On the Use of a Complex Frequency for the Description of Thermoacoustic Engines

HAL Id: hal-02057411

https://hal-univ-lemans.archives-ouvertes.fr/hal-02057411

Submitted on 5 Mar 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Matthieu Guédra, Guillaume Penelet

To cite this version:

Matthieu Guédra, Guillaume Penelet. On the Use of a Complex Frequency for the Description of Thermoacoustic Engines. Acta Acustica united with Acustica, Hirzel Verlag, 2012, 98 (2), pp.232-241.

�hal-02057411�

M.Guedra

a,∗

, G. Penelet

a

.

a

Laboratoired'Aoustiquede l'UniversitéduMaine, UMR CNRS6613, Avenue OlivierMessiaen, 72085 Le

Mans Cedex 9, FRANCE

∗

Corresponding author,E-mail: matthieu.guedra.etuuniv-lemans.fr,fax.: +33-2-43-83-35-20

Abstrat

Inthispaper,aformulationisproposedtodesribetheproessofthermoaoustiampliationinthermoaousti

engines. This formulation is based on the introdution of aomplex frequeny whih is alulated from the

transfer matries ofthe thermoaousti oreand itssurrounding omponents. The real partof this omplex

frequenyrepresentsthefrequenyofself-sustainedaoustiosillations,whileitsimaginarypartharaterizes

the ampliation/attenuationof thewavedue to thethermoaousti proess. This formalisman be applied

to any typeof thermoaousti engineinluding stak-basedorregenerator-based systemsas well as straight,

losed looporoaxialdut geometries. Itanbeapplied tothealulationofthethresholdofthermoaousti

instability,but itis alsowell-suitedfor thedesriptionof thetransient regimeof waveamplitudegrowthand

saturationduetononlinearproesses. All oftheabovementionedaspetsaredesribedinthispaper.

PACS numbers: 43.35.Ud

1 Introdution

Thermoaousti engines belong to a type of heat engines in whih aousti work is produed by exploiting

thetemperaturegradientbetweenahotsoureandaoldsink[1,2℄. Typialarrangementsofthermoaousti

enginesareshowninFig. 1. Itonsistsbasiallyofanaoustiresonatorpartiallylledwithapieeofanopen-

ellporousmaterial,oftenreferredtoasastakoraregenerator. Animportanttemperaturegradientisimposed

along this stak/regenerator, sothat above aritial temperature gradient, aoustimodes of the resonator

an beome unstable and the thermoaousti proess results in the onset of self-sustained, large amplitude

aoustiosillations. Suhkind ofengineshavebeenextensively studied in thepastdeades [3℄, leadingto a

deeperunderstandingoftheiroperationandtothebuildingofafewdeviessuhasthermoaoustiallydriven

thermoaoustirefrigeratorsorthermoeletrigenerators. Theseengineshaveinterestingfeaturesinherenttothe

abseneofmovingparts(pistonsandrankshafts)whihanbeadvantageouslyusedforindustrialappliations

at moderate power densities (typially up to a few kilowatts). It is however worth noting that the design

of thermoaoustiengines is a tedious task whih omprisesan important partof unertainties, beause the

operationof theseenginesis bynaturenonlinear,andbeausetheexistingeientprototypesinlude various

elementslikeowstraighteners,taperedtubes,membranesorjetpumpswhiharediulttomodelaurately.

ManyresearhersusethefreelyavailablesoftwarepakagealledDeltaECdevelopedatLosAlamosNational

Laboratory[4℄forthe designofthermoaousti systems. This omputerodeis averypowerfultoolwhih is

mainlybasedonthelinear(andweaklynonlinear)thermoaoustitheoryinthefrequenydomain. Besidesthe

limitations of this omputer ode for large aoustiamplitudes requiring proper aount of nonlinear eets,

another of its harateristis is that it is expressed in the Fourierdomain, so that it desribes steady state

onditions : the steady-state aousti pressure amplitude is obtained from a temperature eld whih itself

is ontrolled by theaousti eld due to aoustially indued heat transport. The multi-parameter shooting

method whihisemployedinthisomputerodeiswellsuitedforthepreditionofanequilibriumstateabove

the threshold ofthermoaousti instability. However, itis not primarily devoted to thedetermination ofthe

000000 000 111111 111

0000 00 1111 11

Thermoaousti ore

(a)

(b)

x_l x_h x_r

x x_l

0 x_h x_r L x

x

to seondary aousti load

0(L) T_h

T_h T_c

T_c

Figure 1: Simplieddrawingsofastanding waveengine(a)andatravellingwaveloopengine(b), possibly

oupledwithaseondaryaoustiload.

thresholdonditionitself(i.e. therequiredexternalthermalationabovewhihthermoaoustiosillationsbegin

to growupwithtime). Moreover,under someirumstanes,thetransientproessleadingtothesteady state

asewhentheengineis operatedslightlyabovethethresholdofthermoaoustiinstability,whereompliated

eets maybeobserved. Forexample,theexisteneofahysteretiloop[5,6℄in theonsetanddampingofthe

engine,ortheperiodiswithon/oofthermoaoustiinstability[7,8℄havebeenreportedforbothstandingand

travellingwaveengines. Insuhsituations,thexedexternalthermalationonthesystemdoesnotorrespond

to auniquesteadystatesolutionfortheaoustipressureamplitude.

Thoughusefuldesigntoolsarenowadaysavailable,anauratedesriptionofthermoaoustienginesisstill

needed, and an important researheort hasbeen devoted to the desription ofthe onset of thermoaousti

instability and to its saturationby nonlineareets. Various analytial [9,10, 11, 12℄ and numerial models

[13,14℄havebeenproposedin theliterature,whihareyetlimitedtothedesriptionofsimplethermoaousti

deviesofpartiulargeometry. Inthisontext,theaimofthispaperistoproposeageneralmodellingapproah

whih is mainly based onthe transfermatries formalism. As in previous analytialworks [9, 12℄ the model

presentedin this papertakesadvantageofthesigniantdierenebetweentheinstabilitytimesaleandthe

periodof aoustiosillations,whihis exploited herebythe introdution ofaomplexfrequeny, sometimes

used for the treatmentof transientosillatory motions (note that the introdution of omplex frequenyhas

already been proposed in a onferene by J.E. Parker et al. [15℄ to treat thermoaousti osillations, and

also in a similar network approah by Q. Tu et al. [16℄). Depending on its sign, the imaginarypart of this

omplexfrequenyrepresentsanampliationoranattenuationoeient,whihisalulatedfromthelinear

thermoaousti theory applied to the thermoaousti system under onsideration. The analytial treatment

presented here is neessarily based on substantial approximations but, as will be disussed in this paper, it

is well suited to arry out extensive parametri studies of both the transient and steady states. Moreover,

this modelhassomeinterestingsimilaritieswiththeomputerodeDeltaEC inthesense that itonsistsof a

multiportnetworkapproahwhihiswell-suitedforthedesriptionofompliatedaoustinetworksinluding

thermoaoustiores,dutswithonstantorvaryingross-setions,grids,membranes,T-juntionset...The

works presented in this paper basially onsist of a generalization of previous works [10, 17℄ and its main

originalityis thus primarily toproposeto thereaderarather simplemodellingof any kindofthermoaousti

engine in order to determineits onset onditions orto desribe the transient regime leading to steady state

sound intheframeofweaklynonlineartheory.

In setion 2 the multiport network modelling of thermoaousti engines is presented, whih leads to the

analytial expression of the omplex frequeny from the transfer matries of the thermoaousti ore and

its surrounding omponents. In setion 3, this formalism is applied to the determination of the onditions

orrespondingtotheonsetofthermoaoustiinstabilityintheasesofastandingwaveengineandofalosed-

loop travelling wave engine. Setion 4is devoted to the desriptionof basi onepts onerning the use of

this approah to study the transient regime leading to steady state sound (or to more ompliated dynami

behaviorsofthethermoaoustiosillator)inthermoaoustisystems.

2 Theory

Thermoaousti engines are generally made up of a dut network inside whih the thermoaousti ore is

inserted. Thetermthermoaoustiorerefersheretotheheterogeneouslyheatedpartofthedevieinwhih

theampliationofaoustiwavesoperates: itisbasiallyomposedofanopenellporousmaterial-referred

to asthe stak (δκ ∼ r^{) or the}regenerator(δκ >> r^{) depending on thevalue of the average radius} r ^{of its}

poresrelativetothethiknessδκ^{oftheaoustithermalboundarylayer-alongwhiha}temperaturegradient is imposed using heat exhangers. As illustrated in Fig. 1, the great variety of thermoaousti engines an

be shematially separated into two dierent lasses. The rst lass of engines (Fig. 1-(a)) refers to some

onventionalwaveguidearrangementensuringtheresonaneofagasolumn. Amongthis lassofenginesare

the stak-basedstanding wave engines whih were extensively studied during the past deades, but also the

ryogeni devieswhere Taonis osillationsmay our[1℄. Theseond lass of engines(Fig. 1-(b)) refersto

somewaveguidearrangementswhere alosed-looppathexists, allowingthedevelopmentof travellingaousti

wavesrunningalongtheloop. Amongthislassofenginesarethestak-basedtravellingwaveenginerststudied

byYazakiet al. [18℄,thethermoaousti-Stirlingheatengine[19℄rstsuessfullydesignedbyBakhausetal.

[20℄,theregenerator-basedo-axialengines[21℄wherethefeedbakloopisformedbyloatingasmalldiameter

thermoaoustioreintoalargerdiameterwaveguide,andalsoasamatterofinterestsomekindsoffree-piston

Stirlingengines.

Whateverthespeigeometryoftheengineunderonsideration,allofthesedeviesusethefatthatwhen

astrong temperaturegradient isimposed alongthe stak/regenerator,partof the heat supplied isonverted

into aoustiwork insidethestak/regenerator. This thermoaoustiampliationproessresultsintheonset

ofself-sustained,largeamplitudeaoustiwavesosillatingatthefrequenyofthemostunstableaoustimode

of theomplete devie. Inthe following, theonset of thisthermoaousti instabilitywill bedesribed bythe

introdutionofaomplexfrequeny,therealpartofwhihdesribesthefrequenyofaoustiosillationsand

the imaginary partof whih desribes the waveamplitude growth orattenuation. The analytial treatment

presentedhereanbeapplied to any kindofthermoaousti engine(andalso tofree pistonStirling engines),

but it is onvenient here for larity to separate the ases where there exists or not a losed loop path for

the aoustiwaves. Forthe sakeof simpliity, the rstlass of engine will bereferredto asstandingwave

engine,whiletheseondonewillbereferredtoastravellingwaveengine. Thedesriptionoftheaoustield

will beoperated in thefrequeny domainin the frameof the linearapproximation. Assumingthat harmoni

planewavesarepropagatingalongtheenterlineoftheduts,theaoustipressurep(x, t)^{andaoustivolume}

veloityu(x, t)^{arewritten as}

ζ(x, t) =ℜ

ζ(x)ee ^−jωt

, ⁽¹⁾

where ζ ^{may be either} p^or u^, ζ˜^{denotes theomplex amplitudeof} ζ^, ℜ()^{denotes the realpart ofa omplex}

number,andx^{denotesthepositionalongthedutaxis(seeFig1).}

AsshowninFig. 1,theapparatusonsistsofathermoaoustioreonnetedatbothsidesto straight(or

urved)duts. Thepropagationofaoustiwavesthroughthethermoaoustioreisdesribedasanaoustial

two-portrelatingtheomplexamplitudes ofaoustipressureand volumeveloityat bothsides:





 e p(xr) e u(xr)





=







Tpp Tpu

Tup Tuu





 ×





 e p(xl) e u(xl)





,

= TTC ×





 e p(xl) e u(xl)





. ⁽²⁾

Thetransfermatrixofthethermoaoustiore,TTC^{,dependsonthegeometrialand}thermophysialproperties

ofitsomponents. ItalsodependsonthetemperaturedistributionTm(x)^{alongthestak(}x∈[xl, xh]^)andthe

thermal buertube (x∈[xh, xr]^{), and onthe angular frequeny}ω^{. Ifthe imposed} temperature distribution is known, thetransfer matrixTTC ^{an beobtained} theoretially [1, 2, 17℄, but it analso beobtained from

experimentsundervariousheating onditions[22℄.

2.1 Derivation of the harateristi equation

2.1.1 Standingwave engines

Theaseofastandingwaveengineisrstonsideredhere.IfthematrixTTC^{isknown,thetheoretialmodelling}

oftheompletedevierequiresknowledgeoftheaoustipropagationthroughtheomponentssurroundingthe

thermoaousti ore. This an be realized by deriving the expressions of the reeted impedanes Zl,r =

˜

p(xl,r)/˜u(xl,r) ^{at both sides of the} thermoaousti ore. For instane, if a standing-wave devie asthe one

depitedinFig. 1(a)isonsidered,writingthelossypropagationofharmoniplanewavesatangularfrequeny

ω ^{in thedutsofrespetivelengths}xl^andL−xr ^{leadsto the}expressionsofthereetedimpedanes

Zl= p(xe l)

eu(xl) = Z0+jZctan(kxl)

1 +jZ0Zc⁻¹tan(kxl), ⁽³⁾

Zr= p(xe r)

eu(xr) = ZL−jZctan(k(L−xr))

1−jZLZc⁻¹tan(k(L−xr)), ⁽⁴⁾

where

k= ω c0

s

1 + (γ−1)fκ

1−fν

(5)

and

Zc=ρ0c0

S p 1

(1−fν)(1 + (γ−1)fκ)

(6)

are theomplexwavenumberandtheharateristiimpedane ofthedut,respetively. InEqs. (5) and(6),

ρ0 ^{istheuid densityat room}temperature,c0 ^{istheadiabatisound speed,}γ ^{isthespei heatratioofthe}

uid, S ^{is thedut} ross-setion, and the funtions fκ ^and fν ^{haraterize the thermaland visous oupling}

betweentheosillatinguidand thedutwalls[2,23℄. InEqs. (3) and(4), Z0 ^andZL ^{standfor theaousti}

impedanesat positions x = 0 ^and x=L^, respetively. They anbe, for instane, the radiation impedane of an open pipe [24℄, the innite impedane of arigid wall, orthe aousti impedane of an eletrodynami

alternator, depending on theongurationof the standing-waveengine. Finally, ombiningEqs. (3) and (4)

withEq. (2)andsolvingtheassoiatedsystemoftwoequationsleadstotheequation

ZlTpp+Tpu−ZlZrTup−ZrTuu= 0. ⁽⁷⁾

Eq. (7)istheharateristiequationwhihaountsforboththeproessesoperatingthroughthethermoaous-

tioreandthedissipative/reativeproessesoperatingin itssurroundingomponents. Thisequationmustbe

satisedtodesribetheomplete devieproperly.

2.1.2 Travelling wave engines

If thease of atravelling waveengineis nowonsidered,and if thematrix TTC ^{is known, itis also possible}

to derivea harateristi equation similar to Eq. (7). This impliesto desribeaousti propagation at both

sides of the thermoaousti ore. The basi idea is to makeone loop in the devie - eah of the individual

omponents being haraterized by its transfer matrix - so that after one loop, the harateristi equation

will ensure that one arrives at the same starting point. More preisely, there exists on the rst hand the

equation haraterizing thepropagation throughthethermoaousti ore, Eq. (2), and on theother hand,it

ispossibleto obtainanadditionalrelationbetweentheaoustipressureandaoustivolumeveloityatboth

sidesofthethermoaoustiorebymeansofthetotaltransfermatrixTsur^{oftheomponents}surroundingthe

thermoaoustiore:





 p(xe l) e u(xl)





= Tsur ×





 p(xe r) e u(xr)





. ⁽⁸⁾

Forinstane, ifthepartiulargeometryof Fig. 1-(b)isonsidered,andiftheeets oftheloopurvatureare

negleted,thematrixTsur ^{anbewritten as}

Tsur=Tl×Tload×Tr, ⁽⁹⁾

where thematries

Tl,r=







cos(kdl,r) jZcsin(kdl,r) jZc⁻¹sin(kdl,r) cos(kdl,r)





 ⁽¹⁰⁾

haraterize lossy propagation through the duts of respetive lengths dl = xl ^and dr = L−xr ⁽L ^{is the}

unwrappedlengthofthelosed-loop),andwherethematrix

Tload=







1 0

−Yload 1





 ⁽¹¹⁾

aounts for the eet of the seondary aousti load, by means of its reeted aousti admittane Yload

(this aoustiloadan be aseondaryaoustiresonator[20℄, an eletrodynami alternator [25℄oranyother

omponentharaterizedbyitsreetedadmittaneYload^).

Finally,ombiningEq. (8)withEq. (2)leadstothefollowingharateristiequation:

det (TTC×Tsur−I2) = 0, ⁽¹²⁾

where I2^{standsfortheidentitymatrix}2×2^,anddet()^denotesthedeterminantofamatrix.

2.2 Determination of the omplex frequeny

Theproperdesriptionofthethermoaoustidevierequirestosatisfytheorrespondingharateristiequation

f(ω, Tm) = 0 ⁽¹³⁾

where thefuntionf ^{orrespondstothe}left-hand-sideof Eq. (7)orEq. (12),dependingonthesystemunder onsideration. Itisimportanttopointoutthatalloftheaboveequationsarederivedinthefrequenydomain,

and due to this, it is impliitly assumed from Eq. (1) that the thermoaousti system operates on steady

state: this meansthat theangular frequenyω ^{is purelyreal. Infat,theonlyondition forwhih Eq. (13)}

anbesatisedis that thetemperaturedistribution Tm(x)^{is xed in suh awaythat there exists avalueof}

the angularfrequenyω ^{whih anelsthefuntion} f^{. Undersuh aonditiontheaoustiwavesareneither}

ampliednorattenuated,andsinenonlineareetssaturatingthewaveamplitudegrowtharenotonsidered

here,thesolutionsω ^andTm ^{orrespondto thethresholdof}thermoaoustiinstability. However,asitwill be proposedinthefollowing,onemaydesribefromEq. (13)thewaveamplitudegrowthouringaftertheonset

ofthermoaoustiinstabilityunderthequasi-steadystateassumption. Todothis,lettheangularfrequenybe

allowedto haveanimaginarypartǫg^:

ω= Ω +jǫg, ⁽¹⁴⁾

sothattheaoustipressure

p(x, t) =ℜ(p(x)ee ^−jωt) =e^ǫgtℜ(p(x)ee ^−jΩt). ⁽¹⁵⁾

isassumedtoosillateatfrequenyΩ =ℜ(ω)^,whiletheattenuation/growthofthesoundwaveisharaterized by the thermoaoustiampliation oeient ǫg^{. However,}ǫg ^{is assumed to be small ompared to the real}

partΩ^{ofangularfrequeny,whihmeansthattheamplitudeofthewavevariesslowlyatthetimesaleoffew}

aoustiperiods,diminishingorgrowingdependingonthesignofǫg^{,inasmuhasthe}temperaturedistribution

Tm(x)^{staysonstantat thesaleofafewaoustiperiods.}

Underthisquasi-steadystateassumption,ǫg<<Ω^{,andforaonstant}temperaturedistributionTm^(atthe

time-saleofafewaoustiperiods)itispossibletosolveEq. (13)usingonventionalnumerialmethods,and

tond aouple(Ω, ǫg)^whihharaterizesboththefrequenyofaoustiosillationsandthewaveamplitude growth/attenuation. Theadvantagesofthisformulationarethatitiswell-suitedforthepreditionoftheonset

ofthermoaoustiinstability(aswillbeshowninthenextsetion)butalsomoregenerally,aswillbedisussed