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CopyrightAmericanSocietyofHeating,RefrigerationandAirConditioningEngineers,Inc.2003
[Headnote]
ABSTRACT
Fourtwo-variableparameterestimationmodelsforevaluationofthermalresponsetestdataarecomparedwhenappliedonthesametemperatureresponsedata.Twomodelsarebasedonline-sourcetheory,thethirdmodelisacylinder-source-basedsolution,andthefourthisanumericalone-dimensionalfinitedifferencemodel.Thedatasetscontainmeasuredtemperatureresponse,heatload,andundisturbedgroundtemperaturefromthreethermalresponsetests,togetherwithphysicaldataofthetestedboreholeheatexchangers(BHE).ThemodelsestimategroundthermalconductivityandthermalresistanceoftheBHEandarecomparedregardingtestlengthanddataintervalused.Forthethreedefineddatasets,thelinesourceapproximationmodelshowstheclosestagreementwiththemeasuredtemperatureresponse.Thecylindersourceandnumericalmodelsshowsensitivitytotheinclusionofearlydata.Arecommendedminimumresponsetestdurationof50hoursisconcludedfromthemodelcomparison.
INTRODUCTION
Duringathermalresponsetest,adefinedthermalloadisappliedtoaboreholeheatexchangerandthetemperaturedevelopmentoftheinletandoutlettemperaturesaremeasuredovertime.Thistemperatureresponseallowsextrapolationofthethermalbehaviorinfuturetime.Onepossibleconceptualmodelfortheinterpretationistoassumethegroundtobeaconductivemediumandtodeterminetheapparentthermalconductivityandotherthermalparametersofthismedium.Thetestmaybeconductedusingatransportabledevicethatisbroughtonsitetotheborehole.
Sinceitsintroductionin1995-1996,thisin-situmethodhasspreadtomostcountrieswhereboreholesinthegroundareusedasaheatsource/sinkonalargerscale.Themethodservesprimarilytoassessthegroundthermalconductivityandperformanceofdifferentboreholeheatexchangerdesigns,whichareimportantforoptimaldesignandqualitycontrol.Themethodisdescribedinseveralpapers,e.g.,Gehlin(1998),Austin(1998),Austinetal.(2000),ShonderandBeck(2000),andKavanaughetal.(2000).TheprincipleofathermalresponsetestsetupisoutlinedinFigure1.
Theboreholetemperatureresponseisthetemperaturedevelopmentovertimeoftheheatcarrierfluidcirculatingthroughtheboreholeheatexchangerwhenaknownheatingorcoolingloadisimposed.Byevaluatingtheincreasingfluidtemperatureversustime,informationaboutthethermalpropertiesinandaroundtheboreholeisobtained.Alowthermalconductivityis,e.g.,indicatedbyamorerapidtemperatureresponse.Theresponsealsogivesinformationaboutthetemperaturedifferencebetweentheheatcarrierfluidandthesurroundinggroundcausedbytheheattransfer,i.e.,thethermalresistanceoftheboreholeheatexchanger.
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Figure1Thermalresponsetestsetup.
Severalanalyticalandnumericalmethodsareusedfortheevaluationofresponsetesttemperaturedata.Thedifferentmodelsrequiresomewhatdifferentsetsofinputdata.Variousanalyticalmethodsforevaluationofboreholeresponsetestdataarediscussedbelow.
EvaluationMethods
Anumberofmethodshavebeenappliedovertheyearsforthesimulationofboreholeheatexchangerperformance.Bothanalyticalandnumericalmodelshavebeenusedandreportedinseveralpapers,reports,andbooks.Here,thefocusisonmodelsfortheevaluationofthermalresponsetestdatafordetermininggroundthermalconductivityandevaluationoftheefficiencyoftheboreholeheatexchanger.
Thethermalresponsetestmethodisbasedontheso-calledsingleprobemethodfordeterminingthethermalconductivityofsolidmaterialsinalaboratoryenvironment(StalhaneandPyk1931).Initialanalyseswerebasedontheline-sourceapproximation,whichdoesnotconsiderthethermalpropertiesoftheprobematerial.In1954,Blackwellpresentedananalyticalsolutionincludingboththeprobematerialandapossiblecontactresistanceattheprobesurface.Inprinciple,thismethodmakesitpossibletoshortenthemeasurementperiods,especiallyforlargeprobediameters.Attemptstodetermineboththermalconductivityanddiffusivitysimultaneouslybytakingthecontactresistanceintoaccountwerenotsuccessful(Blackwell1954;
Becketal.1956).Thedeterminationofthethermaldiffusivitywasfoundtobeverysensitivetothecontactresistance.Sundberg(1988)developedadetailedFEMmodeloftheprobeinordertoshortenthemeasurementperiod.Hefoundthatboththermalconductivityanddiffusivitywereheavilyinfluencedduringtheinitialtimeperiodbysmallchangesintheprobeproperties.
Analyticalmodels,suchastheline-sourceandcylinder-sourcetheories,requireseveralsimplifyingassumptionsregardingthegeometryoftheboreholeandheatexchangerpipes.Forthepurposeofthethermalresponsetestevaluation,theheatflowtoorfromtheboreholemayberepresentedasaninfinitelylongheatsourceorsinkinthegroundwithnegligibleinfluenceofheatflowsinadirectionalongtheboreholeaxis.Inthegroundoutsidetheboreholeitiscommonpracticetoassumethatthethermalprocessdependsonlyontheradialdistancefromtheboreholeaxis.Theone-ortwo-dimensionalheatflowprocessfromthecirculatingfluidtotheboreholewallisassumedtoberepresentedbyathermalresistancethatcharacterizesthetemperaturelossbetweenheatcarrierfluidandboreholewall.Somemodelsalsoincludethethermalmassofthematerialsintheborehole.
IngersollandPlass(1948)appliedtheline-sourcemodeltothedesignofgroundloopheatexchangers.Mogensen(1983)proposedtouseaboreholesimilartotheprobetoestimatethegroundthermalconductivityfromanexperimentalfieldtest.ThismethodisnowcommonlyusedforthermalresponsetestevaluationinEurope.
ThecylindersourceapproachmodelsthegroundloopheatexchangerasacylinderbyintroducinganequivalentdiametertorepresentthetwopipesofasingleU-pipeheatexchangerasasinglecoaxialpipe.CarslawandJaeger(1959)developedanalyticalsolutionswithvaryingboundaryconditionsforregionsboundedbycylindergeometry.DeermanandKavanaugh(1991)andKavanaughandRafferty(1997)describetheuseofthecylinder-sourcemodelindesigninggroundloopheatexchangers.Theeffectivethermalconductivity(anddiffusivity)ofthegroundformationiscomputedbyreversingtheprocessusedtocalculatethelengthofthegroundloopheatexchanger.Basedonashort-termin-situtest,themeasuredeffectivethermalresistanceofthegroundofadailyheatpulseisfittedtoavaluecomputedfromadimensionlesscylinder-sourcefunctionbyvaryingthethermalconductivityanddiffusivityoftheground.
Numericalmodelscanbedesignedtohandledetailedrepresentationsoftheboreholegeometryandthermalpropertiesofthefluid,pipe,boreholefilling,andground,aswellasvaryingheattransferrates.Themoreextensivesetofrequiredinputdataoftenmakethesemodelsmoredifficultandtime-consumingtousethantheanalyticalmethods,whichsometimesmaybeimplementedassimplespreadsheetapplications.
Berberichetal.(1994)describearesponsetesttypeofmeasurementingroundwater-filledductsinwater-saturatedclaystonewheretemperaturesensorswereplacedalongtheboreholewall.Themeasureddatawereanalyzedwithbothananalyticalline-sourcemodelandanumericaltwo-dimensionalfinitedifferencemodelusingparameterestimationwithgroundthermalconductivityandvolumetricheatcapacityasvariables.Thenumericalmodelcalculatestheheatflowsinboththeverticalandtheradialdirectionsforaboreholeoffinitelength.Theresultsfromthenumericalanalysesresultedin5%lowerthermalconductivityvaluesthantheanalyticalresults.Berberichetal.(1994)attributethisdifferencetoendeffectsofthefinite-lengthborehole,i.e.,increasedheattransfernearthetopandbottomoftheborehole.
ShonderandBeck(1999)developedaparameter-estimation-basedmethod,whichisusedincombinationwithaone-dimensionalnumericalmodel.Thismodelissimilartoacylinder-sourcerepresentationinthatitrepresentsthetwopipesoftheU-tubeasasinglecylinder.However,itaddstwomorefeatures-athinfilmthataddsaresistancewithoutheatcapacityandalayerofgrout,whichmayhaveathermalconductivityandheatcapacitydifferentfromthesurroundingsoil.Thismodelaccommodatestime-varyingheatinput.
Atransienttwo-dimensionalnumericalfinitevolumemodelinpolarcoordinatesforresponsetestevaluationisreportedinAustin(1998)andAustinetal.(2000).ThegeometryofthecircularU-tubepipesisapproximatedby"
pie-sectors,"
overwhichaconstantfluxisassumed.TheconvectionresistanceduetotheheattransferfluidflowinsidetheU-tubesisaccountedforbyusingfluidpropertiesthroughanadjustmentontheconductivityofthepipewallmaterial.AthoroughdescriptionofthenumericalmodelisfoundinYavuzturketal.(1999).Themodelhassincebeenimprovedbyintroducingaboundary-fittedgridsystemthatismoreflexibleandbetterrepresentstheU-tubepipegeometry(Spitleretal.2000).Themodeliscomparedwiththeline-sourceandcylinder-sourcemodelsinAustin(1998).
TABLE1
SummaryofRequiredInputtotheFourAnalysisModels
Smith(1999)