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Analysisand performanceof a large thermoacousticengineG.W. SwiftCondensedatter ndThermal hysicsroup,os/l/amosational aboratory,os/l/amos,New Mexico 87545( Received7November;cceptedorpublicationMay 1992)Measurementsndanalysisf a 13-cm-diamhermoacousticnginerepresented.t itsmostpowerful peratingoint, sing 3.8-bar elium,heengine elivered30W to anexternalacousticoad, onvertingeat o deliveredcousticowerwithanefficiencyf 9%. At lowacousticmplitudes,here linear) hermoacousticheorysexpectedo apply,measurementsof emperatureifferencend requencygree ith hepredictionsf theoryo within %,over onditionspanningactors f 4 in mean ressure,0 n pressuremplitude,infrequency,nd3 in gas ound peeds.utmeasurementsf thesquaref pressuremplitudeversus eater ower iffer rom hepredictionsf theory y 20%, twice heestimateduncertaintyn the esults. t higher ressuremplitudesup o 16%of themean ressure),evenmore ignificanteviationromexistinghermoacousticheorysobserved.everal ausesof thisamplitude-dependenteviationre dentified,ncludingesonance-enhancedarmoniccontentn theacoustic ave, nda new, irst-orderemperatureefectn thermoacousticeatexchangers.hese ausesxplain ome, utnotall,of heamplitude-dependenteviationfthehigh-amplitudeeasurementsromexistinglinear) heory.PACS numbers:43.35.Ud, 43.25.Vt

INTRODUCTIONThe emperaturescillationhataccompanieshepres-sure scillationn a sound ave susually nimportant,x-ceptas t contributeso the attenuationof sound n bulk andat boundaries.However,under somecircumstanceshe tem-peratureoscillation nd in particular ts interactionwithboundaries--thermoacousticrocesses--cane harnessedto produce owerful, seful hermodynamicffects, uchas

theefficientonversionf heat o acousticower.Thisphenomenon,irststudied century go,wasex-plained ualitativelyyRayleighasdue o oscillatoryher-malexpansionndcontractionf thegas or liquid),whichin turn s due o oscillatory otion f thegas long n m-posedemperatureradient. owever.,uantitativelyccu-rate understanding as not achieved ntil Rott's3-8carefultheoreticaltudies,irstvalidated xperimentallyat zerotemperature radient)by Merkli andThomann. Rott's he-oryhasbeenurther onfirmedxperimentallyn cryogenichelium as yYazaki tal., n airby MiillerandLang, Mand n high-pressureelium asustbelow oom empera-tureby Hofler.2 In thesehreeexperiments,ualitativeagreementithall eaturesfRott's heorywas onvincing-ly good.Thermoacousticmeasuremenllsavealsobeenana-lyzed roma porous-mediumointof view.3In the workof Yazakiet al., a known emperatureprofilewas mposed n a uniform-diameterube ull of heli-um gas. he emperatureangewithinwhich hegas pon-taneouslyscillated,nd heoscillationrequency, erede-termined,and the resultswere comparedo calculationsbased nRott's heory.Measurementspannednextreme-ly broad ange f emperatureatioandgeometryatios, ndhence he overallagreement etweenmeasurementnd he-oryprovidedtrong onfidencen Rott's heory.Althoughmuchof the emperature-ratioataagreed ithcalculations

to within 5%, somedisagreed y over30%; most requencydata were n 2% agreement, ut somedisagreed y 8%. Noquantitativecomparisons f pressureamplitude, acousticpower,or thermalpowerwere made.Miiller andLang Muilta thermoacousticngine, om-pletewith stackand two heat exchangers, singair aswork-ing substance.he observedemperature ifference etweenthe heat exchangers t the onset of oscillationswas 50%higher hancalculated ia Rott's heory.At temperature if-ferences igher han onset, he rate of increase f pressureamplitude n time (related o the acoustic owerproduced)was 2/3 of the calculated value.

Hofler'shermoacousticefrigerator used 0-bar eli-um at pressure mplitudes p to 3% of meanpressure. heobserved atio of temperatures f the cold and hot heat ex-changers iffered rom calculations asedon Rott's theoryby 2% to 9%, so that the temperaturedifferencedifferedfrom calculations y 10% to 20%. The agreementwasbestat low pressure mplitudes. owerswerepresentedn termsof coefficient f performance, or which measurements if-fered rom calculations y roughly 10%.Today,aswe contemplate pplications f thermoacous-tics that are very large (suchas heat-driven onarprojec-tors Mndcryogeniquefiers5)or could njoywidespreaduse (suchas in residential efrigeration nd air condition-ing6),t is mportanto try to establishetter uantitativeagreementbetween heory and measurement, o enableac-curate,confident esign f new apparatus. hus, we under-took the measurements described here.The experimentalhermoacousticngine,describedndetail n thenextsection, asdesignedor useprimarilywithhigh-pressureeliumgas. t operated ith thermalpenetra-tion depths f the orderof 1/2 mm, largeenough hat heat

exchangersnd stackcouldbe constructedasilywith an1551 J.Acoust.oc. m. 2 3),September992 0001-4966/92/091551-13500.801992 cousticalocietyfAmerica 1551

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accurately haracterized eometry.ts overalldiameterwaslargeenoughhat resonatorosses id not absorb largefractionof the acoustic ower,so that the efficiency asreasonably igh.In Sec. I, we present esultsof measurements ith theengineat low pressure mplitudes less han 3% of meanpressure).With respecto operatingemperature nd fre-quencywe indexcellent greement ith Rott's heory.Butmeasurementsf thesquare f thepressuremplitude ersusheaterpowerdiffer systematicallyrom calculationsy20%, twice the estimated ncertaintyn the results.Themeasurementseremadeover a broadrangeof operatingconditions,panning factorof 4 in meanpressure,factorof6 n frequency,factor f 3 n gas ound peed,nd nclud-inganacousticoadwhose owerdissipationouldbevariedfrom zero o a valuecomparableo that in the resonator ndheatexchangers.he overallagreement etweenmeasure-mentandcalculationt owamplitudesa principalesult fthe presentwork,reconfirming ott's heoryand demon-strating he validity of our implementation f it. But thesignificantisagreementn pressuremplitude ersus eaterpowerat low amplitudeemains nexplained.We anticipate hat the applicationsmentioned bovewill bemostpractical t thehighest ossiblecousticmpli-tudes, ecause f theattendant ighpowerdensities. ence,wedesignedurexperimentalngineo becapable f operat-ingat pressuremplitudesf theorderof 10%ofmeanpres-sure.At suchhighamplitudesndhighReynolds ndMachnumbers, eviationsrom the linear,acoustic-approxima-tion theoryof Rott are o be expected, nd ndeedYazaki etal. 7andMiillerandLang observedomplexonlinearphenomenat suchamplitudes. ur high-amplitude ea-surements, resentedn Sec. II, which eachpeakpressureamplitudes s high as 16% of the meanpressure,re alsorich andcomplex, ndraiseseveral uestionsor further e-search.

In our high-amplitude easurements,he argest on-zero-amplitudeffectwas hermoacousticeat ransport ythe firstharmonic f the standing ave,whoserequencywas near that of the second normal mode of the resonator.As discussedn Sec. II A, shifting he secondmode's re-quencyby modifying he resonator's eometry uppressedtheharmonic, ssentiallyliminatinghis argesource f ex-cess eat ransport.

We also explored thermal contact in the heat ex-changers t high amplitude, s discussedn Sec. II B. Wediscovered time-averageemperature ifference etweengasandheatexchangermetal hat is firstorder n the acous-tic amplitude, nd we provide n explanationor thisphe-nomenon.We alsooperated he enginewith gasdisplace-mentamplitudesearandexceedinghe engthof the heatexchangers,indingsomeevidence f reduced hermalcon-tact in that regime.After taking hese ffectsnto account, ur high-ampli-tudemeasurementsf heaterpowerversus ressure mpli-tude agreewith the predictions f linear theoryas well athighamplitude s heydo at low amplitude. ut our high-amplitudemeasurementsf the temperature ifference e-tweenheat exchangerstill differ from the predictions f

linear heoryby amplitude-dependentmounts pproachinga factor of 2. We cannotyet assign his phenomenono anyparticularoneof manypossible onlinear ffects.Finally, at the endof thesemeasurementse usedwhatwe had learned o configure he engine or high efficiencyandpower. n 13.8-bar eliumgas,with harmonic uppres-sion, he enginedelivered 30 W of acoustic ower o theacousticoad, requiring7000 W of heaterpowerat a tem-perature ear700C, nddumpingtswaste eat o waterat35 C.This is an efficiency f 9%, which s 13% of the Car-not efficiency t these emperatures. he peakpressure m-plitudeat thisoperating ointwas0.9 bar, 6% of the meanpressure. emperaturedifference, oad power, efficiency,andsquare f pressure mplitudewerewithin 15% of valuescalculatedusingRott's linear theory; operating requencywaswithin 2%. This mpressive erformance odeswell formany applications f thermoacoustics,specially ince hepresent nginewasnot optimized or eitherhigh efficiencyor high power.At thisoperating oint,our calculations how hat only60% of the acoustic owerproduced n the stack s actuallydelivered o the load. Viscousand thermal processesn theheatexchangersbsorb 5% of theproduced ower, nd heremaining 5% isabsorbed y suchprocessesn hesurfacesof the resonator. t is apparent hat studies eading o im-provedheatexchanger esign an mprove he performanceof thermoacousticystems ignificantly.I. APPARATUS

The apparatus sed or thesemeasurementss sketchedschematicallyn Fig. 1. Essentially, t comprised helium-filled cylindrical resonator of 12.7-cm i.d. and 4.32-mlength,containing thermoacousticngine earoneendandwith an acoustic load attached at the other end. In this sec-tion, eachcomponentwill be describedn turn, beginning tthe left end of Fig. 1 and moving o the right.The leftmostsegment f the resonator, f inside ength61.8 cm and inside diameter 12.8 cm, extended from the leftend of the apparatus o the unction between he stackandthe cold heatexchanger.t consisted fa 6.6-mm-wallstain-less-steelipe,with a 1.9-cm-thick apwelded o the eft end.It was thermally insulatedby several ayersof ceramic (alu-minosilicate) fiber blanket, with a total thickness of about 8cm. The first27.9 cm of thissegment, ontaining othingbutheliumgas,will be referred o as he hot duct.Several -mm-thick copperbarswerestrapped nto he hot duct,axially, ohelp t reach nternal hermalequilibriummorequickly hanwaspossiblewith bare stainless teel.The next 6.0cm of this nsulated tainless-steelipecon-tained he hot heat exchanger,n which heat wassupplied,electrically, o the thermoacousticngine.As shown n Fig.2, the hot heatexchanger onsisted f 36 electrical esistanceheaters mbedded n a matrix of nickel ins, esulting n some400 rectangularchannels or the helium, each 6.0 cm long(along he resonator xis), 0.97 mm thick,and ypically1.27cm wide exceptnear the pipe wall. The total cross-sectionalareaopen o thegaswas50.8cm , 39.3%of thepipearea.To construct the hot heat exchanger,we began with0.46-mm-thick nickel sheet (the fin material) and 0.96-mm-

552 J. Acoust.Soc. Am., Vol. 92, No. 3, September1992 (. W. Swift:Large thermoacoustic ngine 552



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ElectricalHot heatexchanger/power leads!nsulatio, Stacks. ,

W,t T. T/"' Tc

Cooling water pipeCold heat exchanger Needle valves Tank

FIG. 1.Schematic,ot oscale, fapparatussedn thesemeasurements.mall ircles,abeled longhebottom f hedrawing, howocationsf empera-ture and pressure ensors.

thick nickel sheet (determining he spacingbetween ins).These were stackedappropriatelyand furnace brazed to-getherusingpurecopper s he brazealloy, o form a rectan-gular block containinghundredsof channelswith cross ec-tion 0.97 X 1.27 mm. The block was then machined down to12.8-cm diameter and 6.0-cm length, and slipped nto thestainless-steel ipe. Next, 36 parallel holes were drilledthrough the assembly, ransverse o the pipe axis, with 6holes equally spaced n each of the 6 regionsbetween insshown n Fig. 2. Thin-wall stainless-steelubeswerepressedinto each of the 36 holesand welded o the pipe at each oftheir ends.Finally, the 36 resistance eatercartridges,6.3-mm diameter,were slipped nto the thin-wall tubes.The heaters were connected, hrough low-resistancetrimming resistors, o three mechanically oupledvariableautotransformers, nd then to the three-phase 0-Hz build-ing power. The trimming resistorswere chosenso that theheat per unit volume delivered o the heat exchangerwasspatiallyuniform, n spiteof the different engths and resis-tances) of different heaters.A three-phaseWattmeter en-

CENTRAL CROSS SECTIONElectric t.,acehlnneCross section shown at right

FIG. 2. Scaledrawingof hot heat exchanger, howingan end view and alateral, central cross section.

abled asymeasurementf he otalpower H deliveredothe hot heat exchanger, ith an accuracyof -F 2%.Three ype-K thermocouplesshown n Fig. 1 enabledmeasurementf the temperature ear he hot heatexchang-er, with an accuracy f -F 1 C.One of the thermocoupleswas ocated etweenhehot heatexchangernd he stack, tthe centerof the pipe;we denote ts temperature implyasTH. A second hermocouple, lsobetween he hot heat ex-changer nd the stack,was ocated n the middleof oneof theoff-centerin sets, t a radiusof 4.3 cm n the pipe;we denoteits emperatures T. We observedhat T- TH[ wasalwaysess han9 C,and ypically3 C, ndicative f goodlateraluniformityn the apparatus. he third thermocouplewas n the hot duct,near he hot heatexchanger, nd at thecenterof the pipe;we denote ts temperature y TductWe H observed .uct-H -- TH to be near he amplitudeof the expect-edadiabaticemperature scillationsn thehotduct (cf. par-cel c, Fig. 14, Ref. 1 .

The stack illed the remaining27.9 cm of the insulatedstainless-steelipe. t consisted f a series f 6 pieces f com-mercially8 vailable oneycomb,achpiece12.8cm n di-ameterand 4.62 cm long, made of 0.10-mm-thick stainless-steel foil with a nominal 1.02-mm honeycombcell size.Shallow slotswere cut into the piecenearest he hot heatexchanger o accommodatehe two thermocouples nd topreventblockageof any honeycomb ellsby the many sub-stantiallysolid egions f the hot heat exchanger.Although our calculationsshow that parallel-platestacks roducesignificantly igherefficiency nd/or powerdensity,we selected his honeycomb tack for our experi-mentsbecauset was nexpensive nd its cellswere uniform.However, the photograph n Fig. 3 shows hat the cellsofthis honeycombare not perfect 1.02-mm hexagons.Wemade severalsimplemeasurementso characterize heir ac-tual geometry.From the weightand overall dimensions feach piece,and the densityof stainless teel,we found that81% of the areawasopen.A 0.914-mm-diampin fit into allcells,but a 0.940-mm-diampin fit into only half of them.Closeexamination f Fig. 3 shows he hexagonso be slight-

1553 J.Acoust.oc. m., ol. 2,No. ,September992 G. W. Swift:Large thermoacoustic ngine 1553



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..

.,,:.

FIG. 3. Photograph f honeycomb tack.

ly elongated,with an inside lat-to-flat separationof 1.06mm n the ongdirection,nda typical ydraulicadius9of(1/4) X0.97 mm. Below, we will model these cells as 1.0-mm-diam circles, a dimension which is in the middle of thisrange of measurements.

Next along he apparatuswas he cold heat exchanger,with a geometry,shown n Fig. 4, similar to that of the hotheatexchanger, ut with flowingwater eplacinghe electricheaters.t extended .1 cm along he engthof the apparatus.Its 48 3-mm-i.d. copper ubes, n 8 banks of 6 tubes each,were soft-soldered into its 12.7-cm-i.d. brass case and to itsmany copper ins,ofO.25-mm thicknessand .81-mm sepa-ration.A total of 61.6 cm2 (49% of the casenside rea) wasopen to the helium gas, n rectangularchannels .81 mmthick and 10.3mm wide exceptnear he brass ase.Shallowcutoutson the side acing he stackprevented lockageofany stack cellsby the otherwisesolid portionsof the heat

WoterutletJonifoldCopper tubeBross cose

Heliumchonnel

O-ring groove-

Woter inlet _]manifold I IFIG. 4. End and ateralviewsof coldheatexchanger. ll dimensionsre toscale, xcepthat he in spacing asbeen xaggeratedy a factorof 2 forclarity.1554 J. Acoust.Soc. Am., Vol. 92, No. 3, September 1992

exchanger. he cold heat exchanger asewas sealed o thestainless-steelipehousinghe stackwith a rubberO-ting. Athermocouple n the water in the manifold feeding he 48tubesmeasuredhe temperatureTc.Ourbuilding'sooling atersystemupplied00m3/sof water o the heatexchanger.We estimate hat, at thisrate,the temperature ise as the water passed hrough he heatexchangerwas 0.3 C/kW, and the temperaturedifferencebetween he copper ubesand the water flowing throughthem was 3 C/kW. We alsocalculatea 1 C/kW tempera-ture difference etween he middleof each in and ts nearbycopper ubesdue to the finite thermal conductance f thecopper ins hemselves.We will neglect hesesmallandunin-teresting emperaturedifferenceshroughout his work.The remaining3.65-m lengthof the apparatus,whichwe will call the coldduct, consisted f two pieces in series)of 12.7-cm-i.d.,6-mm-wall aluminumpipe. The two pieceswere oined ogether, o the coldheatexchanger ase, nd tothe coldendcap (seebelow) with rubberO-tings.Four cool-ing water acketssurrounded bouthalf the total lengthsofthesepipes,being ocatedwherever hey did not interferewith the pipes'assortedlanges, ittings,and plugs n obso-lete holes.These served o keep the cold duct near roomtemperature venwhen 100 W of acoustic owerwasdissi-pated n it.The fight endof the coldduct wasclosedwith a 3.8-cm-thickbrasslange.A dynamic ressureaugewassealedinto the flangeso hat it measuredhe operating requencyfand heendpressuremplitude E- t wascalibratedbothabsolutely sing eciprocity in 6.9-barand 13.8-barheliumgas,at 118 Hz) and by comparison in air at 122 Hz) to apiezoresistiveauge hat had beencalibrated gainst mer-cury manometer. ts voltagewasread by a lock-in amplifierconfigured sa sharply uned ac voltmeter.We estimate heaccuracyof the dynamic pressuremeasurementso be4-2%.

Mean pressure , in the apparatuswasalsomonitoredat the fight end, usinga Bourdon-tube augeconnectedothe apparatus hrough a high-impedance apillary. Thegauge had a specifiedaccuracy of _ 0.04 bar, and wascheckedwith this accuracyagainsta dead-weight ester.Finally, variablecousticoad,whose ower sspa-tion could be measured ccurately,was alsoattached o thefightendof thecoldduct. t consistedf a 22 900-cm ankconnected through three parallel, water-cooled needlevalves o the fight end,so hat acoustic owercouldbe dissi-pated n the valves y the acousticlow hrough hem.Thus,the variable oadwasessentiallyhe acoustic quivalent f anelectricalR C circuit, with variableR. The principles f itsuseare discussedn detailelsewhere?Basically,measure-mentsof the dynamicpressure r in the tank, and knowl-edgeof the tank'svolume,permitscalculation f the oscilla-tory volumetric lowrate Uinto the tank; he time average ftheproduct PE -- Pt) U thengiveshedissipatedower.To obtain the measurements discussed below with thisapparatus,we set he power o the heaterat the desired al-ue,waitedor heengineoreach teadytate, nd ecordedall pressures,emperatures,requency, ndheaterpower.Toconfirm he approach o a steadystate,we monitoredPe on

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a chart ecorder, aitinguntil he slowdrift n Pz wasnolonger discernible ompared o its fluctuationson a timescale fabout 0minutes.Theseluctuations,fmagnitude1%, werecomparablen sizeand time scale o fluctuationsnthe 60-Hz line voltagen our laboratory.) t the highestpowers,steadystate was reached n about 2 h, while at thelowest owersmanyhourswere equired. ata weredeliber-ately taken in an erratic order, so that the small observedscattern thedatashown elows ndicative f he ong-termreproducibilityof the system.Nonlinearity n the variable oad presented n unsu-spected nd interestingcomplication.Because he oscilla-tory low n the oad's alveswas urbulent,2a given alvesetting bsorbedmorepowerper amplitude-squaredt lowamplitude han at high amplitude. hus, as the amplitudechanged lowlyduring he approacho steady tate, he m-pedance f the oadchangedoo,so hat oneor two readjust-ments f thevalveswere equired uring achapproachosteady tate o maintain target oad mpedance.Onlya nuisancet highamplitudes,hiseffect revent-ed steady-stateperation t low amplitudesnd/or openload-valveettings.f the hreshold f nstability asslowlycrossed,y either owering he heator openinghe valves,the pressuremplitude egan o oscillate, ith a periodofsome ensof seconds.he envelope f thisoscillation rewexponentially,ntil n a fewminutes scillationstoppedn-tirely.Theenginehen urned nandoffwitha period f 1 hor more. In this state, TH first slowly rose, as the heaterheatedhehotheatexchangerndnearby arts; ventually,theengineegan scillating,uicklyeachingmplitudesarabove bar; inally, hese igh-amplitudescillationsuick-ly cooledTH, and heoscillationstopped.II. ANALYSIS OF LOW-AMPLITUDE MEASUREMENTSThe low-amplitudemeasurementsaken with heliumgasn theengine representedn Figs.5-7. Fivemean res-sures, panningearly factorof4, wereused.n the igures,asa functionof the heatsupplied t the heater,we show hepressuremplitude,emperatureifference,nd requencythat heengineeachedn steadytate.Generally,peratingfrequency as oughly ndependentf meanpressurendheaterpower;pressure mplitude osequicklywith both

O3

6 19.2bar /v ] 3.8 bar /A 9.6 bar [] 6.9 bar o 5.2 bar /

QH w) oooFIG. 5. Square fendpressuremplitudeersuseater oweror ivediffer-ent meanpressuresf helium.Pointsare measurements;inesare calcula-tions.

500 o

2000 QH W) 1000FIG. 6. Temperaturedifference ersusheaterpower for the samemeanpressuresf heliumas n Fig. 5. Pointsare measurements;inesare calcula-tions. Kinks visible n lineshere and elsewhere re only artifactsof the cal-culations.

meanpressure ndheaterpower,and emperature ifferencewaspressure nd heat ndependent t highpressure, ut rosewith increasing eat and decreasing ressure t low pres-sure.

The lines n the figures re the resultsof our calculationsbasedon the theory of Rott. The agreement etween hesecalculationsand the measurementss a principal result ofthis work. Calculatedand measured aluesof frequency ndtemperaturedifference re in agreement, ut measured al-uesof P are 20% lower han calculated alues.As dis-cussedn the previoussection,we have only _ 2% uncer-tainty in measurement of heater power and _+4%uncertaintyn measurementf P ; these re nsufficientoexplain his 20% disagreement. encewe now consider heprinciplesand approximations sed n the calculations.The calculations re a straightforward mplementationof the theoryof Rott,3-8asreviewed y Swift. A detailedoutlineof the calculationmethod s n the Appendix.Briefly,in eachsegment duct, heat exchanger, r stack) of the ap-paratus,a solution o the appropriateone-dimensional ave125

v A v

1200 io (w) loooFIG. 7. Operating requency ersus eaterpower or the samemeanpres-sures f heliumas n Figs.5 and6. Pointsare measurements;inesare calcu-lations.

1555 J. Acoust. oc.Am.,Vol.92, No.3, September 992 G.W. Swift: arge hermoacousticngine 1555



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equation s found, with pressures nd volume velocitiesmatched at the junctions betweensegments.Within thestack, the wave-equation olutionmust be found simulta-neouslywith that of the enthalpy-flowequationsince hetemperature rofile s not knowna priori. Overall, given hegeometry f the apparatus,heproperties3of its materialsand of the gasused, he heaterpower,and the cold tempera-ture, the calculation ields he operating requency, he tem-peratureprofile,and he amplitude nd phase f the oscilla-tory pressureeverywhere. n this work, we will focusprimarily n tspredictionsf Tn - Tc andP 2.Full appreciation f the resultsdisplayed n Figs. 5-7requiresan understanding f someof the approximationsinvolved in the calculations. One of these involves the treat-ment of the heat that leakeddirectly from the heater o theroom, amounting o typically 70 W in the data displayed nthe figures.To determine his room heat leak, we recordedTn - TcvsQn or owenougheatsandwith hevariableacousticoad ully open) so hat the engine id not oscillate.Subtractinghecalculated eat-exchanger-to-heat-exchang-er thermal conductances of the helium and stainless-steelstackmaterial (which were already ncluded n our code'senthalpy flow calculations), we found that the room heatleak wasvery well approximated y

Qroom (0.189 W/K) ( TH -- Tc )+ (0.00077W/K 2) Tn -- Tc 2.

We incorporated his expressionnto all our calculations, yassuminghat the enthalpy low in the enginewasequal oQH -- Qroom.his is the only part of the calculation hat isobtained from our measurements.In anotherapproximation, e assumehat the rregulargeometryof the stack,shown n Fig. 3, can be modeledas a

number f 1.0-mm-diamircles. rnottet al.24 nvestigatedthe sensitivityof thermoacoustic erformanceon stackchannelgeometry, or channelshapes ncludingcircular,rectangular, nd riangular.Focusing n the functionshownhere n Eqs. (A5) and (A8), which ncorporateshe channelgeometry nto the calculation, hey showed hat circlesandsquares f equal hydraulic adius9 (i.e., circle diameterequal to squareedge ength) differ thermoacoustically yless han 8%. Sincehexagons re more nearlycircular hansquaresare, we expect the hexagon-likechannelsof ourstack o differ thermoacousticallyrom circular channelsofequal hydraulic radius by even less.Arnott et al. furthershowed hat the energy lowsshould hen be computedonthe basis f total openarea,sowe pick the numberof circularchannels n the calculations o hat they have he same otalarea as the actual open area of the stack. Exact choiceofcircle diameter used in the calculations is also not critical.For example, t 13.8bar and 1000W, we ind hat ncreasingcircle diameter 5% results in calculated values ofp2 -, Tn - Tc, andf only0.6%, 2%, and 5 X 10 5 larger,respectively. hus, the imperfectgeometryof our stackdoesnot precludeaccuratemodeling n any obviousway; we ex-pect t contributes n error of less han 8% to the calculatedvalueof P 2.We also assume hat the roughly 13-to-1 aspect-ratiorectangular hannels f our heatexchangers anbe modeled

ashaving nfiniteaspect atio. Again relyingon the work ofArnott et al., we find that this simplification roduces rrorsin the calculated ossesn the heat exchangers f only a fewpercent,huscontributingess han 1% to theerror n calcu-lated system erformance. The error in the real part of thewavevectorn the heat exchangers ue to this assumptionslarger,almost10%; but thiserror affectshe calculatedeso-nant frequencynegligibly,since the heat exchangers remuch shorter han the total lengthof the resonator.)Uncertainties n all other dimensions fthe apparatuscontributensignificantlyo error n thecalculatedalue fP . Uncertaintiesn thegas ropertieslso ontributenlysmall uncertaintiesn calculated esults,For example,n-cludinghesecondirialcoefficientorheliumn ourcode'sequationf statencreasesfndTn -- Tc by0.5%,and n-creases by0.1%,at 13.8barand1000W. Andchangingthevalues f viscositynd hermal onductivityy heiresti-mateduncertainties f -k-_% and 2%, respectively,e-sults n no more han0.5% changesn calculatedesults.We alsomade all calculations resentedn the figuresbased n Tc = 303K, even hough heexperimentalalue

varied as much as 1% around this value due to day-to-daydrifts n ourbuilding'shilledwater upplyemperature.Abouthalf of the disagreementetweenmeasuredndcal-culated requenciesn Fig. 7 is due to this simplification.Calculation at 13.8 bar and 1000 W shows hat increasingTc by 1% resultsn a 1% decreasen P 2, a 1% increaseTn - Tc, anda 0.5% ncreasen , all small noughffectsthat we will ignore hemhere.Simply dding ll of theuncertaintiesn measurementsand alculationsfP vsOndiscussedboveields totalof 16.5%.Adding hem n quadratureields % as heprob-able maximum error between measurementsand calcula-tions.Thus, he combined ncertaintiesn measurementndcalculationare insufficient o account or the 20% disagree-mentdisplayedn Fig.5. Theknown ncertaintiesrise ri-marilyrom he4% uncertaintyn themeasurementfP and he 8% uncertainty ue o modelinghe stackchannelsaal-mm circlesWecannotexplain his disagreement. edonotsuspecthatRott's heory r our mplementationf tis ncorrect. huswe suspecthat some rtifactof the appa-ratus s morecritical o performancehan expected.We operated he engine n other configurationsofurther est greementithcalculations.eremovedneofthe two aluminum ipes omprisinghe coldduct, herebyshorteningheapparatusy 1.83m. With 13.8-bar eliumgas,heenginehenoperatedt 209Hz, ascalculatedy hecode,with P 20% belowcalculated alues, nd Tn - Tc5% higherhancalculatedalues. eturningo theoriginal,full-length oldduct,we substituted.9-bar rgon or thehelium, hus educingheoperatingrequencyo 38 Hz. Inthe ow-amplitudeimit, heengine peratedt thecalculat-ed requencyowithin1%,andwithP 25%below alcu-latedvalues, ndTn - Tc 4% above alculated alues. Wewill discussigher-amplitudeperation ithargon t engthin the next section. )Finally, n Figs.8 and 9 we presentmeasurementsndcalculationsncluding he effects f the variableacousticload.Note hat these ataextendo 2 imeshigher eater

1556 J. Aoust.o. m.,Vol.92, No.3, September 992 G.W. Swift: arge hermoaousticngine 1556



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0.15

0.006OO

//" no load

ulload(b)

2500 , 2000(W)FIG. 8. (a) Squarefendpressuremplitude,nd b) temperatureiffer-ence, ersus eaterpower or 6.9-barhelium,with andwithoutan acousticload.Unloadedata elow 000W were lso resentednFigs. and .Thepower issipatedn he oads 540W/bar )P 2 . Pointsremeasurements;lines are calculations.

k_

500.(b)

2000 ..... Q, (W) 6000FIG. 10. (a) Square f endpressure mplitude, nd (b) temperature iffer-ence,versus eaterpower or 13.8-barheliumat heaterpowers p to 5400W. Data below2500 W were alsopresentedn Fig. 9. Pointsare measure-ments; inesare calculations.

powershan hose fFigs. and6.Theacousticower eliv-ered o he oad anbeobtainedymultiplyingheconstantgivenn thecaptiony hesquaref hepressuremplitudein hecorrespondingigure;fficiencys hen ound ydivid-ing by heaterpower.0.5

0.0400

no loadfh111loadload, i

(b)

2000 , I(W) 2000FIG. 9. (a) Squarefend ressuremplitude,nd b) temperatureiffer-ence, ersus eater ower or 13.8-bar elium,with andwithoutanacousticload. nloadedata elow 000W were lso resentednFigs. and .Thepowerdissipatednder ull load s (490/W/bar2)p 2 Points remeasure-'ments; ines are calculations.

Both unloadedand oaded, he agreement etweenmea-surements nd calculations n Figs. 5 and 6 is as good asbefore n the imit of low amplitude;but deviates ignificant-ly more at higheramplitude,with a substantiallyinear tem-perature rror,anddownwardurvaturen theP measure-ments (most visible in the no-load data). We deferdiscussion f thesehigh-amplitudedeviationsuntil the nextsection.The low-amplitudeagreement rovides urther con-firmation of the conclusions discussed above.III. SOME EFFECTS AT NON-NEGLIGIBLE AMPLITUDE

As mentioned at the end of the last section, he behaviorof the engineat high amplitude deviatedmeasurably romthe predictionsof Rott's theory. An exampleof this devi-ation is displayed in Fig. 10, showing agreementwhichsteadilydecreases t higher amplitudes. n this section,wewill begin o explore he causes f the deviation.Clearly,nonlinearphenomena re to be expected,withthesepressure mplitudes ear 10% of the meanpressure,and acoustic eynolds umbers5approaching000.Thesmoothnessf the data n Fig. 10 suggestshat the nonlinearphenomenaof interesthere do not have a suddenonset (butseebelow). Hence,we will maintaina perturbation-theoryfocus.However,sincea rigorous xtension f Rott's theoryto higherorder than the first (acoustic pproximation)ap-pearsexceedingly hallenging,we proceedhere guidedbyexperiment nd intuition.Many of the interpretations ndconclusionsresentedn subsection below equire urtherexperimental onfirmation, nd presentconsiderableheo-retical challenges.

1557 J.Acoust.oc. m., ol. 2,No. ,September992 G.W. wift:argehermoacousticngine 1557Downloaded 15 May 2013 to 130.209.6.43. Redistribution subject to ASA license or copyright; see http://asadl.org/terms


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0.02

0.00

[]

'x xx

[] 6.9 bar helium, no load 6.9 bar helium, full loadv 13.8 bar helium, no load 13.8 bar helium, full load+ 6.9 bar argon, no load

With inserts, no load'o 6.9 bar heliumx 6.9 bar argon0.00 (pE/pm) 0.02

FIG. 11. First harmonicamplitudeversus quareof fundamental mpli-tude.Botharenormalized y meanpressure. ppersets f data (a subset fthosepresentedn previousigures)are with no cold-ductnserts.n thelowerdata sets, old-duct nserts ignificantly uppresshe harmonic.

A. Harmonic contentExperimentally, he mostobvious onlineareffectpres-ent was the high harmoniccontent n the pressure scilla-tions n the resonator. or example, t the highestpowers,the amplitudePE2 at twice he fundamental'srequencywas1/3 of the amplitudePt of the fundamental both measuredwith the same transducer, at the cold end of the resonator,

using ockinamplifiers).Higher harmonicsweresuccessive-ly smallerby about he same actor.Figure 11 displays e2,whichwasproportionaloP . Remarkably,heproportion-alityconstantetweenE2/P,and Pt/P, )2 s thesamefor a broad range of operatingconditions.Whatever thesource fPE2, it is present, nd arge,sowe mustaccount orits thermoacoustic ffects: t will transportheat along thestack.

Because he time average of products ike cos(cot)X cos(co2t)s zero for co%w2,consideration f Rott's theo-ry shows hat, in the presenceof two superposed oundwaves, he total heat low s simply he sumof the heat flowsdue to each soundwave separately.Thus, we propose hefollowingpartial model or the engine's ehaviorat high am-plitudeswith significantharmoniccontent.Nonlinear pro-cessesomehow enerate he harmonics rom the fundamen-tal mode, thereby absorbing acoustic power from thefundamental. This additional acoustic load on the funda-mentalmoderequires higher Tu -- Tc; additionally, heharmonicsndependentlyarry dditionaleatQn rom otto cold (or, in principle, rom cold to hot).Readerswhoare amiliarwith theshort-enginepproxi-mation,presentedn Secs.IB and V A of Ref. 1, will ap-preciate hat this modelpredicts he correct unctional ormfor the deviations f the data n Fig. 10 from the linear heo-ry. The excesseat lowcarriedby Pe2 will benearlypropor-tional o (P2)2 andhenceo (Pe)4,as s thecasen Fig.10(a). Furthermore, he acoustic owerrequired o main-tainP2 will alsobeproportionalo (Pe2 2 andhenceo(P ), so hat heexcesscousticower bsorbedrom he

fundamentalmustshare his proportionality.his impliesthat heexcessemperatureifferenceequired y he unda-mental hould eproportionalo P , whichs tself earlyproportionaloQn.This ependencesvisiblenFig.10(b).To test his modelagainst ur data,we used he samecomputer ode s n Sec. I, runningt for both he irstandsecondmodes.n the calculations, e constrainedhe ampli-tude of the secondmoderelative o that of the first accordingto the data n Fig. 11,and, or simplicity, ssumedhat theacoustic ower equired y the secondmodewasdeliveredto it at the closed end of the cold duct. We assumed that anequal mount f acousticowerwas emovedrom he irstmode at the same ocation. (This locationwas picked foreaseof calculation--not becauset is the expected ocationfor conversionf power rom the first o the secondmode.Thecombinedode uns henpredicted vs Qn andTn -- Tc vsQnwith heshapeshowny heexperimentaldata n Fig. 10,andwithP magnitudesloseo thedata,and Tn -- Tc magnitudesoughly third of the way romthe original, inear heory o the data.Encouraged y thesequalitatively orrect esults,we

5OOO

06OO

(o) //

Helium:/ / -- No inserts

/// --onserts .rn%/ + xr --- -+ No inserts x Inserts

i

(b)

200o.oo o.(1(RE/Pro)FIG. 12.Comparisonf measurementsndcalculations,:9-bar eliumand rgon, ithandwithoutold-ductnsertsosuppresshe irst armon-ic.Addition f he nsertsmprovedothperformancendagreementithcalculations.

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6000modifiedheapparatuso suppressE2.We put wo 76-mm-diam nserts, achsharpenedn each nd o a 30point, ntothe cold duct, near the velocity antinodesof the secondmode.The exact ocations nd lengthsof the insertswerechosen o leave the fundamental'srequencyunchangedwhile shifting he frequency f the secondmodedown 9%.(This shiftwasof the orderof 2/Q, whereQ is the qualityfactorof the secondmode,calculatedncluding heeffects fthe heat exchangersnd of the stackand its temperaturegradient. With the nsertsn place,nonlinear rocessestillgeneratedPE2 from the fundamental,at twice the funda-mental's requencyjbut 2fwas no longernear a resonancefrequency, o he amplitudeof Pc2 wasreducedby aboutafactorof 5, asshown n Fig. 11.Thus,we wouldexpect 25-fold decreasen the excesshermoacousticeat ransportedby the secondharmonic.Figure 12 comparesmeasurements nd calculations,with and without he nserts.Calculations sed he original,linear theory, just as described n Sec. I; the insertsshiftthese alculatedesults lightly, y ncreasinghepowerdis-sipatedn thecoldductat a givenpressuremplitude, ndby

changinghe ratio of Pe to pressure mplitude t the stack.The insertsdramatically reduced he measuredheat carriedby the engine,and alsoreduced he measuredemperaturedifference.This confirms he ability of resonantharmonicsto carry significantamounts of thermoacousticheat, andsuggestshat high-performancehermoacoustic nginesshould ave esonatorsailored o suppressarmonics.With the insertsn place, he 15% agreementn Fig.12 a) between easurementsnd alculationsfP vsQin helium saboutasgood t highamplitudess t wasat lowamplitudes20%) in Fig. 5. Thesamewas rueat thehigh-power, heavily oadeddatum discussed t the end of the in-troduction.ighamplitudegreementfP vsQ forar-gon sbetterwith he nsertshanwithout, utnotasgood sat low amplitudes.B. Temperature difference

With theamplitude f thesecond armonic uppressedwith the cold-ductnserts,he fractional isagreemente-tween measured and calculated heat flows in helium is com-parable t high and ow acoustic ressure mplitudes. utthe measuredemperature ifferenceshownn Fig. 12 b)arestillmuch argerat highamplitudeshanat low ampli-tudes.n thissection e study hreecandidate xplanationsfor this remainingamplitude-dependentisagreement.Noneof thecandidatesccounts ell or these ata;all invitefurther research.

First, the realization hat the peak-to-peak isplace-mentamplitude f thegas n theheatexchangersxceedshelengthof theheatexchangerst P/Pm >0.04suggestshatour theoreticalreatment f the heat exchangerss ductswith complexwavevectorsmay be much too naive, andfurther hat at the highest mplitudeshe gasmightbe inverypoor hermal ontactwith the heatexchangers.hus,to test whether mproved hermalcontact n the heat ex-changers ould educe he excessemperature ifferencenthe measurements,we constructed a secondcold heat ex-changer, dentical o the first, and installed t in serieswith

5OO

(o)

x....;;-;;' Argon,nserts:..-..... coldeatxch.1 cold heat exch.i i

I

200 ' '0.00 (RE/pro)2 0.03FIG. 13.Comparisonf measurementsndcalculations,.9-bar rgon ndwith cold-duct nserts,with 1 and 2 cold heat exchangers. ddition of thesecond eatexchanger urt performance ut had ittle effecton agreementwith calculations.Short vertical inesmark where the peak displacementamplitude qualshe heat-exchangerength.

the first, effectively oubling he lengthof the cold heat ex-changer.Measurements and calculations with one and two coldheatexchangersre comparedn Fig. 13, for 6.9-barargon.The additionof the second eat exchanger id not improveperformance, r agreementwith theory,evenat displace-ment amplitudesmuchhigher han the lengthof the singleheatexchanger.We find t remarkablehat there s no con-sistentsignof transition n behaviornear the amplitudes,markedby shortvertical ines,where he peakdisplacementamplitude quals he heat-exchangerength. These esultsalsosuggesthat a reductionn heatexchangerengthmightimprove he engine's erformance.

A second andidate xplanationor the observed xcesstemperature ifference t highamplitude lso equires on-sideration f a new effect n the heatexchangers.or clarity,let x be the directionalong he axisof the apparatus.Mostprevious hermoacousticheoryhas used he assumptionthat the temperature in the gas is of the formtotT(x,y,z,t) = Tsox) + T(x,y,z)e' , with the mean (i.e.,time-averaged) as emperaturendependent f lateral di-mensions ,z and equal to the adjacentsolid temperatureTsox). With thisassumption,he timeaverage f the heattransfererunitarea o hesolid ,z Tiao, 0. Here,K isthegas onductivity,y.zs he wo-dimensionalradientntheyzplane, heoverbar enotesimeaverage, nd hederiv-ative s evaluated t the gas-solidboundary.Thus, the pre-viously ssumedormof T(x,y,z,t) isappropriateor ateral-ly thermally solated egionssuch as the stack, whosesteady-state ondition implies zero time-averagedheattransferbetweensolidand adjacentgas.In contrast,n the heatexchangers,here s a significant

1559 J. Acoust. oc.Am.,Vol.92, No.3, September 992 G.W. Swift:Large hermoacousticngine 1559



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nonzero ime-averaged eat transferbetweensolid and gas.Hence, he assumed orm for T(x,y,z,t) must be reconsi-dered.For clarity, consider he hot heat exchanger,wheretime-averagedeat0n is transferredromsolid o gas.(Here,weneglectheparts f Qn that low o the oom,down the duct wall, and along x due to ordinary conduc-tion.) Denote by A the effectivesurfacearea of gas-solidcontactcrosshichQn s ransferredn theheat xchang-er, and assume parallel-plate eometrywith y = 0 in thecenterof the gaschannels, o that the z dependence an beneglected.heheatQn canonly low romsolido gasfO, =AK T/Ylso.With this nonzeroime-averagedtemperature radient n the gas,we then expect

T(x,y,z,t) = Tsox) + Tl(x,y)ei't-{6T(x,y),with 6T(x,y)--Qn(y--yo)?AK ear the boundarytY = Yo.We expect he latter expressiono be approximatelytrue for (y- Yo) 6, we expectaspatial-averagedemperature differencebetween gas andsolid

(6T)= 1 6T(x,y)dy .Yo ao AKNote hat (6T) isnotan oscillatory uantityreit sa spatial-and time-averagedemperature ifference etweengasandsolid. In the hot heat exchanger, he solid is, on average,hotter than the gas; n the cold heat exchanger, he solid s,on average,colder than the gas.

To estimate he magnitudeand amplitudedependenceof (6T), wemustestimate . At thehighest coustic mpli-tudes,A shouldbe the actual surfacearea of gas-solidcon-tact in the heat exchanger. In this case, we expect(6T) oc 2 sincen oc Butat owacousticmplitudes,we expect hat only that part of the heatexchangerwithin adistanceof about 2xl of the stack,wherex is the displace-mentamplitude,contributes ffectivelyo the time-averagedheat transfer (cf. Sec. II A, Ref. 1 . Thus A is reduced fromits high-amplitude alueby roughly2x/Lhx, whereLhx isthe engthof the heat exchanger long he axisof the appara-tus.Since cc EandQnoc , we hen xpectt owam-plitudes 6 T ) ocQn/x oc e. This irst-order ime-averagetemperature ifferences nterestingn viewof the explicitlysecond-orderemperaturedifference nd wall heat flux ex-pressionsonsideredreviously.'9

To look for these effects, we installed a new, 0.5-mm-diam sheathedhermocouple,whose emperaturewe denote.gas n the gasbetween he stackand the cold heat ex-y c ,changer.t was ocated adially3.5cm from hecenterof theengine, ndaxially n theheatexchangermm rom heheatexchanger-stacknterface.Although t casuallyouched neheat exchangerin a few mm from its tip, the tip itselfwasfreely suspendedn the gas,so its temperature houldbesome patialaverage f the time average f the gas empera-ture. t is not smallenough o probe hey dependencef T.

4O

oo.ox

PE/PFIG. 14.Temperatureifference ithincoldheatexchangerersus res-sure mplitude,howingn nitiallyinear elation. hesymbolsave hesamemeaning s n previousigures.

As discussedn Sec. , the thermalcontactbetweenhecooling ater at Tc and he inssexcellent,oTc isessen-tially the solid emperature. hus, Tgasc -- Tc shouldbe anapproximateeasuref (6T). Figure 4displays easure-ments fthisquantity ersus e?Pm,nitially howinglin-ear dependence,s predictedwo paragraphsbove.Athigheramplitudes,gsc -- Tc growsmorequickly han in-early, howinghe educedhermal ontactxpecteds hegeometricalas-fin ontactreabecomeselevant.Thiseffectsclearly oosmall o accountor muchof theexcessemperatureifferencen Figs.12and13.Neverthe-less,heeffect xists, nd s argeenougho beof concernnsome roposedpplications,speciallyigh-amplitude,ow-temperature-differencehermoacousticefrigerators6andheatpumps, here, t Pe/P,,, 0.1, a temperatureiffer-ence near 20 C could exist in each of the two heat ex-changers.eadersamiliarwith heshort-enginepproxi-mation can easilyshow, ollowing he argument hreeparagraphsbove,hatat lowamplitudeseshould ave(6 T ) T,,,Pe/P,,,,of thesame rderof magnitudes hemeasurementsn Fig. 14.At thehighestmplitudesresentedn Figs.12and13,theReynoldsumbern thestacks of theorderof 300 orheliumand 1000 or argon.Here weuse heviscous enetra-tiondepth s hecharacteristicengthorcomputingeyn-oldsnumber; seof thehoneycombhannel iameteresultsin Reynolds umbers and 5 times arger, espectively.Thus,we mustconsiderurbulence sa third possibleauseof the observed igh temperature ifferences.urbulentflow woulddissipate xtraacoustic ower,acting ike anadditional, mplitude-dependentoadon the engine. uchan additional oad has a much greatereffecton Tn- TcthanonQn vsP (cf.Figs. and9), consistentith hemeasurementshown n Figs. 12 and 13.With the additional realization that the gas must turnsharp ornerst theheat xchanger-stackndheat xchang-er-ductunctions,o avoid heblocked reas f theheatex-changers, ecanmake crude stimate f theexpectedr-derof magnitudef turbulent ressurerops ndattendantdissipatedowers.n unidirectional,ullydevelopedurbu-lent low hrough sharp orner,hepressurerop sof the

1560 J. Acoust. oc.Am.,Vol.92, No.3, September 992 G.W. Swift:Large hermoacousticngine 1560



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order fpu ,with thedensitynduthevelocity,nd encethe issipatedowerisof he rderfApu,where s hecross-sectionalreaof he low. f weassumehisexpressionis alsoapplicableo oscillatorylow,we find or Pm= 6.9barandPE/Pm= 0.1 thatE- 1 kW for argon nd3 kW forhelium.Thesepowers re n the correct atio,andonlyoneorder of magnitude oo large, to account or the observed80C (argon) and 250C (helium, extrapolated) xcesstemperature ifferencest PE/Pm--0.1shown n Fig. 12.We take his ough greementsevidenceor heplausibilityof turbulence s an explanation or the excessive aluesofTn -- Tc observed.Unfortunately, he amplitudedependence f the datadoesnot support his interpretation.Miiller and Lang,TMwho also observedarge excessemperature ifferences thighamplitudes, howedhat a temperature ifferencehatincreasesroportionalo PE is characteristicf fully devel-oped urbulence, hilean ncreasesP r s characteristicfnonlinearaminar ffects,ndan ncreasesP r scharacter-isticof partiallydevelopedurbulence hose patialextentgrowswith amplitude. They founda clearcubicdepend-encen theirmeasurements.Our data,shownn Figs.12 b)and 13(b), do not fit any of thesescenarios onvincingly.The excessemperature ifferencesppear ubstantiallyin-earwhenplotted gainst r asshown, ut theargon ataclearly how noffset f (Pr/Pm) ,0.004 eforeheexcesstemperature ifference eginsncreasing. hus,we havenoquantitative videncehat turbulencesresponsibleor theseexcessemperaturedifferences.ACKNOWLEDGMENTS

We aregrateful o ChrisEspinoza, ndy Fusco, ndBillWard for constructionf manyof the componentsf theengine,nd o DaveGardner ndBillWard orexploratorymeasurementsith earlierversions f the engine.We thankDickMartin orsome f hecomputerrogrammingnd orcritical onversations.hisresearch assupportedy theAdvancedndustrial onceptsivisionn theU.S.Depart-mentof Energy's ffice f Industrial echnologies.APPENDIX

Here, we presentdetailsof our thermoacoustic alcula-tion techniques. irst, we discusshe FORTRAN odeused ogeneratehe curvesn the figures f the mainbodyof thepaper.Second, e present simpler, pproximate alcula-tion echniquehat susefulordiscoveringrends ndpre-dicting approximatebehavior. We assumehere that thereader is familiar with the fundamentals of Rott's work. 3-8'1

Letx denotehedistancelongheaxisof heapparatusof Fig. 1, with x = 0 at the closedend of the hot duct. Webeginherewithguessedaluesor , Tn, and hepressureamplitudePu (taken to be real), and withdpl ico2(-- 1 tSKP= (A1)dx 2a 2

to accountorabsorptionf sound y hegas' hermal ene-tration epth t hex = 0 end ap.Here = x/-- 1, co 2rrf,

7/is the ratio of isobaric o isochoric pecific eats, SKs thethermal enetrationepth, nda is hesound peed.3To integratealong the hot duct, we set dT/dx = 0 inRott'swave quation6 or circular eometry,inding

1 dplp,(x) p(x 0)cos(kx)-xx x= 0)sin(kx),(A2)

with the complexwavevectork givenbyk=co(lq-itSv-i y_l)tS),A3)2 R - 2for duct radiusR >>Sand tSv,where S s the viscous ene-tration depth.This yieldsp and dp/dx at the end of the hotduct adjacent o the hot heat exchanger. f the duct is notcircular (as in the case of the cold duct with inserts), R isreplacedby twice the area dividedby the perimeter.Entering he hot heat exchanger rom the hot duct, weimposecontinuity of pressureand of volumetric velocity.The latter condition takes the form

d/Ol) (d/Ol)Aduct1 - 1 -i)tSv/RdX hx , dX duct Zhx (1 --f,hx (A4)Here, A is openarea, and

f,.h,- anh1+ )Yo/tS, (A5)( 1 + i)yo/tS,reflectingour assumptionhat the heat exchanger's arallelrectangulargaschannels an be modeledwith sufficient c-curacy as infinite parallel-plategeometrywith separation

We then usewavepropagation hrough he hot heat ex-changer ccording o Rott's waveequation or parallel-plategeometry nd dT/dx = 0. The solution s of the form of Eq.(A2) but withk= o/1+-7/-)f (A6)1--fv 'wheref is givenby Eq. (AS), andf by a similarexpressionwith tS eplacing S.This yieldspanddp,/dx at the far endof the hot heat exchanger.Leaving he hot heat exchanger nd entering he stack,we again mposecontinuityof pressure nd volumetricve-locity. The latter condition now takes he form

(X) =(dpl} hx1--fv,x). (A7)tack !kdx [hxAstac ( 1 --fv,stack)In the stack, which we model as an assembly f circularchannels f radius o, v takeson the form

f,= 2,1,[i- 1)ro/tS (AS)( -- 1) ( ro/tS,Jo[ -- 1)ro/tS]where J is a Bessel function.

The heart of our complete hermoacoustics ode is afourth-orderRunge-Kutta 7 integration f Rott's waveequation ndenthalpy-flowquation8 n the stack, sde-scribed n some detail in the appendixof Ref. 1. It usesh2 H -- room s the (x-independent)otalenthalpy

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flow hroughhestack, ue o considerationf energylowsintoandoutof a control olume nclosinghehotendcap,hot duct,hot heatexchanger,nd an arbitraryengthofstack djacento the hot heatexchanger.his ntegrationgenerateshe mean emperaturend (complex)pressureamplitudehroughouthestack, tarting ith heirvalues tthe hot end of the stack.The calculationhenproceedsrom he coldendof thestack hrough he coldheat exchangernd coldduct,ac-cording o the sameprinciples s used n the hot heat ex-changerndhotduct:uniform-temperatureavepropaga-tion with complexwavevector, nd with continuityofpressure nd of volumevelocityat the unctions. If mode-shiftingnserts re usedn the coldduct, hese rinciplesmustbeapplied newat each hange f area.We treated he30 tapersat the inserts'endssimply as discontinuouschangesn areaat the apers'midpoints.The procedure utlined o far thus indsTc and thecomplexmpedancet theendof hecoldduct,having tart-ed with assumedalues or , Ta, andPa. Generally, hevalues f Tc andcoldend mpedanceound herewill notbe

thedesired alues,ndicatingncorrectnitialguessesor ,Ta, andPa. A shooting-methodlgorithm7 henadjusts,Ta, andPa, re-runninghe entire ntegration, ntil the de-sired alues f Tc andcomplex nd mpedancere eached.The code hen prints out whatever esults he user sinterestedn, typicallyemperature,ressuremplitude,m-pedance, nd acoustic ower low at various ocations, ndfrequency.Thecalculationechniqueescribedbove,hough p-parentlyquiteaccurate, s too cumbersomeo use n the ini-tial stagesf design f a new hermoacousticngine.n fact,it requireseasonablyoodguessesor frequency,ressureamplitude, nd hot temperature: ith poorguessest canconvergen hewrong esonator ode, r ntegratetswayto valuesof temperature elow absolute ero, and crash.Thus, simpler,houghmore pproximate,echniquesre-quired or generalexploration f designs nd for initialguessesor the moresophisticatedechnique.We call ourapproximateechnique he short-engine pproximation,sincet containss tscorewhatwas eferredoby hatnamein Ref. 1. t requires ightsteps,whichweoutline elow.Weusually se he short-enginepproximationn spreadsheetsoftware,ut t canalso eworked utwitha pocket alcula-tor, or embeddedn graphics oftwareo quicklyexploretrends f dependencesf performancen geometrynd helike.We have ound his echniqueo predict emperaturedifferencesithinabout 0%, and requenciesithin4%; itusually redicts heat low oo arge,by asmuch sa factorof 2.

We beginwitha chosen eometry,as,meanpressure,andcold emperature.n the irststep,we compute fromthe resonator eometry,gnoring he stackand heat ex-changersndassuminghat heentire esonatorsat Tc. Ifthe resonator assubstantiallyniform ross ection,his sassimple sdividingwice he esonatorength y hesoundspeedat Tc. Otherwise,we model t as a seriesof uniform-cross-sectionucts,mposingontinuity f pressurendvol-umevelocity t eachunction.We ncludehe maginary art

of any load mpedance suchas radiation mpedance) f ap-propriate.(Use of Tc for the entire resonator n this step, nsteadof taking the hot end to be at Ta, is actuallya very goodapproximation. he compressibilityf a gas s temperature-independent, hile he nertiadepends tronglyon tempera-ture. Thus he resonancerequencys substantiallyndepen-dent of the temperature at the pressuremaxima of thestandingwave, and dependsmostly on the temperatureatthe velocity maxima.In step ,wecompute re/P , whereWre s hepow-er dissipatedn the resonator, nd Pa is the pressure mpli-tude at the hot end. As above,we ignore he presence f thestackand heat exchangers, nd use Tc throughout. Vre iscomputedusingEq. (89) or (90) of Ref. 1, as appropriate.(The small ncreasen accuracy n this step hat would bepossiblehrough nclusionof Ta is seldomworth the effort,in view of the inaccuracyof steps6 and 8 below.In step ,weobtainWload/P, whereWloads hepowerdelivered o the load. This requiresknowledgeof the loadcharacteristics,uchas radiation mpedancef the engine sintended as a sound source.

In step4, we guessTa, and calculate l/Pa and u 1/Paat the midpointsof eachheat exchanger nd the stack.Weuse ossless ave propagation,startingwith Pa at the hotend, using Tn in the hot duct and hot heat exchanger,( Tn q- Tc )/2 in thestack, ndTc in thecoldheatexchang-er. At each unction we usecontinuityof pressure nd vol-ume velocity.In step wecalculate m,,,/PandWc,,,/P,whereWnh,,ndWc are hepowersissipatedn hehot nd oldheatexchangers,espectively.We useEq. (89) of Ref. 1, andthe u's andp's computedpreviously.Next, in step 6 we set the sum of all these works(W e q-Wload-Wnhq-WCh/P equalo Wstack/P/,whereWstacis heexpressionorshort-engineork ivenbyEq. (80) of Ref. 1. UsingEqs. (74) and (75) of Ref. 1, wesolve or TH. Note that TH is independent f PH.In step7, we teratesteps through6 if the resultingTHwasmuch different rom the initial guess.Finally, in step8 we pick PH and use he short-engineheat equation,Eq. (76) of Ref. 1, to compute he required

heat flow.IG. W. Swift,"Thermoacousticngines," . Acoust.Soc.Am. 84, 1145(1988).2LordRayleighJ. W. Strutt),TheTheory fSound Dover,NewYork,1945), 2nd ed., Vol. 2, Sec. 322.3N.Rott,"Damped nd hermally riven cousticscillationsn wide ndnarrow tubes,"Z. Angew. Math. Phys.20, 230 (1969).4N. Rott, "Thermally riven cousticscillations,art I: Stabilityimitfor helium," Z. Angew. Math. Phys.24, 54 (1973).5N. Rott,"Thermally riven cousticscillations,art II: Second-orderheat flux," Z. Angew. Math. Phys.26, 43 (1975).6N.RottandG. Zouzoulas,Thermally riven cousticscillations,artIV: Tubeswith variablecross-section,". Angew. Math. Phys.27, 197(1976).7G.ZouzoulasndN. Rott,"Thermally riven cousticscillations,artV: Gas-liquidoscillations," . Angew.Math. Phys.27, 325 (1976).8U.A. MiillerandN. Rott,"Thermally riven cousticscillations,artVI: Excitationand power,"Z. Angew.Math. Phys.34, 609 (1983).9p.Merkli andH. Thomann, Thermoacousticffectsn a resonantube,"J. Fluid Mech. 70, 161 (1975).

1562 J. Acoust. oc.Am.,Vol.92, No.3, September992 G.W. Swift: arge hermoacousticngine 1562



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lT. azaki,. Tominaga,nd . Narahara,Experimentsn hermallydrivencousticscillationsfgaseouselium,".LowTemp. hys. 1,45(1980).U. A.Miiller nd .Lang,Experimenteit hermischetriebenenas-Fliissigkeits-Schwingungen,". Angew.Math.Phys. 6, 358 1985).:T.Hofler,Thermoacousticefrigeratoresignnd erformance,"h.D.dissertation,hysicsepartment,niversityfCaliforniatSanDiego,1986;Conceptsor hermoacousticefrigerationnd practicalevice,"Proceedingsf the 5th nternational ryocooler onference,8-19Au-gust1988,Monterey,CA, p. 93.3A. . Atchley,. E.Bass,. J.Hofler,ndH.-T.Lin, Studyfa ther-moacousticrimemoverbelowonset f self-oscillation,". Acoust.Soc.Am. 91, 734 (1992).14T. .Gabrielson,Radiationrom submergedhermoacousticource,"J. Acoust. oc.Am. 90,2628 1991; W. C. WardandM. A. Merrigan,"Recent evelopmentsn thermoacousticallyrivenow-frequencyro-jectors,"o bepublishedn Proc3rd nt. Workshop nTransducersorSonic ndUltrasonics, rlando, L, 6-8 May 1992.SG.W. Swift, . Radebaugh,ndR. A. Martin, Acousticryocooler,"U.S.PatentNo. 4,953,3664 September990).16S.. Garrett ndT. J. Hofler,Thermoacousticefrigeration,"roc.Technology001:heSecondational echnologyransfer onferenceandExposition,December991, an ose A.;Popular ci. 39(1 , 44(1991).7T. azaki, .Takashima,nd . Mizutani,Complexuasiperiodicndchaotictatesbservedn hermallynducedscillationsfgas olumns,"Phys.Rev. Lett. 58, 1108 (1987).8Kentuckyetals, ewAlbany,ndiana.9The ydraulicadiusfa channelsconventionallyefineds hechan-nel's ross-sectionalrea ivided y tsperimeter.hus,hehydraulica-

dius of a circular channel s 1/4 the circle'sdiameter; hat of a squarechannel s 1/4 its edge ength.:Model 02A05, CBPiezotronics,nc., Depew,New York.:G. W. Swift,A. Migliori,S. L. Garrett,andJ. C. Wheatley, Twometh-ods or absolute alibrationof dynamicpressureransducers," ev. Sci.Instrum. 53, 1906 (1982).::A.M. Fusco,W. C. Ward,andG. W. Swift, Two-sensorowermeasure-ments n lossyducts,"J. Acoust.Soc.Am. 91, 2229-2235 (1992).:3Forhelium,weuse dealgas aluesor density, pecificeats, ndsoundspeed. or viscosity/z, e fit a power aw to the table n TherrnophysicalProperties f Matter. the TPRC Data Series, ditedby Y. S. Touloukian(Plenum, New York, 1970), finding z=(0.4120X10 -6 kg/m-S)T '684,ith T in Kelvin.For thermal onductivity, we it a powerlaw to the table in C. Y. Ho, R. W. Powell, and P. E. Liley, "Thermalconductivity f the elements," . Phys.Chem. Ref. Data 1, 279 (1972),findingK--(0.002567 W/m-K)T '76.or stainless teel,we. useK = ( 14.9W/m-K) + (0.0124W/m-K:)( T-273 K), also rom Toulou-kian's book.

:4W.P. Arnott,H. E. Bass, ndR. Raspet, Generalormulation f ther-moacousticsor stacks avingarbitrarilyshaped ore cross ection.s,".Acoust. Soc. Am. 90, 3228 ( 1991 .:sP.MerkliandH. Thomann,Transitiono turbulencen oscillatingipeflow," J. Fluid Mech. 68, 567 (1975); M. Hino, M. Sawamoto, and S.Takasu,"Experiments n transition o turbulencen an oscillatory ipeflow," ibid. 75, 193 (1976).:6Equation19), Ref.3;or Eq. (A19), Ref. 1.:7B.Carnahan, . A. Luther, ndJ.O. Wilkes,4pplied urnencalMethods(Wiley, New York, 1969), or any modernnumerical-methodsext.:SReference, Eq. (10), or Ref. 1,Eq. (A30).

1563 J.Acoust.oc.Am., ol. 2,No. , September992 G.W.Swift: argehermoacousticngine '1563

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