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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 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 capillary- tube-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
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