THE NATURAL HEAT ENGINE

THE NATURALHEAT ENGINE

by John C. Wheatley, Gregory W. Swift, and Albert Migliori

Heat engines are a compromisebetween the crisp ideals dis-cussed in thermodynamictextbooks and the clanking,

hissing realities of irreversible processes,This compromise produces wonderful ma-chines, such as the automobile engine andthe household refrigerator. In designingreal devices, the goal is not to approachthermodynamic ideals by reducing ir-reversibilities but to balance cost, effi-ciency, size, power, reliability, simplicity,and other factors important to the needs ofparticular applications.

Simplicity is the most striking feature ofa natural engine, a reciprocating heat en-gine with no moving parts. As we will see,the basic operating cycle of the naturalengine is so straightforward it can be ap-plied to a wide variety of systems withworking media that range from air toparamagnetic disks.

Although the natural engine is new inconcept, the underlying thermodynamicprinciples and processes are shared withconventional engines, such as the Stirlingand Rankine engines. To set the stage fornatural engines, we will first discuss a fewconventional idealized thermodynamiccycles and the practical engines they sug-gest.

Conventional Heat Enginesand Cycles

In principle, any idealized thermody-namic heat engine cycle is functionally

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reversible in the sense that it can be madeto operate in either of two modes: primemover or heat pump* (Fig. 1). In a primemover, heat flows from high to lowtemperatures, and the engine converts aportion of that heat to work. In a heatpump, the flows of heat and work arereversed; that is, work done on the enginecauses it to pump heat from low to hightemperatures. Few practical engines arefunctionally reversible. The internal com-bustion engine is a prime mover only; thehousehold refrigerator is a heat pumponly: neither engine is ever operated inboth modes,

Figure 1 shows how the first and secondlaws of thermodynamics place an upperlimit on the efficiency of a prime mover-(the fraction of the heat input converted towork), The efficiency of a thermodynam-ically reversible cycle-that is, one inwhich all parts of the system are always inthermodynamic equilibrium—is equal tothat upper limit. (One statement of thesecond law of thermodynamics is that allreversible engines operating between thesame two temperatures have the same effi-ciency.) Figure 1 also shows the upper

*A prime mover is often called an engine and aheat pump a refrigerator. Here we use the termengine to denote both thermodynamic func-tions, and our use of the term heat pump in-cludes the refrigerator. Strictly speaking, however, the purpose of a heat pump is to reject heatat the higher temperature, whereas the purposeof a refrigerator is to extract heat at the lowertemperature.

limit for the coefficient of performance(C. O. P.) of a heat pump (the amount ofheat rejected at the higher temperature perunit of work). Both theoretical limits de-pend only on the temperatures involved.

Carnot. The most fundamental enginecycle operating between two temperaturesis the functionally and thermodynamicallyreversible cycle propounded by SadiCarnot in 1824. The cycle consists of alter-nating adiabatic and isothermal steps(Fig. 2). During an adiabatic step, no heat

remains constant. Thus any flow of workcauses a corresponding change in the tem-perature of the working medium. Duringan isothermal step, the temperature re-mains constant, and flows of entropy,work, and heat occur.

In the Carnot cycle, the entropy changeof one isothermal step exactly balances theentropy change of the other isothermalstep. Over a complete cycle, no entropy isgenerated. If- an engine could be made tofollow a Carnot cycle, its efficiency wouldequal the theoretical upper limit given inFig. 1. Although the upper limit applies toany reversible engine, this efficiency isusually called the Carnot efficiency.

Building an engine that approximates aCarnot cycle requires that all processes inits cycle are carried out very near equi-librium. If not , the result ing ir-reversibilities due to temperature andpressure gradients generate entropy andcause a loss of efficiency, For example, the

Fall 1986 LOS ALAMOS SCIENCE

The release of acoustic energysimple natural heat engine, thetube, made evident by the whiteat the upper end. The device consa two-piece copper tube, closedbottom, and a short set of fiberplates that run parallel to the tubein the region of the flanges. The atic energy results spontaneously wtemperature gradient is applied ithe plates. In this case, the gradieproduced by holding one end of tuwhile immersing the other end (frin liquid nitrogen

by aHoflerplumeists ofat theglass

‘s axisc o u s -hen aacrossnt wasbe tubeosted)

LOS ALAMOS SCIENCE Fall 1986 3

The Natural Heat Engine

temperature differences across the heat ex-changers that move heat in or out of theengine are frequently a source of ir-reversibility that greatly cuts efficiency.(See "The Fridge” for a quantitative ac-counting of this and other losses in a prac-tical heat pump. )

Although one may approach near-equi-librium conditions by designing the engineso as to reduce these gradients, the endresult is a very slow cycling of the engineand a very low power output. An impor-tant point (originally made by F. L.Curzon and B. Ahlborn and generalized byS. Berry, J. Ross, and their collaborators)is that Carnot-like cycles operating be-tween two temperatures with imperfectheat exchangers have quite different effi-ciencies depending on whether work percycle or power is being maximized, Realengines, especially high-speed reciprocat-ing engines, cannot approximate Carnot’scycle closely,

Stirling. The Stirling engine, invented in1816 by the Reverend Robert Stirlingsome eighteen years before Carnot’s ideaswere published and originally called thehot-air engine, is a reciprocating enginethat is functionally reversible and, in prin-ciple, thermodynamically reversible. Theideal Stirling cycle has the Carnot effi-ciency. From a practical standpoint, im-plementing the Stirling cycle suffers fromsome of the problems of implementing theCarnot cycle. However, the introductionof a second thermodynamic mediumprovided the means by which high-speedStirling engines of good efficiency could bebuilt.

The Stirling cycle (the solid black curvein Fig. 3) differs from the Carnot cycle inthat the adiabatic steps are replaced withsteps that are reversible by virtue of beinglocally isothermal, This type of cycle isachieved by using two thermodynamicmedia. The first is the working fluid,which typically can be either a gas or aliquid, (There are Stirling cycles that usesolids, but we do not discuss them here.)continued on page 6

4

Fig. 1. (a) A heat engine operating as limit for W/Qh, the efficiency of the en-prime mover converts some of the heatthat is flowing from a hot temperature Th

to a cold temperature TC into work. Thefirst law of thermodynamics tells us thatQ h, the heat that passes into the engineat the hot temperature, equals Q c, theheat put back into the environment atthe cold temperature, plus W, the workdone by the engine. The second lawtells us that the entropy per cycle gener-ated by the system must be positive or,at best, zero. Since the engine is as-sumed to be in a steady state, the en-tropy change in the environment due tothe heat flow out of the engine, Q c/Tc, isgreater than or equal to that due to theheat f low into the engine , Q h / Th .Together, these two laws give an upper

gine. Note that a prime mover can onlyapproach its highest efficiency of unitywhen T c << T h. (b) In a heat engineoperating as heat pump, all flows ofheat and work are reversed. Thus workdone on the engine causes it to drawheat out of the environment at the coldtemperature and place it into the en-vironment at the hot temperature. Con-sideration here of the first and secondlaws leads to an upper limit on the coef-ficient of performance (C.O.P.), Q h/W,which is the reciprocal of the efficiencyof a prime mover. (For a refrigerator, theC.O.P. is better defined as the ratio ofthe heat extracted at the lower tempera-ture to the work done on the machine,that is, Qc/W.)



The Fridge

The basis for the household refriger-ator is the Rankine cycle, which, asshown in the figure, duplicates a

portion of the Carnot cycle in that it hasone adiabatic step and two isothermalsteps. A key feature of this cycle is a phasechange in the working fluid. and the twoisothermal steps correspond to condensa-tion of the fluid at T h and evaporation atT c. Also, the engine operates with continu-ous flow rather than by reciprocating: theworking fluid cycles through its variousthermodynamic states by being forcedaround a closed loop.

This cycle has intrinsic irreversibilitiesassociated with the free expansion of theliquid and the cooling of the gas to thetemperature at which condensation oc-curs. Thus one expects the Rankine cycleto have less than ideal Carnot effi-ciency-even before accounting for suchlosses as those due to temperature dif-ferences at the heat exchangers. Neverthe-less. Rankine engines remain the design ofchoice in many applications because theyare simple and powerful. Many refriger-ators will run thirty years with little or nomaintenance, and overall cost is low.

The Rankine cycle can also be used inan air-to-air heat pump. Table 1 illustratesthe effects of various irreversibilities onthe coefficient of performance for such apump—one designed to keep a house at20°C when outside air is 5°C so that,ideally, T h – T c is 15°C and the Carnotcoefficient of performance is 19.5. Thelargest drop in the the estimated coeffi-cient of performance occurs when idealheat exchangers are replaced by practicalheat exchangers—ones both small enoughto get through the door of a house andcheap enough to cost less than the house.A small, cheap heat exchanger can onlytransfer large amounts of heat if a largetemperature difference occurs across it.The net effect in our example is that the

I CondenserI

0\I Evaporator I

The Rankine cycle, used in the house- steps and, on the compression side, anhold refrigerator, is based on a liquid- adiabatic step. The two parts of the cy-gas phase change. The cycle is shown cle (shown in red) that differ from thehere superimposed on the phase dia- Carnot cycle—the cooling of the gas atgram for the working fluid; a schematic constant pressure to the condensationof the heat pump is also shown. The temperature T h and the free expansionRankine cycle resembles the Carnot cy- of the liquid—are intrinsically ir-cle in that there are two isothermal reversible. A

Table 1

Losses in the coefficient of performance (C. O. P.) due to irreversibilities for an air-to-air heat pump (adapted from Hear Pumps by R. D. Heap, 1983).

Cycle Irreversibilities

Carnot none 5 20 19.5

Carnot real heat exchangers – 5 45 6.4

Rankine real heat exchangers. intrinsic – 5 45 5.1irreversibilities

Rankine real heat exchangers. intrinsic – 5 45 4.0irreversibilities, compressor losses

Rankine real heat exchangers. intrinsic –5 45 3.0irreversibilities, compressorlosses, miscellaneous

temperature difference. T h – T c of theworking fluid increases from 15°C to 50°C,causing the coefficient of performance forthe Carnot cycle to drop from 19.5 to 6.4.

The C.O.P. drops to 5.1 when one takesinto account the intrinsic irreversibilitiesof the Rankine cycle. Further decreasesoccur because of losses in the compressor

(due to friction and the imperfect con-version of electrical power to shaft power)and miscellaneous losses (such as power torun the fans, the thermostat, and the con-trols). The final C. OP. for a practical,operating Rankine heat pump is 3.0, morethan a factor of 6 lower than the C.O.P. foran ideal engine. ■



continued from page 4The working fluid is displaced at constantvolume through a regenerator containingthe second medium, which is typically asolid. The second medium can be metalplates or just the walls of the vessel, but itsheat capacity should be large compared tothat of the working fluid. A small tempera-ture gradient exists along the length of theregenerator, the total temperature changebeing the temperature difference betweenthe hot and cold heat exchangers at theends of the regenerator. If wc ensure goodthermal contact between the two thermo-dynamic media (say by making the dis-tance between any fluid element and itsadjacent regenerator plate small enough).the fluid can temporarily store heat in theregenerator and recover it later undernearly reversible isothermal conditions.Of course, the steeper the gradient alongthe regenerator or the faster the displace-ment of the working fluid through the heatexchangers and the regenerator, the greaterthe irreversible losses.

During one part of the cycle, fluid entersthe cold end of the regenerator, picks upheat from the second medium, and exitshot. During another part of the cycle, fluidenters the hot end of the regenerator, de-posits heat in the second medium, andexits cold. The net heat stored in the sec-ond medium over a complete cycle is zero(provided, as is the case for an ideal gas,the specific heat of the fluid does notdepend on pressure). The regenerator,therefore, enables us to change the temper-ature of the working fluid from the tem-perature of the hot reservoir T h to thetemperature of the cold reservoir T c a n dback again without the adiabatic ex-pansions and compressions of the Carnotcycle. In other words, locally isothermalreversible steps have replaced theadiabatic reversible steps for changing thetemperature of the working fluid. As aresult, the efficiency of the Stirling cycle isthe same as that of the Carnot cycle.

But what about the Stirling engine?Typically, Stirling engines do not follow aSt i r l ing cycle but ra ther fol low an

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Fig. 2. Temperature-entropy and pres- work continues to be done by the fluid,sure-volume diagrams for the prime-mover and heat-pump modes of aCarnot cycle. When the engine is operat-ing as a prime mover, the first part of theexpansion stroke is the addition of heatto the engine at T h. Because this pro-cess is isothermal, the heat energy isused to expand the working mediumand do work on the surroundings. In thesecond step, further expansion occursadiabatically, that is, with no addition ofheat or change in entropy. Because

the temperature of the medium mustdrop. The third step is isothermal com-pression in which heat is rejected fromthe engine to the lower temperature T c

and the entropy drops. Finally, anadiabatic compression raises the tem-perature of the medium. The Carnot cy-cle for a heat pump is just the reverse ofthat for a prime mover. The areaenclosed by the pressure-volume dia-grams equals the net work done by or onthe engine in a full cycle.

adiabatic pseudo-Stirling cycle (the dashedcurves in Fig, 3), This confusing nomen-clature is illustrative of the compromisesmade between the concept of’ a thermody-namic cycle and the construction of anoperating engine. Unfortunately, becausethe same person’s name can become at-tached to both the cycle and the engine,confusion abounds,

What changes the Stirling cycle to apseudo-Stirling cycle is related to the tem-

perature of the working fluid at the heatexchangers. An adiabatic compressionwarms the fluid prior to its displacementthrough the hot heat exchanger and intothe regenerator, and. at the other end ofthe cycle, an adiabatic expansion cools theliquid prior to its displacement in the op-posite direction. These adiabats partiallyreplace the isotherms of the original cycle.necessitateing extension of the constant-volume displacement steps.



Fig. 3. (a) The ideal Stirling heat-pump the use of a second thermodynamic me- isothermal conditions, cooling the fluid

cycle (black) consists of isotherms and dium in the regenerator. The first step of from Th to T c. In the third step, adiabatic. .constant-volume steps. The adiabaticpseudo-St ir l ing cycle replaces theisotherms with adiabats and extensionsof the constant-volume displacementsteps (dashed curves). It is the pseudo-Stirling cycle that frequently serves asthe basis for practical Stirling engines.(b) One variation (the Rider form) of aStirling engine following the adiabaticpseudo-Stirling cycle. All such enginesare based on the ideal of localisothermal steps made possible through

the cycle depicted here is adiabaticcompression in the cylinder on the rightthat raises the temperature of the fluidabove T h. In the second step, bothpistons move, displacing the fluid to theleft. The heat Q h generated by the com-pression is rejected in the heat ex-changer on the right. Because of thesmall longitudinal temperature gradientand good lateral thermal contact alongthe regenerator, heat is transferred be-tween the two media under essentially

expansion cools the fluid in the leftcylinder. Constant-volume displace-ment of the fluid to the right then causesheat Q c to be drawn in at the left heatexchanger and the heat stored in thesecond medium during step 2 to be re-turned to the fluid. Irreversibility occursat the beginning of both constant-vol-ume displacements (dashed red in part(a)) when the fluid at one temperaturecontacts the heat exchanger at a dif-ferent temperature.

L0S ALAMOS SCIENCE Fall 1986 7


Since the above alterations introduceintrinsic irreversibilities, the maximum ef-ficiency possible for the pseudo-Stirlingcycle is lower than that for the true Stirlingcycle. In particular, fluid that has beenwarmed by adiabatic compression (and

pushed into the hot heat exchanger duringthe displacement step, where it makesthermal contact irreversibly with the ex-changer at temperature Th. The same typeof irreversibility occurs in the other heatexchanger after the adiabatic expansionstep. Such effects are departures from theideal of locally isothermal conditions.

Although a Stirling engine is not as sim-ple conceptually as a Carnot engine. prac-tical Stirling engines that operate at mod-erately high frequencies can indeed bebuilt. As before, other irreversible lossesoccur because there must be significanttemperature differences to drive heatthrough the heat exchangers. Also, if theworking fluid is a liquid (see “The LiquidPropylene Engine”), an additional type ofirreversibility arises: the specific heat of aliquid is pressure-dependent, making therecovery of heat in the regenerator im-perfect. This irreversibility is not an in-trinsic feature of the cycle but is a materialproperty that cannot be avoided. As such,it is of a more fundamental nature than thelimitation, say, of the heat exchangers.

Phasing of the various moving parts in aheat engine is another factor necessary toits operation. Although the engine de-picted in Fig. 3 is a heat pump, if thephasing of the two pistons is altered so thatexpansion occurs on the hot-temperatureside when most of the fluid is hot andcompression occurs on the low-tempera-ture side when most of the fluid is cold,heat flow will be reversed and the enginewill become a prime mover. As we shallsee, both phasing and the second thermo-dynamic medium are of key importance innatural heat engines also, although thereare significant differences in the way inwhich the second medium is used.

Internal Combustion. One way to cir-

cumvent the loss of efficiency from ir-reversibilities at the heat exchangers is togenerate the heating or cooling effects in-side the engine rather than outside. In1893 Rudolf Diesel envisioned such anengine and, in fact, intended it to follow aCarnot cycle of adiabats and isotherms.His idea was to provide the heat for theisothermal expansion by burning coal dustthat was injected into the engine at just theproper rate to maintain isothermal condi-tions. Cooling for the isothermal com-pression was to be provided by sprayingwater into the chamber. So far, no one,including Diesel, has been able to imple-ment this cycle, and we are once againconfronted with confusing nomenclature:the modern Diesel engine does not followthe Diesel cycle,

The idea of internal combustion, ofcourse, survived, and modern Diesel en-gines work very well indeed. But internalcombustion introduces new practical ir-reversiblities, For example, the additionand burning of the fuel in a typical pistonengine causes differences between thepressure and temperature in the cylinderat the end of the cycle and at the begin-ning. A considerable irreversible loss oc-curs as heat and pressure are vented in theexhaust. Thus, internal combustion en-gine cycles differ from the Carnot andStirling heat engine cycles describedearlier in that the working medium is notreturned to its original state.

Nevertheless, the use of phased andcontrolled internal combustion eliminatesthe problem of bringing heat in through afirewall. The diesel and gasoline internalcombustion engines are used today be-cause they are simple, both in principleand in practice, their power density is veryhigh. and their efficiency is relativelygood, sometimes very good. Practicaldiesel engines approach a level of effi-ciency in which the useful work is nearlyhalf the heating value of the fuel.

Otto and Brayton. Two common heatengine cycles that will help illuminate thecharacteristics of a natural engine are the

Fig. 4. The Otto (black) and Brayton (red)heat engine cycles, which consist of twoadiabatic steps that alternate with twononadiabtic steps—the latter steps be-ing the addition or removal of heat atconstant volume in the Otto cycle and atconstant pressure in the Brayton cycle.Only the prime mover mode is shown.Note that for both cycles the highesttemperature T h equals the temperatureT1 at the upper extreme of the adiabatic

expansion step but that the coldest tem-perature T C is lower than the tempera-ture T 2 at the lower extreme of theadiabatic expansion step.

Otto cycle (black curve in Fig. 4) and theBrayton cycle (red curve). Each of thesecycles, both of which are typically im-plemented irreversibly, has two adiabaticsteps and two nonadiabatic steps. In theOtto cycle, the nonadiabatic steps are theaddition and removal of heat at constantvolume; in the Brayton cycle, these stepsare carried out at constant pressure.

If the working fluid is an ideal gas, both

where y is the ratio of the specific heat atconstant pressure to that at constant vol-

Fall 1986 LOS ALAMOS SCIENCE8


ume. What is interesting about this for-mula is that efficiency for these cycles isdetermined by geometry (the ratio V1/V2

of the volumes at the extremes of theadiabatic expansion step) and by a fluid

temperatures of the hot and cold re-servoirs.

Since for an ideal gas the quantity

efficiency can also be expressed in terms ofthe temperatures, T 1 and T 2, at the ex-tremes of the adiabatic expansion step:

(2)

The diagrams for the Otto and Braytoncycles show that in both cycles T1 equalsTh but Tc is lower than T2. This differenceis due to further cooling, after theadiabatic expansion step, along anonadiabatic step (removal of heat at con-stant volume in the Otto cycle and atconstant pressure in the Brayton cycle). Ifwe now examine the limiting case of zeroheat transferred during the nonadiabaticsteps, we see that T2 approaches Tc and theefficiency approaches the Carnot effi-ciency. Of course, at the same time, thearea enclosed by either cycle, and thus thework output, shrinks to zero.

We will find that all of these features ofthe Otto and Brayton cycles have counter-parts in the natural engine.


One guiding principle in the develop-ment of most heat engine cycles has beento minimize irreversibilities because theygenerate entropy and decrease efficiency.In the development of practical engines,however, irreversibilities are often de-liberately introduced to increase power,decrease maintenance, or simplify designand manufacture, enabling one, for exam-ple, to build small engines, or high-speedreciprocating engines, or cheap engines.

On the other hand, irreversibilities play

a more fundamental role in the naturalheat engine. Rather than tolerating ir-reversibilities for the sake of expediency,the natural heat engine takes advantage ofthem. For example, heat conductionacross a temperature gradient is central tothe operation of a natural heat engineknown as the acoustic heat engine.Without this irreversibility, the enginewould not work. The result of such anapproach is a significant leap in simplicityand, for certain applications, a leap inpower and efficiency.

Thus, whereas engines that approx-imate, say, the Stirling cycle are in-trinsically reversible (though possibly ir-reversible in practice), natural heat en-gines are intrinsically irreversible—theycannot work if irreversibilites areeliminated. Nature abounds with usefulirreversible processes, so, for the sake of ashort, appropriate, and easily rememberedname, we call intrinsically irreversible en-gines natural engines.

Acoustic Engines. Work in Los Alamoson natural engines began with an acousticheat-pumping engine. Our work, however,was preceded by two conceptually relateddevices, which we will describe without,for the moment, explaining their some-what surprising behavior.

W. E. Gifford and R. C. Longsworthinvented what they called a pulse tube(Fig. 5a). Part of this closed tube was fittedwith a set of Stirling-type regeneratorplates intended to promote locallyisothermal processes along their length,and part of the tube was left empty. Pulseswere produced at the regenerator end ofthe tube by switching between high- andlow-pressure gas reservoirs at a rapid rate(1 hertz). The extreme inner end of theregenerator plates got very cold, whereas aheat exchanger withdrew heat at the emptyend of the tube. The pulse tube demon-strated the pumping of heat with acousticenergy in the presence of a second thermo-dynamic medium.

The other significant precursor to ourwork, and one of which we were initially

unaware, was the half-wave resonator of P.Merkli and H. Thomann (Fig. 5b). In thisapparatus, a piston drives pressure fluc-tuations in air at nearly half-wave reso-nance in a simple closed tube. Merkli andThomann observed that the center of thetube cooled, whereas the ends of the tubewarmed. At first, these results seemsurprising. Naively, one might expectheating everywhere rather than cooling inone region. Further, the cooling occurredin the center, which, at a quarter of anacoustic wavelength, is coincident with amaximum, or antinode, in acoustic veloc-ity and thus where one would surely ex-pect a warming due to viscous scrubbingof the air on the walls.

The first acoustic heat pump built atLos Alamos used a speaker at one end of aclosed tube to drive the acoustic resonanceand has a stack of fiber glass plates posi-tioned toward the opposite end (Fig. 5c).The plates constitute a second thermody-namic medium but not a Stirling-like re-generator because they are spaced so farapart that locally isothermal conditions donot prevail. With such an arrangement, itis easy to produce a 100-centigrade-degreetemperature difference across a 10-cen-timeter-long stack of plates in only a min-ute or so.

Subsequently, Tom Hofler built a de-vice (opening photograph and Fig. 6) toshow his Ph.D. candidacy committee atthe University of California, San Diego.The device, which we call the Hofler tube,consists of a quarter-wave acoustically res-onant metal tube closed at one end and astack of fiber glass plates that run parallelto the axis of the tube. Short copper stripsglued at each end of each fiber glass plateprovide heat exchange by making contactwith two flanges encircling the tube.

If the closed end of the tube is heated,say by holding it in a warm hand. and itsopen end is cooled by dipping it in liquidnitrogen, the resulting steep temperaturegradient causes the air in the tube tovibrate, and the person holding the tubewill feel his or her whole arm begin toshake. When the tube is removed from the



liquid nitrogen, the sound of the acousticoscillations is very intense. Peak-to-peakpressure oscillations at the closed endhave been found to be as high as 13 percent of the atmospheric pressure! Thus,the tube operates as a prime mover, andheat is converted to acoustic work.

How do this and other acoustic engineswork? The Hofler tube is the grandchild ofthe Sondhauss tube, famous inthermoacoustics and e x p l a i n e dqualitatively by Lord Rayleigh over a hun-dred years ago. Theoretical understandingof these and related devices has beenpromoted by Nikolaus Rott in a series ofpapers published over the last fifteenyears. The same conceptual foundationcan be used to understand quantitativelynot only the Hofler tube but the otheracoustic devices mentioned above as well.

As mentioned before, an important fac-tor in tbe operation of traditional enginesis phasing: pistons and valves have tomove with correct relative timing for theworking medium to be t ranspor tedthrough the desired thermodynamic cycle.The natural engine contains no obviousmoving parts to perform these functions,yet the acoustic stimulation of heat flowand the generation of acoustic work pointto some type of cycling, or timed phasingof thermodynamic processes.

The key to phasing in natural engines isthe presence of two thermodynamicmedia. In the Hofler tube, gas was the firstmedium, the fiber glass plates were thesecond. Consider a parcel of gas thatmoves back and forth along the plates atthe acoustic frequency. As it moves, theparcel of gas will experience changes intemperature. Part of the temperaturechanges come from adiabatic compressionand expansion of the gas by the soundpressure and part as a consequence of thelocal temperature of the plate itself. T h eheat flow from gas to plate that occurs as aconsequence of these temperature d i f -ferences does not produce instantaneouschanges. Rather a thermal lag in the heatflow between the two media creates thephasing between temperature and motion

Fig. 5. (a) At the left end of the Gifford driven on the left by a reciprocatingand Longsworth pulse tube, pressurepulses in a gas are generated at 1 hertz(Hz) by switching between high-pres-sure (5 bars) and low-pressure (1 bar)reservoirs. In conjunction with two sec-ond thermodynamic media (a Stirling-type regenerator and the walls of theopen section of the tube), the pulsescause heat to be pumped from themiddle of the tube to the far right. (b)The half-wave resonator heat pump ofMerkli and Thomann is a simple closedtube whose acoust ic resonance i s

piston. Contrary to one’s intuition, thecenter of the tube, where the acousticvelocity is greatest, cools rather thanwarms. (c) The first acoustic heat pumpbuilt at Los Alamos contains a stack offiber glass plates and helium gas as theworking fluid. The quarter-waveacoustic resonance is driven on the leftby a speaker. The stack acts as a sec-ond thermodynamic medium but is not aStirling-like regenerator because thewide spacing of the plates does notpromote locally isothermal conditions.



that is needed to drive the engine througha thermodynamic cycle. This is why anatural but irreversible process—heatflow across a temperature difference-isintrinsic to the operation of the engine.

An interesting contrast exists betweenthe Stirling engine and natural engines. Inthe Stirling engine, good thermal contactbetween the working fluid and the secondmedium helps ensure reversible operationand high efficiency, In the acoustic heatengine, poor thermal contact is necessaryto achieve the proper phasing betweentemperature and motion of the workingfluid.

One additional condition is importantto the operation of the acoustic heatengine: thermodynamic symmetry alongthe direction of relative motion must bebroken. The concept of thermodynamicsymmetry is fundamental. jet concep-tually simple. In the natural engine, thetwo thermodynamic media are undergo-ing reciprocating relative motion along

is thermodynamic symmetry. But if thelateral interaction changes with the long-itudinal coordinate, the symmetry is saidto be broken. Where the symmetry isbroken there is always some thermody-namic consequence, such as a change oftemperature or a beat flow to an externalreservoir.

Thermodynamic symmetry can bebroken in a variety of ways. For example,in the heat pump depicted in Fig. 5c, it isbroken geometrically at the longitudinalends of the fiber glass plates. In our de-scription of some variable stars as naturalengines, it is broken by changes in opacity ❑

that alter the effective thermal contact be-tween the stellar matter and the radiationfield. It can also be broken dynamically ❑

by, for example, nonlinear localization of-

the acoustic energy in the primary me-dium.

The dramatic effects of breaking ther- ❑

modynamic symmetry can be shown ex-perimentally by fixing several thermocou-

broken) changes rapidly and by largeamounts, whereas the temperature ofother thermocouples further in along theplates changes only by small amounts. Inan acoustic natural engine, the heat ex-changers are, of course, located at posi-tions where thermodynamic symmetry isbroken.

Before explaining in more detail theoperation of the acoustic heat engine, wesummarize by pointing out that all naturalheat engines possess the following ele-ments:

two or more thermodynamic media inreciprocating relative motion,

an irreversible process that causes phas-ing of a thermodynamic effect with re-spect to the motion, and

broken thermodynamic symmetryalong the direction of relative motion.

one direction and are interacting thermo- ples to the central plate of a simple The Cycle. Figure 8 displays the cycles ofdynamically in a direction transverse, or acoustic heat pump (Fig. 7). When the an acoustic engine serving as prime moverlaterally, to the motion. If the lateral inter- acoustic driver or speaker is turned on. the and as heat pump and also follows a typi-action does not change as we move in the temperature of thermocouples at the ends cal parcel of gas as it oscillates alongsidedirection of relative motion, we say there (where thermodynamic symmetry i s one of the fiber glass plates. In a real

Fig. 6. The Hofler tube, a simple ture gradient is applied across the tube. Thermal contact between theacoustic prime mover that consists of plates, the air in the tube vibrates plates and the tube at both ends of thetwo thermodynamic media—air and strongly. The plates are 1.65 cm long, stack is provided by thin copper stripsfiber glass plates—inside a quarter- 0.38 mm thick, and spaced 1 mm apart. that run along the longitudinal edges ofwavelength acoustically resonant tube The stack of plates, here seen from the each plate and into the thick encirclingclosed at one end. If a steep tempera- side, is placed about midway in the copper flanges.

LOS ALAMOS SCIENCE Fall 1986 I I


acoustic engine, the oscillations are sinus-oidal, producing elliptical cycles. For sim-plicity we consider square-wave. orarticulated, motion so that the basic ther-modynamic cycle can be pictured as con-sisting of two reversible adiabatic stepsand two irreversible constant-pressuresteps, as in the Brayton cycle.

Just as in the Stirling engine. relativephasing of motion (steps 1 and 3 in Fig. 8)and heat transfer (steps 2 and 4) de-termines whether the acoustic engine is aprime mover or a heat pump. In the Riderform of a Stirling engine. phasing is ef-fected externally by altering the order inwhich pistons are moved. In an acousticengine, however, phasing is a result of thenatural time delay in the diffusion of heatbetween the two thermodynamic media.The sign of the relative phasing. and thusthe mode of the natural heat engine. isdetermined by the magnitude of the tem-perature gradient along the fiber glasspla tes—a remarkable qual i ty and asubstantial gain in simplicity.

During the compressional part of theacoustic standing wave, the parcel of gas isboth warmed and displaced along theplates. As a result, two temperatures areimportant to that parcel: the temperatureof the gas after adiabatic compressionalwarming and the temperature of the partof the plate next to the gas parcel aftercompression (and displacement). If thetemperature of the gas is higher than thatof the plate, heat will flow from the gas tothe plate. If the temperature of the gas islower heat flows in the opposite directionfrom plate to gas. Both heat and workflows can thus be reversed and the engineswitched between functions by altering thesize of the temperature gradient. A zero orlow gradient is the condition for a heatpump: a high gradient is the condition fora prime mover. This engine is intrinsicallyirreversible but, functionally reversible.

The gradient that separates the twomodes is called the critical temperature

perature change along the plate justmatches the temperature change due to

12

4

3

2

Fig. 7. The temperature change T – T initial symmetry is broken geometrically.of thermocouples placed in a stack ofplates of a simple acoustic heat pumpshows the effect of symmetry breaking.Application of acoustic power to thetube at time zero immediately produceslarge changes at the two ends (posi-tions 1 and 4) where thermodynamic

Much smaller changes occur at themiddle of the stack (position 2) and rel-atively close to the end (position 3) thatare a consequence of a weak dynamicsymmetry breaking due to viscosity andthe nonuniformity of the acoustic pres-sure and velocity fields.

adiabatic compression, and no heat flowsbetween the gas and the plate. (Because oflosses in a real engine, the maximum tem-perature gradient that can be produced by

and the minimum gradient needed todrive a prime mover is somewhat greater

Thermoacoustic Couple. The thermo-acoustic couple is a simple thermoacousticdevice. A calculation of the properties ofthe thermoacoustic couple demonstrates agood deal of the physics of naturalthermoacoustic engines and can be donequantitatively from first principles (see“The Shor t Stack”) . When sui tablycalibrated, the device can also be used as aprobe to measure both acoust ica l lystimulated heat flow and acoustic power.

Typically, such a probe is a single shortthin plate of the type used in an acoustic

engine (or a small stack of such plates) thatcan be moved to various longitudinalpositions in an acoustical! resonant tube.A speaker at the open end of the tubedrives the acoustic oscillations.

The material of the plate has a largethermal conductance so that no substan-tial temperature gradient can build upalong its length, ensuring that the coupleoperates under a low temperature gradientas a heat pump, As the probe is moved tovarious locations in the standing acousticwave, it measures a flow of’ heat generatedby its presence by detecting a small tem-perature drop across its length.

Data taken with such a probe (Fig, 9) fita simple sine curve whose period is halfthe wavelength of the acoustic standingwave. By noting how the sign of the tem-perature difference varies with respect tothe plate’s location in the sound wave, wesee that heat always flows in the direction



LOS ALAMOS SCIENCE Fall 1986

Fig. 8. The thermodynamic cycles (top)of the gas parcels in an acoustic heatengine consist of reversible adiabaticsteps and irreversible constant-pres-sure steps (the acoustic mode is heresimplified to articulated rather thansinusoidal motion). This cycle is identi-cal to the Brayton cycle. If we follow aparcel of gas as it moves alongside afiber glass plate, we see that the prime-mover mode (red) occurs when the tem-perature rise seen by a gas parcel onthe adjacent plate due to displacementof the gas along the gradient (xl VT) islarger than the temperature rise of thegas due to adiabatic compression heat-ing of the gas (T1). The heat-pump mode(black) occurs under the opposite con-ditions, that is, when the gradient on theplate is zero or low. In the prime-movermode, the pressure (p + p 1) during theheat-flow expansion step is larger thanthe pressure (p) during the heat-flowcompression step, so net work is addedto the acoustic vibration. All flows arereversed in the heat-pump mode, andwork is absorbed from the acoustic

—

of the closest pressure antinode. This ef-fect is expected from the description of theheat pump in Fig. 8 because a parcel of gasmoving in the direction of a pressure anti-node is compressionally warmed and willtransfer heat to the low-gradient plate; aparcel moving toward a pressure node iscooled by expansion and will draw heatfrom the plate. This explains the surpris-ing results of the half-wave resonator heatpump of Merkli and Thomann (Fig. 5b).

At both the pressure antinodes and thepressure nodes, heat flow in the coupledrops to zero. This effect occurs becausethe pressure and the gas velocity in a reso-nant acoustic wave are spatially 90 degreesout of phase. Thus, a pressure antinode isalso a velocity node, and heat flow dropsto zero because there is no displacement ofgas. On the other hand. a pressure nodehas zero heat flow because no compression

continued on page 16

13


THE SHORT STACK

To calculate thermodynamic effi-ciency for an acoustic heat engine.we need to know the hydrodynamic

heat flow and the work flow. A heuristicderivation of these two quantities and theresulting efficiency for the particular caseof a short stack follow. We then brieflydiscuss the effects of viscosity.

Heat Flow

Consider a slack of plates in a heatengine whose length is short compared tothe acoustic wavelength and to the dis-tance from the stack to the end of the tube.If that length is short enough. wc canignore the change in the longitudinalacoustic velocity magnitude u l and thechange in the dynamic, or acoustic, pres-sure magnitude p 1 with respect to long-itudinal distance x (measured from theend of the acoustically resonant tube).Further, if we ignore the effects of fluidviscosity, u 1 does not depend on lateraldistance from the plates. Next, we can takethe lateral distance between plates to belarge compared to the thermal penetration

transfer in the fluid during a given cycle ofthe acoustic wave). Thus, and effects weestimate for a stack of plates will be thesame as the a single plate having the sameoverall perimeter II (measured transverseto the flow).

The adiabatic temperature change T 1

accompanying the pressure change p 1 canbe derived from thermodynamics and is

(1)

where T m is the mean absolute tempera-

cient, pm is the mean density, and cP is thespecific heat at constant pressure.

The change of entropy for a parcel os-cillating in the manner depicted in Fig. 8of the main text is just the lateral heat flowfrom the second medium divided by Tm or

the change in the fluid temperature due tothat heat flow. The volume transport ratefor that part of the fluid that is thermody-

estimate the flow of hydrodynamically

two quantities times Tm; that is,

Now from Fig. 8 we also see that

( )(3)

along the plate and x1 is the fluid displace-

is the critical gradient, so

(4)

Combining these equations and defin-ing the temperature gradient ratio parame-

the hydrodynamic heat flow as

heat parameter of the fluid. The presence

the thermodynamically active area in aplane perpendicular to the longitudinalacoustic motion. The formula shows that

gradient, as for a heat pump; when l_= 1,

flows down the temperature gradient. asfor a prime mover.

Work Flow

Now that we have estimated the heatflow, we need to calculate the work flow,which is given by the work per cycle (the

ume diagram in Fig, 8 of the main text)times the rate at which that work occurs(the angular acoustic frequency w), T h e

to the net work is just

(6)

Eq. 3. V, the total volume of gas that isthermodynamically active, is given by

We can now simply put these piecestogether and, using Eqs. 1, 3, 6. and 7.write down the work flow as

From thermodynamics we know that

(9)



where the quantity y — 1 is what we call thework parameter of the fluid, and a is the

speed of sound, so we can rewrite theexpression for work flow as

acoustic wave.

(Eqs. 5 and 10) have a very similar struc-ture, which is expected since they areclosely related thermodynamically. The

mula for Q, and the work parameter y – 1appears in the formula for W. Both Q andW are quadratic in the acoustic amplitude

through unity.

Efficiency

A quantitative evaluation of W and Qfor this case of the short stack but forsinusoidal p I and u 1 would give the sameresults except each formula has a numeri-cal coefficient of 1/4. Thus the efficiency nof a short stack with no viscous or long-itudinal conduction losses is

For our standing acoustic wave, u 1=

is the distance of the stack from the end ofthe tube. Then the efficiency can be rewrit-ten simply as

efficiency is simply

(13)

Thus, in either case, efficiency dependsonly on geometry and fluid parameters,just as for the Brayton and Otto cyclesdiscussed in the text. The temperatures T h

and TC do not enter.As the actual temperature gradient ap-

proaches the critical temperature gradient

proaches zero, so that even at the acousticangular frequency w the heat transfer rateand the power output approach zero, justwhat is needed to give the Carnot effi-ciency in the Brayton and Otto cycles.What happens in this engine? We use Eqs.1, 4, and 9 and the fact that u l = x 1w torewrite the efficiency formula (Eq. 11) ingeneral as

(14)

we have at the critical temperature gra-dient

(15)

What About Viscosity?

So far we have assumed that the work-ing fluid is inviscid. What if it is not? Weknow how to do the theory quantitativelyfor this more general case, but the resultingexpressions for Q and W are terribly com-plicated and opaque. We can simplify

them by assuming that the Prandtl num-ber (the square of the ratio of the viscous

(16)

To lowest order, then, the effect of vis-cosity on heat flow is just to decrease Q bya term proportional to the viscous pene-tration depth. This simply means that vis-cosity prevents a layer of fluid of thickness

acoustically and contributing to theacoustically stimulated heat transport.Similarly, the work flow is decreased by a

the energy lost from the acoustic wave dueto viscous drag on the plate.

For simplicity in Eqs. 16 and 17 we have

though another effect of viscosity is tomake the concept of a critical temperaturegradient less well defined. In fact, withviscosity present there is a lower criticalgradient below which the engine pumpsheat and a higher critical gradient abovewhich the engine is a prime mover. B e -tween these two gradients the engine is in auseless state. using work to pump heatfrom hot to cold.

The Prandtl number for helium gas isabout 0.67, so that viscous effects are verysignificant for our gas acoustic engines(and, in fact, Eqs. 16 and 17 are ratherpoor approximations). On the other hand.the Prandtl number for liquid sodium isabout 0.004, so that viscous effects aremuch smaller. ■



continued from page 1 3

or expansion takes place. In other words,acoustic heat flow depends on both theacoustic pressure and the fluid velocity.

Figure 9 also illustrates the dramaticeffect the positioning of a plate in theacoustic wave has on the operation of anatural heat engine, A plate or stack ofplates placed completely within a quarterof a wavelength of the end of the tubeoperates in the manner depicted in Fig. 8.If that same stack is repositioned in thesecond quarter of a wavelength, the pic-torial analysis of Fig. 8 still applies, but thedirections of all heat flows and long-itudinal temperature gradients are re-versed. A stack that extends beyond anadjacent node-antinode pair, however. hasheat flows that counter each other, cancel-ing part of the overall transport of heatfrom one end of the plates to the other,

Also important is the stack’s positionwithin a given node-antinode pair sepa-r a t e d b y a q u a r t e r o f a n a c o u s t i cwavelength (Fig. 10). For an engine in theheat-pump mode, a stack close to a pres-sure ant inode—say, the end of thetube—can develop steep temperature gra-dients. Why? In such a region the acousticpressure change in a parcel of gas is largeand thus the rise in temperature fromcompressional warming is large. This re-gion is also near a velocity node, so dis-placement of the gas parcel is small. Largetemperature changes over small displace-ments, of course, result in large tempera-

which bounds the region between the heat-pump and prime-mover modes, is largeclose to a pressure antipode.)

As a plate or stack of plates is movedaway from the pressure anti node, the tem-perature gradient developed becomessmaller. At a quarter of a wavelength, no

positioning effect is important in the de-sign of a refrigerator, because, togetherwith the length of the plates, it places anupper limit on the maximum temperaturedrop possible across the stack.

Positioning also affects the losses that

16

A

I I I >

h

Acoustic Standing Wave

of Heat

characterize an engine. For example, astack close to a pressure antinode is closeto a velocity node, and viscous losses willbe small at that position. However, be-cause temperature gradients are steepthere, losses from ordinary diffusivethermal conduction in the plates andworking fluid will be increased. The prob-lem of ordinary conduction losses isespecially critical for an engine acting asprime mover because such an engineneeds a temperature gradient higher than

set of plates is a tradeoff between viscouslosses, losses from longitudinal conduc-tion, the desired temperature span acrossthe engine. and power output.

Heat and Work. We can now betterunderstand the natural acoustic engine byexamining what happens near a short platepositioned between a node and an anti-node (Fig. 11). If the displacement of agiven parcel of gas is small with respect to



measured across a thermoacoust iccouple as a function of the plate’s posi-tion in the acoustic standing wave. Notethat heat flows toward the closest pres-sure antinode, making that end of thecouple hottest. However, at both thepressure antinodes and nodes there is

were obtained f o r a n acoustic

POSITIONING EFFECTS

Fig. 10. Close to a pressure antinode atypical parcel of gas experiences largechanges in pressure p 1 and thus largechanges in temperature 11 due to com-pressional heating. At the same time,displacement x l of the parcel is small,

since x 1 is proportional to the distance x

In the heat pump mode, the maximumtemperature gradient that can be de-

flow between gas parcel and platestops when that gradient is reached),which means that close to a pressureantinode we can expect large tempera-ture gradients. Further from the pres-sure antinode, pressure and tempera-ture changes become smaller whereasdisplacements become larger, so themaximum temperature gradient that canbe developed is smaller. A


the length of the plate, there will be anentire train of adjacent gas parcels, eachconfined in its cyclic motion to a shortregion of length x l and each reaching thesame extreme position as that occupied byan adjacent parcel half a cycle earlier.What is the net result of all these individ-ual cycles on the flow of heat and work?

If the motion of the parcels is sinus-oidal, only those about a thermal penetra-tion depth* from the nearest plate arcthermoacoustically effective. Parcels closeto a plate transfer heat to and from theplate in a locally isothermal and reversiblemanner, just like the fluid in the re-generator of a Stirling engine. Parcels faraway have no thermal contact and arcsimply compressed and expandedadiabatically and reversibly by the soundwave. However, parcels that are at about athermal penetration depth from a platehave good enough thermal contact to ex-change some heat with the plate but, at thesame time. are in poor enough contact toproduce a time lag between motion andheat transfer.

During the first part of the cycle for theheat-pump mode, the individual parcelswill each move a distance x 1 toward thepressure antinode and deposit an amountof heat Q at that position on the plate.During the second half of the cycle, eachparcel moves back to its starling positionand picks up the same amount of heat Qfrom the plate, But this heat was depositedthere a half cycle earlier by an adjacentparcel of gas, In effect, an amount of heatQ is merely passed along the plate fromone parcel of gas to the next in the direc-tion of the pressure antinode. Thus, as inthe Stirling engine, the second thermody-namic medium is used for the temporarystorage of heat.

At the ends of the plates, the thermody

teristic length describing heat diffusion throughthe gas during one period of the acoustic cycle.

thermal diffusivity of the gas and f is the fre-quency of the sound.

namic symmetry is broken. Parcels of gasthat move farther from the end of the platethan a thermal penetration depth idlethrough part of their cycle without accept-ing or rejecting heat. For example. if aparcel of gas at the end closest to theantinodc is in equilibrium with the plateon one half of the cycle but then moves outof the range of thermal interaction, it hasnowhere to deposit the heat resulting fromits adiabatic warming. AS this parcel com-pletes its cycle, it cools adiabatically backto the temperature of the plate. The heattransferred to the plate from the next adja-cent parcel down the line is un-compensated, so there is a net heat trans-fer to the plate on that end, and the tem-perature of the plate increases there. Insimilar fashion. heat drawn from the endclosest to the node is not replaced. and thatend cools. We can take advantage of thenet effect-a flow of heat from one end tothe other—by bringing the ends of theplates into contact with heat exchangers.

During each cycle an individual parcelof gas transports heat Q across only a smalltemperature interval along the plate that iscomparable to the adiabatic temperaturechange T 1. However, because there aremany parcels in series, the heat Q is shut-tled down the stack, thereby traversing thetemperature interval T h – T c, which canbe much larger than T 1. Within the limitsof a quarter of a wavelength, the flow ofheat is not a strong function of plate length(in fact, for a stack much shorter than aquarter of a wavelength, heat flow does notdepend on plate length at all).

If, on the other hand, we examine thistrain of gas parcels with respect to the flowof work, we realize that each parcel has anet effect. For example, a parcel of gas nearthe plates in an engine operating in theheat-pump mode absorbs net work be-cause its expansion is at a lower pressurethan the corresponding compression. Butsince the same is true for every parcel inthe train. the total work done on the gas isroughly proportional to plate length (for avery short stack, work flow is proportional

to plate lenght).

17


Efficiency. A calculation of heat andwork flows for an acoustic heat enginewith a short stack close to the end of theresonator tube and no viscous losses (see“The Short Stack”) yields a limiting effi-ciency given by

(3)

where T m is the mean absolute tempera-

and x is the distance of the plates from the

ACOUSTIC ENGINE GAS MOTION

Fig. 11. An acoustic heat engine can be oscillatory motion and deposit heat at the net result is that an amount of heat Qthought to have a long train of adjacent the other extreme. However, idling is passed from one end of the plate togas parcels, all about a thermal penetra- parcels at both ends oscillate without the other. Adjacent work contributionstion depth from the plate, that draw heat removing or depositing heat. Adjacent do not cancel, so that each parcel of gasfrom the plate at one extreme of their heat flows cancel except at the ends; contributes to the total work.

18 Fall 1986 LOS ALAMOS SCIENCE


netic engine is based on the adiabat icchange of temperature with magnetization.

The primary medium of our hypotheti-cal apparatus (Fig. 12) consists of a stackof magnetic disks.* Each disk has a highinternal thermal conductance. but each isalso thermally insulated from the others sothat a large temperature gradient can besustained in the longitudinal direction.

The collection of disks is placed in atube whose walls constitute the secondmedium. Like the first medium, the sec-ond has a high lateral thermal conduc-tance. a large heat capacity, and a low,longitudinal thermal conductance. Thus,it, too, can sustain a large temperature

cooler’s driver and hot heat exchanger.

Fig. 12. A hypothetical natural magneticengine in which the primary mediumconsists of magnetic disks placed in thefringing field at the side of a permanentmagnet. This placement allows an ex-ternal mechanism to displace the disksin a reciprocating fashion in the pres-ence of a magnetic field gradient. A gas


surrounding the disks makes thermalcontact with the second medium (thewalls of the tube), but conductivity of thegas i s poor enough to create thenecessary phasing for the engine. Inboth media, thermal conductance ispoor longitudinally so that large temper-ature gradients can be supported. A

gradient parallel to the direction of rela-tive motion between the two media.

The device is positioned between thepoles of a permanent magnet in such a waythat the disks of the primary medium arein the nonuniform fringing field at theside. The disks arc linked mechanically toan external mechanism so that they can bemoved in a reciprocating fashion. A gasfills the small annular space around themagnetic disks. providing lateral thermalcontact with the second mediurn, but thiscontact is poor enough to create thenecessary phasing for the engine. There isalso some means for heat exchange withexternal reservoirs at each end of the sec-ond medium.

If we follow an element of the firstthermodynamic medium in a magneticengine through an articulated cycle (Fig.13), we see that the various steps arcanalogous to those of an acoustic heatengine. For example, in the first step of aheat-pump cycle, the clement is metedquickly and adiabatically to a region ofhigher magnetic field. As a result. its tem-perature rises. (Temperature changes of afew degrees per tesla are typical for fer-

romagnetic and strongly paramagneticmaterials. ) In the second step, the elementthermally relaxes, its temperature adjust-ing to that of the adjacent region in thesecond medium, which, in the heat-pumpmode. means that heat flows from the firstmedium to the second. As the elementcools. its magnetization increases. Thethird step is motion back to a region oflower field; the fourth is another thermalrelaxation. As in the case of acoustic en-gines, the phasing between motion andheat transfer is a result of the natural timedelay caused by diffusion of heat betweenthe two media.

Some Applications ofNatural Engines

What happens now if these ideals of’natural engines arc put into practice?What are the clanking, hissing realities ofreal natural engines?

19


Cryocooler. As part of his Ph.D. thesis,Tom Hofler designed and built a devicecalled the cryocooler (Fig. 14) in which thenumerical aspects of the design were basedon the general thermoacoustic theory ofRott for ideal gases.

The cryocooler is an acoustic coolingdevice with a number of important fea-tures. Perhaps the most important is thefact that the acoustic resonance is drivenfrom the hot end of the stack. All the earlycooling engines were arranged with thestack near the closed end of the acousticresonator tube and with the acousticdriver at the opposite end. Very large tem-perature differences (about 100 centigradedegrees) could be easily induced across theentire stack this way, but the cold end wasseldom less than 20 degrees below ambienttemperature. The problem was that thedriver (at ambient temperature) and thecold end of the stack maintained mod-erately good thermal contact with one an-other by means of acoustic streaming. Thisphenomenon is a second-order, steadycirculatory flow of the working gas that issuperimposed on the oscillatory motion,The effect of the acoustic streaming was touse up a substantial amount of the refriger-ation available at the cold end of the stacktrying to cool down the driver.

Now while it is necessary for work toflow into the stack to pump heat, we re-alized that it is of no real importancewhether that flow occurs at the cold or thehot end. Putting the driver at the hot, orclosed end, means that none of the avail-able refrigeration is used to cool the driver.Thus, with the “closed” end replaced by amovable piston acting at high dynamicpressure and low displacement, perform-ance is improved,

The stack with its heat exchangers wasplaced rather close to the driver piston,that is, rather close to a pressure anti node.As noted earlier (see Fig. 10). such a regionhas the high critical temperature gradientneeded for a heat pump. Of course, someseparation between driver and stack isnecessary because acoustically stimulatedheat transfer is proportional to the dis-

20

placements of the parcels of gas.Such a configuration means the re-

mainder of the resonator tube can beat thecold temperature, allowing it to be just athermally insulated straight tube roughlyhalf an acoustic wavelength long. How-ever, losses due to the dynamical effects ofviscosity and thermal conduction alongthe walls of the resonator reduce the ex-ternally available refrigeration. Roughlyhalf this loss could be eliminated by usinga quarter-wavelength resonator with oneend open, but an open end eliminates theuse, say, of several atmospheres of heliumas the working fluid and revives theoriginal heat load problem of acousticstreaming—here between the driver andthe atmosphere. Moreover, an open end isdownright noisy, radiating useful work outinto the room.

The simple solution is to replace ap-proximately half the half-wavelength res-onator with a closed container of substan-tial volume, Dynamic pressure will besmall in a region of large volume. makingthe losses correspondingly small. The res-

Fig. 13. The cycle shown here for theheat-pump mode of a hypothetical natu-ral magnetic engine is analogous to thecycle for the heat-pump mode of theacoustic heat engine (Fig. 8) with mag-netic field strength H taking the role ofpressure and magnetization m takingthe role of volume. Thus, the cycle con-sists of reversible adiabatic steps andirreversible constant-field steps. For an

Thus, in the first step, when the diskmoves adiabatically to a region ofhigher magnetic field, magnetization re-mains constant and temperature riseswith increasing H. In the second step,heat flows to the lower temperature ofthe second medium, causing the disk tocool at constant H and the magnetiza-tion to increase. The net result of all foursteps is the transport of heat up thegradient as a result of the work (whichwill equal m lH 1) needed to move thedisk through its cycle. ➤




onator below the stack was modifiedfurther by decreasing the diameter of theconfining tube and shortening its length.This last modification, at first sight, wouldappear to be of negative value as onewould expect viscous losses to go up; but,for small decreases in neck diameter, dy-namic thermal-conduction losses go down

Fig. 14. The driver in the acousticcyrocooler is an ultralight aluminumcone attached to the voice coil of acommercial loudspeaker. The secondthermodynamic medium, rather than be-ing a set of parallel plates, consists of asheet of Kapton rolled about a verticalrod and spaced with 15-roil nylon fishingline aligned vertically. Copper heat ex-changers are attached at both ends.The form of the bulb and neck, includingthe constriction, were chosen to reduceviscous and thermal losses by reducingsurface area. The device is drawn to

Fig. 15. Experimental data for thecryocooler of Fig. 14, obtained with amean helium pressure of about 10 barsand acoustic frequencies in the range of540 to 590 Hz. For thermal isolation theengine was placed in an evacuated ves-sel and surrounded by superinsulation.The frequency was adjusted e lec-tronically so the dynamic pressure andvelocity were always in phase at thedriver. Part (a) shows how the tempera-ture difference between the hot heatexchanger at approximately 26°C andthe cold heat exchanger increases withrelative dynamic pressure amplitude(the ratio of the acoustic pressureamplitude p I at the pressure antinode tothe mean pressure p). No heat load wasapplied to the cold heat exchanger. Part(b) shows how, for a relative dynamicpressure amplitude of 0.03, the temper-ature difference gradually drops withincreasing refrigeration load at the cold

faster than viscous losses go up, and thereis a net decrease in the overall losses.These surprising qualities explain the gen-eral shape and configuration of thecyrocooler.

Performance is rather good (Fig. 15). Asthe relative dynamic pressure amplitudeincreases, the temperature difference thatcan be pumped up for zero external heatload increases, eventually topping out atabout — 100°C when the acoustic pressureamplitude is about 2 or 3 per cent of them e a n p r e s s u r e . A t t h a t p o i n t , t h ecryocooler can handle a significant re-frigeration load and still maintain a ratherlow temperature. This type of refrigerationcapability is very suitable for cooling in-struments and sensors.

A Heat-Driven Acoustic Cooler. Natu-ral acoustic engines are functionally re-versible: they can be either prime moversthat use heat to produce sound or heatpumps that use sound to refrigerate. Whynot combine these two functions in onedevice and usc heat to cool? Such an en-gine would have heat flow through thewalls but no external flow of work.

A key problem in the design of a heat-driven acoustic cooler is where to positionthe two sets of plates—one set acting asprime mover, the other acting as refriger-ator. Ideally, the refrigerator plates shouldbe positioned as they are in the cryocooler.that is, close to the end of the tube wherethe velocity of the gas and the viscous

it would also be good to keep viscouslosses low for the prime-mover plates, it ismore important to have these plates near a

enough for the stack to develop adequatepower. These considerations imply thatthe refrigerator stack needs to be closer tothe end of the tube than the prime-moverstack. However. such a configurationwould put the hottest region (the hot endof the prime mover) next to the coldestregion (the cold end of the refrigerator),creating a difficult thermal-design prob-lem.

21


Fig. 16. The upper set of plates in thiscooler is a prime mover that draws heatfrom a heater (at about Th = 390°C) andrejects waste heat to cooling coils (at T a

= 23°C or room temperature), generat-ing acoustic work. The lower engineuses that work to reject heat to thecooling coils (at T a and to draw heatfrom an even lower temperature ( TC =0°C or the ice point). The acoustic tubeis about half a meter in length,terminates in a 2-liter bulb, and containshelium at a pressure of 3 bars that res-onates at a frequency of 585 Hz. Bothsets of plates are made of 10-mil (0.025cm) stainless steel, and the spacing be-tween plates in both sets is 0.08 cm. Thehot heat exchanger is made of nickelstrips; the ambient and cold heat ex-changers of copper. ➤

To avoid large heat inputs to the cooler, constant,

Bob Oziemski adjusting the flow of theworking fluid in the liquid propyleneStirling engine. A

22

the positions of’ the two stacks can bereversed and the various temperatures ar-ranged in decreasing order along the tube.What now becomes paramount is for con-ditions to be such that the prime mover isable to adequately drive the cooler.Amplification of acoustic fluctuations oc-

shorten the prime-mover stack but keepthe temperature difference across the stack

fortunately, this type of change increasesthe heat loss due to conduction down thestack.

away from the pressure anti node at theend of the tube. Because of the interveningprime-mover stack, the refrigerator stackis already much farther from the pressureantinode than in the cryocooler, and addi-tional movement of the prime-mover



BEER COOLER DATA

(a) o 400

■ ■ .m ■ ■ ■ ■

+10

t 1

300

■

stack pushes the refrigerator stack even produce cryogenic temperatures, themore from its optimum position. Onceagain, changing an idea into a practicalheat engine entails a set of compromises.

A schematic of an operational heat-driven cooler in which the prime-moverstack is between the end of the tube andthe refrigerator stack is shown in Fig. 16.Because of the above consider-ations—especially those related to

cannot be expected to cool much belowambient temperature. Although unable to


cooler ought to be able to producetemperatures low enough to cool a can ofbeer. For this reason we have affec-tionately dubbed the engine the "beercooler."

As in the case of the cryocooler, therather complex design was carried out nu-merically, and many of the features impor-tant to the cryocoolcr apply to the beercooler. For example, the resonator issimilar to the resonator in the cryocooler,and the driver is on the “hot” side of the

Fig. 17. For these measurements on thebeer cooler, the refrigerator stack andthe resonator were located in an evacu-ated space for thermal insulation. Part(a) shows how the cooling temperaturedifference (black) across the refriger-ator stack and the driving temperaturedifference (red) across the prime-moverstack vary with the level of oscillation(here given by the square of the relativedynamic pressure amplitude) when noexternal refrigerator load is placed onthe cooler. The level of oscillation isdetermined by the rate at which heat issuppl ied and removed across theprime-mover stack, but, as can be seen,this results in little change in the drivingtemperature difference. At the sametime, the cold temperature drops gradu-ally below the freezing point of water.(b) For a level of oscillation of about 4 X1 0-3 (that is, p 1 is about 0.19 bar), wesee that the beer cooler can handle asmall heat load of 10 watts and still

refrigerator stack, thus reducing viscousand acoustic-streaming losses.

In our model engine the plate materialin the stacks (stainless steel) and the spac-ing of the plates were dictated by ease offabrication as much as by anything else.The refrigerator assembly was placed in anevacuated container but not otherwisethermally insulated. The working fluidwas helium, whose pressure was chosenexperimentally to minimize the cold tem-perature. A major problem, yet to be satis-factorily solved, was efficient exchange ofheat at the heat exchangers—especially thehot exchanger made of nickel.

In spite of the engine’s compromises, itstill sings along. performing rather well(Fig. 17). As the heat supplied to the hotend of the prime mover is increased, thelevel of oscillation increases—the largestpeak-to-peak dynamic pressure amplitudemeasured at the ambient exchangers ex-ceeding a tenth of the mean pressure. Inagreement with our understanding, thetemperature drop across the prime-mover

23


stack does not change much as the dy-namic pressure amplitude increases; thesmall changes seen in the data result fromthe diffusive flow of heat across the gaps ofgas and through the heat exchangers fromthe heat source to the ambient heat ex-changer. Figure 17b shows that the beercooler can manage a 10-watt cooling loadwhile keeping T C 5 centigrade degreesbelow the freezing point of water-arather encouraging result for the first 1abo-ratory model.

A number of issues concerning the prac-tical use of this engine concept and of thecryocooler remain to be resolved. It islikely that the most important is the mat-ter of heat exchange, This problem. aswe’ve mentioned, has always been a keyone in the development of heat en-gines—classical or otherwise.

The Liquid Sodium Acoustic Engine.As man moves from Earth into space, sodoes his need for reliable power. However,differences in the requirements and in theoperating environment in space mayprompt radical changes in the engines thatprovide such power. An idea stimulatedby such differences is the liquid sodiumacoustic engine, which not only is a natu-ral. rather than a conventional, engine butuses a liquid instead of a gas as its workingfluid.

The concept of using a liquid can betraced to a 1931 paper by J. F. J. Malone inwhich he pointed out that certain liquidshave important thermodynamic qualitiesthat make them suitable for use in heatengines, Although concerned about itschemical reactivity, Malone knew thatliquid sodium was one of these “goodliquids,” but materials technology wasthen inadequate for him to consider itsuse.

Today’s materials technology suggestsrevival of these ideas, and we had beenworking on the liquid propylene Stirlingengine (see “The Liquid Propylene En-gine”) as a modern example of an i n -novative but more conventional enginethat uses liquids. Thus, when we learned

24

from then Associa te Direc tor KayeLathrop of the need in space for a reliable,moderately efficient electrical generator. itwas not difficult for us to propose a naturalacoustic engine based on liquid sodium,Especially ideal for this application is thehigh “cold” temperature (at least 400kelvins) of the liquid sodium engine. Thisfact is important because the cold sink forany heat engine in space must ultimatelybe a black-body radiator whose size would

Liquid sodium has many potential ad-vantages as a working substance in a natu-ral engine. The heat and work parametersare acceptably large. For example, at700°C, which is roughly in the middle ofthe temperature range of a possible high-power engine, liquid sodium has a veryhigh expansion coefficient and a large

and y — 1 = 0.43 (compared to amonatomic gas such as helium. for which

For a given Mach number,* the powerdensity in the stack is proportional to pa3,where p is the density of the working fluidand a is the speed of sound. The density ofliquid sodium is about 500 times greaterthan that of helium at the pressures used inour gas acoustic engines, and the speed ofsound is more than a factor of 2 greater.Thus, the power density for a liquid so-dium acoustic engine should be more than1 03 times greater than for a heliumacoustic engine, a definite advantage.

This dramatic increase is not without itsdrawbacks, however. The heat capacityper unit area for the sodium within athermal penetration depth of the second

*The Mach number is the ratio of the fluid speedto the speed of sound in the fluid.

Fig. 18. The temperature drop appliedacross both stacks of molybdenumplates causes the liquid sodium in thisproposed engine to oscillate back andforth between the poles of the magnet.A magnetohydrodynamic effect is usedto convert acoustic to electric energy. ➤


zThe Natural Heat Engine

Chris Espinoza welding heat exchangemanifolds onto the resonator tube of theliquid sodium natural heat engine.

medium is so large that the usual assump-tion of infinite heat capacity of the secondmedium is not valid. As a consequence,the power density drops. Moreover, theacoustic impedance pa of the sodium isrelatively high—roughly equal to that ofsolids—which means that in a sodiumengine motion of the stack and containercan be expected to send heavy vibrationsthroughout the entire engine (unlike thebeer cooler, for example, in which a peak-to-peak dynamic pressure oscillation of 10per cent of the mean pressure producesonly a pleasantly audible tone in theroom). To counter this effect, stiff, high-density materials like molybdenum ortungsten need to be used in the stack, andthe walls of the resonator need to be madeof heavy stainless steel. Even with suchstrong walls, high Mach numbers cannotbe achieved because the high acousticpressures would burst the resonator.

Liquid sodium has other very desirablefeatures. For example, its Prandtl number,which can be thought of as the square ofthe ratio of the viscous penetration depthto the thermal penetration depth, is ex-


tremely low (about 0.004 for sodium at700°C compared to 0.667 for helium gas),The reason for such a low Prandtl numberis that liquid sodium is a metal. As aresult. its kinematic viscosity, is rather nor-mal for a liquid, but, owing to electroniccontributions to the conduction of heat, itsthermal diffusivity is high. The conse-quences arc important. In helium, viscousshear extends into the gas from a bound-ary about as far as the temperature gra-dients that drive the flow of heat. Thisshear drains energy, decreasing efficiencyand making it difficult for a gaseous heatengine to work. The low Prandtl numberof liquid sodium means that heat can betransported between working fluid and theplates for a volume fifteen times largerthan the volume being affected by vis-cosity, and viscous losses arc correspond-ingly small. Again. however, a price mustbe paid: diffusive heat conduction in thesodium down the stack increases.

The fact that liquid sodium is a metalhas yet another important consequence.Electrical current can be generated fromthe sound via magnetohydrodynamiccoupling. Such coupling means electricpower can be produced from heat withoutusing moving parts (ignoring the non-neg-ligible motion of the vessel containing thesodium!). This feature, of course, is one ofthe main reasons for the expected re-

liability of the engine. Figure 18 is aschematic of a possible liquid sodiumprime mover that uses a half-wavelengthresonant tube, two driving stacks (one oneach side of the magnet), and magnetohy-drodynamic power coupling.

To design a model liquid sodium en-gine, we constructed a thermoacoustic the-ory for liquids and then evaluated it nu-merically. The calculated characteristics ofa reasonably designed engine arc given inTable 1. Note that the dynamic pressurealmost equals the mean pressure of thesodium and that efficiency is calculated tobe about 18 per cent (31 per cent of theCarnot efficiency).

A complete engine has not yet beenbuilt, but work (supported by the Divisiono f A d v a n c e d E n e r g y P r o j e c t s i nDOE/BES) has been done separately onthe magnetohydrodynamics and thethermoacoustics. In both cases prelimi-nary results arc encouraging. though tech-nical problems remain.

First, a magnetohydrodynamic con-verter was built that consisted essentiallyof a liquid sodium acoustic resonator witha central rectangular channel for guidingthe sodium in the transverse direction be-tween the poles of a magnet. Electrodes forpicking up the electric current were at-tached to the channel. The device wastested by exciting an acoustic standing

Table 1

Characteristics of a reasonably designed liquid sodium prime mover.

Frequency 1000 HzHot temperature 1000 KCold temperature 400 KMean pressure 200 barsDynamic pressure 198 barsPlate spacing 0.0373 cmPlate thickness 0.0280 cmDistance of hot end from tube end 8.65 cmLength of stack 8.0 cmAverage Qh 300 W/em2

Average W 55.1 W/cm2

0.1840.307

25


wave (by temporarily putting electricpower into the magne tohydrodynamicconverter!) and then letting the energystored in the acoustic resonance flowthrough the converter into a resistive loadacross the e lect rodes . The eff ic ien-

cy-defined as the ratio of the measuredelectric energy delivered to the load to thecalculated stored acoustic energy-is al-ready quite high in this first prototype(Fig. 19) and a number of improvementsare possible. From a technological point ofview, it is very significant that the max-imum efficiency is still reasonable in amagnetic field of only 0.9 tesla, suggestingthat a permanent magnet is appropriatewith a consequent simplification and de-crease in weight.

The thermoacoust ic pr ime m o v e rtested had a single stack of molybdenumplates (Fig. 20) inside a straight half-wavelength tube. For this test the cold heatexchanger was filled with pressurizedwater at 125°C and the hot heat exchangerwith heated sodium at various tempera-tures ranging from 440°C to 645°C. Al-though the test was preliminary, it wassuccessful. The application of varioustemperature drops across the stack re-sulted in the data of Fig. 21 and, above a350°C drop, in an obvious acoustic vibra-tion of the entire assembly.

We obtained the rate of heat supplied tothe engine Q h by monitoring the flow rateand the inlet and outlet temperatures ofthe sodium flowing through the hot heatexchanger. For a low temperature drop(AT) across the stack, the heat flowthrough the engine is due solely to thesimple conduction of heat by the sodium,molybdenum, and stainless steel. How-

to increase dramatically>; above the valuefor simple conduction. This result agreeswith the fact that acoustic oscillations at

520°C the resonator was oscillating athigh enough amplitude that the sound inthe room was unpleasantly loud and theapparatus was vibrating strongly.

26

MAGNETOHYDRODYNAMIC COUPLING EFFICIENCY

●

A

:

1 10 100

External Resistive Load (ohms)

Fig. 19. These initial data demonstrate holding the liquid sodium is 1.2 cm thickthe efficiency with which acoustic in the direction of the magnetic field, 7.6energy in liquid sodium was converted cm thick in the direction of electric cur-to electric energy via magnetohydrody- rent flow, and 31 cm long; however, onlynamic coupling as a function of the re- 20 cm of that length is actually in con-sistence of an external load and for tact with the electrodes. The centralthree different magnetic field strengths. channel is part of a l-m-long acousticIn the apparatus used to obtain this resonator filled with liquid sodium at adata, the central rectangular channel

The magnetohydrodynamic converter used to test power coupling for the liquidsodium heat engine.



Heat Exchanger Tubes

LIQUID SODIUM PRIME MOVER PARTS

Fig. 20. Our first operating liquid sodiumprime mover has a single stack ofmolybdenum plates (left) that were fab-ricated at Los Alamos from a solid rodusing electric discharge machining.Plate thickness is 0.3 mm; spacing be-tween plates is 0.38 mm; the length ofthe stack is only 5.2 cm so that theengine would oscillate at a reasonablylow AT. The transverse tubes of the twoheat exchangers (one set can be seenat the end of the cylindrical section of

the resonator tube on the right) weremade from stainless steel hypodermicneedles. Hot liquid sodium at varioustemperatures (Th is circulated throughone heat exchanger and pressurizedhot water (TC = 125°C) through the other.The stack just fits inside the half-wavelength resonator tube, which has alength of 106 cm. The plates are posi-tioned in the tube at about x = L/14 fromthe end. The acoustic resonant fre-quency is 906 Hz.

Acoustically .StimulatedHeat Flow

●

●

●

Normal Conduction


Fig. 21. The data shown here (dots) rep-resent heat flow Q h into the liquid so-dium prime mover at the hot heat ex-changer as a function of the tempera-

the solid curve represents the calcu-lated flow of heat due to conduction with

heat flow is due to normal conductionacross the stack. At high AT, however,the sharp rise in Q h is indicative of

There arc some disagreements betweenthe experimental results and our theoreti-cal calculations. A calculation for theparticular geometry and acoustic fre-quency of the device predicts lhat it

rather than 350°C. We also expected a

520°C, whereas the measured value was

2600 watts. We do not yet understandthese quantitative disagreements but arcextremely encouraged by the initial suc-cess of the engine.

Molecular Natural Engines. Heat en-gines of any sort transform energy betweenthe random thermal motion of atoms andthe coherent motion needed for usefulwork. The concepts of heat and tempera-ture—implicit to the understanding ofheat engines—arc statistical in nature.Hence, for these variables to be well de-fined, a system must have large numbersof atoms, But what is the smallest systemthat will still allow us to apply these con-cepts?

If we take the error in statistical quan-tities in thermodynamics to be approx-imately the reciprocal of’the square root ofthe number of degrees of freedom and ifwe assume for a heat engine that errors of afew per cent are tolerable, then only sev-eral hundred to a few thousand atoms aresufficient. Rather nice systems of thismesoscale size, consisting of large organicmolecules. are commen. Furthermore,such systems behave much like a purelyclassical collection of’ masses and springs.

Within such systems, nonlinear inter-molecular potentials give rise tophenomena directly related to the thermalexpansion needed for heat engines. In-tramolecular and intermolecular interac-tions provide connections between regionsof irrational energy that, if large enough,can be considered to be heat reservoirs. Inother words, all the ingredients for anengine are present.

Do such engines then exist? And, if so,do they serve a useful function in nature,say perhaps as tiny engines in biochemicalsystems? Will the concepts of natural en-gines apply not only to these small sizesbut also to the high frequencies associatedwith molecular vibrations?)

Heat pumping might occur in meso-scale systems if coherent vibratory motion

27


Fig. 22. Localization of acoustic energyw a s s t u d i e d b y c o u p l i n g t w e l v en o n l i n e a r H e l m h o l t z r e s o n a t o r stogether in a ring and measuring thed i r e c t i o n o f h e a t f l o w w i t h t h ethermoacoustic couples positioned inthe coupling tubes and measuring thelevel of vibration in individual re-sonators with the pressure sensors. Theconstruction of the neck of each re-s o n a t o r ( s e e d e t a i l s b e l o w t h edodecagon) introduces a nonlinearitybecause v ibrat ing gas that rushesthrough the neck causes the Kapton toflex, altering the resonant frequency ofthat resonator. The entire system isdriven by a loudspeaker at the center. ➤

can first be established and then survivelong enough to have a significant effect.Also, if the concepts of temperature andtemperature gradients are to be useful,then the mean free paths of the heat-carry-ing excitation should be small comparedto the classical thermal penetration depthand to the size of the mesoscale object.Using an angular frequency of 1011 Hz, weestimate the penetration depth to be about14 angstroms. Hence, mesoscale objectsperhaps 50 to 100 angstroms in size andvibrating at frequencies of order 1010 H zmight be large enough and slow enough tobe natural engines, providing their level ofcoherent excitation is high enough.

Rather than building an object of- suchsmall size. the same effects may be realizedin a natural way via a concept fromnonlinear science—the solitary wave. Anacoustic heat engine with its stack of platescentered at a pressure anti node will pumpheat from both ends of the stack towardthe middle. If we alter this idea by using acontinuous stack that has, owing to dis-persive and nonlinear effects, a localized,or solitary, vibrational disturbance in thelongi tudina l d i rec t ion , then hea t i spumped from the wings to the center ofthe disturbance. Because the stack is con-tinuous, the thermodynamic symmetry isnot broken geometrically rather it is

28 Fall 1986 LOS ALAMOS SCIENCE


broken dynamically. We call such a devicea nonlinear natural engine. In principle.such a localized disturbance could be avibrational excitation of a mesoscale ob-ject.

Localized waves in lower-dimensionalvibrational systems have received a greatdeal of theoretical attention because oftheir potential application to biologicalprocesses. However, macroscopic model-ing experiments in a water wave trough atthe University of California. Los Angeles.have been very valuable in developinginsight about solitary waves. (Anoutstanding example is the Wu-ton, a non-propagating soliton in water surfacewaves. ) As a result, we decided to build anacoustical model that might give insightinto how a coherently vibrating molecularsystem might behave. If such objects areindeed found to be real, we believe thefield of potential applications will be muchbroader than just lower-dimensional sys-tems.

Our apparatus, which we call thedodecagon, has been likened to a 12-ele-ment benzene ring. It consists of a circle oftwelve coupled acoustic Helmholtz re-sonators with a nonlinear element in-cluded within each resonator (Fig. 22). Weintroduce the nonlinearity by building theresonator from two bulbs connected by aneck with a thin Kapton plastic film thatflexes with changes in pressure. To pre-vent the neck from flapping at the acousticfrequency or its harmonics, we loaded theplastic film with oil.

According to one mathematical analy-sis, localization of energy can occur if theresonant frequency of any given resonatordecreases as the amplitude increases. Inour resonators, as the dynamic pressure ofthe acoustic wave increases, the velocity offluid through the neck increases. whichmeans, from Bernoulli’s principle, that theaverage pressure there decreases. TheKapton neck then flexes inward, reducingthe cross-sectional area and, thus, the reso-nant frequency (which is proportional tothe square root of the area).

We installed a thermoacoustic couple in


each tube linking resonators to measurethe direction of heat flow and also put adynamic pressure sensor in each resonatorto measure its level of vibration. Thewhole system was driven symmetricallyfrom the center by an acoustic driver.When we drove the system at a frequencyless than the low-amplitude resonance fre-quency. localization of energy did occurabove a certain threshold amplitude.Further, heat was pumped toward the re-gion of high amplitude. But the localiza-tion was stronger and occurred at a muchlower drive amplitude than expected. Thislocalization was also attended by a low-frequency modulation-typically at 1/11or 1/12 of the drive frequency but oftenwith components a factor of 100 or moretimes lower than the drive.

What happened? Our resonatorsperformed as expected so far as alterationof the resonant frequency was concerned.However, we had unwittingly introduced asecond set of vibrational systems into theexperiment: plate-like vibrations on theKapton-oil system, We believe that undersuitable conditions the driver resonantlyexcites the Kapton-oil system and inducesthe film to make a hysteretic transition toa different geometry that facilitates thelocalization.

Our acoustical model experiments havebeen helpful in inspiring thought on mo-lecular-scale or mesoscale systems.Localization of energy and heat pumpingdid occur. More important, though, at-tending and preceding the localization, weobserved behavior that changed on an en-tirely different time scale than the acousticphenomenon.

We conjecture that in molecular andmesoscale systems it is important to havetwo or more interacting. or coupled,“fields.” These coupled fields could besome of the normal optical vibrationalmodes of a molecular system. Inparticular, torsional or Vibrational modesof motion are almost certainly couplednonlinearly with the longitudinal modesof motion. We expect a time-dependentconfirmational change, say in the long-

itudinal field, to attend localization ofvibrational energy. The fundamental mo-lecular vibrational frequencies arc of theorder of 1012 Hz or greater. but a time-dependent confirmational change in thelongitudinal field could beat a much lowerfrequency-possibly low enough to createnatural engine effects. These conjectureshave motivated us to begin experimentalwork with a number of other collaboratorson the general question of the localizationof vibrational energy in materials. This is acase where we think we know what we arelooking for, but we don’t know what wewill find.

Thus, whether shaking loudly in thelaboratory or, perhaps, vibrating sound-lessly in a molecule, natural engines maybe a widespread phenomenon of generalimportance. Not only are natural enginessimple, they use a necessary thermody-namic evil—irreversibilities—as apositive feature of the engine. We hope anunderstanding of these concepts will servemankind well in his quest for appropriateengines and will help us to comprehendbetter the behavior of molecular vibra-tional systems. ■

Further Reading

John Wheatley, T. Hofler, G. W. Swift, and A.Migliori. 1985. Understanding some simplephenomena in thermoacoustics with applica-tions to acoustical heat engines. AmericanJournal of Physics 53:147.

J. C. Wheatley, T. Hofler. G. W. Swift, and A.Migliori, 1983. An intrinsically irreversiblethermoacoustic heat engine. Journal of theAcoustic Society of America 74:153.

G. W. Swift, A. Migliori, T. Hofler, and JohnWheatley. 1985. Theory and calculations for anintrinsically irreversible acoustic prime moverusing liquid sodium as primary working fluid.Journal of the Acoustic Society of America78:767.

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The Liquid PropyleneEngine

An ideal use of geothermal energy isto warm buildings by extractingh e a t f r o m g r o u n d w a t e r a t

temperatures of only about 10°C. This ap-plication involves the pumping of largeamounts of heat across small temperaturedifferences (of the order of 30°C). An effi-cient way to effect such heat transfer isfrom one liquid to another. As a result, aheat pump that appears well suited for thispurpose is a conventional reciprocatingheat engine using a liquid for a workingsubstance.

We have been studying just such anengine—a Stirling engine that uses liquidpropylene as its working fluid. Our dis-cussion of this device will both contrastthe simplicity of natural engines with thecomplexity of more traditional enginesand, more important, will introduce theuse of a liquid as a thermodynamic work-ing substance. (The section in the mainarticle called “ T h e L i q u i d S o d i u mAcoustic Engine” discusses a natural heatengine that uses a liquid as its primarythermodynamic medium.)

It is a common misconception thatliquids behave much like an idealizedhydraulic fluid, with density independentof temperature and pressure, In fact,especially near the critical point (where theliquid and gaseous phases become indis-tinguishable), a typical real liquid is some-what compressible, has a large thermalexpansion coefficient (comparable to orlarger than that of an ideal gas!), and hasother attractive thermophysical proper-

30

Fig. 1. In this propylene-to-water heatexchanger, made up of a stack of hun-dreds of stainless steel sheets copper-brazed together at Los Alamos, thepropylene flows in at the top right of thestack and across through the propylenemanifolds and channels, then moves upand out through the other propyleneduct. The arrow in the figure traces thepath through just one of the sets ofchannels and manifolds; similar flow oc-curs through the other, lower propylenechannels and manifolds. At the sametime, water flows in and up through onewater duct and across the stack (butthrough alternate sets of plates andacross the plates in a direction perpen-dicular to the corresponding propyleneflow) until it returns, exiting through theother water duct. Because of the in-timate thermal contact between fluidand stainless steel, heat can be trans-ferred at a rate of 230 W/°C. ➤

ties. These facts were first appreciated byJohn Malone, who in the 1920s built sev-eral Stirling prime movers that used liquidwater with pressures as high as 700 bars asthe working substance. We chose liquidpropylene (C3H 6) for our work because itscritical temperature is just above roomtempera ture and i t s Prandt l number(which can be thought of as a measure ofthe material’s viscous losses in relation toits thermal transport capacity) is lower

than that of other fluids with similar criti-cal temperatures,

A major advantage of a liquid workingsubstance is that liquids have a very largeheat capacity per unit volume comparedto gases, making it possible to build effi-cient and compact heat exchangers andregenerators. This point is illustrated bythe compact propylene-to-water heat ex-changer we have developed for our engine(Fig. 1). The exchanger is made of hun-



dreds of chemically milled stainless-steelsheets copper brazed together (several ofthe individual plates are shown on thecover). Although the exchanger (4 by 4 by9 centimeters in size) entrains only a fewcubic centimeters of propylene, it transfersheat between the two fluid streams at arate of 230 watts per °C with only a fewwatts of power required to pump the fluidsthrough the exchanger.

Another advantage of a liquid workingsubstance is that liquids are typicallymuch less compressible than gases. Thusthe large pressure amplitudes needed topump large amounts of heat can beachieved with only small displacements ofa piston, even for a substantial volume ofentrained liquid in the thermal elements.


Because of this quality it is possible tobuild a high-power engine that uses a shortstroke, making the mechanical elementsvery efficient without corn promising onthe size and efficiency of the thermal ele-ments.

Our Laboratory-scale Iiquid-propyleneStirling engine (Fig. 2) uses the same con-figuration of parts shown in Fig. 3 of themain article (the Rider form of the Stirlingengine), except that we have four suchassemblies, These assemblies operatef rom a common crankshaf t and aremechanically phased 90 degrees apartso that the shaft torque oscillations areminimized, eliminating the need for a bigflywheel. Although much of the wiring inFig. 2 is for diagnostic purposes, the

Fig. 2. The heat engine shown here con-sists of four Stirling engines of the Riderform operating from a commoncrankshaft but phased 90 degreesapart. The working medium is liquidpropylene, and heat exchange betweenwater and the propylene takes place inthe stainless-steel exchangers de-picted in Fig. 1.4

photograph, when contrasted with photo-graphs of natural engines (See the mainarticle) is nevertheless a dramatic rep-resentation of the complex of a moreconventional reciprocating engine.

In its heat-pump mode, our engine useswork supplied by an electric motor totransfer heat from a source at or belowroom temperature to a heat sink consistingof flowing water at or above mom temper-ature. For convenient measurement, thelow-temperature source is an electricheater. Mean pressure, oscillating pressureamplitude, volumetric displacement, shaftrotation frequency f, and hot and coldtemperatures are all independently con-trollable. We can measure both the rate atwhich heat is pumped away from the heat

In addition. our laboratory engine hasvalves that quickly change it from theordinary heat-pump configuration to onein which there is no flow of propylenethrough the regenerators and heat ex-changers, even though crankshaft andpiston motion, pressure amplitudes,temperatures, and so forth remain thesame. This feature allow’s uS to accurately

quired to pump the heat, with the back-ground torques due to bearing and sealfriction, piston blowby, and the likeeliminated.

Large amounts of heat can be pumpedby the engine (Fig. 3a)-around 1300watts at a crankshaft rotation frequency of4.5 Hz—and the data points match verywell curves predicted from theory for theparticular geometry of’ the engine and forthe use of propylene as the working fluid.

31


PROPYLENE ENGINE DATA

(a)14001-

Oscillating Pressure Amplitude

(b)

0 1 2 3 4

Crankshaft Rotation Frequency (Hz)

Crankshaft Rotation Frequency (Hz)

Fig. 3. (a) The rate at which thepropylene engine pumps heat Q as afunction of crankshaft rotation fre-quency f at two different oscillatingpressure amplitudes agrees very wellwith theoretical curves predicted fromthe physical properties of propyleneand the geometry of the engine. (b) The

as a function of f, is just that part of thetorque needed to pump the heat. In bothgraphs the blue data points representno temperature difference across theregenerators, whereas the red data

The lines drawn on Fig. 3b represent thetorque required by an engine with theCarnot efficiency to pump the observedamount of heat added to the torque as-sociated with just the viscous losses ofpushing the fluid through the regeneratorsand heat exchangers. Our measuredtorque differences agree well with thesetheoretical curves.

Our laboratory engine is very far from apractical, economically useful device. Itsscale and most of its design are ap-propriate for experimental measurementsand for the understanding of- principles.not for optimized efficicncy or low manu-facturing or operating costs in a specificapplication. But, as expected, we are learn-ing that liquids are good heat engine work-ing substances, Liquid engines may ul-timately be of great technological im-portance.

We arc also learning much about thepractical details of the use of liquids inengines. For example, we suspect that thenext logical step in the development ofpractical liquid engines is to abandon thereciprocating Stirling engine entirely. In-stead, we would use the liquid in, say, aBrayton engine with rotary compressorsand expanders. Such a configurationwould reduce losses from such things asbearing and seal friction that, until now,we have regarded as qui te unin ter -esting. ■



John C. Wheatley (1927-1986) joined Los Ala-mos in 1981. During his tenure here, heperformed experiment; on novel heat enginesand on the fundamentals of thermal andstatistical physics. He received his B.S. in elec-trical engineering in 1947 from the Universityof Colorado and his Ph.D. in physics in 1952from the University of Pittsburgh. He waselected a member of the National Academy ofSciences in 1975 and appointed to the Academyof Finland in 1980. His many honors includethe two top awards given by the low-tempera-ture physics community: the Simon MemorialPrize and the Fritz London Memorial Award.At the time of his death, he was the first jointFellow of the University, of California, Los An-geles, and Los Alamos National Laboratory

Albert Migliori earned his B.S. in 1968 fromCarnegie-htellon University and his Ph.D. inphysics in 1973 from the University of Illinois.where he studied superconducting thin films.He then joined Los Alamos as a pos-doctoralfellow and studied high-field and self-fieldbehavior of hard type II superconductors. In1975 he was awarded a National Science Foun-dation Fellowship to study internal and surfacemagnetic fields in current-carrying supercon-ductors with the Mossbauer effect. In 1976 hebecame a staff member of the Condensed Mat-ter and Thermal Physics Group.

Gregory W. Swift is a staff- member in theCondensed Matter and Thermal PhysicsGroup, where he has been working on novelheat engines, acoustics, and superfluid helium-3since 1981. He received his B.S. in physics andmathematics from the University, of Nebraskaand his Ph.D. in physics from the University ofCalifornia, Berkeley. From 1983 to 1985 he heldan Oppenheimer Fellowship at Los Alamos.


THE NATURAL HEAT ENGINE

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