʹͺ͵Ǥͳͷ

ܸሶ ஼௢௡ௗ ܸሶ ா௩௔௣ (4.20) ൅ ͲǤͲͺͷ͵͹͹͹ ௪ െ ͲǤͲͳ͵ͷͳͺ͹ ௪ ൱ ͲǤͲͲͲͻͶ͸͵ͷ͵ ͲǤͲͲͲͻͶ͸͵ͷ͵

ൈ ʹͳͲ͹ʹǤ

Inthisstudy,theheatpumphasequalflowratesatthecondensersideandthe evaporatorside.Thebaselineconditionwaschosenwithequalflowratesaswell(Table

4.1).TheCoefficientofPerformance(COP)wascalculatedfromEqns.(4.19)and(4.20) andcomparedwiththecaseheatpumpperformancedata.Thecomparisonisfairly good(seeresults).However,whentheflowratesareunequal,theresultsarenotin agreementwiththemanufacturer’sdata.

39 Shenoy(2004)indicatedtheuseofcomputermodelofthemanufacturerto generatethetechnicaldatatablewithafewmeasuredpointsobtainedfromthetesting process.Inpreparingthecaseheatpumpmodel,samplepointswerecarefullychosen byexaminingthereversiblecycleCOP(Cengel&Boles2002):

ܶܭு ܥܱܲ௥௘௩ ൌ ǡ (4.21) ܶܭு െܶܭ௅

where,ܥܱܲ௥௘௩isthereversiblecycleCOP;ܶܭுistheabsolutetemperatureofthehighͲ temperaturereservoir;ܶܭ௅istheabsolutetemperatureofthelowͲtemperature reservoir.Thehightemperaturecorrespondstotheenteringtemperatureatthe condenser,thelowtemperaturetotheenteringtemperatureattheevaporator.Among theperformancedataofthecaseheatpump,thepointscorrespondingtoܥܱܲ௥௘௩in

negativeoratunreasonablehighvalues(uptohundredseventhousands)wererejected.

Withtheelectricpowerinputandheatoutputrate,theCOPofheatpump,

ܥܱܲு௉,iscalculatedas:

ܳሶ ஼௢௡ௗ ܥܱܲு௉ ൌ Ǥ (4.22) ܹሶ ு௉

4.3.2. Flowratesandenteringtemperaturesofheattransferfluids

ሶ ஼௢௡ௗ ሶ ா௩௔௣ Theflowratesܸ௪ andܸ௪ areinputparametersoftheheatpumpmodel.

ሶ ஼௢௡ௗ ሶ ா௩௔௣ Ͳ3 3 Inthisstudy,ܸ௪ =ܸ௪ =9.46×10 m /s,referredtothecaseheatpump performancedata.

40 Theenteringtemperaturesofheattransferfluidsaredeterminedbythe operationofotherdevicesconnectedtotheheatpump.Theenteringtemperaturesare theinputsoftheheatpumpmodel.Attheheatpumpevaporatorside,theentering

ௐ temperatureistakenthetemperatureofthewaterͲlayerܶ௪ intheicestoragetank

(Section4.4.1)as:

ா௩௔௣ ௐ ܶܭ௜௡ǡ௪ ൌ ʹ͹͵Ǥͳͷ ൅ ܶ௪ Ǥ (4.23)

Atthecondenserside,thewaterstoragetankandtheradiantflooraretheheatsinks.

Thetemperatureofwaterenteringthecondensereitheristhetemperatureofwater

ோி ேଵ leavingtheradiantfloorܶ௢௨௧ǡ௪(Section4.6.1),orthewaterͲnodetemperatureܶ௪ in

thewaterstoragetank(Section4.5.1),orthemassͲweightedaveragetemperatureof thosetwoas:

஼௢௡ௗ ܶܭ௜௡ǡ௪

ோி ݎݐ݂݈݋݋݊ܽ݅݀ܽݎݐݏ݊݅ܽ݃ܽ ͹͵Ǥͳͷ ൅ ܶ௢௨௧ǡ௪ǡʹ ۓ ேଵ ۖ ݐܽ݊݇݁݃ܽݎݐ݋ݏݎݐݓܽݐ݁ݏ݊݅ܽ݃ܽ ͹͵Ǥͳͷ ൅ ܶ௪ ǡʹ ൌ ோி ோி ஼௢௡ௗ ோி ேଵ ǡ ሶ ௪ ܶ௢௨௧ǡ௪ ൅ ሺ݉ሶ ௪ െ݉ሶ ௪ ሻܶ௪݉ ۔ ݐܽ݊݇ ݁݃ܽݎݐ݋ݏݎݓܽݐ݁݀݊ܽݎݐ ݂݈݋݋݊ܽ݅݀ܽݎ ݐ ܾ݋ݐ݄ݏ݊݅ܽ݃ܽ ͹͵Ǥͳͷ ൅ ஼௢௡ௗ ǡʹۖ ሶ ௪݉ ە

(4.24)

where,

஼௢௡ௗ ஼௢௡ௗ ஼௢௡ௗ ݉ሶ ௪ ൌߩ௠௘௔௡ǡ௪ܸ௪ሶ ǡ (4.25)

ோி ேଵ ܶ௢௨௧ǡ௪isthetemperatureofwaterleavingtheradiantfloor;ܶ௪ isthetemperatureof

ோி thewaterͲnodeܰͳofthewaterstoragetank;݉ሶ ௪ isthemassflowrateofwater

஼௢௡ௗ throughtheradiantfloor;݉ሶ ௪ isthemassflowrateofwaterthroughtheheatpump

஼௢௡ௗ condenser;ߩ௠௘௔௡ǡ௪isthemeandensityofwaterthroughthecondenser.

41 ஼௢௡ௗ Thedensityߩ௠௘௔௡ǡ௪iscalculatedatthemeanwatertemperatureoverthe

certainperiodoftimeݐ௞(݇=1…݇),asof

σ௞ ܶ஼௢௡ௗ ௞ୀଵ ௔௩௚ǡ௪௧ ܶ஼௢௡ௗ ൌ ೖǡ (4.26) ௠௘௔௡ǡ௪ ݇

where,

஼௢௡ௗ ஼௢௡ௗ ܶ ൅ܶ௢௨௧ǡ௪ ܶ஼௢௡ௗ ൌ ௜௡ǡ௪ Ǥ (4.27) ௔௩௚ǡ௪ ʹ

Thetimeintervalݐ௞(݇=1…݇)denotesthetimeintervalwhentheheatpumpisin

஼௢௡ௗ operation;thetemperatureܶ௔௩௚ǡ௪istheaveragetemperatureofwaterthroughthe condenser.

஼௢௡ௗ Theaveragetemperatureܶ௔௩௚ǡ௪ isstepfunction.Themeantemperature ௧ೖ

஼௢௡ௗ ܶ௠௘௔௡ǡ௪thereforeonlycorrespondstothetimespanswhentheheatpumpisturnedon.

Inthemodel,themeanistheinputparameteroftheheatpumpmodel.Theinputtothe

஼௢௡ௗ modeledcasewasbasedontheEqns.(4.26)and(4.27).Thestudiedcaseusedܶ௠௘௔௡ǡ௪=

44.1Ԩ,whichisthemeanvalueduringtheheatingseasonwhentheheatpumpison operation.

4.3.3. Energybalanceandleavingtemperaturesofheattransferfluids

Theheatinputrateattheevaporator,ܳሶ ா௩௔௣,isderivedfromtheheatpump energybalanceequationas:

ܳሶ ா௩௔௣ ൌܳሶ ஼௢௡ௗ െܹሶ ு௉Ǥ (4.28)

42 ஼௢௡ௗ Thetemperatureofwaterleavingthecondenser,ܶ௢௨௧ǡ௪,iscalculatedas:

ሶ ஼௢௡ௗ ஼௢௡ௗ ஼௢௡ௗ ܳ ܶ௢௨௧ǡ௪ ൌܶ ൅ ǡ (4.29) ௜௡ǡ௪ ݉ሶ ஼௢௡ௗܥ ஼௢௡ௗ ௪ ௣௠௘௔௡ǡ௪

஼௢௡ௗ where,݉ሶ ௪ isthemassflowrateofwaterthroughthecondenser;theconstant pressurespecificheatܥ ஼௢௡ௗ iscalculatedatthetemperatureܶ஼௢௡ௗ . ௣௠௘௔௡ǡ௪ ௠௘௔௡ǡ௪

ா௩௔௣ Thetemperatureofwaterleavingtheevaporator,ܶ௢௨௧ǡ௪,iscalculatedas:

ሶ ா௩௔௣ ா௩௔௣ ா௩௔௣ ܳ ܶ ൌܶ െ ǡ (4.30) ௢௨௧ǡ௪ ௜௡ǡ௪ ݉ሶ ா௩௔௣ܥ ா௩௔௣ ௪ ௣௠௘௔௡ǡ௪

where,

ா௩௔௣ ா௩௔௣ ሶ ா௩௔௣ ݉ሶ ௪ ൌߩ௠௘௔௡ǡ௪ܸ௪ ǡ (4.31)

݉ሶ ா௩௔௣isthemassflowrateofwaterthroughtheevaporator;ܥ ா௩௔௣ isthemean ௪ ௣௠௘௔௡ǡ௪

ா௩௔௣ constantpressurespecificheatofwaterthroughtheevaporator;ߩ௠௘௔௡ǡ௪isthemean

densityofwaterthroughtheevaporator.

ா௩௔௣ Thewaterproperties,thedensityߩ௠௘௔௡ǡ௪andconstantpressurespecificheat

ܥ ா௩௔௣ ,arecalculatedasatthemeanwatertemperatureoverthecertainperiodof ௣௠௘௔௡ǡ௪

timeݐ௞(݇=1…݇),asof

σ௞ ܶா௩௔௣ ௞ୀଵ ௔௩௚ǡ௪௧ ܶா௩௔௣ ൌ ೖǡ (4.32) ௠௘௔௡ǡ௪ ݇

where,theaveragetemperatureofwaterthroughtheevaporator

43 ா௩௔௣ ா௩௔௣ ܶ ൅ܶ௢௨௧ǡ௪ ܶா௩௔௣ ൌ ௜௡ǡ௪ Ǥ (4.33) ௔௩௚ǡ௪ ʹ

ா௩௔௣ Themeantemperatureܶ௠௘௔௡ǡ௪istheinputparameteroftheheatpumpmodel.The

ா௩௔௣ studiedcaseusedܶ௠௘௔௡ǡ௪=3.9Ԩ,whichisthemeanvalueduringtheheatseason

whentheheatpumpisturnedon.

4.3.4. Icemassfractionatevaporatoroutlet

Icemassfraction,ߦ,isthefractionalmassflowrateofsolidwater(ice)inthe waterthroughtheevaporator.Themassflowrateofsolidwater,liquidwater,andthe overallwithoutdistinguishingphasesaresubjecttothefollowingrelations:

ா௩௔௣ ா௩௔௣ ݉ሶ ௪௜ ൌߦ݉ሶ ௪ ǡ (4.34)

ா௩௔௣ ா௩௔௣ ݉ሶ ௪௙ ൌ ሺͳെߦሻ݉ሶ ௪ ǡ (4.35)

ா௩௔௣ ா௩௔௣ where,݉ሶ ௪௜ isthemassflowrateofsolidwater(ice)leavingtheevaporator;݉ሶ ௪௙ is

themassflowrateofliquidwaterleavingtheevaporator.

Theheattransferrateattheevaporatorcanbeformulatedas:

ܳሶ ா௩௔௣ ൌ݉ሶ ா௩௔௣ܮ ൅݉ሶ ா௩௔௣ܥ ா௩௔௣ ൫ܶா௩௔௣ െܶா௩௔௣ ൯ǡ ௪௜ ௪ ௪௙ ௣௔௩௚ǡ௪ ௜௡ǡ௪ ௢௨௧ǡ௪௙ (4.36)

where,ܮ௪isthespecificenthalpyoffusionofwater,333606J/kg.Theconstant pressurespecificheatܥ ா௩௔௣ iscalculatedattheaveragetemperatureofwateratthe ௣௔௩௚ǡ௪ evaporator,

44 ா௩௔௣ ா௩௔௣ ܶ௜௡ǡ௪ ൅ܶ௢௨௧ǡ௪௙ ܶா௩௔௣ ൌ ǡ (4.37) ௔௩௚ǡ௪ ʹ

ா௩௔௣ where,ܶ௢௨௧ǡ௪௙isthetemperatureofliquidwaterleavingtheevaporator.When

ா௩௔௣ solidificationoccurs,theliquidwatertemperatureisnotedbyܶ௢௨௧ǡ௪௙,whilethesolid

ா௩௔௣ watertemperatureisnotedbyܶ௢௨௧ǡ௪௜.Theliquidwatertemperatureisestimatedas

0.01Ԩ.Thesolidwatertemperatureisestimatedas0ԨregardlessofsubͲcooling

(Tamasauskasetal.2012).

BysubstitutingEqns.(4.34)and(4.35)ontoEqn.(4.36),themassfractionis

derivedas:

ா௩௔௣ ܳሶ ா௩௔௣ ா௩௔௣ ா௩௔௣ െܥ௣ ൫ܶ െܶ ൯ ݉ሶ ா௩௔௣ ௔௩௚ǡ௪ ௜௡ǡ௪ ௢௨௧ǡ௪௙ ߦൌ ௪ Ǥ (4.38) ܮ െܥ ா௩௔௣ ቀܶா௩௔௣ െܶா௩௔௣ ቁ ௪ ௣௔௩௚ǡ௪ ௜௡ǡ௪ ௢௨௧ǡ௪௙

4.3.5. Isentropicefficiencyofcompressor

Theisentropicefficiencyofcompressorisameasureofthedeviationofthe

compressoroperationfromthecorrespondingidealprocess.Thecompressorisentropic

஼௢௠௣ efficiency,ߟ௜௦௘௡ ,iscalculatedas:

ܹሶ ு௉ ߟ஼௢௠௣ ൌ ௜௦௘௡ǡ (4.39) ௜௦௘௡ ܹሶ ு௉

ሶ ு௉ where,ܹ௜௦௘௡istheisentropiccompressionpower.

ሶ ு௉ Theisentropicpowerܹ௜௦௘௡isobtainedbymodelingthevaporcompression

refrigerationcycle.Fig.4.4illustratesthesingleͲstagevaporcompressionrefrigeration

45 cycleintemperatureͲentropy(TͲS)diagram.Thepoints1,2,4…8arethekeystatesof therefrigerantundertheoperationofheatpump.IncounterͲclockwisetheactual processisrepresentedby1ї2forthecompressioninthecompressor,2ї4ї5ї6for

thedeͲsuperheating,condensation,andsubͲcoolinginthecondenser,6ї7forinthe expansionvalve,7ї8ї1forthevaporizationandsuperheatingintheevaporator.The pressuresofvaporizationandcondensationareindicatedbyܲ௘andܲ௖,alongwithܶ௘ andܶ௖asofthecorrespondingtemperatures.Theprocessofisentropiccompressionis representedbythetransformation1ї3.Theisentropicpoweriscalculatedin

ሶ ஼௢௠௣ ܹ௜௦௘௡ ൌ݉ሶ ௥൫݄௥ǡଷ െ݄௥ǡଵ൯ǡ (4.40)

where,

ܹሶ ு௉ ݉ሶ ௥ ൌ ǡ (4.41) ሺܲ௖ െܲ௘ሻݒ௥ǡଵ

݉ሶ ௥isthemassflowrateofrefrigerant;݄௥ǡଷisthespecificenthalpyofrefrigerantatthe

state3;݄௥ǡଵisthespecificenthalpyofrefrigerantatthestate1;ݒ௥ǡଵisthespecific

volumeofrefrigerantatthestate1.Eqn.(4.39)to(4.41)arereferredtoCengel&Boles

(2002).

Byconsultingthetechnicalinformationofthecaseheatpump(GeoSmart2012), therefrigerantR410Awasusedinthisstudy,alongwiththeoperatingparameters presentedinTable4.2.Wherethecondition,thetemperatureofwaterenteringthe condenserat21.1Ԩǡwasdeterminedreferringtotheleavingtemperatureofwaterat theradiantfloor(SectionͶǤ͸Ǥͳ).

Fig.4.4TͲSdiagramofsingleͲstagevapourcompressionrefrigerationcycle

Table4.2Operatingparametersofcaseheatpump(GeoSmart2012) Temperatureofwaterenteringcondenserat21.1Ԩ Temperatureofwaterenteringevaporator,Ԩ tч10 1021.11 Suctionpressure,kPa 468.84 520.55 572.27 Dischargepressure,kPa 1727.14 1825.39 1923.64 Superheattemperature difference,Ԩ 4.72 5.97 7.22

4.3.6. Heatoutput

Theheatoutputfromtheheatpumpistransferredtothewaterstoragetank and/ortheradiantfloor.Whenonlythewaterstoragetankneedsheatfromtheheat pump,thefollowingconditionapplies:

ܳሶ ு௉ସோி ൌͲǡ (4.42)

ܳሶ ு௉ସௐ் ൌܳሶ ஼௢௡ௗǤ (4.43)

Whenboththewaterstoragetankandtheradiantfloorneedheat,theneedof theradiantfloorissatisfiedfirstwhilethewaterstoragetankissuppliedwiththe excessiveheat.Theheattransferratefromthecondenserܳሶ ஼௢௡ௗisnotedasܳሶ ு௉ସௐ்for

47 thetransfertothewaterstoragetank,andܳሶ ு௉ସோிforthetransfertotheradiantfloor.

Theheatdistributioniscomputedusingthefollowingrelations:

ܳሶ ு௉ସோி ൌܳሶ ோி ቊ ௗ௘௠ௗ ǡ ݂݅ ܳሶ ோி ൏ܳሶ ஼௢௡ௗ ሶ ு௉ସௐ் ሶ ஼௢௡ௗ ሶ ோி ௗ௘௠ௗ ܳ ൌܳ െܳௗ௘௠ௗ (4.44) ܳሶ ு௉ସோி ൌܳሶ ஼௢௡ௗ ቊ ǡ ݂݅ ܳሶ ோி ൒ܳሶ ஼௢௡ௗ ܳሶ ு௉ସௐ் ൌͲ ௗ௘௠ௗ

ሶ ோி where,ܳௗ௘௠ௗisthedemandedheattransferrateoftheradiantfloor.

4.3.7. Exergyanalysisofheatpump

Thissectionpresentstheexergyanalysisoftheheatpumpoperationby regardingtheheatpumpasathermodynamicopensystem.Theexergybalance equationisestablishedconsideringtheuseofelectricity,theflowsofwater,andthe exergydestroyedduringtheoperation.

Exergyofwaterflowsthroughevaporatorandcondenser

Thenetexergytransferratebywaterflowthroughtheevaporator,ܺሶ ா௩௔௣,is

estimatedas

ሶ ா௩௔௣ ா௩௔௣ ா௩௔௣ ா௩௔௣ ா௩௔௣ ா௩௔௣ ா௩௔௣ ܺ ൌ݉ሶ ௪ ߰௜௡ǡ௪ െ൫݉ሶ ௪௙ ߰௢௨௧ǡ௪௙ ൅݉ሶ ௪௜ ߮௢௨௧ǡ௪௜൯ǡ (4.45)

ா௩௔௣ ா௩௔௣ where,߰௜௡ǡ௪ isthespecificexergyofwaterenteringtheevaporator;߰௢௨௧ǡ௪௙isthe

ா௩௔௣ specificexergyofliquidwaterleavingtheevaporator;߮௢௨௧ǡ௪௜isthespecificexergyof

thesolidwaterleavingtheevaporator.

48 Thenetexergytransferratebywaterflowthroughthecondenser,ܺሶ ஼௢௡ௗ,is

estimatedas

ሶ ஼௢௡ௗ ஼௢௡ௗ ஼௢௡ௗ ஼௢௡ௗ ܺ ൌ݉ሶ ௪ ൫߰௢௨௧ǡ௪ െ߰௜௡ǡ௪ ൯ǡ (4.46)

஼௢௡ௗ ஼௢௡ௗ where,߰௢௨௧ǡ௪isthespecificexergyofwaterleavingthecondenser;߰௜௡ǡ௪ isthespecific

exergyofwaterenteringthecondenser.

Theexergyistransferredthroughthecirculationloopܮ͵fromtheheatpump condensertotheradiantfloorand/orthewaterstoragetank.Theexergytransferrate totheradiantfloorisnotedasܺሶ ு௉ସோி,andthattothewaterstoragetankasܺሶ ு௉ସௐ்.

Therefore,

ܺሶ ஼௢௡ௗ ൌܺሶ ு௉ସோி ൅ܺሶ ு௉ସௐ்Ǥ (4.47)

Exergydestructionandexergeticefficiency

ሶ ு௉ Theexergydestructionratebytheheatpumpoperation,ܺௗ௘௦௧,isobtainedfrom

theexergybalanceequation:

ሶ ு௉ ሶ ு௉ ሶ ா௩௔௣ ሶ ஼௢௡ௗ ܺௗ௘௦௧ ൌܹ ൅ܺ െܺ ǡ (4.48)

ு௉ Theexergeticefficiencyoftheheatpumpoperation,ߟூூ ,iscalculatedas:

ሶ ு௉ ு௉ ܺௗ௘௦௧ ߟூூ ൌͳെ Ǥ (4.49) ܹሶ ு௉ ൅ܺሶ ா௩௔௣

49 4.4. Mathematicalmodeloficestoragetank Thestoragetank(Fig.4.5)containswaterasthestoragemedia,allowingthe coexistenceofliquidwaterandsolidwater.Thestoragetankisoperatedbetweena heatsourceandaheatsink.Themathematicalmodelsimulatestheicestoragetank operationtocalculatethetemperatureofwaterandthemassoficeinthestoragetank.

Fig.4.5Schematicoficestoragetankmodel Inthecaseofonlyliquidphase,thestoragemediaisonehomogeneouswaterͲ layer.Inthegeneralcaseoftwophases,thereisonehomogeneouswaterͲlayerbelow onehomogeneousiceͲlayer.Whentheicestoragetankisundertheheattransferwith theheatsourceand/ortheheatsink,thereisflowofwaterthroughthestoragemedia fromtoptobottom.Inthegeneralcase,thewaterflowcontainssolidwater(ice)and liquidwater.Theliquidwateriscalledpassingflowinhere,whichacrosstheiceͲlayer flowintothewaterͲlayer,whiletheiceaccumulatestheiceͲlayer.Theheattransferof

theicestoragetanktakesplace:(i)acrossthecontactsurfacebetweentheiceͲlayerand

waterͲlayer;(ii)throughthestoragetanksurfacewiththesurroundingenvironment;(iii)

50 bythepassingflowwiththeheatsourceandheatsink.Thissectionpresentstheice storagetankmodelbyTamasauskasetal.(2012)withmodifications.

4.4.1. EnergybalanceoficeͲlayerandwaterͲlayer

TheenergybalanceequationoftheiceͲlayerisasfollows:

ሶ ூ ሶ ூ ሶ ூ ሶ ூ ூ்௅ ܳ௦௧௢ ൌܳ௜௡௧ ൅ܳ௘௫௧ ൅ܳ௙ െܮ௪݉ሶ ௪௜ ǡ (4.50)

ሶ ூ ሶ ூ where,ܳ௦௧௢istheenergychangerateoftheiceͲlayer;ܳ௜௡௧istheheattransferrateof

ூ theiceͲlayerwiththewaterͲlayeracrossthecontactsurfaceinbetween;ܳሶ௘௫௧isthe

ሶ ூ heattransferoftheiceͲlayerwiththesurroundingsacrossthetanksurface;ܳ௙isthe

heattransferoftheiceͲlayerwiththepassingflow;ܮ௪isthelatentheatoffusionof

ூ்௅ water;݉ሶ ௪௜ isthemassflowrateofsolidwater(ice)fromtheheatsink.

TermsinEqn.(4.50)havefollowingformulations,

݀݉ூ ܳሶ ூ ൌെ ௜௖௘ ܮ ǡ (4.51) ௦௧௢ ݀ݐ ௪

ሶ ூ ூ் ௐ ூ ܳ௜௡௧ ൌ݄௦௟௨௥௥௬ܣ௖௡௧൫ܶ௪ െܶ௜௖௘൯ǡ (4.52)

ሶ ூ ூ் ூ ூ ூ் ூ ܳ௘௫௧ ൌ൫ܷ௩௦ ܣ௩௦ ൅ܷ௛௦ܣ௖௡௧൯൫ܶ௜௔ െܶ௜௖௘൯ǡ (4.53)

ܳሶ ூ ൌ݉ሶ ூ்௅ܿ ூ்௅ ܶூ்௅ ൅݉ሶ ூ்ௌܿ ூ்ௌ ܶூ்ௌ െ൫݉ሶ ூ்௅ ൅݉ሶ ூ்ௌ൯ܿ ூ் ܶூ் Ǥ ௙ ௪௙ ௣௜௡ǡ௪௙ ௜௡ǡ௪௙ ௪ ௣௜௡ǡ௪ ௜௡ǡ௪ ௪௙ ௪ ௣௙ǡ௪ ௙ǡ௪ (4.54)

ூ ூ்௅ where,ݐisthetime;݉௜௖௘isthemassoftheiceͲlayer;݉ሶ ௪௙ isthemassflowrateof

ூ்ௌ liquidwaterfromtheicestoragetankheatsink;݉ሶ ௪ isthemassflowrateofwater

ூ ௐ fromtheicestoragetankheatsource;ܶ௜௖௘isthetemperatureoftheiceͲlayer;ܶ௪ isthe

ூ் temperatureofthewaterͲlayer;ܶ௜௔ istheairtemperatureofthesurroundings;ܶ௙ǡ௪is

thetemperatureofthepassingflow;݄௦௟௨௥௥௬istheheattransfercoefficientbetweenthe

ூ் slurryiceandliquidwater;ܷ௩௦ isthethermaltransmittanceofverticalsurfaceoftheice

ூ storagetank;ܷ௛௦isthethermaltransmittanceofthehorizontalsurfaceoftheiceͲlayer;

ூ் ூ ܣ௖௡௧isthecontactsurfaceareabetweentheiceͲlayerandthewaterͲlayer;ܣ௩௦isthe verticalsurfaceareaoftheiceͲlayer.

TheenergybalanceequationofthewaterͲlayerisasfollows:

ሶ ௐ ሶ ௐ ሶ ௐ ሶ ௐ ܳ௦௧௢ ൌܳ௜௡௧ ൅ܳ௘௫௧ ൅ܳ௙ ǡ (4.55)

ሶ ௐ ሶ ௐ where,ܳ௦௧௢istheenergychangerateofthewaterͲlayer;ܳ௜௡௧istheheattransferofthe

ௐ waterͲlayerwithiceͲlayeracrossthecontactsurfaceinbetween;ܳሶ௘௫௧istheheat

ሶ ௐ transferofthewaterͲlayerwiththesurroundingsacrossthetanksurface;ܳ௙ istheheat

transferofthewaterͲlayerwiththepassingflow.

TermsinEqn.(4.55)havefollowingformulations,

ௐ ௐ ௐ ݀ ቀ݉௪ ܿ௣ ܶ௪ ቁ ܳሶ ௐ ൌൌ ௪ ǡ (4.56) ௦௧௢ ݀ݐ

ሶ ௐ ூ் ூ ௐ ܳ௜௡௧ ൌ݄௦௟௨௥௥௬ܣ௖௡௧൫ܶ௜௖௘ െܶ௪ ൯ǡ (4.57)

ሶ ௐ ூ் ௐ ௐ ூ் ௐ ܳ௘௫௧ ൌ൫ܷ௩௦ ܣ௩௦ ൅ܷ௛௦ܣ௕ ൯൫ܶ௜௔ െܶ௪ ൯ǡ (4.58)

ܳሶ ௐ ൌ൫݉ሶ ூ்௅ ൅݉ሶ ூ்ௌ൯ܿ ூ் ܶூ் െ ሺ݉ሶ ூ்௅ ൅݉ሶ ூ்ௌሻܿ ௐܶௐǤ ௙ ௪௙ ௪ ௣௙ǡ௪ ௙ǡ௪ ௪ ௪ ௣௪ ௪ (4.59)

ௐ ௐ where,݉௪ isthemassofthewaterͲlayer;ܷ௛௦isthethermaltransmittanceofthe

ௐ horizontalsurfaceofthewaterͲlayer;ܣ௩௦istheverticalsurfaceareaofthewaterͲlayer;

ூ் ܣ௕ isthebaseareaoftheicestoragetank.

52 4.4.2. Masscontentinicestoragetank

ூ் Thetotalwatermassinsidetheicestoragetank,݉௪ ,consistsofthemassofthe twolayers,

ூ் ூ ௐ ݉௪ ൌ݉௜௖௘ ൅݉௪ Ǥ (4.60)

Themassfractionoficeintheicestoragetank,ߛ,istheratioasof

݉ூ ௜௖௘ (4.61) ߛൌ ூ் ǡ ݉௪

where,

ௐ ூ் ߩ௪ ߩ௜௖௘ ூ் ݉௪ ൌ ௐ ܸ ǡ (4.62) ߛ௠௔௫ߩ௪ ൅ ሺͳെߛ௠௔௫ሻߩ௜௖௘

andfortherectangulartank,

ܸூ் ൌܪூ்ܹூ்ܮூ்ǡ (4.63)

ௐ ߩ௪ isthedensityofthewaterͲlayer,estimatedwiththemaximumtemperatureofthe

ௐ waterͲlayer,denotedasܶ௠௔௫;ߩ௜௖௘isthedensityoftheiceͲlayer,estimatedas916.2

3 kg/m foriceat0°C;ߛ௠௔௫isthemaximummassfractionoficeintheicestoragetank;

ܸூ்isthevolumeoftheicestoragetank;ܪூ்istheheightoftherectangulartank;ܹூ்

isthewidthoftherectangulartank;ܮூ்isthelengthoftherectangulartank.

Themaximummassfractionoficeߛ௠௔௫,themaximumtemperatureofwaterͲ

ௐ ூ் ூ் ூ் layerܶ௠௔௫,andthethreedimensionsoftankܪ ǡܹ ,andܮ ,aretheinput parameters.Forinstance,inthecasestudy,ܪூ்=2m,ܹூ்=2m,andܮூ்=3.5m.The maximummassfractionoficeߛ௠௔௫issetfortheoperationalrequirement,sincecertain amountofliquidwatershouldbesecuredforthecirculationswiththeheatsourceand

53 ௐ heatsink.ReferringtoSunwell(2012),ߛ௠௔௫=0.7.Themaximumtemperatureܶ௠௔௫is

setfromtheeconomicconsiderations.Highwatertemperaturedamagesthe possibilitiesofwaterundergoingphasechange,whichdecreasesthethermalstorage

capacity.Besides,theheatpumpactsastheheatsinkoftheicestoragetank.Heat

pumpsusuallygetanoperationrange,intermsoftheenteringtemperatureatthe

ௐ evaporator.BasedontheheatpumpperformancedatafromGeoSmart(2012),ܶ௠௔௫=

37.7°C.

4.4.3. Heattransferbetweenlayers

TheheattransferbetweentheiceͲlayerandthewaterͲlayeristreatedasnatural

convectionofwarmwaterunderneathacoldplate(Kreith&Bohn2001),duetothe smallmagnitudeofthepassingflow.Theintensityofheattransferbetweeniceand watervaries(Maciejewski1996).Theheattransfercoefficientbetweeniceandwateris

calculatedas(Kreith&Bohn2001),

ௐ തܰݑതതത௦௟୳௥௥௬݇௪ ݄ ൌ ǡ (4.64) ௦௟௨௥௥௬ ܮூ

ௐ where,ܰݑ௦௟௨௥௥௬istheNusseltnumberoficeslurry;݇௪ istheconductivityofthewaterͲ layer;ܮூisthecharacteristiclength,takenasthewetperimeteroftheiceͲlayer,

ܹூ்ܮூ் ܮூ ൌ Ǥ (4.65) ʹሺܹூ் ൅ܮூ்ሻ

TheNusseltnumberpiecewisedependsontheRayleighnumber(Kreith&Bohn2001),

54 Ͳǡ ܴܽ ൑ͳͲହ ௦௟௨௥௥௬ ۓ ଵ ൗସ ହ ଻ (ݑ௦௟௨௥௥௬ ൌ ͲǤͷͶܴܽ௦௟௨௥௥௬ ǡͳͲ൏ܴܽ௦௟௨௥௥௬ ൑ͳͲ ǡ (4.66ܰ ଵ ۔ ൗଷ ଻ ͲǤͳͷܴܽ௦௟௨௥௥௬ ǡܴܽ௦௟௨௥௥௬ ൐ͳͲە

where,ܴܽ௦௟௨௥௥௬istheRayleighnumberforiceslurry,

ூ ௐ ܴܽ௦௟௨௥௥௬ ൌܩݎܲݎ௪ ǡ (4.67)

and

ௐ ௐ ௐ ூ ூଷ ݃ߚ௪ ߩ௪ ൫ܶ௪ െܶ ൯ܮ ܩݎூ ൌ ௜௖௘ Ǥ (4.68) ௐଶ ߤ௪

ூ ௐ ܩݎ istheGrashofnumberoftheiceͲlayer;ܲݎ௪ isthePrandtlnumberofthewaterͲ

ௐ layer;݃isthegravityacceleration;ߩ௪isthedensityofthewaterͲlayer;ܶ௪ isthe

ௐ temperatureofthewaterͲlayer;ߤ௪ istheviscosityofthewaterͲlayer.

TheareaofcontactsurfacebetweentheiceͲlayerandthewaterͲlayeris calculatedas:

ூ் Ͳǡ ݂݅ ߛ൏Ͳ ܣ௖௡௧ ൌ൜ ூ் ǡ (4.69) ܣ௕ ǡ ݂݅ ߛ൐Ͳ

where,

ூ் ூ் ூ் ܣ௕ ൌܹ ܮ ǡ (4.70)

forarectangularstoragetank.

4.4.4. Heattransferwithsurroundings

Theheattransferbetweentheicestoragetankanditssurroundingenvironment

takesplacearoundthestoragetanksurface.Neglectingthethermaltransmittanceof

55 thetankshell,thethermaltransmittanceofverticalsurfaceoftheicestoragetankis

calculatedwith:

ூ் ͳ ܷ௩௦ ൌ ǡ ͳ ஺௜௥ (4.71) ூ் ൅ܴ௩௦ ܷ௜௡௦௨

ூ் where,ܷ௜௡௦௨isthethermaltransmittanceofinsulationattachedtotheicestoragetank;

஺௜௥ 2 ܴ௩௦ isthesurfacefilmresistanceofstillairunderhorizontalheatflow,0.12m ͼԨ/W

(ASHRAE2009c).

ThethermaltransmittanceofthehorizontalsurfaceoficeͲlayerisapiecewise

functionasfollows:

ூ் ூ ܷ௛௦ǡௗ௢௪௡ǡܶ௜௖௘ ൏ܶ௜௔ ூ ூ ܷ௛௦ ൌ ൞ Ͳǡ ܶ௜௖௘ ൌܶ௜௔ ǡ (4.72) ூ் ூ ܷ௛௦ǡ௨௣ǡܶ௜௖௘ ൐ܶ௜௔

where,

ூ் ͳ ܷ௛௦ǡௗ௢௪௡ ൌ ǡ ͳ ஺௜௥ (4.73) ூ் ൅ܴ௛௦ǡௗ௢௪௡ ܷ௜௡௦௨

ூ் ͳ ܷ௛௦ǡ௨௣ ൌ ǡ ͳ ஺௜௥ (4.74) ூ் ൅ܴ௛௦ǡ௨௣ ܷ௜௡௦௨

ூ் ܷ௛௦ǡௗ௢௪௡isthethermaltransmittanceoficestoragetankhorizontalsurfaceunderheat

ூ் flowdownward;ܷ௛௦ǡ௨௣isthethermaltransmittanceoficestoragetankhorizontal

஺௜௥ surfaceunderheatflowupward;ܴ௛௦ǡௗ௢௪௡isthefilmresistanceofstillairunder

2 ஺௜௥ horizontalsurfaceunderheatflowdownward,0.16m ͼԨ/W(ASHRAE2009c);ܴ௛௦ǡ௨௣is

56 thefilmresistanceofstillairunderhorizontalsurfaceunderheatflowupward,0.11 m2ͼԨ/W(ASHRAE2009c).

RespectingtherelationbetweenthewaterͲlayertemperatureandtheindoorair

temperature,thepiecewisefunctionofthethermaltransmittanceofhorizontalsurface ofthewaterͲlayerisas

ூ் ூ் ௐ ܷ௛௦ǡ௨௣ ൅ܷ௛௦ǡௗ௢௪௡ሺͳെݏ݃݊ሺߛሻሻǡܶ௪ ൏ܶ௜௔ ௐ ௐ ܷ௛௦ ൌ ൞ Ͳǡ ܶ௪ ൌܶ௜௔ ǡ (4.75) ூ் ூ் ௐ ܷ௛௦ǡௗ௢௪௡ ൅ܷ௛௦ǡ௨௣ሺͳെݏ݃݊ሺߛሻሻǡܶ௪ ൐ܶ௜௔

where,thesignfunctionݏ݃݊ሺߛሻreturns1forߛ>0,returns0forߛ=0.

TheverticalsurfaceareaoftheiceͲlayerisestimatedas

ூ ூ் ܣ௩௦ ൌߛ௩௦ǡ (4.76)

andtheverticalsurfaceareaofthewaterͲlayer

ௐ ூ் ܣ௩௦ ൌ ሺͳെߛሻܣ௩௦ǡ (4.77)

where,theverticalsurfaceareaoftheiceͲstoragetankwallsis

ூ் ூ் ூ் ூ் ܣ௩௦ ൌʹሺܹ ൅ܮ ሻܪ Ǥ (4.78)

ூ் Thethermaltransmittanceܷ௜௡௦௨andthetemperatureܶ௜௔ aretheinput

ூ் 2 parameters.Thecasestudyusedܷ௜௡௦௨=0.253W/(m ͼԨ)fortheicestoragetank thermaltransmittance;theresidentialenvironmenttemperatureof18Ԩ(Leckner&

Zmeureanu2011)astheindoorairtemperature.

57 4.4.5. Passingflow

Thepassingflowtemperatureislowerthantheenteringliquidwater temperature,becausetheiceͲlayercoolsthepassingflow.Thetemperatureofthe passingflowiscalculatedasBehschnitt(1996):

݉ሶ ூ்௅ܶூ்௅ ൅݉ሶ ூ்ௌܶூ்ௌ െߝൣ݉ሶ ூ்௅൫ܶூ்௅ െܶூ ൯൅݉ሶ ூ்ௌ൫ܶூ்ௌ െܶூ ൯൧ ூ் ௪௙ ௜௡ǡ௪௙ ௪ ௜௡ǡ௪ ௪୤ ௜௡ǡ௪௙ ௜௖௘ ௪ ௜௡ǡ௪ ௜௖௘ ܶ௙ǡ௪ ൌ ூ்ௌ ூ்௅ ǡ ݉ሶ ௪ ൅݉ሶ ௪௙

(4.79)

where,ߝisthecoolingeffectivenessoftheiceͲlayer,calculatedasTamasauskasetal.

(2012):

ͶʹǤ͵ͷߛǡ Ͳ ൑ ߛ ൏ ͲǤͲʹ ߝൌቐͳʹͳǤͷߛହ െ ͳ͸ͷǤͻߛସ ൅ͺͺǤ͹͹ߛଷ ൅ ʹ͵Ǥ͵ͳߛଶ ൅ ͵Ǥͳʹߛ ൅ ͲǤ͹ͻʹͺǡ ͲǤͲʹ ൑ ߛ ൑ ͲǤͶͶǤ ͳǡ ͲǤͶͶ ൏ ߛ ൑ ͳ

(4.80)

4.4.6. Temperaturesinicestoragetank

ThetemperatureoftheiceͲlayerisestimatedatconstanttemperatureof0Ԩ

ሺTamasauskasetal.2012).ThewaterͲlayertemperatureiscalculatedbysolvingthe energybalanceEqns(4.50)and(4.55).ByemployingbackwardEulermethodto discretizetheordinarydifferentialequations,weobtain:

ூ ூ ݉௜௖௘௧ െ݉௜௖௘௧ ܳሶ ூ ൌെ ೕ ೕషభ ܮ ǡ (4.81) ௦௧௢ οݐ ௪

ቀ݉ௐܿ ௐቁ ቀܶௐ െܶௐ ቁ ௪ ௣௪ ௪ ௧ೕ ௪ ௧ೕషభ ௧ೕషభ (4.82) ܳሶ ௐ ൌ ǡ ௦௧௢ οݐ

where,

(οݐ ൌ ݐ௝ െݐ௝ିଵǤ (4.83

.οݐisthetimestep;ݐ௝isthecurrenttime;ݐ௝ିଵistheprevioustime.RearrangingEqns

(4.81)and(4.82)wehave:

ሶ ூ οݐܳ ூ ூ ௦௧௢ (4.84) ݉௜௖௘௧ ൌ݉௜௖௘௧ െ ǡ ೕ ೕషభ ܮ௪

ሶ ௐ οݐܳ ܶௐ ൌܶௐ ൅ ௦௧௢ Ǥ ௪ ௧ೕ ௪ ௧ೕషభ (4.85) ቀ݉ௐܿ ௐቁ ௪ ௣௪ ௧ೕషభ

ThetemperatureofthewaterͲlayeriscalculatedbysimultaneouslysolvingabovetwo

ௐ ூ் ூ equationswith݉௪ =݉ Ͳ݉௜௖௘substituted.

.Thetimestepοݐistheinputparameter,thecasestudyusedοݐ=900s

4.4.7. InterchangeofenergyinsolidͲliquidequilibrium

InterchangeofenergyforsolidͲliquidequilibriumisaproceduretoamendthe discretizedsolutionsfortheoccasionsinvolvingthephasechanging.Phasechange occursduringtheheatcharginganddischargingoftheicestoragetank.Inthenumerical solutionofthediscretizedenergybalanceequationsoftheicestoragetank,thefinal momentsofphasechangemaytakeplaceduringoneofthecalculationtimeperiod.In

thefinalmomentstheicecouldcompletelydisappeared,orthewatereventually

reachedthesolidificationtemperatureof0Ԩ.Thecalculationtimeperiodreferstothe timestepusedtodiscretizetheordinarydifferentialequationsoftheenergybalances.

Thediscretizedsolutioncannottellthefinalmoments,butonlygivenforthefixedtime

step,e.g.ݐ௝andݐ௝ିଵ.Ifthemomentstakeplacebetweenݐ௝andݐ௝ିଵ,thecalculations

59 ூ ௐ shouldabletohandle,orthenumericalsolutionsas݉ <0orܶ௪ <0arepossibleto ௜௖௘௧ೕ ௧ೕ

present.Smallertimestepsmitigatebutdonoteliminatethepotentialproblem.

Withtheprocedureemployedinthisthesis,thelimitstatesofheatchargingand

dischargingarecounted.Reachingthelimitstates,theamountofheatthatisunableto

bechargedontotheiceͲlayerischargedontothewaterͲlayerinstead;andtheamount

ofheatthatisunabletobedischargedfromthewaterͲlayerisdischargedfromtheiceͲ

layer.Twonewvariablesareintroducedforcountingthedispatch:(i)Theheattransfer rate,denotedbyܳሶ ூଶௐ,istheinterchangeofenergyfromtheiceͲlayertothewaterͲ layer;(ii)Theheattransferrate,denotedbyܳሶ ௐଶூ,istheinterchangeofenergyfromthe

waterͲlayertotheiceͲlayer.

I. TheiceͲlayer

TheprocedurestartswiththepreͲestimationoftheenergychangerateofthe

ሶ ூ iceͲlayer,ܳ௦௧௢௣௥௘,with:

ሶ ூ ሶ ூ ሶ ூ ሶ ூ ூ்௅ ܳ௦௧௢௣௥௘ ൌܳ௜୬௧ ൅ܳ௘௫௧ ൅ܳ௙ െܮ௪݉ሶ ௪௜ Ǥ (4.86)

Becauseofnotknowingifexistsenoughicetofulfilltheheatcharging,thecalculation byEqn.(4.50)isdistinguishedbythepreͲestimatedquantity.

ூ ThepreͲestimationofthemassoftheiceͲlayer,݉௜௖௘௣௥௘,attimeݐ௝iscalculated with:

ூ ሶ௦௧௢ οݐܳ ூ ூ ௣௥௘ (4.87) ݉௜௖௘௣௥௘ ൌ݉௜௖௘௧ െ Ǥ ௧ೕ ೕషభ ܮ௪

60 ூ ூ ூ IfthepreͲestimatedmass݉௜௖௘௣௥௘ш0,thenatthetimeݐ௝,݉௜௖௘௧ ൌ݉௜௖௘௣௥௘ ;if ೔ ௧ೕ

݉ூ <0,݉ூ issetequaltozero. ௜௖௘௣௥௘ ௜௖௘௧ೕ

TheheatstoragecapacityoftheiceͲlayeriscalculatedwithEqn.(4.81).Theheat

transferrateܳሶ ூଶௐiscalculatedasthedifferencebetweenthepreͲestimateddemand forheatstorageandthecapacityoftheiceͲlayeras:

ሶ ூଶௐ ሶ ூ ሶ ூ ܳ ൌܳ௦௧௢௣௥௘ െܳ௦௧௢Ǥ (4.88)

II. ThewaterͲlayer

Theheattransferrateܳሶ ூଶௐisaddedtoEqn.(4.55):

ሶ ௐ ሶ ௐ ሶ ௐ ሶ ௐ ሶ ூଶௐ ܳ௦௧௢௣௥௘ ൌ൫ܳ௜௡௧ ൅ܳ௘௫௧ ൅ܳ௙ ൯൅ܳ Ǥ (4.89)

ௐ ThepreͲcalculationofthetemperatureofthewaterͲlayerattimeݐ௝,ܶ௪ ௣௥௘,ispreͲ

calculatedas:

ௐ ሶ௦௧௢ οݐܳ ௐ ௐ ௣௥௘ ܶ௪ ௣௥௘ ൌܶ௪ ௧ ൅ Ǥ (4.90) ௧ೕ ೕషభ ቀ݉ௐܿ ቁ ௪ ௣௪ ௧ೕషభ

ௐ ௐ ௐ ௐ Ifܶ௪ ௣௥௘ ш0,congelationdoesnotcommence,andhenceܶ௪ ௧ =ܶ௪ ௣௥௘ ;ifܶ௪ ௣௥௘<0, ௧ೕ ೕ ௧ೕ

ܶௐ issetequaltozero.Withthetemperatureܶௐ ,theheatstoragecapacityofthe ௪ ௧ೕ ௪ ௧ೕ

waterͲlayeriscalculatedwithEqn.(4.82).Theheattransferrateܳሶ ௐଶூiscalculatedas:

ሶ ௐଶூ ሶ ௐ ሶ ௐ ܳ ൌܳ௦௧௢௣௥௘ െܳ௦௧௢ǡ (4.91)

61 whichcorrespondstotheamountoficefrozenfromwaterinextra.ThemassoficeͲ layer,therefore,shouldbe:

݉ூ ൌ݉ூ ൅݉ூ ǡ ௜௖௘௧ೕ ௜௖௘௣௥௘ ௜௖௘௘௫௧௥௔ (4.92)

ூ where,݉௜௖௘௘௫௧௥௔istheextramassofice,

ሶ ௐଶூοݐܳ ூ (4.93) ݉௜௖௘௘௫௧௥௔ ൌ Ǥ ܮ௪

4.4.8. Enteringconditionsofheattransferfluid

Theenteringconditionsofheattransferfluidaredeterminedbytheoperationof otherdevicesconnectedtotheicestoragetank.Theenteringconditionsaretheice storagetankmodelinputs,relatedtothestoragetankheatsourceandheatsink.

Thesolarthermalcollectoristheheatsourceoficestoragetankinthesolar

combisystem.Theleavingconditionsoftheheattransferwaterfromthesolarthermal

collectoraretheenteringconditionstotheicestoragetank:

ூ்ௌ ஼௢௟௟ ݉ሶ ௪ ൌ݉ሶ ௪ ǡ (4.94)

ூ்ௌ ஼௢௟௟ ܶ௜௡ǡ௪ ൌܶ௢௨௧ǡ௪ǡ (4.95)

ܿ ூ்ௌ ൌܿ ஼௢௟௟ Ǥ ௣௜௡ǡ௪ ௣௢௨௧ǡ௪ (4.96)

Theheatpumpistheheatsinkoficestoragetankinthesolarcombisystem.The leavingconditionsoftheheattransferwaterfromtheheatpumpevaporatorarethe enteringconditionstotheicestoragetank:

ூ்௅ ா௩௔௣ ݉ሶ ௪௜ ൌ݉ሶ ௪௜ ǡ (4.97)

ூ்௅ ா௩௔௣ ݉ሶ ௪௙ ൌ݉ሶ ௪௙ ǡ (4.98)

ூ்௅ ா௩௔௣ ܶ௜௡ǡ௪௙ ൌܶ௢௨௧ǡ௪௙ǡ (4.99)

ܿ ூ்௅ ൌܿ ா௩௔௣ Ǥ ௣௜௡ǡ௪௙ ௣௢௨௧ǡ௪௙ (4.100)

4.4.9. Exergyanalysisoficestoragetank

Thissectionpresentstheexergyanalysisoftheicestoragetankoperationby

regardingtheicestoragetankasathermodynamicopensystem.Theexergybalance equationisestablishedconsideringtheflowofwater,theheattransferbetweenthe storagetankandsurroundings,theinterͲlayerheattransferbetweenthetwocontrol

volumes,andtheexergydestroyedduringtheoperation.

Exergytransferwithheatsourceandheatsink

Whenthesolarthermalcollectorisoperatedagainsttheicestoragetank,the netexergytransferratefromtheheatsource,ܺሶ ூ்ௌ,is:

ܺሶ ூ்ௌ ൌܺሶ ஼௢௟௟ସூ்Ǥ (4.101)

Whentheheatpumpisturnedon,thenetexergytransferratetotheheatsink,ܺሶ ூ்௅,is:

ܺሶ ூ்௅ ൌܺሶ ா௩௔௣Ǥ (4.102)

Exergytransferwithsurroundings

ூ் Theexergytransferbetweentheicestoragetankandthesurroundings,ܺሶ௘௫௧,is

calculatedwith:

63 ூ் ூ ௐ ܺሶ௘௫௧ ൌܺሶ௘௫௧ ൅ܺሶ௘௫௧ǡ (4.103)

where,

ூ ூ ܺሶ௘௫௧ ൌܳሶ௘௫௧ȁ߬௜௔ȁǡ (4.104)

ௐ ௐ ܺሶ௘௫௧ ൌܳሶ௘௫௧ȁ߬௜௔ȁǡ (4.105)

ூ ௐ ܺሶ௘௫௧istheexergytransferrateoftheiceͲlayerwiththeindoorsurroundings;ܺሶ௘௫௧isthe

exergytransferrateofthewaterͲlayerwiththeindoorsurroundings;߬௜௔isthequality factorofconductionͲconvectionwithrespecttotheindoorairtemperatureܶܭ௜௔,

ܶܭͲ ߬݅ܽ ൌͳെ Ǥ (4.106) ܶܭ݅ܽ

்௄ ForthequalityfactorofconductionͲconvection,generallyexpressedas߬=1Ͳ బ, ்௄

whenܶܭ>ܶܭ଴,thequalityfactorispositive,indicatingtheconductionͲconvection embodieswarmexergywithrespecttothereferencestate.Whenܶܭ<ܶܭ଴,thequality

factorisnegative,indicatingcoldexergyembodied.Whenܶܭ=ܶܭ଴,thequalityfactoris

zero,indicatingnoexergyistransferredduetotheconductionͲconvection.Bytakingthe qualityfactorinabsolutevalue,thesignconventionofthequalityfactoriswithdrawn, sothatnoconflictionwouldbecausedagainstthesignconventionofenergytransfer process.

ExergydestroyedbyinterͲlayerheattransfer

TheheattransferbetweentheiceͲlayerandwaterͲlayerduetothetemperature

differencedegradesthequalityoftheheat.TheinterͲlayerexergydestructionrate,

ሶ ூ் ܺ௜௡௧ǡௗ௘௦௧,iscalculatedas:

64 ሶ ூ் ሶ ூ ሶ ௐ ܺ௜௡௧ǡௗ௘௦௧ ൌܺ௜௡௧ െܺ௜௡௧ǡ (4.107)

ሶ ூ ሶ ௐ where,ܺ௜௡௧istheexergytransferrateoftheiceͲlayerwiththewaterͲlayer;ܺ௜௡௧isthe

exergytransferrateofthewaterͲlayerwiththeiceͲlayer.Furthertransforming,

ሶ ூ் ሶ ூ ௐ ூ ܺ௜௡௧ǡௗ௘௦௧ ൌܳ௜௡௧൫ห߬ หെห߬ ห൯ǡ (4.108)

where,߬ௐisthequalityfactorofthetransferredheatwithrespecttothewaterͲlayer

ௐ temperature,ܶܭ௪ ,as:

ܶܭ ௐ ଴ (4.109) ߬ ൌͳെ ௐ Ǣ ܶܭ௪

߬ூ isthequalityfactorofthetransferredheatwithrespecttotheiceͲlayertemperature,

ூ ܶܭ௜௖௘,as:

ூ ܶܭ଴ ߬ ൌͳെ ூ Ǥ (4.110) ܶܭ௜௖௘

Changeofexergyduringheatstorageprocessinicestoragetank

Theexergyvariationratewithintheicestoragetank,οܺሶ ூ்,iscalculatedasthe

sumofthatofthetwolayers,

οܺሶ ூ் ൌοܺሶ ூ ൅οܺሶ ௐǡ (4.111)

where,οܺሶ ூistheexergyvariationrateoftheiceͲlayer,

ூ ூ ூ ቀ݉௜௖௘ െ݉௜௖௘ ቁ ߮௜௖௘ ௧ೕ ௧ೕషభ (4.112) οܺሶ ூ ൌ Ǣ ߂ݐ

οܺሶ ௐistheexergyvariationrateofthewaterͲlayer,

65 ௐ ௐ ௐ ௐ ݉௪ ௧ ߰௪ ௧ െ݉௪ ௧ ߰௪ ௧ οܺሶ ௐ ൌ ೕ ೕ ೕషభ ೕషభ Ǣ (4.113) ߂ݐ

ூ ௐ ߮௜௖௘isthespecificexergyoficeintheiceͲlayer;߰௪ isthespecificexergyofwaterinthe

waterͲlayer.

Exergydestructionandexergeticefficiency

ሶ ூ் Theexergydestructionratebytheicestoragetankoperation,ܺௗ௘௦௧,isobtained fromtheestablishedexergybalance:

ሶ ூ் ሶ ூ்ௌ ሶ ூ்௅ ሶ ூ் ሶ ூ் ሶ ூ் ܺௗ௘௦௧ ൌܺ െܺ ൅ܺ௘௫௧ ൅ܺ௜௡௧ǡௗ௘௦௧ െ߂ܺ Ǥ (4.114)

ூ் Theexergeticefficiencyoftheicestoragetankoperation,ߟூூ ,iscalculatedas:

௧ೕ ூ் σ ܺௗ௘௦௧ ߟூ் ൌͳെ ଴ Ǥ (4.115) ூூ ௧ೕ ூ்ௌ σ଴ ܺ

௧ೕ ூ் σ଴ ܺௗ௘௦௧isthesumofexergydestructionoftheicestoragetanksincethestarttimeof

operation,where,

ூ் ሶ ூ் (ௗ௘௦௧ ൌܺௗ௘௦௧߂ݐǡ (4.116ܺ

௧ೕ ூ்ௌ σ଴ ܺ isthesumofexergytransferfromtheheatsourceoficestoragetanksincethe starttimeofoperation,where,

(ூ்ௌ ൌܺሶ ூ்ௌ߂ݐǤ (4.117ܺ

4.5. Mathematicalmodelofwaterstoragetank Thewaterstoragetank(Fig.4.6)containswaterasthethermalstoragemedia.

Twoimmersionheatexchangersaremountedinsidethestoragetank,aspartoftwo

66 circulationloopsconnectingthestoragetanktotheheatsourcesandheatsinks.Inside thecirculationloops,wateristheheattransferfluid.Heattransfersbetweentheheat transferfluidandthestoragemediabytheimmersionheatexchangers.Coldwaterfrom thecitymainflowsintothestoragetanktocompensatethehotwaterdrawnoff.

WheneverthedrawͲofftemperaturebelowthesettemperature,aninlineheateronthe drawͲoffpipelinestartsuptoreheatthehotwaterusingelectricity.Themathematical modelsimulatesthewaterstoragetankoperationtocalculatethetemperatureofwater inthestoragetankandtheelectricpowerinputtotheinlineheater.

Fig.4.6Schematicofwaterstoragetankmodel ThemodelingofwaterstoragetankisbasedonDuffie&Beckman(2006)and

Cruickshank&Harrison(2010).Thestoragemediaisregardedasfivedistinct

homogeneouslayersinequalmass.EachlayerisidentifiedasonewaterͲnode,denoted

asܰ݅,݅=1,2…5insequencefromthebottomtotop.Oneimmersionheatexchanger

67 ܪܺͳlocatesatthewaterͲnodeܰͳ,theotherimmersionheatexchangerܪܺʹlocatesat thewaterͲnodeܰ͵.TheheattransferofwaterͲnodestakesplaceacrossthecontact surfacebetweenadjacentwaterͲnodes,tanksurface,andimmersionheatexchanger surfaces.ThecoldwaterportlocatesatthewaterͲnodeܰͳ,thehotwaterportlocates atthewaterͲnodeܰͷ.Thewaterflowcreatedbytheleavinghotwaterandentering coldwateriscalledjetflowinhere,whichisinvolvedwiththetransferofheatwiththe waterͲnodesaswell.Themodelingassumesthatthejetflowtakesplaceaswater movingfromthelowernodetotherightabovehighernode,whichisapatternofrelay inthebottomͲupdirection.Apremiseimpliedisthattheamountofjetflowatonetime stepdoesnotexceedthemassoftheonewaterͲnode.

4.5.1. TheenergybalanceofwaterͲnodes

EnergybalanceisestablishedforthewaterͲnodeܰ݅as:

ሶ ே௜ ሶ ே௜ ሶ ே௜ ሶ ே௜ ሶ ே௜ ܳ௦௧௢ ൌܳு௑ ൅ܳ௜௡௧ ൅ܳ௘௫௧ ൅ܳ௙ (4.118)

ሶ ே௜ ሶ ே௜ where,ܳ௦௧௢istheenergychangerateofthewaterͲnodeܰ݅;ܳு௑istheheattransfer

ሶ ே௜ rateoftheimmersionheatexchanger;ܳ௜௡௧istheheattransferrateofthewaterͲnode

ே௜ ܰ݅withtheadjacentnodesacrossthecontactsurfaceinbetween;ܳሶ௘௫௧istheheat

ሶ ே௜ transferrateofthewaterͲnodeܰ݅withthesurroundingsacrossthetanksurface;ܳ௙ is

theheattransferrateofthewaterͲnodeܰ݅withthejetflow.

TermsinEqn.(4.118)havefollowingformulations,

ே ே௜ ே௜ ݀ ቀ݉௪ܿ௣ ܶ௪ ቁ ܳሶ ே௜ ൌ ௪ ǡ (4.119) ௦௧௢ ݀ݐ

ሶ ே௜ ሶ ே௜ ሶ ே௜ ሶ ே௜ ܳு௑ ൌܳௐ்ௌଵ ൅ܳௐ்ௌଶ ൅ܳௐ்௅ଵǡ (4.120)

݇ௐ்௜ାଵ ௐ் ௐ் ே௜ାଵ ே௜ ۓ ൫ܣ௕ ൅ ܣ௖௦ǡ௦௛௘௟௟൯൫ܶ௪ െܶ௪ ൯ǡ ݅ ൌ ͳ ௐ்௜ାଵܮ ۖ ௐ்௜ ۖ ௐ் ௐ் ே௜ିଵ ே௜ ݇ ۖ ௖௦ǡ௦௛௘௟௟൯൫ܶ௪ െܶ௪ ൯ ൅ܣ௕ ൅ܣ൫ ۖ ܮௐ்௜ ሶ ே௜ ܳ௜௡௧ ൌ ௐ்௜ାଵ ǡ (4.121) ௐ் ௐ் ே௜ାଵ ே௜ ݇ ۔ ௖௦ǡ௦௛௘௟௟൯൫ܶ௪ െܶ௪ ൯ǡ ݅ ൌ ʹǡ ͵ǡ Ͷܣ௕ ൅ܣௐ்௜ାଵ ൫ ۖ ܮ ۖ ௐ்௜݇ ۖ ௐ் ௐ் ே௜ିଵ ே௜ ۖ ൫ܣ௕ ൅ ܣ௖௦ǡ௦௛௘௟௟൯൫ܶ௪ െܶ௪ ൯ǡ ݅ ൌ ͷ ௐ்௜ܮ ە

ௐ் ே௜ ௐ் ௐ் ே௜ ൫ܷ௩௦ ܣ௩௦ ൅ܷ ܣ ൯൫ܶ െܶ௪ ൯ǡ ݅ ൌ ͳǡ ͷ ܳሶ ே௜ ൌቊ ௛௦ ௕ ௜௔ ǡ (4.122) ௘௫௧ ௐ் ே௜ ே௜ ܷ௩௦ ܣ௩௦ ൫ܶ௜௔ െܶ௪ ൯ǡ ݅ ൌ ʹǡ ͵ǡ Ͷ

஽ுௐ ெ௔௜௡௦ ெ௔௜௡ ே௜ ே௜ ݉ሶ ௪ ቀܿ௣ ܶ௪ െܿ௣ ܶ௪ ቁ ǡ݅ൌͳ ሶ ே௜ ௪ ௪ ܳ௙ ൌ ቐ Ǥ (4.123) ݉ሶ ஽ுௐ ቀܿ ே௜ିଵܶே௜ିଵ െܿ ே௜ܶே௜ቁ ǡ ݅ ൌʹǡ͵ǡͶǡͷ ௪ ௣௪ ௪ ௣௪ ௪

ሶ ே௜ where,ܳௐ்ௌଵistheheattransferratefromtheheatsourcethroughtheheatexchanger

ሶ ே௜ ܪܺͳtothewaterͲnodeܰ݅;ܳௐ்ௌଶistheheattransferratefromtheheatsource

ሶ ே௜ throughtheheatexchangerܪܺʹtothewaterͲnodeܰ݅;ܳௐ்௅ଵistheheattransferrate

ே fromthewaterͲnodeܰ݅totheheatsinkthroughtheheatexchangerܪܺͳ;݉௪isthe

஽ுௐ ே௜ massofthewaterͲnode;݉ሶ ௪ isthemassflowrateofdomestichotwaterdrawn;ܶ௪

ே௜ାଵ isthetemperatureofthewaterͲnodeܰ݅;ܶ௪ isthetemperatureoftheofthewaterͲ

ே௜ିଵ ெ௔௜௡ nodeܰ݅+1;ܶ௪ isthetemperatureoftheofthewaterͲnodeܰ݅Ͳ1;ܶ௪ isthe temperatureofwaterfromthecitymain;݇ௐ்௜isthethermalconductivityofwater

ௐ்௜ାଵ ௐ் givenbyEqn.(4.140)(similarto݇ );ܷ௩௦ isthethermaltransmittanceofvertical

ௐ் surfaceofthewaterstoragetank;ܷ௛௦ isthethermaltransmittanceofthehorizontal surfaceofthewaterstoragetank;ܮௐ்௜isthelengthgivenbyEqn.(4.144)(similarto

69 ௐ்௜ାଵ ௐ் ௐ் ܮ ሻ;ܣ௕ isthebaseareaofthewaterstoragetank;ܣ௖௦ǡ௦௛௘௟௟isthecrossͲsectional

ே௜ areaofthewaterstoragetankshell;ܣ௩௦ istheverticalsurfaceareaofthewaterͲnode

ܰ݅.

4.5.2. Heattransferofimmersionheatexchangers

Theheattransferbyconductiontakesplaceinthewaterstoragetankbetween theadjacentnodes,infunctionalofthetemperaturegradientandthermalconductivity.

Thedependentquantitiesgenerallyareavailableforthemodeling.Incontrast,theheat transferbyconvectionismorecomplex,andthemodelingdoesnothaveanapproach

widelybeingacceptedforallsituations.However,theconvectioncannotbeabandon.

Withoutemployingcertaincomputationalapproach,largetemperaturedifference

presentamongtheexchangernode(thenodewiththeheatexchanger)andnonͲ

exchangernode(thenodewithouttheheatexchanger).

Becausetheconvectioncausedbythefluidmechanismofbuoyancyeffectis

unknown,anewconceptofsoͲcalledcontinuousnodeisconceivedformodeling stratifiedthermalstoragetank.Theconceptispresentedinthissection,withaseriesof

definitionincludingnodeexchangeheat,nodeexchangecontribution,andnode exchangemassforexplicatethecontinuousnodecomponents.Theapplication

approachisgivenbycaseofthisstudy.

70 Theheattransferofimmersionheatexchangerܪܺwiththenodeܰ݅isdefined

ሶ ே௜ asnodeexchangeheat.Denotingthenodeexchangeheatratebyܳு௑,theoverallheat

transferrateofimmersionheatexchangerܪܺbyܳሶ ு௑,herehas

௡ ே௜ ு௑ ෍ܳሶு௑ ൌܳሶ ǡ ሺ݅ൌ ͳǥ݊ሻǡ (4.124) ௜ୀଵ

where,݊isthenumberofnodesinthestratifiedthermalstoragetank.Theratioofthe

nodeexchangeheatratetotheoverallheattransferrateofimmersionheatexchanger

ܪܺisdefinedasnodeexchangecontribution,denotingbyߙே௜,

ܳሶ ே௜ ߙே௜ ൌ ு௑ Ǥሺ݅ൌͳǥ݊ሻ (4.125) ܳሶ ு௑

Forasingleheatexchangerܪܺ,thesummationofallnodeexchangecontributions

equalsto1,

௡ ෍ߙே௜ ൌͳ ൫Ͳ൑ߙே௜ ൑ͳ൯ǡ (4.126) ௜ୀଵ

where,ߙே௜=0indicatesthatthenodeܰ݅isnotinfluencedbytheheattransferof

immersionheatexchangerܪܺ;ߙே௜>0indicatesthatthenodeܰ݅isinfluencedbythe

heattransferofimmersionheatexchangerܪܺ;ifߙே௜=1,thenodeܰ݅istheonlyone

nodethatundergoestheheattransferwiththeimmersionheatexchangerܪܺ.

Thecontinuousnodeisalumpednodeaggregatedwiththenodesthat

undergoingthetransferofheatwiththeimmersionheatexchanger.Byassigningthe

indices݌ǥݍtothecomponentnodes,thecontinuousnodeiscomposedofthenodeܰ݅

(݅=݌ǥݍ),andthecomponentnodeexchangercontributionissubjectto:

71 ߙே௜ ൐ͲǤ ሺ݅ൌ ݌ǥݍሻ (4.127)

Becausethecomponentnodesaggregateintoonecontinuousbody,eachhasanequal nodeexchangecontribution.Themagnitudeofthenodeexchangecontributionis, therefore,thereciprocalofthecomponentnumber,

ͳ ߙே௜ ൌ Ǥሺ݅ൌ݌ǥݍሻ (4.128) ݍെ݌൅ͳ

Thecontinuousnodeincludestheexchangernode,and,maybe,nonͲexchangernode.

Beingwiththeimmersionheatexchanger,anexchangernodeisalwaysunderthe influenceoftheheattransferand,therefore,Eqn.(4.127)isappliedandanexchanger nodeisalwaysthecontinuousnodecomponent.AsforthenonͲexchangernode,it couldbeornotbeinvolvedintheheattransferofimmersionheatexchanger.The situationforanonͲexchangernodeinvolvedthetransferofheatandbecomesthe continuousnodecomponentis:(i)itsadjacenthighertemperaturenodeinpositiveߙே௜

underaheatchargingprocess,or(ii)itsadjacentlowertemperaturenodeinpositive

ߙே௜underaheatdischargingprocess.

Duringaheatchargingprocess,thenodeexchangeheatrateiscalculatedas:

ே௜ ே௜ ு௑ ܳሶு௑ ൌߙ ܳሶ Ǥሺ݅ൌͳǥ݊ሻ (4.129)

Bothofthequantitiesattherightsideareknown.Duringaheatdischargingprocess,the

nodeexchangemassisdefinedasthepartialmassofnodeܰ݅thatisuponthe

ே௜ discharge,denotedby݉ு௑.Themagnitudeofthenodeexchangemassisbetween0 andthemassofthenode,inclusive.Thenodethatdoesnotbelongtothecontinuous

72 nodehasthenodeexchangemassofzero.Thecomponentnodesofcontinuousnode

ே haveequalnodeexchangemassforeach,denotedby݉ு௑,

ሶ ு௑ ே ܳ οݐ ݉ு௑ ൌ ǡሺ݅ൌͳǥ݊ሻ (4.130) σ௡ ቂݏ݃݊൫ߙே௜൯ ቀܥ ே௜ܶே௜ െܥ ு௑ܶு௑ቁቃ ௜ୀଵ ௣ ௣௜௡ ௜௡

ே௜ ு௑ whereܶ isthetemperatureofthenodeܰ݅;ܶ௜௡ isthetemperatureoftheheat transferfluidenteringtheimmersionheatexchangerܪܺ.Andthedischarging temperatureoftheimmersionheatexchangeroperationisthearithmeticmean temperatureofthecomponentnodes,calculatedas:

௤ ே௜ ே௜ σ௜ୀ௣ ݏ݃݊൫ߙ ൯ܶ ܶு௑ ൌ Ǥሺ݅ൌ݌ǥݍሻ (4.131) ௢௨௧ ݍെ݌൅ͳ

Thenodeexchangemassfeaturestheheatdischargingprocess,usedtocalculatethe nodeexchangemassofnodeܰ݅,

ே௜ ே௜ ே ݉ு௑ ൌݏ݃݊൫ߙ ൯݉ு௑Ǥሺ݅ൌͳǥ݊ሻ (4.132)

Whereupon,thenodeexchangeheatiscalculatedas:

ே௜ ே௜ ே௜ ு௑ ு௑ ݉ு௑ ቀܥ௣ ܶ െܥ௣ ܶ௜௡ ቁ ܳሶ ே௜ ൌ ௜௡ Ǥሺ݅ൌͳǥ݊ሻ (4.133) ு௑ οݐ

Abriefsummaryisgivenfortheaboveconceptdevelopment:Thesimplest

situationforacontinuousnodebeingformediswithoneexchangernode;theutmostis withallnodesinastoragetank.Withinacontinuousnode,thetemperatureofan exchangernodeisthehighestduringtheheatcharging,thelowestduringtheheat discharging.IfanonͲexchangernodejoinsacontinuousnodedependson:(i)the

73 temperaturedifferencewiththeadjacentnodes;and(ii)theoperationstatusofan

immersionheatexchanger,inheatchargingorheatdischarging.Eachnodeinastorage

tankisappointedanodeexchangecontribution.Theheatcharging/dischargingimpacts

madebyanimmersionheatexchangeroneachofthenodesarecalculated.

Thefollowingsubsectionsarepresentedhowtheconceptisapplied.Thewater storagetankhastwoimmersionheatexchangers,ܪܺͳandܪܺʹ,operatedinthree ways:theimmersionheatexchangerܪܺͳinheatchargingmode;theimmersionheat exchangerܪܺʹinheatchargingmode;theimmersionheatexchangerܪܺͳinheat

dischargingmode.

I. Theimmersionheatexchangerࡴࢄ૚intheheatchargingmode

ሶ ே௜ Theheattransferrateܳௐ்ௌଵiscalculatedas:

ሶ ே௜ ே௜ ሶ ௐ்ௌଵ ܳௐ்ௌଵ ൌߙௐ்ௌଵܳ ǡ (4.134)

ሶ ௐ்ௌଵ ே௜ where,ܳ istheheatchargingrateoftheimmersionheatexchangerܪܺͳǢߙௐ்ௌଵis

thenodeܰ݅exchangecontributionwithrespecttotheheattransferrateܳሶ ௐ்ௌଵ.

ே௜ Thenodeexchangecontributionߙௐ்ௌଵisdeterminedwiththewaterͲnode temperaturesintheheatchargingmode,showninTable4.3.

Table4.3NodeexchangecontributionsofimmersionheatexchangerHX1inheatchargingmode Conditions Decisions ேଵ ேଶ ேଵ ே௜ ܶ௪ ൏ܶ௪ ߙௐ்ௌଵ=1andߙௐ்ௌଵ=0( ݅=2,3…5) ேଵ ேଶ ேଷ ே௜ ே௜ ܶ௪ ൒ܶ௪ ൏ܶ௪ ߙௐ்ௌଵ=1/2( ݅=1,2)andߙௐ்ௌଵ=0(݅=3,4,5) ேଵ ேଶ ேଷ ேସ ே௜ ே௜ ܶ௪ ൒ܶ௪ ൒ܶ௪ ൏ܶ௪ ߙௐ்ௌଵ=1/3( ݅=1,2,3)andߙௐ்ௌଵ=0(݅=4,5) ேଵ ேଶ ேଷ ேସ ேହ ே௜ ேହ ܶ௪ ൒ܶ௪ ൒ܶ௪ ൒ܶ௪ ൏ܶ௪ ߙௐ்ௌଵ=1/4( ݅=1,2…4)andߙௐ்ௌଵ=0 ேଵ ேଶ ேଷ ேସ ேହ ே௜ ܶ௪ ൒ܶ௪ ൒ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଵ=1/5( ݅=1,2…5)

74 II. Theimmersionheatexchangerࡴࢄ૛intheheatchargingmode

ሶ ே௜ Theheattransferrateܳௐ்ௌଶiscalculatedas:

ሶ ே௜ ே௜ ሶ ௐ்ௌଶ ܳௐ்ௌଶ ൌߙௐ்ௌଶܳ ǡ (4.135)

ሶ ௐ்ௌଶ ே௜ where,ܳ istheheatchargingrateoftheimmersionheatexchangerܪܺʹ;ߙௐ்ௌଶis

thenodeܰ݅exchangecontributionwithrespecttotheheatchargingrateܳሶ ௐ்ௌଶ.

ே௜ Thenodeexchangecontributionߙௐ்ௌଶisdeterminedwiththewaterͲnode temperaturesintheheatchargingmode,showninTable4.4.

Table4.4NodeexchangecontributionsofimmersionheatexchangerHX2inheatchargingmode Conditions Decisions ேଶ ேଷ ேସ ேଷ ே௜ ܶ௪ ൐ܶ௪ ൏ܶ௪ ߙௐ்ௌଶ=1andߙௐ்ௌଶ=0( ݅=1,2,4,5) ேଵ ேଶ ேଷ ேସ ே௜ ே௜ ܶ௪ ൐ܶ௪ ൑ܶ௪ ൏ܶ௪ ߙௐ்ௌଶ=1/2( ݅=2,3)andߙௐ்ௌଶ=0(݅=1,4,5) ேଵ ேଶ ேଷ ேସ ே௜ ே௜ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൏ܶ௪ ߙௐ்ௌଶ=1/3( ݅=1,2,3)andߙௐ்ௌଶ=0(݅=4,5) ேଶ ேଷ ேସ ேହ ேଷ ே௜ ܶ௪ ൐ܶ௪ ൒ܶ௪ ൏ܶ௪ ߙௐ்ௌଶ=1/2( ݅=3,4)andߙௐ்ௌଶ=0(݅=1,2,5) ேଵ ேଶ ேଷ ேସ ேହ ே௜ ே௜ ܶ௪ ൐ܶ௪ ൑ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଶ=1/3( ݅=3,4,5)andߙௐ்ௌଶ=0(݅=1,2) ேଵ ேଶ ேଷ ேସ ேହ ே௜ ேହ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଶ=1/4( ݅=1,2…4)andߙௐ்ௌଶ=0 ேଶ ேଷ ேସ ேହ ே௜ ே௜ ܶ௪ ൐ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଶ=1/3( ݅=3,4,5)and ߙௐ்ௌଶ=0(݅=1,2) ேଵ ேଶ ேଷ ேସ ேହ ே௜ ேଵ ܶ௪ ൐ܶ௪ ൑ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଶ=1/4( ݅=2,3…5)andߙௐ்ௌଶ=0 ேଵ ேଶ ேଷ ேସ ேହ ே௜ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൒ܶ௪ ൒ܶ௪ ߙௐ்ௌଶ=1/5( ݅=1,2…5) III. Theimmersionheatexchangerࡴࢄ૚intheheatdischargingmode

Theheattransferrateܳሶ ௐ்௅ଵiscalculatedas:

ே௜ ே௜ ே௜ ௐ்௅ଵ ௐ்௅ଵ ݉ௐ்௅ଵ ቆܥ௣ ܶ௪ െܥ௣ ܶ௜௡ǡ௪ ቇ ௧ೕ ௪ ௧ ௧ೕషభ ௜௡ǡ௪ ௧ೕ (4.136) ே௜ ೕషభ ௧ೕ ܳሶௐ்௅ଵ ൌ ǡ ௧ೕ οݐ where,

݉ே௜ ൌݏ݃݊ቀߙே௜ ቁ ݉ே ǡ (4.137) ௐ்௅ଵ௧ೕ ௐ்௅ଵ௧ೕ ௐ்௅ଵ௧ೕ

and

ሶ ௐ்௅ଵ οݐܳ ே ௧ೕ ݉ௐ்௅ଵ ൌ ǡ ௧ೕ (4.138) ே௜ ே௜ ே௜ ௐ்௅ଵ ௐ்௅ଵ σ ቈݏ݃݊ ቀߙ ቁ ቆܥ௣ ܶ௪ െܥ௣ ܶ ቇ቉ ௐ்௅ଵ௧ೕ ௪ ௧ೕషభ ௜௡ǡ௪ ௜௡ǡ௪ ௧ ௧ೕషభ ௧ೕ ೕ

ሶ ௐ்௅ଵ ௐ்௅ଵ ܳ istheheatdischargingrateoftheimmersionheatexchangerܪܺͳ;ܶ௜௡ǡ௪ isthe

ே௜ temperatureofwaterenteringtheimmersionheatexchangerܪܺͳ;݉ௐ்௅ଵisthenode

ሶ ௐ்௅ଵ ே௜ ܰ݅exchangemasswithrespecttotheheatdischargingrateܳ ;ߙௐ்௅ଵisthenode

ሶ ௐ்௅ଵ ே ܰ݅exchangecontributionrespectingtheheatdischargingrateܳ ;݉ௐ்௅ଵisthe nodeexchangemassrespectingtheheatdischargingrateܳሶ ௐ்௅ଵ.

ே௜ Thenodeexchangecontributionߙௐ்௅ଵwasdeterminedwiththewaterͲnode temperaturesfortheheatdischargingmode,showninTable4.5.

Table4.5NodeexchangercontributionsofimmersionheatexchangerHX1inheatdischargingmode Conditions Decisions ேଵ ேଶ ேଵ ே௜ ܶ௪ ൐ܶ௪ ߙௐ்௅ଵ=1andߙௐ்௅ଵ=0( ݅=2,3…5) ேଵ ேଶ ேଷ ே௜ ே௜ ܶ௪ ൑ܶ௪ ൐ܶ௪ ߙௐ்௅ଵ=1/2( ݅=1,2)andߙௐ்௅ଵ=0(݅=3,4,5) ேଵ ேଶ ேଷ ேସ ே௜ ே௜ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൐ܶ௪ ߙௐ்௅ଵ=1/3( ݅=1,2,3)andߙௐ்௅ଵ=0(݅=4,5) ேଵ ேଶ ேଷ ேସ ேହ ே௜ ேହ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൑ܶ௪ ൐ܶ௪ ߙௐ்௅ଵ=1/4( ݅=1,2…4)andߙௐ்௅ଵ=0 ேଵ ேଶ ேଷ ேସ ேହ ே௜ ܶ௪ ൑ܶ௪ ൑ܶ௪ ൑ܶ௪ ൑ܶ௪ ߙௐ்௅ଵ=1/5( ݅=1,2…5) Asforthedischargingtemperatureoftheimmersionheatexchangerܪܺͳ,the calculationisas:

௤ ே௜ ே௜ σ௜ୀ௣ ݏ݃݊ ቀߙௐ்௅ଵ ቁ ܶ௪ ௐ்௅ଵ ௧ೕ ௧ೕషభ (4.139) ܶ௢௨௧ǡ௪ ൌ ǡ ௧ೕ ݍെ݌൅ͳ

ௐ்௅ଵ where,ܶ௢௨௧ǡ௪ isthetemperatureofwaterleavingtheimmersionheatexchangerܪܺͳ.

76 SubstitutingEqns.(4.134),(4.135),and(4.136)toEqn.(4.120),thedetermined

ே௜ ே௜ ே௜ nodeexchangecontributionsߙௐ்ௌଵ,ߙௐ்ௌଶ,andߙௐ்௅ଵ,basedonthenode temperaturesofprevioustimeܶே௜ ,willbeusedtocalculateܳሶ ே௜ ,theheattransfer ௪ ௧ೕషభ ு௑

rateoftheimmersionheatexchangerofnodeܰ݅.Thenodetemperaturesatthecurrent time,ܶே௜ ,iscalculatedbysolvingtheenergybalanceequationsofthefivenodesof ௪ ௧ೕ thewaterstoragetank.

Theapproachpresentedabovewasusedtoestimatetheimmersionheat exchangeroperationontothestratifiednodes.Theconceptofcontinuousnodewas developedbyacknowledgingthatthefluidmotionofconvectiontakesplaceinthe presenceoftemperaturedifference.Theexistenceofflowmotioncanberecognizedby thetemperaturedifferencesamongthenodes:Fromthepointofviewoffluid mechanics,thetemperaturedifferenceindicatesthediffusionandcollisionsinthefluid, especiallywhenlargetemperaturedifferenceinplace,thebuoyancyisbustlingaboutto

homogenizethedensityoffluid(Turner1973).Withtheconceptofcontinuousnode,to

modeltheflowmotionisavoided.Instead,bydesignatingthermodynamicsystemina

flexibleway,theapproachreplacestheneedformodelingtheconvection.Fromthe

thermodynamicpointofview,howthecontinuousnodeconceptworkscanbe explicatedthisway:Theestablishmentofenergybalanceequation(4.118)upon controlledmassimpliesthat,onenodeisonethermodynamicsystem,andthestorage tankoperationisrepresentedbyseveralthermodynamicsystems.Whenthecontinuous nodehasmorethanonecomponentnodes,themultiplethermodynamicsystemsofthe

77 componentnodes(eachinoneͲnodeͲmagnitude)areregardedasonethermodynamics system(inmultipleͲnodesͲmagnitude)undergoingtheheatchargingordischarging process.Inthisperspective,thenodeexchangecontributionisassigned.Afterwards,the multiplethermodynamicsystemsareresumed,theenergybalanceequation(4.118)is validandtheunknowntemperatureofeachnodecanbesolved.

4.5.3. Heattransferbetweennodes

TheinterͲnodalheattransfertakesplaceunderthetemperaturedifference

betweenadjacentwaterͲnodesintheverticaldirectionbyconduction.Inaddition,the

heatconductionthroughthetankshelliscalculated.Itwasassumedthattheinner verticalsurfaceoftankshellhasthetemperatureequaltothetemperatureofthe waterͲnodeatthesameheight.

Thethermalconductivity݇ௐ்௜isanareaͲweightedpropertyas

ௐ் ௐ் ௐ்௜ ௐ் ௐ்௜ ݇௦௛௘௟௟ܣ௖௦ǡ௦௛௘௟௟ ൅݇௔௩௚ǡ௪ܣୠ ݇ ൌ ௐ் ௐ் ǡ݅ൌʹǡ͵ǥͷǡ (4.140) ܣ௖௦ǡ௦௛௘௟௟ ൅ ܣ௕

ௐ் ௐ்௜ where,݇௦௛௘௟௟isthethermalconductivityofthewaterstoragetankshell;݇௔௩௚ǡ௪isthe thermalconductivityofwaterwithrespecttothetemperature,

ܶெ௔௜௡௦ ൅ܶே௜ ௪ ௪ ǡ݅ൌͳ ۓ ௐ்௜ ʹ ܶ௔௩௚ǡ௪ ൌ Ǥ (4.141) ே௜ିଵ ൅ܶே௜ܶ۔ ௪ ௪ ǡ݅ൌʹǡ͵ǥͷ ʹ ە

ௐ் ௐ் Foraverticalcylindertank,theareasܣ௖௦ǡ௦௛௘௟௟ǡܣ௕ arecalculatedas:

78 ௐ் ௐ் ௐ் ܣ௖௦ǡ௦௛௘௟௟ ൌߨܦ ߜ௦௛௘௟௟ǡ (4.142)

ߨ ଶ ܣௐ் ൌ ܦௐ் ǡ (4.143) ௕ Ͷ

ௐ் ௐ் where,ܦ isthediameterofthecylinderwaterstoragetank;ߜ௦௛௘௟௟isthethicknessof

thewaterstoragetankshell.

ThedistancethatcorrespondstotheinterͲnodalheatconductioniscalculatedin

ܮௐ் ൅ܮௐ் ܮௐ்௜ ൌ ே௜ିଵ ே௜ ǡ݅ൌʹǡ͵ǥͷǡ (4.144) ʹ

where,

ே ௐ் ݉௪ ܮே௜ ൌ ே௜ ௐ்ǡ (4.145) ߩ௪ ܣ௕

ௐ் ܮே௜ isthethicknessofthewaterͲnodeܰ݅.ForafiveͲnodeswaterstoragetankmodel,

ͳ ݉ே ൌ ൫ߩௐ்ܣௐ்ܪௐ்൯ǡ (4.146) ௪ ͷ ௪ ௕

ௐ் ௐ் where,ߩ௪ isthedensityofwaterinthewaterstoragetank,estimatedat99Ԩ;ܪ is

theheightofthewaterstoragetank.

ௐ் ௐ் ௐ் ௐ் Thethermalconductivity݇௦௛௘௟௟andtankdimensionsߜ௦௛௘௟௟ǡܦ ǡܪ arethe

ௐ் inputparameters.Forinstance,inthecasestudy:݇௦௛௘௟௟=13.64W/(mͼԨ)forstainless

ௐ் ௐ் ௐ் steelat40Ԩ,andߜ௦௛௘௟௟=0.002m,ܦ =1.3m,ܪ =1.657m(Consolar2012).

4.5.4. Heattransferwithsurroundings

Theheattransferbetweenthewaterstoragetankandthesurrounding environmenttakesplacethroughthestoragetankwalls.Thethermaltransmittanceis

79 calculatedwiththethermaltransmittanceofthestoragetankinsulationandthesurface filmresistanceofstillair.

Thethermaltransmittanceofwaterstoragetankhorizontalsurfaceunderheat

ௐ் flowupwardisdenotedbyܷ௛௦ǡ௨௣;thethermaltransmittanceofwaterstoragetank

ௐ் horizontalsurfaceunderheatflowdownwardisdenotedbyܷ௛௦ǡௗ௢௪௡.Thecalculations

ௐ் ௐ் ௐ் forܷ௛௦ǡ௨௣andܷ௛௦ǡௗ௢௪௡,andܷ௩௦ aswell,aresimilartoEqns.(4.71),(4.74),and(4.73),

ௐ் respectively(Section4.4.4),exceptusingܷ௜௡௦௨(thethermaltransmittanceofinsulation

ூ் attachedtothewaterstoragetank)insteadofusingܷ௜௡௦௨.Thethermaltransmittanceof

horizontalsurfaceofthewaterstoragetankiscalculatedinapiecewisefunctionwith

respecttotherelationbetweenthebottomandtopnodestemperaturesandtheindoor airtemperatureas:

ௐ் ேଵ ேହ ܷ௛௦ǡ௨௣ǡܶ௪ ൏ܶ௜௔ ݋ݎ ܶ௪ ൐ܶ௜௔ ௐ் ேଵ ேହ ܷ௛௦ ൌ ൞ Ͳǡ ܶ௪ ൌܶ௜௔ ݋ݎ ܶ௪ ൌܶ௜௔ Ǥ (4.147) ௐ் ேଵ ேହ ܷ௛௦ǡௗ௢௪௡ǡܶ௪ ൐ܶ௜௔ ݋ݎ ܶ௪ ൏ܶ௜௔

Foracylinderstoragetank,theverticalsurfaceareaofthewaterͲnodeܰ݅is calculatedas:

ே௜ ୛் ௐ் ܣ௩௦ ൌߨܦ ܮே௜ Ǥ (4.148)

ௐ் Thethermaltransmittanceܷ௜௡௦௨istheinputparameter.Inthecasestudy,

ௐ் 2 ܷ௜௡௦௨=0.286W/(m ͼԨ).

80 4.5.5. Domestichotwatermakeup

Duetothedischargeofhotwater,heatoutputratefromthewaterstoragetank iscalculatedas:

ܳሶ ௐ்௅ଶ ൌ݉ሶ ஽ுௐ ቀܿ ேହܶேହ െܿ ெ௔௜௡௦ܶெ௔௜௡௦ቁǡ (4.149) ௪ ௣௪ ௪ ௣௪ ௪

where,

஽ுௐ ெ௔௜௡ ஽ுௐ ݉ሶ ௪ ൌߩ௪ ܸ௪ሶ Ǥ (4.150)

ܳሶ ௐ்௅ଶistheheatoutputrateofthewaterstoragetankontoheatingwaterforthe

஽ுௐ domesticdrawͲoff;ܸ௪ሶ isthevolumeflowrateofthedomestichotwaterdrawͲoff.

ெ௔௜௡ Thetemperatureܶ௪ istheinputvariable.Forinstance,thecasestudyused thecorrelateddatabyHugo(2008)basedonthemeasuredtemperatureofthecity mainofMontrealbyDumas&Marcoux(2004),

ெ௔௜௡ ڮ ሺʹ߱ܰሻ ൅ ݃ଶ ሺʹ߱ܰሻ ൅ ሺ߱ܰሻ ൅ ݃ଵ ሺ߱ܰሻ ൅ ݂ଶ ௪ ൌ݁൅݂ଵܶ (4.151) ൅ ݂ଵ଻ ሺͳ͹߱ܰሻ ൅݃ଵ଻ ሺͳ͹߱ܰሻǡ

where,߱ǡ ݁ǡ ݂௜ǡ݃௜(݅=1,2…7)arethecoefficientscitedinTable4.6.

஽ுௐ Theflowrateܸ௪ሶ istheinputvariable.ThecasestudyuseddatabyUlrike

Jordan&Vajen(2001)withaveragedailydomestichotwaterconsumptionof266L

(Aguilaretal.2005);ThepublishedconsumptionrateinoneͲminutetimescalewas transformedto15mintimescale.

81 Eqn.(4.123)calculatestheheattransferrateduetothehotwaterdrawͲoff.The formulationimpliesapremisethatthemassofthejetflowisnotgreaterthanthemass ofthewaterͲnode.Thestudiedcasesatisfiedthepremise,thatis,

஽ுௐ ே (ሶ ௪ οݐ ൑ ݉௪ǡ (4.152݉

.where,οݐisthetimestep,οݐ=900s

Table4.6Coefficientsofcitymaintemperatureofcasestudy Units Units ߱ 0.0086904 dimensionless ݁ 11.490452 Ԩ

݂ଵ 0.1839946 Ԩ ݃ଵ 0.0905672 Ԩ

݂ଶ Ͳ6.7763674 Ԩ ݃ଶ Ͳ7.3992997 Ԩ

݂ଷ Ͳ0.0965715 Ԩ ݃ଷ Ͳ0.2033946 Ԩ

݂ସ 0.0843686 Ԩ ݃ସ 0.8829336 Ԩ

݂ହ Ͳ0.0031887 Ԩ ݃ହ Ͳ0.0370572 Ԩ

݂଺ Ͳ0.0969744 Ԩ ݃଺ 0.1161188 Ԩ

݂଻ 0.0140706 Ԩ ݃଻ Ͳ0.0229117 Ԩ

଼݂ Ͳ0.3362503 Ԩ ଼݃ Ͳ0.0142773 Ԩ

݂ଽ Ͳ0.0277629 Ԩ ݃ଽ Ͳ0.0179754 Ԩ

݂ଵ଴ Ͳ0.0362475 Ԩ ݃ଵ଴ Ͳ0.1390454 Ԩ

݂ଵଵ Ͳ0.0136723 Ԩ ݃ଵଵ Ͳ0.0541225 Ԩ

݂ଵଶ Ͳ0.0187207 Ԩ ݃ଵଶ 0.1353561 Ԩ

݂ଵଷ Ͳ0.0066015 Ԩ ݃ଵଷ Ͳ0.0206411 Ԩ

݂ଵସ Ͳ0.0893228 Ԩ ݃ଵସ 0.0843321 Ԩ

݂ଵହ Ͳ0.0449833 Ԩ ݃ଵହ Ͳ0.0122686 Ԩ

݂ଵ଺ 0.1727085 Ԩ ݃ଵ଺ 0.0686489 Ԩ

݂ଵ଻ 0.0242816 Ԩ ݃ଵ଻ 0.0008067 Ԩ

4.5.6. Temperaturesinwaterstoragetank

ThewaterͲnodetemperatureiscalculatedbysolvingtheenergybalance formulatedforthewaterstoragetank.EmployingbackwardEulermethodtodiscretize theEqn.(4.119),weobtain:

ቀ݉ே ܿ ே௜ቁ ቀܶே௜ െܶே௜ ቁ ௪ ௣௪ ௪ ௧ೕ ௪ ௧ೕషభ ௧ೕషభ (4.153) ܳሶ ே௜ ൌ Ǥ ௦௧௢ οݐ

Byrearrangingwehave:

ሶ ே௜ οݐܳ ܶே ൌܶே ൅ ௦௧௢ Ǥ ௪ ௧ೕ ௪ ௧ೕషభ (4.154) ቀ݉ே ܿ ே௜ቁ ௪ ௣௪ ௧ೕషభ

4.5.7. Electricpowerofheater

Theelectricpoweroftheheateriscalculatedas:

ܹሶ ஺௨௫ ൌ݉ሶ ஽ுௐ ቀܥ ஺௨௫ ܶ஺௨௫ െܥ ேହܶேହቁǡ (4.155) ௪ ௣௪ǡ௦௘௧ ௪ǡ௦௘௧ ௣௪ ௪

஺௨௫ ஺௨௫ where,ܹሶ istheelectricpoweroftheauxiliaryheater;ܶ௪ǡ௦௘௧isthesetpoint temperatureofdomestichotwater.

஺௨௫ Thesettemperatureܶ௦௘௧ istheinputparameter.Forinstance,thestudiedcase

஺௨௫ setܶ௦௘௧ =40.5Ԩ(CaliforniaEnergyCommission2005).

4.5.8. Heattransferwithheatsourceandheatsink

Theheattransferratesofthewaterstoragetankwiththeheatsourcesandheat sinkaretheinputofthewaterstoragetankmodel.Determinedbytheoperating conditionsofthedevicesconnectedtotheimmersionheatexchangers,theheat transferratesareasbelow.

Fortheimmersionheatexchangerܪܺͳ,theheattransferrateintheheat

chargingmodeiscorrespondedtotheheattransferfromtheheatpumpas:

83 ܳሶ ௐ்ௌଵ ൌܳሶ ு௉ସௐ்Ǥ (4.156)

Intheheatdischargingmode,theenteringconditionoftheheattransferwateristaken astheleavingconditionfromtheradiantfloor,

ௐ்௅ଵ ோி ܶ௜௡ǡ௪ ൌܶ௢௨௧ǡ௪ǡ (4.157)

ோி where,ܶ௢௨௧ǡ௪isthetemperatureofwaterleavingtheradiantfloor.Theheatdischarging rateiscorrespondedtotheheattransfertotheradiantflooras:

ܳሶ ௐ்௅ଵ ൌܳሶ ௐ்ସோிǡ (4.158)

where,

ሶ ௐ்ସோி ሶ ோி ܳ ൌܳௗ௘௠ௗǤ (4.159)

ሶ ௐ்ସோி ሶ ோி ܳ istheheattransferratefromthewaterstoragetanktotheradiantfloor;ܳௗ௘௠ௗ istheheattransferratedemandedbytheindoorspaceontotheradiantfloor.

Fortheimmersionheatexchangerܪܺʹ,theheatchargingrateiscorresponded totheheattransferfromthesolarthermalcollectoras:

ܳሶ ௐ்ௌଶ ൌܳሶ ஼ை௟௟ସௐ்Ǥ (4.160)

4.5.9. Heatingloadofwaterstoragetank

Theheatingloadofwaterstoragetankhastwocomponents:theheat dischargingofheatexchangerܪܺͳandthedomestichotwatermakeup.Therefore,the heatoutputrateofwaterstoragetank,ܳሶ ௐ்௅,iscalculatedas:

ܳሶ ௐ்௅ ൌܳሶ ௐ்௅ଵ ൅ܳሶ ௐ்௅ଶǤ (4.161)

4.5.10. Exergyanalysisofwaterstoragetank

Thissectionpresentstheexergyanalysisofthewaterstoragetankoperationby regardingthewaterstoragetankasathermodynamicopensystem.Theexergybalance equationisestablishedconsideringtheflowofwater,theheattransferbetweenthe storagetankandsurroundings,theinterͲnodalheattransferbetweenadjacentwaterͲ nodes,andtheexergydestroyedduringtheoperation.

Exergytransferwithheatsourceandheatsink

Thenetexergytransferratefromthewaterstoragetankheatsources,ܺሶ ௐ்ௌ,is

calculatedas:

ܺሶ ௐ்ௌ ൌܺሶ ௐ்ௌଵ ൅ܺሶ ௐ்ௌଶ (4.162)

where,ܺሶ ௐ்ௌଵisthenetexergytransferrateoftheimmersionheatexchangerܪܺͳ;

ܺሶ ௐ்ௌଶisthenetexergyinputrateoftheimmersionheatexchangerܪܺʹ;botharethe inputofthewaterstoragetankmodel.Whenthewaterstoragetankreceivesexergy fromtheheatpump,

ܺሶ ௐ்ௌଵ ൌܺሶ ு௉ସௐ்Ǣ (4.163)

Whenwaterstoragetankreceivesexergyfromthesolarthermalcollector,

ܺሶ ௐ்ௌଶ ൌܺሶ ஼௢௟௟ସௐ்Ǥ (4.164)

Theexergytransferrateܺሶ ு௉ସௐ்iscalculatedwith:

ܺሶ ு௉ସௐ் ൌܺሶ ஼௢௡ௗ െܺሶ ு௉ସோிǤ (4.165)

85 Thenetexergytransferratetothewaterstoragetankheatsink,ܺሶ ௐ்௅ଵ,isthe

inputofthewaterstoragetankmodel.Whenthewaterstoragetankoutputexergyto

theradiantfloor,

ܺሶ ௐ்௅ଵ ൌܺሶ ௐ்ସோிǤ (4.166)

where,ܺሶ ௐ்ସோிistheexergytransferratefromthewaterstoragetanktotheradiant floor.

Exergytransferwithsurroundings

ௐ் Theexergytransferbetweenthewaterstoragetankandthesurroundings,ܺሶ௘௫௧ , iscalculatedas:

ହ ௐ் ே௜ ܺሶ௘௫௧ ൌ෍ܺሶ௘௫௧ǡ (4.167) ௜ୀଵ

where,

ሶ ே௜ ሶ ே௜ ܺ௘௫௧ ൌܳ௘௫௧ห߬௜௔หǡ (4.168)

ே௜ ܺሶ௘௫௧istheexergytransferrateofthewaterͲnodeܰ݅withtheindoorsurroundings.

ExergydestroyedbyinterͲnodalheattransfer

Duetothetemperaturedifferenceamongtheadjacentnodes,theinterͲnodal

ሶ ௐ் exergydestructionrateofthewaterstoragetank,ܺ௜௡௧ǡௗ௘௦௧,iscalculatedas:

ସ ሶ ௐ் ሶ ே௜ ே௜ ே௜ାଵ ܺ௜௡௧ǡௗ௘௦௧ ൌ෍หܳ௜௡௧ǡே௜ାଵ൫ห߬ หെห߬ ห൯ห (4.169) ௜ୀଵ

where,

86 ݇ௐ்௜ାଵ ܳሶ ே௜ ൌ ൫ܣௐ் ൅ ܣௐ் ൯൫ܶே௜ାଵ െܶே௜൯ǡ (4.170) ௜௡௧ǡே௜ାଵ ܮௐ்௜ାଵ ௕ ௖௦ǡ௦௛௘௟௟ ௪ ௪

ܶܭ ே௜ ଴ (4.171) ߬ ൌͳെ ே௜ǡ ܶܭ௪

ܶܭ ே௜ାଵ ଴ (4.172) ߬ ൌͳെ ே௜ାଵǡ ܶܭ௪

ሶ ே௜ ே௜ ܳ௜௡௧ǡே௜ାଵistheheattransferrateofthewaterͲnodeܰ݅withthenodeܰ݅+1,߬ isthe

qualityfactorofconductionͲconvectionwithrespecttothewaterͲnodeܰ݅absolute

ே௜ temperatureܶܭ௪ .

Changeofexergyduringheatstorageprocessinwaterstoragetank

Theexergyvariationratewithintheicestoragetank,οܺሶ ௐ்,iscalculatedasthe

sumofthatofthewaterͲnodes,

ହ ே ே௜ ே௜ ݉௪ ቀ߰௪ ௧ െ߰௪ ௧ ቁ ߂ܺሶ ௐ் ൌ෍ ೕ ೕషభ ǡ (4.173) ߂ݐ ௜ୀଵ

where,߰ே௜ isthespecificexergyofwaterinthewaterͲnodeܰ݅. ௪ ௧ೕ

Exergydestructionandexergeticefficiency

ሶ ௐ் Theexergydestructionratebythewaterstoragetankoperation,ܺௗ௘௦௧,is obtainedfromtheestablishedexergybalance:

ሶ ௐ் ሶ ௐ்ௌ ሶ ௐ்௅ ሶ ௐ் ሶ ௐ் ሶ ௐ் ܺௗ௘௦௧ ൌܺ െܺ ൅ܺ௘௫௧ ൅ܺ௜௡௧ǡௗ௘௦௧ െ߂ܺ Ǥ (4.174)

ௐ் Theexergeticefficiencyoftheicestoragetankoperation,ߟூூ ,iscalculatedas:

௧ೕ ௐ் σ ܺௗ௘௦௧ ߟௐ் ൌͳെ ଴ ǡ (4.175) ூூ ௧ೕ ௐ்ௌ σ଴ ܺ

௧ೕ ௐ் ௧ೕ ௐ்ௌ Thecalculationsforσ଴ ܺௗ௘௦௧,andσ଴ ܺ aresimilartotheEqns.(4.116)and(4.117), respectively.

Exergyoutputforheating

Theexergyoutputrateofthewaterstoragetankontoheating,ܺሶ ௐ்௅,is

calculatedas:

ܺሶ ௐ்௅ ൌܺሶ ௐ்௅ଵ ൅ܺሶ ௐ்௅ଶǡ (4.176)

where,

ௐ்௅ଶ ஽ுௐ ஽ுௐ ெ௔௜௡௦ ܺሶ ൌ݉ሶ ௪ ൫߰௪ െ߰௪ ൯ǡ (4.177)

ܺሶ ௐ்௅ଵistheexergyoutputrateoftheimmersionheatexchangerܪܺͳ;ܺሶ ௐ்௅ଶisthe

஽ுௐ exergyoutputratefordomestichotwatermakeup;߰௪ isthespecificexergyofwater

ெ௔௜௡௦ withrespecttothetemperatureofthedomestichotwaterdrawnͲoff;߰௪ isthe

specificexergyofwaterwithrespecttothetemperatureofthecitymain.

4.6. Mathematicalmodelofradiantfloor Theradiantfloor(Fig.4.7)takesheatfromtheheatsourcetoemittothe residentialspace,whichistheheatsinkoftheradiantfloor.Themathematicalmodel simulatestheradiantflooroperationtocalculatetheradiantfloorheatsupplytothe residentialspace.

Fig.4.7Schematicofradiantfloormodel

4.6.1. Heattransferflow

ோி Themassflowrateofwaterthroughtheradiantfloor,݉ሶ ௪ ,iscalculatedas:

ܳሶ ோி ݉ሶ ோி ൌ ǡ (4.178) ௪ ܥ ோி ܶோி െܥ ோி ܶோி ௣௜௡ǡ௪ ௜௡ǡ௪ ௣௢௨௧ǡ௪ ௢௨௧ǡ௪

ሶ ோி ோி where,ܳ istheheatinputratetotheradiantfloor;ܶ௜௡ǡ௪isthetemperatureofwater

ோி enteringtheradiantfloor;ܶ௢௨௧ǡ௪isthetemperatureofwaterleavingtheradiantfloor.

AccordingtoASHRAE(2004),theminimumfloorsurfacetemperatureis19Ԩ.

Thetemperatureofwaterleavingtheradiantfloorissetat20Ԩ.

4.6.2. Supplyconditions

Thesupplyconditionsaretheinputsoftheradiantfloormodel,determinedby theleavingconditionsofthedevicesconnected.Theheattransferratefromtheradiant floorheatsourceisas:

89 ݏݑ݌݌݈݅݁ݏ ݐܽ݊݇ ݁݃ܽݎݐ݋ݏ ݎሶ ௐ்ସோிǡ ݓ݄݁݊ ݓܽݐ݁ܳ ܳሶ ோி ൌቊ Ǥ (4.179) ݏݑ݌݌݈݅݁ݏ ሶ ு௉ସோிǡ ݓ݄݁݊ ݄݁ܽݐ ݌ݑ݉݌ܳ

Thetemperatureofwaterenteringtheradiantfloorisas:

ௐ்௅ଵ ݏݑ݌݌݈݅݁ݏ ݐܽ݊݇ ݁݃ܽݎݐ݋ݏ ݎோி ܶ௢௨௧ǡ௪ ǡ ݓ݄݁݊ ݓܽݐ݁ ܶ௜௡ǡ௪ ൌቊ ஼௢௡ௗ Ǥ (4.180) ݏݑ݌݌݈݅݁ݏ ௢௨௧ǡ௪ǡ ݓ݄݁݊ ݄݁ܽݐ ݌ݑ݉݌ܶ

4.6.3. Exergyanalysisofradiantfloor

Thissectionpresentstheexergyanalysisoftheradiantflooroperationby regardingtheradiantfloorasathermodynamicopensystem.Theexergybalance equationisestablishedconsideringtheflowofwater,theheattransferbetweenthe radiantfloorandindoorspace,andtheexergydestroyedduringtheoperation.

Exergyofwaterflowthroughradiantfloor

Theexergytransferratebywaterflowthroughtheradiantfloor,ܺሶ ோிௌ,is

calculatedas:

ሶ ோிௌ ோி ோி ோி ܺ ൌ݉ሶ ௪ ൫߰௜௡ǡ௪ െ߰௢௨௧ǡ௪൯ǡ (4.181)

ோி ோி where,߰௜௡ǡ௪isthespecificexergyofwaterenteringtheradiantfloor;߰௢௨௧ǡ௪isthe

specificexergyofwaterleavingtheradiantfloor.

Theexergytransferrateܺሶ ோிௌisnotedasܺሶ ு௉ସோிfortheheatpumpastheheat source,andܺሶ ௐ்ସோிforthewaterstoragetanksupplyingtheradiantfloor.

Exergyflowintoindoorspace

Theexergyoutputrateoftheradiantfloor,ܺሶ ோி௅,iscalculatedas:

90 ሶ ோி௅ ሶ ோி ܺ ൌܳ ห߬௜௔หǤ (4.182)

Destroyedexergyandexergeticefficiency

ሶ ோி Theexergydestructionratebytheradiantflooroperation,ܺௗ௘௦௧,isobtained fromtheexergybalanceequation:

ሶ ோி ሶ ோிௌ ሶ ோி௅ ܺௗ௘௦௧ ൌܺ െܺ ǡ (4.183)

ோி Theexergeticefficiencyoftheradiantflooroperation,ߟூூ ,iscalculatedas:

ܺሶ ோி ߟோி ൌͳെ ௗ௘௦௧Ǥ (4.184) ூூ ܺሶ ோிௌ

4.7. Mathematicalmodelofcirculatingpump Thecirculatingpump(Fig.4.8)driveswaterflowalongthecirculationloop.The mathematicalmodelsimulatesthecirculatingpumpoperationtocalculatetheelectric powerinput.

Fig.4.8Schematicofcirculatingpumpmodel

91 4.7.1. Electricpower

Thepumpperformanceiscalculatedusingpolynomialmodel(ASHRAE2009a).

Electricpowerinputiscalculatedusingaregressionmodelwiththefollowing formulation:

ଶ ଷ ସ ହ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ܹ ൌ ൬ܿ଴ ൅ܿଵܸ௪ ൅ܿଶܸ௪ ൅ܿଷܸ௪ ൅ܿସܸ௪ ൅ܿହܸ௪ (4.185) ሶ ௉௨௠௣଺ ሶ ௉௨௠௣଻ ሶ ௉௨௠௣଼ ௉௨௠௣ ൅ܿ଺ܸ௪ ൅ܿ଻ܸ௪ ൅଼ܸܿ௪ ൰ ܨ ǡ

௉௨௠௣ where,ܹሶ istheelectricpowerinputtothecirculatingpump;ܿ଴ ǥ଼ܿarethe

ሶ ௉௨௠௣ coefficients;ܸ௪ isthevolumeflowrateofwaterthroughthecirculatingpump;

ܨ௉௨௠௣istheflowfraction,

ܸሶ ௉௨௠௣ ܨ௉௨௠௣ ൌ ௪ ǡ (4.186) ሶ ௉௨௠௣ ܸ௠௔௫

ሶ ௉௨௠௣ andܸ௠௔௫ isthemaximumvolumeflowrateofwaterthroughthecirculatingpump.

ሶ ௉௨௠௣ Thecoefficientsܿ଴ ǥ଼ܿandthequantityܸ௠௔௫ aregivenforaspecificpumpby

theuserextractedfromthemanufacturecatalogue.Fig.4.9showsthecoefficientsand theperformancecurveofthepump,WiloͲStar16F(Wilo2012),usedinthecasestudy.

Eqn.(4.187)isthespecificformulationpreparedtomodelthecasepump,usedontoall

pumpsofthesolarcombisystem.Regardingtheflowfraction,aminimumisrequiredfor

thesakeofavoidinglowflowdamage,suchas,temperaturerise,radialbearingloads,

axialthrust,etc.Recommendedminimumflowraterangesfrom10%to80%oftheflow atbestefficiencypoint;however,thesametypeofpumpmayhavethe

92 recommendationinarangeforagivenapplication(Volk2005).Thecasestudysetthe

௉௨௠௣ minimumasܨ௠௜௡ =0.1.

3RZHU :

í )ORZ P V [ ܿ଴, ܿଵ, ܿଶ, ܿଷ, ܿସ, ܿହ, ܿ଺, ܿ଻, ଼ܿ, 3 2 6 3 9 4 12 5 15 6 18 7 21 8 24 W Wͼs/m Wͼs /m Wͼs /m Wͼs /m Wͼs /m Wͼs /m Wͼs /m Wͼs /m 14.918 Ͳ0.55325 Ͳ5.423 1.9018 Ͳ0.76084 Ͳ0.76548 0.74376 0.1408 Ͳ0.15351 Fig.4.9Performancecurveofcasepump ଶ ଷ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ሶ ௉௨௠௣ ܹ ൌ ൬ͳͶǤͻͳͺ െ ͲǤͷͷ͵ʹͷ ܸ௪ െ ͷǤͶʹ͵ ܸ௪ ൅ ͳǤͻͲͳͺܸ௪

ሶ ௉௨௠௣ସ ሶ ௉௨௠௣ହ ሶ ௉௨௠௣଺ െ ͲǤ͹͸ͲͺͶܸ௪ െ ͲǤ͹͸ͷͶͺܸ௪ ൅ ͲǤ͹Ͷ͵͹͸ܸ௪ (4.187)

௉௨௠௣ ଻ ଼ ܸሶ ൅ ͲǤͳͶͲͺܸሶ ௉௨௠௣ ൅ ͲǤͳͷ͵ͷͳ ܸሶ ௉௨௠௣ ൰ ௪ Ǥ ௪ ௪ ͲǤͲͲͳͲͶͳ

ሶ ௉௨௠௣ ሶ ௉௨௠௣ଵ ሶ ௉௨௠௣ଶ ሶ ௉௨௠௣ଷ Theflowrateܸ௪ isnotedbyܸ௪ ǡܸ௪ andܸ௪ forthepumpܲͳ,

ܲʹ,andܲ͵connectedtothecirculationloopܮͳ,ܮʹ,andܮ͵,respectively.Also,the

ሶ ௉௨௠௣ ሶ ௉௨௠௣ଵ ሶ ௉௨௠௣ଶ ሶ ௉௨௠௣ଷ powerinputܹ isnotedbyܹ௪ ǡܹ௪ andܹ௪ .Forthesolar

combisystem,theoverallelectricpowerinputofthecirculatingpumpsiscalculatedas:

ሶ ௉௨௠௣ ሶ ௉௨௠௣ଵ ሶ ௉௨௠௣ଶ ሶ ௉௨௠௣ଷ ܹ௦௨௣௣ ൌܹ ൅ܹ ൅ܹ Ǥ (4.188)

93 4.7.2. Flowrates

Thevolumeflowrateistheinputofthecirculatingpumpmodel,determinedby theoperationofthecirculationloopconnected.Onthecirculationloopܮͳ,theflow

rateisthatthroughthesolarthermalcollector,

ሶ ௉௨௠௣ଵ ሶ ஼௢௟௟ ܸ௪ ൌܸ௪ Ǥ (4.189)

Onthecirculationloopܮʹ,theflowrateisthatthroughtheheatpumpevaporator,

ሶ ௉௨௠௣ଶ ሶ ா௩௔௣ ܸ௪ ൌܸ௪ Ǥ (4.190)

Onthecirculationloopܮʹ,theflowrateisthatthroughtheheatpumpcondenser,

ሶ ௉௨௠௣ଷ ሶ ஼௢௡ௗ ܸ௪ ൌܸ௪ Ǥ (4.191)

4.7.3. Exergyanalysisofcirculatingpump

Fortheoperationofcirculatingpumpܲͳ,ܲʹ,andܲ͵,theexergydestruction

ሶ ௉௨௠௣ଵ ሶ ௉௨௠௣ଶ ሶ ௉௨௠௣ଷ rate,denotedbyܺௗ௘௦௧ ,ܺௗ௘௦௧ ,andܺௗ௘௦௧ ,arecalculatedasthefollowing:

ሶ ௉௨௠௣ଵ ሶ ௉௨௠௣ଵ ܺௗ௘௦௧ ൌܹ Ǥ (4.192)

ሶ ௉௨௠௣ଶ ሶ ௉௨௠௣ଶ ܺௗ௘௦௧ ൌܹ Ǥ (4.193)

ሶ ௉௨௠௣ଷ ሶ ௉௨௠௣ ܺௗ௘௦௧ ൌܹ Ǥ (4.194)

Forthesolarcombisystem,theoverallexergydestructionofcirculatingpump

ሶ ௉௨௠௣ operations,ܹௗ௘௦௧ ,iscalculatedas:

ሶ ௉௨௠௣ ሶ ௉௨௠௣ ܹௗ௘௦௧ ൌܹ௦௨௣௣ Ǥ (4.195)

94 4.8. Mathematicalmodelofsystem Themathematicalmodelofsystemisdevelopedbycouplingallthe mathematicalmodelsofthecomponentdevices.Forthewholecombisystem,the thermodynamicsystemboundarywasdrawnaroundthecombisystemphysicalparts(II inFig.4.10)toisolatethecombisystemfromtheouterandindoorenvironment,which arethethermodynamicsystemsurroundings.

Fig.4.10Thermodynamicsystemboundaryofsolarcombisystem

4.8.1. Heatingload

ሶ ௌ௬௦ Thecombisystemheatoutputrateforheating,ܳ௟௢௔ௗ,includesthatforthe

preparationofdomestichotwaterandthatforspaceheating,

ሶ ௌ௬௦ ሶ ௐ்௅ଶ ሶ ோி ܳ௟௢௔ௗ ൌܳ ൅ܳ Ǥ (4.196)

4.8.2. Energysupplyandloss

ሶ ௌ௬௦ Theenergysupplyrateofthecombisystem,ܧ௦௨௣௣,iscalculatedas:

95 ሶ ௌ௬௦ ሶ ௌ௬௦ ሶ ௌ௬௦ ܧ௦௨௣௣ ൌܧ௦௢௟ ൅ܹ௘௟௘ ǡ (4.197)

ሶ ௌ௬௦ where,ܧ௦௢௟ isthesolarenergysupplyratetothecombisystem,

ሶ ௌ௬௦ ሶ ஼௢௟௟ ܧ௦௢௟ ൌܳ௦௢௟ Ǣ (4.198)

ሶ ௌ௬௦ ܹ௘௟௘ istheelectricpowerinputratetothecombisystem,

ሶ ௌ௬௦ ሶ ஺௨௫ ሶ ு௉ ሶ ௉௨௠௣ ܹ௘௟௘ ൌܹ ൅ܹ ൅ܹ௦௨௣௣ Ǥ (4.199)

ሶ ௌ௬௦ Theenergylossrateofthecombisystem,ܧ௟௢௦௦,istheenergylossrateatthe solarthermalcollector,

ሶ ௌ௬௦ ሶ ஼௢௟௟ ܧ௟௢௦௦ ൌܧ௟௢௦௦ Ǥ (4.200)

4.8.3. Heattransferwithenvironment

ሶ ௌ௬௦ Theheattransferrateofthecombisystemwiththeenvironment,ܳ௘௫௧ ,includes

thatoftheicestoragetankandthatofthewaterstoragetank,

ሶ ௌ௬௦ ሶ ூ் ሶ ௐ் ܳ௘௫௧ ൌܳ௘௫௧ ൅ܳ௘௫௧ Ǥ (4.201)

4.8.4. Heatstorage

ሶ ௌ௬௦ Theenergychangerateofthecombisystem,ܳ௦௧௢ ,includesthatintheice

storagetankandthatinthewaterstoragetank,

ሶ ௌ௬௦ ሶ ூ் ሶ ௐ் ܳ௦௧௢ ൌܳ௦௧௢ ൅ܳ௦௧௢ Ǥ (4.202)

96 4.8.5. Systemperformance

Theperformanceofthesolarcombisystemisexaminedoverthewhole simulationperiodusingindicators:(i)theCoefficientofPerformance,(ii)thesolar energycontribution,and(iii)thesolarenergyfraction.

SystemCOP

ThecombisystemCOPiscalculatedastheoverallcombisystemheatingload,

௧ ௌ௬௦ ௧ ௌ௬௦ σ௧ୀ଴ ܳ௟௢௔ௗ,dividedbytheoverallcombisystemelectricityconsumption,σ௧ୀ଴ ܹ௘௟௘ ,as:

σ௧ ܳௌ௬௦ ܥܱܲ ൌ ௧ୀ଴ ௟௢௔ௗǤ (4.203) ௧ ௌ௬௦ σ௧ୀ଴ ܹ௘௟௘

where,

ௌ௬௦ ሶ ௌ௬௦ (௟௢௔ௗ ൌܳ௟௢௔ௗοݐǡ (4.204ܳ

ௌ௬௦ ሶ ௌ௬௦ (௘௟௘ ൌܹ௘௟௘ οݐǤ (4.205ܹ

Solarenergycontribution

Thesolarenergycontributionofthecombisystem,ܵܥா ,iscalculatedasthe

௧ ௌ௬௦ overallsolarenergysuppliedtothecombisystem,σ௧ୀ଴ ܧ௦௢௟ ,dividedbytheoverall

௧ ௌ௬௦ energysuppliedtothecombisystem,σ௧ୀ଴ ܧ௦௨௣௣,as:

σ௧ ܧௌ௬௦ ܵܥ ൌ ௧ୀ଴ ௦௢௟ ǡ (4.206) ா ௧ ௌ௬௦ σ௧ୀ଴ ܧ௦௨௣௣

where,

ௌ௬௦ ሶ ௌ௬௦ (௦௢௟ οݐǡ (4.207ܧ௦௢௟ ൌܧ

97 ௌ௬௦ ሶ ௌ௬௦ (௦௨௣௣οݐǤ (4.208ܧ௦௨௣௣ ൌܧ

Solarenergyfraction

Solarenergyfractionevaluatesthecontributionofsolardevicesofasolarsystem.

Theevaluationofthesolarenergyfractionrequiresthecomparisonwithareference systemthatdoesnotincludeanysolardevicebutmeetsthesameheatingneedsasthe solarsystemdoes.Inthisstudy,anallelectricalheatingsystemisusedasthereference system.

Thesolarenergyfractionofthecombisystem,ܵܨா,iscalculatedastheoverall

௧ ஼௢௟௟ solarenergysuppliedbythesolarthermalcollector,σ௧ୀ଴ ܳ ,totheenergyusedby

௧ ௥௘௙ௌ௬௦ thereferencesystem,σ௧ୀ଴ ܧ௦௨௣௣ ,as:

݋݈݈ܥ ݐ ܳ σݐൌͲ ܵܨ ൌ ǡ (4.209) ݏݕ݂ܵ݁ݎ ݐ ܧ ݑ݌݌ݏܧ σݐൌͲ

where,

(஼௢௟௟ ൌܳሶ ஼௢௟௟οݐǡ (4.210ܳ

௥௘௙ௌ௬௦ ሶ ௥௘௙ௌ௬௦ (௦௨௣௣ οݐǤ (4.211ܧ௦௨௣௣ ൌܧ

4.9. Mathematicalmodelofcontrolsystem Themathematicalmodelsimulatestheoperationcontrolsofthevalves, circulatingpumps,andauxiliaryheater.Thecontrolsystemsimulationisaccomplished bycustomizingthecontrolledvariables,setpoints,andorganizingthecontrollogic.

4.9.1. Controlledvariables

Thecontrolsystemsendsoutoperationinstructionsbycontrolsignals.Inthe

controlsystemmodeling,thecontrolsignalsaresimulatedusingcontrolledvariables.

Controlledvariablesarecustomizedforthecombisystemtocontroltheflowsofheat transferfluids,tooperatetheauxiliaryheater.Thedevelopedmathematicalmodelsof

combisystemdevicesareintegratedwiththecontrolledvariables,particularlyintothose modelsectionsthatarerelatedtotheheattransferfluidflowsandtheelectricpowerof theauxiliaryheater.Table4.7showsthecontrolledvariableswithrespecttothedevices beingcontrolled(seeFig.4.1).

Table4.7Controlledvariablesofdevices Devices Controlledvariables Circulatingpumpܲͳ ߠ஼௢௟௟ Circulatingpumpܲʹ ߠு௉ Circulatingpumpܲ͵ ߠு௉ǡߠௐ்ସோி ThreeͲwayvalveܸͳ,LͲtype ߠ஼௢௟௟ସௐ்ǡߠ஼௢௟௟ସூ் ThreeͲwayvalveܸʹ,TͲtype ߠு௉ସௐ்ǡߠு௉ସோிǡߠௐ்ସோி Bypassvalveܸ͵ ߠு௉ǡߠௐ்ସோி Auxiliaryheater ߠ஺௨௫ I. Thecirculationloopࡸ૚

Thesolarthermalcollectorandthetwostoragetanksarelinkedbythe circulationloopܮͳ,whichhasthecirculatingpumpܲͳandthevalveܸͳ.Whenthe circulationloopisonoperation,heatistransferredfromthesolarthermalcollectorto thestoragetanks.ThevalveܸͳissimulatedasathreeͲwayvalveofLͲtype,sothatone

99 ofthestoragetanks,eitherthewaterstoragetankortheicestoragetank,canreceive

theheattransferred.

Thecontrolledvariableࣂ࡯࢕࢒࢒isdefinedforthecontrolofcirculatingpumpܲͳ:

ߠ஼௢௟௟=1forturningonthepump,ߠ஼௢௟௟=0forturningoff.Thecontrolled variableࣂ࡯࢕࢒࢒૝ࢃࢀisdefinedforthecontrolofvalveܸͳtoconnectthesolarthermal collectorwiththewaterstoragetank,ߠ஼௢௟௟ସௐ்=1foropenedconnection,ߠ஼௢௟௟ସௐ்=0

forclosed.Thecontrolledvariableࣂ࡯࢕࢒࢒૝ࡵࢀisdefinedforthecontrolofvalveܸͳto connectthesolarthermalcollectorwiththeicestoragetank:ߠ஼௢௟௟ସூ்=1foropened

connection,ߠ஼௢௟௟ସூ்=0forclosed.

Withthecontrolledvariablesdefined,Eqn.(4.5)ismodifiedas:

ேଵ ஼௢௟௟ସௐ் ܶ௪ ǡߠ ൌͳ ܶ஼௢௟௟ ൌቊ ௧ିο௧ Ǥ (4.212) ௜௡ǡ௪ ௧ ௐ ஼௢௟௟ସூ் ܶ௪ ௧ିο௧ǡߠ ൌͳ

II. Thecirculationloopsࡸ૛andࡸ૜

Theoperationofthecirculationloopܮʹisrelatedtotheoperationofthe circulationloopܮ͵,duetobothareconnectedtotheheatpump.Thecontrolled variableࣂࡴࡼisdefinedforthecontrolofheatpump:ߠு௉=1forturningup,ߠு௉=0for

shuttingdown.Thecirculatingpumpsܲʹandܲ͵areoperatedtogetherwiththeheat pump,andthereforebotharecontrolledbythecontrolledvariableߠு௉.

Theheatpump,thewaterstoragetank,andtheradiantfloorarelinkedupby thecirculationloopܮ͵,whichhasthecirculatingpumpܲ͵,andthevalveܸʹ.Thevalve

100 ܸʹissimulatedasathreeͲwayvalveofTͲtype.Thecirculationloopܮ͵hasonebypass

fortheswitchofheatsources.Bycontrollingthebypassvalveܸ͵,eithertheheatpump orthewaterstoragetankisswitchedtobetheheatsourceofthecirculationloopܮ͵.

Theoperationisasolesourcesituation:Whenthecirculationloopܮ͵isonoperation withtheheatpumpastheheatsource,heatistransferredfromtheheatpumptoeither thewaterstoragetank,ortheradiantfloor,orboth.Whenthecirculationloopison

operationwiththewaterstoragetankastheheatsource,heatistransferredfromthe waterstoragetanktotheradiantfloor.Theoperationofcirculationloopܮ͵ synchronizedtheoperationofthecirculatingpumpܲ͵.

Thecontrolledvariablesࣂࡴࡼ૝ࢃࢀ,ࣂࡴࡼ૝ࡾࡲ,andࣂࢃࢀ૝ࡾࡲaredefinedforthe specifieddeviceconnectionsalongthecirculationloopܮ͵,eitheroneofcontrolled variablesinfluencestheoperationsofvalveܸʹandvalveܸ͵atthesametime.The controlledvariableࣂࡴࡼ૝ࢃࢀisdefinedfortheconnectionbetweentheheatpumpand thewaterstoragetank:ߠு௉ସௐ்=1foropenedconnection,ߠு௉ସௐ்=0forclosed.The controlledvariableࣂࡴࡼ૝ࡾࡲisdefinedfortheconnectionbetweentheheatpumpand radiantfloor:ߠு௉ସோி=1foropenedconnection,ߠு௉ସோி=0forclosed.Thecontrolled

variableࣂࢃࢀ૝ࡾࡲisdefinedfortheconnectionbetweenthewaterstoragetankand radiantfloor,ߠௐ்ସோி=1foropenedconnection,ߠௐ்ସோி=0forclosed.

Thecontrolledvariablesߠு௉andߠௐ்ସோிneverbothequalto1atthesametime,

becausetheheatpumpdoesnotworksimultaneouslywiththehotwaterstoragetank

101 toheattheradiantfloor.Forthecontrolledvariablesߠு௉ସோிandߠு௉ସௐ்,ifoneorboth

equalto1,thenߠு௉=1.

Withthecontrolledvariablesdefined,Eqns.(4.23)and(4.24)ismodifiedas:

ா௩௔௣ ௐ ܶܭ ൌ ʹ͹͵Ǥͳͷ ൅ ܶ௪ ǡ (4.213) ௜௡ǡ௪ ௧ ௧ିο௧

and

ܶܭ஼௢௡ௗ ௜௡ǡ௪ ௧

ʹ͹͵Ǥͳͷ ൅ ܶோி ǡߠு௉ସௐ் ൌͲܽ݊݀ߠு௉ସோி ൌͳ ௢௨௧ǡ௪ ۓ ͹͵Ǥͳͷ ൅ ܶேଵ ǡߠு௉ସௐ் ൌͳܽ݊݀ߠு௉ସோி ൌͲʹ ۖ ൌ ௪ ௧ିο௧ Ǥ ோி ோி ஼௢௡ௗ ோி ேଵ ሶ ௪ ܶ௢௨௧ǡ௪ ൅൫݉ሶ ௪ െ݉ሶ ௪ ൯ܶ௪ ௧ିο௧ ு௉ସௐ் ு௉ସோி݉ ۔ ͹͵Ǥͳͷ ൅ ஼௢௡ௗ ǡߠ ൌͳܽ݊݀ߠ ൌͳʹۖ ሶ ௪݉ ە

(4.214)

III. Theheater

Thecontrolledvariableࣂ࡭࢛࢞isdefinedforthecontroloftheauxiliaryheater:

ߠ஺௨௫=1forturningon,ߠ஺௨௫=0forturningoff.

4.9.2. Setpoints

Thesetpointistheconstantassignedtoasettingvariableforspecifingthe designed/expectedoperationstaterepresentedbythevariable.Anydepartureofthe currentoperationstatefromtheexpectedstatewouldmakethecontrolsystemtosend newcontrolinstructionforadjustment.ThecontrolsettingsarepresentedinTable4.8,

where,thesetpointsaregivenbyquantitiessofortheusertoassignvalues.

102 Table4.8Settingsofsolarcombisystemcomponentmodels Devices Settingvariables Setpoints ோி ோி Radiantfloor ܶ௜௡ǡ௪ ܶ௖௢௠ ௐ ௐ Icestoragetank ߛǡ ܶ௪ ߛ௠௔௫ǡܶ௠௔௫ ேଵ ேଷ ேହ ௐ் ௐ் ௐ் Waterstoragetank ܶ௪ ǡܶ௪ ǡܶ௪ ܶ௦௘௧ ǡܶ௩௔௥ ǡܶ௠௔௫ I. Theradiantfloor

Thesupplytemperaturetotheradiantfloorislimitedtoalowerbound.The settingis

ோி ோி ܶ௜௡ǡ௪ ൒ܶ௖௢௠ǡ (4.215)

ோி where,ܶ௖௢௠isthecomforttemperatureofradiantfloor.Inthestudy,thevaluewas

ோி ܶ௖௢௠=25Ԩ.Thesettingwasreferredtotheoptimaltemperatureforsedentaryinterms ofthermalcomfortbyOlesen(1977).

II. Theicestoragetank

Themassfractionoficeintheicestoragetankislimitedtoanupperlimit.The settingis

ߛ൑ߛ௠௔௫ǡ (4.216)

where,ߛ௠௔௫isthemaximummassfractionoficeintheicestoragetank.Inthecase study,ߛ௠௔௫=0.7±0.01,asthemaximumvalueof0.7,allowingthevariationof0.01in

thesimulation.

Thetemperatureofliquidwaterintheicestoragetankislimitedtoanupper limit.Thesettingis

103 ௐ ௐ ܶ௪ ൑ܶ௠௔௫ǡ (4.217)

ௐ where,ܶ௠௔௫isthemaximumtemperatureofthewaterͲlayerintheicestoragetank.In

ௐ thecasestudy,ܶ௠௔௫=37.8±1Ԩ,asthemaximumtemperatureof37.8Ԩ,allowingthe

variationof1Ԩinthesimulation.Thesettingcompliedwiththeheatpumpcatalogue intermsoftheoperationtemperaturesoftheheatsource(GeoSmart2012).Tolimit withamaximumwasalsoregardedinfavorofmakinguseofthermalcapacityofwater underphasechanging,sincethestoragetankisoperatedasacoldenergystore.

III. Thewaterstoragetank

Temperaturesinthewaterstoragetankaresubjecttoamaximum,denotedby

ௐ் ܶ௠௔௫.Thebasicsettingsarethewatertemperatureofthreenodesinthewaterstorage tankasfollows.

ேଵ ௐ் ௐ் ܶ௪ ൒ܶ௦௘௧ െܶ௩௔௥ ǡ (4.218)

ேଷ ௐ் ܶ௪ ൒ܶ௦௘௧ ǡ (4.219)

ௐ் ேହ ௐ் ௐ் ܶ௦௘௧ ൑ܶ௪ ൑ܶ௦௘௧ ൅ܶ௩௔௥ ǡ (4.220)

ௐ் ௐ் where,ܶ௦௘௧ isthetemperaturesetforthewaterstoragetank;ܶ௩௔௥ isthevariation

allowedwithrespecttothewaterstoragetanksettingtemperature.

ௐ் ௐ் ௐ் Inthecasestudy,ܶ௠௔௫=60Ԩ,ܶ௦௘௧ =40Ԩ,andܶ௩௔௥ =5Ԩ.Withthemaximum limit,theoperationrangeforthebottom,middle,andtopsectiontemperaturesare35Ԩ

ேଵ ேଷ ேହ чܶ௪ ч60Ԩ,40Ԩчܶ௪ ч60Ԩ,and40Ԩчܶ௪ ч45Ԩ,respectively.Forthetop waterͲnodeܰͷ,theupperboundistomeetthetemperaturerequirementofdomestic hotwateruse,thesafetyrequirementofnotexceed50Ԩfromscald,aswellas

104 avoidingtheLegionellapneumophilatobe“inthe60Ԩrange”(ASHRAE2007).Forthe

bottomwaterͲnodeܰͳ,thesettingwasreferredtothesupplytemperature requirementoftheradiantfloor.Theimpactmadebythelowtemperatureoftheheat sourceofheatpumpwasalsotakenintoaccount.ForthemiddlewaterͲnodeܰ͵,the settingwasinbetweenofthetopandthebottom,thelowerboundwassetthesameas ofthetopnode.

4.9.3. Controllogic

Thecontrollogicismodelingthedecisionmakingatthecontrolhubof mechanicalsystembasedonsettingvaluesandthedeviationsofcorresponding measurestakenbysensors,beforecontrolactuatorssettingoutcontroldirectivesto regulatethebehaviorofsystemdevices(ASHRAE2011).Table4.9sortedthecontrolled

variablesforthecombisystemcomponentdevices.Eachofthedevicesisunderthe

operationcontroldirectivesdeliveredbythelistedcontrolledvariables.

Table4.9Controlledvariablesofsolarcombisystemcomponent Devices Controlledvariables Solarthermalcollector ߠ஼௢௟௟(ߠ஼௢௟௟ସௐ்ǡߠ஼௢௟௟ସூ்) Heatpump ߠு௉(ߠு௉ସௐ்ǡߠு௉ସோி) Icestoragetank ߠ஼௢௟௟ସூ்ǡߠு௉ Waterstoragetank ߠ஼௢௟௟ସௐ்ǡߠு௉ସௐ்ǡߠௐ்ସோி Radiantfloor ߠு௉ସோிǡߠௐ்ସோி Fig.4.11illustratesthecontrollogicofthenonͲheatingseasonoperationofthe combisystem.Fig.4.12illustratesthecontrollogicoftheheatingseasonoperationof thecombisystem.Thecontrollogicdirecttheintegralresponseofthesolar

105 combisystemoperationbasedontheconditionsof:(i)therelationsofthecurrentvalues ofsettingvariablesandthesetpoints,(ii)theavailabilityofinsolation,and(iii)the

residentialheatingdemands.Theconditionsarerepresentedbydiamondconditional

boxesonthefigures.Tohavethegraphicpresentationsinconcise,acoupleofpremises

arenotdetailed:(i)ܩሶ ൐Ͳforߠ஼௢௟௟=1,alsoߠ஼௢௟௟ସௐ்=1orߠ஼௢௟௟ସூ்=1,thatis,onlywhen

insolationisavailable,thesolarthermalcollectormightstartontocollectsolarheat.(ii)

ௐ ௐ ஼௢௟௟ସூ் ܶ௪ ൏ܶ௠௔௫forߠ =1,thatis,onlywhentheicestoragetankwaterͲlayer temperaturebelowitssetpoint,theheatispossiblyallowedtotransferfromthesolar thermalcollectortotheicestoragetank.

Controldirectivesarefirmedbyassigningvaluestothecontrolledvariables, whichareenclosedintherectangularboxesandtrapezoidalboxesonthefigures.The controldirectivesaredecisionsmadefortheconditionsbasedonTrueorFalse,denoted byTandFonthefigures.Thesecontroldirectiveswouldcontroltheauxiliaryheater

(circledbythelineinovalonthefigures),manipulatetheoperationsofcirculationloop

ܮͳ,ܮʹ,andܮ͵bycontroltheoperationsofcirculatingpumpsandvalves.Control directivesenclosedinthetrapezoidalboxesareflexiblesituationswithonedecisionin higherpriorityovertheother.Flexibilityisgrantedregardingtheheattransferfromthe solarthermalcollectortooneofthetwostoragetanks,whenneitherofthestorage tanksinurgeofheatcharging.

106 Thecontrollogicisnotexclusivetoobtainthesameseriescontroldirectives.

However,anychangeisbelievedverylikelytoyielddifferentresults.Thepresented controllogicisusedinthisstudy.

ܰͳ ܹܶ ܹܶ ݐ െܾܶܽ݊݀ T݁ݏݓ ൒ܶܶ ܰ͵ ܹܶ ݐ݁ݏݓ ൒ܶܶ ܰͷ ܹܶ ܹܶ ݐ ൅ܾܶܽ݊݀݁ݏݓ ൒ܶܶ

ܦܪܹ T ܸݓሶ ൐Ͳ ܰͷ ܹܶ ݐ F݁ݏݓ ൒ܶܶ

ݑݔ ൌͲ ܰ͵ ܹܶ Tܣߠ ݐ݁ݏF ܶݓ ൒ܶ ܰͷ ܹܶ ݓ ൒ܶܶ ݐ݁ݏ ݑݔ ൌͳܣߠ F ߠܥ݋݈݈Ͷܫܶ ൌͲ

T ܥ݋݈݈ Ͷܹܶ ߛ൏ߛ݉ܽݔ ߠ ൌͳ

F ߠܪܲͶܹܶ ൌͳ

ߠܥ݋݈݈Ͷܫܶ ൌͳ ߠܥ݋݈݈ Ͷܹܶ ൌͲ

ߠܪܲͶܹܶ ൌͲ

Fig.4.11FlowchartofoperationcontrolalgorithmfornonͲheatingseason

107

T ܦܪܹ I ߛ൏ߛ݉ܽݔ T ܸݓሶ ൐Ͳ ܰͷ ܹܶ ݐ݁ݏݓ ൒ܶܶ F ݑݔ ൌͲ Fܣߠ ߠܪܲͶܴܨ ൌͲ ݑݔ ൌͳܣߠ ߠܥ݋݈݈Ͷܫܶ ൌͳ ߠܪܲͶܹܶ ൌͲ

ߠܥ݋݈݈ Ͷܹܶ ൌͲ II ܹܶܮͳ ܴܨ T T ܶ݋ݑݐ ǡݓ ൒ܶܿ݋݉ ሶ ܴܨ ܰ͵ ܹܶ ܰͳ ܹܶ ܹܶ ݐ െܾܶܽ݊݀ T݁ݏݐ ܶݓ ൒ܶ݁ݏ൐Ͳ ܶݓ ൒ܶ ݀݉݁݀ܳ ܰͷ ܹܶ ܰ͵ ܹܶ ݐ݁ݏݐ ܶݓ ൒ܶ݁ݏݓ ൒ܶܶ F ܶܰͷ ൒ܹܶܶ ൅ܹܶܶ ܾ݀݊ܽ ݐ݁ݏ ൌͳ ݓ ܨߠܹܶͶܴ

ߠܹܶͶܴܨ ൌͲ F III

ܹܶܮͳ ܴܨ ݓ ൒ܶܿ݋݉ Tܶ T T ܴܨ ܰͳ ܹܶ ܹܶ ሶ ൐Ͳ ܶݓ ൒ܶ െܶܳ ܹܶ ͵ܰ ܾ݀݊ܽ ݐ݁ݏ ݀݉݁݀ ݐ݁ݏݓ ൒ܶܶ ܰͷ ܹܶ ݐ݁ݏݓ ൒ܶܶ F IV F F ߠܥ݋݈݈ Ͷܹܶ ൌͳ prior over ߠܥ݋݈݈Ͷܹܶ ൌͲ ܥ݋݈݈Ͷܫܶ ߠ ൌͳ ߠܥ݋݈݈Ͷܫܶ ൌͳ ܥ݋݈݈Ͷܫܶ prior over ߠ ൌͳ ߠܥ݋݈݈ Ͷܹܶ ൌͳ

ߠܪܲͶܹܶ ൌͲ

ߠܪܲͶܹܶ ൌͳ

T ሶ ܴܨ ܪܲͶܴܨ ܳ݀݁݉݀ ൐Ͳ ߠ ൌͳ ܥ݋݊݀ ܴܨ ݋ݑݐ ǡݓ ൐ܶܿ݋݉ܶ

F ߠܪܲͶܴܨ ൌͳ

T ܹܶͶܴܨ ሶ ܴܨ ߠ ൌͳ ܳ݀݁݉݀ ൐Ͳ

F ߠܹܶͶܴܨ ൌͲ Fig.4.12Flowchartofoperationcontrolalgorithmforheatingseason

108 ThecontrolforthenonͲheatingseasonoperation(Fig.4.11)isrelativesimple.

Duringthen,thewaterstoragetanktakesontheheatdistributioncenterofthe

combisystem.Incontrast,thecontrolfortheheatingseasonoperation(Fig.4.12)is

relativecomplex.Toensuretheresidentialheatingdemandsbeingmet,thethermal storagecapacityoftheicestoragetankiscrucial.OnFig.4.12,thetopleftpartiscircled bydashedoutlinenotedwiththecalloutI,whichcorrespondsthesituationofice storagetanklackedofheatstoragecapacity,featuredbyߛшߛ௠௔௫.Thispartofcontrol flowsindicate:Iftheicestoragetankiscrippled,soistheheatpump,andthewater storagetankbecomesthesoledevicethatcanrespondtotheheatingdemands.Insuch situation,thecombisystemiscompletelyclimateͲdependent.Withonlywaterstorage tankandsolarthermalcollectorbefunctional,thecombisystemsuppliesheatfarfrom enoughasbeingdemanded.

Iftheicestoragetankhasthecapacitytostoreheat,thecontrollogicwouldflow tothesituationthathasthewaterstoragetankstatusbeingfocused,notedbythe calloutII,III,andIVonFig.4.12.FortheconditionincalloutII,thewaterstoragetankis sufficientwithheatinstorage,andwouldsupplyheatifneededbutnottakeinany.The conditionincalloutIIIrepresentsthebottomlineofwaterstoragetankinheat reservation,forspaceheating.Reachingthisbottomline,theheatpumpistobe employedforheatinputtothewaterstoragetank.TheconditionincalloutIV representsthatthewaterstoragetankcanbeusedtoheattheradiantfloor.Andatthis situation,thesolarthermalcollectorisemployedastheheatsourceofwaterstorage

tank.Thecontrolstrategysuggestedinhereisthat,usingthewaterstoragetankto

109 respondtheheatingdemandsofradiantflooratthehigherprioritythanusingtheheat pump.Thereasonisthatunlikethewaterstoragetank,theheatpumpoutputislackof flexibility.

OnFig.4.12,theconditionregardingheatsupplytotheradiantfloor,as formulatedbyEqn.(4.215),isdistinguishedintoconditionsoftwoforms:

ௐ்௅ଵ ோி ܶ௢௨௧ǡ௪ ൒ܶ௖௢௠ǡ (4.221)

forthewaterstoragetankonthesupply,and

஼௢௡ௗ ோி ܶ௢௨௧ǡ௪ ൒ܶ௖௢௠ǡ (4.222)

fortheheatpumpcondenseronthesupply.

4.10. Integrationofmathematicalmodelsincompletecomputermodel Thesolarcombisystemmathematicalmodelisbuiltfromintegratedcomponent models,includingthecontrolsystemmodel,thesystemmodel,andthecomponent equipmentmodels.Thecomponentequipmentmodels,includingthesolarthermal collectormodel,heatpumpmodel,icestoragetankmodel,waterstoragetankmodel, radiantfloormodel,andcirculatingpumpmodel,arecouplingonewiththeother

throughtherelationsthatrepresenttheheattransferwaterbehaviorswithinthe circulationloops.Theintegrationofcontrolsystemmodelwiththecoupledcomponent equipmentmodelsispracticedbynotonlyincludingthecontrolalgorithm,butalsoby

110 modifyingthoserelationsregardingcirculationloopswithappropriatecontrolled variablestorepresentthecontrolledheattransferthroughthecirculationloops.

Thesolarcombisystemcomputermodelisachievedbyimplementingthesolar combisystemmathematicalmodelontoacomputerprogram.Where,interfacewas createdforinputs.Thesolarcombisystemoperationcanbemodeledbyrunning

computersimulationwiththecomputermodel.ThisresearchusedtheEESprogram

(Klein2012)fortheimplementation.EESisageneralpurposeequationͲsolvingprogram handlingcouplednonͲlinearalgebraicanddifferentialequations.TheprogramhasbuiltͲ infunctionscalculatingthethermodynamicandtransportpropertiesofsubstances.

Suchfeaturesmaketheprogramsuitableforthisstudytouse.

ReadingpreviousvaluesinquasiͲstaticapproach

ThecombisystemthermodynamicmodelisdevelopedinquasiͲstaticapproach.

Thisimpliesthethermodynamicprocessofthecombisystemisregardedassteadystate

overeachofthetimeintervalοݐ.Becauseofthermalstorageprocessinvolved,asgiven byEqns.(4.87),(4.90),and(4.154),thestateofonetimeintervalisrelatedtothestate oftheprevioustimeinterval.Thisrequiresthecomputermodelshouldbeabletocarry

thepreviousstateovertothecurrenttimeforthethermalstorageprocesstobe

computed.

Twovariablesareintroducedforthepurposeofcontinuouscomputation:ܴݑ݊

asthesequencingnumberofsimulationruns,whichstartsfrom1attheinitiation;ܶ݅݉݁

111 forthecombisystemoperationtimesimulated,whichistheinputstartingfrom0sat theinitiation.Thefollowingrelationissubjected:

ܶ݅݉݁ (ݑ݊ ൌ ൅ͳǤ (4.223ܴ οݐ

,Forexample,thecasestudyhasοݐ=900s,withܶ݅݉݁=0,900,1800…input,ܴݑ݊=1,2

3…areobtained.

Certaintypesofvariables

Afewvariablesarefunctionalforthecomputermodeltocompute,presentedin

Table4.10.Thesevariablesaregroupedintothreetypesbytheirfunctions.Thetime variablesaretheinputvariablesofthecombisystemmodel,varyingateachtimestep.

Thedifferentialvariablesarethevariablesfromthedifferentialequationsofstorage

tankmodels.Thistypeofvariablesneedstheinitialvalueinput,thecasestudyused:

ூ ௐ ே௜ ݉௜௖௘=0,ܶ௪ =18Ԩ,andܶ௪ =18Ԩ.Thetracingvariablesarethevariablesbeingtraced

forthepreviousvalues.Thosedifferentialvariablesarebelongingtothistypetoo.

MultiplesoͲcalledParametricTablesarecreatedforthesevariables.Thenumber ofparametrictablesdependsonthenumberofsimulationruns.Becauseeachtable, named,canholdalimitednumberofrows,onerowcorrespondingtoonesimulation run.TheparametrictablesarejointlyusedwiththebuildͲinTableValueFUNCTION,

(ሻǡ (4.224݈ܾ݁ܽ݅ݎ݋ݓǡ ݒܽݎ ሺݐܾ݈ܽ݁ǡ ൌ ݈ܾ݁ܽ݅ݎݒܽ

݋ݓdenotesthenumberofݎ;where,ݐܾ݈ܽ݁denotesthenameoftheparametrictable

isthenameofthevariableincludedinthe݈ܾ݁ܽ݅ݎtherowintheparametrictable;ݒܽ

,݋ݓasݎparametrictable.Byassigningtheargument

112 (݋ݓ ൌ ܴݑ݊ െ ͳǡ (4.225ݎ thefunctionreadsandassignsthepreviousvalueofthegivenvariablefromthe specifiedparametrictable.

Table4.10VariablesinparametrictablesofEESͲbasedsolarcombisystemmodel Symbols Descriptions Type ܩሶ Insolationontiltedpanel

ܶ௢௔ Outdoorairtemperature ܶெ௔௜௡ Citymainwatertemperature ௪ Timevariable ஽ுௐ ܸ௪ሶ Domestichotwatervolumeflowrate ሶ ோி ܳௗ௘௠ௗ Spaceheatingdemand ܶ݅݉݁ Simulationtime ݉ூ IceͲlayermass ௜௖௘ Differentialvariable ܶௐ WaterͲlayertemperature ௪ Tracingvariable ே௜ ܶ௪ WaterͲnodetemperature ߛ Icemassfraction ௐ ݉௪ WaterͲlayermass ோி ݉ሶ ௪ Watermassflowrateofradiantfloor Tracingvariable ߠு௉ସௐ் Controlledvariableofheatpumpchargingwaterstoragetank ߠு௉ସோி Controlledvariableofheatpumpchargingradiantfloor

Modelparameters

ThesoͲcalledLookupTablesarecreatedtostorethemodelparametersinputby theuser.Onetableisusedforonecomponentequipment,plusacommononeforthe

simulationoperation(includingparameterssuchasthetimestepοݐ).ThebuiltͲin

LookupFUNCTIONisusedfortheprogramtoreadandassigntheparametervalues,

(ሻǡ (4.226ݎݐ݁݁݉ܽݎ݋ݓǡ ݌ܽݎ ሺݐܾ݈ܽ݁ǡ ൌ ݎݐ݁݁݉ܽݎ݌ܽ

݋ݓdenotesthenumberoftheݎ;where,ݐܾ݈ܽ݁denotesthenameofthelookuptable

.istheparametertobelookedupforvaluesݎݐ݁݁݉ܽݎtablerow;݌ܽ

113 Programming

Atthisstage,eachofthecomponentmodelsneedstobebrokendowntoparts foreachparttobewrittenasaprocedure.Theprocedures,alongwithsomeuserͲ writtenfunctions,becomethesubroutinesofthecombisystemcomputermodel.The subroutines,eithersavedasstandͲaloneEESlibraryfilesunderparticulardirectoryor

arrangedwithinasoͲcalledEquationWindow,aretobeusedbythemainroutineofthe combisystemcomputermodel.Themainroutineiswrittenintheequationwindow, withfunctionsanddirectivescompiledinsequence.Arrangingthesequenceis necessaryforsolvingtheunknownwiththeknowninready(appearedearlier).The programmedmodelcanpreͲcalculate,suchastheimminentcapacitiesofcomponent equipment.Thiswasusedbythecontrolalgorithmforadjustingthedeviceoperation.

Valueswerereadfromthelookuptablesandparametrictablesusingthespecific functions,thedirectivescallproceduresandexportresults.

Solving

RunningsimulationisexecutedupontheparametrictablesusingthesoͲcalled

SolveTableCOMMAND.Whilethesolvingisprocessedrowbyrowontheparametric tables,thecombisystemoperationissimulatedfromonetimesteptothenext.During eachrunofsimulation,thevariablesintheparametrictables(excepttheinputtime

variables)arewrittenwithresultsthatarejustsolvedbygoingthroughthemainroutine tocomputeforalltheunknowns.Thesolvingoperationemploysiterativecomputation technique,startingfromaguessvalueforanunknown,andterminatingtillthestop criteriabeingreached.Thecasestudyappliedthedefaultsettingofcriteriatolerances

114 ofEES:numberofiterations>100,relativeresiduals<10Ͳ6,changeinvariables<10Ͳ9,

elapsedtime>3600s.

4.11. Modelvalidationandverification Aftercompletingthemodelimplementation,modelvalidationandverification wereconductedfollowingtwoofthefourprocessespresentedbySargent(2011)(Fig.

4.13):theconceptualmodelvalidationandthecomputerizedmodelverification.The formerverifiesthesolarcombisystemmathematicalmodel’srepresentationisforthe intendedpurposeofthemodelingofthesystemoperation.Thelatterverifiesthe

programmingandimplementationintheEESprogramwascorrect.

Fig.4.13Simplifiedversionofthemodelingprocess(Sargent2011) Themathematicalandlogicalrelationshipsofthecombisystemcomponent modelsandtheoverallcombisystemmodelwereexaminedforvalidation.The

115 componentenergymodels,eitherfrompreviousstudies,currentindustrypracticeor newdevelopmentsunderwentthecheckupoftheSecondLawofthermodynamics.

WheneverthemodelsfailedtocomplywithboththeFirstLawofthermodynamicand theSecondLawofthermodynamic,themodelsweremodified.

Forcomputermodelverification,structuredwalkthroughsandcorrectness

proofswereperformed,asofthestatictestingapproach.Dynamictestingwasexecuted undersimulationtestswithvarioussimulationtimespans,forday,forseason(i.e., heatingseasonandnonͲheatingseason),andforacompletesimulationyear.Detailed outputfromthetestsimulationsweretracedforinvestigatingtheinputͲoutputrelations,

themodelinternalconsistency,etc.Energybalancewascheckedforeachcomponent equipmentmodels,aswellasforthecompletecombisystemmodel.Effortsweremade tolookforthehiddenerrors,forinstancecheckingdimensionlessnumbers,suchas thermalefficiency,whichbearsvaluesinarangebetween0and1.Thefeatureof checkingunitconsistencyintheEESprogramwasusedtoreduceerrorsforthiscomplex combisystemmodel.

116

5. Resultsanddiscussions ThecasestudyhouseisinMontreal,Canada.TheheatingseasonisfromOctober

17toApril30,followedbythenonͲheatingseasonfromMay1toOctober16.The designindoortemperatureofthehouseis18Ԩ.Thebuildinggeometryandphysical characteristicscanbefoundinLeckner&Zmeureanu(2011).Thesolarcombisystem computermodelwassimulatedtomeetthecasehouseresidentialheating requirementsduringayear.Theinputspaceheatingdemandwasfromthestudyby

Leckner&Zmeureanu(2011)withaseasonaloverallof8,323kWhandapeakof12.992 kWinJanuary.TheinputdomestichotwaterconsumptionwasfromthestudybyUlrike

Jordan&Vajen(2001)withadailyaverageof266L(Aguilaretal.2005).Theinputcold watertemperaturewasfromthestudybyHugo(2008),whocorrelatedthetemperature ofthecitymainofMontrealmeasuredbyDumas&Marcoux(2004).

Table5.1presentsthecapacityandcontrolsettingsofthesolarcombisystem

equipmentinthesimulatedcase.Thewaterstoragetank(Consolar2012),heatpump

(GeoSmart2012),andsolarthermalcollector(SunEarth2012)areproductsavailablefor

purchase;thesizeoficestoragetankwasstudiedascustomized.Theircapacitiesand thenodesettingtemperatureswerefinalizedafteranalysesperformedusingthe

developedcomputermodel.Thedecisionwasmadebasedon:tomeetthecaseheating

requirementsusingequipmentinthelowestcapacities.

117 Table5.1Capacitiesandcontrolsettingsofsolarcombisystemsimulatedcase Componentequipment Capacity Settings Minimumsupplytemperature25Ԩ(Olesen1977) RadiantfloorloopͲ Returntemperature20Ԩ(ASHRAE(2004) Nodessettingtemperature40±5Ԩ Maximumallowabletemperature60Ԩ(ASHRAE2007) Waterstoragetank 2.2m3 AuxiliaryheaterstartͲuptemperature40.5Ԩ (CaliforniaEnergyCommission2005) Maximumicefraction70%(Sunwell2012) Icestoragetank 14m3 Maximumwatertemperature37.8Ԩ(GeoSmart2012) 12.95kW Heatpump Ͳ COP3.1 Tiltangle60°(Hugo2008) Solarthermalcollectors 27.58m2 OrientationdueSouth Theinitialconditionsofthesimulationwerethetemperaturesinthestorage tanks,whichweretakenastheindoorairtemperature18Ԩ.Theimpactoftheinitial valueswaseliminatedbysimulatingthecombisystemoperationforasimulationtimeof threecontinuousyears.Resultsofthethirdyearwereusedfortheanalysis.

5.1. Energyperformanceofsolarcombisystem

5.1.1. Overviewofannualenergyflow

Thesolarcombisystemusedsolarheatandelectricitytomeettheresidential heatingneeds.Theannualenergyflowsamongthesystemcomponentsareplottedin

Fig.5.1.Therectangularboxesbycontinuouslinerepresentthesystemcomponents.

Therectangularboxesbydottedlinerepresentthesupplyofenergy(atthetop)andthe heatingloads(atthebottom).Theboxesinirregularshaperepresenttheoutdoor environmentandtheindoorsurroundings.Thearrowsconnectingtheboxesshowthe directionoftheflowpaths.

118

Fig.5.1Flowchartofenergyflowsofsolarcombisystemannualoperation SystemCOP:3.98.Solarenergycontribution:89.5%.Solarenergyfraction(reference):1.3. TheannualenergyusewasquantifiedinkWh.Theamountofchangeofenergy wasindicatedforthestoragetanks,positiveforenergystoredandnegativeforenergy extracted.Similarconventionwasusedfortheindoorsurroundings.Theoperation temperatureswerelabeled.Forinstance,inthecaseofsolarcollectors,ܶത௢௨௧Τܶത௜௡isthe

119 meanoutlettemperaturefollowedbythemeaninlettemperature.Inthecaseof outdoorenvironment,ܶ௠௔௫Τܶ௠௜௡istheupperlimitandthelowerlimitofoutdoor temperature.Somecaseswerelabelledwithonetemperature:meantemperatureof theicestoragetankwaterͲlayer,meantemperatureofthewaterstoragetanknodes,or supplytemperaturealongtheflowpath.

5.1.2. Combisystemenergyperformance

CombisystemCOP

ThesystemCOPwascalculatedastheratioofthesystemheatingloadtothe systemelectricityconsumption.Annualoperationofthecombisystemused2,956kWh ofelectricitymeetingtheheatingneedsof11,763kWh.Thecombisystemhadanannual systemCOPof3.98.

ݐ݁݉ ݄݁ܽݐ݅݊݃ ݈݋ܽ݀ݏݕݏ ൌ ܱܲܥݐ݁݉ݏݕܵ ݑ݉݌ݐ݅݋݊ݏݐݕ ܿ݋݊݅ܿ݅ݎݐ݁݉ ݈݁݁ܿݐݏݕݏ (5.1) ͳͳǡ͹͸͵ ൌ ൌ͵Ǥͻͺ ʹǡͻͷ͸

Solarenergyfractioncomparedtoreferencesystem

AcomparednonͲsolarsystemissupposedtosatisfythesameheatingneedsas

ofasolarsystem(Hastings&Wall2007).Thereferencesystemcomparedwasconsisted ofelectricwaterheaterandelectricbaseboardheater(Table5.2).

120 Bycomparingthetwosystems,thesolarenergyfractionofasolarsystemis calculatedasaratioofthesolarenergycollectionofasolarsystemtotheenergyused bythereferencesystem,

ݐ݁݉ݏݕݏݎ݋݈ܽݏ ݕ ܿ݋݈݈݁ܿݐ݅݋݊ ݋݂݃ݎ݁݊݁ ݎ݋݈ܽݏ ሻ ൌ ܨݐ݅݋݊ሺܵܿܽݎݕ݂݃ݎ݁݊݁ݎ݋݈ܽܵ ݐ݁݉ݏݕݏ ݁ܿ݊݁ݎ݂݁݁ݎ ݕܾ ݀݁ݏݕ ݑ݃ݎ݁݊݁ ா (5.2) ͳͶǡͳʹͳ ൌ ൌͳǤ͵ ͳͲǡͺͷͲ

Thesolarcombisystemhadanannualsolarenergyfractionof1.3.Aprilhadthe solarenergyfractionpeakof5.39,andtheminimumappearedinDecemberasof0.52.

Table5.2Referencesystemenergyusevs.solarenergycollectionofsolarcombisystem Reference system Solarcombisystem Electricity Electricity consumption consumption ofbaseboard ofwater Totalenergy Collectedsolar Solarenergy heater, heater, use, radiation, fraction, kWh kWh kWh kWh dimensionless January 2,331.04 277.51 2,608.55 1,707.84 0.65 February 1,721.11 262.53 1,983.64 2,222.92 1.12 March 949.38 306.54 1,255.92 2,807.75 2.24 April 182.27 283.44 465.72 2,511.75 5.39 May 0.00 213.88 213.88 433.03 2.02 June 0.00 156.79 156.79 342.73 2.19 July 0.00 117.67 117.67 298.52 2.54 August 0.00 100.80 100.80 274.45 2.72 September 0.00 145.52 145.52 350.41 2.41 October 171.68 171.73 343.41 1,018.88 2.97 November 976.15 218.15 1,194.30 974.43 0.82 December 1,991.79 271.83 2,263.62 1,178.00 0.52 Year 8,323 2,526 10,850 14,121 1.30

Solarenergycontribution

Among28,023kWhofenergy,asumoftheannualincidentsolarenergy(25,067 kWh)andtheannualelectricityconsumption,theannualincidentsolarenergy contributionwas89.5%.

121 ݕ݃ݎ݁݊݁ ݎ݋݈ܽݏ ݐ݊݁݀݅ܿ݊݅ ሻ ൌ ൈ ͳͲͲ ܥݑݐ݋݊ ሺܾܵ݅ݎݕܿ݋݊ݐ݃ݎ݁݊݁ݎ݋݈ܽݏݐ݊݁݀݅ܿ݊ܫ ݕ ݅݊݌ݑݐ݃ݎ݁݊݁ ݐ݁݉ݏݕݏ ா (5.3) ʹͷǡͲ͸͹ ൌ ൈ ͳͲͲΨ ൌ ͺͻǤͷΨ ʹͺǡͲʹ͵

Among17,077kWhofenergy,asumoftheannualsolarenergycollection

(14,121kWh)andtheannualelectricityconsumption,theannualcollectedsolarenergy contributionwas82.7%.

ݕ ܿ݋݈݈݁ܿݐ݅݋݊݃ݎ݁݊݁ ݎ݋݈ܽݏ ሻ ൌ ൈ ͳͲͲ ܥݑݐ݋݊ ሺܾܵ݅ݎݕܿ݋݊ݐ݃ݎ݁݊݁ݎ݋݈ܽݏ݋݈݈݁ܿݐ݁݀ܥ ݕ ݅݊݌ݑݐ݃ݎ݁݊݁ ݐ݁݉ݏݕݏ ா (5.4) ͳͶǡͳʹͳ ൌ ൈ ͳͲͲΨ ൌ ͺʹǤ͹Ψ ͳ͹ǡͲ͹͹

Energyefficiency

Thesystemoperationused28,023kWhofenergytomeettheheatingneedsof

11,763kWhoveroneyear.Theannualsystemenergyefficiencyof42.0%wascalculated astheratiooftheannualheatingloadtotheannualenergyinput:

ݐ݁݉ ݄݁ܽݐ݅݊݃ ݈݋ܽ݀ݏݕݏ ݕ݂݂݁݅ܿ݅݁݊ܿݕ ൌ ൈ ͳͲͲ݃ݎݐ݁݉݁݊݁ݏݕܵ ݕ ݅݊݌ݑݐ݃ݎ݁݊݁ ݐ݁݉ݏݕݏ (5.5) ͳͳǡ͹͸͵ ൌ ൈ ͳͲͲΨ ൌ ͶʹǤͲΨ ʹͺǡͲʹ͵

Ifcountingtheenergyusesincetheheatinputbythesolarthermalcollector,to supplytheheatingloadledtheenergyefficiencyof68.9%(among17,077kWh).

122 5.1.3. ElectricityconsumptionandcombisystemCOP

Table5.3presentstheelectricityconsumptionofheatpumpandcirculating pumps,alongwiththesystemCOPcalculatedastheaverageofthesystemCOPsover thecorrespondingtimeperiod.

Table5.3ResultsofelectricityconsumptionandsystemCOPofsolarcombisystemannualoperation Heatpump, Circulatingpumps,kWh SystemCOP, kWh P1 P2 P3 Overall dimensionless January 854.89 1.58 1.54 1.73 4.85 3.07 February 564.02 1.84 1.03 1.18 4.04 3.55 March 264.50 2.21 0.48 0.59 3.29 4.84 April 99.43 2.36 0.19 0.23 2.78 4.94 MayͲ0.78 ͲͲ0.78 449.77 JuneͲ0.67 ͲͲ0.67 420.69 JulyͲ0.47 ͲͲ0.47 491.50 AugustͲ0.55 ͲͲ0.55 380.45 SeptemberͲ0.51 ͲͲ0.51 553.23 October 64.64 1.16 0.12 0.17 1.44 6.54 November 367.64 1.27 0.65 0.83 2.75 3.31 December 714.98 1.30 1.29 1.49 4.08 3.19 Year 2,930 15 5 6 26 3.98 Theheatpumpused2930kWhofelectricityintheyear,withthehighest

consumptioninJanuaryfollowedbyDecember.Thecirculationpumpsܲʹ&ܲ͵were

operatedduringJanuarytoAprilandOctobertoDecemberintheheatingseason.The

circulationpumpܲͳhasbeenoperatedthroughouttheyear.Theauxiliaryheater,not

shown,wasneverusedwithoutanyconsumption.Intheheatingseason,themonthly averagedsystemCOPwasover3fromJanuarytoAprilandfromOctobertoDecember.

FromMaytoDecembermonthlyaveragedsystemCOPrepresentedlargevalues,dueto smallamountofelectricitywasconsumedonoperatingthecirculatingpumpܲͳ.The

monthlyaveragedsystemCOPinOctoberwaslargerthanintherestheatingseason

months,becausehalfofthemonthwasinnonͲheatingseason.

123

5.1.4. Majorflowsofenergy

TheenergyflowsofsolarcombisystemannualoperationshowninFig.5.1are detailedbymonthinthefollowedTable5.4.

Table5.4Resultsofenergyflowsofsolarcombisystemannualoperation

Energyinput,kWh Heat transferred Heatlostto Incident Heating withindoor outdoor Heat solar load, surroundings, environment, storage, heat Electricity Overall kWh kWh kWh kWh January 2,823.18 859.75 3,682.93 2,640.14 66.91 1,115.34Ͳ10.49 February 3,680.97 568.06 4,249.02 2,014.13 25.57 1,458.05 798.38 March 4,714.15 267.79 4,981.94 1,295.93 Ͳ51.51 1,906.41 1,724.80 April 4,015.50 102.22 4,117.72 504.91 Ͳ59.81 1,503.75 2,046.47 May 1,200.62 0.78 1,201.40 351.81 Ͳ65.85 767.59 15.36 June 948.72 0.67 949.39 279.79 Ͳ64.76 605.99Ͳ1.82 July 725.36 0.47 725.83 233.21 Ͳ68.20 426.83Ͳ2.88 August 758.99 0.55 759.53 208.46 Ͳ68.79 484.53Ͳ2.80 September 875.64 0.51 876.16 284.46 Ͳ62.49 525.23 3.46 October 1,783.80 66.08 1,849.88 431.87 Ͳ53.31 764.91 598.34 November 1,530.94 370.39 1,901.33 1,224.29 17.47 556.51 135.25 December 2,008.93 719.06 2,728.00 2,294.88 41.29 830.94Ͳ360.62 Year 25,067 2,956 28,023 11,764 Ͳ343 10,946 4,943

Heatingloadandenergyuse

Thecombisystemsupplied11,764kWhofheatingloadforthespaceheating

(8,323kWh)andthepreparationofdomestichotwater(3,440kWh)duringanannual operation.Fig.5.2presentsthemonthlyprofileofspaceheatingloadandwaterheating load.Thesimulatedspaceheatingloadmatchedtheinputheatingdemandappeared fromOctobertoApril,correspondingtheheatingseasonstartedfromOctober17to

April30.Thewaterheatingloadcontinuedthroughoutthe12months.

124 VRODUKHDW HOHFWULFLW\ ZDWHUKHDWLQJ VSDFHKHDWLQJ

(QHUJ\N:K

-DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.2Energysupplyandheatingloadofsolarcombisystem Thetotal28,023kWhofenergyinputtothesystemincluded2,956kWhof

electricityand25,067kWhofincidentsolarheatduringtheoperationperiods.The

profileofmonthlyenergyusefromthetwosourcesisalsopresentedinFig.5.2.

Heattransferbetweencombisystemandindoorenvironment

Thecombisystemlost343kWhofheattotheindoorenvironmentthroughoutan entireyear.Thelosswastheresultsoftransferofheatbetweenthewaterstoragetank andthesurroundings,andbetweentheicestoragetankandthesurroundings.Fig.5.3 showsthemonthlyprofileoftheheattransfer,wherethepositivesrepresenttheheat

transferredfromtheindoorenvironmenttothestoragetanks,thenegativesrepresent theheattransferredfromthestoragetankstotheindoorenvironment.

125

LFHVWRUDJHWDQN ZDWHUVWRUDJHWDQN RYHUDOO

+HDWN:K

í -DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.3Heattransferbetweenstoragetanksandindoorsurroundings (positivevaluesrepresenttanksinheatgain;negativevaluesrepresenttanksinheatloss)

Heatlosstooutdoorenvironment

Thecombisystemlost10,946kWhofheattotheoutdoorenvironmentduring theannualoperation.Thelosswasaccountedfortheamountofsolarheatfailedtobe absorbedbythesolarthermalcollectors.

Heatstorage

Thecombisystemusingthewaterstoragetankandtheicestoragetankstored

4,943kWhofheatthroughoutanentireyear.Fig.5.4presentsthestorageofheatby month,wherethepositivesrepresenttheheatstoredinthestoragetanks,the negativesrepresenttheheatreleasedfromthestoragetanks.Themonthlyprofile

showstheicestoragetankinactiveofstoringandreleasingheatfromJanuarytoMay andfromSeptembertillDecemberduringtheheatingseason.Themostamountofheat storedappearsattheicestoragetankinApril,themostamountofheatreleasedatthe

126 icestoragetankinDecember.Theresultsshowedsmallermagnitudeofstorageofheat

tookplaceinthewaterstoragetank,incontrasttotheicestoragetank.

LFHVWRUDJHWDQN ZDWHUVWRUDJHWDQN

+HDWN:K

í -DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.4Heatstorageofsolarcombisystem (positivevaluesrepresentheatstored;negativevaluesrepresentheatreleased)

5.2. CombisystemoperationinnonͲheatingseason ThecombisystemhadnonͲheatingseasonoperationof169days,fromMay1to

October16,producingthedomestichotwater.ThenonͲheatingseasonoperationis discussedwiththehourlyresultsofthreecontinuouscalendardaysfromJuly20toJuly

22.

Heatingloadanduseofheat

Fig.5.5presentstheheatingloadofthesolarcombisystemfordomestichot waterproduction.Thewaterstoragetank,wheretopreparethedomestichotwater,

127 hadtwoheatsources:theheatpumpandthesolarthermalcollectors.Buttheresults showedthattheidlestatusofheatpumpduringthenonͲheatingseason.Thisindicated theuseofsolarthermalcollectorsinnonͲheatingseasonwassufficient.Theheatcharge fromthesolarthermalcollectorstothewaterstoragetankispresentedinFig.5.6.

+HDWN:K

-XO\-XO\-XO\ Fig.5.5HeatingloadofsolarcombisystemfromJuly20toJuly22

+HDWN:K

-XO\-XO\-XO\ Fig.5.6SolarheatcollectionchargingwaterstoragetankfromJuly20toJuly22

128 Fig.5.7presentsthesolarradiationandtheheatflowsatthesolarthermal collectors.Theavailablesolarincidentisreferredtotheincidentsolarenergyonthe surfacesofsolarthermalcollectorsregardlessthecollectoroperation.Theutilizedsolar incidentisreferredtotheincidentsolarenergyonthesurfacesofsolarthermal collectorsduringtheonoperationofsolarthermalcollectors.Whenthesolarthermal collectorswereon,solarheatwascollectedchargingtothestoragetanks.Theresults showtheheatchargingonlytookplacefromthesolarthermalcollectorstothewater storagetank,nothingfortheicestoragetank.

SRWHQWLDOVRODUUDGLDWLRQ LQFLGHQWVRODUUDGLDWLRQ FKDUJLQJLFHVWRUDJHWDQN FKDUJLQJZDWHUVWRUDJHWDQN

+HDWN:K

-XO\-XO\-XO\ Fig.5.7SolarheatcollectionandstoragefromJuly20toJuly22 ThesolarthermalcollectorshavebeenoperatedwithintwohoursonJuly20and within3hoursonJuly22,nooperationonJuly21.Thissuggestedthatthedomestichot waterproductiononJuly21usedtheheatstoredinthewaterstoragetank.Afterthe releaseofheatfromstorage,thesolarthermalcollectorscollectedheatonJuly22for

129 thewaterstoragetank,morethanonJuly20.Thisshowedhowthesolarthermal collectoroperationservingtheshortperiodheatstorageoperationofwaterstorage tank.

Heatstorage

Fig.5.8presentsthechangeofheatinthewaterstoragetank.Thepositive valuesindicatethewaterstoragetankwasundertheheatcharging,thenegativevalues indicatethewaterstoragetankwasundertheheatdischarging.

+HDWN:K

-XO\-XO\-XO\ Fig.5.8HeatstorageofwaterstoragetankfromJuly20toJuly22

Storagetemperature

Fig.5.9presentsthetemperaturevariationoffivewaterͲnodes(ܰͳtoܰͷ)inthe

waterstoragetank.AfterthedomestichotwaterproductiononJuly21,the

temperatureofnodeܰͳdroppeddownto35.5Ԩ,nearthesetpointof35Ԩ.Thisled thesolarthermalcollectorstostartoperationonthenextdayJuly22assoonassolar

heatwasavailable.Thecombisystemcontrolsystemallowedthechargeofheattothe

130 waterstoragetankwhenthetemperatureinwaterstoragetankisbelow60Ԩ.Forthis

reason,asthetemperaturereached62ԨatthenodeܰͶand61Ԩatthenodeܰͷ,the solarthermalcollectorsstoppedtheonoperation(Fig.5.7)thoughsolarheatwasstill

available.Thetemperatureprofileindicatedthermalstratificationinthewaterstorage

tank.

& °

7HPSHUDWXUH QRGH1 QRGH1 QRGH1 QRGH1 QRGH1 -XO\-XO\-XO\ Fig.5.9NodetemperatureofwaterstoragetankfromJuly20toJuly22

Electricityuse

Fig.5.10presentstheelectricityusedbythecombisystemoperation.The electricityconsumptionwassolelyduetotheoperationofpumpܲͳtocirculatewater alongtheloopܮͳforthetransferofheatfromthesolarthermalcollectorstothewater storagetank.

131

(OHFWULFLW\:K

-XO\-XO\-XO\ Fig.5.10ElectricityconsumptionofsolarcombisystemfromJuly20toJuly22

5.3. Combisystemoperationinheatingseason Thecombisystemhadtheheatingseasonoperationof196daysfromOctober17

toApril30.Thecombisystemcoveredtheheatingneedsforthedomestichotwaterand thespaceheating.Theoperationatthepeakspaceheatingistobeoverviewedinthis section,followedbytheheatingseasonoperatingconditiondiscussedwiththehourly

resultsofthreecontinuouscalendardaysfromJanuary11toJanuary13.

Peakspaceheating

Theresidentialspaceheatingdemandreachedthepeakof12,992Wat7:15a.m. onJanuary12,andtherewasalsodomestichotwateruseatthismoment(Fig.5.11).

132

Fig.5.11Flowchartofenergyflowsofsolarcombisystematspaceheatingpeak Aboveflowchartpresentstheoperationtemperaturesalongwiththeenergy flowratesinW.Atthispeaktime,theoutdoortemperaturewasͲ26.4Ԩ,citywater temperaturewas3.5Ԩ,andtheindoortemperaturewas18Ԩ.Thecombisystemused

133 electricity(0.56W)torunacirculationpumpforthespaceheating,whichwassupplied bytheradiantfloorusingtheheatfromwaterstoragetank(12,992W).Thewater storagetankalsocoveredtheheatmeetingthedomestichotwaterneeds(609W).The

supplytemperaturewas47.7Ԩfortheradiantfloorand44.4Ԩforthedomestichot

water.Thetotalheattransferratefromthewaterstoragetankfortheheatingwas

13,601W.

Terrell(1979)hasreportedthepossibilityofasolarheatingsystemsatisfyingthe heatingdemandduringthecoldestmonthswithoutdirectlyusingsolarenergy.Instead,

thesolarenergystoredfromtheprevioustimewasreleasedtomeettheheatingneeds, andthestoretemperaturewasreducedbysuchoperation.Thereportedresultswere confirmedbytheresultsshowninthissimulatedcase:Thecombisystemsatisfiedthe heatingdemandattheresidentialpeakspaceheatingtimeusingtheheatstoredfrom theprevioustime.

Thesurroundingsofthecombisystemunderwenttheheattransferwiththe storagetanks.Thewaterstoragetanklostheat(71W)tothesurroundings,whereasthe icestoragetankgainedheat(159W)fromthesurroundings.Thewaterstoragetank

operationatthispeakdecreasedtheheatstoredattherateof13,672W.Atthispeak,

thesolarthermalcollectors,heatpump,andtheauxiliaryheaterwereidle.

Heatingload

Fig.5.12presentsthecombisystemheatingloadforthespaceheatingandthe domestichotwater.Thespaceheatingwassuppliedbytheradiantfloorusingtheheat

134 fromheatpumpandwaterstoragetank.Thedomestichotwaterwaspreparedinthe waterstoragetank.Thespaceheatingloadexhibitedsimilarpatternsofvariation throughthehoursofeachday.Theheatprovidedbywaterstoragetankforthespace heatingwasnolessthantheheatprovidedbyheatpump,whenthespaceheatingload wererelativelow.Whenthespaceheatingloadwashigh,theheatpumpprovidedmost amountofheatforthespaceheating.Thisindicatedtheoperationdifferencebetween theheatpumpandthewaterstoragetank,intermsofmatchingthespaceheating demandrespectingitsowncapability.

ZDWHUKHDWLQJ VSDFHKHDWLQJXVLQJZDWHUVWRUDJHWDQN VSDFHKHDWLQJXVLQJKHDWSXPS

+HDWN:K

-DQ-DQ-DQ Fig.5.12HeatingloadofsolarcombisystemfromJan.11toJan.13

Heatpumpoutput

Fig.5.13presentstheuseoftheheatoutputfromheatpump.Theheatoutput wastransferredtothewaterstoragetankandtheradiantfloor.Withinthehourswith higheroutput,theheatpumpprovidedtheradiantfloormoreheatthandoesthewater

135 storagetank.Onthecontrary,thewaterstoragetankreceivedheatfromtheheatpump morethantheradiantfloordidwithinthehoursthattheheatpumpmadeoutputs.

ZDWHUVWRUDJHWDQN UDGLDQWIORRU

+HDWN:K

-DQ-DQ-DQ Fig.5.13HeatoutputofheatpumpfromJan.11toJan.13

Solarheatcollection

Solarheatwasthesolesourceprovidingthecombisystemwithheat.Fig.5.14 presentsthesolarradiationandtheheatflowsatthesolarthermalcollectors.

Solarheatwasavailablearoundthemiddlehouseeachday,andthesolarthermal collectorswereonoperationduringthen.Thecollectedheatwasinputtotheice

storagetankandthewaterstoragetankforstorage.Theresultsshowthattheice storagetankreceivedmoreheatthanthewaterstoragetankduringthesedays.The

seasonalpeakspaceheatingtookplaceonJanuary12.Theresultsshowthatthesolar thermalcollectorshadonoperationduringmostofthetimewhensolarheatwas available.

136

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+HDWN:K

-DQ-DQ-DQ Fig.5.14SolarincidenceandsolarheatcollectionfromJan.11toJan.13

Heatstorage

I. Waterstoragetank

Fig.5.15presentstheheatchargingtothewaterstoragetank,bytheheatpump andthesolarthermalcollectors.Thewaterstoragetankstartedtotakeinheatatevery earlymorningfromtheheatpump.Aroundthemiddlehoursofeachday,therewas heattransferredfromthesolarthermalcollectors.

Fig.5.16presentsthechangeofheatinthewaterstoragetank,exhibitedthe shorttermheatstorage.Thepositivevalueswerecorrespondedtotheheatgains,the negativevalueswerecorrespondedtotheheatlosses.Thegreatestmagnitudeofheat gainwasonJanuary12,whichoccurredhoursafterthepeakspaceheatingload.

137

KHDWSXPS VRODUFROOHFWRU

+HDWN:K

-DQ-DQ-DQ Fig.5.15HeatinputtowaterstoragetankfromJan.11toJan.13

+HDWN:K

-DQ-DQ-DQ Fig.5.16HeatstorageofwaterstoragetankfromJan.11toJan.13 II. Icestoragetank

Theicestoragetankisthethermalenergystorageoperatedbetweenthesolar thermalcollectorsandtheheatpump.Fig.5.17presentstheinputandoutputofheatof

138 icestoragetank.Theresultsshowedtheicestoragetankoutputheatmorefrequently thaninput.

KHDWLQSXW KHDWRXWSXW

+HDWN:K

-DQ-DQ-DQ Fig.5.17HeatinputandheatoutputoficestoragetankfromJan.11toJan.13 Fig.5.18presentsthechangeofheatintheicestoragetank.Comparedtothe waterstoragetank,theicestoragetankhadlargermagnitudeofchangeofduringthe day.Thissuggestedtheicestoragetankwasoperatedwithlargerstoragecapacitythan thewaterstoragetank.

139

+HDWN:K

í -DQ-DQ-DQ Fig.5.18HeatstorageoficestoragetankfromJan.11toJan.13

Storagetemperature

I. Waterstoragetank

Fig.5.19presentsthetemperatureofnodes(ܰͳ ǥ ܰͷ)inthewaterstoragetank.

Mostofthetemperaturesfellinbetween42.9Ԩand49.3Ԩ.Thelargesttemperature dropoccurredinJanuary12downto36.7Ԩ.Thehourlyresultsshowedthenodeܰͷ hadthesteadiesttemperaturesamongallthenodes.Thenodeܰͳhadthemost

dynamicchangeoftemperature,whichwasduetothecharginganddischarging

operationoftheimmersionheatexchangerܪܺͳ.Thetemperaturevariationofeach

nodeduringthisperiodwas:nodeܰͷin2Ԩ,nodeܰͶin3Ԩ,nodeܰ͵in4.2Ԩ,node

ܰʹin4.4Ԩ,andnodeܰͳin12.6Ԩ.Thermalstratificationexistedlocallyatsometime, forinstance,amongthenodesܰʹ,ܰ͵,ܰͶ,andܰͷfrom18:00onJanuary12to6:00on

January13.

140

QRGH1 QRGH1 QRGH1 QRGH1 QRGH1 & °

7HPSHUDWXUH

-DQ-DQ-DQ Fig.5.19NodetemperatureofwaterstoragetankfromJan.11toJan.13 II. Icestoragetank

Fig.5.20presentsthetemperatureofthewaterͲlayeroficestoragetank.The watertemperatureremainedalmostconstantat0.1ԨǤRelatedtothewaterͲlay temperature,themassfractionoficeintheicestoragetankispresentedonFig.5.21.

Themassfractionoficewas0.45atthebeginningofJanuary11,reached0.53atthe endofJanuary13.Theresultsshowedthecoexistenceoftwophasesofwaterintheice storagetankduringthisperiod.Theicestoragetankhadthemaximumicemassfraction of0.7inthissimulatedcase.AttheendofJanuary13,theicestoragecapacityofice storagetankwasusedupto75.7%.

141

7HPSHUDWXUH

-DQ-DQ-DQ Fig.5.20WaterͲlayertemperatureoficestoragetankfromJan.11toJan.13

0DVVIUDFWLRQGLPHQVLRQOHVV

-DQ-DQ-DQ Fig.5.21IcemassfractionoficestoragetankfromJan.11toJan.13

Electricityuse

Fig.5.22presentsthetotalelectricityconsumptionofthecombisystemduring thethreedayinheatingseason.Theelectricitywasusedtooperatingtheheatpump,

142 andthecirculatingpumpsܲͳ,ܲʹ,andܲ͵.Theheatpumpused121kWhofelectricity, thelargestamountintheoverall.Thecirculatingpumpsܲͳ,ܲʹ,andܲ͵usedelectricity of0.15kWh,0.218kWh,and0.237kWh,respectively.

(OHFWULFLW\N:K

-DQ-DQ-DQ Fig.5.22ElectricityconsumptionofsolarcombisystemfromJan.11toJan.13

5.4. Responseforspaceheating Asspaceheatingdemandfortheindoorspace,theradiantfloorwasprovided heatfromtwoheatsources,theheatpumpandthewaterstoragetank.Thissection presentsthemanagementofthetwoheatsourcestorespondtheradiantfloorand meetthespaceheatingdemand,whichistheoperationofcirculationloopܮ͵.

Table5.5presentsthespaceheatingdemandandtheheatsupplytoradiant

floor.Theresultsshowedthespaceheatingdemandsweresatisfiedcompletely,among

theheatpumpsupplied30.4%andthewaterstoragetanksupplied69.6%.Theresults

143 wereduetotheuseofwaterstoragetankwascontrolledtorespondtheradiantfloor needsforheatatthehigherpriority(4.9.3).Theadvantageofthestrategywasthe spaceheatingdemandwasmatchedbyoperatingthecirculatingpumpܲ͵totransfer thedemandedamountofheattotheradiantfloor.Unlikely,theheatpump(with

constantwaterflowsattheevaporatorandcondenser)lackedofflexibility,intermsof heatoutputamount.Notusingthisstrategy,theoperationwouldtendtooverheatthe space,oroverheatthewaterstoragetankandfurthercausethecombisystemshut

down.

Table5.5Resultsofspaceheatingdemandandheatsupplytoradiantfloor Spaceheating Heatsupply,kWh demand, Byheat Fromwater kWh pump storagetank Overall October 171.68 23.05 148.63 171.68 November 976.15 220.76 755.39 976.15 December 1,991.79 653.39 1,338.39 1,991.79 January 2,331.04 875.18 1,455.86 2,331.04 February 1,721.11 539.30 1,181.82 1,721.11 March 949.38 187.15 762.24 949.38 April 182.27 29.18 153.09 182.27 Season 8,323 2,528 5,795 8,323 Ͳ30.4% 69.6% 100%

5.5. Operationofheatpumpwithicestoragetank

Heatpumpperformance

Table5.6presentstheheatpumpoperationresults,includingtheenergyflows andtheoperationCOPovertheheatingseason.Amongtheheatpumpoutputof9326 kWh,thewaterstoragetankreceived73%(6,799kWh),morethantheradiantfloor.

TheseasonalaverageCOPwas3.19.

144 Table5.6Monthlyresultsofheatpumpoperation Energyinput,kWh Heatoutputfromcondenser,kWh Heatat Toradiant Towater COP, Electricity evaporator floor storagetank Overall dimensionless October 64.64 187.65 23.05 229.24 252.29 3.93 November 367.64 801.98 220.76 948.86 1,169.62 3.19 December 714.98 1,455.79 653.39 1,517.37 2,170.77 3.04 January 854.89 1,611.14 875.18 1,590.85 2,466.04 2.89 February 564.02 1,201.15 539.30 1,225.87 1,765.17 3.13 March 264.50 804.02 187.15 881.38 1,068.52 4.06 April 99.43 334.77 29.18 405.02 434.20 4.41 Year 2,930 6,397 2,528 6,799 9,327 3.19 Fig.5.23presentsthedailymeanCOPofheatpumpthroughoutanentire heatingseason.Thedailymeantemperatureofwaterenteringtheevaporatoris presentedinFig.5.24,inFig.5.25presentsthedailymeantemperatureofwater enteringthecondenser.Resultsshownwiththethreefiguressuggestedtheimpact madeontheheatpumpCOPfromtheinlettemperatureattheevaporatorside.

&23GLPHQVLRQOHVV

1RY 'HF -DQ )HE 0DU $SU 0D\ Fig.5.23HeatpumpCOPinheatingseason,dailymean

145

& °

7HPSHUDWXUH

1RY 'HF -DQ )HE 0DU $SU 0D\ Fig.5.24Temperatureofwaterenteringheatpumpevaporator,dailymean

& °

7HPSHUDWXUH

1RY 'HF -DQ )HE 0DU $SU 0D\ Fig.5.25Temperatureofwaterenteringheatpumpcondenser,dailymean

Stateofwaterinicestoragetank

Fig.5.26presentsthemassfractionoficeintheicestoragetank.Theice appearedinNovember,presentingastableexistencesinceearlyDecember.Throughout onemonthofaccumulation,themassfractionoficeclimbeduptothepeakof0.69at

146 thelateJanuary.TheicemeltedawaybymidͲFebruary.Afterthenandtilltheendofthe

heatingseason,theicestoragetankcontainedonlyliquidwater.

,FHPDVVIUDFWLRQGLPHQVLRQOHVV

1RY 'HF -DQ )HE 0DU $SU 0D\ Fig.5.26Icemassfractionoficestoragetankinheatingseason,dailymean

5.6. Operationofsolarcollectorwiththermalstorage Solarheatwasanexclusiveheatsourcetothecombisystem.Table5.7presents

theenergyflowsregardingtheoperationofsolarthermalcollectors,aswellasthe

thermalefficiencyofsolarthermalcollectors.Duringthetimewhenthesolarthermal

collectorswereonoperation,theannualincidentsolarenergywas25,067kWh,the

annualsolarheatcollectionwas14,121kWh,while10,946kWhofsolarheatfailedto becollectedandlosttotheoutdoorenvironment.Theoperationofsolarthermal collectorshadanannualaveragethermalefficiencyof0.51.

147 Table5.7Monthlyresultsofsolarcollectoroperation Solarheatcollection,kWh Heatlossto Thermal Incidentsolar Inwater Inice outdoor efficiencyof radiation, storage storage environment, solarcollector, kWh tank tank Overall kWh dimensionless January 2,823.18 232.74 1,475.10 1,707.84 1,115.34 0.58 February 3,680.97 289.61 1,933.30 2,222.92 1,458.05 0.57 March 4,714.15 272.66 2,535.08 2,807.75 1,906.41 0.54 April 4,015.50 119.03 2,392.72 2,511.75 1,503.75 0.57 May 1,200.62 433.03 0.00 433.03 767.59 0.31 June 948.72 342.73 0.00 342.73 605.99 0.31 July 725.36 298.52 0.00 298.52 426.83 0.36 August 758.99 274.45 0.00 274.45 484.53 0.31 September 875.64 350.41 0.00 350.41 525.23 0.35 October 1,783.80 217.84 801.05 1,018.88 764.91 0.51 November 1,530.94 99.18 875.26 974.43 556.51 0.62 December 2,008.93 173.25 1,004.75 1,178.00 830.94 0.55 Year 25,067 3,103 11,017 14,121 10,946 0.51

Distributionofsolarheatcollection

Amongtheannualsolarheatcollection(14,121kWh),thewaterstoragetank received3,103kWhofheatthroughoutanentireyear,countedfor22%.Theicestorage tankreceived11,017kWhofheatduringtheheatingseason(fromMay1toOctober16), countedfor78%.Theoperationoficestoragetankhandledthemajorityofsolarheat storage.

Amongtheannualheatstoredinthewaterstoragetank(3,103kWh),thewater

storagetankstored1,870kWhofheatintheheatingseason.Inadditiontothe11,017 kWhofheatstoredintheicestoragetankintheheatingseason,theoverallsolarheat

storageduringtheheatingseasonwas12,887kWh,countedfor87%oftheannualsolar heatcollection.Therest23%ofsolarheatcollectionwasmadeduringthenonͲheating season,whichwasallchargedtothewaterstoragetank.

148 Thermalefficiencyofsolarcollectoroperation

Thetwostoragetanks,withdifferentlevelsofoperationtemperature,made differentimpactontheperformanceofsolarthermalcollector.Table5.8presentsthe thermalefficiencyofsolarthermalcollectorbystoragetanks,toshowtheimpactmade

bytheenteringtemperatureofcollectorworkingfluid.

Table5.8Monthlyresultsofthermalefficiencyofsolarthermalcollector Units:dimensionless Withwater Withice storagetank storagetank January 0.30 0.66 February 0.34 0.62 March 0.33 0.58 April 0.31 0.59 May 0.31 Ͳ June 0.31 Ͳ July 0.36 Ͳ August 0.31 Ͳ September 0.35 Ͳ October 0.32 0.59 November 0.26 0.69 December 0.28 0.63 Year 0.32 0.62 Operatedwiththewaterstoragetank,thesolarthermalcollectorhadtheannual meanthermalefficiencyof0.32.Operatedwiththeicestoragetank,thesolarcollector thermalefficiencyrangedfrom0.58inMarchto0.69inNovember,leadingtheannual meanthermalefficiencyof0.63.Thethermalefficiencyofsolarthermalcollectorrose

abouttwotimeswhenagainstacoldthermalstorage,comparedtowithawarm

thermalstorage.

149

5.7. Operationofwaterstoragetank

Temperature

Thewaterstoragetankwasmodeledasastratifiedthermalstoragetank.Table

5.9presentsthetemperaturesoffivenodes(ܰͳtoͷ)inthewaterstoragetank,along

withthehotwaterdrawͲofftemperature.ThetemperatureprofileispresentedonFig.

5.27.Theresultsshowedthatthenodesܰ͵,ܰͶ&ܰͷhavehighermonthlyaverage temperaturesfromMaytoNovemberthanothermonths.

Table5.9Monthlyresultsofwaterstoragetanknodetemperatures Hotwater Nodetemperature,Ԩ drawͲoff temperature, N1 N2 N3 N4 N5 Mean Ԩ January 45.3 44.9 45.3 45.1 44.7 45.1 44.8 February 45.2 44.8 45.4 45.3 45.0 45.1 45.1 March 43.8 44.8 46.1 45.9 45.4 45.2 45.5 April 41.7 44.1 45.8 45.9 45.4 44.6 45.5 May 31.3 46.0 57.6 59.7 59.5 50.9 59.6 June 34.5 47.9 58.7 60.3 59.9 52.3 60.0 July 37.8 49.7 59.1 60.6 60.2 53.5 60.2 August 39.7 51.1 59.7 60.7 60.0 54.3 60.2 September 35.9 48.1 58.0 60.2 60.0 52.4 60.0 October 39.0 45.6 51.9 53.5 53.4 48.7 53.3 November 46.3 45.1 45.2 45.0 44.6 45.2 44.6 December 45.6 44.7 44.9 44.8 44.5 44.9 44.5 Year 40.5 46.4 51.5 52.3 51.9 48.5 51.4

150

QRGH1 QRGH1 QRGH1 QRGH1 QRGH1

& °

7HPSHUDWXUH

-DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.27NodetemperatureofWaterstoragetank,monthlymean

Heatinputandoutput

Thetemperaturedistributioninthewaterstoragetankwastheoutcomeofthe heattransferthatthestoragetankunderwent.Table5.10presentstheheatflowsofthe waterstoragetankoperation.Theannualheatinputwas9,902kWh.Theannualheat outputwas9,236kWh.Duetothetemperaturedifferencewiththesurroundings,the

storagetanklost667kWhofheatthroughoutanentireyear.

151

Table5.10Monthlyresultsofheatflowsofwaterstoragetank Heatinput,kWh Heatoutput,kWh Heattransfer From with Heat solar Byheat To Forhot surroundings, storage, collector pump Overall space water Overall kWh kWh January 232.74 1,590.85 1,823.59 1,455.86 309.10 1,764.96 Ͳ51.21 7.43 February 289.61 1,225.87 1,515.49 1,181.82 293.02 1,474.84 Ͳ46.33 Ͳ5.68 March 272.66 881.38 1,154.04 762.24 346.55 1,108.79 Ͳ51.17 Ͳ5.91 April 119.03 405.02 524.05 153.09 322.63 475.73 Ͳ48.11 0.21 May 433.03 0.00 433.03 0.00 351.81 351.81 Ͳ59.46 21.76 June 342.73 0.00 342.73 0.00 279.79 279.79 Ͳ60.36 2.58 July 298.52 0.00 298.52 0.00 233.21 233.21 Ͳ64.96 0.35 August 274.45 0.00 274.45 0.00 208.46 208.46 Ͳ66.50 Ͳ0.51 September 350.41 0.00 350.41 0.00 284.46 284.46 Ͳ60.91 5.03 October 217.84 229.24 447.08 148.63 260.18 408.81 Ͳ56.82 Ͳ18.56 November 99.18 948.86 1,048.04 755.39 248.14 1,003.53 Ͳ49.98 Ͳ5.47 December 173.25 1,517.37 1,690.62 1,338.39 303.09 1,641.48 Ͳ50.88 Ͳ1.74 Year 3,103 6,799 9,902 5,795 3,440 9,236 Ͳ667 Ͳ0.51

5.8. Heattransitviathermalstorage Theheattransitisreferredtotheoveralltransferofheatthroughheatcharging andheatdischarging.Thetwothermalstoragetanksofcombisystembridgedthetime gapofheatbetweentheavailableandinneedy.Fig.5.28presentstheheatchargesand dischargesofthewaterstoragetankovertheyear.Theannualgrossofheattransitof thewaterstoragetankiscalculatedasasumoftheheatchargedtothewaterstorage

tankandtheheatdischargedfromthewaterstoragetank;theamountwas19,138kWh.

Fig.5.29presentstheheatchargesanddischargesoftheicestoragetankovertheyear.

Theannualgrossofheattransitoftheicestoragetankiscalculatedasasumoftheheat chargedtotheicestoragetankandtheheatdischargedfromtheicestoragetank;the amountwas17,414kWh.

152

KHDWFKDUJLQJ KHDWGLVFKDUJLQJ

+HDWN:K

-DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.28Heattransitviawaterstoragetank,byweek KHDWFKDUJLQJ KHDWGLVFKDUJLQJ

+HDWN:K

-DQ )HE 0DU $SU 0D\ -XQH -XO\ $XJ 6HS 2FW 1RY 'HF Fig.5.29Heattransitviaicestoragetank,byweek Theannualgrossofheattransitviatheicestoragetankwas1,724kWhlessthan

thewaterstoragetank.Countingthetwostoragetankstogether,theoverallheattransit was36,552kWh.Amongthisamount,thewaterstoragetankhasbeeninchargeof

153 52.4%,theicestoragetank47.6%.Theresultsshowedthesignificantperformanceof thewaterstoragetankinthesimulatedcombisystemannualoperation.

5.9. Exergyperformanceofsolarcombisystem

CombisystemexergeticCOP

TheexergeticCOPiscalculatedastheratiooftheexergyofheatingload(507 kWh)totheauxiliaryelectricityconsumption(2,956kWh).Thecombisystemhadan annualexergeticCOPof0.17.

Solarexergyfractioncomparedtoreferencesystem

Table5.11presentstheexergyusedbythespecifiedreferencesystem(i.e., electricheatingsystem),andtheexergycollectionofthecombisystemsolarthermal collectors.

Table5.11Referencesystemexergyusevs.combisystemsolarexergycollectionofsolarcombisystem Referencesystem Solarcombisystem Electrical exergyuseof Electrical baseboard exergyuseof Totalexergy Collected Solarexergy heater, waterheater, use, solarexergy, fraction, kWh kWh kWh kWh dimensionless January 2,331.04 277.51 2,608.55 49.85 0.02 February 1,721.11 262.53 1,983.64 95.67 0.05 March 949.38 306.54 1,255.92 226.53 0.18 April 182.27 283.44 465.72 209.86 0.45 May 0.00 213.88 213.88 76.27 0.36 June 0.00 156.79 156.79 61.23 0.39 July 0.00 117.67 117.67 53.70 0.46 August 0.00 100.80 100.80 49.49 0.49 September 0.00 145.52 145.52 62.63 0.43 October 171.68 171.73 343.41 91.19 0.27 November 976.15 218.15 1,194.30 41.64 0.03 December 1,991.79 271.83 2,263.62 49.53 0.02 Year 8,323 2,526 10,850 1,068 0.10

154 Overtheentireyear,thesolarcollectorscollected1,068kWhofthermalenergy.

Comparedto10,850kWhofexergyofheatingloadbythereferencesystem,the combisystemhadasolarexergyfractionof0.10.Themaximumsolarexergyfraction appearedinAugustasof0.49.Theminimumsolarexergyfractionof0.02appearedin

JanuaryandDecember.

Solarexergycontribution

Among26,421kWhofexergy,sumoftheannualincidentsolarexergy(23,465 kWh)andtheannualelectricalexergy,theannualincidentsolarexergycontributionwas

88.8%.

Among4,024kWhofexergy,sumoftheannualsolarexergycollection(1,068 kWh)andtheannualelectricityconsumption,theannualcollectedsolarexergy

contributionwas26.5%.

Exergyefficiency

Table5.12presentstheexergydestructionestimatedbymonthlyandannual operationofthecombisystem.Thecombisystemoperationovertheyeardestroyed

15,657kWhofexergyoutofthetotal26,421kWhofexergysupplyincludingthe incidentsolarexergyandelectricity.LessexergyweredestroyedduringMaythrough

Octoberthantheothermonths.Amongthecomponentequipment,themostamountof exergywasdestroyedbyoperatingthesolarthermalcollector,accountedfor77.6%.

Othercontributionswere:theheatpump11.2%,theradiantfloor4.3%,thewater

155 storagetank3.8%,theicestoragetank2.9%,andthecirculatingpumps0.2%.The estimatedexergeticefficiencyispresentedinTable5.13.

Table5.12Monthlyresultsofexergydestroyedbysolarcombisystemoperation Units:kWh Ice Water Solar Solar Heat Auxiliary storage storage Radiant combisystem collectors pump Pumps heater tank tank floor January 2,390.27 1,548.87 512.22 4.85 0 21.93 111.51 190.87 February 2,590.32 1,985.22 335.54 4.04 0 35.82 94.76 134.94 March 2,845.44 2,401.82 160.30 3.29 0 140.03 68.58 71.42 April 2,420.35 2,141.41 62.53 2.78 0 172.75 27.54 13.34 May 361.54 329.09 Ͳ 0.78 0 2.51EͲ02 31.65 Ͳ June 285.56 259.60 Ͳ 0.67 0 1.23EͲ02 25.28 Ͳ July 247.54 225.75 Ͳ 0.47 0 6.43EͲ03 21.31 Ͳ August 227.53 207.43 Ͳ 0.55 0 3.22EͲ03 19.55 Ͳ September 289.35 265.39 Ͳ 0.51 0 1.58EͲ03 23.44 Ͳ October 992.11 862.60 38.84 1.44 0 49.06 27.04 13.12 November 1,246.33 870.53 216.00 2.75 0 22.11 54.18 80.76 December 1,760.89 1,053.20 423.84 4.08 0 20.00 95.58 164.18 Year 15,657 12,151 1,749 26 0 462 600 669 100% 77.6% 11.2% 0.2% 0.0% 2.9% 3.8% 4.3%

Table5.13Annualresultsofexergeticefficiencyofsolarcombisystemoperation Solar Solarthermal Icestorage Waterstorage combisystem collectors Heatpump tank tank Radiantfloor 40.7% 9.3% 42.6% 19.0% 61.0% 17.8% Iftheexergydestroyedduringthesolarthermalcollectoroperationisexcluded fromtheanalysis,theexergydestroyedwassummedto3,506kWhandthe correspondingexergyefficiencywas12.9%(among4,024kWh).

156

6. Conclusions Thisthesishaspresentedthedevelopmentofasolarcombisystemcomputer model,implementedinEESprogram.Theworkpresentedinthethesisincludes:(i)A methodologyforthedevelopmentofasolarcombisystemcomputermodel;(ii)The mathematicalmodelsofthesolarcombisystemmodelcomponents;(iii)Thefeaturesof aEESͲbasedcomputermodelofasolarcombisystembeingdeveloped;(iv)Asimulated solarcombisystemoperationcaseresults;and(v)Theanalysisofthesimulatedsolar combisystemoperatingconditionsbasedontheSecondLawofthermodynamics.

ThemodeldevelopmentbasiswastheSecondLawofthermodynamics,which

waschosenfortheabilitytorepresenttheenergyandthequalityoftheenergy

relatedtothesolarcombisystemoperation.

Themathematicalmodelofthesolarcombisystemwasdevelopedincomponent

approachasofmodulartype,whichlefttheopportunityforthedeveloped

modulestobeintegratedformodelsofdiverseHVACsystems.

ThemathematicalmodelofsolarcombisystemwasimplementedintheEES

program,whichwaschosenforitsequationͲbasedscheme,aswellasthebuiltͲin

functionsforcalculatingthethermodynamicpropertiesofworkingfluids.

Thecompletecomputermodelofsolarcombisystemissuitableformodelingthe

selectedsolarcombisystemoperationasafunctionofthespaceheatingdemand,

domestichotwaterconsumption,outdoordryͲbulbairtemperature,andsolar

157 radiation,andsubjecttotheoperatingcontrolalongwiththesimulationtime

stepassignedbytheuser.

Operationfeatures

TheenergyͲconsumingequipmentofthesolarcombisystemincludesthesolar thermalcollector,circulatingpumps,heatpump,andauxiliaryheater.Thesolarthermal collectorandtheheatpumparetheprimaryenergyͲconsumingcomponents,their energyconsumptionsaredeterminednotonlybytheheatingrequirementsandclimate condition,butalsoaffectedbytheenteringstateofheattransferfluid.Thestatesatthe

equipmentinletsrelyontheoperationofthermalstoragetanksundertheoperating

controlperformedonthecompletesolarcombisystem.

Thesimulatedsolarcombisystemprovided3,440kWhofheatfortheannual

domestichotwateruseofatypicaloneͲstoreydetachedhouse,andmetthe

residentialspaceheatingdemandof8,323kWhthroughoutaheatingseasonof

196dayswiththeoutdoortemperaturethelowestatͲ28.1ԨinMontreal

Canada.

ThedrawͲofftemperatureofthedomestichotwaterwas51.4Ԩinannual

average.Thesupplytemperatureoftheradiantfloorwas37.7Ԩinseasonal

average,whilethereturntemperaturebeingcontrolledatconstant20Ԩ.

Satisfyingtheannualheatingneedsof11,764kWhintotal,thesolar

combisystemreceived14,121kWhofsolarheatfromthesolarthermal

collectors,consumed2,956kWhofelectricitytorunawaterͲtoͲwaterheat

158 pumpandthreecirculationpumpswhereaswithoutstartinguptheauxiliary

electricheater.

Thewaterstoragetank(warmthermalstorage)hadanannualaverage

temperatureof48.5Ԩ.

Theicestoragetank(coldthermalstorage)hadanannualaveragetemperature

of13.7Ԩ.

Energyperformance

Thesolarcombisystemoperationrepresents:

a. AnannualsystemCOPof3.98;

b. Anannualsolarenergyfractionof1.3comparedtoanelectricheating

system;

c. Anannualincidentsolarenergycontributionof89.5%andanannual

collectedsolarenergycontributionof82.7%;

d. Anannualsystemenergyefficiencyof42.0%regardingtheincidentsolar

heatduringthecollectorsonoperationasthethermalenergyinput,and

of68.9%regardingthesolarheatcollectionasthethermalenergyinput.

Thesolarthermalcollectoroperationrepresents:

a. Anannualaveragethermalefficiencyof0.51;

b. Anannualaveragethermalefficiencyof0.62operatedagainstthecold

thermalstorage;

c. Anannualaveragethermalefficiencyof0.32operatedagainstthewarm

thermalstorage.

159 TheheatpumphadaseasonalaverageCOPof3.19.

Exergyperformance

Thesolarcombisystemdestroyed15,657kWhofexergyovertheannual

operationandrepresents:

a. AnannualexergeticCOPof0.17;

b. Anannualsolarexergyfractionof0.10comparedtoanelectricheating

system;

c. Anannualincidentsolarexergycontributionof88.8%andanannual

collectedsolarexergycontributionof26.5%;

d. Anannualsystemexergyefficiencyof40.7%regardingtheincidentsolar

exergyduringthecollectorsonoperationasthethermalexergyinput,

andof12.9%regardingthesolarexergycollectionasthethermalexergy

input.

Theannualaverageexergyefficiencywas:

a. 61.0%forthewaterstoragetank;

b. 19.0%fortheicestoragetank;

c. 9.3%forthesolarthermalcollectors;

d. 42.55%fortheheatpump;

e. 17.82%fortheradiantfloor.

160

6.1. Futurework I. Modelvalidation

Toimprovethedevelopedsolarcombisystemmodel,validationofthe componentmodelsandthewholesolarcombisystemmodelcanbecarriedout,if installationorexperimentalsetupisavailable.

II. Analysisofsensitivity

Aninterestingissuefoundduringthemodelsimulationisthatthesolar combisystemcomponentcapacities,thesystemoperatingcontrol,andthecombisystem

heatingperformancearecloselyrelated.Toinvestigatethethreeaspectstheimpact

addedbyoneontotheotherwillprovidevaluableinformationforthesystem

application,andfavorthesolarcombisystemoperationtowardsbetterperformance.

III. SolarcombiͲplus,highsolarcombiͲplus,andoptimization

Theresultsshowedthedrawbackofoversizingofthesolarcombisystemforthe

simulatedcase,withoutcoolingloadinthenonͲheatingseason.Acasewithsolar coolingcanbesimulatedusingthedevelopedmodelwithcertainmodificationto investigateasolarcombiͲplussystemoperation,toseeifabletoachievethe performanceasahighsolarcombiͲplussystemdoes,andfurtherconductoptimization study.Thestudiedsolarcombisystemseemedhasthepotentialbecausethe

161 configurationhasbothsensiblethermalstorageandlatentthermalstorage,plusthe

configurationisnotonlycomplexbutalsoflexible.

IV. SecondLawmodelLibraryofHVACcomponents

SimilarmodeldevelopmentcanbedoneontootherHVACcomponentsfora

SecondLawmodellibraryofsimulationprograms.

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176 Appendices

AppendixA

ApplicationconstraintsofEq.(4.3)

Torevealthepotentialofthesituations(1)ߟ஼௢௟௟<0and(2)ߟ஼௢௟௟>1,atheoretical

approachoffindingsolutionstoquadraticispresentedbelow.Abnormalresultswould

occur,if

஼௢௟௟ ஼௢௟௟ ଶ ܶ௜௡ǡ௪ െܶ௢௔ ൫ܶ௜௡ǡ௪ െܶ௢௔൯ ܽ଴ ൅ܽଵ ൅ܽଶ ൏Ͳ ሺ݅ሻ ܩ ܩ (1) ଶ ܶ஼௢௟௟ െܶ ൫ܶ஼௢௟௟ െܶ ൯ ܽ ൅ܽ ௜௡ǡ௪ ௢௔ ൅ܽ ௜௡ǡ௪ ௢௔ ൐ͳ ሺ݅݅ሻ ଴ ଵ ܩ ଶ ܩ

஼௢௟௟ whichcouldbefurthertransformedas(denotingܶ௢௔toܶ௔,ܶ௜௡ǡ௪ toܶ௜௡)

ܽଶ ଶ ʹܽଶ ܽଵ ܽଶ ଶ ܽଵ ܶ௔ െ ቀ ܶ௜௡ ൅ ቁ ܶ௔ ൅ ቀ ܶ௜௡ ൅ ܶ௜௡ ൅ܽ଴ቁ ൏ Ͳ ሺ݅ሻ ܩ ܩ ܩ ܩ ܩ (2) ܽ ʹܽ ܽ ܽ ܽ ଶ ܶ ଶ െ ቀ ଶ ܶ ൅ ଵቁ ܶ ൅ ቀ ଶ ܶ ଶ ൅ ଵ ܶ ൅ܽ െͳቁ ൐ Ͳ ሺ݅݅ሻ ܩ ௔ ܩ ௜௡ ܩ ௔ ܩ ௜௡ ܩ ௜௡ ଴

ܻ଴ ൏Ͳ ሺ݅ሻ (3) ܻଵ ൐Ͳ ሺ݅݅ሻ

where,

ܽଶ ଶ ʹܽଶ ܽଵ ܽଶ ଶ ܽଵ ܻ଴ ൌ ܶ௔ െ ቀ ܶ௜௡ ൅ ቁ ܶ௔ ൅ ቀ ܶ௜௡ ൅ ܶ௜௡ ൅ܽ଴ቁ ሺ݅ሻ ܩ ܩ ܩ ܩ ܩ (4) ܽ ʹܽ ܽ ܽ ܽ ܻ ൌ ଶ ܶ ଶ െ ቀ ଶ ܶ ൅ ଵቁ ܶ ൅ ቀ ଶ ܶ ଶ ൅ ଵ ܶ ൅ܽ െͳቁ ሺ݅݅ሻ ଵ ܩ ௔ ܩ ௜௡ ܩ ௔ ܩ ௜௡ ܩ ௜௡ ଴

fitsinageneralquadraticformof

177 ଶ ܻ଴ ൌ ܣܶ௔ ൅ܤܶ௔ ൅ܥ଴ ሺ݅ሻ ଶ (5) ܻଵ ൌ ܣܶ௔ ൅ܤܶ௔ ൅ܥଵ ሺ݅݅ሻ

whichbearsthediscriminant

ଶ ܤ െͶܣܥ଴ ሺ݅ሻ οൌ ቊ ଶ (6) ܤ െͶܣܥଵ ሺ݅݅ሻ

with

ܽଶ ܣ ൌ ǡ (7) ܩ

ʹܽଶ ܽଵ ܤൌെ൬ ܶ ൅ ൰ǡ (8) ܩ ௜௡ ܩ ܽ ܽ ܥ ൌ ଶ ܶ ଶ ൅ ଵ ܶ ൅ܽ ሺ݅ሻ ଴ ܩ ௜௡ ܩ ௜௡ ଴ ܽ ܽ (9) ܥ ൌ ଶ ܶ ଶ ൅ ଵ ܶ ൅ܽ െͳ ሺ݅݅ሻ ଵ ܩ ௜௡ ܩ ௜௡ ଴

Whenοш0,quadraticfunctionܻ଴andܻଵhastwodistinctroots

െܤ േ ξο ܶ ൌ Ǥ (10) ௔ଵǡଶ ʹܣ

QuadraticfunctionsinEqn.(4)areillustratedinFig.1usingthecasesolar collectorcoefficients,whereܽଶ<0,ܣ<0and,therefore,thecurvesopenupward.If quadraticdiscriminantο>0,twodistinctrootsexistastheplottedsituation.Eqn.(3)ሺ݅ሻ istruewhenܶ௔<ܶ௔ଵ,orܶ௔>ܶ௔ଶ;Eqn.(3)ሺ݅݅ሻistruewhenܶ௔ଵ<ܶ௔<ܶ௔ଶ.Underthe

applicationofthesolarcollectoroperation,onlyܶ௔ଵismeaningful,sinceonecan

calculateܶ௔ଶatamuchvaluethanthatoftheoutdoorairmayreachinreality,suchas

200Ԩ.Theweightedcurvesthatoverlayinguponܻ଴andܻଵhighlightwhereܻ଴ ൏Ͳ

andܻଵ ൐Ͳ,respectively.Similarsituationforο=0,buttwodistinctrootsbecomingone.

Ifο<0,therootsdonotexist,thequadraticcurvesfallbelowtheaxisܶ௔.Therefore,Eqn.

178 (3)ሺ݅ሻalwaysholdsbutnotሺ݅݅ሻ.Amongthesituationsexplicated,wheneverEqn.(3)

truejustifiestheinvalidapplicationofthermalefficiencyofsolarcollector,whichshould alwayslieinarangeof0чߟ஼௢௟௟ч1.

Fig.1Solutionstoquadraticinapplicationsofsolarthermalefficiency Theapplicationconstrainsofsolarthermalcollectorproductusedinthecase studywereexaminedinthequadraticapproach.Fig.2showstheupperandlower boundsoftheoutdoorairtemperaturewithrespecttothesolarinsolationfrom0W/m2

2 to1200W/m whilevariousenteringtemperatures,ܶ௜௡,between0.01Ԩto70Ԩ.For instance,thevalidapplicationofthermalefficiencyshouldbewiththeoutdoor temperatureranginginͲ20Ԩ<ܶ௔<40Ԩwhentheenteringtemperatureat20Ԩand

thesolarinsolationdensityat200W/m2.

Abidingbythelowerboundsensuresthermalefficiencyofsolarthermalcollector

ߟ஼௢௟௟>0,bytheupperboundsensuresߟ஼௢௟௟<1.Alongtheverticalaxisdownward,the boundsofoutdoortemperaturedropandlowerboundsgoalongwithlowerentering temperatures.Alongthehorizontalaxistotheleftside,theoutdoortemperatureis

179 constrainedtighterwithinnarrowrangesasthesolarinsolationdecreasing.But, constraintsarerelievedprovidingthesolarinsolationhigherthan425W/m2.

Fig.2Upperandlowerboundsofoutdoortemperatureofcasesolarthermalefficiency

180

AppendixB

Inputandoutputofthecomponentmodels

Thesolarthermalcollectormodel:

Symbols Descriptions Units

Inputs ܽ଴ Coefficientofsolarcollectorthermalefficiency dimensionless 2 ܽଵ Coefficientofsolarcollectorthermalefficiency W/(m ͼԨ) 2 2 ܽଶ Coefficientofsolarcollectorthermalefficiency W/(m ͼԨ ) ஼௢௟௟ 2 ܣ௣௔௡௘௟ Grossareaofsinglesolarthermalcollectorpanel m ݊஼௢௟௟ Numberofsolarthermalcollectorpanel dimensionless ሶ ஼௢௟௟ 3 2 ܸ௨௡௜௧ Volumeflowrateperunitcollectorarea m /(sͼm ) Powerdensityofinsolationupontiltedcollectorpanelsurface, ܩሶ W/m2 timevariable

ܶ௢௔ Outdoorairtemperature,timevariable Ԩ Temperatureofwaterenteringsolarthermalcollector, ܶ஼௢௟௟ Ԩ ௜௡ǡ௪ associatedvariable ஼௢௟௟ Outputs ܶ௢௨௧ǡ௪ Temperatureofwaterleavingsolarthermalcollector Ԩ ஼௢௟௟ ݉ሶ ௪ Massflowrateofwaterthroughsolarthermalcollector kg/s ஼௢௟௟ 3 ܸ௪ሶ Volumeflowrateofwaterthroughsolarthermalcollector m /s ܳሶ ஼௢௟௟ Solarheatabsorptionrateofsolarthermalcollector W

181 Theheatpumpmodel:

Symbols Descriptions Units Volumeflowrateofwaterthroughevaporatorunderbaseline Inputs ܸሶ ா௩௔௣ m3/s ௕௔௦௘ condition,inputparameter Volumeflowrateofwaterthroughcondenserunderbaseline ܸሶ ஼௢௡ௗ m3/s ௕௔௦௘ condition,inputparameter Electricpowerofheatpumpunderbaselinecondition,input ܹሶ ு௉ W ௕௔௦௘ parameter Heatoutputrateatcondenserunderbaselinecondition,input ܳሶ ஼௢௡ௗ W ௕௔௦௘ parameter

ܾ଴ǡܾଵǡǥܾସ Coefficientsofheatpupelectricpower,inputparameterͲ

݀଴ǡ݀ଵǡǥ݀ସ Coefficientsofheattransferrateatcondenser,inputparameterͲ ா௩௔௣ 3 ܸ௪ሶ Volumeflowrateofwaterthroughevaporator,inputparameter m /s ஼௢௡ௗ 3 ܸ௪ሶ Volumeflowrateofwaterthroughcondenser,inputparameter m /s ா௩௔௣ ܶ௜௡ǡ௪ Temperatureofwaterenteringevaporator,associatedvariable Ԩ ஼௢௡ௗ ܶ௜௡ǡ௪ Temperatureofwaterenteringcondenser,associatedvariable Ԩ ா௩௔௣ ܶ௠௘௔௡ǡ௪ Meantemperatureofwaterthroughevaporator,inputparameter Ԩ ஼௢௡ௗ ܶ௠௘௔௡ǡ௪ Meantemperatureofwaterthroughcondenser,inputparameter Ԩ

οܶ௦௨௣௥௛ Superheattemperaturedifference,inputparameter Ԩ

ܲ௖ Pressureofcondensation,inputparameter Pa

ܲ௘ Pressureofvaporization,inputparameter Pa ሶ ோி ܳௗ௘௠ௗ Spaceheatingdemandontoradiantfloor,inputtimevariable W ா௩௔௣ Outputs ܶ௢௨௧ǡ௪௙ Temperatureofliquidwaterleavingevaporator Ԩ ஼௢௡ௗ ܶ௢௨௧ǡ௪ Temperatureofwaterleavingcondenser Ԩ ா௩௔௣ ݉ሶ ௪ Massflowrateofwaterthroughevaporator kg/s ா௩௔௣ ݉ሶ ௪௙ Massflowrateofliquidwaterleavingevaporator kg/s ா௩௔௣ ݉ሶ ௪௜ Massflowrateofsolidwater(ice)leavingevaporator kg/s ஼௢௡ௗ ݉ሶ ௪ Massflowrateofwaterthroughcondenser kg/s ா௩௔௣ 3 ܸ௪ሶ Volumeflowrateofwaterthroughevaporator m /s ஼௢௡ௗ 3 ܸ௪ሶ Volumeflowrateofwaterthroughcondenser m /s ܹሶ ு௉ Electricpowerofheatpump W ܳሶ ஼௢௡ௗ Heatoutputrateatcondenser W ܳሶ ு௉ସோி Heatoutputratefromheatpumptoradiantfloor W ܳሶ ு௉ସௐ் Heatoutputratefromheatpumptowaterstoragetank W

182 Theicestoragetankmodel:

Symbols Descriptions Units Inputs ܮூ் Lengthofrectangularstoragetank m ܹூ் Widthofrectangularstoragetank m ܪூ் Heightofrectangularstoragetank m Thermaltransmittanceofinsulationattachedtoicestorage ܷூ் W/(m2ͼԨ) ௜௡௦௨ tank ௐ ܶ௠௔௫ MaximumtemperatureofwaterͲlayer Ԩ

ߛ௠௔௫ Maximumicemassfractioninicestoragetank dimensionless οݐ Timestep s

ܶ௜௔ Indoorairtemperature Ԩ Massflowrateofwaterenteringfromicestoragetankheat ݉ሶ ூ்ௌ kg/s ௪ source,associatedvariable Massflowrateofsolidwaterfromicestoragetankheatsink, ݉ሶ ூ்௅ kg/s ௪௜ associatedvariable Massflowrateofliquidwaterenteringfromicestoragetank ݉ሶ ூ்௅ kg/s ௪௙ heatsink,associatedvariable Temperatureofwaterenteringfromicestoragetankheat ܶூ்ௌ Ԩ ௜௡ǡ௪ source,associatedvariable Temperatureofwaterenteringfromicestoragetankheat ܶூ்௅ Ԩ ௜௡ǡ௪௙ sink,associatedvariable

ூ்ௌ Constantpressurespecificheatofwaterenteringfromice ܿ௣ J/(kgͼԨ) ௜௡ǡ௪ storagetankheatsource,associatedvariable

ூ்௅ Constantpressurespecificheatofliquidwaterenteringfrom ܿ௣ J/(kgͼԨ) ௜௡ǡ௪௙ icestoragetankheatsink,associatedvariable ௐ Outputs ܶ௪ TemperatureofwaterͲlayer Ԩ

183 Thewaterstoragetankmodel:

Symbols Descriptions Units Inputs ܪௐ் Heightofwaterstoragetank m ܦௐ் Diameterofcylinderwaterstoragetank m ௐ் ߜ௦௛௘௟௟ Thicknessofwaterstoragetankshell m ௐ் ݇௦௛௘௟௟ Thermalconductivityofwaterstoragetankshell W/(mͼԨ) Thermaltransmittanceofinsulationattachedtowaterstorage ܷௐ் W/(m2ͼԨ) ௜௡௦௨ tank οݐ Timestep s

ܶ௜௔ Indoorairtemperature Ԩ ஽ுௐ 3 ܸ௪ሶ Volumeflowrateofdomestichotwaterdrawn,inputvariable m /s ெ௔௜௡ ܶ௪ Temperatureofwaterfromcitymain,inputvariable Ԩ Heatchargingrateofimmersionheatexchangerܪܺͳ,associated ܳሶ ௐ்ௌଵ W variable Heatchargingrateofimmersionheatexchangerܪܺʹ,associated ܳሶ ௐ்ௌଶ W variable ሶ ோி ܳௗ௘௠ௗ Spaceheatingdemandontoradiantfloor,inputvariable W ே௜ Outputs ܶ௪ TemperatureofwaterͲnodeܰ݅ Ԩ ௐ்௅ଵ ܶ௢௨௧ǡ௪ Temperatureofwaterleavingimmersionheatexchangerܪܺͳ Ԩ ܳሶ ௐ்௅ଵ Heatdischargingrateofimmersionheatexchangerܪܺͳ W ܳሶ ௐ்௅ଶ Heatoutputrateofwaterstoragetankfordomesticwaterheating W ܳሶ ௐ்௅ Heatoutputrateofwaterstoragetankforheating W ܳሶ ௐ்ସோி Heattransferratefromwaterstoragetanktoradiantfloor W

Theradiantfloormode:

Symbols Descriptions Units Heattransferratefromwaterstoragetanktoradiantfloor, Inputs ܳሶ ௐ்ସோி W associatedvariable Heattransferratefromheatpumptoradiantfloor,associated ܳሶ ு௉ସோி W variable Temperatureofwaterleavingimmersionheatexchangerܪܺͳ, ܶௐ்௅ଵ Ԩ ௢௨௧ǡ௪ associatedvariable Temperatureofwaterleavingheatpumpcondenser,associated ܶ஼௢௡ௗ Ԩ ௢௨௧ǡ௪ variable ோி Outputs ݉ሶ ௪ Massflowrateofwaterthroughradiantfloor kg/s ܳሶ ோி Heatinputratetoradiantfloor W

184 Thecirculatingpumpmodel:

Symbols Descriptions Units

Inputs ܿ଴ Coefficientofelectricpowerinputtocirculatingpump W 3 ܿଵ Coefficientofelectricpowerinputtocirculatingpump Wͼs/m 2 6 ܿଶ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 3 9 ܿଷ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 4 12 ܿସ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 5 15 ܿହ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 6 18 ܿ଺ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 7 21 ܿ଻ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m 8 24 ଼ܿ Coefficientofelectricpowerinputtocirculatingpump Wͼs /m ሶ ௉௨௠௣ 3 ܸ௠௔௫ Maximumvolumeflowrateofwaterthroughcirculatingpump m /s ௉௨௠௣ଵ 3 ܸ௪ሶ Volumeflowrateofcirculatingpumpܲͳ,associatedvariable m /s ௉௨௠௣ଶ 3 ܸ௪ሶ Volumeflowrateofcirculatingpumpܲʹ,associatedvariable m /s ௉௨௠௣ଷ 3 ܸ௪ሶ Volumeflowrateofcirculatingpumpܲ͵,associatedvariable m /s Outputs ܹሶ ௉௨௠௣ଵ Electricpowerinputtocirculatingpumpܲͳ W ܹሶ ௉௨௠௣ଶ Electricpowerinputtocirculatingpumpܲʹ W ܹሶ ௉௨௠௣ଷ Electricpowerinputtocirculatingpumpܲ͵ W ሶ ௉௨௠௣ ܹ௦௨௣௣ Overallpowerinputoncirculatingpumps W

185 Thecontrolsystemmodel:

Symbols Descriptions Units Powerdensityofinsolationupontiltedcollectorpanelsurface, Inputs ܩሶ W/m2 inputvariable ሶ ோி ܳௗ௘௠ௗ Spaceheatingdemandontoradiantfloor,inputvariable W ஽ுௐ 3 ܸ௪ሶ Volumeflowrateofdomestichotwaterdrawn,timevariable m /s ߛ Icemassfractionoficestoragetank,associatedvariable dimensionless

ߛ௠௔௫ Maximumicemassfractioninicestoragetank,inputparameter dimensionless ௐ ܶ௪ IcestoragetankwaterͲlayertemperature,associatedvariable Ԩ ௐ் ܶ௦௘௧ Temperaturesetforwaterstoragetank,inputparameter Ԩ Variationallowedrespectingtosettingtemperatureofwater ܶௐ் Ԩ ௩௔௥ storagetank,inputparameter ேଵ ܶ௪ TemperatureofwaterͲnodeܰͳ,associatedvariable Ԩ ேଷ ܶ௪ TemperatureofwaterͲnodeܰ͵,associatedvariable Ԩ ேହ ܶ௪ TemperatureofwaterͲnodeܰͷ,associatedvariable Ԩ Temperatureofwaterleavingimmersionheatexchangerܪܺͳ, ܶௐ்௅ଵ Ԩ ௢௨௧ǡ௪ associatedvariable ஼௢௡ௗ ܶ௢௨௧ǡ௪ Temperatureofwaterleavingcondenser Ԩ ோி ܶ௖௢௠ Comforttemperatureofradiantfloor,inputparameter Ԩ Outputs ߠ஼௢௟௟ Controlledvariableforoperatingcirculatingpumpܲͳ dimensionless ControlledvariableforthreeͲwayvalveܸͳ,respectingthe ߠ஼௢௟௟ସூ் dimensionless operationbetweensolarthermalcollectorandicestoragetank ControlledvariableforthreeͲwayvalveܸͳ,respectingthe ߠ஼௢௟௟ସௐ் operationbetweensolarthermalcollectorandwaterstorage dimensionless tank ߠு௉ Controlledvariableforoperatingheatpump dimensionless ControlledvariableforthreeͲwayvalveܸʹ,respectingthe ߠு௉ସௐ் dimensionless operationbetweenheatpumpandwaterstoragetank ControlledvariableforthreeͲwayvalveܸʹ,respectingthe ߠு௉ସோி dimensionless operationbetweenheatpumpandradiantfloor Controlledvariableforbypassvalveܸ͵,respectingthe ߠௐ்ସோி dimensionless operationbetweenwaterstoragetankandradiantfloor ߠ஺௨௫ Controlledvariableforoperatingauxiliaryheater dimensionless

186

AppendixC

EmpireECͲ24SRCCOGͲ100certification

Source:SunEarth(2012)

187

AppendixD

RegardingcoefficientsidentificationwithStatGraphicsforGeoSmartHS050

ࢃሶ ࡴࡼ I. Theplotof forEqn.(4.19)regression ૜૜ૢ૙

RͲSquared=99.7768percent

StandardErrorofEst.=0.0136088

TheRͲSquaredstatisticindicatesthatthemodelasfittedexplains99.7768%of

ௐሶ ಹು thevariabilityin .Thestandarderroroftheestimateshowsthestandarddeviation ଷଷଽ଴ oftheresidualstobe0.0136088.

188

ࡽሶ ࡯࢕࢔ࢊ II. Theplotof forEqn.(4.20)regression ૛૚૙ૠ૛

RͲSquared=98.6571percent

StandardErrorofEst.=0.0158837

TheRͲSquaredstatisticindicatesthatthemodelasfittedexplains98.6571%of

ொሶ ಴೚೙೏ thevariabilityin .Thestandarderroroftheestimateshowsthestandarddeviation ଶଵ଴଻ଶ oftheresidualstobe0.0158837.

189 III. GeoSmartHS050performancedata

Source:GeoSmart(2012)

190

AppendixE

WiloͲStar16Fperformancecurve

Source:Wilo(2012)

191