thermal modeling and performance analysis of a ...ju/paper/paper-thermoacoustic... · thermal...

Thermal modeling and performance analysisof a thermoacoustic refrigerator

David G. Holmberg and G. S. ChenInstitute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan

H. T. LinVehicle Engineering Department, Chung Cheng Institute of Technology, National Defense University,Tai-Hsi, Tao-Yuan 33509, Taiwan

Andrew M. WoInstitute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan

~Received 9 January 2001; revised 20 March 2003; accepted 15 May 2003!

A heat-driven thermoacoustic refrigerator has been designed and tested. A detailed thermal model ofthe device is presented. Energy balances within the system are discussed using external, heatexchanger, and stack control volumes in order to clarify the relationships of work and heat fluxesbelow and above onset. Thermal modeling is discussed as a tool for performance analysis as wellas for determining system heat losses and finding input heat flows required by a thermoacousticcode. A method of using the control volume balance equations to find stack work and deviceefficiencies is presented. Experimental measurements are compared toDELTAE thermoacousticmodeling predictions. Modeling results show that viscous losses within the system have a significantimpact on the device performance as well as on the ability ofDELTAE to accurately predictperformance. Modeling has led to an understanding of system performance and highlighted losssources that are areas for improvement in a redesign. ©2003 Acoustical Society of America.@DOI: 10.1121/1.1590971#

PACS numbers: 43.35.Ud@SGK#

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I. INTRODUCTION

The basic science of using a temperature gradient aa surface to generate sound~or vice versa! has been pre-sented well by Swift in his review article~Swift, 1988!.Since that time, the development of numerical thermoacotic ~TA! modeling codes~especiallyDELTAE, Ward and Swift,1996! has aided in a rapid increase in the number of reseaprograms working on thermoacoustic systems. Morecently, Swift has combined additional material into book fomat ~Swift, 2002!, giving a review of the current state of TAknowledge and research while pointing out the challenahead.

Thermal modeling of thermoacoustic devices~i.e., mod-eling heat movement throughout a system and not just instack! is not often discussed in the literature. In Swift’s itroductory thermoacoustics book~2002!, methods for mea-suring insulation heat loss and heat removal in water-cooflows are discussed. Backhaus and Swift~2000! give a moredetailed discussion of pertinent heat losses in thStirling-TA engine and how they were calculated. Howevthe relationships of these losses to thermoacoustic heat fland the use of thermal modeling for performance analysisnot discussed explicitly. Adeff and Hofler~2000! discuss atransient method used to find the load on their heat pustack. The arrangement of their device, with the heat pustack located on the opposite end of their half-wave resotor, allowed them to use lumped capacitance analysis tothe loading on the heat pump stack from transient solar teTheir approach allows finding actual cooling power, butnot useful in the case of the present apparatus~heat-driven

782 J. Acoust. Soc. Am. 114 (2), August 2003 0001-4966/2003/

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thermoacoustic apparatus, HDTA!, nor of many other de-vices.

This paper seeks to address a proper understandinthe place of a thermal system model relative to numerthermoacoustic modeling of a TA device, and to documthe use of thermal modeling as a tool for obtaining stawork and thermal efficiencies from experimental temperatand heat-flow measurements. After a discussion of thermodeling, the design of the HDTA setup is presented. Thcontrol volume analysis is used to show interaction of thmal and acoustic fluxes and demonstrate how work termsbe found from the thermal model. Finally, HDTA measurments are compared toDELTAE predictions with discussion othermal and viscous loss estimation.

II. THERMAL MODELING

An accurate thermal model not only translates inppowers and measured temperatures into stack fluxes veheat losses~insulation and others!, but by itself it can be usedto find stack work fluxes~acoustic power! even without pres-sure measurement, at least for some device configuratio

Thermal modeling of any system has its foundationfirst-law energy balances of different system control volum~e.g., a heat exchanger or stack!. The heat fluxes across thcontrol volume boundaries include solid conduction, convtion, and radiation. In TA devices above onset, these sacontrol volume balances have additional acoustic wfluxes in them as well as changes in some of the therconduction terms. Any heat-transfer textbook~e.g., Incroperaand DeWitt, 1985! will discuss conduction~Fourier’s law!,

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convection, and radiation flux estimation, as well as theplications of nonconstant material properties, inhomoneous materials, and application of finite-element methfor more complex control volumes. More detail on contaresistance estimation can be found in Madhusudana~1996!.Property data sources include references such as the T~Touloukian and DeWitt, 1970! for older materials. The website atmatweb.comis an excellent database of recent dataa variety of materials.

An accurate thermal model must be detailed: accudevice dimensions, contact resistances, thermal conductvariation with temperature, internal and external radiatand convection, and consideration of nonuniform~i.e., 3D!temperature profiles within system elements. The tempture distribution within a part should be estimated, andnecessary measured or computationally modeled. Somemate of the temperature variation is useful at least for plament of thermocouples, or interpretation and error analyof thermocouple signals. Generally, considering all of thelements in the design phase can lead to a better designmore accurate measurements.

In its typical form, a thermal model is a system of equtions with the first-law balances of several control volumwithin a device. Below onset, an accurate thermal modelbalance. That is, supplying measured power and temperadata as inputs to the model will result in all control volumshowing a balanced condition. The external balanceshow that the combined input and load power is all seexiting the device, either in the cooling water stream orinsulation/convection loss. An internal part balance wshow that the sum of all fluxes crossing the control voluboundary equals zero.

Below-onset data are useful for tuning the model paraeters necessary to find heat losses above onset. Ideadevice is operated at the same thermal conditions abovebelow onset so that thermal losses are the same and caseparated from TA fluxes. In the case of HDTA, five beloonset data sets~using lower mean gas pressure to avoid onat higher temperatures! were used to tune values of somuncertain thermal properties: insulation thermal resistanthermal conductivity of some materials at elevated tempetures, contact resistance between flanges, etc. Additimeasurements~temperature surveys at different points ondividual parts, heat flux probe measurements! were made tohelp understand temperature profiles and thermal losTuning involved finding a set of property values that resulin all control volumes balancing for a given data set, athen repeating this for all below-onset data sets. Mean vaof the below-onset property values were used for aboonset data sets.

The thermal model was implemented in a spreadshformat with measured power levels and temperaturesthermal properties as inputs, and the values of each convolume balance equation and subequations as output.model includes TA fluxes only as a by-product, appearinga result of the model for above-onset data sets. See furdiscussion below.

J. Acoust. Soc. Am., Vol. 114, No. 2, August 2003

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III. DELTAE THERMOACOUSTICS DESIGN SOFTWARE

Design of the HDTA apparatus and performance anasis were aided byDELTAE ~Design Environment for Low-amplitude ThermoAcoustic Engines, Ward and Swift, 199!,an executable code that takes device parameters~physicaldimensions and material properties! of basic acoustic andthermo-acoustic elements and then integrates along thenected elements to predict ideal device performance baselow-amplitude ~linear! thermo-acoustic equations. Greflexibility is provided for applying boundary conditions sucthat any relevant variables can be fixed. Both the umanual and Swift’s recent book~Swift, 2002! provide guid-ance in using the code.

Besides the linear acoustics limitation,DELTAE has twoother limitations relevant to this paper. First, whileDELTAE

accounts for conduction losses in the solid stack materialgas within the stack, it does not account for thermal condtion losses through the device body to ambient or shlosses around the stacks through the solid body of the deshell. These thermal losses effectively decrease performaand must be accounted for separately. Second,DELTAE doesnot automatically account for flow effects such as turbulflow disturbance present due to discontinuities betweenvice elements, or bulk fluid motion streaming effects whiwill tend to reduce performance.

IV. THERMOACOUSTIC APPARATUS DESIGN

The thermoacoustically driven and cooled heat-drivthermoacoustic apparatus~HDTA! is a nominal quarter-length resonator device with two stacks sandwiched by thheat exchangers, and a compliance on the bottom, FigHeat (QIN) at the hot end~HXF! generates 264-Hz sound ithe prime mover stack~stack 1!. This acoustic energy drivethe heat pump stack 2 which pumps energy from the cheat exchanger~HXL ! to the midheat exchanger~HXM !,where heat from the hot end and cold end is removedcirculating room-temperature water. The device was initiaconceived with solar application in mind which led to a hoend design with the hot heat exchanger copper fins~HXF!extending from the stack to the copper end block. Thisrangement would allow sunlight, gas-fired, or electricagenerated heat to be absorbed in the copper end capdirectly conducted to the stack.

DELTAE optimization of HDTA involved the sizing ofvarious subcomponents of the device: heat exchangstacks, and ducts. Various device parameters are presentTable I. The gas~He 60%–Ar 40%! was selected to mini-mize Prandtl number, thus decreasing viscous los~Belcheret al., 1999!. The mean pressure~2 bar! was chosento fit an overall desired length of the device to the staceramic structure and working gas. The stack material~Corn-ing Celcor® 400-cpsi square cell extruded ceramic! was cho-sen based on its strength, low thermal conductivity, higtemperature capability, and availability.

The heat exchangers are all standard copper fin desA 0.7-mm-thick stainless-steel shell surrounding HXF apart of stack 1 minimizes thermal structure losses. Theflon rings serve the dual purpose of insulation between m

783Holmberg et al.: Thermal modeling and performance analysis

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flanges and heat exchangers as well as gripping the staWater flowing through brass tubes passing through the Hfins ~three tubes in either direction! circulates in a counterflow arrangement to minimize temperature nonuniformDifferential temperature measurement across the waterand outlet and flow rate measurement give the heat remoThe cold heat exchanger has a flat ring-shaped electric heattached around the flange to provide loading (Qload) capa-bility.

Bolts pass through the stack/heat-exchanger pile, Figholding the pile together. This pile and bolt design hasundesirable side effect of a complicated thermal mowhich must account for conduction through the bolts (Qbolt)from hot end to cold end with heat loss along the lengththe bolt with various contact resistances. The resonatorand compliance were designed for simplicity; the abrresonator-duct/compliance~RC! transition has a significanflow pressure loss penalty.

The HDTA setup was instrumented with thermocoup~1-mm diameter, SS sheathed! in each heat exchanger as weas the stainless-steel flanges above HXM~SShot! and belowHXL ~SScold!. Thermocouple accuracy at room temperatuis approximately 0.2 °C, which is less than the temperat

FIG. 1. HDTA thermoacoustics refrigerator schematic~to scale!. HXF cop-per fins extend from endcap to stack 1. HXM heat exchanger is cooledcirculating room temperature water~in six 3.2-mm brass tubes!. HXL heatexchanger has an electric heater for loading purposes.

784 J. Acoust. Soc. Am., Vol. 114, No. 2, August 2003

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variation seen in the parts at operating conditions: tempture nonuniformity in HXM was observed to be large radally due to the bolt influence, and circumferentially variedapproximately 10 °C on the external surface~coldest at thewater inlets and outlets!, but thermocouple temperatures nethe interior wall agreed within 1 °C at operating conditionPressure measurements were made in a port just below Hin the resonator wall. HDTA power inputs have an accura~based on voltage, resistance, and current measuremen! of1%. HDTA sensors and power levels~current and voltage!are monitored usingLABVIEW PC-based software and datacquisition hardware.

When tested at elevated temperatures, approximatelywere required to reach thermal equilibrium at the first powset point, with approximately 1-h settling time betweenpoints. The below-onset data series, run for thermal moing purposes, had no external HXL heat loading (Qload

50 W). Above-onset data were collected withQload set to 0W and 10 W.

V. DEVICE THERMAL MODEL

The thermal model recognizes below-onset heat fluin seven control volumes: the external balance, three hexchangers, two stainless-steel flanges~above HXM and be-low HXL !, and the bolts. The external control volume surounds the device~although cutting through the resonatduct!, giving the summation of all fluxes entering or exitinthe device as shown in Fig. 2, left side. The heat flux, ‘‘Q,’’and work flux, ‘‘W,’’ terms seen in Fig. 2, will be discussebelow.Solid arrowsarebelow-onsetfluxes that appear in thethermal model andopen arrowsare TA fluxes that appearabove onset~and do not appear explicitly in thermal modeas discussed below!. Control volumes for the three heat exchangers and two stacks are also shown in Fig. 2.

The stainless-steel flange control volumes are not shoin detail in Fig. 2~and will not be discussed below! becauseno TA terms enter the control volume. The bolt control vo

TABLE I. Physical parameters of the HDTA device and gas.

Parameter Value Units

Mean pressure 2.00 barOperating frequency 264 HzGas mixture He0.6 Ar0.4Typical DT across hot stack 1 500 KGas therm. pen. depth~300 K, 800 K! 0.312, 0.511 mmGas visc. pen. depth~300 K, 800 K! 0.137, 0.317 mmStack thermal conductivity (ks) 1.46 W/mKStack specific heat (cp) 1000 J/kgKStack material density~r! 2510 Kg/m3

Stack square cell web thickness (2l ) 0.18 mmStack cell web separation (2a,2b) 1.10 mmStack thermal penetration depth (ds) 0.0265 mmHot stack 1 length 33.7 MmCold stack 2 length 20.0 mmHXF length 35.0 mmHXM length 6.0 mmHXL length 5.0 mmHXF, HXM fin separation 3.0 mmHXL fin separation 4.0 mmHX ~all! fin thickness 1.0 mm

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FIG. 2. HDTA control volumes for Eqs.~1!–~6!. Arrows show heat andwork flows as given in Eqs.~1!–~6!, with direction giving sign. Solidarrows represent below-onset heat flows~explicit in thermal model!.Open arrows are above-onset acoustic work or viscous loss terms.bolt schematic is representative of the 16 bolts around the device circference. Fluxes related to the SShot, SScold, and bolt control volumenot shown.


ume includes 16 bolts in contact with each steel flange, hexchanger, and Teflon ring. This is modeled using 1D finelements starting at the head at SShot down and endingthe nut at SScold. The model includes conduction fluxacross contact resistances between the bolt and each laydifferent temperatures along the length of the bolt. Becathe Teflon internal temperature distributions are unknowthe Teflon was treated as a thermal resistor between asegment and either a heat exchanger or a steel flange.bolt equations matrix was solved iteratively as a first stepbalancing the thermal model. The equations are not psented here as they are numerous and no TA terms arvolved.

Each control volume in the thermal model producesbalance equation stating that fluxes crossing the bound~dotted line of Fig. 2! must sum to zero. Below onset, onthe fluxes shown as solid arrows in Fig. 2 exist andthermal model~which only recognizes these fluxes! will bal-ance. Above onset, thermoacoustic terms enter the convolume balances~open arrows in Fig. 2!, including: workfluxes, changing stack conduction fluxes, surface visclosses, and flow losses. The two stack control volumes,seen in Fig. 2, have nontrivial balances only above onbelow onset they are merely thermal conduction pathstween heat exchangers and are only included in the thermodel as such.

Following are the first-law energy balance equationseach control volume that includes thermoacoustic~TA!fluxes. Arrow directions in Fig. 2 indicate positive sign forgiven flux. All viscous work losses are treated as heat inpat the location of the loss. Prime mover stack 1 work (WTA1)is defined as positive leaving the control volume~produced!,while stack 2 work (WTA2) is positive entering the controvolume ~consumed!.

A. External balance

05QIN1Qload2Qout-water2Qroom-loss2QTAvisc-lossRC, ~1!

QIN[electric power input at HXF

~Vrms~corrected for lead resistance!* I rms),

Qload[electrical load at HXL,

Qout-water[energy leaving in water cooling at HXM,

Qroom-loss[sum of losses to room air via convection,

including insulation loss,

QTAvisc-lossRC[TA work dissipated in resonator and com-

pliance due to:~a! surface viscous losses,

and~b! abrupt resonator-compliance

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B. HXF balance

05QIN2Qinsul2QstructureHXF2Qcond,HXF-12QTA,HXF-1

1QTAvisc-lossHXF, ~2!

QIN[electric power input at HXF,

Qinsul[energy loss through hot-end insulation,

QstructureHXF[conduction loss through stainless-steel

shell surrounding stack 1 plus a small

radiation loss to cooler regions in view,

Qcond,HXF-1[conduction from HXF into stack 1, equal to

Qcond,1avgbelow onset,

QTA,HXF-1[TA heat flow from HXF to stack 1,

QTAvisc-lossHXF[TA work that is converted to heat due to

viscous losses in HXF.

C. Stack 1 balance

05Qcond,HXF-12Qcond,1-HXM1QTA,HXF-12QTA,1-HXM

2WTA1 , ~3!

Qcond,HXF-1[conduction from HXF into stack 1, equal to

Qcond,1avgbelow onset,

Qcond,1-HXM[conduction from stack 1 into HXM, equal

toQcond,1avgbelow onset,

QTA,HXF-1[TA heat flow from HXF to stack1,

QTA,1-HXM[TA heat flow from prime mover stack 1 to

HXM,

WTA1[stack work produced in stack 1.

D. HXM balance

05Qcond,1-HXM1Qcond,2-HXM1QstructureHXM2Qout-water

1QTA,1-HXM1QTA,2-HXM1QTAvisc-lossHXM, ~4!

Qcond,1-HXM[conduction loss from stack 1 to HXM,

Qcond,2-HXM[conduction loss from stack 2 to HXM,

QstructureHXM[sum of internal structural~Teflon casing

and bolts! fluxes into HXM,

Qout-water[energy leaving in the cooling water flow at

HXM,

QTA,1-HXM[TA heat flow from prime mover stack 1 to

HXM,

QTA,2-HXM[TA heat flow from heat pump stack 2 to

HXM,


QTAvisc-lossHXM[TA work that is converted to heat due

to surface viscous losses in HXM.

E. Stack 2 balance

05Qcond,HXL-22Qcond,2-HXM1QTA,HXL-2 2QTA,2-HXM

1WTA2 , ~5!

Qcond,HXL-2[conduction from HXL into stack 2, equal


Qcond,2-HXM[conduction from stack 2 into HXM, equal


QTA,HXL-2[TA heat flow from HXL into stack 2,

QTA,2-HXM[TA heat flow from heat pump stack 2

to HXM,

WTA2[ work consumed in stack 2, used to pump heat

from HXL to HXM.

F. HXL balance

05Qload1QstructureHXL2Qcond,HXL-22QTA,HXL-2

1QTAvisc-lossHXL, ~6!

Qload[electrical power external load to heat pump

stack 2,

QstructureHXL[sum of internal structural~Teflon casing

and bolts! fluxes into HXL,

Qcond,HXL-2[conduction from HXL into stack 2,

QTA,HXL-2[TA heat flow from HXL into stack 2,

QTAvisc-lossHXL[TA work that is converted to heat due to

surface viscous losses in HXL plus mino

flow losses at HXL fin ends.

Each of these control volume equations can be reranged by bringing the TA terms to the other side of tequal sign, thus separating TA terms from the below-onterms recognized by the thermal model. However, care mbe taken in dealing with stack conduction terms. Below oset, the conduction flux~loss! in a stack is constant along thlength of the stack, producing a linear temperature profilethermal conductivity is constant. This flux~stack 1! is de-fined as

Qcond,1avg[~THXF2THXM !/Rstack1,~7!

Qcond,2avg[~THXL2THXM !/Rstack2,

whereRstack1 andRstack2 are the total thermal resistancesthe stacks in units of~K/W! and include conduction througthe stack solid material and gas volume~with thermal con-ductivity taken at the mean stack temperature!, and across


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gaps at either end of the stacks~where gas properties artaken at the heat exchanger temperatures!. Therefore,Qcond,avgcan be calculated based on thermal propertiesheat exchanger temperatures below and above onset. Aonset, however, there is some deviation from a linear profor a long stack~i.e., for a stack that does not meet thmathematical ‘‘short’’ stack condition! due to a continualflow of energy from the stack surface to the oscillating flo~or vice versa! along the stack’s length. This means thsome of the energy that enters stack 1 by conduction insolid and gas (Qcond,HXF-1) is later converted to work, or thaenters in the TA heat flow (QTA,HXF-1) later becomes pureconduction loss. In other words, while below onsQcond,1avg5Qcond,HXF-15Qcond,1-HXM, above onset thesequalities do not hold.

For analysis purposes, it is desired to separate heatfor work production from the conduction loss. For this pupose the average conduction flow based on the thermodel,Qcond,1avg, is subtracted from the total heat flow etering stack 1 at HXF to give the heat used in work prodtion. That is

QTAin,1[QTA,HXF-11Qcond,HXF-12Qcond,1avg. ~8!

If the prime mover stack is viewed as a heat engine, tQTAin,1 is equivalent to the heat flow entering from the hreservoir at HXF and used to produce work,WTA1 , withrejected heat,QTAout,1, to the cold reservoir at HXM.QTAin,1

is essentiallyQIN ~power in at HXF! with all conductionlosses~to room, through structure and stack to HXM! re-moved.QTAout,1 and the stack 2 equivalents are defined a

QTAout,1[QTA,1-HXM1Qcond,1-HXM2Qcond,1avg, ~9!

QTAin,2[QTA,HXL-2 1Qcond,HXL-22Qcond,2avg, ~10!

QTAout,2[QTA,2-HXM1Qcond,2-HXM2Qcond,2avg. ~11!

Then, substituting Eqs.~8!–~11! into Eqs. ~1!–~6! andrearranging gives

@EXT balance#5QIN1Qload2Qout-water2Qroom-loss

5QTAvisc-lossRC, ~12!

@HXF balance#5QIN2Qinsul2QstructureHXF2Qcond,1avg

5QTAin,12QTAvisc-lossHXF, ~13!

@STK1 balance#5Qcond,1avg2Qcond,1avg

505QTAin,12WTA12QTAout,1, ~14!

@HXM balance#5Qcond,1avg1Qcond,2avg1QstructureHXM

2Qout-water

52~QTAout,11QTAout,2

1QTAvisc-lossHXM!, ~15!

@STK2 balance#5Qcond,2avg2Qcond,2avg

505QTAin,21WTA22QTAout,2, ~16!

@HXL balance#5Qload1QstructureHXL2Qcond,2avg

5QTAin,22QTAvisc-lossHXL, ~17!


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where each balance refers to the summation of fluxes inthermal model. Below onset, the TA terms~second line ofeach of the above equations! will all equal zero so that thebalances will equal zero~to the accuracy of the model!. Thethermal model only recognizes the below-onset terms,thus sees the balance as written in the first line of each etion above. Above onset the TA terms will not equal zeand thus the thermal model energy balance will not eqzero. This imbalance of the thermal model at a particucontrol volume is therefore seen to be equal to the sum ofTA terms at that particular control volume.

For the HDTA device, an accurate thermal model wneeded for finding the real heat load at HXL. While thistrue for any device, HDTA’s bolt design called for carefanalysis. Figure 3 shows the various HXL heat flows at dferent operating temperatures, here for below-onset@Fig.

FIG. 3. Heat flows from HXL control volume as measured for~a! below-onset data and~b! above-onset low load (Qload50) data. Here,Qbolt-HXL

1Qwall-HXL5Qstructure-HXL of Fig. 2, and@HXL balance# equals the first-lawenergy imbalance for below-onset data~a! and equals QTAin,2

2QTAvisc-lossHXL for above-onset data~b!. Lines in~a! are straight line fits todata through origin. Lines in~b! are for guiding the eye only.


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3~a!# and above-onset@Fig. 3~b!# data.Qbolt-HXL is the fluxfrom the bolts into HXL either directly across the bolt threto copper contact resistance, or indirectly through the Teresistor.Qwall-HXL is the sum of the fluxes from HXM and thlower steel flange~SScold, Fig. 2! entering HXL through theTeflon. The sum of Qbolt-HXL and Qwall-HXL equalsQstructureHXL seen here. In Fig. 3~a!, below-onset data showthat asQIN is increased at HXF,THXL increases, which leadto a competing effect where the hot bolt is conducting heaHXL while heat is simultaneously transferred from HXthrough Teflon to HXM and SScold.

Figure 3~b! gives the same fluxes for above-ons(Qload50) data. In this case HXL is cooled below rootemperature and the previously negative wall flux is nseen to reverse, leading to a largely increasedQstructureHXL.The @HXL balance# is seen to increase and is now equalQTAin,22QTAvisc-lossHXL @Eq. ~17!# for this above-onset dataThe difference between the@HXL balance# and QstructureHXL

values is equal toQcond,2avg(Qload50). The total load onHXL ( QTAin,2) reaches 7.4 W at theQIN5380-W setting,with Qbolt-HXL the largest component. IncreasingQload ~forthe 10-W series, not shown! raisesTHXL and therefore re-ducesQstructureHXL.

Concerning accuracy of thermal model estimated fluxthe plot of @HXL balance# in Fig. 3~a! shows representativerror levels for below onset data, i.e., all control volumgenerally balanced within half a watt. However, half a wata significant percentage of the low fluxes seen at HXL. Ufortunately, uncertainties for above-onset fluxes are higThe control volumes for SShot and SScold~which have noTA terms present! should balance above onset as wellbelow. However, an imbalance at SScold was seen to gwith decreasingTHXL ~increasingQIN) for above-onset datawith a maximum imbalance of 3.2 W at the minimumTHXL

@whereQIN5380 W, Fig. 3~b!#. SShot showed some smalleimbalance also. This forces the conclusion that TA effewithin the device are changing thermal conditions signcantly away from the below-onset case: for example, cooof HXL at the center~and heating of HXM! could cause asignificant change in the 3D temperature distribution withthe heat exchangers that did not exist below onset, and wwould likely not be seen by the limited number of thermcouples. The changed temperature profiles will affect strtural fluxes and thus balances as seen at SShot and SSThe conclusion is that uncertainties onQstructureHXL must beassumed to be of similar magnitude, i.e., approximately 1near onset up to 3 W at lowestTHXL , or an uncertainty ofabout 50% onQstructureHXL and thus onQloaddE.

The complicating effects of the bolts can also be seenconsidering the effect of a changingTHXL on the HXF heatbalance. The bolt heat loss, and thus the structure losHXF, is influenced byTHXL , an effect that was observeexperimentally. The presence of the bolt, therefore,change the heat input toDELTAE and the total heat loss evewhile THXF , Troom, andTHXM are unchanged. This indicatethe need for a loss correlation that includesTHXL rather thana more simple function of (THXF2THXM). In any case, a losscorrelation should account for loss components relative


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appropriate temperature differences, and a careful thermodel accomplishes this.

VI. USING THERMAL MODEL FOR PERFORMANCEANALYSIS

A thermal model provides the ‘‘room-loss’’~insulationand convection! terms and structural losses~heat that by-passes the stacks through device shell!, and based on thesgives the required inputs to a thermoacoustics code sucDELTAE, where forDELTAE ~that does not account for roomloss or structure loss!

QINdE[QIN2Qinsul2QstructureHXF,~18!

QloaddE[Qload1QstructureHXL.

The thermal model also provides a tool to evaluate stem performance. Temperature and power measurementthe inputs to the thermal model. The results of the thermmodel, combined with pressure measurements and numemodeling, give device performance. The thermal model pvides inputs to a thermoacoustics code, but may be ableitself to provide stack work measurements.

To see the power of the thermal model in performananalysis, consider the following use of the balance equati@Eqs.~12!–~17!#. First, the@EXT balance# could give an ex-perimental measure of the viscous loss in the resonatorcompliance which is equal to the imbalance in the extercontrol volume. This is true because the external control vume is drawn cutting through the resonator below HXL~Fig.2!, with heat loss to the resonator based on the temperadifference between HXL and the compliance~at room tem-perature!, and also because the resonator and complianceuninsulated with sufficient room convection to remove thenergy. However, in practice, uncertainty for the insulatiloss is greater than for the viscous loss component, andthe external balance is used to find insulation loss estima

Second, stack work terms can be obtained from E~13!–~17!. The viscous loss terms must be either estimaor neglected. For this example, viscous losses are assunegligible, with the resulting equations

@HXF balance#5QTAin,1 , ~138!

@STK1 balance#505QTAin,12WTA12QTAout,1, ~148!

@HXM balance#52QTAout,12QTAout,2, ~158!

@STK2 balance#505QTAin,21WTA22QTAout,2, ~168!

@HXL balance#5QTAin,2 . ~178!

For each of these equations, the left-hand-side balancknown from the thermal model when above-onset dataentered as inputs. Unfortunately~for the HDTA thermalmodel!, there are only five equations but six unknowns. Wthe current HDTA hardware configuration, both stackscooled by one heat exchanger~HXM !. Separating the stackand using two room-temperature heat exchangers~e.g., byplacing stacks on opposite ends of a half-wavelength resotor as done by Adeff and Hofler, 2000! would allow separa-tion of QTAout,1 andQTAout,2, and solution of the stack workterms. WhileQTAin,1 is the actual heat used to produce wor


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QTAin,2 is the actual heat removed from HXL solely by thefforts of WTA2 acting in the heat pump stack, i.e., the trload on HXL.

In essence, the work estimates found in this wayvalid even at high pressure fluctuation levels becauseare based on measurements, not computations. The accof these work estimates depends on the accuracy of themal model as well as estimates of viscous losses.

One method of estimating viscous losses requires bthe aid of a thermoacoustic code as well as pressure msurement. Thermoacoutic viscous surface losses are clated in a code such asDELTAE. Acoustic power dissipated bviscosity acting at interior surfaces is proportional to tfluctuating velocity squared, which~assuming acoustic frequency, mean pressure, and temperature match! is propor-tional to the fluctuating pressure squared~Swift, 1988!. Thus,surface viscous losses can be taken fromDELTAE multipliedby the ratio (upumeasured/upuDELTAE)2. Flow viscous losses caalso be estimated using nonoscillating pipe flow ‘‘minoloss’’ coefficients according to the method of Swift~2002!,which at least gives some estimate of flow losses. The actic power dissipated due to these minor losses is shown tproportional to fluctuating velocity cubed,uUu3, which bythe same argument allowsDELTAE results to be used withcorrection by (upumeasured/upuDELTAE)3.

Having QTAin,1 and QTAin,2 allows for ‘‘lossless’’ ~noconduction loss! estimates of stack performance. For stac

h1-lossless5WTA1

QTAin,1, ~19!

and for stack 2

COPlossless5QTAin,2

WTA2. ~20!

Even for the HDTA device, where the work terms cannbe isolated, several useful thermal efficiencies can be conered. First, there is the gross performance criterion

h therm-gross5Qload/QIN , ~21!

which gives the external power load moved for a giventernal power input, including all losses. A second thermefficiency is

h therm-dE5QloaddE/QINdE, ~22!

which is the amount of energy removed at HXL for a givheat input to stack 1 at HXF subtracting out room and strture losses, but not removing viscous and stack conduclosses.QloaddE and QINdE are also the inputs required bDELTAE. Finally, based on the HDTA HXF and HXL balances, and with minor terms estimated with the aidDELTAE, a lossless thermal efficiency is

h therm-lossless5QTAin,2 /QTAin,1 ~23!

which gives an indication of the efficiency that couldachieved with the current physical geometry if perfect marials ~nonconducting except as needed for thermoacouperformance in the stack! were used such that no conductiolosses were present.


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Therefore, depending on hardware configuration amagnitude of viscous losses, useful measures of deviceformance can be found without any acoustic modelingmeasurements. A thermoacoustics code such asDELTAE

serves best in a design capacity, rather than in performaanalysis. However, it is also a powerful tool for separatiout thermoacoustic terms that the thermal model cannotferentiate, as noted above for aiding in estimating viscolosses.

Work terms presented in the Results section are tadirectly from the results of aDELTAE model, which is limitedby both the accuracy of the thermal model~supplying inputs!as well as of the thermoacoustics model of the device. THDTA DELTAE model includes ‘‘minor’’ flow loss terms aHXF, HXM, HXL, and at the resonator/compliance~RC!junction. Of these, only the RC loss is large. Minor-loss cefficients were taken~with some engineering judgment required! from Idelchik’s handbook~1994!, and applied ac-cording to the recommendations of Swift~2002!.DELTAE calculations were run withQIN fixed andQload as atarget allowingTHXF , THXL , and pressure fluctuation level tvary.

VII. RESULTS AND DISCUSSION

The goal of system thermal modeling as well as of thmoacoustic modeling is to understand the system andlyze performance. In the case of the HDTA device, modelhas led to an understanding of system performance and hlighted loss sources that are areas for improvement in adesign. A good example of this was given in Sec. V conceing the heat flows into HXL~Fig. 3!. The thermal modelclearly shows the influence of the bolt design on the actloading on the HXL heat exchanger, even showing the cnection between HXL temperature and the heat input at HThe thermal model provided insulation and stack conductlosses as well as the required heat inputs toDELTAE.

Results ofDELTAE modeling, using thermal model provided heat inputs, show thatDELTAE does not accurately predict device performance, although results are reasonablehelpful for understanding the system. Figure 4 shows pdicted versus measured prime mover stack temperatureferentials, showing thatDELTAE predictions are about 20%above measurements for theQload50 W data.DELTAE pre-dicted frequency, Fig. 5, is correspondingly high. Pressfluctuation levels show a crossing trend, Fig. 6.

These differences inDELTAE and measurements are molikely due to several factors. First, it is suspected thatabrupt transition between resonator duct and compliancea strong influence on actual device performance that ismodeled byDELTAE. The abrupt transition may produceresonator effective duct length different from that whichDEL-

TAE calculates, which would affect predicted frequency ahot-end temperatures. The drop in predicted pressure fluction levels below measurements in Fig. 6~max pressure fluc-tuation level in Figs. 4–6 wasupu/Pm55.3%) suggests thathe calculated resonator/compliance~RC! viscous losses areoverpredicted at higher pressure fluctuation levels, likely dto an overprediction of the minor flow loss pressure drestimate inDELTAE which is based on nonoscillating flow


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correlations, and increases asuUu3. The viscous surface los~normally calculated byDELTAE! in the compliance may alsobe overly large sinceDELTAE applies simple conservation omomentum at the RC interface, while the actual fluid motin the compliance is certainly not simple uniform oscillatinflow. Finally, it is possible that the thermal model is overpdicting QstructureHXL, as mentioned in Sec. V. This increasload at HXL would drive theDELTAE model closer to onsewith higher resultingTHXF .

The results above demonstrate that there are elementhe HDTA device that are not accurately modeled inDELTAE.However, understanding these elements~e.g., the fluid dy-namics of the compliance! is not required. Instead,DELTAE

results give work and viscous loss estimates that can betogether with thermal modeling results. This in turn is heful for pinpointing loss sources leading to device improvment.

FIG. 4. Measured prime mover stack temperature difference andDELTAE

predicted temperature difference versusQINdE @DELTAE input power, Eq.~18!# for low load (Qload50 W) data. Data withQload510 W showedgreater disagreement.

FIG. 5. Measured TAFA resonance frequency andDELTAE predicted fre-quency versusDELTAE input power @Eq. ~18!# for low load (Qload50 W)data.


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Some device performance information is presentedFigs. 7 and 8. Figure 7 shows the heat terms from the Hcontrol volume for the different low load (Qload50) datasets showing relative magnitudes of heat flows. The insution loss is seen to be relatively high and an easy targetimprovement. Stack and structure losses also may have rfor improvement. Utilizing a more efficient stack~e.g., par-allel plate! could make better use of the available he(QTAin,1) for work production.

Figure 8 shows the work balance in HDTA, where twork balance states that all work produced in stack 1 (WTA1)is consumed within the device either usefully asWTA2 orwasted on viscous losses. Clearly,QTAvisc-lossHXF, while thesmallest band of Fig. 7, is actually the largest viscous losthe system, and therefore HXF redesign is a serious cadate for improvement. The other obvious drain on workthe RC viscous flow loss that could be eliminated with

FIG. 6. Measured TAFA normalized pressure fluctuation levels andDELTAE

predicted values for low load (Qload50 W) data.

FIG. 7. Stacked area plot giving heat terms from the HXF control volufor the low load (Qload50) data. The heat available for work productionstack 1 (QTAin,1) and viscous loss increase more rapidly withQIN than otherterms.


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smooth transition from the resonator duct to the complianNote that all TA viscous loss and work fluxes in Figs. 7 a8 were estimated inDELTAE.

Figure 9 shows the results of the thermal efficiencdiscussed earlier@Eqs.~23!–~25!#. Gross efficiency is abou3% for the 10-W load case, but could be higher at higloads. TheDELTAE efficiency ~with structural and insulationheat losses removed! reaches above 10%. The high efficiecies for theQload510 W andQIN5260 W setting occur whenTHXL is 8 °C above THXM. The conduction-lossless efficiencis highest at the point where useful input power (QTAin1) is atits lowest while HXL load is still high~and thereforeTHXL

.THXM). However, the main reason forh therm-losslessbeing

FIG. 8. Stacked area plot giving terms in work balancWTA15WTA21viscous losses. Viscous losses in HXF (QTAvisc-lossHXF) and inthe resonator-duct and compliance (QTAvisc-lossRC) are seen to be larger thathe stack 2 work (WTA2).

FIG. 9. Thermal efficiency results corresponding to Eqs.~23!–~25!.


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significantly aboveh therm-dE is the removal of stack 1 conduction losses, a nonrealistic possibility. Nonetheless, ththermal model results give targets to shoot for and cleashow where losses exist.

VIII. CONCLUSION

The design, modeling, and performance of the NatioTaiwan University’s Institute of Applied Mechanics anChung Cheng Institute of Technology thermoacousgroup’s heat-driven thermoacoustic refrigerator have bpresented. Thermal modeling, based on control volumevarious components in the device, has been discussedtool for determining system losses, for finding input heflows to a thermoacoustic code, and for performance ansis. The relationships of external input powers to the inheats required byDELTAE have been discussed relative to tactual heat flows in the prime mover and heat pump stathat are used for work production and heat pumping, resptively. The relationships of heat losses and acoustic wflows have been derived along with a method for findistack work from the thermal model. The present modelhas proven useful in identifying and understanding thermand viscous losses in the system, specifically in the bostack/heat-exchanger pile design, in the hot heat exchadesign, and in the abrupt transition between resonator dand compliance. These specific loss sources are believehave contributed toDELTAE’s poor prediction of device performance, and elimination of these losses in a redesign colead to significant improvements in the device.

ACKNOWLEDGMENTS

The authors would like to acknowledge the help of MJoseph F.S. Lee in the construction of the thermoacoudevice, and to thank G. Swift and A. Atchley for their advicand encouragement. Partial funding of this work was sported by the National Science Council of the RepublicChina under Grants NSC 89-2212-E-014-021 and NSC2811-E-002-004.

Adeff, J. A., and Hofler, T. J.~2000!. ‘‘Design and construction of a solarpowered, thermo-acoustically driven, thermoacoustic refrigerator,’’ psented at ASA Atlanta meeting, June 2000, and available online at: htwww.physics.nps.navy.mil/hofler/stadtar.htm

Backhaus, S., and Swift, G. W.~2000!. ‘‘A thermoacoustic-Stirling heatengine: Detailed study,’’ J. Acoust. Soc. Am.107~5!, 3148.

Belcher, J. R., Slaton, W. V., Raspet, R., Bass, H. E., and Lightfoot~1999!. ‘‘Working gases in thermoacoustic engines,’’ J. Acoust. Soc. A105~5!, 2677–2684.

Idelchik, I. E. ~1994!. Handbook of Hydraulic Resistance, 3rd ed.~BegellHouse, New York!.

Incropera, F. P., and DeWitt, D. P.~1985!. Fundamentals of Heat and MasTransfer, 2nd ed.~Wiley, New York!.

Madhusudana, C. V.~1996!. Thermal Contact Conductance~Springer, NewYork!.

Swift, G. W. ~1988!. ‘‘Thermoacoustic engines,’’ J. Acoust. Soc. Am.84~4!,1145–1180.

Swift, G. W. ~2002!. Thermoacoustics~Acoustical Society of America, NewYork!.

Touloukian, Y. S., and DeWitt, D. P.~1970!. Thermophysical Properties oMatter, TPRC Data Series~IFI/Plenum, New York!.

Ward, B., and Swift, G. W.~1996!. ‘‘Design Environment for Low-amplitude Thermoacoustic Engines,’’ Los Alamos National LaboratoLA-CC-93-8.


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