Experimental study of a radial mode thermoacousticprime mover
Jay A. LightfootDynetics, Inc., 1000 Explorer Boulevard, Huntsville, Alabama 35759
W. Patrick ArnottAtmospheric Science Center, Desert Research Institute, P.O. Box 60220, Reno, Nevada 89506
Henry E. Bass and Richard RaspetNational Center for Physical Acoustics and Department of Physics and Astronomy,University of Mississippi, University, Mississippi 38677
~Received 10 June 1998; revised 10 December 1998; accepted 11 December 1998!
The primary goal of this paper is to examine an expemental radial wave thermoacoustic engine, taking intocount existing radial wave thermoacoustic theory andcommon imperfections of nonuniform pore size and teperature discontinuities between the stack and heat exchers. Swift briefly mentioned thermoacoustics in the radmode of a cylindrical resonator,1 and developed the radiamode thermoacoustic wave equation. Arnottet al. derivedcoupled first-order differential equations for pressure aspecific acoustic impedance in a stack with a temperagradient, and pressure and impedance translation equafor open resonator sections and heat exchangers.2 Numericalimplementation of these equations as described in Reallows for the prediction of the onset temperature differen(DTonset), the temperature difference across the stackwhich acoustic oscillations are observed.
DTonset is an important quantity in thermoacoustic egines for several reasons. The total power generated bprime mover is the power generated by the stack~which isproportional to the temperature difference across the stDT! minus the thermal and viscous losses in the stackprime mover will begin to make sound when the total powgenerated in the stack overcomes other losses in theexchangers and the resonator.3,4 Therefore,DTonsetis propor-tional to the ratio of the acoustic power dissipated inentire system~including the stack, heat exchangers, and aexternal load! to the acoustic power generated by the staSecond, from an experimental vantage point,DTonset mea-surements are relatively easy to make, so that predictions
2652 J. Acoust. Soc. Am. 105 (5), May 1999 0001-4966/99/105(5
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measurements can be compared to determine the accurathe numerical model.
An outgrowth of this research was the need to elucidexperimental deviations from the theoretical ideal—a disbution of pore sizes within the stack rather than a unifopore size and temperature discontinuities between the sand the heat exchangers. Elementary methods of accounfor these complications are presented which bring expment and theory into agreement.
I. RADIAL WAVE THERMOACOUSTIC PRIME MOVERDESIGN
A schematic of the resonator portion of the radial thmoacoustic prime mover is shown in Fig. 1. The outer ri~A! is constructed of steel with inner and outer diameters148.6 cm and 156.2 cm. The top and bottom lids~B! have anouter diameter of 158.75 cm and a thickness of 2.54 cm.center hole in B, for access to the thermoacoustic elemehas a diameter of 35.6 cm. All seals were made with o-rinThe caps~C! are constructed of stainless steel and havgroove cut at the location of the stack, leaving a1
4-in. plateover this region, to reduce thermal conduction of heat inradial direction across the stack region of the resonaWhen the caps~C! are placed into the holes in B, the innefaces of the caps are flush with the inner faces of the lThe height of B, and thus the inner height of the resonato10.2 cm. The assembled resonator has a mass of aboutkg. It should be noted that measures can be taken to sigcantly reduce the mass of the resonator.
The heat exchangers5 are constructed of four standar14-in. copper tubes bent into circles, with copper plates~each
having four 14-in. pre-punched holes to fit snugly over th
tubing! fed onto the copper tubes. Steel washers were plabetween the copper plates to maintain the correct spacThe individual copper plates are 10 cm tall, 1.3 cm wi~radial dimension!, and 0.76 mm thick. The washer spaceare also 0.76 mm thick, and the porosity or ratio of open ato total area of the heat exchangers is 0.32, where blockby heat exchanger tubing and washer spacers has beeninto account. The inner~hot! heat exchanger was heateddissipation of electrical current in 80% Nickel, 20% chrmium wire inserted into the copper tubing and electricainsulated from the tubing with temperature resistant cerabeads. The outer~cold! heat exchanger was cooled using twater. The hot heat exchanger has an outer diameter ofcm and an inner diameter of 21.2 cm when fitted to the stawhile the cold heat exchanger has an outer diameter of 2cm and an inner diameter of 26.3 cm when fitted to the sta
The silicon bonded mica paper stack, with inner aouter diameters of 23.8 cm and 26.3 cm, was sandwicbetween the two heat exchangers. The thickness of thevidual stack pieces is 0.015 cm. Mica was chosen becausits high temperature tolerance~up to 773 K! and its lowthermal conductivity of 0.163 W/~m*K !. The stack wasformed by placing 183 of these pieces on top of each owith 8 smaller mica washers between each of the larger mpieces to maintain the proper spacing. The spacer waswere 0.64 cm in diameter and 0.038 cm thick. The ovestack height was 10.2 cm. In an attempt to further maintproper spacing in the stack, 0.015 cm diameter temperaresistant teflon thread was used between each pair of mwasher spacers to provide a total of 16 support locatiequally spaced about the circumference of the stack.
A line drawing of the entire system is shown in Fig.The Endevco 8510B microphone used for detection ofacoustic wave was located at a pressure antinode next toouter wall of the resonator. It would be preferable to havemicrophone at the center of the resonator, since the higsound pressure levels~SPL! occur there, but high tempera
FIG. 1. Unassembled view of the resonator used for the large radial wthermoacoustic prime mover. Pieces C are stainless steel caps whichaccess to the inner thermoacoustic elements. Pieces A and B are plaibon steel.
2653 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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tures precluded such a placement of the microphone.other inlets to the resonator~i.e., plumbing, heating, gasvalve, and thermocouples! are located at the pressure nodeorder to reduce losses due to any leaks at these juncture
II. MEASUREMENTS AND ASSOCIATED ERRORS
All temperature measurements were taken with typethermocouples having a precision of61 K. Test measure-ments with multiple thermocouples of the heat exchantemperature varied by a maximum of 2 K, therefore tempeture measurements were taken using a single thermocofor each heat exchanger. The thermocouples were placedtween plates of the heat exchangers and coupled to theexchangers using high temperature–high thermal conducity paste. At DTonset the sound in the tube, initiated bysmall tap on the resonator, increased instead of decayThe error associated with the measurement ofDTonset is es-timated to be62 K ~the difference between the earlieDTonset after tapping and theDTonset obtained without tap-ping!, so that the total error in temperature measurement65 K.
SPL was measured with an Endevco 8510B pieresistive microphone which was calibrated using a pistphone. The microphone output was routed to a dynamicnal analyzer for viewing the SPL as a function of frequenThe error associated with SPL measurements was less0.2 dB, and the frequency step size was 0.25 Hz.
III. EXPERIMENTAL RESULTS AND COMPARISON TOTHEORY
Several questions required measurements with the rawave prime mover. The first had to do with theoretical pdictions; can we predict the behavior of radial wave primmovers using radial wave thermoacoustic theory? The sond was whether the radial modes reduce harmonic gention. Originally we were interested in the effect of slopestacks in radial wave thermoacoustic engines, but theoryshown that the effect in radial prime movers with a natuslope between plates is minimal.6
A. Initial comparison of experimental and theoreticalonset temperatures
Measurements ofDTonsethave been made on the abovdescribed radial wave prime mover. Thermocouples wplaced in the heat exchangers~attempts to measure the temperature at the face of the stack gave ambiguous resultsto the short radial length of the stack and an inabilityprecisely position the thermocouple!. Figures 3 and 4 showthe lowest experimentalDTonset results ~solid circles!achieved as a function of ambient pressure, for air and argrespectively. The ambient temperature in both cases wasK. Also shown are the theoretical predictions for the hot stemperature~dashed line! assuming that the stack facemaintain the same temperature as theheat exchangers. Forthe present discussion, the dotted lines in Figs. 3 anshould be ignored. It is obvious that theory and experim
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2653Lightfoot et al.: Thermoacoustic prime mover
FIG. 2. Line drawing of the radialwave thermoacoustic prime mover.
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are not at all in agreement, with the experimental onset tperatures being significantly higher than predicted for bgases.
There are several complicating features of our radialgine. First, for computations the stack was assumed to hafixed pore size or plate spacing, but measurements showa distribution of pore sizes exists. Second,DTonset is sensi-tive to the ‘‘fit’’ of the heat exchangers around the stackfact which is enhanced by the short radial length of the st
2654 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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as will be shown later. The heat exchanger ‘‘fit’’ is not adifficult in the plane wave case, since the manufacture oflat heat exchanger surface is easily achieved so thatstack surfaces can be flush against heat exchangers witalteration of the stack pore shape, so that direct thermal ctact is sustained between the stack and the heat exchaSuch a fit is more difficult in the radial case since the hexchanger surfaces must maintain perfect curvature tovide the best possible thermal contact between stack and
2654Lightfoot et al.: Thermoacoustic prime mover
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FIG. 3. Onset temperatures for the radial wave prime mover with air asworking fluid, assuming no temperature discontinuity between the stackheat exchangers. Solid line shows cold heat exchanger temperature; dline is the predicted hot heat exchanger temperature for a stack with a spore size; dotted line is the predicted hot heat exchanger temperature fostack with the pore distribution shown in Fig. 5; and circles show expmental results.
FIG. 4. Onset temperatures for the radial wave prime mover with argothe working fluid, assuming no temperature discontinuity between the sand heat exchangers. Solid line shows cold heat exchanger temperdashed line is the predicted hot heat exchanger temperature for a stacka single pore size; dotted line is the predicted hot heat exchanger temture for the stack with the pore distribution shown in Fig. 5; and circshow experimental results.
2655 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
exchanger. Attempts to force the heat exchangers againsstack, thus creating the necessary curvature in the heachangers, tend to distort the stack plate spacing. Sinceheat exchanger is much more rugged than the stack, a pis reached where the stack conforms to the heat exchashape via warping of the stack plates rather than the hexchanger conforming to the stack shape. For identicalcompositions and pressures, results forDTonset varied de-pending upon the proximity of the heat exchangers tostack and the stack distortion.
B. Nonuniform stack plate spacing
As mentioned previously, attempts were made to matain a constant plate spacing throughout the stack by umica washer spacers and later by adding teflon threadmaintain the correct spacing. In hindsight, it would habeen much better to use thicker~and thus more rigid! micaplates in the stack at the expense of reducing the stackrosity ~this would also have prevented additional distortiof the stack spacing due to pressure from the heat exchers!. In order to characterize the present stack, measuremof the plate spacing were taken over the entire height ofstack, assuring that the average spacing is equal to the onally expected constant spacing. Radial variations in pspacing were negligible. Over 500 measurements were mto ensure an accurate representation of the axial/azimupore size distribution. The measurements were by necestaken without the heat exchangers in place, such that no pwarping was introduced by pressure from the heat exchaers. Ideally, measurements would have been repeatedthe heat exchangers in place. However, sinceDTonset mea-surements in the following sections were taken for casewhich the heat exchangers were pressed as tightly as posagainst the stackwithout distorting it, pore measurements othe stack alone should be sufficient. Figure 5 shows the msured distribution in terms of the percent of the total oparea in the stack occupied by each gap size.
Prediction ofDTonset for a stack with a distribution ofpore sizes is accomplished by treating the various pore sas having parallel impedances. The electrical analog wobe parallel resistors. Beginning with a known impedance apressure amplitude at the hot, rigid end of the resonator,
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FIG. 5. Area distribution of plate spacing in the radial prime mover sta
2655Lightfoot et al.: Thermoacoustic prime mover
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pressure and impedance at the hot face of the stack madetermined using translation theorems, Eqs.~5! and ~6! ofRef. 2. The boundary conditions at the hot face of the stare continuity of pressure and volume velocity, so
P1 j5PH , ~1!
and
(j
AjV1j5AHVH , ~2!
where j represents quantities for a particular pore size, sscript 1 denotes that the quantity is evaluated at the hotof the stack,P1 j is the complex acoustic pressure at the hside of an individual pore,PH is the pressure at the hot facof the stack determined by translation,Aj is the stack crosssection occupied by a particular pore size,AH is the resona-tor cross-sectional area at the hot face of the stack,V1j is thebulk particle velocity averaged over the cross-section ofstack having a given plate spacing at the hot side, andVH isthe velocity at the hot face of the stack determined by tralation.
To obtain a full numerical solution to the problem,would be necessary to guess the complex pressurePC andthe various pore velocities at the cold side of the stack, ingrate backward to the hot side of the stack checking to sethe conditions in Eqs.~1! and ~2! had been met, and repethe process with new guesses until the solution convergThis would require scanning an extremely large paramspace. Therefore, the method of Raspetet al. for approximat-ing thermoacoustic calculations using a single step7 has beenutilized to reduce computation times. The single step metwas shown to be accurate forukLu<0.3, whereL is the stacklength andk is the wave number. The radial prime movcertainly meets this criterion sinceukLu'0.065.
Assumptions are that the acoustic pressure is consacross the pore cross section and is a function only ofradial distance along the pore, and that the transverse veity is small relative to the radial velocity. The relevant equtions describing the fluid motion in the pore are taken froEqs.~7! and ~8! of Ref. 2 @with porosity accounted for sucthat v r(r )5Vj /V j , where V j is the porosity for a givenpore size andVj5Pj /Zj ] and are given by
Vj5V jF j~l j !
ivr0
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and
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ivr0kj
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wherekj2 is given by Eq.~4! of Ref. 2. The finite difference
forms of these equations are given by
iC j
V1 j1V2 j
25
P1 j2P2j
L~5!
and
2656 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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iC j
V1 j2V2 j
L1 iC jD j
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where
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The subscript 2 refers to pore quantities at the cold sidethe stack, subscript 1 to pore quantities at the hot side ofstack, subscriptj to pores of a particular size, and all parameters are evaluated at the center of the stack. Note thavolume coefficient of expansion,b, in Eq. ~8! has been re-placed by the ideal gas result, 1/T.8 Equations~5! and ~6!may be combined to eliminateV2 j so we are left withV1 j interms ofP1 j andP2 j
V1 j51
iC jLH P1 jF12
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22
kj2L2
4 G2P2 jF12
D jL
21
kj2L2
4 G J . ~9!
Now we need only to guess the magnitude and phase ofP2 j ,and use the known pressureP1 j to determineV1 j . Thesevalues ofV1 j are then tested to see if the condition in Eq.~2!holds. If the requirement is not met,P2 j is adjusted and theprocess is repeated. When the requirement is met, Eq.~5! isrearranged to giveV2 j using the now known values forV1 j ,P1 j , andP2 j . The pressure and velocity at the cold sidethe stack are then determined by requiring conservationpressure and volume velocity
PC5P2 j ~10!
and
ACVC5(j
AjV2 j , ~11!
wherePC is the pressure in the tube at the cold face ofstack,AC is the resonator cross-section at the cold face ofstack, andVC is the area averaged particle velocity in thtube at the cold face of the stack.
Using the method outlined above,DTonset was deter-mined for a stack having a distribution of pore sizes. Tdotted lines in Figs. 3 and 4 show corrections, usingmeasured pore distribution of Fig. 5, to the constant poreresults represented by dashed lines for the hot side tempture at onset in air and argon, respectively. Physically,reason for the increasedDTonsetas a result of the pore distribution is due to the fact that only the pore spacings inmiddle of the distribution have the correct ratio of penettion depth to pore size, while the pores with other sizesnot acting as efficiently. It is obvious that the pore distribtion in the stack explains a significant portion of the discreancy between experiment and theory, but further interpretion is required to account for the remaining disparity.
2656Lightfoot et al.: Thermoacoustic prime mover
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C. Temperature discontinuities between the heat
exchanger and the stack face
As the stack in a thermoacoustic prime mover becomvery short, the temperature gradient necessary for onseoscillations increases, which in turn raises the amount of hflowing via thermal conduction from the hot side to the coside of the stack. In addition, any small gap betweenstack and the heat exchanger becomes a more apprecpercentage of the stack length, so that heat which is cducted across the gap may no longer be ignored by assuthat the stack face and the heat exchanger maintain the stemperature. Rather, it becomes necessary to accounconduction through the fluid across gaps between the sand heat exchanger as well as for normal thermal conducthrough the stack, when determining the temperatures afaces of the stack which ultimately cause onset.
A simple model may be used to explain the remaindifference between experiment and theory, and why theference is much more noticeable for the radial engine tfor a plane wave device. Consider the heat exchangstack–heat exchanger pictured in Fig. 6. Perfect insulahave been placed on top and bottom to confine heat flowthe horizontal direction. The hot and cold heat exchangare held at temperaturesTHHX andTCHX . Rather than assuming that the stack face and the heat exchanger are in phycontact and at the same temperature, they are separatedsmall gap of lengthl filled with gas. The stack length idenoted byL. Assumptions are: heat is transported onlyconduction,kC ~the thermal conductivity of the gas in thgap near the cold heat exchanger! is evaluated at a temperature of TCHX , kH ~the thermal conductivity of the gas in thgap near the hot heat exchanger! is evaluated at a temperature ofTHHX , kS ~the thermal conductivity of the solid stacmaterial! is assumed to be constant and is evaluated atemperature of12 (THHX1TCHX). Our goal is to find the hotand cold side stack temperatures,THS andTCS, in terms of
FIG. 6. Illustration of the gaps present between the radial stack andexchangers. The upper plot shows a representative temperature distribover the heat exchanger–gap–stack–gap–heat exchanger system. Teture is constant in the heat exchangers, while the gradient is larger ingaps than in the stack.
2657 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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the known quantitiesl , L, TCHX , THHX , kC , kH , and kS .Assuming linear temperature gradients within the threegions and requiring heat flow to be continuous at all bouaries, application of the 1D steady state heat equation yi
THS5kHTHHX1kCTCHXkCTCS
kH~12!
and
TCS5@kCkH1kCkS~ l /L !#TCHX1kHkSTHHX~ l /L !
kHkC1kHkS~ l /L !1kCkS~ l /L !.
~13!
Examining Eqs.~12! and ~13!, it is evident that asl /L→0,THS→THHX andTCS→TCHX as expected. In the case othe radial wave prime mover,L51.25 cm and a good average estimate forl ~determined by physical measurementthe gaps between the heat exchangers and the stack! is l
50.45 mm, producing a ratio ofl /L50.036. For the typical5.08 cm plane wave stack of Ref. 9, a conservative estimwould bel 50.025 mm so thatl /L50.002.
Figures 7 and 8 show deviations of the stack face teperatures from the heat exchanger temperatures over a rof heat exchanger temperature differences for the aboveues ofl /L for air and argon. From these figures we see tthe radial prime mover does a poor job of establishingexpected temperature difference across the stack due toshort radial length of the stack and the presence of nnegligible gaps between the stack and heat exchanger. F150 K difference across the heat exchangers, the actualference across the stack is only 106 K for air and 94 Kargon, while a conservative estimate for the 5.08 cm stacRef. 9 gives differences of 147 K for air and 144 K fo
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FIG. 7. Demonstration of the temperature discontinuity between the sand heat exchanger face forl /L50.036. The various temperature locationare shown pictorially in Fig. 6.
2657Lightfoot et al.: Thermoacoustic prime mover
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argon. These results indicate that assuming the heatchanger temperature was equivalent to the stack faceperature was not appropriate.
Previously, it was mentioned that experimental measuments ofDT varied depending upon the ‘‘fit’’ of the heaexchangers to the stack. Measurements of the gap betwthe heat exchangers and the stack range from 0 to 0.6when the heat exchangers are tightened and pressed agthe stack. The upper end of this range can be reduceabout 0.4 mm by lightly hammering the heat exchanagainst the stack. Measurements have been made for a fo~hammered! and an unforced~only tightened and pressed! fitof the heat exchangers against the stack. In both casel
should be close to the average measured gap value, altha slight variance due to azimuthal heat flows in the fluid athe stack may do a better job of modeling the experimeresults. Therefore, an effective gap length,l eff , is defined tosimulate the effect that the entire gap distribution hasDTonset.
In order to compare experiment with theory using tmeasured distribution of pore sizes and accounting for abetween the stack and heat exchangers,TCS and THS arecalculated using the measured values forTCHX andTHHX fora guessedl eff in air, the initial guess being the average mesured gap size.TCS is then used as the cold side stack teperature in theDTonsetcalculation to determine the predictehot side temperature required for onset, denoted byTHS calc.This predicted value is compared withTHS. l eff is then ad-justed and the process repeated untilTHS calc and THS bestmatch up over a range of pressures. Admittedly, this procis backward, sincel eff is adjusted to give the best possibresults for a given gas~air!. However, if the theory is correctthen the resultingl eff for one gas should be applicableother gases as well. Therefore, thel eff determined for air is
FIG. 8. Demonstration of the temperature discontinuity between the sand heat exchanger face forl /L50.002. The various temperature locatioare shown pictorially in Fig. 6.
2658 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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used to determineTCS and THS for the prime mover filledwith argon, and the resultingTHS is compared withTHS calc
for argon.Figures 9 and 10 showTCHX , TCS, THS, THHX , and
THS calc for air and argon, respectively. Note that measutemperatures are represented by the squares and cirwhile triangles represent calculated values. The experimeresults are the same as those shown in Figs. 3 and 4,again represent the configuration giving the lowestDTonset.Keep in mind thatTCHX andTHHX are experimental resultshowever, the comparison will be between the calculatfrom-experimentTHS and the predicted hot side stack temperatureTHS calc. In the upper plot of Fig. 9, we see thatchosen value ofl eff50.2 mm produces nice agreement fair. Applying this value ofl eff to the argon data, we see ithe upper plot of Fig. 10 thatTHS andTHS calcare within 5 Kof each other over the pressure range tested, and appearfollowing the same general trend. It should be mentionedthis point that it would have been nice to extend the pressrange over which measurements were made, however, duthe large flat surface area of the resonator, increasingpressure beyond 6 psig would have placed an exorbiamount of force upon the bolts holding the resonatorgether, and could have caused the steel plates to bowward.
Measurements were also made which gave higher otemperatures due to a looser fitting of the heat exchangFigures 11 and 12 show the results for air and argon, restively. l eff50.41 mm provides good agreement betweenTHS
andTHS calc in air. Applying this l eff to the argon data also
ck
FIG. 9. Air results forl eff50.2 mm. The upper plot is a blown up versioof THS andTHS calc in the lower plot, whereTHS calc is the numerically pre-dicted onset temperature usingTCS as the temperature at the cold side of thstack. Measured temperatures are represented by circles and squarestriangles represent calculations. Errors in the lower plot are smaller thandata points.
2658Lightfoot et al.: Thermoacoustic prime mover
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gives good results, such thatTHS andTHS calc are within 8 Kof each other, and again the general trend of the curvesthe same. The agreement is not as good as in the prevcase, due to a larger distribution of gaps between the sand heat exchangers. In particular, we have assumed thal eff
is the same at the hot and cold sides of the stack. For a lofit of the heat exchangers on the stack, this assumptioncomes less valid than in the case where the heat exchanwere fit as tightly as possiblewithout disfiguring the stack.
D. Radial prime mover harmonic generation
In most cases, the linear acoustic wave equation suffito describe the behavior of a system. However, whenacoustic pressure amplitude becomes large, higher oterms must be taken into account. Coppens and Sandercounted for nonlinear effects for finite-amplitude standiwaves in a resonance tube by modifying the linear equatof state and continuity to allow for second-order effects10
Chen included an additional second-order term in the mmentum equation for the same problem and found therection to be small,11 as would be expected since good agrement was shown between theory and experiment usingCoppens and Sanders approximation.
The general behavior of a prime mover was effectivdescribed by Atchleyet al.12
‘‘Once onset of self oscillation is reached, the acous-tic amplitude in the tube immediately assumes alarge value, typically about 1% of the ambient pres-sure. The observed waveform is noticeably nonsinu-
FIG. 10. Argon results forl eff50.2 mm. The upper plot is a blown upversion ofTHS and THS calc in the lower plot, whereTHS calc is the numeri-cally predicted onset temperature usingTCS as the temperature at the colside of the stack. Measured temperatures are represented by circlesquares, while triangles represent calculations. Errors in the lower plosmaller than the data points.
2659 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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FIG. 11. Air results forl eff50.41 mm. The upper plot is a blown up versioof THS andTHS calc in the lower plot, whereTHS calc is the numerically pre-dicted onset temperature usingTCS as the temperature at the cold side of thstack. Measured temperatures are represented by circles and squarestriangles represent calculations. Errors in the lower plot are smaller thandata points.
FIG. 12. Argon results forl eff50.41 mm. The upper plot is a blown upversion ofTHS and THS calc in the lower plot, whereTHS calc is the numeri-cally predicted onset temperature usingTCS as the temperature at the colside of the stack. Measured temperatures are represented by circlesquares, while triangles represent calculations. Errors in the lower plosmaller than the data points.
soidal. As more energy is supplied to the hot end ofthe stack, the temperature of that end increases onlyslightly while the acoustic amplitude in the tube in-creases rapidly... Unfortunately, as the acoustic am-plitude increases, an increasing fraction of the acous-tic energy appears as higher harmonics–harmonicdistortion increases.’’
The authors presented results for a constant cross-seplane wave prime mover immediately after onset and awhen the hot end temperature was raised 43 K beyondnecessary temperature for onset. In both cases, nonligeneration of higher harmonics was demonstrated. Theperimental results for the relative amplitudes of the harmics to the fundamental in the prime mover just beyond onprovided nice agreement with the theory of Ref. 10.
It is desirable in a prime mover to minimize the genetion of higher harmonics so that more acoustic energychanneled into the fundamental. In a plane wave resonaone of the ways to accomplish this is to vary the crosection of the tube.13 This detuning causes the natural modof the resonator to be anharmonic. Gaitan and Atchle14
following the method outlined by Coppens and Sanders15 ina later paper, investigated higher harmonic generationtubes with harmonic and anharmonic natural modes forplication to thermoacoustic engines. The anharmonic tuwere made by varying the cross-section in the center oftube to a size different than at the ends. Experimental resfor the amplitudes of the harmonics for a given amplitudethe fundamental were in excellent agreement with theoryboth tube types. In the case of the harmonic tube, the amtude of the first harmonic was only 10–20 dB below tfundamental, depending upon the acoustic pressure amtude of the fundamental. However, in two separate anhmonic tubes~one with a larger center cross-section thanthe ends and one with a smaller center cross-section! theamplitude of the first harmonic was reduced 30–40 dB bethe fundamental.
The generic nonlinear wave equation presented in Reis
(n
S cn2¹22
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nv
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]
]t D pn
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]t2F S u
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1g21
2 S p
r0c2D 2G , ~14!
wherecn and Qn are the sound speed and quality factorthe nth resonance of the tube,v is the angular driving fre-quency~the fundamental resonance for a prime mover!, pn isthe acoustic pressure of thenth harmonic of the driving fre-quency,u andp are the total acoustic velocity and pressuand are functions of radial locationr in radial systems~orlongitudinal positionz in plane systems! and timet. Equation~18! is valid near resonance and assumes the total stanwave to be of the form
p5 (n51
`
pn5( r0c2MRn cos~nkz!sin~nvt1fn!,
~15!
2660 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
iono
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et
-isr,-s
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ltsfrli-
li-r-t
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1
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ng
whereM is the peak Mach number of the fundamental,Rn isthe nondimensional amplitude of thenth harmonic,k is thewave number of the driving frequency, andfn is the tempo-ral phase of thenth harmonic. For a radial system, Eq.~15!would haveJ0 in place of cosine. The harmonic amplitudeRn, are shown in Refs. 14 and 15 to be directly proportioto
Rn}Qn cosS tan21S 2Qn
nvvn
vnD D , ~16!
wherevn denotes normal modes of the resonator, so thatharmonic amplitudes are maximized when the normal mocorrespond exactly to the harmonics of the fundamentalquency. Therefore, a detuned resonator leads to a reduin the amplitudes of the harmonics.
With the existing radial wave prime mover, we are limited to examination of the behavior of the system just aboonset. The reason for this is that when attempts are madincrease the temperature beyond onset, the following cyensues. When the necessary temperature differencreached, sound is produced. The system responds witincreased acoustic pressure amplitude and thus increacoustic heat transport from the hot side to the cold sidethe stack. The cold heat exchanger, unable to sustain itsperature with the increased heat load, begins to heat upthat the temperature difference across the stack falls bethe necessary difference for oscillations to be maintainWhen the acoustic wave ceases, the acoustic heat tranalso ceases, and the cold heat exchanger returns to its onal cooler temperature. Once again the temperature difence reaches the critical onset value, the acoustic wavregenerated, and the cycle is repeated. A similar effectobserved by Olson and Swift in a plane wave prime move16
Although it would be nice to get a comparison of plane aradial prime movers at lower and higher acoustic pressamplitudes, the results for temperatures just above oshould give some insight into the reduction of higher hmonic generation which occurs in radial systems as copared to constant cross-section plane systems.
Figure 13 shows a spectrum from the previously dscribed radial wave prime mover. The hot heat exchantemperature is 427 K and the ambient heat exchanger tperature is 293 K, giving a temperature difference of 134The fundamental frequency of oscillation is 288 Hz in airatmospheric pressure. The sound pressure level~SPL! at themicrophone is 153 dB. The SPL at the center of the resonis estimated to be 161 dB. The first radial wave harmooccurs at 526 Hz, but there is no noticeable signal atfrequency. The first nonlinear harmonic occurs at twicefundamental, or 584 Hz. There is a significant harmonic geration at this frequency, with an SPL of 115 dB. Howevthis is 38 dB below the fundamental. These results are vsimilar to those of Ref. 14 for a detuned plane wave resotor driven by a piston. In both cases, the first harmonicnearly 40 dB below that of the fundamental, and there isnoticeable sound production at the other normal modes oftubes. In the radial prime mover, the second and third nlinear harmonics occur at three and four times the fundam
2660Lightfoot et al.: Thermoacoustic prime mover
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tal, 876 Hz and 1.168 kHz, and are 63 dB and 75 dB bethe fundamental, respectively.
For comparison, a constant cross-section plane wprime mover was constructed and the spectrum is showFig. 14. A schematic of the prime mover, which is similardesign to that described by Belcher,9 is shown in Fig. 15.The hot heat exchanger temperature was 410 K and thebient heat exchanger temperature was 300 K for a tempture difference of 110 K. The frequency of operation wassame as in the radial case, 288 Hz, and the thermoacoelements were located at the position predicted to givelowest temperature difference necessary to produce sounceramic square pore stack and parallel-plate copper heachangers were used. The microphone was located at the
FIG. 13. Acoustic spectrum from the radial wave prime mover in airatmospheric pressure.
FIG. 14. Acoustic spectrum from the plane wave prime mover in airatmospheric pressure.
2661 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
w
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. Ax-m-
bient end of the tube. The sound pressure level~SPL! at themicrophone was 156 dB. The first harmonic occurs at twthe fundamental frequency, or 576 Hz. There is a large ctribution at this frequency, with an SPL of 137 dB, only 1dB below the fundamental~compared to a first harmoniwhich was 38 dB below the fundamental in the radial primmover!. The first through sixth harmonics give noticeabcontributions in the plane engine compared to only the fithrough third harmonics in the radial engine. The resultsthe plane wave prime mover are similar to those of Ref.for a prime mover just beyond onset, and Ref. 14 for a hmonic resonator driven by a piston. In all three cases,first harmonic has an amplitude about 20 dB below the fdamental with more harmonics having significant amplitudthan in the radial and detuned plane resonators.
The reduction in higher harmonic generation for thedial engine is understood qualitatively by considering tproximity of the nonlinear higher harmonics to the natumodes or overtones of the resonator. In general, the nonear harmonics have a very narrow bandwidth12 while theovertones have a wider bandwidth. In the case of the plresonator the nonlinear harmonics are not exactly the sas the overtones, since a very slight detuning of the ovtones occurs due to dissipation in the resonator and the tmoacoustic elements; however, the two are in the samecinity so that the nonlinear harmonics certainly fall withthe bandwidth of the overtones, thus enhancing the higharmonic generation. The radial engine and detuned presonators, by contrast, have no overtones in the vicinitythe nonlinear harmonics so that these systems do not enhthe generation of higher harmonics. A convenience of adial system is that no variation in tube shape is necessarmake the resonator anharmonic.
IV. CONCLUSIONS
The primary goal of this research was to test the radwave thermoacoustic theory. Although some satisfaction
t
t
FIG. 15. Plane wave model with dimensions similar to the proposed laradial wave thermoacoustic prime mover. All lengths shown are in thetical direction. The tube diameter is 8.5 cm.
2661Lightfoot et al.: Thermoacoustic prime mover
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lost in not being able to directly compare measured ontemperatures with predicted values due to unforeseen phcal constraints on the system~i.e., significant gaps betweestack and heat exchanger!, with these constraints included,has been shown that the existing radial wave theory is arate and useful for prediction of engine performance.
A secondary goal was to test the generation of higharmonics in a radial mode prime mover. The expectedsult was attained. Comparison of harmonic generation inradial prime mover and a similar plane prime mover showthat the radial prime mover does not enhance the generaof higher harmonics, since the resonator harmonics are nthe vicinity of multiples of the fundamental. In additionsound pressure levels of the higher harmonics in the raprime mover were found to be similar to those produceda detuned plane prime mover.
We identified several areas in which care shouldtaken in future research, and developed some useful toolunderstanding these problem areas. In radial engines, splates which are rigid enough to provide a constant pspacing should be used. Since the radial stack was charaized by a distribution of pore sizes rather than a conspore size, a method was developed for analyzing pore dibutions in the stack. This should be a useful first step inanalysis of inhomogeneous stacks~i.e., steel wool, fiberglassetc.!. In addition, some difficulties with short stacks habeen discovered and understood. It was shown that the fithe heat exchanger to the stack is much more important wshort stacks are used.
ACKNOWLEDGMENT
The authors wish to gratefully acknowledge the suppof the Office of Naval Research.
2662 J. Acoust. Soc. Am., Vol. 105, No. 5, May 1999
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1G. W. Swift, ‘‘Thermoacoustic engines,’’ J. Acoust. Soc. Am.84, 1145–1180 ~1988!.
2W. P. Arnott, J. A. Lightfoot, R. Raspet, and Hans Moosmu¨ller, ‘‘Radialwave thermoacoustic engines: Theory and examples for refrigeratorshigh-gain narrow-bandwidth photoacoustic spectrometers,’’ J. AcoSoc. Am.99, 734–745~1996!.
3J. Wheatley, ‘‘Intrinsically irreversible or natural engines,’’ inFrontiers inPhysical Acoustics~Elsevier, New York, 1986!.
4W. P. Arnott, J. R. Belcher, R. Raspet, and H. E. Bass, ‘‘Stability analyof a helium-filled thermoacoustic engine,’’ J. Acoust. Soc. Am.96, 370–375 ~1994!.
5The heat exchangers were constructed by W. Patrick Arnott, Robertbott, and Michael Ossofsky at the Desert Research Institute, Reno, N
6J. A. Lightfoot, ‘‘Thermoacoustic engines in alternate geometry resotors,’’ Ph. D. Dissertation, The University of Mississippi~1997!.
7R. Raspet, J. Brewster, and H. E. Bass, ‘‘A new approximation methodthermoacoustic calculations,’’ J. Acoust. Soc. Am.103, 2395–2402~1998!.
8F. Reif, Fundamentals of Statistical and Thermal Physics~McGraw-Hill,New York, 1965!, pp. 166–169.
9J. R. Belcher, ‘‘A study of element interactions in thermoacousticgines,’’ Ph.D. Dissertation, The University of Mississippi~1996!.
10A. B. Coppens and J. V. Sanders, ‘‘Finite-amplitude standing wavesrigid-walled tubes,’’ J. Acoust. Soc. Am.43, 516–529~1968!.
11R. Chen, ‘‘Time averaged pressure distributions for finite amplitude staing waves,’’ Master’s Thesis from Pennsylvania State University, Graate Program in Acoustics~December 1994!
12A. A. Atchley, H. E. Bass, and T. J. Hofler, ‘‘Development of nonlinewaves in a thermoacoustic prime mover,’’ inFrontiers of NonlinearAcoustics: Proceedings of 12th ISNA,edited M. F. Hamilton and D. T.Blackstock~Elsevier, New York, 1990!, pp. 603–608.
13G. W. Swift, ‘‘Analysis and performance of a large thermoacoustic egine,’’ J. Acoust. Soc. Am.92, 1551–1562~1992!.
14D. F. Gaitan and A. A. Atchley, ‘‘Finite amplitude standing wavesharmonic and anharmonic tubes,’’ J. Acoust. Soc. Am.93, 2489–2495~1993!.
15A. B. Coppens and J. V. Sanders, ‘‘Finite-amplitude standing wawithin real cavities,’’ J. Acoust. Soc. Am.58, 1133–1140~1975!.
16J. R. Olson and G. W. Swift, ‘‘Similitude in thermoacoustics,’’ J. AcousSoc. Am.95, 1405–1412~1994!.
