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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.
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.
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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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