Numerical investigation of heated gas flow in a thermoacoustic device

articleinfoabstract

Self-sustainedoscillationinastandingwavethermoacousticdeviceisreproducedviacomputational?uiddynamicssimulations,andheated?owbehaviorinthedeviceisexploredusingtheresultsobtained.Thestraight-typethermoacousticdeviceiscomposedoftworesonancetubes,twoheatexchangersandastack.Inthesesimulations,boththeacousticcharacteristicsandthetemperature?eldduringself-sustainedoscillationareconsidered.Therefore,toreproducetheself-sustainedoscillatory?ow,anacous-ticsignalisinjectedintothecomputationaldomainasatriggerpulse.Fromtheresults,oscillatory?uidmotionaroundtheengineisinvestigated,andcharacteristics,includingtheacoustic?eldorwork?ow,theenergydissipation,theworksourceandotherassociatedaspects,areestimated.Theresultsagreewellwiththoseoflineartheory,althoughhighenergydissipationcausedbyvortexgenerationisobservedneartheengine.Thisresultveri?esthecomputational?uiddynamicssimulationresults.Thetemperature?eldaroundtheengineisalsoinvestigated.Theresultsshowoccurrenceofasymmetricaltemperatureoscillationswithintheheatexchangers.Thisbehaviorcannotbepredictedusinglinearthe-orybecausethenon-uniformtemperaturegradientintheengineunitistransferredinstream-wisebyconvection.Finally,amodi?cationtoconventionallineartheoryissuggestedtoreproducethisbehavior.

ó2016ElsevierLtd.Allrightsreserved.

Articlehistory:

Received10March2016Revised1July2016

Accepted16August2016

Availableonline18August2016Keywords:

ThermoacousticsHeatexchanger

Computational?uiddynamicsOscillatory?ow

1.Introduction

ThethermoacousticdevicereviewedbySwiftin1988combinestheadvantagesoftheinherentthermalef?ciencypropertiesoftheStirlingcyclewiththeabilitytoworkusingaminimumnumberofmovingparts[1].Swiftetal.demonstratedthefeasibilityofthedeviceusinganinexpensiveprototype[2],andsincethen,ther-moacousticdeviceshaveattractedconsiderableattentionasappli-cationsofrenewableheatenergy.Manybasicandpracticalstudiesofthesedeviceshavebeenperformedinrecentyears.

Thermoacousticdevicesworkusingthermoacousticphenom-ena.Theoriestoexplainthesephenomenawereoriginallyformedbasedonacoustictheory[3–7],andthelineartheoriesdevelopedbyRott[8,9]andTijdeman[10]arestillcommonlyappliedtoCorrespondingauthor.

E-mailaddresses:(S.Hasegawa).

kuzuu@tokai-u.jp

(K.

Kuzuu),

s.hasegawa@tokai-u.jp

developmentsinthesedevices.Forexample,theDesignEnviron-mentforLow-amplitudeThermoacousticEngines(DELTAE),whichisanumericalanalysiscodedevelopedbySwiftetal.[11],usesthelineartheoryofRott[8]andisusefulforthermoacousticdevicedesign.

Lineartheoryisthussigni?cantwhendealingwiththermoa-cousticphenomenathatoccurinathermoacousticdevice.However,somephenomenacannotcurrentlybeexplainedusinglineartheory.Itisnotactuallypossibletopredictnon-linearphe-nomenaliketwoorthree-dimensionalvortexgeneration.Heattransferintheheatexchangerisanotherproblemtobesolvedinthis?eld,andtheproblemmustbestudiedusingmethodsbeyondlineartheory.Manystudiesofheattransferinthermoacousticdeviceshavebeenperformedrecently,andtheycanbeclassi?edintoseveraldifferenttypesofapproaches.

The?rstapproachcombineslineartheoryandnumericalcalcu-lations.Piccoloetal.introducedasimpleenergyconservationmodelcoupledwithclassicallinearthermoacoustictheory,and

http://dx.doi.org/10.1016/j.applthermaleng.2016.08.0931359-4311/ó2016ElsevierLtd.Allrightsreserved.

1284K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

Nomenclature

uihuirppmp1uupuTTmT1Tw

THw,TCwnitDtfVSA

qqm

density

mean(time-averaged)density?owvelocityvector

velocityamplitudeatcross-sectionabsolutepressure

mean(time-averaged)pressurepressureamplitudeatcross-sectionphaseofacousticwave

phasedifferencebetweenpressureandvelocityabsolutetemperature

mean(time-averaged)temperaturetemperatureamplitudeatcross-sectionwalltemperature

walltemperaturesofHEXandCEX,respectivelynormalvectorofsurfaceelementtime

periodofacousticwavefrequencyofacousticwave

volumeelementofcontrolvolumesurfaceelementofcontrolvolumeareaofcross-section

gasconstant

sijshearstresstensor

heat?uxqj

lviscositymkineticviscositykthermalconductivityathermaldiffusivitycspeci?cheatratiorPrandtlnumberdijKroneckerdeltajimaginaryunitxangularfrequency

vm,vathermoacousticfunctionIwork?ow

Wm,Wpkineticandpotentialenergydissipation

Wprog,Wstandtravellingandstandingwavecomponentsofwork

source,respectively

eiinternalenergy/viscousdissipationfunctionCpspeci?cheatatconstantpressure

R

usedittoestimatetheheattransferpropertiesofthermoacoustic

heatexchangerscomposedofseveralparallelplates[12,13].Inanothermethod,deJongetal.proposedaheattransfermodelforone-dimensionaloscillatory?ow.Thismodelcanbeappliedtoparallel-platethermoacousticheatexchangers,andwasusedtoinvestigatetheirheattransferproperties[14].

Experimentalapproachescanalsobeusedtoinvestigateheattransfer.WakelandandKeolianetal.investigatedtheheattransferofparallel-plateheatexchangersinoscillatory?owenvironments[15].Theyestimatedheatexchangereffectivenessbasedonitstemperature?eld,whichwasmeasuredusingthermistorprobesplacedattheheatexchangerexitsandentrances.Theyalsocom-paredtheresultsoftheirstudywiththoseobtainedusingtheDEL-TAEcode[11].Additionally,Jaworskietal.investigatedtheheattransferpropertiesofparallel-plateheatexchangersthroughacetone-basedplanarlaser-induced?uorescence(PLIF)measure-ments[16,17],andcomparedtheirmeasuredresultswithnumer-icalresults[18,19].

The?nalapproachisbasedoncomputational?uiddynamics(CFD).Thisapproachisadvantageousbecausetheassumptionsinlineartheoryareexcluded,andbecausetheheattransferproper-tiesofathermoacousticdevicecanbecalculateddirectlyfromitstemperature?eld.Somenumericalsimulationsofoscillatory?owinheatedpipeswereperformedbyZhaoandCheng[20,21].Whiletheirsimulationswereforoscillatory?owinducedbyacousticwavepropagation,theirresultsprovidethecharacteristicfeaturesofanoscillatorytemperature?eldinthetube.Inanothernumeri-calsimulation,Caoetal.investigatedtheenergy?uxdensityinathermoacousticcoupleunderacousticstandingwaveconditions[22].Inthisstudy,theyestimatedtheeffectsofthedisplacementamplitudeonheattransfer.Thestudyissigni?cantforheattrans-ferestimationbecausethedisplacementamplitudeinRott’stheory[8,9]isassumedtobenegligiblewhencomparedwiththedevicelength.IshikawaandMeealsostudiedthe?ow?eldsandenergytransportnearthermoacousticcouplesthroughnumericalsimula-tionsusinga2DfullNavier–Stokessolver[23].MarxandBlanc-Benonperformednumericalsimulationsofathermoacousticrefrigeratorthatconsistedofaresonatorandaparallelplatestack[24].Theycomparedtheirresultswiththosepredictedbylineartheory,andshowedthatthereisadifferenceinmeantemperaturebetweenthe?uidandtheplate.MohdandJaworskialsoinvesti-gatedtheoscillatory?owandheattransferofparallelheatexchangers,andcomparedtheirresultswithexperimentaldata[25].Usingbothnumericalresultsandexperimentaldata,theydemonstratedtheeffectofthetemperature?eldonoscillatory?owandthedependenciesofheattransferontheReynoldsnumber.

Additionally,self-sustainedoscillationisalsoreproducedanddiscussedwithrespecttoCFDsimulationsofthermoacousticphe-nomena.HantschkandVortmeyersimulatedself-sustainedoscilla-tioninaRijketube[26].Whilethesimulatedtubeincludesonlyheatingelements,ratherthanheatexchangers,theydiscussednon-linearityintheheattransferprocess.Recently,otherCFDsim-ulationsofself-sustainedoscillationinthermoacousticengineshavebeenperformed.Spoelstraetal.simulatedatravelling-wavethermoacousticengine,andshowedstrongnon-lineareffectsforhigh-amplitudethermoacousticsystems[27].Zinketal.showedthetransitionfrominitialdisturbancetoself-sustainedoscillationinathermoacousticengine,andexploredtheeffectsofacurvedresonator[28].Daietal.simulatedself-sustainedoscillationina300Hzstandingwavethermoacousticengine[29]byvisualizing?ow?eldsattheendsofthestack,anddiscussedmulti-dimensionaleffectsthatoccurredbecauseoftheabruptchangein?owarea.

Asdescribedabove,manystudiesofthermoacousticdeviceshavebeenperformedusingavarietyofapproaches.CFDsimula-tionsattractparticularattentionbecauseoftheirabilitytorepro-ducenon-lineareffectsinthermoacousticphenomena,andthetechniquehasprogressedsuchthatthenon-linearityof?owbehaviorinself-sustainedoscillationcanbediscussed.However,whiletheacoustic?eldsobtainedfromthesesimulationsarecom-parablewiththoseobtainedusingconventionallineartheory,thenon-linearityofheated?owbehavior,whichmustaffectthesys-temheattransfer,hasnotbeendiscussedadequatelytodate.

Inthisstudy,toinvestigatesuchheated?owbehavioroccurringinathermoacousticdeviceingreaterdetail,unsteadyCFDsimula-tionsofself-sustainedoscillatory?owareperformed.Forthispur-pose,afull-scaledevicemodelwassetupandtheengineunitwasgivenasuitabletemperaturepro?le.Usingthissetup,self-sustainedthermoacousticoscillationcanbereproduced

K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–12931285

numerically.Alinearanalysisbasedonconventionalthermoacous-tictheoryisperformedsimultaneously,andtheresultsarecom-pared.Inparticular,ratherthansimplycomparetheresultsfortheacousticcharacteristics,therelationshipbetweentheacousticcharacteristicswhentreatedusinglineartheoryandthe?ow?eldsobtainedfromCFDisclari?ed,andthelinearandnon-linearchar-acteristicsofoscillatory?owsarediscussed.Finally,thetempera-ture?eldbehaviorthatcannotbeexplainedusinglineartheoryisinvestigated.2.Calculations

2.1.CalculationmethodinCFD

CFDsimulationsarecarriedoutusingtheLS-FLOWunstruc-turedcompressible?owsolverdevelopedbytheJapanAerospaceeXplorationAgency(JAXA)[30].Thesolverisbasedonthree-dimensionalunsteadycompressibleNavier–Stokesequations.Thebasicequationsareasfollows.

@

??VQdVttSeFeàFvTdS?0e1T2

6

q3

2

3

2

6qux776qU

6quxUtnx7603

sntsntsn7Q?6

666

qu7p776xxxyxxzxx7

7y74qu7;F667667z7e?566quyUtnyp4qu7zUtnzp7;Fv?566sxynytsyynytszyny4s7xznztsyznztszznz75

EeEtpTUeuisijàqjTnjU?u2xnxtuynytuznz;

u3xu5;s2??i?64uy7

ij?à@u??

iu3leráUTdijtl

t@ujji

zE?

1à1

pt1

qU22;

qi?àkrT;

whereq,ui,p,T,landkarethedensity,thevelocityvector,thepressure,thetemperature,theviscosityandtheheatconductivityofthegas,respectively,andniisthevectornormaltothevolumeelementsurface.Themainnumericalschemesusedinthissimula-tionaregiveninTable1.2.2.CFDcalculationmodel

Fig.1showsthecalculationmodelusedinthissimulation.Theengineunithasthreecomponents:hotandcoldheatexchangers(HEXandCEX),andastack(STK).Eachcomponentcontainssix?atplatesandsevenchannels.Thecomputationaldomainincludesthebufferregion.TheindividualpartsizesandboundaryconditionsareshowninFig.1(a).Themodelistwo-dimensional;thisisachievedbyprovidingsymmetricalconditionsinthezdirection.Thewalltemperatures,excludingthoseofthesix?atplatesof

Table1

NumericalschemesusedinCFDsimulations.CalculationNameofscheme

TimeintegrationThreepointsbackwardstepapproximationImplicitsolutionLU-SGS[31]

InterpolationMUSCLschemebyGreen-GaussmethodNumericalConvectiveSLAU[32]

?ux

term

Viscousterm

Wang’smethod[33]

theengineunit,areall?xedatroomtemperature(298.15K).The

temperaturesintheengineunitare423.15KforHEXand298.15KforCEX.Alineardistributionrangingbetween423.15Kand298.15KisusedfortheSTKtemperature.Theworkinggasisair,theworkingpressureisp=1.01325?105Pa,andthetemper-atureT=298.15K.

Inthissimulation,toinduceself-sustainedoscillatory?ow,thepressuredisturbancewasinjectedasatriggerpulsefromtheopenend.Thesimulationsactuallystartfromastaticinitialconditionandcontinueforashorttime.Then,one-halfcycleofasinusoidal

acousticwaveisinjectedasadisturbancewith^p

%283Paandf=21.2Hzfromtheopenendintothebufferregion.Thesupplyofthisacousticwaveisthenterminated.

Themeshcon?gurationiscomposedofhexahedralcellsthatwereconvertedfromnon-uniformCartesiangrids.Theconcen-tratedmeshstateoftherepresentativepartsisshowninFig.1(b).Fortwo-dimensionalcalculations,thenumberofdivisionsinthez-directionissetatone.Thetotalnumberofcellsisapproxi-mately300,000,andtheminimummeshsize,whichcorrespondstothedistancefromtheboundarywalltotheadjacentmesh,is0.0237mm.Thetimestepusedforthissimulationis2.0ls,whichcorrespondstoaCourant–Friedrichs–Lewy(CFL)numberof30basedonthesoundvelocity.

2.3.Numericalcalculationsbasedonlineartheory

UsingRott’slineartheory[8],thebasicequationsforone-dimensionalthermoacousticoscillatory?owwithinatubecanbedevelopedandcanthusbeexpressedassimultaneousdifferentialequationswithrespecttovelocityandpressure.

@p1?àjxqm

huimr

;e2T

@hui !rjx?àp1àcà1e1àvmTp1tvàv1@Tm

huimTmr;

me3T

wherevmandvaaretheviscousandthermalthermoacousticfunc-tions,respectively,andhavetwo-dimensionalformulationsasfollows.

?tanhee1tjTp?????????

v?

tanhee1tjT?????????

m

xsxsamT

m

;vpa

T

a

e4T

sm?r20=2m;sa?r20=2a

Here,r0isone-halfofthetwo-dimensionalchannelwidth.

Theseequationsarecalculatednumerically.Forexample,Uedaetal.proposedanumericalmethodbasedonthefourth-orderRunge–Kuttamethodandcalculatedthecriticaltemperatureratioforself-sustainedthermoacousticoscillation[34].

Inthelinearanalysis,thepressureamplitudeattheantinodepoint(closedend)isgivenasaboundarycondition.Avalueof495PaiscalculatedusingCFD.Forthetemperatureconditions,thetimeandsection-averagedtemperatureobtainedfromCFDisused.TheseconditionsleadtothepropertiesshowninTable2.3.Resultsanddiscussion

3.1.Productionofself-sustainedoscillation

Tocon?rmself-sustainedoscillation,timevariationofthephys-icalvaluesafterterminationofthetriggerinjectionisobserved.Fig.2showsthevariationsintheaxialvelocityandpressureat(x,y)=(1.04,0.0).Theacousticsignalinjectiontimeist=2.054–2.078s.Asshowninthe?gure,thephenomenonisconsideredto

1286K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

Table2

Propertiesofairwithintheengine.T(K)402298.15

q(kg/m3)

0.87821.1847

m(m2/s)

2.6299-051.5561-05

smx

5.6179.493

resonancetube(x=1.045and2.00m).Bothpro?lesagreewiththeresultsoflineartheory.Thisimpliesthatthefundamentaloscillatory?owbehaviorobeyslineartheorywithinthedevice.3.3.Acousticproperties

Inthermoacoustictheory,theacousticpowerisexpressedasawork?ow.FromtheCFDresults,wecancalculatethisvalueusingEq.(5).

beinaperiodicsteadystateafter4sandistheninself-sustainedoscillation.

Theenlargedgraphshowstheperiodaroundtheoccurrenceofthetriggerpulseandtheperiodicsteadystate.

3.2.Veri?cationofCFD

Oscillatory?owbehaviorinthedeviceisinvestigated,andtheCFDresultsarecomparedwiththosefromlineartheory.Here,weshowthevelocitypro?lesattwodevicecross-sections.Thevelocitypro?leofoscillatory?owvarieswithtime.Fig.3shows12phasesinasinglecycle.InFig.3,thevelocityamplitudeisthesection-averagedvalueatthemidpointoftheSTK(x=1.045m),andthecorrespondingdisplacementamplitudeisalsoshown.Fig.4showsthevariationsinthevelocitypro?les.Eachgraphcorrespondstopro?lesonthesectionincludingtheSTKandthe

xI??

2Z

t

tt2p=x

ftAepàpmTuxdAgdt

e5T

InEq.(5),thework?ow,I,isthetime-averagedvalueoverasin-gleacousticwavecycle.

Fig.5showsacomparisonoftheCFDsimulationresultswiththosefromlinearanalysis.Here,thetemperaturegradientusedinthelinearanalysisisbasedontheCFDresults.Inthe?gure,theCFDresultsalmostagreewiththepowergainobtainedinthelinearanalysis.However,theCFDpowergainisapproximately9%smallerthanthatdeterminedbylinearanalysis.ThismaybecausedbydifferencesbetweentheCFDsimulationsandthelinearanalysisintermsoftemperatureconditionsateachsection.In

fact,

K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

1287

whilethetemperatureisassumedtobecommonateachsectionoftheenginechannelsinthelinearanalysis,thetemperaturediffersineverychannelintheCFDsimulations.Thisdifferenceisdescribedindetaillater.

Theenlargedgraphshowsthecharacteristicsaroundtheengineunit.

3.4.Energydissipationandworksource

Inthissection,wecalculatetheenergydissipationandworksourcefromtheCFDresultsandcomparethemwiththeresultsfromlineartheory.

First,weconsidertheenergydissipationandtheworksourceasthermoacousticproperties.Toexplainthecauseofthework?ow,Tominaga[35]introducedtheconceptofenergydissipationandtheworksourcebasedonlineartheoryof?owinatube.Inthistheory,energydissipationisclassi?edintermsofthekineticandpotentialenergydissipations,orWmandWp,respectively.Addition-

ally,theworksourceisdividedintotwoparts,i.e.,thetravellingandstandingwavecomponents,WprogandWstand,respectively.Theseparametersarede?nedasfollows.

!Ajxqm

Wm?àRejhuirj2

2m !

áAjxà

1tecà1Tvajp1j2Wp?àRe

mWprog

!AevàvT@Tm

jp1jjhuirjcosupu?Re

mm !AevàvT@Tm

jp1jjhuirjsinupu?Im

mme6T

e7T

e8T

Wstande9T

Here,wemustextracttheenergydissipationandthework

sourcefromtheCFDresults.

Numerical investigation of heated gas flow in a thermoacoustic device.doc下載

1288K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

The?rstlawofthermodynamicsiscommonlygivenas

Kineticenergy(viscous)dissipation:

dQ?deitpdv

e10T

dQistheheatenergyaddedtothe?uidelementfromanexter-nalsystem,deiistheincrementintheinternalenergy,andpdvcor-respondstotheworkinwhichtheelementactsfortheexternalsystem.GiventhatdQistheheatenergyaddedtothe?uidelementbyheatconduction,theenergyequationthatappliestotwo-dimensional?uidmotionisobtainedasfollows.

@u@u@v@v/?sxxtsyxtsxytsyy?l

????????!????2????2!22

@u@v@v@u2@u@v

tt?2tàt

3e12T

&????????'????

Dei@@T@@T@u@v

àktàkteàpTtq?à????

@u@u@v@vtsyxtsxytsyytsxx

x

?h/it?2Z

t

tt2p=x

ftA/dAgdt

e13T

e11T

@u@tw?àepàpaT?

Summationofpotentialenergydissipationandworksource:

?

epàpaT

?

Dqe14T

Intheaboveequation,thethirdtermcorrespondstokineticenergydissipationbyviscosity,whilethesecondtermcanberegardedasasummationofthepotentialenergydissipationandtheworksource.Byintegratingthesetermswithrespecttobothtimeandcross-section,theenergydissipationandtheworksourcepropertiescanbeobtainedasfollows.

x?hwit?2Z

t

tt2p=x

ftAwdAgdte15T

Fig.6showstheenergydissipationsandtheworksourcecalcu-latedusingtheaboveequations.Theworksourceintheresonancetubecanbeneglectedbecauseofthelackofaheatsource.

From

K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

1289

this,itcanbesaidthatboththeviscousandpotentialenergydis-sipationsalmostfollowlineartheoryintheresonancetube.

Additionally,weshowthekineticenergydissipationwithinandneartheengine.InCFD,thisdissipationiscalculatedusingEq.(12).TheenlargedgraphinFig.6showsacomparisonoftheresultsofCFDandlinearanalysis.Thegraphshowsgoodagreement,butdif-ferencescanbeobservedneartheheatexchangerextremities(x=1.00and1.09m).Thismaybetheeffectofvortexgenerationaroundthecornersoftheengineplates.However,thework?owisnotsostronglyaffectedbythisdifferencebecausethetotalenergylossisdecidedbytheintegrationinthex-directionandthepeakvalueislimitedtoquiteanarrowarea.

Next,theeffectsofpotentialenergydissipationandtheworksourcearoundtheengineunitareinvestigated.InCFD,thiseffectisdescribedusingEq.(14),andisthesummationofthepotentialenergydissipationandtheworksource.Therefore,theanalyticalresultsforcomparisonmustbeexpressedusingthesummationofWp,WprogandWstandfromEqs.(7)–(9).Fig.7comparesthe

1290

K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

resultsfromCFDandlineartheory.Theyshowalmostperfectagreement.

Comparedvalues:hwitinCFDandWp+Wprog+Wstandintheanalysis.

Fromthermoacoustictheoryinatube,thework?owmustbeconsistentwiththeintegrationoftheabovevalues,asfollows:

ZI?

x

à

á

WmtWptWprogtWstanddx

e16T

Fig.8comparestheresultsoftheintegrationinEq.(16)abovewiththework?owcalculatedfromEq.(5),andshowsgoodagree-mentbetweentheseresults.TheworksourceandthedissipationarethusconsideredtohavebeencorrectlyestimatedusingtheCFD?owdata.

3.5.Flowbehaviorinatemperature?eld

Finally,thetemperature?eldaroundtheengineunitisinvestigated.

InthelineartheoryofRott[8],itisassumedthatthetime-averagedtemperatureofthe?uidineachsectioncoincideswiththewalltemperature.However,becauseofheattransferbetweenthe?uidandthesolidwall,theaveraged?uidtemperaturemaypossiblydifferfromthewalltemperature.Infact,whilethewalltemperatureoftheHEX(423.15K)isgivenasshowninFig.1,the?uidtemperatureintheCFDsimulationislowerthanthewalltemperature.Fig.9showsthetimeandsection-averagedtempera-turedistributionsinthex-directionaroundtheengine.Whitecir-clesdenotetheresultsforeachsectionalongthedevice,andintheengineregion(x=1.00–1.09m),thetemperaturedistributionforeachchannelisalsoshowninthegraph.Eachresultcorrespondstooneofthecenter(blackcircles),bottom(whitetriangles)ortop(blacktriangles)channelsintheengine.Thewalltemperaturesoftheengineplatesareplottedusingadottedline.Thereisadif-ferencebetweenthetemperatureofthe?uidandthatoftheengineplates;speci?cally,the?uidtemperaturesinthebottomandtopchannelsaremuchlowerthanthatoftheengineplates.Thisisbecauseonesideofeachofthesechannelsissetatroom

temperature(298.15K).Asaresult,theeffectivetemperature,whenaveragedoverallenginechannelsections(whitecircles),isfurtherreducedwhencomparedwiththeengineplatetemperatures.

Ontheotherhand,ahigh-temperatureregionontheoutsideoftheHEX(x=0.95–1.00m)canalsobeobserved.ThisiscausedbyleakageofheatedairfromtheHEX;theareaiscalledathermalbufferregion.Therapidincreaseintemperatureatx=1.0mcanbeconsideredtobeaffectedbythewalltemperatureattheengineplateextremity.ThebehavioroftheheatedairisvisualizedinFig.10.Inthis?gure,thehightemperatureregionofthethermalbufferisasymmetricalaroundthecentrallineandde?ectstowardstheupperwall.Thisisbecausethetemperatureisaveragedoveroneperiodofanacousticwaveatmost,despitethe?owbeingdis-turbedbyvortexgeneration.

Next,toinvestigatethetemperature?eldbehaviorindetail,thephasevariationsofthetemperatureareobserved.Theobservedareaisthecentralchanneloftheengine.TheCFDresultsarealsocomparedwiththevaluesthatwerepredictedusingEq.(17)basedonlineartheory.

??

fàf1àf1@Tm

huirT1?e1àfaTp1tà

Cpmjm1àm

1

e17T

InEq.(17),thetemperaturegradientinthex-directionistakenfromthetemperaturedistributiononthecenterchannel(blackcir-cles)showninFig.9.

Fig.11showsthedistributionwithinthecentralsectionofSTK.ThenormalizedtemperatureHmustnowbeintroduced:

??

H?

TàTw

;

THwCw

e18T

agreewiththeresultspredictedusingEq.(17).Theactualtemper-atureoscillationintheCFDresultsisasymmetrical.InEq.(17),thetemperaturegradientisassumedtobeconstantwithinthedis-placementamplitudeofthelocal?uidelement.Inthiscase,theactual?uidelementmovestoanextentof15mm,asshowninFig.3.WhenthetemperaturedistributionshowninFig.9isconsid-ered,theaboveassumptioncannotbeappliedinthisarea.Thismeansthatthetemperaturegradientvariesintheoscillatorycycle.Toimprovethispoint,theconvectivetermofEq.(17)mustthereforebemodi?ed.Here,itisassumedthatthemeantempera-turegradientcanbedescribedusingthelineartransferequation

whereTw,THw,andTCwarethewalltemperaturesofthelocalsec-tion,theHEX,andtheCEX,respectively.Asthegraphshows,theCFDresultscloselyfollowthoseofthelineartheory.Thisisbecausethetemperaturegradientinthex-directionisalmostuniforminthisregion.

Incontrast,attheboundaryareabetweentheheatexchanger(HEXandCEX)andthestack(STK),thesituationisquitedifferent.AsshowninFig.12,thetemperatureoscillationfromCFDdoesnot

@F@Ftux?0;Fex;tT?

@Tm

:e19T

Thesolutiontotheaboveequationis:

Fex;tT?FexàuxtT:e20T

Here,uxtcorrespondstothe?uidelementdisplacementintheoscillatory

?ow.

1292K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–1293

Inconventionallineartheory,oF/oxisequalto0withinthelengthofthedisplacementorder.ThismeansthatthevalueofT1calculatedusingEq.(17)isunaffectedbythetemporalvariationofthetemperaturegradientata?xedpoint.Therefore,thetemper-atureamplitudeatphaseu=xtcanbecalculatedusingEq.(21).

T1euT?Re?T1eju??

e21T

Here,T1isconstantduringthevariationofu.

Inthismodi?cation,thefollowingequationisused:

@Tm

?FexàuxtT?FexàDnTe22T

Dn?Imhui!rjeu

:

e23T

Thetimevariationofthetemperatureamplitudeisthencalcu-latedusingEq.(24).

T1euT?Re?T1euTeju??

e24T

Usingthismodi?cation,theanalyticaltemperaturedistributionthenapproachesthatoftheCFDresults,asshowninFig.13.Thisimpliesthattheconvectivetermusedabovefortemperaturebehaviorintheheatexchangerissubstantial.

4.Conclusions

UsingCFDtechniques,numericalsimulationsoftheoscillatory?owwithinathermoacousticdevicewereperformed.Thisstraight-typedevicegeneratesastandingacousticwave.TheCFDresultswerealsocomparedwiththosegeneratedusinglinearthe-ory.Fromadiscussionoftheseresults,thefollowingconclusionsaredrawn.

??Injectionofanacousticsignalthatactsasatriggerpulseintothedeviceallowsself-sustainedthermoacousticoscillationtobereproducedintheCFDsimulations.

??ThedissipationtermsintheCFDsimulationswereassignedtothethermoacousticpropertiesfromlineartheory,andeachpropertywascomparedfordifferencesbetweenCFDandlinearanalysis.Fromthiscomparison,itwasfoundthattheestimatedacousticpowerobeyslineartheory,althoughnon-lineareffectsappearneartheengineplateextremities.Thisoccursbecausethesephenomenaarelimitedtonarrowregionsunderthecon-ditionsused.

??Thetemperatureoscillationshowsastronglyasymmetricalstructurewithintheengine,unlikethatintheresonancetuberegion.Thisisduetothenon-uniformityofthetemperaturegradientwithintheengine.Additionally,byintroducingaphasetransfertoestimatethetemperaturegradientandthenmodify-ingconventionallineartheory,itwouldbepossibletoimprovetemperature?eldpredictionswithintheheat

exchanger.

K.Kuzuu,S.Hasegawa/AppliedThermalEngineering110(2017)1283–12931293

Acknowledgements

ThisworkwassupportedbytheJapanScienceandTechnologyAgencythroughtheAdvancedLowCarbonTechnologyResearchandDevelopmentProgram.References

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