This is a repository copy of A three-dimensional numerical model of borehole heat exchanger heat transfer and fluid flow.

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1 This is a repository copy of A three-dimensional numerical model of borehole heat exchanger heat transfer and fluid flow. White Rose Research Online URL for this paper: Version: Accepted Version Article: Rees, SJ orcid.org/ and He, M (2013) A three-dimensional numerical model of borehole heat exchanger heat transfer and fluid flow. Geothermics, 46. pp ISSN Elsevier Ltd. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by ing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk

2 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics A"three'dimensional"numerical"model"of"borehole"heat" exchanger"heat"transfer"and"fluid"flow" Simon J. Rees a, Miaomiao He a a)+institute+of+energy+and+sustainable+development,+de+montfort+university,+leicester,+uk.+ CorrespondingAuthor: SimonReessjrees@dmu.ac.uk InstituteofEnergyandSustainableDevelopment,DeMontfortUniversity,TheGateway, Leicester,LE19BH,UK.+44(0) Keywords:boreholeheatexchanger;numericalmodel,conduction,fluidflow " " 1

3 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics A"three'dimensional"numerical"model"of"borehole"heat" exchanger"heat"transfer"and"fluid"flow" " Abstract" CommonapproachestothesimulationofBoreholeHeatExchangersassumeheat transferwithinthecirculatingfluidandgrouttobeinaquasi5steadystateandignore axialconductionheattransfer.thispaperpresentsanumericalmodelthatisthree5 dimensional,includesexplicitrepresentationsofthecirculatingfluidandotherborehole components,andsoallowscalculationofdynamicbehavioursovershortandlong timescales.themodelisformulatedusingafinitevolumeapproachusingmulti5block meshestorepresenttheground,pipes,fluidandgroutinageometricallycorrect manner.validationandverificationexercisesarepresentedthatusebothshort timescaledatatoidentifytransportdelayeffects,andlongtimescaledatatoexaminethe modellingofseasonalheattransferandshowthemodeliscapableofpredictingoutlet temperaturesandheattransferratesaccurately.atlongtimescalesboreholeheat transferseemswellcharacterizedbythemeanfluidandboreholewalltemperatureif thefluidcirculatingvelocityisreasonablyhighbutatlowerflowratesthisisnotthe case.studyoftheshorttimescaledynamicshasshownthatnonlinearitiesinthe temperatureandheatfluxprofilesarenoticeableoverthewholevelocityrangeof practicalinterest.theimportanceofrepresentingthethermalmassofthegroutandthe dynamicvariationsintemperaturegradientaswellasthefluidtransportwithinthe boreholehasbeenhighlighted.implicationsforsimplifiedmodellingapproachesare alsodiscussed. Nomenclature" C P specificheat(kj/kg.k) D+ diameter(m) E East F cellfaceflux(w/m 2 )+ +h+ convectionheattransfercoefficient(w/m 2.K) L+ length(m) n surfacenormalvector(5) P Point(inmesh) Q heattransferrateperunitlength(w/m) r+ radius(m) R thermalresisitance(m.k/w)+ S surfacearea(m 2 ) t++ time(s)+ T+ temperature( C) v surfacevelocityvector(m/s) 2

4 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics v!+ volumeflowrate(m 3 /s) V volume(m 3 ) Greek+Symbols α+ thermaldiffusivity(m 2 /s)+ ξ+ localcoordinates(m) ρ+ density(kg/m 3 ) Γ thermalconductivity(w/m.k) τ" non#dimensional+time+(#)+ subscripts+ i cellindex e+ east+ w+ west+ n+ north+ s+ south+ t+ top+ b+ bottom+ superscripts+ +C+ convection D diffusion n+ convectioncorrelationexponent Numbers+ Nu Nusseltnumber Re Reynoldsnumber Pr Prantlnumber Acronyms+ BHE BoreholeHeatExchanger BiCGSTAB Bi5conjugategradientstabilizedsolver CARM CapacitanceResistanceModel CFD ComputationalFluidDynamics DST DuctStoragemodel EWS Erdwärmesonden GEMS3D GeneralElypticalMulit5blockSolver3D HVAC HeatingVentilationandAir5conditioning RMSE RootMeanSquareError RTD Residencetimedistribution TRCM ThermalResistanceCapacitanceModel TRNSYS TransientSystemsimulationtool TRT ThermalResponseTesting " 1."Introduction" Singlepairsofpipesformedina U loopandgroutedintoverticalboreholesare probablythecommonestformofgroundheatexchangerfoundingroundsourceheat Pumpsystems,andareknownasBoreholeHeatExchangers(BHEs).Thecomponentsof 3

5 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics suchaheatexchangerareillustratedinfig.1.bhesofthistypearenotonlyusedin buildingheatingandcoolingsystemsbutinlargethermalstorageschemesalso.the primaryphysicalphenomenaofinterestinthestudyofheatexchangerperformanceare thedynamicconductioninthepipe,groutandsurroundinggroundaswellasconvection atthepipewall.inreality,theheattransferinthesurroundinggroundmaybeenhanced bygroundwaterflowthroughporousandpossiblyfracturedrock.ifinteractionwiththe heat5pumpsystemanditscontrolsistobeconsideredthenitbecomesnecessaryto considerthephysicsofvariableflowanddiffusionofheatinthecirculatingfluid. Itisnotcommon,noralwaysnecessary,toincluderepresentationofallthese physicalprocessesinbhemodels.thismaybepartlyapracticalconsiderationofwhat physicalparameterdataareavailable(ormeasurable)aswellasthelevelofdetail requiredtomeetthemodellingobjective.modelsofbheshavethreeprinciple applicationsnamely(i)designofbhes determiningtherequiredboreholedepth, numberofboreholesetc.;(ii)analysisofin5situgroundthermalresponsetest(trt) data;and(iii)integratedbuildingandsystemsimulationi.e.withthemodelcoupledto HVACandbuildingthermalmodelstostudyoverallsystemperformance. Anumberofanalytical,numericalandhybridmodelsexistandthefeaturesofa numberarereviewedhere.thesemodelsdiffermostlyaccordingtowhetherthey considerthreespatialdimensions,multipleboreholes,groundwaterconvectionand buoyancyeffects,heterogeneousthermalproperties,groutandpipethermalcapacity andexplicitrepresentationoftransportofheatbythecirculatingfluid. Thequestionofdimensionalityandtowhatlevelofdetailthegout,pipeandfluid componentsarerepresentedbearsarelationshiptoboththetimescalesandlength scalesthathavetobeconsidered.atshorttimescales,beingabletoresolvethedynamic changesintemperaturegradientwithintheboreholeisessentialtodeterminingfluid temperatures.atlongtimescales(anumberofyears),itisnecessarytoconsider conductioninthesurroundinggroundintheaxial(third)dimension.thisisbecause particularlyifanarrayofboreholesisconsidered conductionbelowtheborehole,and towardsthegroundsurfacefurtherfromtheborehole,becomemoresignificantaftera numberofyearsofoperation.thiswasdemonstratedintheearlyworkofeskilson (1987)whoappliedanaxial5radial2Dnumericalmodeltocaptureaxialconduction effectsandananalyticalsuperpositionmethodtoconsiderinteractionbetween 4

6 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics neighbouringboreholesinthehorizontaldirection.theimportanceofaxialheat transferishasbeencommenteduponbymarcotteet+al.+(2010)andhasalsobeen recognisedinmorerecentapplicationofanalyticalfinitelinesourcemodels(zenget+al. 2002,Molina5Giraldoet+al.2011)althoughinthesemodelsinteractionbetween neighbouringboreholesisneglected. Ifonlymediumandlongtimescalesareconsidered astheyareinthe g5 function responsefactormodelsofeskilson(1987)andhellström(1991) thenthe boreholecanbeconsideredasingleresistiveelement.thiscanbearguedtobesufficient forapplicationsofthemodelfordesignpurposeswhereitismoreimportanttoconsider long5termresponses,particularlywhereannualheatingandcoolingdemandsarenot wellbalanced.eskilsonstatedthatg5functiondataderivedusinghisapproachshould onlybeappliedattimescalessuchthatt+>+5rb 2 /α.thislimitmayamounttoanumberof days.ifshortertimescalesaretobeconsidered astheyneedtobewheresystem simulationistheobjective itmaybesufficienttoconsiderheattransferintwo5 dimensions,andpossiblyonlytheradialdirection.atshortertimescales,behaviouris stronglydependentonthedynamicbehaviouroftheboreholepipe,groutandfluid components.hybridapproacheswherebydifferentmodelsareapplieddependingon timescalecanbedevisedtotreatthewholerangeoftimescales.forexample,aone5 dimensionalnumericalmodelwasaddedtotheresponsefactorapproachinthedst model(hellström1991)toallowsimulationinthetrnsyssimulationenvironment (SEL,1997). YavuzturkandSpitler(1999)usedatwodimensionalnumericalmodelofa borehole(yavuzturket+al.1999)tocalculateshorttimescaleresponsesand subsequentlyextendg5functionresponsedatatoallowshortsimulationtimesteps(as shortasafewminutes)tobesimulated.severalstudieshavebeencarriedoutusingthis shorttimestepg5function model(gentryetal.,2006;sankaranarayanan,2005)andthe modelhasbeenimplementedintheenergyplussimulationenvironment(fisheret+al,. 2006).Yavuzturk snumericalmodel(1999)representedthepipesas piesector shapes anddidnotincludeanexplicitrepresentationofthecirculatingfluid.young(2004) soughttoaddressthisbyapplyinga buriedcable analogytoincludetheeffectofthe fluid sthermalcapacity.xuandspitler(2006)soughttosimplifythederivationofshort 5

7 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics timescaleresponsedatafurtherbydevelopinganequivalentone5dimensionalnumerical modelthatincludesthethermalmassofthefluid. Asvariationinfluidtemperatureaccordingtodepthcannotbeconsidered explicitlyintwo5dimensionalmodelssuchasthosediscussedabove,someassumption hastobemadeaboutthefluidtemperaturesassociatedwiththetwopipesandtheir relationshiptotheinletandoutlettemperatures.forexample,bothpipescouldbe assumedtobeatatemperatureequivalenttotheaverageoftheinletandoutlet temperatures.analternativeistoassumeonepipetemperatureisthesameasthatof theinletandtheotherisattheoutlettemperature.theseassumptionscanbeavoidedin athree5dimensionalnumericalmodelwheretemperaturevariationwithdepthcanbe consideredexplicitly. Insomegroundsourceheatpumpsystems,dependingonthedynamicload profile,theminimumandmaximumoperatingfluidtemperaturesareadominant considerationinthedesignandcontrolofthesystem.theseextremetemperaturescan occuronveryshorttimescales,forexample,wherecapacitycontrolisbyswitchingthe heatpumpcyclicallyonandoff.extremetemperaturesmaybeexhibitedontimescales ofafewhoursinintermittentlyoccupiedbuildingssuchaschurches.theimportanceof suchshorttimescaleeffects,andtheirimpactontheoperationofthecontrolsystem, weredemonstratedinasimulationstudyofahybriddomesticsystembykummertand Bernier(2008)andmeasurementsofalargernon5residentialsystembyNaikerand Rees(2011).Itisapparentthatatsuchshorttimescales,orgenerallyathigher frequenciesofinlettemperaturevariation,theresponseatthebheoutletisfarfrom instantaneousandpeaksinoutlettemperaturearebothdampedanddelayed.thisis alsoindicatedintheexperimentaldatapresentedlaterinthispaper.accurately predictingpeakorminimumtemperaturesisthereforeanimportantmodellingissuein someimportantapplications. Dampingoftheinlettemperaturefluctuationscanbeaccountedforpartlybythe observationthat,particularlyinnon5residentialsystemswithlargerdiameter distributionpipes,thetotalthermalcapacityofthefluidintheu5tubesand interconnectingpipesisrelativelylarge probablyofthesameorderasthatofthegrout inalltheboreholes.thephysicalprocessthathasafurthereffectontheshorttimescale responseisthedynamictransportofthecirculatingfluidandthermaldiffusionalongthe 6

8 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics pipes.thiscouldbeexpectedtobeimportantatshorttimescalesifoneconsidersthat thenominaltransittimeofthefluidtravellingthroughtheu5tubecouldbeoftheorder ofafewminuteswithtypicalbhedepthsandpipevelocities.variationsininlet temperaturearediffusedbecausefluiddoesnotcirculateina plug withuniform velocitybutfluidatthecentreofthepipetravelsathighervelocitythanthefluidnear thepipewall.hence,fluidattheoutletwillgenerallyhavebeenmixedwithfluidinthe pipethatenteredtheheatexchangeratanearliertimeandprobablyatadifferent temperature.boththethermalmassofthefluidandthediffusivetransportprocess meanthatswingsininlettemperaturetendtobedamped.sucheffectscanalsobe expectedtobemorenoticeableinsystemswithvariableflow.insuchsystems,the transportdelaycouldbeseveralminuteswhentheflowisreducedtominimumlevels duringpartloadconditions. AnumberofBHEmodelsincludeanexplicitrepresentationoffluidcirculation butstopshortofcompletethree5dimensionaldiscretization.somemodelsmakeother simplificationsinordertolimitthetotalnumberofequationstobesolved.forexample, WetterandHuber sewsmodel(1997)isdiscretizedverticallysothataseriesoffluid nodesareincludedbuttheheattransferinthegroutisrepresentedbyasinglelumped capacitanceandconductionisassumedradialonly.oppeltet+al.(2010)havesoughtto addressthislimitationoftheewsmodelbydividingthegroutintosectorssothateach verticallayerofadoubleu5tubewasrepresentedbyfivelumpedthermalcapacitances. DeCarliet+al.(2010)developedaso5calledCapacityResistanceModel(CaRM)and discretizedtheborehole includingthecirculatingfluid intoseveralslicesalongits depthwitheachslicealsodiscretizedintheradialdirection.theyalsoproposed modifyingtheouterboundaryconditionstoallowwholeboreholearraystobe modelled. Baueret+al.(2011)usedasimplifiedrepresentationoftheboreholecomponents intheformofanetworkofresistancesandcapacitancesinthetrcmmodeland discretizedtheboreholeintheverticaldirectioninasimilarwaytotheewsandcarm models.fluidresponsesandverticaltemperaturegradientscalculatedovershort timescalesusingthismodelcomparedfavourablywiththosefromafullydiscretized finiteelementmodel.thesenetworkorsimplifiedfinitedifferencemodelscouldbe thoughtofasquasi5three5dimensionalandhavetheadvantagethatfluidtransportis 7

9 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics explicitlyrepresentedbutarelimitedinnotbeingabletocalculateaxialheattransfer outsidetheboreholeandhencenotabletorepresentalllongtimescaleeffects. Wherethree5dimensionalnumericalmodelshavebeenapplied,theinteresthas mostlybeeninstudyinglongtimeandspatialscales,forexample,interactioninlarger boreholearrays,heterogeneousgroundpropertiesortheeffectsofgroundwaterflow.in viewofthecomputationaldemandsofsuchmethods,anumberofapproacheshavebeen proposedtoreducethediscretizationoftheboreholecomponents.al5khouryet+al. (2005,2006)developedaspecial1Dheatpipefiniteelementthatconsideredthepipe flowsandconductioninthegroutmaterialusingasingleelement.theboreholefieldsof interestwerediscretizedusingaverticallineofsuchelementscoupledtosurrounding 3Delements.AverysimilarapproachwasdevelopedbyDierschet+al.(2011)to integrateabheinacommercialfiniteelementsoftwarepackageandsimilarly, Signorelliet+al(2007).MottaghyandDijkshoorn(2012)havetakenasimilarapproach exceptthattheboreholeisrepresentedbyafinitedifferencemodelcoupledtothe3d finiteelementsolver.cuiet+al.(2008)usedacommercialfiniteelementsolverbut discretizedthegroutandpipeswitharelativelyfinemeshof3delements.although theirtreatmentofthefluidisnotfullydescribedthepredictionsofheattransferrates andpipewalltemperaturecomparedfavorablywithshorttimescaleexperimental measurements. OneapproachtomodellingBHEwiththeaimofcapturingallthephysicaleffects notedearlier,istouseathreedimensionalnumericalmodelthatdiscretizesthe boreholecomponentsandincludesadiscretedynamicmodelofthecirculatingfluid. Thisistheapproachtakenintheworkreportedhere.Three5dimensionalmodelshave theadvantagethatdynamicfluidtransportalongthepipeloopcanberepresented explicitlyandtemperaturevariationsaccordingtodepthcanbemodelled.inaddition, differentlayersofrockandsoilcanbeexplicitlyrepresentedandclimatedependent boundaryconditionsatthesurfacecanbeapplied.furthermore,heattransferbelowthe boreholearraycanbeexplicitlyconsideredandinitialverticalgroundtemperature gradientscanbeimposed. Three5dimensionalmodelsoffermostgeneralityandpotentiallymostaccurate representationofheattransferbuthavethedisadvantagethatconsiderablecomputing resourcesarerequiredfortransientsimulationoverextendedtimescales.themodel 8

10 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics presentedherehasconsequentlybeenusedtomakegenericstudiesofbhebehaviour andtocalculateresponsefunctiondataoverthefullrangeofshortandlongtimescales. Althoughthemodelhasbeenusedinannualsimulationthiswouldnotbepracticalfor mostusers.themodelhasbeenpresentedinbrieferform,andforparticular applications,inheet+al.(2010)andmorefullyinhe(2012).theintentionhasalsobeen tousethemodelasareferencesothatthelimitationsofsimplertwo5dimensional modelscanbebetterunderstoodandanimprovedmodeldeveloped.thisisreported elsewhere. Inthispaper,wedescribetheunderlyingnumericalmethodandtheapproach takentodiscretizethebheandsurroundingground.themodelhasbeenvalidatedina numberofways.themodel sabilitytoaccuratelycalculatetheconductionaroundthe pipeswithintheborehole,andgenerallytodealwithnon5orthogonalmeshes,isverified withreferencetoanalyticalconductionheattransfersolutions.thelimitationsofthe approachtakentomodelthetransportoffluidalongthepipearealsostudiedwith referencetoananalyticalsolution.morethanoneyearofexperimentaldatahasbeen usedtovalidatetheabilityofthemodeltocalculateseasonalheatbalancesandhigh frequencydatatostudythesignificanceofthefluidcirculationonshorttimescale dynamicresponse.laterinthispaper,wepresentanumericalstudyofboreholevertical temperatureandheatfluxprofilesunderarangeoffluidflowconditions. 2."Model"development" Adynamicthree5dimensionalnumericalmodelofBHEshasbeendeveloped,builtupon afinitevolumesolverknownasgems3d(generalellipticalmulti5blocksolver3d) whichisanin5housecodeimplementedinfortran90.thegems3dsolverhasbeen usedtomodelgroundheatexchangerproblemsinanumberofearlierprojects(denget+ al.,2005;reeset+al.,2002)andfurtherdetailsaregiveninhe(2012).modelverification andvalidationofthissolverhasbeenreportedelsewherebyyoung(2004)andfanand Rees(2009).ThemodelhassimilaritieswiththefinitevolumemodelpresentedbyLi andzheng(2009)exceptthatthemeshextendstoincludethepipeandfluidratherthan stoppingattheoutsideofthepipe. 2.1"The"numerical"method" 9

11 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics TheFiniteVolumeMethodhasbeenusedtodiscretizetheintegralformofthe convection5diffusiontemperatureequation.theapproachtodealingwithnon5 orthogonalcellgeometrieshasbeentodiscretizetheequationinphysicalspaceusingan approachsimilartothatdescribedbyferzigerandperic(2002).theprimaryvariables aredefinedatthehexahedralcellcentroidsonablock5structuredmesh.theintegral formoftheconvection5diffusionequationsolvedhere(leavingasidesourceterms)is:!!"!!!!!"!!!!!!!!!!!!!!!!!!! (1)!! Weexplainthedescretizationoftheconvectiveanddiffusivefluxtermsseparatelyas follows.theadvectionfluxtermindiscreteformisapproximatedbythesumofthe convection(advection)fluxesthrougheachcellfacesuchthat,!!!!!!!!!!!!! (2)! where,i=n,s,w,e,t,bforahexahedralcell.assumingthefluxcanberepresentedbythe valuesoftemperatureandvelocityatthecellfacecentroid,wecanusethefaceareaand normalvectortomakethesecondorderapproximation,!!!!!!!!!!!!!!!! (3) Inthisparticularimplementationoftheheatexchangermodeltheconvection (advection)fluxtermisonlyusedforsimulatingthefluidflowsinthepipesandonlythe velocityinthedownwardandupwarddirectionsintheassociatedcellsisnon5zero. Again,assumingthatthevalueofthetemperatureToveraparticularfaceiswell representedbythevalueatthefacecentroid,thediffusionheatfluxcanbe approximatedas!!!!!!!!!!!!!!!!!!!!!!!! (4) Thisrequiresadiscretemethodforfindingthegradientofthetemperature(!!) ateachcellfaceusingthecellcentroidvalues.atypicalnon5orthogonalcell,withlocal coordinatesattheeastcellface,isillustratedinfig.2.thecoordinatenisdefinedinthe 10

12 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics directionnormaltothefaceatitscentroids,andthecoordinateξisdefinedontheline betweenneighboringcentroidswhichpassesthroughthefaceatpointe. Inordertocalculatethegradientofthevariableatthecellface,thevaluesofthe variableatthecellcentroidsareusedastheyaretheprimaryvariables.thegradientis calculatedusingthevaluestpandteatneighboringcentroidsandthedistancebetween thesepoints,lp,e(infig.2attheeastface!!!!!!!!!!"!"!!! )butthisisonlysecond5 orderaccurateifthegridisorthogonal.inordertopreservesecond5orderaccuracythe calculationofthegradientalongthenormaltothefaceatthecentroidneedstobemade usingthevaluesatpointsp ande.however,thevaluesofthetemperatureatthese pointsarenotcalculatedexplicitlyandhavetobeinterpolatedfromthecellcentroid values.consequentlya deferredcorrection approachisusedtocalculatingthefluxas follows:!!!!!!!!!"!"!!!!!!!!!"!"!!!"!"!!!!"# (5)"" Duringtheiterativesolutionprocess,thetermsinthesquarebracketsare calculatedfromthepreviousestimatesofthevariables.whenthesolutionisconverged thefirstandthethirdtermscanceleachothertoleavethetermthatonlyusesthe gradientalongthefacenormal.centraldifferencingisusedtoestimatethegradients:!"!!!!!!!! and!"!!!!!!!"!"!!!!!!!!!!!!!!!!! (6)"" ValuesatthelocationsP ande areinterpolatedfromthecellcentroidvalues usingthegradientofthevariablewhich,inturn,canbecalculatedfromthefacecentroid valuesbyapplyinggausstheorem(ferzigerandperic,2002).temporaldiscretization canbefirstorsecondorderbackwardsimplicitusingamethodthatallowsforvariable timesteps(singh and Bhadauria, 2009). Thesetsofalgebraicequationsarisingfromthediscretizationonthemulti5block mesharesolvedusinganiterativemethodbasedonthestronglyimplicitprocedure (Stone,1968)adaptedtoallowcommunicationofdataacrossblockboundariesduring theiterativeprocedureandhasbeenfoundtobeveryrobust.abicgstabsolverisalso availableforstronglyconvectiveproblems.usingablock5structuredapproachalso 11

13 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics allowsadegreeofparallelprocessingbythesolvingtheequationsofeachblockwith separatethreads. 2.2"Mesh"generation" Theboreholeheatexchangergeometryhasbeendiscretizedusingathree5dimensional multi5blockboundaryfittedstructuredmeshandthishasbeendefinedusinganin5 houseutility(rees,2009)thatusesatwo5dimensionaldefinitionoftheborehole componentsandextrudesthistoforma3dmeshsuchasthatshowninfig.3.individual blocksdefinethepipesandtwoblocksareusedtodefinethegroutmaterialwithinthe borehole.multipleblocksmaybeusedtodefinethesurroundinggrounddependingon thefarfieldboundaryshapeandalso byrepeatingsimilarboreholeblock arrangements adjacentboreholes.inthispaperonlyasingleboreholeisstudiedanda semi5circularfarfieldboundaryissufficient.(applicationswithmulti5boreholemeshes aredescribedinhe,2012).coarseandfineboreholemeshesareillustratedinfig.4. ThefluidcirculatingintheU5tubeisrepresentedasasinglelayerofcellsinthe meshadjacenttotheinnerpipesurfaces.thisisallthatisrequiredtoimplementafully coupledone5dimensionalmodelofthefluidtransport.thethermalcapacityofthese cellsisadjustedsothatthefluidmassofthewholepipeistakenintoconsideration.the inletboundaryconditionisdefinedbythevelocityandtemperatureatthecellfacesat thetopofthepipeandthefluidvelocityisimposedineachcellalongthelengthofthe pipesaccordingtothetimevaryingmassflowrate.thisapproachamountstoaone dimensionalrepresentationofthefluidflow.thelimitationsofthisapproximationare discussedlater. Heattransferbyconductionwithinthecirculatingfluidisnotsignificant comparedtotheconvectionprocesses.theconvectiveheattransferbetweenthefluid andthepipewallismodeledaccordingtothedittus5boelterequation,!"!!!023!"!!!!"! (7) wheren=0.4forheatingand0.3forcooling.theconvectioncoefficientcanthenbe foundaccordingto,!!!!023!"!!!!"!!! (8)! 12

14 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Theconductanceofthefluidcellsisaccordinglymodifiedtoachievethecorrect relationshipbetweenfluidandpipewalltemperaturesateachtimestepaccordingtothe massflowrate. 3."Model"validation" Theabilityofthemodeltocalculatetransientheattransferratesoverbothshortand longtimescaleshasbeenvalidatedbyacombinationofanalyticalandexperimental analysis.theaccuracyofthefluidtransportmodelhasbeenevaluatedbyreferenceto analyticalsolutionsforadiabaticpipeflow.calculationofconductionwithinthe boreholehasbeenverifiedbyreferencetoanalyticalsolutionsofthesteady5state boreholethermalresistance.finally,experimentaldatahasbeenusedtovalidatethe model spredictionsofheattransferratesonbothshortandlongtimescales. " 3.1"Fluid"transport" Therepresentationofthecirculatingfluidinthismodel,wherebythefluidenteringeach controlvolumeistransportedatthetemperatureupstream,canbeconsideredsimilarto acompartments5in5seriesmodelofpipeflow(wenandfan,1975).fluidtransport modelsofthistypehavebeenwidelyusedinprocessengineeringandtheir characteristicsarewellknown.hanbyet+al.(2002)analysedthistypeofmodelboth withandwithoutconvectiveheattransferandevaluatedafinite5difference discretizationbymakingcomparisonswithanadiabaticanalyticalsolutionbasedon thatofbosworth(1949).thebhemodeloffluidtransporthasbeenassessedina similarmanner. Thetransportpropertiesofapipe(beitheatorachemicalspeciesthatis transported)canbethoughtofintermsofresidencetimedistribution(rtd).thertd isconsideredasthefractionoffluid,whichundergoesastepchangeattheinlet,appears intheoutgoingfluidattimet,anditisrepresentedbythefunctionf(t),illustratedinaf5 Diagram.Theanalysisissimplifiedbyusingdimensionlesstimegivenby, v!t τ = "" " " " " (9)" V TheactualshapeoftheF5Diagramdependsprimarilyonthevelocityprofile.Ingeneral, thefaster5movingfluidnearthepipecentrelinewillarriveattheendofthepipemore 13

15 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics quicklythanthebulkofthefluid.atthesametime,fluidnearthepipewalltravelsata velocitylowerthanthemean.thecirculatingfluidundergoesadiffusionprocesssothat stepchangesininletconditionappearsmoothedattheoutlet.themodel sabilityto reproducethisphenomenoncanbeevaluatedbycomparingthetransienttemperature responseattheoutlettoastepchangeattheinletwiththeanalyticallycalculatedrtd. TheF5DiagrampredictedbythemodelisshowninFig.6.Theseresultshavebeen calculatedusing60cellsalongthelengthofthepipeandusingtwodifferenttemporal discretizationschemes.thesearefirstorderandsecondorderbackwardsimplicit schemes(singh and Bhadauria, 2009).Thesolutionsarecomparedwiththeanalytical solution(hanbyet+al.,+2002;bosworth,1949).thefirstorderschemeslightlyover5 predictsthediffusionoftheflowandthesecondorderschemelessso. Hanbyet+al.(2002)foundthatthepredictedRTDhadsomedependenceonthe numberofcells(compartments)usedtorepresentthepipe.thishasalsobeenfound withthisbhemodel.thiseffectcanbeevaluatedbycalculationofthedifferences betweenthepredictedtemperaturesandtheanalyticalf5function.therootmean SquareError(RMSE)fortwodifferenttemporaldiscretizationschemesovertherange 0.8<τ<1.5fordifferentnumberofcellsisshowninFig.6. Usingthefirstorderbackwardsimplicitschemetheerrordropsquicklyfrom20 to60cells,andapproaches0.08formorethan60cells.usingthesecondorder approximationtheerrorreachesaminimumwith47cells.theseresultsareconsistent withthefindingsofhanbyet+al.(2002).inpracticalcalculations,60 80cellshavebeen usedtodiscretizetheboreholeintheverticaldirectionandthefirstorderschemehas beenadopted(beingmorerobust)sothaterrorsofthistypeintherange couldbeexpected. Itshouldbenotedthatthemodelalwaystendstoover5predictthedegreeof diffusion.althoughtheerrorisgreaterthanonewouldlikeitshouldbenotedthatthis cannotbeaddressedtoanygreatdegreebyhigherorderdifferencingschemes.the modelaccuracyislimitedprimarilybythefactthatthetwo5dimensionalflowinthepipe isassumedtobeone5dimensional.thismightbeaddressedbycfdcalculationofthe pipefluidflow(i.e.solutionofthenavier5stokesequations)butthiswouldadd unreasonablecomputationalburden.someoverestimationofthediffusionprocessis acceptableasfurtherdiffusionoccursinrealboreholearraysbyvirtueofthehorizontal 14

16 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics headerpipes.wesuggestthatothermodelsthatuseaone5dimensionaldiscretizationof thefluid(e.g.1dfiniteelementsorfinitedifferenceapproximations)arelikelytoshow similarlevelsofaccuracyandpossiblymeshdependence. 3.2"Borehole"conduction" Thereisnoexactanalyticalsolutionforthree5dimensionalheattransferinaborehole geometrythatcanbeappliedtotrytovalidateanumericalmodel.however,ifonly conductionintwodimensionsisconsidered,calculationofsteady5stateborehole thermalresistancecanbeusedasametrictovalidatethemodel.young(2004)carried outanextensivestudytocomparethesteadystateboreholeresistancescalculatedusing anumberofmodels,includingpaul smethod(1996),theguando Neal sapproximate diametermethod(1998)andthemultipolemethod(bennetet+al.,1987).theseresults werecomparedwithnumericalresultsusinga2dversionofthegems3dsolver.the multipolemethodcanberegardedasareferencemethodandaccuratetowithin machineprecision.asimilarcomparisonofnumericalmodelresultswithvaluesof boreholethermalresistancecalculatedusingthemultipolemethodispresentedbelow. Thiscomparisonisausefulvalidationexerciseinthatitteststheabilityofthenumerical methodtodealwiththecurvedpipeandgroutboundarygeometry. AssumingtheheattransferofBHEsisinsteady5state,thetotalamountofheat transferrateperunitlengthbetweenthefluidandthegroundcanbeexpressedas:!!!!!!!!! (10) Theboreholeresistancedefinedhereincludestheconvectiveresistancebetween thefluidandtheinnersideofthepipes,theconductiveresistanceofthepipes,andthe conductiveresistanceofthegrout.valuesofboreholethermalresistancecanbefound fromthenumericalmodelbymakingasteady5statecalculationoftheheatfluxacross theboreholewallandboreholetemperatureforagivenfluidandfarfieldtemperature difference.twosinglebheswithdifferentboreholediametersandtwodifferentgrout typeshavebeenstudied.theboreholedimensionsandthermalpropertiesareshownin Table1. Inprinciple,anumericalmethodshouldproduceresultsapproachingthe analyticalvaluesofconductanceasthemeshisrefined.accordingly,toverifythemodel, 15

17 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics boreholethermalresistancehasbeencalculatedwitharangeofmeshdensities.the calculatedboreholethermalresistancesareshownintable2formesheswithnumbers ofcellsinatwo5dimensionalplaneofbetween656and40,448cells(seefig.4).the modelcanbeseentobecapableofmatchingwiththeanalyticalvaluesfoundusingthe multipolemethod(bennetet+al.,1987)towithin0.1%.variationofmeshdensityfrom 656to40,448cellsshowsverysmallchangesintermsofboreholethermalresistance.In practicalcalculations,usingcoarsermeshestoreducecomputationtimeswouldbe reasonable. " 3.2"Experimental"validation" DataobtainedfromanexperimentalfacilityatOklahomaStateUniversityhavebeen usedtovalidatethenumericalcalculationsofoverallheattransferratesandoutlet temperatureresponseovershortandlongtimescales.theexperimentalfacilitywas designedandconstructedtostudyhybridgroundsourceheatpumpsystems(hern, 2004)andthemeasureddatafromthegroundloophavebeenusedhere.Thesamedata sethasbeenusedinagroundheatexchangerinter5modelcomparativestudybyspitler et+al.(2009).theboreholedimensionsandpropertiesareshownintable3.theground thermalconductivityvaluewastakenasthemeanofthreevaluesdeterminedby ThermalResponseTests. IntheseexperimentstheinletandoutletfluidtemperaturesoftheBHEswere measuredat1minuteintervalsover18months.thethreeboreholesarespacedfar enoughapartsothatnothermalinteractioncouldbeexpectedduringthisinitial operatingperiodandsothedatacanbeinterpretedasrepresentingthebehaviorofa singleborehole.inthefollowingvalidationexercisesthemeasuredinlettemperature andmassflowratehavebeenusedasmodelboundaryconditionsandoutlet temperaturesandheattransferrateshavebeencomparedwithvaluespredictedbythe model "Validation"over"short"timescales" Inordertostudythepredictionsofshorttimescaleresponseandexaminethe significanceoffluidtransporteffects,minutelydataforthefirstmonthoftheexperiment havebeencomparedwithpredictedvalues.duringtheexperimentstheheatpumpwas 16

18 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics switchedonandoffintermittentlyandthecirculatingpumprancontinuously.data showingtwocyclesofoperationinthe15 th dayofoperation(march15,2005)are showninfig.s7and8.whentheheatpumpswitcheson(time15:11)theinlet temperaturefallsquicklybyapproximately3k.theexperimentalresultsshowthat thereisnoresponseobservableattheoutletuntil5minuteslater.laterintheoperating cycletheoutlettemperaturefallsatasimilarratetothatoftheinlet.attheendofthe operatingcycle(15:50)theinlettemperatureshowsasharpincreaseandasimilardelay intheoutlettemperatureresponsecanbeobserved.thedelayintheresponseisofthe samemagnitudeasthenominaltransittimeoftheu5tubewhich,attheflowratein question,is4.4minutes.verysimilartrendscanbeseeninthesubsequentcycleof operationshowninfig.8. Theoutlettemperaturepredictedbythenumericalmodelcanbeseento demonstrateverysimilardelaysinresponseatboththebeginningandtheendofheat pumpoperation.theresponseattheendofheatpumpoperationisslightlymore dampedthanthatshownintheexperimentaldata.duringtheoperatingperiodthe outlettemperaturepredictionfollowstheexperimentaldataclosely.therootmean SquareError(RMSE)forthepredictedoutlettemperaturedatashowninFigs.7and8is 0.324Kand0.320Krespectively. Thesignificanceofmodelingthefluidcirculationhasbeenhighlightedby includingdatafromarelatedtwo5dimensionalmodelinfigs.7and8.thismodelis reportedindetailinhe(2012)andusesthesamenumericalsolverandsimilarmesh densitybutonlyonecellindepth.theoutlettemperatureinthisandother2dmodels necessarilyrespondsinstantly(althoughdampedbythethermalmassofthegrout)to changesininlettemperature.theoutlettemperaturepredictedbythis2dmodelcan, accordingly,beseentorespondtothechangesininlettemperaturewithoutany observabledelay.othermodelsthatdonotexplicitlyrepresentthefluidflowcouldbe expectedtorespondinasimilarmannertothis2dmodel.thesignificanceofthese effectscouldbeexpectedtobegreateratlowervelocitiesandinvariableflowrate systems(heet+al.,2010) "Validation"over"Long"Timescales" 17

19 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Inordertostudythepredictionsofheatrejectionandtemperatureresponseoverlong timescales,hourlydataforthewhole18monthsoftheexperimenthavebeenusedas boundaryconditionstothemodel.thepredictionsofmeanmonthlyheatexchanger outlettemperatureareshownalongwiththerelatedexperimentaldatainfig.9.the RMSEinthemonthlymeanoutlettemperaturepredictionoverthisperiodis0.57K.The differencesaregreatestinthemonthofdecember(1.4k)andthefollowingaugust (1.5K).InothermonthstheerrorismuchlessthantheRMSEvalue. PredictedMonthlynetheatexchangesarecomparedwithvaluesderivedfrom theexperimentaldatainfig.10.inmostmonthsthepredictedvaluesarelessthanthe experimentalvalues.theexceptionstothisareinthemonthsofaprilofthefirstyear andmarchofthesecondyear.thegreatestdiscrepanciesareduringthewinterperiod. Similarcomparisons,foranumberofmodels,werepresentedbySpitleret+al. (2009).Mostofthemodelstestedshowedthemostsignificantdifferencesalsooccurred inthemidwinterperiod.onepossibleexplanationofthistrendofferedbytheauthors wasthatthisperiodcorrespondedwiththeoccasionswhereheatpumpoperationwas moreintermittenti.e.operatingcycleswereshorterandtherewerenoloadsforlonger periods.theyalsonotethatcyclicoperationwasnotwellrepresentedinthehourlydata whenthecycleswerelessthanonehourinduration.wesuggestthatonefurther complicatingfactormaybetheinfluenceofthehorizontalheaderpipesconnectingthe threeboreholes.althoughthesepipesareshortrelativetothetotalpipelengththeyare morelikelytobeinfluencedbyatmosphericconditions.allthemodels,includingtheone presentedhere,ignoreanyinfluenceofheattransferatthegroundsurfaceandall horizontalpipes. 4."Vertical"temperature"and"heat"flux"profiles" Oneofthefeaturesandadvantagesofathreedimensionalmodelsuchasthis,isthat temperatureandheatfluxvariationswithdepthareexplicitlycalculated.wehave, furthertothevalidationexercisesreportedabove,madeanumericalstudyofthe behaviorofabhewithrespecttotemperatureandheatfluxvariationwithinthe boreholeandtheirvariationwithdepth.ofparticularinteresthavebeenthe relationshipsbetweenfluidtemperaturesandtheinletandoutlettemperatures, variationinfluxandtemperatureattheboreholewallandalsothesignificanceofthe 18

20 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics heatfluxesbetweenthetwolegsoftheu5tube.variationswithrespecttofluid circulationrateshavebeenoffurtherinterest. TheseaspectsofBHEbehaviorarepartlyofinterestasdifferentassumptionsare madeabouttheserelationshipsinsimplermodels.again,somedistinctioncanbemade betweenshortandlongtimescalebehaviorandthefollowingdiscussionisseparated accordingly.inthesestudiesasinglebhewithaboreholediameterof150mmanda depthof100mhasbeensimulated.thegroutthermalconductivitywas0.75w/m.kand otherparametersareasshownintable1.thesurroundinggroundtoadiameterof4 metreshasbeenincludedinthesimulationdomain. 4.1"Long"timescale"profiles" ToinvestigatethermalbehaviorofaBHEatlongtimescalesinagenericmannerwehave madeaseriesofcalculationswithsteady5stateboundaryconditions.theseconditions arealsorepresentativeofconditionsatlongtimescaleswherestepresponseisbeing considered.inthesecalculationstheinletandfar5fieldboundarytemperaturesandthe fluidflowratehavebeenfixed.a10kdifferencebetweeninletandfarfieldtemperature hasbeenappliedandcalculationsmadeforfluidmeanvelocitiesintherange m/s.afluidvelocityof0.6m/sisthoughttoberepresentativeofawelldesignedbhe withreasonablepressuredrops.1.0m/sisthoughttobetowardstheupperlimitof whatmaybefoundinpractice.thisvelocityistypicalofbuildingsystemsbutmayresult inunreasonablepressuredrops.flowvelocitiesatthelowerendofthisrangeareof interestastheymayoccurinsystemswithvariablespeedpumping.thislowerlimit shouldstillresultinfullyturbulentflow.evenlowerflowratesthatresultinlaminar flowmayoccurinvariableflowsystems.theflowratesandreynoldsnumbersand convectioncoefficientsthatapplyoverthisvelocityrangeareshownintable4. ThefluidtemperaturevariationsalongthetwolegsoftheU5tubeareshownin Fig.11andthecorrespondingtemperatureattheboreholewallinFig.12.Thisborehole walltemperaturehasbeenfoundfromtheaveragearoundthecircumferenceofthe boreholeatagivendepth.overmuchoftheflowraterangethefluidtemperature profiles(fig.11)areapproximatelylinearandthefluidtemperatureatthebottomofthe boreholeisrepresentativeofthemean.atlowerflowratestheprofileisnoticeablynon5 linear,particularlyatthebottomofthevelocityrange(0.2m/s).atthisflowratethereis 19

21 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics agreatertemperaturechangealongthepipewithdownwardflow(thepipeformingthe inlet)thantheother. Atthehighestflowratethetemperatureattheboreholewall(Fig.12.)variesvery little.atlowerflowrates( m/s)non5linearvariationofboreholetemperature withdepthisnoticeable.thecorrespondingheatfluxesacrosstheboreholewallare showninfig.13.inthistypeofcalculationtheheatfluxesaredrivenbythetemperature differencesbetweenthefluidandtheboreholewall.althoughthereisanapproximately linearvariationoffluidtemperatureathigherflowrates,themeanfluidtemperatureat aparticulardepthisnearlythesame.hence,thefluxalongtheboreholeathigherflow ratesisnearlyconstant.atlowflowratesthisisnotthecaseandanoticeablynon5linear profilecanbeseen. Duetothetemperaturedifferencebetweentheupwardanddownwardfluidin adjacentpipes,someheatistransferreddirectlyfromonepipetotheother,ratherthan theboreholewall.(thisinter5tubeheatfluxissometimesreferredtoasthe short5circuit heatflux.)theinter5tubeheatfluxcouldbeexpectedtobeproportionaltothe temperaturedifferencebetweentheupwardanddownwardflowingfluidsatagiven depthinsteadyconditions.thisisreflectedintheheatfluxesshowninfig.14.thisflux isnaturallyhighestatthetopoftheboreholewherethisfluidtemperaturedifferenceis greatestandlowestatthebottom.fig.11showedthegreatesttemperaturedifferences nearthetopoftheboreholeatthelowestflowrate.theinter5tubefluxis,accordingly, showninfig.14tobehighestatthelowestflowrate.atthehighestflowratethisflux representsapproximately2%ofthetotalboreholeheattransferandatthelowestflow ratethisisincreasedto10%. Someobservationsregardingmodelingpracticecanbemadebasedonthese results.modelsthatdonotincludeanexplicitrepresentationofthecomponentsofthe boreholebutdefinetherelationshipbetweenthefluidtemperatureandboreholewall temperaturebyasinglethermalresistance(e.g.g5functionmodelssuchasthatin EnergyPlus(Fisheret+al.2006),oftenassumethefluidtemperaturetobethemeanof theinletandoutletvaluesandthatconditionsareconstantalongtheborehole.the linearandsymmetricaltemperatureprofileandtheconstantboreholewallconditions shownintheresultsforhigherflowratessuggestthatthisisareasonableassumption. 20

22 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Atlowerflowratesthisisnotthecaseandanothermodelingapproachshouldprobably beconsidered. Inatwo5dimensionalnumericalmodel,forexamplethatofYavuzturket+al. (1999),someassumptionhastobemadeaboutthetemperatureboundarycondition appliedateachpipe.thesamemeantemperatureisfoundwhetherthepipe temperaturesareassumedtobetheinletandoutlettemperatures,orwhethertheyare bothassumedtobeatthemeancondition.however,oneassumptionwillresultinan over5predictionoftheinter5tubefluxandtheotherazerointer5tubeflux.neitherof theseassumptionsseemsappropriateinlightoftheresultsreportedabove.one interpretationofthelinearprofilesshowninsomeresultswouldbethatthe temperatureboundaryconditionsappliedatthepipesshouldcorrespondtothathalf wayalongtheborehole.thiswouldbeequivalentto¼and¾ofthedifferencebetween theinletandoutlettemperaturesrelativetotheinlet.again,thismaynotbeappropriate atlowflowrates. 4.2"Short"timescale"profiles" Ithasalreadybeenshownthat,atshorttimescales,thedynamicresponseofthegrout andthetransportoffluidaroundthepipescanhaveasignificanteffectontheresponse of the heat exchanger. Short timescale variations in temperature and heat flux profile have been studied by making transient calculations and applying a step boundary conditiontotheinlettemperature.inthesecalculationstheinletisincreased10kabove theinitialstateandresultsarepresentedforarangeofnominalcirculatingfluidtransit times up to 3.0. A step response boundary condition is of practical interest because manysystemsarecontrolledbyturningtheheatpumponandoffintermittently. Thetemperatureprofilesvarywithtimebutalsowithfluidcirculatingvelocityor flowrate.resultsforfluidvelocitiesof1.0m/s,0.6m/sand0.2m/sareshowninfigs.15 17respectively.Inallcasesthefluidtemperatureprofileattheearliesttimes(τ=0.3) changes much more, and in a noticeably non5linear manner, in the downward flowing tube. As time progresses(towards τ=3) the fluid temperature profile can be seen to developintoamorelinearandequallydividedform.inthecasewithlowfluidvelocity (Fig.17)thenon5linearnatureofthetemperatureprofilepersists. 21

23 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Thedynamicchangesinfluidtemperatureprofileattheseshorttimescalesarea functionofthesimultaneoustransientconductionthroughthepipeandgroutandalso thetransportoffluidalongthepipe.thecouplingbetweenthefluidandthepipe/grout isalsochangedslightlybythereductioninheattransfercoefficientatlowerflowrates. At normalized times of 0.3 and 0.6 there has been very little increase in the outlet temperature.thisisconsistentwiththeresidencetimedistributionshowninfig.5and thereforesuggeststhatthetransportdelayaccountsformuchoftheseeffectsatthevery shortesttimescales.astimeprogressesdynamicconductionbetweenthefluidandthe boreholebecomesthemoredominanteffect. The borehole heat flux profile for the case with a fluid velocity of 0.6 m/s is shown in Fig.18. It is noticeable that the profiles are less linear than they were in the steady5state(fig 13). The heat fluxes change very little during the initial stages of the stepchange.atthelatesttimeshown(τ=3)thefluxeshavereachedapproximately20% ofthesteady5statevalue.itcanalsobeseenthatthevariationinfluxalongtheborehole wall(ataparticulartime)isnotassignificantasthevariationinfluidtemperatures. Thesetrendscanbeexplainedbyconsideringtheroleofthethermalmassofthe grout between the pipes and the borehole wall. The thermal mass of the grout damps outanddelaystheeffectoftheshorttimescalechangesininletfluidtemperature.the factthatthetemperaturesinonelegoftheu5tubecanbeverydifferentthanthoseinthe otheratshorttimescalesisconsequentlynotreflectedinthetemperaturevariationsat theboreholewallinthesecalculations. It is interesting to consider thebhe overall heat transfer rate. This could be consideredbyreferencetothesummationoftheheatfluxesalongtheboreholewallbut alsobyreferencetotheinlet5outletfluidtemperaturedifferenceataparticulartime.if thefluxesattheboreholewallareconsidered(fig.18)thefluxrisesslowlytowardsthe steady value. In contrast, if the fluid temperatures are considered, as the outlet temperature duetothetransportdelay doesnotincreaseforsomeminutestheheat transferratedefinedbythefluidheatbalancewillappearverylargeand,infact,larger than the steady5state value. At later times the heat transfer rate at the borehole wall approachesthevalueindicatedbythefluidheatbalance onedecreaseswhiletheother increases. 22

24 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Inthediscussionoftheinter5tubeheatfluxesatlongtimescalesitwasnotedthat themagnitudeofthefluxcorrespondedtothedifferencesbetweenthefluidtemperature inthetwolegsoftheu5tubesothatlargerfluxeswerefoundinthecaseswithlower fluid velocities. As the fluid temperature differences are relatively large at the start of thestepchangeininlettemperature(figs.15517)theinter5tubefluxmaybethoughtto be correspondingly large. However, this is not the case. The dynamic change in inter5 tubefluxhasbeenshownforthreefluidvelocitiesinfig.19.itisapparentthat,although the temperature differences between the adjacent fluid is relatively large at the beginning,heathastobeabsorbedbythegroutpresentbetweenthepipesbeforethe inter5tube heat flux increases. This suggests that, if short timescale effects are to be considered properly, it is important that dynamic temperature gradients and the thermalmassofthegroutbetweenthepipesareconsideredinanymodel. 4."Conclusions" Usingamulti5blockmeshtorepresenteachcomponentoftheboreholeheatexchanger in three5dimensions and applying a Finite Volume numerical method it has been possible todevelop a borehole heat exchanger model that can represent both conduction and fluid circulation processes over both short and long timescales. The model s main purpose, in light of the computational demands of three5dimensional numericalcalculations,istoderivestepresponsedataforothermodelingapproaches,to act as a reference model and for use in investigations of three5dimensional and fluid flowphysicalphenomena. The model s ability to model the conduction between the components of the borehole andthegroundhasbeenverifiedbyreferencetothermalresistancescalculatedusingan analytical solution. The model has been shown to be able to reproduce the analytical resultstothreesignificantfigures. Comparisonswithanalyticalresultshavealsobeenusedtoverifythecalculations offluidtransportwithinthepipes.calculationofthepipefluidtransportanddiffusion processes with a high degree of accuracy probably requires the pipe to be fully discretizedandtheapplicationofcfdmethods.however,thiswouldaddconsiderably tothe model s computational demands. This modelusesasinglelayerofcellsto representthefluidandamountstoaone5dimensionaltreatmentoffluidtransport.this 23

25 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics approach,whenresidencetimedistributionsarecompared,showsthatdiffusionisover5 predictedbyapproximately8%.experimentalvalidationshowsthatthemodelisableto capturetheeffectsoffluidtransportanddiffusionsatisfactorily. ModelvalidationexerciseshavebeencarriedoutwithaBHEexperimentaldata setcontainingmeasurementsmadeatrelativelyhighfrequency.shorttimescaleeffects havebeeninvestigatedusingdatacollectedduringcyclicoperationoftheexperimental heat pump. The behavior of the system has been reproduced very well, including the delayed response due to the transient circulation of fluid in the U5tube.Beingableto model such effects is important if interaction between the BHE and the heat pump system controls are to be considered. The model s ability to predict seasonal heat transferhasbeeninvestigatedusinghourlydatafromtheexperimentoveraperiodof 18 months. The model is able to predict average outlet temperature very accurately. Monthly net heat transfer has been calculated and shows good agreement with the experimentaldataandcomparesfavorablywithothermodels. Numerical study of borehole vertical temperature and heat flux profiles has highlighted the importance of modeling variations in heat transfer with depth and the importance of modeling the dynamic interaction between the pipes and the grout at short timescales. A number of phenomena seem to depend on the flow rate or fluid transit time. In particular, at long timescales borehole heat transfer seems well characterized by the mean fluid and borehole wall temperature if the fluid circulating velocityisreasonablyhighbutatlowerflowratesthisisnotthecase.studyoftheshort timescaledynamicshasshownthatnonlinearitiesinthetemperatureandheatfluxesare noticeable over the whole velocity range of practical interest. The importance of representingthethermalmassofthegroutandthedynamicvariationsintemperature gradientaswellasthefluidtransportwithintheboreholehasbeenhighlighted. Acknowledgements" The authors would like to thank J.D. Spitler and S. Hern of the Building, Thermal and Environmental Systems Research Group at Oklahoma State University for provision of theexperimentaldata. 24

26 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics " References" Al5Khoury,R.,Bonnier,P.G.,Brinkgreve,R.B.J.,2005.Efficientfiniteelementformulationfor geothermal heating systems Part I: Steady state. International Journal for Numerical MethodsinEngineering63, Al5Khoury,R.,Bonnier,P.G.,2006.Efficientfiniteelementformulationforgeothermalheating systemspartii:transient.internationaljournalfornumericalmethodsinengineering67, Bauer,D.,Heidemann,W.,Diersch,H.J.G.,2011.Transient3Danalysisofboreholeheatexchanger modelling.geothermics40, Bennet,J.,Claesson,J.,Hellstrom,G.1987.MultipoleMethodtoComputetheConductiveHeat FlowsToandBetweenPipesInaCompositeCylinder.UniversityofLund,Sweden. Bosworth,R.C.L.,1949.Distributionofreactiontimeforturbulentflowincylindricalreactors. PhilosophicalMagazineSeries40, Cui,P.,Yang,H.,Fang,Z.,2008.Numericalanalysisandexperimentalvalidationofheattransfer ingrouhdheatexchangersinalterhantiveoperationmodes.energyinnbuildings40, DeCarli,M.,Tonon,M.,Zarrela,A.,Zecchin,R.,2010.Acomputationalcapacityresistancemodel (CaRM)forverticalground5coupledheatexchangers.RenewableEnergy35, Deng,Z.,Rees,S.J.,Spitler,J.D.,2005.Amodelforannualsimulationofstandingcolumnwell groundheatexchangers.internationaljournalofheating,ventilation,air5conditioning,and RefrigerationResearch11(4), Diersch,H.5J.G.,Bauer,D.,Heidemann,W.,Ruhaak,W.,Schatzl,P.,2011.Finiteelementmodeling ofboreholeheatexchangersystemspart1.fundamentals.computers&geosciences37, Eskilson,P.1987.ThermalAnalysisofHeatExtractionBoreholes.DoctoralThesis,Universityof Lund. Fan,D.,Rees,S.J.,2009.ModelerreportforBESTESTcasesGC10a5GC80c:Gems3Dversion2.07. Leicester:DeMontfortUniversity. Ferzinger,J.H.,Peric,M.,2002.ComputationalMethodsforFluidDynamics,Springer,Berlin. Fisher,D.E.,Rees,S.J.,Padhmanabhan,S.K.,Murugappan,A.,2006.Implementationand ValidationofGround5SourceHeatPumpSystemModelsinanIntegratedBuildingand SystemSimulationEnvironment.InternationalJournalofHVAC&RResearch12(3a), Gentry,J.E.,Spitler,J.D.,Fisher,D.E.,Xu,X.,2006.Simulationofhybridgroundsourceheatpump systemsandexperimentalvalidation,7thinternationalconferenceonsystemsimulationin Buildings,Liege. Gu,Y.,O'Neal,D.,1998.Developmentofanequivalentdiameterexpressionforverticalu5tubes usedinground5coupledheatpumps.ashraetransactions104(2), Hanby,V.I.,Wright,J.A.,Fletcher,D.W.,Jones,D.N.T.,2002.Modelingthedynamicresponseof conduits.internationaljournalofhvac&rresearch8(1),

27 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics He,M.,2012.Numericalmodellingofgeothermalboreholeheatexchangersystems.Doctoral Thesis,DeMontfortUniversity. He,M.,Rees,S.J.Shao,L.,2010.Simulationofadomesticgroundsourceheatpumpsystemusing a three5dimensional numerical borehole heat exchanger model. Journal of Building PerformanceSimulation4, Hellstr!m,G.,1991.GroundHeatStorage:ThermalAnalysisofDuctStorageSystems:PartI Theory.DoctoralThesis,UniversityofLund. Hern,S.A.,2004.Designofanexperimentalfacilityforhybridgroundsourceheatpump systems.mastersthesis,oklahomastateuniversity. Kummert,M.,Bernier,M.,2008.Sub5hourlySimulationofResidentialGroundCoupledHeat PumpSystems.BuildingServicesEngineeringResearchandTechnology29, Li,Z.,Zheng,M.,2009.DevelopmentofanumericalmodelforthesimulationofverticalU5tube groundheatexchangers.appliedthermalengineering29, Marcotte,D.,Pasquirer,P.,Sheriff,F.,Bernier,M.,2010.Theimportanceofaxialeffectsfor boreholedesignofgeothermalheat5pumpsystems.renewableenergy35, Molina5Giraldo,N.,Blum,P.,Zhu,K.,Bayer,P.,Fang,Z,2011.Amovingfinitelinesourcemodelto simulateboreholeheatexchangerswithgroundwateradvection.internationaljournalof ThermalSciences50, Mottaghy,D.,Dijkshoorn,L.,2012.Implementinganeffectivefinitedifferenceformulationfor boreholeheatexchangersintoaheatandmasstransportcode.renwableenergy.inpress. doi: /j.renene Naiker,S.S.,Rees,S.J.,2011.MonitoringandPerformanceAnalysisofaLargeNon5domestic GroundSourceHeatPumpInstallation.ProceedingsofCIBSE+Technical+Symposium+2011,De MontfortUniversity,Leicester,6thand7thSeptember2011. Oppelt,T.,Riehl,I.,Gross,U.,2010.ModellingoftheboreholefillingofdoubleU5tubeheat exchangers.geothermics39, Paul,N.D.,1996.Theeffectofgroutthermalconductivityonverticalgeothermalheatexchanger designandperformance.mastersthesis,southdakotastateuniversity. Rees,S.J.,Spitler,J.D.,Xiao,X.2002.Transientanalysisofsnow5meltingsystemperformance. ASHRAE+Transactions108(2), Rees,S.J.,2009.ThePGRID3Dparametricgridgenerationtooluserguide,version1.2.Leicester: DeMontfortUniversity. Sankaranarayanan,K.P.,2005.Modeling,VerificationandOptimizationofhybridgroundsource heatpumpsysteminenergyplus.mastersthesis,oklahomastateuniversity,stillwater. SEL,1997.Atransientsystemssimulaionprogram,user'smanual,version14.2,SolarEnergy Laboratory,UniversityofWisconsin5Madison. Signorelli,S.,Bassetti,S.,Pahud,D.,Kohl,T.,2007.Numericalevaluationofthermalresponse tests.geothermics36, SinghA.K.,BhadauriaB.S.,2009.Finitedifferenceformulaforunequalsub5intervalsusing lagrange sinterpolationformula.internationaljournalofmathematicsanalysis17(3),

28 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Spitler,J.D.,Culline,J.,Bernier,M.,Kummert,M.,Cui,P.,Liu,X.,Lee,E.,Fisher,D.E.,2009. Preliminaryintermodelcomparisonofgroundheatexchangersimulationmodels.11th InternationalConferenceonThermalEnergyStorage5Effstock2009.Stockholm,Sweden. Stone,H.L.,1968.Iterativesolutionofimplicitapproximationsofmultidimensionalpartial differentialequations.siamjournalofnumericalanalysis5(3), Wen,C.Y.,Fan,L.T.,1975.ModelsforFlowSystemsandChemicalReactors,NewYork,Marcel Dekker,Inc. Wetter,M.,Huber,A.,1997.TRNSYSType451:VerticalBoreholeHeatExchangerEWSModel, Version3.15ModelDescriptionandImplementingintoTRNSYS.TranssolarGmbH,Stuttgart, Germany. Xu,X.,Spitler,J.D.,2006.Modelingofverticalgroundloopheatexchangerswithvariable convectiveresistanceandthermalmassofthefluid.proceedingsofthe10thinternational ConferenceonThermalEnergyStorage5Ecostock2006,Pomona,NJ. Yavuzturk,C.,Spitler,J.D.,1999.AShortTimeStepResponseFactorModelforVerticalGround LoopHeatExchangers.ASHRAETransactions105(2), Yavuzturk,C.,Spitler,J.D.,Rees,S.J.1999.ATransientTwo5DimensionalFiniteVolumeModel for the Simulation of Vertical U5Tube Ground Heat Exchangers. ASHRAE Transactions 105(2), Young,T.R.,2004.Development,Verification,andDesignAnalysisoftheBoreholeFluidThermal MassModelforApproximatingShortTermBoreholeThermalResponse.MastersThesis, OklahomaStateUniversity. Zeng,H.Y.,Diao,N.R.,Fang,Z.H.,2002.Afiniteline5sourcemodelforboreholesingeothermal heatexchangers.heattransfer Asianresearch31(7),

29 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Figures" Fig."1.AsingleU5tubeboreholeheatexchanger.Depthsaretypically505150mandbore diameters mm. Fig."2.Adiagramofatypicalnon5orthogonalcellshowingthelocalvariablesand coordinatesdefinedadjacentacellface.theeastfaceishighlighted. 28

30 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Fig."3.Thenumericalmeshusedtorepresentasingleboreholeheatexchanger. Fig."4.Ahorizontalplanethroughexamplesofmulti5blockstructuredmeshes representingasinglebhe. " 29

31 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics " Fig."5."Thepredictedresidencetimedistributioncomparedwiththeanalyticalsolution. Resultsforfirstandsecondordertemporaldiscretizationschemesareshown. " Fig."6.DifferencesbetweenthepredictedandanalyticalRTDaccordingtothenumberof cellsusedtodiscretizethepipe. " 30

32 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics " " Fig."7."Experimentalandpredictedheatexchangertemperatures(15 th dayofoperation, 15:00516:30). Fig."8."Experimentalandpredictedheatexchangertemperatures(15 th dayofoperation, 17:00518:30). 31

33 PublishedinGeothermics+(2013)+46: DOI: /j.geothermics Fig."9.Measuredandpredictedmonthlymeanboreholeoutletfluidtemperatures. " " Fig."10.Measuredandpredictedmonthlynetheatexchange(MWhpermonth).Positive valuesrepresentheatrejection. 32