General Formulation of Thermoacoustics for Stacks Having Arbitrarily Shapedpore Crosssections A) W

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General formulation of thermoacoustics for stacks having arbitrarily shapedpore crosssections a) W. Pat Arnott,Henry E. Bass,and Richard Rasper NationalCenter for PhysicalAcoustics, Physical Acoustics Research Group, Departmentof Physics and Astronomy, University of Mississippi,University, Mississippi 38677 (Received4 March 1991;revised 14 July 1991;accepted 2 August1991 ) Theoreticaltreatments of thermoacousticshave been reported for stackswith circularpore and parallelplate geometries. A generallinear formulation is developedfor gas-filled thermoacousticelements such as heat exchangers,stacks, and tubeshaving pores of arbitrary cross-sectionalgeometry. For compactnessin the following,F representsthe functionalform of the transversevariation of the longitudinalparticle velocity. Generally, F is a functionof frequency,pore geometry,the responsefunctions and transportcoefficients of the gasused, and the ambientvalue of the gasdensity. Expressions are developedfor the acoustic temperature,density, particle velocity, pressure, heat flow, and work flow from knowledgeof F. Heat and work flowsare comparedin the shortstack approximation for stacksconsisting of parallelplates, circular, square,and equilateraltriangular pores. In this approximation,heat andwork flowsare found to be greatestfor the parallelplate stack geometry. Pressure and specificacoustic impedance translation theorems are derivedto simplifycomputation of the acousticalfield quantities at all pointswithin a thermoacousticengine. Relations with capillary-pore-basedporous media modelsare developed. PACS numbers: 43.35.Ud, 43.28.Kt

INTRODUCTION point of viewenhances the understandingof thermoacous- In a broadview, thermoacousticscan be regardedas the ticsand is very helpfulin evaluatingpractical devices. studyof effectsdue to the interactionof heat and sound..A An exposedview of a thermoacousticelement is shown large and growing subbranchis concernedwith thermoa- in Fig. 1(b). Thermoacousticelements consist of a parallel cousticsin fluid-filled (gas and liquid) resonatorsthough combinationof manyelementary capillary tubes or pores.In observations of heat-driven oscillations in tubes date back to Fig. 1(b) the poreshave square cross sections. The theory at leastthe late 18thcentury. A full, linear,theoretical inves- for a thermoacousticheat engine is built up from knowledge tigationof theseoscillations was performed first by Rott.• The reciprocalmode of operation,which uses a soundwave in a resonatorto transport heat from cold to hot as in a cold hot refrigerator,has also been of recentinterest. This thermoa- acoustic heat heat cousticstreaming has its analogy in acousticstreaming, driver exch stack exch which is the D.Ci transportof momentumby an acoustic wave.Merkli andThomann 2 foundexperimental verifica- tion for their theoryof thermoacousticstreaming in a driven resonance tube resonancetube. Rott and Merkli andThomann were mainly (a) interestedin thermoacousticeffects in a singletube having a circularcross section. Rott and Zouzoulas 3 also investigated thermally driven acousticoscillations for circular tubeswith variable cross-sectional area. lb) exposed view of a thermoacousticelement WheatIcy, Cox, Swift, Hofler, and others have deve- lopedthe connectionbetween the acousticalportion of ther- moacousticsand a broaderthermodynamics point of view. Swift4 hasreviewed much of thiswork, from fundamentals to state-of-the-art. Thermoacoustic elements such as heat (c) exchangersand a stack,as shownin Fig. 1(a), are usedto singlearbitrary-perimeter tube of a investigateprime moversand heat pumps.In the thermo- thermoacoustic element dynamic point of view heat exchangersand stacks are thoughtof asheat reservoirs and engines, respectively. This FIG. 1. (a) Geneticarrangement used in thermoacousticheat engines. Thermoacousticelements are the heat exchangers, resonator sections, and stacks.(b) An exposedview of a thermoacousticelement consisting of a Thiswork was presented, in part, at the120th Meeting of the Acoustical parallelcombination of squarecapillary tubes. (c) A singlearbitrary-per- Societyof America[J. Acoust.Sec. Am. Suppl.I 88, S96(1990)]. meter capillarytube for usein a thermoacousticelement.

3228 J. Acoust.Sec. Am. 90 (6), December1991 0001-4966/91/123228-10500.80 @ 1991Acoustical Society of America 3228 of thermoacousticeffects in a singlecapillary tube. Both acousticimpedance. A numericalanalysis technique for the Rott• and Swift4 consideredthermoacoustic effects for presentformulation is givenin Sec.I G. An applicationof acousticoscillation between parallel plates. the theoryis givenin Sec.II whereheat and work flowsare The initialintent of thisstudy was to investigatether- computedin theshort stack approximation for stacks having moacousticeffects in a stackhaving square pores as shown in a varietyof porecross sections. Here, the emphasisis on Fig. 1(b). Inexpensivesources of squarepore stack material investigatingthe effectspore geometryhave on heat and areceramic monolithic catalyst supports often used in auto- work flows. mobilecatalytic converters. 6 This ideal-geometry material was previouslyused to verify first-principlestheory for soundpropagation in porousmedia. 7 Sinceceramic has a I. PROPAGATION IN THERMOACOUSTIC ELEMENTS lowthermal conductivity in comparisonto mostmetals, it is Thename stack was originally descriptive of theparallel attractivefor useas a stackas parasitic heat loss due to ther- plate arrangementused for the thermoacousticelement mal conductionreduces efficiency. which possiblyhas a temperaturegradient down it. The In thispaper, thermoacoustics is investigated for stacks poresin the parallelplate arrangement are described in their havingarbitrary pore geometries (parallel plates, rectangu- transversedirection as parallel plates and as straight tubes in lar pores,equilateral-triangle pores, circular pores, etc.). In their longitudinaldirection [the z directionin Fig. 1(c)]. particular,our interest is in thefollowing question: What are For arbitrarypore geometries, the stackor heatexchangers theminimum necessary calculations to describethe acoustics can be describedas a sectionof a porousmedium. In this ofgas-filled thermoacoustic elements made of arbitrary-peri- section,the fluid fieldequations and assumptionsnecessary metercapillary tubes? An exampleof an arbitrary-perimeter to treat the generalcase are established.An equationfor the capillarytube is shown in Fig.1 (c). Rott5 pursuedthis ques- pressurein a singlepore is established.Enroute, reference is tion to the pointof computingthe acousticalquantities for madeto the terminologyused in acousticalmodeling of por- parallelplate and circular pore geometries. Here, the acous- ousmedia. With the specificacoustic impedance assumed tic fieldquantities and the second-order energy flow are con- knownat the hot endof the stack,impedance and pressure sideredfor arbitraryperimeter pores. Heat andwork flows translation theorems are derived for the stack. Heat and are comparedin the shortstack approximation for stacks work flowsare computedfor arbitrarypore geometries and havingthe aforementionedpore geometries to investigate areexpressed in termsof pressureand specific acoustic impe- the effectsof pore geometry.In addition,connections are dance. establishedbetween thermoacoustic theory and capillarytube-basedporous media theory. An analogousinvestiga- A. Fluid field equations and assumptions tion hasrecently been performed for porousmedia by Stin- The transversecoordinates in a pore are taken to be x son? andy, and the longitudinalcoordinate is z as shownin Fig. Oncethe acousticalproperties of the separatethermoa- 1(c). The ambienttemperature is takento be a functionofz cousticelements have been determined,the elementsmust in the stack.Assumed is that the porewalls are of sufficiently beconnected in seriesinside of a resonatoras shown in Fig. highheat capacity and thermalconductivity, in comparison l(a). Previously,numerical integration of the acoustical to that of the gas,that the pore wall temperatureis locally equationswas used to computefield quantities in the stack unaffectedby temperaturevariations in the gascaused by an since,in general,a temperaturegradient exists from one end acousticwave. Also assumedare that constantfrequency tothe other.•'4 The physical parameters ofambient density, pressurevariations exist in the pore and that the pore walls viscosity,sound speed, thermal conductivity, etc., are tem- are rigid and nonporous.The pore is taken to be infinitely peraturedependent and thus depend on locationwithin the longin thez direction.With theseassumptions, the task is to stack.Here, specific acoustic impedance and pressure trans- derivethe pore acousticfield to first order in the acoustic lationtheorems are developed to computeall acousticalfield variables. quantitiesand energyflow at each point in the resonance The fluid quantitiesin a pore approximatedto first tubeshown in Fig. 1(a). Translationtheorems are relations order are betweenspecific acoustic impedance or pressureat locationz andthe value of thesequantities at a differentlocation z - d. P(z,t) = Po + P• (z)exp( - i•ot), (1) Usingthe translation theorem approach it iseasy to analyze complicatedsystems, e.g., a resonatorcontaining a refrigera- v(x,y,z,t)-- [v•(x,y,z) + oz(x,y,z)•]exp(- loot),(2) tion stack,a primemover stack, and heat exchangers. T(x,y,z,t) = To(z) + T• (x,y,z)exp( -icot), (3) SectionI A containsthe basicfluid equationsand as- s(x,y,z,t) = So(z) + s• (x,y,z)exp( -- icot), (4) sumptions.The forceequation is consideredin Sec.I B and the transversetemperature profile is givenin Sec.I C. A p(x,y,z,t) =Po (z) + p• (x,y,z)exp( -- icot). (5) differentialequation is givenfor thethe acoustic pressure in Sec.I D. SectionsI A-I D applyto a singlecapillary tube of In order,Eqs. ( 1)-(4) are the approximationsto first order arbitrary perimeter.The specificacoustic impedance and for pressure,particle velocity, temperature, entropy, and pressuretranslation theorems are developedin Sec.I E for density.Acoustic waves of radianfrequency to are assumed. thermoacousticelements. Expressions are givenin Sec.I F Where shown,subscript 0 indicatesambient values and sub- for heatand work flowsin termsof pressureand specific script 1 the acoustic or first-order values. In Eq. (2),

3229 J. Acoust. $oc. Am., Vol. 90, No. 6, December 1991 Arnott ota/.: Thermoacoustics of stacks 3229 v, (x,y,z) and v• (x,y,z) are the transverseand longitudinal to)t/a or2r = 2•/2R/• forthe ratio of the pore radius to the componentsof particlevelocity. The ambienttemperature thermal boundarylayer thickness.Use of the Prandtl num- To(z) is assumedto dependon z. Note the definitionto be berNp• = *lcp/trgives the relation Ar - AN•/a For definite- usedfrequently below: Toz--=dTo (z)/dz. The ambientden- ness,take R to be twicethe ratio of the transversepore area sity in Eq. (5) alsodepends on positionz. Becauseof the to the poreperimeter so for a circularor squarepore, R isthe ambienttemperature gradient physical parameters includ- poreradius. This valueof R is twicethe hydraulicradius of ing density,viscosity r/, thermal conductivityto, adiabatic the pore.In thispaper the A and•'r notationis used. soundspeed c, and the coefficientof thermal expansion /• = -- ( Sp/•T)v/poalso depend on position. The Navier-Stokesfluid field equationsare givenfor examplein Reft9. The Navier-Stokesequations can be sim- B. Transverse velocity profile in a pore plifiedto modelsound propagation in narrowtubes. Physi- To obtaina solutionto theequation of motion,Eq. (6), cally, the transversevariation (in the tube crosssection) of the z componentof velocityis takento be particlevelocity, temperature, etc., is muchgreater than the longitudinal variation (down the tube) due to the close v=(x,y,z)- F(x,y;A) dP•(z) (11) proximityof wallswhere boundary conditions must be met. iOpo dz For constantfrequency waves, the approximateset of fluid From the equationof motion,Eq. (6), F(x,y,g ) satisfies fieldequations to firstorder are F(x,y;A)+ (R :/i;c2)V•F(x,y;A) = 1, (12) dP] (z) - icopov=(x,y,z) - k VV•vz(x,y,z), (6) subjectto theboundary condition that F(x,y;A) iszero at the dz porewalls. This is the boundarycondition on particlevelo- c•[Po (Z)Vz (x,y,z) ] cityat thepore wall. As will becomeapparent in thefollowing, - icop•(x,y,z) -+ c•z thisis theonly differential equation which needs to besolved for determiningthefirst-order acoustic quantities and second- + Po(z)V•'% (x,y,z) = 0, (7) orderheat and workflows. p] (x,y,z)= --Po (z)/•T• (x,y,z)+ (y/c2)Pt(z), (8) In anticipationof laterdevelopments, averages over the st(x,y,z) = (q/T o) T, (x,y,z)-- (• /Po)Pt (z), (9) pore cross section, defined for example by vz(z) = A - •.fv•(x,y,z)dx dy, whereA is the area of the pore and crosssection, are taken.Denote by v=(z) andF(A) the area averageof v• (x,y,z) and F(x,y;A) over the crosssection of - icopo(z)cn T• (x,y,z)+ Po(z)%Vz (x,y,z) Toz the pore.The averagedequation of motionfor the fluid can : -- irvinToPI (2) q-trV2•Tt (xd',z), (10) now be expressedsimply as wherethe transversegradient and Laplacianoperators are Vz(z) - dP•(z)/dz F(A ). (13) definedby V• = a•/cJx• + c•/cJy•yand V2• = (• 2/cJx2+ • 2/ iCOpo( z ) c•y:),cs isthe isobaric heat capacity per unit mass, and )/is theratio of specificheats. In order,these equations approxi- Capillary-tube-basedporous media modelingn'•: intro- matelyexpress the z componentof the equationof motion, ducesa complexdensity • (z;A) at thispoint which is defined as continuityor massconservation, equations of statefor den- sityand entropy, and heat transfer. Except for the Tozterms •(z;A ) = Po(z)/F(A ). (14) in Eq. (10), theseare the equationsfor the low reduced- . ß 10 The complexdensity is the apparentdynamical density of frequencyapproximation givenby Zwikkerand Kosten n the fluid in the pore. in their solutionfor the propagationof soundin circular pores.In the derivationof Eq. (10), the convectivederiva- C. Transverse temperature profile in a pore tive for the entropyis evaluatedusing the equationof state The equationfor excesstemperature T• (x,y,z) in the Eq. (9) and the relationv,Vs= v=(x,y,z)dso(z)/dz=% porefluid is givenby Eq. (10). Algebraicrearrangement X Vz(x,y,z) Toy/To. Equation(10) expressesthat the tem- followed by use of the thermodynamic relation peratureat a fixed positionchanges due to motion of the ambientfluid, due to compressionof thegas, and due to heat To/•:/%= (y - 1)/c a for the first term on the right and by conduction.Further discussionof theseequations and the useof Eq. ( 11) resultsin validityof the approximationsmay be foundin Refs.4, 8, T• (x,y,z) + (R •//2 ' •r)V•Tt • (x,y,z) and 10. Severalvariations in notationshouM be noted.First, the _ 7/--1 Pt (z) -- Tø=F(x,y;A) dP•(z____•) (15) viscousand thermal boundary layer thicknesses are given by C•po]• porO • dz 6• = ( 2•]/COpo) •/2 and 6• = (2to/COpe%)•/•. Swift 4 writes mostequations in termsof 6• and6•. Tijdemanm andAtten- AssumeTm (x,y,z) can be written in the form borough•a introducea dimensionless"shear wave number" T, (x,y,z)= G, (x,y;Ar)[ (7/- 1)/C2po•]P• (z) A = R(poW/V)m era = 2•/2R/6•, whereR isa character- Toz dP• (z) istictransverse dimension of the pore.They alsouse the di- - O• (x,y;A•gr) -- -- (16) mensionlessthermal disturbance number A r = R(poCOC•/ poCO2 dz

3230 J. Acoust.Sec. Am., Vol. 90, No. 6, December1991 Arnottota/.: Thermoacousticsof stacks 3230 Theexcess temperature changes due to compressionand ex- p•(Z) =po•(Z',ftr)P 1(z) pansionof thegas and from the second term, displacement of lTo. F( & ) -- NoF(,t ) dP• (z) gaswhich can havedifferent ambient temperatures on ac- (25) countof the temperaturegradient. '1 --No dz Effectson T• (x,y•) of thermalconductivity and viscosity are accountedfor in the dimensionlessfunctions The motivationfor averagingthe densityover the crosssec- Ga( x,y• r) andGb ( x,y;,t•tr ). For aninviscid, nonthermal- tion of the porebecomes apparent in the nextsection. ly conductinggas G a = Gb = 1. Use of Eq. (16) splitsEq. (15) into two equationscorresponding to the two driving D. Pressure equation in a pore terms The continuityequation, Eq. (7), alongwith the equa- Ga(x,y;Ar) + (R :/bl •r)?,Ga(x,y;Ar) 2 = 1, (17) tion of motion,Eq. (13), and the equationof statefor the (Jb(x,y',)[,• r) + (R 2/Lg•-)7•Gb (x,y',,•,,•r) density,Eq. (25), can be combinedto yield an equationfor the pressurein a pore.When averagingthe continuityequa- = F(x,y;A). (18) tionover the pore area the integral A - lœ•.v• (x,y•z)dxdy The boundarycondition is T• (x,y,z) = 0 for x andy on the is encountered.Use of the divergencetheorem in the x,y poreboundary; therefore, Ga (x,y',A r) = 0 = Gb (x,y'•,,t r) plane gives A - on theboundary are the boundaryconditions for thesefunc- X (x,y,z)dS = 0 since% (x,y,z) = 0 onthe perimeter$ hav- tions.Comparing F.q. (17) with Eq. (12), the solutionfor ing outwardnormal n. Consequently,the cross-sectionally Eq. (17) followsimmediately: averagedcontinuity equation for the poreis G• (x,3r•r) = F(x,y;Ar). (19) To obtainthe solutionfor G• (x,y) in Eq. (18), thedifferen- - (z) tial equationresulting from useof Eq. (19) in Eq. (17) along = --irop• (z)+ Pc (z) ___d o(z) - t•o(z) To, v,(z) = O, with the differentialequation for F(x•v;A) in Eq. (12) canbe used to show that (26) Go(x,Y•r) = [ F(x,y'•r) -- NprF(x,y71)]/( 1 -- Npr). wheredpo (z)/dz = --/•(z) Pc(z) To, wasused on the se- (20) condform. The o, and dv,/dz termscan be evaluatedusing Eq. (13): The excesstemperature is T, (x,y,z)= [ (y -- 1)/C•po]• ]F(x,yv[r)P, (z) --irop• (z)+Pc ko (z•) dz d (F(/I)LOo ( z ) dP•dz(z) ) To, F(x,y;/•r)--NwF(x,y;J.) dP•(z) F(,t) dP• -- --/•To, -- O. (27) porO• 1 -- Np• dz ko dz (21) MultiplyingEq. (27) by kaF(,t) and usingEq. (25) for The equationof state for the acousticdensity fluctu- p• (z) the equationfor pressureis ation,Eq. (8), canbe combinedwith the expressionfor the acoustictemperature, Eq. (21), to give p• (x,y,z)= (1/c•)(?'-- (?'-- 1)F(x•v•r))P• (z) F(A)Podzd (F(/t)\' •oodPl(Z) az ') + 2a(A•r) •dP,(z) + k(/•/•T)•P• (z) = 0, (28) fiTo•F(x,y;,t•) -- Np•F(x,y;A)dP• (z) where o • 1 - N• dz (22) The firstterm on the right of Eq. (22) canbe usedto definea a(A'Ar)=--•\tTo, [ ] ---•'v• - 1.)' (29) complexcompressibility. and When To• is zero, Eq. (22) can be usedto definea complexcompressibility which is usefulin porousmedia theory. k(2,Ar)•- --ro • -- 1 [Y- (r- 1)F(2r)]. (30) Denote by p• (z), F(Ar), and F(A) as the averageof cz F(/D p• (x,y,z), F(x•v;Ar) andF(x,y;A) overthe crosssection of In the absence of a temperature gradient To• = 0 so the pore.Following the capillary-tubeapproach of porous a ()[,g r) = 0. The complexwave numberin the poreis then mediamodeling, I• thecomplex compressibility isdefined as givenby ___k, whichis theusual form found in porousmedia modeling.7'n Theform of the equation for pressure isremin- = (z) (23) iscentof the time analogof a dampedharmonic oscillator; PcP, (z) ' however,here a and k are complexquantities. and fromEq. (22), with To, = 0, is givenby •(z;Ar) = (l/pod)[?'-- (y-- l)F(Xr) ]. (24) E. Specific acoustic impedance and pressure translation theorems Thecross-sectionally averaged density can then be expressed It is appropriateto establishthe terminologyused in this as section.The specificacoustic impedance of an acousticalme-

3231 J. Acoust. Soc. Am., Vol. 90, No. 6, December 1991 Arnott eta/.: Thermoacousticsof stacks 3231 dium is equalto the ratio of the total acousticpressure and dP• (z) P• (z) -- - ik(z)Z•., (z) -- (35) totalparticlevelocity. For example,in a fluid layer, the total dz Z(z) acousticpressure is a combinationof a downgoingwave and The definedquantities k, a, and Z•.t in Eqs. (33) and (35) an upgoingwave. The acousticimpedance is equal to the dependon positionbecause of the temperaturegradient. ratio of the total acousticpressure and the totalvolume velo- Expressions(33) and (35) are a set of coupledfirst- city.For porousmedia the appropriateboundary conditions order differentialequations which can be readily solvedus- at a surfaceare continuity of pressure,and continuity of ing numerical techniques.The well-known fourth-order volumevelocity or equivalentlycontinuity of acousticimpe- Runge-Kutta algorithm is a recommendednumerical dance.•2For adiabaticsound, the characteristicor intrinsic method.Assume given values ofP• (z) andZ(z) at position impedanceis equalto Po c wherePo is the ambientdensity z asshown schematically in Fig. 2. Then pressureand speci- and c is the adiabaticsound speed. fic acousticimpedance are determinedat z -- d from useof To this point, propagationin a singlepore of infinite lengthhas beenconsidered. Consider now a poroussample the algorithm Pt (z -- d) = RK [Pt (z),Z(z) ] and Z(z -- d) = RK[Z(z) ] whereRK is symbolicnotation for [e.g., Fig. l(b)] consistingof a parallel combinationof the Runge-Kutta algorithm.Thus thismethod of determin- manyidentical single straight pores [e.g., Fig. 1(c) ] of finite ing the pressureand impedanceis similarto Rayleigh'sim- lengthand denoteby N the numberof poresper unit areain pedancetranslation theorem. the crosssection. Define by V•ba bulk particlevelocity aver- For thermoacousticelements not having temperature aged over unit crosssection of poroussample, by Aresthe gradients(such as heat exchangers and sections of the reson- areaof the resonatorat the faceof the sample,and by A the ator), Rayleigh'simpedance translation theorem •3 canbe areaof a singlepore in thesample. Volume velocity is A written for porousmedia as = N A Ares vz = f• Ares v• wheref• = NA is the porosityof the sample.Thus, in the analysisof thermoacousticelements Z(z)cos( kd) - iZ•.• sin(kd) with many pores,v• is replacedwith Vzb/• in all of the single Z(z - d) = Zint (36) Ziat cos(kd) - iZ(z)sin(kd) poreequations. At boundaries,Pt (z) and V•o(z) or equiva- lently Z(z) = P(z)/Vzo (z) are continuous,where Z(z) is The pressuretranslation theorem is the specificacoustic impedance. Rayleighdeveloped an impedancetranslation theorem P•(z -- d) = P•(z) {cos( kd) - i[ Z•,t/Z(z)] sin(kd)}. (37) for homogeneousfluid layers. • Theimpedance translation theoremrelates the specificacoustic impedance at one side In Eqs. (36) and (37), k is the complexwave number for of a layerto that at the other.In thismanner, one may apply heatexchangers or opensections of the tubeand canbe de- the theoremas many times as necessary to computethe spe- terminedfrom the porousmedia expression, Eq. (30). Use cificacoustic impedance at any surfacein the layeredmedia. of these translation theorems will be discussed in Sec. I G. This translationtheorem is applicableto heat exchangers Knowledgeof heatand work flowis centralto thermoa- and resonatorsections, but is not applicableto the stackbe- coustics.In the next section, heat and work flows are eva- causethe physicalparameters such as density, sound speed, luated for arbitrary pore perimetersand are expressedin etc., dependon z in a continuousmanner on accountof the termsof pressureand specificacoustic impedance. temperaturegradient. Impedanceand pressuretranslation theorems,which take into accountthe dependenceof physi- F. Heat and work flow cal parameterson position,will be derivedfor the stack. The relevantexpressions are the averageforce equation The time averagedenergy flow to secondorder (sub- for a bulk samplefrom Eq. (13) and v• (z) = V•o(z)/f•, script2) is4 iwpo dP• (z) -- V•o (z) = F(2) --, (31 (z) = Q• (z) + W• (z) - Q•o•(z), (38) • dz the definitionof specificacoustic impedance, where time-averagedheat flow due to hydrodynamictran- Z(z) = P• (z)/V•o (z), (32) sportis andthe expression for pressure,Eq. (28). EliminatingP, (z) and V•o(z) from theseexpressions using a proceduresimilar to that of Ref. 14 gives

z-d z z+e dZ(z)•zz-- ik(z)Zmt (z)(1 Zi•t(z)•,z(z)] + 2a(z)Z(z), (33) wherea(z) and k(z) are givenin Eqs. (29) and (30) and subsections of a thermoacoustic element Zmt = poro/[llF( 2 ) k ] (34) is analogousto the intrinsicor characteristicimpedance of a FIG. 2. Subsectionof a thermoacousticelement, shown here having square pore capillary tubes. For the stack thermoacousticelement, an arbitrary porousmedium.•2 The combinationof Eqs.(31 ) and (32) numberof subsectionsmay be usedin spanningthe temperaturedifference gives at the ends of the stack.

3232 d. Acoust.Sec. Am., Vol. 90, No. 6, December 1991 Arnott eta/.: Thermoacousticsof stacks 3232 QA(z)-

-- l•To vz (x,y,z)P l*(z) ] dxdy; (39) the heat flow due to conductiondown a temperaturegradient is (47) •to•(z) -- 11&.• To•+ (I - l•)&•s,a•To•; (40) and and the time-averagedwork flow (or power) is W• (z) - -- • A•2 im(Pl•(Z)P.___•(z) \ poo F(A)).(48) •w, (z) - •'•• '4res Relf•v•(x,y,z)P•'(z)dxdy. (41)Equations (47) and (48) are generalexpressions for heat Here,.4• is the cross-sectionalarea of the resonancetube at andwork flows with thefunctional form ofF{A) dependent point z, ,4 is the cross-sectionalarea of a singlepore, gl is on the particular pore geometry. It can be shown that porosity,•c• andg,,• arethe thermal conductivity of the Eqs.(38),(47), and (48) arethe same as Swiff's 4 equation gasand stack, * indicatescomplex conjugation, and Re indi- (A30) for the specialcase of parallelplate geometries and catesthe realpart of theexpression. The product(11.4r•) is assumingthe e• = 0 (appropriatefor gasthermodynamics) cross-sectionalope.,.n area of th._e tube at positionz. in Swift'stheory. To aid in comparingresults, note that To determineQ2 (z) and W2(z) useis madeof Eq. (21) F*(A) = I -f,, andF*(A r) = 1 -f• wheref, andf• are used in Swift's notation. for T1 (x,y,z) and Eq. (11) for o•(x,y,z) in Eqs. (39) and (40). Resultingexpressions are Since impedance and pressure translation theorems havebeen derived for arbitrary locationswithin the resonator, and specificallyfor the stackelement, it is usefulto ex- {•(z)DM,• 2 poCvA imla F(x,y;A) COpo plx(2) pressheat and work flowsin termsof the specificacoustic impedanceZ(z) and pressureP, (z). From the definitionof specificacoustic impedance, Z(z) = Pt (z)/[ flv• (z) ], and X(•2•o;F*(x,Y',Ar)P•'(z) Eq. (13) for o,(z), To• F*(x,y;Ar) -- NvrF*(x,y;•) dP• (z) iopoP• (z) poCO' 1 - Nw -- -- P• (z) -- (49) dz •F(A )Z(z) XP•*•(z))dx dy-[3To • (z), (42)Heat and work flows are .--- Ar• /?To IP, (z) l• whereP• (z) = dP• (z)/dz and Q• (z) .... 2 1 + N.• IZ(z)l • W•--ß(z) = fi-•'4• AI Im• F(x,y;A) O•poPt•(z)P•(z)dxdy. (43) X{Re[Z(z)*(F;-• )+ Np•)] Recallthe definitions Ar = N mk and To• poc• 1 /?To rico IF(A)[: F(A)= • F(x,y;X)dxdy. (44) The followinggeneral integral result, which is provenat the end of this section, ImIF* (Xr) + No•F(A) ]} _ •T ø•z (z), (50)

and 1f• F(x,y.•)F.(x,y.fir)dx dy I•2(z)= (A•e•/2)[lP,(z)WlZ(z)lqReZ(z).(51) = [F(A)Np•+F*(Ar)]/(Np• + 1), (45) In Eqs. (50) and (51 ), Pt (z) andZ(z) areglobal variables and the integralthat may be evaluatedusing Eq. (45) in that they dependon the detailedarrangement of all ele- mentsof the thermoacousticengine, and F(A) and F(Ar) dependon thelocal properties of thegas and stack geometry. I•= 3 f• F( x,Y;A )F * ( x,y;A )dx dy The remainderof this sectionis devotedto proof of Eq. (45). From Eq. (12), Eq. (45) can be written as = lim I• = ReF(A), (46) Npr• I I,=•F(x,y;•)(l+ iX•rR---•-2V•F*(x,y;Ar) )dxdy. are necessaryto evaluateEqs. (42) and (43). By making use (52) of these integrals and the thermodynamic relation Useof the vectoridentity for the divergenceof theproduct of y -- 1 = • • Toc•/%, heat and work flows are a scalarand a vectorand the definitionofF(A) give

3233 J. Acoust. Soc. Am., Vol. 90, No. 6, December t 991 Arnott ot at: Thormoacousticsof stacks 3233 To determinethe frequencyresponse, one starts with a known valueof specificacoustic impedance at the rigid end of the tube in Fig. 1(a). Specificacoustic impedance here canbe evaluated using the expression for theboundary layer impedanceof a rigidwall. n Useof Eq. (36) determinesZ firstat thehot-heat exchanger, open-tube interface, and then at the stack-hot-heatexchanger interface. The valueof Z at this interfaceis the startingvalue for the Runge-Kutta al- (53) gorithmsolution of Eqs. (33). Then, useof Eq. (36) deter- minesZ first at the cold-heatexchanger, open-tube inter- wherethe gradientoperator in thex,y planeis V,. The diver- face,and finally at the positionof the acousticdriver. For gencetheorem in the x,.Vplane can be appliedto the second known acousticdriver response(e.g., constant displace- integral in Eq. (53): ment,etc.), the pressurecan be determinedabsolutely at the driver locationusing the driver responseand the calculated • ?•.[F(x,y;•)V•F*(x,y;,•r)]dx dy expressionfor specificacoustic impedance. It shotfidbe not- edthat all variablesin Eqs. ( 33) and (36) areto beevaluated at the local positionof the thermoacousticelement or sub- =;s n'[F(x'Y;2)VrF*(x'Y;/lr) ]dS=0, (54)section. The translationtheorem approach for computing wheredS is an elementof perimeteron the poreof arbitrary acousticalquantities is a superiorway of performingcalcula- shape,and n is the outward normal at the pore wall. The tions for resonatorscontaining many thermoacousticele- integral in Eq. (54) is zero by the boundary condition ments. F(x,y;A) -- 0 for x andy on $. Thus The secondexample considered is evaluationof a ther- moacousticrefrigerator. Though the typical configuration 4 I• = F(A ) usedfor a refrigeratorhas the acousticdriver at thehot (am- /t iA• V,F(x,y;•. ) bient) exchanger,the arrangementshown in Fig. 1(a) will 'V ,F* (x,y;Ar ) dx dy. (55) sufficefor the presentdiscussion. The driver is assumedto deliversufficient acoustic power to the tubethat heatis ther- Now, Eq. (12) for F(x,y;A) could have been usedin Eq. moacousticallytransported from the cold to the ambient (45), rather than for F*(x,y;A r) as wasdone to obtainEq. heatexchanger. Experimentaland theoreticalß evidence ß •s in- (55). Had this beendone, Eq. (55) would be dicates that the ambient temperature distribution in the stackis, in general,different from that which would occur =v*(xT)+ñ R2f v,v(x,y?) fora simpletemperature gradient wit.[h no acoustic transport of heat. Indeedthe secondterm of Q2(z) in Eq. (50) hasa ßV•F*(x,y'•r)dxdy. (56) temperaturegradient multiplied by a factor dependingon the pressureand impedancethat actsas a dynamicalcoeffi- Eliminating the common integral betweenEqs. (55) and cient of thermal conductivity. (56) givesEq. (45). Steady-stateequilibrium of the refrigeratoris achieved when the ambient temperaturethroughout the systemis constant.No net heat is.,.absorbed intothe walls of the stack G. Numericalevaluation of an engine's performance (i.e., heatengine) so H• (z) is a constantin the stack.4'ls The first and simplestsituation considered is the fre- Quantitiesassumed known are the pressureand specific quencyresponse of a nondriventube below the onset of oscil- acousticimpedance at theright end of the tube,the hot-heat lation. An examplebased on Fig. 1(a) will sufficeto demon- exc._.hangertemperature, and the steady-state constant value stratethe procedurefor computingthe frequencyresponse of H2 in thestack. The unknownquantities of interestare the using the impedancetranslation theorem. The acoustic cold-heatexchanger temperature, the power deliveredby driver is consideredto deliver constantamplitude acoustic the acousticdriver, and the net heat flow from the cold-heat oscillationsof radianfrequency co of insufficientamplitude exchanger.One may computethe Carnot coefficientof per- to acousticallystimulate the transportof heatfrom cold to formance(COP) from the hot- and cold-heatexchanger hot. An ideal microphonelocated near the driver is usedto temperatures.• This COP maybe comparedto the coeffi- determine the frequency responseP• (z •- 0,•o). cient of performancecomputed from the ratio of the net heat Assumea steady-stateequilibrium in which a tempera- flow from the coldexchanger and the powerdelivered by the ture gradientexists across the stackas a resultof heat input driver. 4 at thehot exchangerand heat removal at thecold exchanger. Specificacoustic impedance and pressureat the stack- The opentube sections of the resonatorare taken to be at the hot-heatexchanger interface can be determinedfrom useof sametemperature as the nearestheat exchanger.The am- translationtheorems in Eqs. (36) and (37). Then,withH 2 a bienttemperature of the stackwill not, in general,vary lin- knownconstant, Eq. (38) alongwith Eqs. (40), (50), and earlyfrom thecold to hot enddue to the temperaturedepen- (51) are usedto solvefor To,(z), the ambienttemperature dence of the stack and gas thermal conductivity. The gradientin the stack.Runge-Kutta integrationof the cou- ambienttemperature is assumedknown at all points. pled set of three first-orderequations, To• (z), P1 (z) from

3234 J. Acoust.Soc. Am., Vol. 90, No. 6, December 1991 Arnott otal.: Thermoacousticsof stacks 3234 Eq. (35), and Z(z) from Eq. (33), is usedto determinethe r= •Toctan[ko(L--z)] ambienttemperature distribution To (z) in the stack and To ofl(7 - 1) thusthe cold-heat exchanger temperature. Translation theo- remscan then be usedto computethe pressureand imped- = Tot% tan [kø (L -- z) ] (60) anceelsewhere in thetube. Equation (51 ) isused to compute oToc the powerdelivered by the driver. By energyconservation, In the inviscidapproximation for whichNpr = 0 and thedifference in•-/2 between the cold heat exchanger-stack F(2) = 1, interfaceand the open-tube-cold heat exchanger interface is the amountof heat flowingout of the cold heat exchanger. A• Pi (L)2 sin[2ko(œ-z)] 02 (z) = -- --/•/T o This methodof analyzingthermoacoustic refrigerators was 2 poc 2 first described in Ref. 15. XlrnF*(Ar) (1 -- F). (61)

II. HEAT AND WORK FLOWS IN THE SHORT STACK Physically,the termIm F* (Ar) isa measureof thedynami- APPROXIMATION FOR VARIOUS STACK GEOMETRIES cal thermal interactionbetween the gas and solid. Recall thatAr ----R (poCO%/•c)]/2, where R istwice the ratio of ca- Heat and work flowsare comparedfor stackshaving pillary porearea to poreperimeter. The functionF(x,y;Ar) differentpore geometries.In additionto the squarepore is a solutionto the partialdifferential equation, Eq. (12), for stackshown in Fig. 1(b), parallelplate, circular, and equi- a particularpore geometry and F(Ar) is theaverage of this lateraltriangular capillary tubes will beconsidered. quantityover the porecross section. According to Eq. (61), Theshort stack approximation was used by Swift 4 toget stacksmade of poresfor whichIm F* (g r) is a largevalue an interpretableanalytical expression for energyflow. Fig- will resultin the greatestheat flow. ure3 showsthe arrangement for theshort stack approxima- Work flow is givengenerally by Eq. (51 ). No work is tion.The stack is assumed to beshort enough that the empty donein the regionto the right of the stackin Fig. 3, which tubestanding wave is unaffected.The temperaturediffer- canbe verifiedreadily by usingthe impedanceof Eq. (57) in encebetween opposite ends of the stack is assumedto be Eq. (51 ). In thisregion, pressure and velocityhave standing muchless than the averagetemperature at the stackcenter wavephasing. To computethe work donein the stack,use is so that the thermophysicalquantities are approximately madeof the impedancetranslation theorem to getthe impe- constantand are evaluated at theaverage temperature. Stack danceat the left sideof the stack.In the shortstack approxi- porosityis fl. mation, kod•oM/c• 1; hence,Z(z- d) can be approxi- Pressureand specificacoustic impedance at z are, from matedfrom a simplefinite difference approximation of Eq. Eqs. (36) and (37), (33), Z(z) = ipoCeot[k o (L -- z) ], (57) and Z(z--d) = Z(z)I --2ad F(A)•Z(z) P] (z) = P] (L)cos[ko(L --z)] + idZ(z)k2F(2)fl.}. (62) •ipocVo cos[ko (L -- z) ]/sin(koL), (58) po co / wherethe wavenumber in the emptytube is ko. Useof Eqs. After somemanipulation, work flow to firstorder in kod is (57) and (58) in the heatflow equation,Eq. (50), gives '• •4resPt (L) 2 //To • (z) /l•flP• (L )2 To/• •od Q2(z) ------2 Po% 2 poC I +Npr X ImF* (At)cos• [ko (L -- z) ] X sin[2kø(L--z)]im(F;(•)2 X(1--I' Im[F(Ar)/F(A)] X( 1-- F Im[F(Ar)F(A)](1 Im[F(Ar)+NprF*(A)] --Npr)' )' (59) + W2•,• (z), (63) where where work due to viscosityis •a•i•(z).4•fl P, (L) 2•adsin2[ko(L z)] 2 poc2 X{ImF*(A)/[F(A)F*(A)]}. (64) i STACKI Work flow in the inviscidapproximation is V(0,t) = Vo exp-i o•t IV2(z) = (Ar•fl/2)P, (L )2(TolS'Zcod /poCt, ) Xlm F*(Ar)cosZ[ko(L--z)] (1 - F). d (65)

FIG. 3. Arrangementfor the shortstack approximation. Equations(63) and (65) arethe acousticpower absorbed by

3235 d. Acoust. Sec. Am., VoL 90, No. 6, December 1991 Arnott et aL: Thermoacoustics of stacks 3235 lowsone to computethe optimaloperating frequency from therelation Ac = (pococphc)•/2R.In other words, you can getabout 10% moreheat and work flowsin thermoacoustics by choosingto make your stackfrom parallelplates rather (a) (b) thanthe otherpore geometries. The functionalform ofF(2) for the variouspore geometriesis givenbelow. Swift4 andRott s haveworked out the parallel plate geo- metrystack. Parallel plates have also been of interestin porous media.12'•6 Denote by 2a the separationdistance betweenplates. To be consistentwith the definition of the characteristicpore radius R asbeing equal to twicethe trans- (d) (c) versepore area dividedby the pore perimeter,take R - 2a. They axisis centered at themidpoint between plates with the FIG. 4. (a) Parallelplate, (b) circular,(c) rectangular,and (d) equilateral z axisextending in the longitudinaldirection. The function triangularcapillary tube geometries considered for the shortstack approxi- F(y;A) that satisfiesthe differentialequation for the trans- mation.For example,(c) correspondsto the stackshown in Fig. 1(b). versedependence of the equationof motion,Eq. (12), is F(y;A) = 1- cosh(x/- iA/2 y/a) (66) cosh(x/-- i•./2) the stack,with and without gasviscosity, and when these quantitiesare negative,it indicatesthat acousticpower is The average of F(y;A) over the pore is defined as beingproduced by the stack.Stacks made of poresfor which F( A ) = 1/ ( 2a) œF(y;A) dy and is Im F* (At) is a large valuewill resultin the greatestwork flow. F(A) = 1 - (2/Ax[ - i)tanh(x/- iA/2). (67) Work and heatflows are to be comparedfor the various Rott has workedout the cylindricalpore • geometry pore geometriesshown in Fig. 4(a)-(d). In the inviscid stack.One type of porousmedia theory is based upon propa- short stack approximation,pores with a large value of gationin a singlecylindrical capillary tube. ••'•2'•6 Denote by I m F(2) * will havethe greatestheat and work flowsas indi- a the radiusof the cylindricalpore. The characteristicpore cated by Eqs. (61) and (65). According to Fig. 5, which radiusfor cylindricalpores is R ----a. The radial coordinate showsthe real and imaginaryparts of F(Z) for the various of a cylindrical coordinatesystem centered at the middle of pore geometries,the parallel plate geometryhas the largest the circularpore is r. The functionF(r;g) whichsatisfies the valueofIm F(A)*. The valueoccurs for Ac•3.2, whichal- differentialequation for the transversedependence of the equationof motion, Eq. (12), is F(r;A) = 1 -- Jo(x/7gr/a)/Jo (•r{A ) ß (68)

1.0 The averageof F(r;2) over the pore is definedas F(2) = 2/a 2 yF(r;A) r dr and is F(A)= 1- (2/x/72)[J•(x/7A)/Jo(x/7A)]. (69) Previouswork has emphasizedthermoacoustic stacks 0.5 consistingof parallelplate pores and cylindricalpores. The basicequations are given here for stacksconsisting of rectan- gularpores. Denote by 2a and2b thelength along the x andy axesof the rectangularpore crosssection. Take the coordinate systemorigin to be at the lower left cornerof the rectan- 0.0 gularpore. It is convenientto definea functionY,,, (2) as Y,,,(A) = 1 + (i•r2/A2) [ (b 2rn2q- a2n2)/(aq- b)2]. (70) The characteristictransverse dimension equal to twice the

-0.5 pore area divided by the pore perimeteris R = 2ab/(a + b). (71) 0 2 4 6 8 10 12 14 A seriessolution for the functionF(x,y;A) which satisfies thedifferential equation for thetransverse dependence of the FIG. 5. Real and imaginarypart of the F(2) for parallel plates,circular equationof motion,Eq. (12), is7'8 pores,square pores, and equilateral triangle pores. Heat andwork flowsare proportionalto the imaginarypart of [F(/I) ] in the shortstack approximation thus indicatingthe parallelplate stackis the bestchoice for optimal thermoacousticengines. Also shownis the boundarylayer approximation F(x,y;A)= •16 rnCod d sin(mrrx/2a)sin(nrry/2b) m nY.•.(A) of F(,•) fbr all pore geometries,valid in the limit •. • •. (72)

3236 J. Acoust. Soc. Am., Vol. 90, No. 6, December 1991 Arnott eta/.: Thermoacoustics of stacks 3236 In all sumsgiven in thissection, m andn extendover positive pointsin the resonatorshown in Fig. 1(a). With thisap- oddintegers. For a rectangularpore the average is defined by proach,analysis of complicatedarrangements of thermoa- F( A ) = 1/ ( 4ab) œF( x,y;• ) dx dy, wherethe integral extends cousticelements can be evaluatedreadily and in a unified overthe entirecross section of the pore.Then, manner.Work and heat flowswere expressedin termsof specificacoustic impedance and pressureto takeadvantage 64 1 of these theorems. F(A) = •r4 ,.•d rn2n2Yr•, (A) (73) Finally, the functionF(/I) is also the key element of capillary-porebased porous media models. •2.•5 Factors are The functionF(x,y;A) in Eq. (17) wasobtained from a solu- usedin thesemodels to scaleproperties of randommedia to tionby Han l* forthe same differential equation as Eq. (12). circularpores. Thus the scalingfactors and methodology of Equilateraltriangular pores have been recently investi- porousmedia modeling can be readily adapted to beuseful in gatedby Stinson. • Thepore geometry isshown in Fig.4 (d). thermoacoustics.After all, thermoacoustic elements are The characteristicdimension is R = a/3 •/• and nothingmore than sections of porousmedia, with theadded richnessof ambienttemperature gradients. F(A)=1 •2•--iX coth 3A_ . +---73A ACKNOWLEDGMENTS This work was supportedby the Officeof Naval Re- In theboundary layer approximation appropriate for all search.We are gratefulto Mike Stinsonfor the triangular poregeometries in thelimit A--, oo, pore solution.We acknowledgeinsightful conversations F(A) = 1 -- (•/A)(l +i) = I -- (•/R)(1 +i), (75) with Greg Swift, AnthonyAtchley, and Tom Hofler. where •5is the viscousor thermal boundarylayer thickness and R is twice the pore area dividedby the pore perimeter. Equation(75) caneasily be derived from thelarge A limit of Eqs. (67), (69), or (74). Physically,26/R is the areaof the N. Rott, "Thermoacoustics,"Adv. Appl. Mech.20, 135-175(1980). boundary layer divided by the pore cross-sectionalarea. 2p. Merkli and H. Thomann, "Thermnacousticeffects in a resonance tube," $. Fluid Mech. 70, 161-177 (1975). Therefore,it isexpected and noteworthythat poreswith the N. Reit andG. Zouzoulas,"Thermally driven acoustic oscillations. Part sameA havethe samecomplex wave number in the wide- IV. Tubeswith variablecross-section," Z. Angew. Math. Phys.27, 197- tubelimit; i.e., A-, oo.J3 Figure 5 givesa graphicalillustra- 224 (1976). tion as to when the boundarylayer approximationis useful nG. W. Swift,"Thermoacoustic engines," J. Acoust.Sac. Am. 84, 1145- 1180 (1988). for a particularpore geometry. It alsoillustrates why it was N. Rott,"Damped and thermally driven acoustic oscillation in wideand necessaryfor Rott• to improveupon the boundary-layer narrowtubes," Z. Angew.Math. Phys.20, 230-243 (1969). theoryof thermoacoustics. Theceramics are manufactured by, among others, Coming Incorporated, IndustrialProducts Division, Coming, NY 14831.For a discussionof theirproperties, see $. J. Burtonand R. L. Gatten,•4doanced Materials in Catalysis{Academic, New York, 1977), Chap. 10. H. S. Roh,W. P. Amott, I. M. Sabatier,and R. Raspet,"Measurement IlL CONCLUSION andcalculation of acousticpropagation constants in arraysof smallair- Linearthermoacoustics for gas-filledstacks has been re- filledrectangular tubes," J. Acoust.Sac. Am 89, 2617-2624 { 1991); W. P. Amott, J. M. Sabatier,and R. Raspet,"Sound propagation in capil- ducedto calculationof a singlefunction F. The function lary-tube-typeporous media with smallpores in the capillarywalls," I. F(x,y;A) givesthe transversevariation of the longitudinal Acoust. Sec. Am. 90, 3299-3306. particlevelocity v z (x,y,z). It satisfiesthe partialdifferential aM. R. Stinson,"The propagationof planesound waves in narrowand equation, Eq. (12), and the boundary condition wide circular tubes,and generalizationto uniform tubesof arbitrary cross-sectionalshape," J. Acoust.Sac. Am. 89, 550-558 (1991). F(x,y;A) = 0 for x andy on the poreperimeter. The average 9A. L. Fetterand •. D. Walecka,Theoretical Mechanics of Particlesand of F(x,y;A) over the porecross section is F(A). The para- Continua (McGraw-Hill, New York, 1980). meterA is proportionalto the ratio of the porehydraulic •OH.Tijdeman, "On the propagation of soundwaves in cylindricaltubes," radiusand the viscousboundary layer thicknessthat would J. Sound Vib. 39, 1-33 (1975). I• C. Zwikkerand C. Kosten,Sound •4bsorbing Materials (Elsevier, Am- beappropriate for a flatpore boundary. Similarly, A r ispro- sterdam, 1949). portionalto the ratio of porehydraulic radius and the ther- •ZK. Attenborough,"Acoustical characteristics of porousmaterials," mal boundarylayer thickness. The formofF(r) dependson Phys.Rep. 82, 179-227 (1982). a•A.D. Pierce,•4coustics: AnIntroduction to ItsPhysical Principles and,4p- pore geometry.All first-orderacoustical field quantities plications( AmericanInstitute of Physics,New York, 1989). (Sec. I A-D) and the secondorder energyflux (Sec. I F) •nL. M. Brekhovskikh,Watws in LayeredMedia (Academic,New York, canbe evaluatedusing this function. 1980), p. 21 I. The generalframework was used in Sec.II to investigate •J. WheatIcy,T. Hofler,G. W. Swift,and A. Migliori,"An intrinsically irreversiblethermoacoustic heat engine,"J. Acoust.Sac. Am. 74, 153- the optimalchoice of capillarytube geometryfor stacks. 170 (1983). Heat and work flows evaluated in the inviscid, short stack •aM. A. Blot,"Theory of propagationof elasticwaves in a fluid-saturated approximation,are approximately10% greaterfor the par- poroussolid," J. Aeoust.Sec. Am. 28, 168-191 (1956). allel plate stackgeometry than for the circular, square,and •?L. S. Hah, "Hydrodynamicentrance lengths for incompressiblelaminar flow in rectangularducts," J. Appl. Mech. 27, 403--409(1960). equilateral-trianglepore geometries. •aM. R. Stinsonand Y. Champoux,"Assignment of shapefactors for po- Impedanceand pressuretranslation theorems were de- rous materials having simple pore geometries,"J. Aeoust. Sac. Am. velopedin Sec.I E for determiningthese quantities at all Suppl.I 88, Sl21 (1990).

3237 J. Acoust.Sec. Am., Vol. 90, No.6, December1991 Amottoral.: Thermoacoustics of stacks 3237