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A pistonless Stirling enginc The traveling wave heatenginePeter H. CeperleyDepartment f Physics, eorgeMasonUniversity, airfax, Virginia22030(Received11 April 1979;acceptedor publication May 1979)The propagation f acoustical aves hrougha differentially eated egeneratoresults n gas n theregenerator ndergoing Stirling thermodynamic ycle. One directionof wave propagation esults namplification f the wavesand conversion f thermalenergy nto acoustical nergy.The oppositedirection esults n acoustical nergybeingused o pumpheat. The ideal gain and maximum nergyconversionatesare derived n this paper.Low powergain measurementseremadewhichverify hederived ainequation. ractical ngines nd heatpumps sing hisprinciple re discussed.PACS numbers: 43.28.Kt, 43.88.Ar

INTRODUCTIONUnfortunately, most machines capable of convertingthermal energy into mechanical energy are quite com-plex. However, if one accepts acoustical energy as

being a fluctuating form of mechanical energy, thereis a class of very simple devices, called singing pipes(shown n Fig. 1), capable ofsuch conversion. Theseuse standing waves in pipes to force gas to undergo acycle of compression, heating, expansion, and coolingsimilar to that of normal heat engines, and similarlyconvert heat into mechanical energy. In a singing pipe,the heat is normally supplied by a flame heating an ob-ject in the resonant tube and the mechanical energyproducedgoes into maintaining the standingwave. Sing-ing pipes are usually thought of as demonstration de-vices, serving no useful purpose other than to make aloud noise. However in 1948 and 1952 Bell TelephoneLaboratories received patents'2 on singingpipescoupled to acoustical-to-electrical transducers whichcould produce useful electrical power. While these de-vices were attractive because of their simplicity, theywere inefficient because of their use of standing waves.This inefficiency is explained later.

In this paper, we shall explore a different, but simi-lar class of acoustical heat engines, based on travelingwaves, which promise higher efficiencies. Thesetraveling wave heat engines use a Stirling thermodyn-amic cycle, which is reversible, allowing the enginesto also serve as heat pumps.

HOT COLD

FIG. 1. A singing pipe. Sound is produced when the closedend is placed in a flame. This device uses the acousticalstanding waves set up in the pipe to force the gas there toundergo a cycle of compression, heating, expansion, andcooling, similar to that in a normal heat engine. In thiscase the thermal energy is converted into acoustical energywhich maintains the standing waves.

I. THE ENERGY CONVERSION PROCESSThe basic energy conversion process involves anacoustical traveling wave propagating through a dif-ferentially heated regencrater, as shown in Fig. 2;

The regencrater consists of a casing packed with metalor ceramic parts, small enough to insure that gas inany part of the regencrater is essentially at the tem-perature of the packing at that point, but not so fine asto cause excessive attenuation of the acoustical waves.This trade-off limits the power density as discussedbelow. A continuous emperature gradient is set upalong the length of the regencrater by external sourceswhich heat one end and cool the other. When a waveforces a volume of gas to move towards the hot end,it is heated by the hotter regencrater there, when it ismoved towards the colder end, it is likewise cooled.

The pressure and velocity that a traveling wave im-parts to the gas volume it is propagating through isshown in Fig. 3 as a function of time. For a wavetraveling from cold to hot througha differentially heatedregencrater, this would cause (1) a build-up of pres-sure (compression), (2) then a flow of gas towards thehot end (heating), (3) followed by a drop in pressure(expansion),and (4) finally a flow of gas towards hecool end (cooling). Since his is the same type of cyclea gas volume would undergo in a standard Stirling en-gine,4's one would expect a similar conversion of therm-al energy into mechanical energy. However, since theacoustical wave is responsible for the compression, ex-pansion, and gas movement, the mechanical energyproduced by the cycle in the traveling wave heat enginewill amplify the wave.

AC OU S TICAL AMPLIFIEDTRAVELING WAVE WAVE

COLD HOTFIG. 2. Traveling wave heat engine. This device is similarto a singing pipe except in its use of traveling waves instead ofstanding waves. Acoustical raveling waves propagatingthrough he differentially heated regencrater from cold to hotare amplified. This device is reversible and promises higherefficiency than the singing pipe.

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GASELOCITYPRESSURECOMPRESSION EXPANSION

HEAT NG COOL NGFIG. 3. Pressure and velocity as a function of time for gasthrough which a traveling wave is propagating. For gas in aproperly orientated, differentially heated regenerator, positivevelocity results in gas moving to the hot end and being heated,while negative velocity results in gas being cooled. As shown,the gas undergoes a cycle of compression, heating, expansion,and cooling, similar to that occurring in a regular Stirling en-gine.

In the case of acoustical waves traveling in the re-verse direction, from hot to cold through the regenera-tot, the flow of gas relative to the pressure wave isreversed. In this case, acoustical energy is convertedinto thermal energy, increasing the thermal gradient.This action allows this device to pump heat.

In a traveling wave heat pump, the waves pump heatin a direction opposite to their propagation direction:waves go from hot to cold and heat is pumped from coldto hot. In the traveling wave heat engine, the heat flowdirection is still opposite to the direction of wave prop-agation: waves go from cold to hot and heat goes fromhot to cold. This action is analogous to that occurringin a positive displacement water pump: the pump's ro-tational speed and direction determine the rate and di-rection of the water flow through it. In a traveling waveheat engine, the traveling waves determine the directionand rate of heat flow through the device. In the case ofthe pump, the pressure of the water on the input ascompared with that on the output determines whetherthe pump pumps the water or the water drives thepump. Similarly, in the case of the traveling wave heatengine, the temperature of the acoustical input end ascompared with the output end, determines the directionof energy transformation between acoustical and thermalforms. If the acoustical input is colder than the output(as in the case of the heat engine), the thermal po-tential drives or amplifies the waves. On the otherhand, if the input is hotter than the output (as in thecase of thd heat pump), the reverse thermal poten-tial bucks or attenuates the waves and draws energyfrom them, as is necessary for pumping heat.

The tr:veling w:ve he:t engine might be consideredsimilar to some Stirling engines which use a column ofliquid as the piston? The traveling wave heat engine'suse of an air column as a piston should be an improve-ment since it lacks the problems typically associated

GASELOCITY/ . PRESSURE

COMPRESSION EXPANSIONAND ANDHEATING COOLINGFIG 4. Pressure and velocity as a function of time in a gasin which a standing wave exists. For gas in a differentiallyheated regenerator, positive velocity results in gas moving tothe hot end and being heated, while negative velocity resultsin gas being cooled. As shown, in this cycle compression andheating occur sixnultaneously and expansion and cooling arealso simultaneous. In order for this system to convert thermalenergy into acoustical energy, a regenerator with poor heattransfer must be used to delay the heating and cooling pro-cesses so that heating follows compression and cooling followsexpansion. This renders the process irreversible and fairlyinefficient.

with liquids such as confinement, maintenance of theliquid, and corrosion, and should have less viscouslosses.

II. THERMODYNAMIC CYCLE OF SINGING PIPESThe pressure and velocity peaks of standing waves

such as those in a singing pipe are phased to occur oneafter the other such that the waves force the gas in thetemperature gradient regions of the pipes to undergoheating simultaneously with compression and coolingsimultaneouslywith expansion shown n Fig. 4). Sucha cycle would not convert heat into acoustical powerwere it not for thermal delay in the heating and coolingprocesses which allow some of the heating to occurafter the compression and some of the cooling to occurafter the expansion. In fact, in a standing wave heatengine, best operation occurs when the thermal delaycauses 90 phase lags2 in the heating and cooling pro-cesses. These delays are paramount to regeneratorineffectiveness and decrease the efficiency of the de-vice, as well as rendering it irreversible. Since thetraveling wave heat engine does not rely on such therm-al delay for proper phasing, it can capitalize on highregenerator effectiveness, is capable of higher ef-ficiences, and is also reversible. On the other hand,the addition of a small component of properly phasedstanding waves to a traveling wave heat engine mightimprove that engine's efficiency by correcting for smallinevitable thermal delays in its heat exchange pro-cesses.

III. THEORETICAL GAIN OF A TRAVELING WAVEHEAT ENGINE

In the limit of a very short regenerator (l

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F- 1 FI I . iI I I I II I II I I I IJ SOURCEI I REGENERATORI I IL_ / L

I

II RtmIII

i

IERMINATINI IL j

FIG. 5. A lumped parameter model of astanding wave heat engine including a sourceand termination of the waves.

pressures. Using the ideal gas law (so volumetric flowvaries as T/P), we get the current gain of the regen-erator:

g-= o/I, = ToP,/T,P, = (To/T,)[1 - (R/R,)]- ,where Tt and To are the input and outlet (absolute)temperatures, the regenerator packing resistance R= (Pt -Po)/li, and the impedance f the tubingR=Pt/I .If the packing is chosen such that R

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FROMMICROPHONES TUNED

POWERTOPEAKERFIG. 8. Electronics used in gain measurements.

flow with a well developed emperature profile. Thefollowing calculation will consider the flow in one chan-nel of the regenerator of Fig. 6. The velocity is givenbyv -(2x/o)2], (1)

where is the average velocity, x is the distance fromthe channel center, and D is the channel width. Ne-glecting longitudinal heat conduction, the temperaturedistribution must satisfy

kd2TpCv =2 ,where k is the thermal conductivity, p is the gas densi-ty, and C is the heat capacity at constant pressure. As-suming a solution of the form

T xz=X x+ dd---zT),fwhere X is the deviation of the gas temperature fromthe wall temperature at a certain position z along thechapel and (dT/dz) is the temperature gradient alongthe regenerator wall, we get

8k 'The average value of X is

- (2)The resistance to flow is

&P l dP _ l dZvRp A A dz A 'where A is the regenera[or cross sectional area and lis its length. Sincedv/ z: 12/D from Eq. (1), weget

REGENERATOR ELECTRICALHEATINGELEMENT

FIG. 9. Regenerator used in gain measurements. A porousretainer keeps the steel wool packing in place but is trans-parent to waves. The heating elements at both ends allowseither end to be heated while the other remains cool. Silicondiode temperature probes are attached to the regenerator endsto monitor the temperatures there.

R,= 12gl/D2A (3)Finally, using Eqs. (2) and (3) we get that the number ofheat transfer units6 NTU is given by

NTU=A_T: R c__X 6P Rt 'whereP, = pC/K 0.7 is the Prandaltnumber,R= Pc/A is the characteristic impelnee of the regenera-tor, c is the velocityof sound,andAT = (dT/)l isthe total temperature drop from one end of the regen-erator to the other. Solving for the sound intensity andassuming minimumNTU andR/R bothequal o 5,we get (pc) a,

z z (330)= 100 kW/m 2

ile this is less than the limitation due to turbulence,it still makes for an attractive engine. It would meanthat an engine with a 10-in. i.d. regenerator (cross-sectional area of 0.05 m ) would be able to handle asound power of 5 kW. With a temperature drop of200 and a gain of 1.2, the enginecouldgenerate 1kW which would be adequate for a stationary engine.R desired, higher power might be achieved with highertemperature drops, multiple regenerators, specialpiping and resonant transformers to increase the ef-fective impedance of the engine, higher pressure gas,or use of helium or hydrogen as working fluids.V. GAIN MEASUREMENTS

The acoustical gain in a differentially heated regen-erator has been measured at relatively low acousticalpower levels. The experimental apparatus is shown in

TABLE I. Measured ow power gains for acousticalwaves traveling through he regenerator shownin Fig. 9 with various temperature gradients.Temperatures

Input Output Difference(C) (C) (C)

Gain=powerout/(powern-power eflected)Measured gainTheoretically Measured normalized byexpected gain gain first entry90 90 0 1.00 0.81 1.00

150 90 -60 0.86 0.70 0.8690 150 60 1.16 0.90 1.11

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Fig. 7. It consists of a loud speaker with damping, aflexible tube (to carry the sound, but not wall vibra-tions), anda series of rigid tubes. The first tube thereflectometer) has two identical microphones mountedon it spaceda quarter wavelengtl part (for the 190-Hzsound used). Each microphone is coupled to the re-flectometer tube via a 2 cm length of l-ram i.d. tubing.The sound from the speaker passes through the flexibletube, on through he reflectometer, on'throughhe re-generator column, and finally to a termination tubewhich has a third microphone and an acoustical termi-nating resistor. The outputsof the three microphonesare connected o a switch box (see Fig. 8) which allowseach in turn to be connected up to electronics whichmeasure the amplitude and phase of each microphonesignal relative to that of the signal driving the speaker.Use of the amplitude and phase of the first two signalsallows for the calculation of the amplitudes and phasesof the wave traveling from speaker to regenerator (theincident wave) and the wave reflected off the regenera-tor (the reflected wave). The third microphone allowsthe calculation of the wave transmitted through theregenerator (the transmittedwave). The microphonesare calibrated by removing the regenerator and firstputting a plug in its place so that the incident and re-flected waves are equal with a known phase relation-ship, and secondly by connecting he termination tubedirectly onto the reflectometer tube so that the incidentand transmitted wave are equal and the reflected wavezero. After calibration, the regenerator is insertedand its effects on the waves are measured. It is im-portant to measure the reflected wave because the re-generator and temperature gradient in it can cause re-flections which can affect the gain of the regenerator.

The regenerator used is shown n Fig. 9. It waspacked with 00-grade steel wool. Heating coils wereused to apply heat to either end of the regenerator.Temperatures of the regenerator ends were measuredby attaching sinall calibrated silicon diodes to the endsand measuring he diodes' forward voltage drop.9Table I shows the results. Listed are the theoretical-ly expected ain, the measuredgain [powerout/(powerincident-power reflected)], and the relative measuredgain, all three with no differential heating, with heating

of the acoustical input end, and finally, with heating ofthe acoustical output.

COOL HOTREGENERATOR

FIG. 10. An acoustical oscillator consisting of a travelingwave heat engine and a positive feedback pipe. Some soundcan be coupled out to power an electrical transducer and makeelectricity, for example.

With no temperature gradient, the gain was 0.81, in-dicating a 19%attenuationof insertion loss due to flowresistance. A -60 C emperature gradient (soundgo-ing from hot to cold), and a + 60 Cgradient producedgains of 0.70 and 0.90, respectively. When comparedwith the 0.81 no gradient transmission, they had rela-tive gains of 0.81 and 1.11, which are very close to thetheoretically expectedgains of 0.86 and 1.16. Theseresults indicate that the heat transfer in the regeneratorwas more than adequate for the power levels used(probablyabout 80 dB), and that less densepackingshould be used which will have smaller insertion loss.

With a better choice of packing, and perhaps agreater temperature gradient, we expect to be able toachieve absolute gains of greater than 1.0. With suchgains, we could make an acoustical oscillator and ex-periment at very high acoustical power levels, such asthose used in practical engines. Achieving this, wehope to measure the efficiency and power density actuallyachievable in these traveling wave heat engines.

Vl. APPLICATIONS OF TRAVELING WAVE HEATENGINES

A traveling wave heat engine is an acoustical ampli-fier, which can be used in various acoustical circuits,for a range of energy conversion applications. For ex-ample, Fig. 10 shows an acoustical oscillator consist-ing of such an engine and a pipe to provide positiveacoustical feedback. Such a device would convertthermal energy into acoustical power which could befurther converted into electrical power via an acousti-cal-to-electrical transducer, similar to the earlierBell Labs singing pipe device, but with higher efficiencybecause of its use of traveling waves. The lack of mov-ing parts, simplicity, and reliability of this device maymake it attractive as a power supply for isolated equip-ment. Figure 11 shows another circuit, a thermallydriven heat pump, which consists of a pair of travelingwave heat engines, one converting heat into acousticalpower which is fed through the second engine acting asa heat pump. This device might find use as a solar-

COOL HOTI REGENERATOR

COLD CIOLREGENERATORFIG. 11. A thermally driven heat pump consisting of tostanding wave heat engines. An external heat source providesthermal energy to drive the top engine, which produces theacoustical waves to power the bottom engine acting as a heatpump. The combination might be used as a refrigerator, airconditioner, or heat pump. The lines represent pipes with thearrows showing the direction of wave propagation.

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driven air conditionerwhere its simplicity, low cost,and long lifetime may be more important than its lowspecific power. Obviously, many other circuits andapplications involving conversion of thermal and me-chanical power are possible.

VII. THE TRAVELING WAVE HEAT ENGINE-AGENERALIZED STIRLING ENGINE

Since the motion and pressure cycles of the gas in theregenerator of a normal Stirling engine are identicalwith those occurring in the regenerator of a travelingwave heat engine,s one might consider a normal Stirlingengine to be a special case of a traveling wave heatengine with its waves being created and absorbed bymoving pistons. Thus, one could consider the travelingwave heat engine, consisting of a differentially heatedregenerator which has the appropriate pressure andmotion cycles characterized by a traveling wave, to bethe essence of a Stirling engine or to be a generalizedStirling engine, with the method for creating and usingthese waves to be specified for the particular type ofStirling engine. The use of waves to characterize thepressure and motion cycles might be particularly il-

luminating in Stirling engines operating at high cyclicrates where the wavelengths become short and thetransit responses and delays important. The transitresponses and delays can be treated as part of the wavepropagation process and can be dealt with using themathematical methods and experimental techniqnesdeveloped for acoustics and microwaves.

R. V. L. Hartlay, U.S. Patent No. 2,549,464 (1951).2W. A. Marrison, U.S. Patent No. 2, 836,033 (1958).3p. H. Ceper[ey, U.S. Patent No. 4,114,380 (1978).4ShouldWe Have a New Engine?, SPL SP 43-17, JPL(California Institute of Technology, Pasadena, CA, 1975).G. Walker, Sci. Am. 234(8), 80 (1973).W. R. Martini, -Developments in Stirling Engines,,, pre-sented at the Annual Winter Meeting of ASME, MDAC PaperWD 1833, Nov. 1972 (unpublished).?W. R. Martini, The Free-Displacer, Free-Piston StirlingEngine... Potential Energy Conse ye r, I E C E C 1975 Record,995 (1975).8Noiseand Vibration Control, editedby L. L. Beranek(McGraw-Hill, New York, 1971), p. 397.9C. J. Koch, Electronics 51, 110 (1976).

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