The Natural Heat Engine

Home , Carnot cycle, Diesel cycle, Engine, Heat engine, Hot air engine, Rankine cycle

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

eat engines are a compromise reversible in the sense that it can be made limit for the coefficient of performance between the crisp ideals dis- to operate in either of two modes: prime (C. O. P.) of a heat pump (the amount of cussed in thermodynamic mover or heat pump* (Fig. 1). In a prime heat rejected at the higher temperature per H textbooks and the clanking, mover, heat flows from high to low unit of work). Both theoretical limits de- hissing realities of irreversible processes, temperatures, and the engine converts a pend only on the temperatures involved. This compromise produces wonderful ma- portion of that heat to work. In a heat chines, such as the automobile engine and pump, the flows of heat and work are Carnot. The most fundamental engine the household refrigerator. In designing reversed; that is, work done on the engine cycle operating between two temperatures real devices, the goal is not to approach causes it to pump heat from low to high is the functionally and thermodynamically thermodynamic ideals by reducing ir- temperatures. Few practical engines are reversible cycle propounded by Sadi reversibilities but to balance cost, effi- functionally reversible. The internal com- Carnot in 1824. The cycle consists of alter- ciency, size, power, reliability, simplicity, bustion engine is a prime mover only; the nating adiabatic and isothermal steps and other factors important to the needs of household refrigerator is a heat pump (Fig. 2). During an adiabatic step, no heat particular applications. only: neither engine is ever operated in Simplicity is the most striking feature of both modes, remains constant. Thus any flow of work a natural engine, a reciprocating heat en- Figure 1 shows how the first and second causes a corresponding change in the tem- gine with no moving parts. As we will see, laws of thermodynamics place an upper perature of the working medium. During the basic operating cycle of the natural limit on the efficiency of a prime mover- an isothermal step, the temperature re- engine is so straightforward it can be ap- (the fraction of the heat input converted to mains constant, and flows of entropy, plied to a wide variety of systems with work), The efficiency of a thermodynam- work, and heat occur. working media that range from air to ically reversible cycle-that is, one in In the Carnot cycle, the entropy change paramagnetic disks. which all parts of the system are always in of one isothermal step exactly balances the Although the natural engine is new in thermodynamic equilibrium—is equal to entropy change of the other isothermal concept, the underlying thermodynamic that upper limit. (One statement of the step. Over a complete cycle, no entropy is principles and processes are shared with second law of thermodynamics is that all generated. If- an engine could be made to conventional engines, such as the Stirling reversible engines operating between the follow a Carnot cycle, its efficiency would and Rankine engines. To set the stage for same two temperatures have the same effi- equal the theoretical upper limit given in natural engines, we will first discuss a few ciency.) Figure 1 also shows the upper Fig. 1. Although the upper limit applies to conventional idealized thermodynamic any reversible engine, this efficiency is cycles and the practical engines they sug- usually called the Carnot efficiency. gest. *A prime mover is often called an engine and a Building an engine that approximates a heat pump a refrigerator. Here we use the term engine to denote both thermodynamic func- Carnot cycle requires that all processes in Conventional Heat Engines tions, and our use of the term heat pump in- its cycle are carried out very near equi- and Cycles cludes the refrigerator. Strictly speaking, how librium. If not, the resulting ir- ever, the purpose of a heat pump is to reject heat reversibilities due to temperature and at the higher temperature, whereas the purpose In principle, any idealized thermody- of a refrigerator is to extract heat at the lower pressure gradients generate entropy and namic heat engine cycle is functionally temperature. cause a loss of efficiency, For example, the

2 Fall 1986 LOS ALAMOS SCIENCE The release of acoustic energy by a simple natural heat engine, the Hofler tube, made evident by the white plume at the upper end. The device consists of a two-piece copper tube, closed at the bottom, and a short set of fiber glass plates that run parallel to the tube ‘s axis in the region of the flanges. The acoustic energy results spontaneously when a temperature gradient is applied iacross the plates. In this case, the gradient was produced by holding one end of tube tube while immersing the other end (frosted) in liquid nitrogen

LOS ALAMOS SCIENCE Fall 1986 3 The Natural Heat Engine

temperature differences across the heat exchangers that move heat in or out of the engine are frequently a source of irreversibility that greatly cuts efficiency. (See "The Fridge” for a quantitative accounting of this and other losses in a practical heat pump. ) Although one may approach near-equilibrium conditions by designing the engine so as to reduce these gradients, the end result is a very slow cycling of the engine and a very low power output. An important point (originally made by F. L. Curzon and B. Ahlborn and generalized by S. Berry, J. Ross, and their collaborators) is that Carnot-like cycles operating between two temperatures with imperfect heat exchangers have quite different effi- ciencies depending on whether work per cycle or power is being maximized, Real engines, especially high-speed reciprocating engines, cannot approximate Carnot’s cycle closely,

Stirling. The Stirling engine, invented in 1816 by the Reverend Robert Stirling some eighteen years before Carnot’s ideas were published and originally called the hot-air engine, is a reciprocating engine that is functionally reversible and, in principle, thermodynamically reversible. The ideal Stirling cycle has the Carnot efficiency. From a practical standpoint, im- Fig. 1. (a) A heat engine operating as limit for W/Qh, the efficiency of the en- plementing the Stirling cycle suffers from prime mover converts some of the heat gine. Note that a prime mover can only some of the problems of implementing the that is flowing from a hot temperature Th approach its highest efficiency of unity

Carnot cycle. However, the introduction to a cold temperature TC into work. The when T c << Th. (b) In a heat engine of a second thermodynamic medium first law of thermodynamics tells us that operating as heat pump, all flows of provided the means by which high-speed Q h, the heat that passes into the engine heat and work are reversed. Thus work

Stirling engines of good efficiency could be at the hot temperature, equals Q c, the done on the engine causes it to draw built. heat put back into the environment at heat out of the environment at the cold The Stirling cycle (the solid black curve the cold temperature, plus W, the work temperature and place it into the en- in Fig. 3) differs from the Carnot cycle in done by the engine. The second law vironment at the hot temperature. Con- that the adiabatic steps are replaced with tells us that the entropy per cycle gener- sideration here of the first and second steps that are reversible by virtue of being ated by the system must be positive or, laws leads to an upper limit on the coef- locally isothermal, This type of cycle is at best, zero. Since the engine is as- ficient of performance (C.O.P.), Q h/W, achieved by using two thermodynamic sumed to be in a steady state, the en- which is the reciprocal of the efficiency media. The first is the working fluid, tropy change in the environment due to of a prime mover. (For a refrigerator, the which typically can be either a gas or a the heat flow out of the engine, Q c/Tc, is C.O.P. is better defined as the ratio of liquid, (There are Stirling cycles that use greater than or equal to that due to the the heat extracted at the lower tempera- solids, but we do not discuss them here.) heat flow into the engine, Q h /Th . ture to the work done on the machine, continued on page 6 Together, these two laws give an upper that is, Qc/W.)

4 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

The Fridge

he basis for the household refrigerator is the Rankine cycle, which, as T shown in the figure, duplicates a portion of the Carnot cycle in that it has Condenser I one adiabatic step and two isothermal I steps. A key feature of this cycle is a phase change in the working fluid. and the two isothermal steps correspond to condensa- 0\ tion of the fluid at T h and evaporation at T . Also, the engine operates with continu- c I Evaporator I ous flow rather than by reciprocating: the working fluid cycles through its various thermodynamic states by being forced The Rankine cycle, used in the house- steps and, on the compression side, an around a closed loop. hold refrigerator, is based on a liquid- adiabatic step. The two parts of the cy- This cycle has intrinsic irreversibilities gas phase change. The cycle is shown cle (shown in red) that differ from the associated with the free expansion of the here superimposed on the phase dia- Carnot cycle—the cooling of the gas at liquid and the cooling of the gas to the gram for the working fluid; a schematic constant pressure to the condensation temperature at which condensation oc- of the heat pump is also shown. The temperature T h and the free expansion curs. Thus one expects the Rankine cycle Rankine cycle resembles the Carnot cy- of the liquid—are intrinsically ir- to have less than ideal Carnot effi- cle in that there are two isothermal reversible. A ciency-even before accounting for such Table 1 losses as those due to temperature differences at the heat exchangers. Neverthe- Losses in the coefficient of performance (C. O. P.) due to irreversibilities for an air- less. Rankine engines remain the design of to-air heat pump (adapted from Hear Pumps by R. D. Heap, 1983). choice in many applications because they are simple and powerful. Many refriger- Cycle Irreversibilities ators will run thirty years with little or no Carnot none 5 20 19.5 maintenance, and overall cost is low. The Rankine cycle can also be used in Carnot real heat exchangers –5 45 6.4 an air-to-air heat pump. Table 1 illustrates Rankine real heat exchangers. intrinsic –5 45 5.1 the effects of various irreversibilities on irreversibilities the coefficient of performance for such a Rankine real heat exchangers. intrinsic –5 45 4.0 pump—one designed to keep a house at irreversibilities, compressor losses 20°C when outside air is 5°C so that, Rankine real heat exchangers. intrinsic –5 45 3.0 ideally, T – T is 15°C and the Carnot h c irreversibilities, compressor coefficient of performance is 19.5. The losses, miscellaneous largest drop in the the estimated coefficient of performance occurs when ideal heat exchangers are replaced by practical temperature difference. T h – T c of the (due to friction and the imperfect con- heat exchangers—ones both small enough working fluid increases from 15°C to 50°C, version of electrical power to shaft power) to get through the door of a house and causing the coefficient of performance for and miscellaneous losses (such as power to cheap enough to cost less than the house. the Carnot cycle to drop from 19.5 to 6.4. run the fans, the thermostat, and the con- A small, cheap heat exchanger can only The C.O.P. drops to 5.1 when one takes trols). The final C. OP. for a practical, transfer large amounts of heat if a large into account the intrinsic irreversibilities operating Rankine heat pump is 3.0, more temperature difference occurs across it. of the Rankine cycle. Further decreases than a factor of 6 lower than the C.O.P. for The net effect in our example is that the occur because of losses in the compressor an ideal engine. ■

LOS ALAMOS SCIENCE Fall 1986 5 The Natural Heat Engine

continued from page 4 The working fluid is displaced at constant volume through a regenerator containing the second medium, which is typically a solid. The second medium can be metal plates or just the walls of the vessel, but its heat capacity should be large compared to that of the working fluid. A small temperature gradient exists along the length of the regenerator, the total temperature change being the temperature difference between the hot and cold heat exchangers at the ends of the regenerator. If wc ensure good thermal contact between the two thermodynamic media (say by making the distance between any fluid element and its adjacent regenerator plate small enough). the fluid can temporarily store heat in the regenerator and recover it later under nearly reversible isothermal conditions. Of course, the steeper the gradient along the regenerator or the faster the displacement of the working fluid through the heat exchangers and the regenerator, the greater the irreversible losses. During one part of the cycle, fluid enters the cold end of the regenerator, picks up Fig. 2. Temperature-entropy and pres- work continues to be done by the fluid, heat from the second medium, and exits sure-volume diagrams for the prime- the temperature of the medium must hot. During another part of the cycle, fluid mover and heat-pump modes of a drop. The third step is isothermal com- enters the hot end of the regenerator, de- Carnot cycle. When the engine is operat- pression in which heat is rejected from

posits heat in the second medium, and ing as a prime mover, the first part of the the engine to the lower temperature T c exits cold. The net heat stored in the sec- expansion stroke is the addition of heat and the entropy drops. Finally, an

ond medium over a complete cycle is zero to the engine at T h. Because this pro- adiabatic compression raises the tem- (provided, as is the case for an ideal gas, cess is isothermal, the heat energy is perature of the medium. The Carnot cy- the specific heat of the fluid does not used to expand the working medium cle for a heat pump is just the reverse of depend on pressure). The regenerator, and do work on the surroundings. In the that for a prime mover. The area therefore, enables us to change the temper- second step, further expansion occurs enclosed by the pressure-volume dia- ature of the working fluid from the tem- adiabatically, that is, with no addition of grams equals the net work done by or on

perature of the hot reservoir T h to the heat or change in entropy. Because the engine in a full cycle.

temperature of the cold reservoir T c and back again without the adiabatic ex- adiabatic pseudo-Stirling cycle (the dashed perature of the working fluid at the heat pansions and compressions of the Carnot curves in Fig, 3), This confusing nomen- exchangers. An adiabatic compression cycle. In other words, locally isothermal clature is illustrative of the compromises warms the fluid prior to its displacement reversible steps have replaced the made between the concept of’ a thermody- through the hot heat exchanger and into adiabatic reversible steps for changing the namic cycle and the construction of an the regenerator, and. at the other end of temperature of the working fluid. As a operating engine. Unfortunately, because the cycle, an adiabatic expansion cools the result, the efficiency of the Stirling cycle is the same person’s name can become at- liquid prior to its displacement in the op- the same as that of the Carnot cycle. tached to both the cycle and the engine, posite direction. These adiabats partially But what about the Stirling engine? confusion abounds, replace the isotherms of the original cycle. Typically, Stirling engines do not follow a What changes the Stirling cycle to a necessitateing extension of the constant- Stirling cycle but rather follow an pseudo-Stirling cycle is related to the tem- volume displacement steps.

6 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

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 Tc. In the third step, adiabatic constant-volume steps. The adiabatic the cycle depicted here is adiabatic expansion cools the fluid in the left pseudo-Stirling cycle replaces the compression in the cylinder on the right cylinder. Constant-volume displace- isotherms with adiabats and extensions that raises the temperature of the fluid ment of the fluid to the right then causes of the constant-volume displacement above T h. In the second step, both heat Q c to be drawn in at the left heat steps (dashed curves). It is the pseudo- pistons move, displacing the fluid to the exchanger and the heat stored in the

Stirling cycle that frequently serves as left. The heat Q h generated by the com- second medium during step 2 to be re- the basis for practical Stirling engines. pression is rejected in the heat ex- turned to the fluid. Irreversibility occurs (b) One variation (the Rider form) of a changer on the right. Because of the at the beginning of both constant-vol- Stirling engine following the adiabatic small longitudinal temperature gradient ume displacements (dashed red in part pseudo-Stirling cycle. All such engines and good lateral thermal contact along (a)) when the fluid at one temperature are based on the ideal of local the regenerator, heat is transferred be- contacts the heat exchanger at a dif- isothermal steps made possible through tween the two media under essentially ferent temperature.

L0S ALAMOS SCIENCE Fall 1986 7 The Natural Heat Engine

Since the above alterations introduce cumvent the loss of efficiency from ir- intrinsic irreversibilities, the maximum ef- reversibilities at the heat exchangers is to ficiency possible for the pseudo-Stirling generate the heating or cooling effects in- cycle is lower than that for the true Stirling side the engine rather than outside. In cycle. In particular, fluid that has been 1893 Rudolf Diesel envisioned such an warmed by adiabatic compression (and engine and, in fact, intended it to follow a Carnot cycle of adiabats and isotherms. pushed into the hot heat exchanger during His idea was to provide the heat for the the displacement step, where it makes isothermal expansion by burning coal dust thermal contact irreversibly with the ex- that was injected into the engine at just the changer at temperature Th. The same type proper rate to maintain isothermal condi- of irreversibility occurs in the other heat tions. Cooling for the isothermal com- exchanger after the adiabatic expansion pression was to be provided by spraying step. Such effects are departures from the water into the chamber. So far, no one, ideal of locally isothermal conditions. including Diesel, has been able to imple- Although a Stirling engine is not as sim- ment this cycle, and we are once again ple conceptually as a Carnot engine. prac- confronted with confusing nomenclature: tical Stirling engines that operate at mod- the modern Diesel engine does not follow Fig. 4. The Otto (black) and Brayton (red) erately high frequencies can indeed be the Diesel cycle, heat engine cycles, which consist of two built. As before, other irreversible losses The idea of internal combustion, of adiabatic steps that alternate with two occur because there must be significant course, survived, and modern Diesel en- nonadiabtic steps—the latter steps be- temperature differences to drive heat gines work very well indeed. But internal ing the addition or removal of heat at through the heat exchangers. Also, if the combustion introduces new practical ir- constant volume in the Otto cycle and at working fluid is a liquid (see “The Liquid reversiblities, For example, the addition constant pressure in the Brayton cycle. Propylene Engine”), an additional type of and burning of the fuel in a typical piston Only the prime mover mode is shown. irreversibility arises: the specific heat of a engine causes differences between the Note that for both cycles the highest liquid is pressure-dependent, making the pressure and temperature in the cylinder temperature T h equals the temperature recovery of heat in the regenerator im- at the end of the cycle and at the begin- T1 at the upper extreme of the adiabatic perfect. This irreversibility is not an in- ning. A considerable irreversible loss oc- expansion step but that the coldest tem- trinsic feature of the cycle but is a material curs as heat and pressure are vented in the perature T C is lower than the tempera- property that cannot be avoided. As such, exhaust. Thus, internal combustion en- ture T 2 at the lower extreme of the it is of a more fundamental nature than the gine cycles differ from the Carnot and adiabatic expansion step. limitation, say, of the heat exchangers. Stirling heat engine cycles described Phasing of the various moving parts in a earlier in that the working medium is not Otto cycle (black curve in Fig. 4) and the heat engine is another factor necessary to returned to its original state. Brayton cycle (red curve). Each of these its operation. Although the engine de- Nevertheless, the use of phased and cycles, both of which are typically im- picted in Fig. 3 is a heat pump, if the controlled internal combustion eliminates plemented irreversibly, has two adiabatic phasing of the two pistons is altered so that the problem of bringing heat in through a steps and two nonadiabatic steps. In the expansion occurs on the hot-temperature firewall. The diesel and gasoline internal Otto cycle, the nonadiabatic steps are the side when most of the fluid is hot and combustion engines are used today be- addition and removal of heat at constant compression occurs on the low-tempera- cause they are simple, both in principle volume; in the Brayton cycle, these steps ture side when most of the fluid is cold, and in practice, their power density is very are carried out at constant pressure. heat flow will be reversed and the engine high. and their efficiency is relatively If the working fluid is an ideal gas, both will become a prime mover. As we shall good, sometimes very good. Practical see, both phasing and the second thermo- diesel engines approach a level of effi- dynamic medium are of key importance in ciency in which the useful work is nearly natural heat engines also, although there half the heating value of the fuel. are significant differences in the way in which the second medium is used. Otto and Brayton. Two common heat engine cycles that will help illuminate the where y is the ratio of the specific heat at Internal Combustion. One way to cir- characteristics of a natural engine are the constant pressure to that at constant vol-

8 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

ume. What is interesting about this for- a more fundamental role in the natural unaware, was the half-wave resonator of P. mula is that efficiency for these cycles is heat engine. Rather than tolerating ir- Merkli and H. Thomann (Fig. 5b). In this determined by geometry (the ratio V1/V2 reversibilities for the sake of expediency, apparatus, a piston drives pressure fluc- of the volumes at the extremes of the the natural heat engine takes advantage of tuations in air at nearly half-wave reso- adiabatic expansion step) and by a fluid them. For example, heat conduction nance in a simple closed tube. Merkli and across a temperature gradient is central to Thomann observed that the center of the temperatures of the hot and cold re- the operation of a natural heat engine tube cooled, whereas the ends of the tube servoirs. known as the acoustic heat engine. warmed. At first, these results seem Since for an ideal gas the quantity Without this irreversibility, the engine surprising. Naively, one might expect would not work. The result of such an heating everywhere rather than cooling in efficiency can also be expressed in terms of approach is a significant leap in simplicity one region. Further, the cooling occurred the temperatures, T 1 and T 2, at the ex- and, for certain applications, a leap in in the center, which, at a quarter of an tremes of the adiabatic expansion step: power and efficiency. acoustic wavelength, is coincident with a Thus, whereas engines that approx- maximum, or antinode, in acoustic veloc- imate, say, the Stirling cycle are in- ity and thus where one would surely ex- (2) trinsically reversible (though possibly ir- pect a warming due to viscous scrubbing reversible in practice), natural heat en- of the air on the walls. gines are intrinsically irreversible—they The first acoustic heat pump built at The diagrams for the Otto and Brayton cannot work if irreversibilites are Los Alamos used a speaker at one end of a cycles show that in both cycles T1 equals eliminated. Nature abounds with useful closed tube to drive the acoustic resonance

Th but Tc is lower than T2. This difference irreversible processes, so, for the sake of a and has a stack of fiber glass plates posi- is due to further cooling, after the short, appropriate, and easily remembered tioned toward the opposite end (Fig. 5c). adiabatic expansion step, along a name, we call intrinsically irreversible en- The plates constitute a second thermody- nonadiabatic step (removal of heat at con- gines natural engines. namic medium but not a Stirling-like re- stant volume in the Otto cycle and at generator because they are spaced so far constant pressure in the Brayton cycle). If Acoustic Engines. Work in Los Alamos apart that locally isothermal conditions do we now examine the limiting case of zero on natural engines began with an acoustic not prevail. With such an arrangement, it heat transferred during the nonadiabatic heat-pumping engine. Our work, however, is easy to produce a 100-centigrade-degree steps, we see that T2 approaches Tc and the was preceded by two conceptually related temperature difference across a 10-cen- efficiency approaches the Carnot effi- devices, which we will describe without, timeter-long stack of plates in only a min- ciency. Of course, at the same time, the for the moment, explaining their some- ute or so. area enclosed by either cycle, and thus the what surprising behavior. Subsequently, Tom Hofler built a de- work output, shrinks to zero. W. E. Gifford and R. C. Longsworth vice (opening photograph and Fig. 6) to We will find that all of these features of invented what they called a pulse tube show his Ph.D. candidacy committee at the Otto and Brayton cycles have counter- (Fig. 5a). Part of this closed tube was fitted the University of California, San Diego. parts in the natural engine. with a set of Stirling-type regenerator The device, which we call the Hofler tube, plates intended to promote locally consists of a quarter-wave acoustically res- The Natural Heat Engine isothermal processes along their length, onant metal tube closed at one end and a and part of the tube was left empty. Pulses stack of fiber glass plates that run parallel One guiding principle in the develop- were produced at the regenerator end of to the axis of the tube. Short copper strips ment of most heat engine cycles has been the tube by switching between high- and glued at each end of each fiber glass plate to minimize irreversibilities because they low-pressure gas reservoirs at a rapid rate provide heat exchange by making contact generate entropy and decrease efficiency. (1 hertz). The extreme inner end of the with two flanges encircling the tube. In the development of practical engines, regenerator plates got very cold, whereas a If the closed end of the tube is heated, however, irreversibilities are often de- heat exchanger withdrew heat at the empty say by holding it in a warm hand. and its liberately introduced to increase power, end of the tube. The pulse tube demon- open end is cooled by dipping it in liquid decrease maintenance, or simplify design strated the pumping of heat with acoustic nitrogen, the resulting steep temperature and manufacture, enabling one, for exam- energy in the presence of a second thermo- gradient causes the air in the tube to ple, to build small engines, or high-speed dynamic medium. vibrate, and the person holding the tube reciprocating engines, or cheap engines. The other significant precursor to our will feel his or her whole arm begin to On the other hand, irreversibilities play work, and one of which we were initially shake. When the tube is removed from the

LOS ALAMOS SCIENCE Fall 1986 9 The Natural Heat Engine

liquid nitrogen, the sound of the acoustic oscillations is very intense. Peak-to-peak pressure oscillations at the closed end have been found to be as high as 13 per cent of the atmospheric pressure! Thus, the tube operates as a prime mover, and heat is converted to acoustic work. How do this and other acoustic engines work? The Hofler tube is the grandchild of the Sondhauss tube, famous in thermoacoustics and explained qualitatively by Lord Rayleigh over a hundred years ago. Theoretical understanding of these and related devices has been promoted by Nikolaus Rott in a series of papers published over the last fifteen years. The same conceptual foundation can be used to understand quantitatively not only the Hofler tube but the other acoustic devices mentioned above as well. As mentioned before, an important factor in tbe operation of traditional engines is phasing: pistons and valves have to move with correct relative timing for the working medium to be transported through the desired thermodynamic cycle. The natural engine contains no obvious moving parts to perform these functions, yet the acoustic stimulation of heat flow and the generation of acoustic work point to some type of cycling, or timed phasing of thermodynamic processes. The key to phasing in natural engines is the presence of two thermodynamic media. In the Hofler tube, gas was the first medium, the fiber glass plates were the second. Consider a parcel of gas that moves back and forth along the plates at Fig. 5. (a) At the left end of the Gifford driven on the left by a reciprocating the acoustic frequency. As it moves, the and Longsworth pulse tube, pressure piston. Contrary to one’s intuition, the parcel of gas will experience changes in pulses in a gas are generated at 1 hertz center of the tube, where the acoustic temperature. Part of the temperature (Hz) by switching between high-pres- velocity is greatest, cools rather than changes come from adiabatic compression sure (5 bars) and low-pressure (1 bar) warms. (c) The first acoustic heat pump and expansion of the gas by the sound reservoirs. In conjunction with two sec- built at Los Alamos contains a stack of pressure and part as a consequence of the ond thermodynamic media (a Stirling- fiber glass plates and helium gas as the local temperature of the plate itself. The type regenerator and the walls of the working fluid. The quarter-wave heat flow from gas to plate that occurs as a open section of the tube), the pulses acoustic resonance is driven on the left consequence of these temperature dif- cause heat to be pumped from the by a speaker. The stack acts as a sec- ferences does not produce instantaneous middle of the tube to the far right. (b) ond thermodynamic medium but is not a changes. Rather a thermal lag in the heat The half-wave resonator heat pump of Stirling-like regenerator because the flow between the two media creates the Merkli and Thomann is a simple closed wide spacing of the plates does not phasing between temperature and motion tube whose acoustic resonance is promote locally isothermal conditions.

Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

that is needed to drive the engine through is thermodynamic symmetry. But if the broken) changes rapidly and by large a thermodynamic cycle. This is why a lateral interaction changes with the long- amounts, whereas the temperature of natural but irreversible process—heat itudinal coordinate, the symmetry is said other thermocouples further in along the flow across a temperature difference-is to be broken. Where the symmetry is plates changes only by small amounts. In intrinsic to the operation of the engine. broken there is always some thermody- an acoustic natural engine, the heat ex- An interesting contrast exists between namic consequence, such as a change of changers are, of course, located at posi- the Stirling engine and natural engines. In temperature or a beat flow to an external tions where thermodynamic symmetry is the Stirling engine, good thermal contact reservoir. broken. between the working fluid and the second Thermodynamic symmetry can be Before explaining in more detail the medium helps ensure reversible operation broken in a variety of ways. For example, operation of the acoustic heat engine, we and high efficiency, In the acoustic heat in the heat pump depicted in Fig. 5c, it is summarize by pointing out that all natural engine, poor thermal contact is necessary broken geometrically at the longitudinal heat engines possess the following ele- to achieve the proper phasing between ends of the fiber glass plates. In our de- ments: temperature and motion of the working scription of some variable stars as natural fluid. engines, it is broken by changes in opacity ❑ two or more thermodynamic media in One additional condition is important that alter the effective thermal contact be- reciprocating relative motion, to the operation of the acoustic heat tween the stellar matter and the radiation engine: thermodynamic symmetry along field. It can also be broken dynamically ❑ an irreversible process that causes phas- the direction of relative motion must be by, for example, nonlinear localization of- ing of a thermodynamic effect with re- broken. The concept of thermodynamic the acoustic energy in the primary me- spect to the motion, and symmetry is fundamental. jet concep- dium. tually simple. In the natural engine, the The dramatic effects of breaking ther- ❑ broken thermodynamic symmetry two thermodynamic media are undergo- modynamic symmetry can be shown ex- along the direction of relative motion. ing reciprocating relative motion along perimentally by fixing several thermocou- one direction and are interacting thermo- ples to the central plate of a simple The Cycle. Figure 8 displays the cycles of dynamically in a direction transverse, or acoustic heat pump (Fig. 7). When the an acoustic engine serving as prime mover laterally, 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 alongside direction of relative motion, we say there (where thermodynamic symmetry is 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 the acoustic prime mover that consists of plates, the air in the tube vibrates plates and the tube at both ends of the two thermodynamic media—air and strongly. The plates are 1.65 cm long, stack is provided by thin copper strips fiber glass plates—inside a quarter- 0.38 mm thick, and spaced 1 mm apart. that run along the longitudinal edges of wavelength acoustically resonant tube The stack of plates, here seen from the each plate and into the thick encircling closed at one end. If a steep tempera- side, is placed about midway in the copper flanges.

LOS ALAMOS SCIENCE Fall 1986 II The Natural Heat Engine

4 acoustic engine, the oscillations are sinusoidal, producing elliptical cycles. For sim- 3 plicity we consider square-wave. or articulated, motion so that the basic thermodynamic cycle can be pictured as consisting of two reversible adiabatic steps and two irreversible constant-pressure steps, as in the Brayton cycle. 2 Just as in the Stirling engine. relative phasing of motion (steps 1 and 3 in Fig. 8) and heat transfer (steps 2 and 4) de- termines whether the acoustic engine is a prime mover or a heat pump. In the Rider form of a Stirling engine. phasing is ef- fected externally by altering the order in which pistons are moved. In an acoustic engine, however, phasing is a result of the natural time delay in the diffusion of heat between the two thermodynamic media. The sign of the relative phasing. and thus the mode of the natural heat engine. is Fig. 7. The temperature change T – Tinitial symmetry is broken geometrically. determined by the magnitude of the tem- of thermocouples placed in a stack of Much smaller changes occur at the perature gradient along the fiber glass plates of a simple acoustic heat pump middle of the stack (position 2) and rel- plates—a remarkable quality and a shows the effect of symmetry breaking. atively close to the end (position 3) that substantial gain in simplicity. Application of acoustic power to the are a consequence of a weak dynamic During the compressional part of the tube at time zero immediately produces symmetry breaking due to viscosity and acoustic standing wave, the parcel of gas is large changes at the two ends (posi- the nonuniformity of the acoustic pres- both warmed and displaced along the tions 1 and 4) where thermodynamic sure and velocity fields. plates. As a result, two temperatures are important to that parcel: the temperature adiabatic compression, and no heat flows engine (or a small stack of such plates) that of the gas after adiabatic compressional between the gas and the plate. (Because of can be moved to various longitudinal warming and the temperature of the part losses in a real engine, the maximum tem- positions in an acoustical! resonant tube. of the plate next to the gas parcel after perature gradient that can be produced by A speaker at the open end of the tube compression (and displacement). If the drives the acoustic oscillations. temperature of the gas is higher than that and the minimum gradient needed to The material of the plate has a large of the plate, heat will flow from the gas to drive a prime mover is somewhat greater thermal conductance so that no substan- the plate. If the temperature of the gas is tial temperature gradient can build up lower heat flows in the opposite direction along its length, ensuring that the couple from plate to gas. Both heat and work Thermoacoustic Couple. The thermo- operates under a low temperature gradient flows can thus be reversed and the engine acoustic couple is a simple thermoacoustic as a heat pump, As the probe is moved to switched between functions by altering the device. A calculation of the properties of various locations in the standing acoustic size of the temperature gradient. A zero or the thermoacoustic couple demonstrates a wave, it measures a flow of’ heat generated low gradient is the condition for a heat good deal of the physics of natural by its presence by detecting a small tem- pump: a high gradient is the condition for thermoacoustic engines and can be done perature drop across its length. a prime mover. This engine is intrinsically quantitatively from first principles (see Data taken with such a probe (Fig, 9) fit irreversible but, functionally reversible. “The Short Stack”). When suitably a simple sine curve whose period is half The gradient that separates the two calibrated, the device can also be used as a the wavelength of the acoustic standing modes is called the critical temperature probe to measure both acoustically wave. By noting how the sign of the tem- stimulated heat flow and acoustic power. perature difference varies with respect to perature change along the plate just Typically, such a probe is a single short the plate’s location in the sound wave, we matches the temperature change due to thin plate of the type used in an acoustic see that heat always flows in the direction

12 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

Fig. 8. The thermodynamic cycles (top) of the gas parcels in an acoustic heat engine consist of reversible adiabatic steps and irreversible constant-pressure steps (the acoustic mode is here simplified to articulated rather than sinusoidal motion). This cycle is identi- cal to the Brayton cycle. If we follow a parcel of gas as it moves alongside a fiber glass plate, we see that the prime- mover mode (red) occurs when the temperature rise seen by a gas parcel on the adjacent plate due to displacement of the gas along the gradient (xl VT) is larger than the temperature rise of the gas due to adiabatic compression heat-

ing of the gas (T1). The heat-pump mode (black) occurs under the opposite conditions, that is, when the gradient on the plate is zero or low. In the prime-mover

mode, the pressure (p + p 1) during the heat-flow expansion step is larger than the pressure (p) during the heat-flow compression step, so net work is added to the acoustic vibration. All flows are reversed in the heat-pump mode, and work is absorbed from the acoustic — of the closest pressure antinode. This effect is expected from the description of the heat pump in Fig. 8 because a parcel of gas moving in the direction of a pressure antinode is compressionally warmed and will transfer heat to the low-gradient plate; a parcel moving toward a pressure node is cooled by expansion and will draw heat from the plate. This explains the surprising results of the half-wave resonator heat pump of Merkli and Thomann (Fig. 5b). At both the pressure antinodes and the pressure nodes, heat flow in the couple drops to zero. This effect occurs because the pressure and the gas velocity in a resonant acoustic wave are spatially 90 degrees out of phase. Thus, a pressure antinode is also a velocity node, and heat flow drops to zero because there is no displacement of gas. On the other hand. a pressure node has zero heat flow because no compression continued on page 16

LOS ALAMOS SCIENCE Fall 1986 13 The Natural Heat Engine

THE SHORT STACK heat parameter of the fluid. The presence the thermodynamically active area in a plane perpendicular to the longitudinal acoustic motion. The formula shows that

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

o calculate thermodynamic effi- where T m is the mean absolute tempera- flows down the temperature gradient. as ciency for an acoustic heat engine. for a prime mover. we need to know the hydrodynamic cient, p is the mean density, and c is the T m P heat flow and the work flow. A heuristic specific heat at constant pressure. derivation of these two quantities and the The change of entropy for a parcel os- Work Flow resulting efficiency for the particular case cillating in the manner depicted in Fig. 8 of a short stack follow. We then briefly of the main text is just the lateral heat flow Now that we have estimated the heat discuss the effects of viscosity. from the second medium divided by Tm or flow, we need to calculate the work flow, which is given by the work per cycle (the the change in the fluid temperature due to Heat Flow that heat flow. The volume transport rate ume diagram in Fig, 8 of the main text) for that part of the fluid that is thermody- times the rate at which that work occurs Consider a slack of plates in a heat (the angular acoustic frequency w), The engine whose length is short compared to estimate the flow of hydrodynamically the acoustic wavelength and to the dis- to the net work is just tance from the stack to the end of the tube. two quantities times Tm; that is, If that length is short enough. wc can ignore the change in the longitudinal (6) acoustic velocity magnitude u l and the change in the dynamic, or acoustic, pressure magnitude p 1 with respect to long- Eq. 3. V, the total volume of gas that is itudinal distance x (measured from the Now from Fig. 8 we also see that thermodynamically active, is given by end of the acoustically resonant tube). Further, if we ignore the effects of fluid (3) () viscosity, u 1 does not depend on lateral distance from the plates. Next, we can take the lateral distance between plates to be large compared to the thermal penetration along the plate and x1 is the fluid displace- We can now simply put these pieces together and, using Eqs. 1, 3, 6. and 7. transfer in the fluid during a given cycle of is the critical gradient, so write down the work flow as the acoustic wave). Thus, and effects we estimate for a stack of plates will be the same as the a single plate having the same (4) overall perimeter II (measured transverse to the flow).

The adiabatic temperature change T 1 Combining these equations and defin- accompanying the pressure change p1 can ing the temperature gradient ratio parame- be derived from thermodynamics and is the hydrodynamic heat flow as From thermodynamics we know that

(1) (9)

Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

where the quantity y — 1 is what we call the them by assuming that the Prandtl num- work parameter of the fluid, and a is the (13) ber (the square of the ratio of the viscous speed of sound, so we can rewrite the expression for work flow as Thus, in either case, efficiency depends only on geometry and fluid parameters, just as for the Brayton and Otto cycles

discussed in the text. The temperatures T h

and T C do not enter. As the actual temperature gradient ap- (16) acoustic wave. proaches the critical temperature gradient

(Eqs. 5 and 10) have a very similar struc- proaches zero, so that even at the acoustic ture, which is expected since they are angular frequency w the heat transfer rate closely related thermodynamically. The and the power output approach zero, just what is needed to give the Carnot effi- mula for Q, and the work parameter y – 1 ciency in the Brayton and Otto cycles. appears in the formula for W. Both Q and What happens in this engine? We use Eqs.

W are quadratic in the acoustic amplitude 1, 4, and 9 and the fact that u l = x 1w to rewrite the efficiency formula (Eq. 11) in To lowest order, then, the effect of vis- through unity. general as cosity on heat flow is just to decrease Q by a term proportional to the viscous pene- (14) tration depth. This simply means that vis- Efficiency cosity prevents a layer of fluid of thickness

A quantitative evaluation of W and Q acoustically and contributing to the for this case of the short stack but for we have at the critical temperature gra- acoustically stimulated heat transport. sinusoidal p I and u 1 would give the same dient Similarly, the work flow is decreased by a results except each formula has a numerical coefficient of 1/4. Thus the efficiency n (15) the energy lost from the acoustic wave due of a short stack with no viscous or long- to viscous drag on the plate. itudinal conduction losses is For simplicity in Eqs. 16 and 17 we have

though another effect of viscosity is to make the concept of a critical temperature gradient less well defined. In fact, with viscosity present there is a lower critical gradient below which the engine pumps heat and a higher critical gradient above For our standing acoustic wave, u 1= which the engine is a prime mover. Be- is the distance of the stack from the end of tween these two gradients the engine is in a the tube. Then the efficiency can be rewrit- useless state. using work to pump heat ten simply as from hot to cold. What About Viscosity? The Prandtl number for helium gas is about 0.67, so that viscous effects are very So far we have assumed that the work- significant for our gas acoustic engines ing fluid is inviscid. What if it is not? We (and, in fact, Eqs. 16 and 17 are rather know how to do the theory quantitatively poor approximations). On the other hand. for this more general case, but the resulting the Prandtl number for liquid sodium is expressions for Q and W are terribly com- about 0.004, so that viscous effects are efficiency is simply plicated and opaque. We can simplify much smaller. ■

LOS ALAMOS SCIENCE Fall 1986 15 The Natural Heat Engine

continued from page 13 or expansion takes place. In other words, acoustic heat flow depends on both the acoustic pressure and the fluid velocity. Figure 9 also illustrates the dramatic effect the positioning of a plate in the A acoustic wave has on the operation of a natural heat engine, A plate or stack of plates placed completely within a quarter of a wavelength of the end of the tube operates in the manner depicted in Fig. 8. If that same stack is repositioned in the second quarter of a wavelength, the pic- I I I > torial analysis of Fig. 8 still applies, but the directions of all heat flows and longitudinal temperature gradients are reversed. A stack that extends beyond an adjacent node-antinode pair, however. has heat flows that counter each other, cancel- ing part of the overall transport of heat h from one end of the plates to the other, Also important is the stack’s position within a given node-antinode pair sepa- rated by a quarter of an acoustic wavelength (Fig. 10). For an engine in the heat-pump mode, a stack close to a pressure antinode—say, the end of the tube—can develop steep temperature gradients. Why? In such a region the acoustic pressure change in a parcel of gas is large Acoustic Standing Wave and thus the rise in temperature from compressional warming is large. This re- of Heat gion is also near a velocity node, so displacement of the gas parcel is small. Large temperature changes over small displacements, of course, result in large tempera- which bounds the region between the heat- pump and prime-mover modes, is large close to a pressure antipode.) characterize an engine. For example, a As a plate or stack of plates is moved stack close to a pressure antinode is close set of plates is a tradeoff between viscous away from the pressure anti node, the tem- to a velocity node, and viscous losses will losses, losses from longitudinal conduc- perature gradient developed becomes be small at that position. However, be- tion, the desired temperature span across smaller. At a quarter of a wavelength, no cause temperature gradients are steep the engine. and power output. there, losses from ordinary diffusive positioning effect is important in the de- thermal conduction in the plates and Heat and Work. We can now better sign of a refrigerator, because, together working fluid will be increased. The prob- understand the natural acoustic engine by with the length of the plates, it places an lem of ordinary conduction losses is examining what happens near a short plate upper limit on the maximum temperature especially critical for an engine acting as positioned between a node and an anti- drop possible across the stack. prime mover because such an engine node (Fig. 11). If the displacement of a Positioning also affects the losses that needs a temperature gradient higher than given parcel of gas is small with respect to

16 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

the length of the plate, there will be an namic symmetry is broken. Parcels of gas measured across a thermoacoustic entire train of adjacent gas parcels, each that move farther from the end of the plate couple as a function of the plate’s posi- confined in its cyclic motion to a short than a thermal penetration depth idle tion in the acoustic standing wave. Note region of length xl and each reaching the through part of their cycle without accept- that heat flows toward the closest pres- same extreme position as that occupied by ing or rejecting heat. For example. if a sure antinode, making that end of the an adjacent parcel half a cycle earlier. parcel of gas at the end closest to the couple hottest. However, at both the What is the net result of all these individ- antinodc is in equilibrium with the plate pressure antinodes and nodes there is ual cycles on the flow of heat and work? on one half of the cycle but then moves out If the motion of the parcels is sinus- of the range of thermal interaction, it has were obtained for an acoustic oidal, only those about a thermal penetra- nowhere to deposit the heat resulting from tion depth* from the nearest plate arc its adiabatic warming. AS this parcel com- thermoacoustically effective. Parcels close pletes its cycle, it cools adiabatically back to a plate transfer heat to and from the to the temperature of the plate. The heat POSITIONING EFFECTS plate in a locally isothermal and reversible transferred to the plate from the next adja- manner, just like the fluid in the re- cent parcel down the line is un- generator of a Stirling engine. Parcels far compensated, so there is a net heat trans- away have no thermal contact and arc fer to the plate on that end, and the tem- simply compressed and expanded perature of the plate increases there. In adiabatically and reversibly by the sound similar fashion. heat drawn from the end wave. However, parcels that are at about a closest to the node is not replaced. and that thermal penetration depth from a plate end cools. We can take advantage of the have good enough thermal contact to ex- net effect-a flow of heat from one end to change some heat with the plate but, at the the other—by bringing the ends of the same time. are in poor enough contact to plates into contact with heat exchangers. produce a time lag between motion and During each cycle an individual parcel heat transfer. of gas transports heat Q across only a small During the first part of the cycle for the temperature interval along the plate that is heat-pump mode, the individual parcels comparable to the adiabatic temperature

will each move a distance x 1 toward the change T 1. However, because there are Fig. 10. Close to a pressure antinode a pressure antinode and deposit an amount many parcels in series, the heat Q is shut- typical parcel of gas experiences large of heat Q at that position on the plate. tled down the stack, thereby traversing the changes in pressure p 1 and thus large During the second half of the cycle, each temperature interval T h – T c, which can changes in temperature 11 due to com- parcel moves back to its starling position be much larger than T 1. Within the limits pressional heating. At the same time, and picks up the same amount of heat Q of a quarter of a wavelength, the flow of displacement xl of the parcel is small, from the plate, But this heat was deposited heat is not a strong function of plate length there a half cycle earlier by an adjacent (in fact, for a stack much shorter than a since x1 is proportional to the distance x parcel of gas, In effect, an amount of heat quarter of a wavelength, heat flow does not Q is merely passed along the plate from depend on plate length at all). In the heat pump mode, the maximum one parcel of gas to the next in the direc- If, on the other hand, we examine this temperature gradient that can be de- tion of the pressure antinode. Thus, as in train of gas parcels with respect to the flow the Stirling engine, the second thermody- of work, we realize that each parcel has a flow between gas parcel and plate namic medium is used for the temporary net effect. For example, a parcel of gas near stops when that gradient is reached), storage of heat. the plates in an engine operating in the which means that close to a pressure At the ends of the plates, the thermody heat-pump mode absorbs net work be- antinode we can expect large tempera- cause its expansion is at a lower pressure ture gradients. Further from the pres- than the corresponding compression. But sure antinode, pressure and tempera- since the same is true for every parcel in ture changes become smaller whereas teristic length describing heat diffusion through the train. the total work done on the gas is displacements become larger, so the the gas during one period of the acoustic cycle. roughly proportional to plate length (for a maximum temperature gradient that can thermal diffusivity of the gas and f is the fre- very short stack, work flow is proportional be developed is smaller. A quency of the sound. to plate lenght).

LOS ALAMOS SCIENCE Fall 1986 17 The Natural Heat Engine

Efficiency. A calculation of heat and work flows for an acoustic heat engine with a short stack close to the end of the resonator tube and no viscous losses (see “The Short Stack”) yields a limiting efficiency 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 Q thought to have a long train of adjacent the other extreme. However, idling is passed from one end of the plate to gas parcels, all about a thermal penetra- parcels at both ends oscillate without the other. Adjacent work contributions tion depth from the plate, that draw heat removing or depositing heat. Adjacent do not cancel, so that each parcel of gas from 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 The Natural Heat Engine

netic engine is based on the adiabatic gradient parallel to the direction of rela- change of temperature with magnetization. tive motion between the two media. The primary medium of our hypotheti- The device is positioned between the cal apparatus (Fig. 12) consists of a stack poles of a permanent magnet in such a way of magnetic disks.* Each disk has a high that the disks of the primary medium are internal thermal conductance. but each is in the nonuniform fringing field at the also thermally insulated from the others so side. The disks arc linked mechanically to that a large temperature gradient can be an external mechanism so that they can be sustained in the longitudinal direction. moved in a reciprocating fashion. A gas The collection of disks is placed in a fills the small annular space around the tube whose walls constitute the second magnetic disks. providing lateral thermal medium. Like the first medium, the sec- contact with the second mediurn, but this ond has a high lateral thermal conduc- contact is poor enough to create the tance. a large heat capacity, and a low, necessary phasing for the engine. There is longitudinal thermal conductance. Thus, also some means for heat exchange with it, too, can sustain a large temperature external reservoirs at each end of the second medium. If we follow an element of the first thermodynamic medium in a magnetic engine through an articulated cycle (Fig. 13), we see that the various steps arc analogous to those of an acoustic heat engine. For example, in the first step of a heat-pump cycle, the clement is meted cooler’s driver and hot heat exchanger. quickly and adiabatically to a region of higher magnetic field. As a result. its temperature rises. (Temperature changes of a few degrees per tesla are typical for fer- romagnetic and strongly paramagnetic materials. ) In the second step, the element thermally relaxes, its temperature adjusting to that of the adjacent region in the second medium, which, in the heat-pump mode. means that heat flows from the first medium to the second. As the element cools. its magnetization increases. The third step is motion back to a region of lower field; the fourth is another thermal relaxation. As in the case of acoustic engines, the phasing between motion and heat transfer is a result of the natural time delay caused by diffusion of heat between the two media. Fig. 12. A hypothetical natural magnetic surrounding the disks makes thermal engine in which the primary medium contact with the second medium (the Some Applications of consists of magnetic disks placed in the walls of the tube), but conductivity of the Natural Engines fringing field at the side of a permanent gas is poor enough to create the magnet. This placement allows an ex- necessary phasing for the engine. In What happens now if these ideals of’ ternal mechanism to displace the disks both media, thermal conductance is natural engines arc put into practice? in a reciprocating fashion in the pres- poor longitudinally so that large temper- What are the clanking, hissing realities of ence of a magnetic field gradient. A gas ature gradients can be supported. A real natural engines?

LOS ALAMOS SCIENCE Fall 1986 19 The Natural Heat Engine

Cryocooler. As part of his Ph.D. thesis, placements of the parcels of gas. Tom Hofler designed and built a device Such a configuration means the re- called the cryocooler (Fig. 14) in which the mainder of the resonator tube can beat the numerical aspects of the design were based cold temperature, allowing it to be just a on the general thermoacoustic theory of thermally insulated straight tube roughly Rott for ideal gases. half an acoustic wavelength long. How- The cryocooler is an acoustic cooling ever, losses due to the dynamical effects of device with a number of important fea- viscosity and thermal conduction along tures. Perhaps the most important is the the walls of the resonator reduce the ex- fact that the acoustic resonance is driven ternally available refrigeration. Roughly from the hot end of the stack. All the early half this loss could be eliminated by using cooling engines were arranged with the a quarter-wavelength resonator with one stack near the closed end of the acoustic end open, but an open end eliminates the resonator tube and with the acoustic use, say, of several atmospheres of helium driver at the opposite end. Very large tem- as the working fluid and revives the perature differences (about 100 centigrade original heat load problem of acoustic degrees) could be easily induced across the streaming—here between the driver and entire stack this way, but the cold end was the atmosphere. Moreover, an open end is seldom less than 20 degrees below ambient downright noisy, radiating useful work out temperature. The problem was that the into the room. driver (at ambient temperature) and the The simple solution is to replace ap- cold end of the stack maintained mod- proximately half the half-wavelength res- erately good thermal contact with one an- onator with a closed container of substan- other by means of acoustic streaming. This tial volume, Dynamic pressure will be phenomenon is a second-order, steady small in a region of large volume. making circulatory flow of the working gas that is the losses correspondingly small. The res- superimposed on the oscillatory motion, The effect of the acoustic streaming was to Fig. 13. The cycle shown here for the use up a substantial amount of the refriger- heat-pump mode of a hypothetical natu- ation available at the cold end of the stack ral magnetic engine is analogous to the trying to cool down the driver. cycle for the heat-pump mode of the Now while it is necessary for work to acoustic heat engine (Fig. 8) with mag- flow into the stack to pump heat, we re- netic field strength H taking the role of alized that it is of no real importance pressure and magnetization m taking whether that flow occurs at the cold or the the role of volume. Thus, the cycle con- hot end. Putting the driver at the hot, or sists of reversible adiabatic steps and closed end, means that none of the avail- irreversible constant-field steps. For an able refrigeration is used to cool the driver. Thus, with the “closed” end replaced by a Thus, in the first step, when the disk movable piston acting at high dynamic moves adiabatically to a region of pressure and low displacement, perform- higher magnetic field, magnetization re- ance is improved, mains constant and temperature rises The stack with its heat exchangers was with increasing H. In the second step, placed rather close to the driver piston, heat flows to the lower temperature of that is, rather close to a pressure anti node. the second medium, causing the disk to As noted earlier (see Fig. 10). such a region cool at constant H and the magnetiza- has the high critical temperature gradient tion to increase. The net result of all four needed for a heat pump. Of course, some steps is the transport of heat up the separation between driver and stack is gradient as a result of the work (which necessary because acoustically stimulated will equal m lH 1) needed to move the heat transfer is proportional to the dis- disk through its cycle. ➤

20 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

onator below the stack was modified faster than viscous losses go up, and there further by decreasing the diameter of the is a net decrease in the overall losses. confining tube and shortening its length. These surprising qualities explain the gen- This last modification, at first sight, would eral shape and configuration of the appear to be of negative value as one cyrocooler. would expect viscous losses to go up; but, Performance is rather good (Fig. 15). As for small decreases in neck diameter, dy- the relative dynamic pressure amplitude namic thermal-conduction losses go down increases, the temperature difference that can be pumped up for zero external heat load increases, eventually topping out at Fig. 14. The driver in the acoustic about — 100°C when the acoustic pressure cyrocooler is an ultralight aluminum amplitude is about 2 or 3 per cent of the cone attached to the voice coil of a mean pressure. At that point, the commercial loudspeaker. The second cryocooler can handle a significant re- thermodynamic medium, rather than be- frigeration load and still maintain a rather ing a set of parallel plates, consists of a low temperature. This type of refrigeration sheet of Kapton rolled about a vertical capability is very suitable for cooling in- rod and spaced with 15-roil nylon fishing struments and sensors. line aligned vertically. Copper heat exchangers are attached at both ends. A Heat-Driven Acoustic Cooler. Natu- The form of the bulb and neck, including ral acoustic engines are functionally re- the constriction, were chosen to reduce versible: they can be either prime movers viscous and thermal losses by reducing that use heat to produce sound or heat surface area. The device is drawn to pumps that use sound to refrigerate. Why not combine these two functions in one device and usc heat to cool? Such an engine would have heat flow through the Fig. 15. Experimental data for the walls but no external flow of work. cryocooler of Fig. 14, obtained with a A key problem in the design of a heat- mean helium pressure of about 10 bars driven acoustic cooler is where to position and acoustic frequencies in the range of the two sets of plates—one set acting as 540 to 590 Hz. For thermal isolation the prime mover, the other acting as refriger- engine was placed in an evacuated ves- ator. Ideally, the refrigerator plates should sel and surrounded by superinsulation. be positioned as they are in the cryocooler. The frequency was adjusted elec- that is, close to the end of the tube where tronically so the dynamic pressure and the velocity of the gas and the viscous velocity were always in phase at the driver. Part (a) shows how the tempera- it would also be good to keep viscous ture difference between the hot heat losses low for the prime-mover plates, it is exchanger at approximately 26°C and more important to have these plates near a the cold heat exchanger increases with relative dynamic pressure amplitude enough for the stack to develop adequate (the ratio of the acoustic pressure power. These considerations imply that

amplitude pI at the pressure antinode to the refrigerator stack needs to be closer to the mean pressure p). No heat load was the end of the tube than the prime-mover applied to the cold heat exchanger. Part stack. However. such a configuration (b) shows how, for a relative dynamic would put the hottest region (the hot end pressure amplitude of 0.03, the temper- of the prime mover) next to the coldest ature difference gradually drops with region (the cold end of the refrigerator), increasing refrigeration load at the cold creating a difficult thermal-design problem.

LOS ALAMOS SCIENCE Fall 1986 21 The Natural Heat Engine

Fig. 16. The upper set of plates in this cooler is a prime mover that draws heat from a heater (at about Th = 390°C) and rejects waste heat to cooling coils (at T a = 23°C or room temperature), generat- ing acoustic work. The lower engine uses that work to reject heat to the cooling coils (at T a and to draw heat from an even lower temperature (TC = 0°C or the ice point). The acoustic tube is about half a meter in length, terminates in a 2-liter bulb, and contains helium at a pressure of 3 bars that res- onates at a frequency of 585 Hz. Both sets of plates are made of 10-mil (0.025 cm) stainless steel, and the spacing between plates in both sets is 0.08 cm. The hot heat exchanger is made of nickel strips; the ambient and cold heat exchangers of copper. ➤

To avoid large heat inputs to the cooler, constant, the positions of’ the two stacks can be fortunately, this type of change increases reversed and the various temperatures ar- the heat loss due to conduction down the ranged in decreasing order along the tube. stack. What now becomes paramount is for conditions to be such that the prime mover is able to adequately drive the cooler. away from the pressure anti node at the Amplification of acoustic fluctuations oc- end of the tube. Because of the intervening prime-mover stack, the refrigerator stack Bob Oziemski adjusting the flow of the is already much farther from the pressure working fluid in the liquid propylene shorten the prime-mover stack but keep antinode than in the cryocooler, and addi- Stirling engine. A the temperature difference across the stack tional movement of the prime-mover

22 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

Fig. 17. For these measurements on the BEER COOLER DATA beer cooler, the refrigerator stack and the resonator were located in an evacu- 400 (a) o ated space for thermal insulation. Part ■ ■ ■ ■ .m ■ ■ (a) shows how the cooling temperature difference (black) across the refrigerator stack and the driving temperature +10 300 difference (red) across the prime-mover t 1 stack vary with the level of oscillation (here given by the square of the relative dynamic pressure amplitude) when no external refrigerator load is placed on the cooler. The level of oscillation is determined by the rate at which heat is supplied and removed across the prime-mover stack, but, as can be seen, this results in little change in the driving temperature difference. At the same time, the cold temperature drops gradually below the freezing point of water. (b) For a level of oscillation of about 4 X -3 10 (that is, p 1 is about 0.19 bar), we see that the beer cooler can handle a small heat load of 10 watts and still

refrigerator stack, thus reducing viscous and acoustic-streaming losses.

■ In our model engine the plate material in the stacks (stainless steel) and the spacing of the plates were dictated by ease of fabrication as much as by anything else. The refrigerator assembly was placed in an evacuated container but not otherwise thermally insulated. The working fluid was helium, whose pressure was chosen experimentally to minimize the cold temperature. A major problem, yet to be satis- stack pushes the refrigerator stack even produce cryogenic temperatures, the factorily solved, was efficient exchange of more from its optimum position. Once cooler ought to be able to produce heat at the heat exchangers—especially the again, changing an idea into a practical temperatures low enough to cool a can of hot exchanger made of nickel. heat engine entails a set of compromises. beer. For this reason we have affec- In spite of the engine’s compromises, it A schematic of an operational heat- tionately dubbed the engine the "beer still sings along. performing rather well driven cooler in which the prime-mover cooler." (Fig. 17). As the heat supplied to the hot stack is between the end of the tube and As in the case of the cryocooler, the end of the prime mover is increased, the the refrigerator stack is shown in Fig. 16. rather complex design was carried out nu- level of oscillation increases—the largest Because of the above consider- merically, and many of the features impor- peak-to-peak dynamic pressure amplitude ations—especially those related to tant to the cryocoolcr apply to the beer measured at the ambient exchangers ex- cooler. For example, the resonator is ceeding a tenth of the mean pressure. In cannot be expected to cool much below similar to the resonator in the cryocooler, agreement with our understanding, the ambient temperature. Although unable to and the driver is on the “hot” side of the temperature drop across the prime-mover

LOS ALAMOS SCIENCE Fall 1986 23 The Natural Heat Engine

stack does not change much as the dy- from then Associate Director Kaye namic pressure amplitude increases; the Lathrop of the need in space for a reliable, small changes seen in the data result from moderately efficient electrical generator. it the diffusive flow of heat across the gaps of was not difficult for us to propose a natural gas and through the heat exchangers from acoustic engine based on liquid sodium, the heat source to the ambient heat ex- Especially ideal for this application is the changer. Figure 17b shows that the beer high “cold” temperature (at least 400 cooler can manage a 10-watt cooling load kelvins) of the liquid sodium engine. This

while keeping T C 5 centigrade degrees fact is important because the cold sink for below the freezing point of water-a any heat engine in space must ultimately rather encouraging result for the first 1abo- be a black-body radiator whose size would ratory model. A number of issues concerning the prac- Liquid sodium has many potential ad- tical use of this engine concept and of the vantages as a working substance in a natu- cryocooler remain to be resolved. It is ral engine. The heat and work parameters likely that the most important is the mat- are acceptably large. For example, at ter of heat exchange, This problem. as 700°C, which is roughly in the middle of we’ve mentioned, has always been a key the temperature range of a possible high- one in the development of heat en- power engine, liquid sodium has a very gines—classical or otherwise. high expansion coefficient and a large

The Liquid Sodium Acoustic Engine. and y — 1 = 0.43 (compared to a As man moves from Earth into space, so monatomic gas such as helium. for which does his need for reliable power. However, differences in the requirements and in the For a given Mach number,* the power operating environment in space may density in the stack is proportional to pa3, prompt radical changes in the engines that where p is the density of the working fluid provide such power. An idea stimulated and a is the speed of sound. The density of by such differences is the liquid sodium liquid sodium is about 500 times greater acoustic engine, which not only is a natu- than that of helium at the pressures used in ral. rather than a conventional, engine but our gas acoustic engines, and the speed of uses a liquid instead of a gas as its working sound is more than a factor of 2 greater. fluid. Thus, the power density for a liquid so- The concept of using a liquid can be dium acoustic engine should be more than traced to a 1931 paper by J. F. J. Malone in 103 times greater than for a helium which he pointed out that certain liquids acoustic engine, a definite advantage. have important thermodynamic qualities This dramatic increase is not without its that make them suitable for use in heat drawbacks, however. The heat capacity engines, Although concerned about its per unit area for the sodium within a chemical reactivity, Malone knew that thermal penetration depth of the second liquid sodium was one of these “good liquids,” but materials technology was *The Mach number is the ratio of the fluid speed then inadequate for him to consider its to the speed of sound in the fluid. use. Today’s materials technology suggests Fig. 18. The temperature drop applied revival of these ideas, and we had been across both stacks of molybdenum working on the liquid propylene Stirling plates causes the liquid sodium in this engine (see “The Liquid Propylene En- proposed engine to oscillate back and gine”) as a modern example of an in- forth between the poles of the magnet. novative but more conventional engine A magnetohydrodynamic effect is used that uses liquids. Thus, when we learned to convert acoustic to electric energy. ➤

24 Fall 1986 LOS ALAMOS SCIENCE zThe Natural Heat Engine

tremely low (about 0.004 for sodium at liability of the engine. Figure 18 is a 700°C compared to 0.667 for helium gas), schematic of a possible liquid sodium The reason for such a low Prandtl number prime mover that uses a half-wavelength is that liquid sodium is a metal. As a resonant tube, two driving stacks (one on result. its kinematic viscosity, is rather nor- each side of the magnet), and magnetohy- mal for a liquid, but, owing to electronic drodynamic power coupling. contributions to the conduction of heat, its To design a model liquid sodium en- thermal diffusivity is high. The conse- gine, we constructed a thermoacoustic the- quences arc important. In helium, viscous ory for liquids and then evaluated it nu- shear extends into the gas from a bound- merically. The calculated characteristics of ary about as far as the temperature gra- a reasonably designed engine arc given in dients that drive the flow of heat. This Table 1. Note that the dynamic pressure shear drains energy, decreasing efficiency almost equals the mean pressure of the and making it difficult for a gaseous heat sodium and that efficiency is calculated to engine to work. The low Prandtl number be about 18 per cent (31 per cent of the of liquid sodium means that heat can be Carnot efficiency). transported between working fluid and the A complete engine has not yet been plates for a volume fifteen times larger built, but work (supported by the Division than the volume being affected by vis- of Advanced Energy Projects i n cosity, and viscous losses arc correspond- DOE/BES) has been done separately on Chris Espinoza welding heat exchange ingly small. Again. however, a price must the magnetohydrodynamics and the manifolds onto the resonator tube of the be paid: diffusive heat conduction in the thermoacoustics. In both cases prelimi- liquid sodium natural heat engine. sodium down the stack increases. nary results arc encouraging. though tech- The fact that liquid sodium is a metal nical problems remain. medium is so large that the usual assump- has yet another important consequence. First, a magnetohydrodynamic con- tion of infinite heat capacity of the second Electrical current can be generated from verter was built that consisted essentially medium is not valid. As a consequence, the sound via magnetohydrodynamic of a liquid sodium acoustic resonator with the power density drops. Moreover, the coupling. Such coupling means electric a central rectangular channel for guiding acoustic impedance pa of the sodium is power can be produced from heat without the sodium in the transverse direction be- relatively high—roughly equal to that of using moving parts (ignoring the non-neg- tween the poles of a magnet. Electrodes for solids—which means that in a sodium ligible motion of the vessel containing the picking up the electric current were at- engine motion of the stack and container sodium!). This feature, of course, is one of tached to the channel. The device was can be expected to send heavy vibrations the main reasons for the expected re- tested by exciting an acoustic standing throughout the entire engine (unlike the beer cooler, for example, in which a peak- Table 1 to-peak dynamic pressure oscillation of 10 Characteristics of a reasonably designed liquid sodium prime mover. per cent of the mean pressure produces only a pleasantly audible tone in the Frequency 1000 Hz room). To counter this effect, stiff, high- Hot temperature 1000 K density materials like molybdenum or Cold temperature 400 K tungsten need to be used in the stack, and Mean pressure 200 bars the walls of the resonator need to be made Dynamic pressure 198 bars of heavy stainless steel. Even with such Plate spacing 0.0373 cm strong walls, high Mach numbers cannot Plate thickness 0.0280 cm be achieved because the high acoustic Distance of hot end from tube end 8.65 cm pressures would burst the resonator. Length of stack 8.0 cm 2 Liquid sodium has other very desirable Average Qh 300 W/em features. For example, its Prandtl number, Average W 55.1 W/cm2 which can be thought of as the square of 0.184 the ratio of the viscous penetration depth 0.307 to the thermal penetration depth, is ex-

LOS ALAMOS SCIENCE Fall 1986 25 The Natural Heat Engine

MAGNETOHYDRODYNAMIC COUPLING EFFICIENCY

wave (by temporarily putting electric power into the magnetohydrodynamic converter!) and then letting the energy ● stored in the acoustic resonance flow A through the converter into a resistive load : across the electrodes. The efficiency-defined as the ratio of the measured electric energy delivered to the load to the 1 10 100 calculated stored acoustic energy-is al- External Resistive Load (ohms) ready quite high in this first prototype (Fig. 19) and a number of improvements Fig. 19. These initial data demonstrate holding the liquid sodium is 1.2 cm thick are possible. From a technological point of the efficiency with which acoustic in the direction of the magnetic field, 7.6 view, it is very significant that the max- energy in liquid sodium was converted cm thick in the direction of electric cur- imum efficiency is still reasonable in a to electric energy via magnetohydrody- rent flow, and 31 cm long; however, only magnetic field of only 0.9 tesla, suggesting namic coupling as a function of the re- 20 cm of that length is actually in con- that a permanent magnet is appropriate sistence of an external load and for tact with the electrodes. The central with a consequent simplification and de- three different magnetic field strengths. channel is part of a l-m-long acoustic crease in weight. In the apparatus used to obtain this resonator filled with liquid sodium at a The thermoacoustic prime mover data, the central rectangular channel tested had a single stack of molybdenum plates (Fig. 20) inside a straight half- wavelength tube. For this test the cold heat exchanger was filled with pressurized water at 125°C and the hot heat exchanger with heated sodium at various temperatures ranging from 440°C to 645°C. Al- though the test was preliminary, it was successful. The application of various temperature drops across the stack re- sulted in the data of Fig. 21 and, above a 350°C drop, in an obvious acoustic vibration of the entire assembly. We obtained the rate of heat supplied to the engine Q h by monitoring the flow rate and the inlet and outlet temperatures of the sodium flowing through the hot heat exchanger. For a low temperature drop (AT) across the stack, the heat flow through the engine is due solely to the simple conduction of heat by the sodium, molybdenum, and stainless steel. How- to increase dramatically>; above the value for simple conduction. This result agrees with the fact that acoustic oscillations at

520°C the resonator was oscillating at high enough amplitude that the sound in the room was unpleasantly loud and the The magnetohydrodynamic converter used to test power coupling for the liquid apparatus was vibrating strongly. sodium heat engine.

26 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

Heat Exchanger Tubes

rather than 350°C. We also expected a

520°C, whereas the measured value was 2600 watts. We do not yet understand these quantitative disagreements but arc extremely encouraged by the initial suc- cess of the engine.

Molecular Natural Engines. Heat engines of any sort transform energy between the random thermal motion of atoms and the coherent motion needed for useful work. The concepts of heat and temperature—implicit to the understanding of heat engines—arc statistical in nature. LIQUID SODIUM PRIME MOVER PARTS Hence, for these variables to be well defined, a system must have large numbers of atoms, But what is the smallest system Fig. 20. Our first operating liquid sodium the resonator tube on the right) were that will still allow us to apply these con- prime mover has a single stack of made from stainless steel hypodermic cepts? molybdenum plates (left) that were fab- needles. Hot liquid sodium at various If we take the error in statistical quan- ricated at Los Alamos from a solid rod temperatures (Th is circulated through tities in thermodynamics to be approx- using electric discharge machining. one heat exchanger and pressurized imately the reciprocal of’the square root of

Plate thickness is 0.3 mm; spacing be- hot water (TC = 125°C) through the other. the number of degrees of freedom and if tween plates is 0.38 mm; the length of The stack just fits inside the half- we assume for a heat engine that errors of a the stack is only 5.2 cm so that the wavelength resonator tube, which has a few per cent are tolerable, then only sev- engine would oscillate at a reasonably length of 106 cm. The plates are posi- eral hundred to a few thousand atoms are low AT. The transverse tubes of the two tioned in the tube at about x = L/14 from sufficient. Rather nice systems of this heat exchangers (one set can be seen the end. The acoustic resonant fre- mesoscale size, consisting of large organic at the end of the cylindrical section of quency is 906 Hz. molecules. are commen. Furthermore, such systems behave much like a purely Fig. 21. The data shown here (dots) rep- classical collection of’ masses and springs.

resent heat flow Q h into the liquid so- Within such systems, nonlinear inter- dium prime mover at the hot heat ex- molecular potentials give rise to changer as a function of the tempera- phenomena directly related to the thermal expansion needed for heat engines. In- Acoustically . the solid curve represents the calcu- tramolecular and intermolecular interac- Stimulated Heat Flow lated flow of heat due to conduction with tions provide connections between regions of irrational energy that, if large enough, heat flow is due to normal conduction can be considered to be heat reservoirs. In across the stack. At high AT, however, other words, all the ingredients for an the sharp rise in Q is indicative of engine are present. ● h

● Do such engines then exist? And, if so, ● do they serve a useful function in nature, Normal Conduction say perhaps as tiny engines in biochemical There arc some disagreements between systems? Will the concepts of natural en- the experimental results and our theoreti- gines apply not only to these small sizes cal calculations. A calculation for the but also to the high frequencies associated particular geometry and acoustic fre- with molecular vibrations?) quency of the device predicts lhat it Heat pumping might occur in mesoscale systems if coherent vibratory motion

LOS ALAMOS SCIENCE Fall 1986 27 The Natural Heat Engine

Fig. 22. Localization of acoustic energy was studied by coupling twelve nonlinear Helmholtz resonators together in a ring and measuring the direction of heat flow with the thermoacoustic couples positioned in the coupling tubes and measuring the level of vibration in individual resonators with the pressure sensors. The construction of the neck of each resonator (see details below the dodecagon) introduces a nonlinearity because vibrating gas that rushes through the neck causes the Kapton to flex, altering the resonant frequency of that resonator. The entire system is driven by a loudspeaker at the center. ➤ can first be established and then survive long enough to have a significant effect. Also, if the concepts of temperature and temperature gradients are to be useful, then the mean free paths of the heat-carrying excitation should be small compared to the classical thermal penetration depth and to the size of the mesoscale object. Using an angular frequency of 1011 Hz, we estimate the penetration depth to be about 14 angstroms. Hence, mesoscale objects perhaps 50 to 100 angstroms in size and vibrating at frequencies of order 1010 Hz might be large enough and slow enough to be natural engines, providing their level of coherent excitation is high enough. Rather than building an object of- such small size. the same effects may be realized in a natural way via a concept from nonlinear science—the solitary wave. An acoustic heat engine with its stack of plates centered at a pressure anti node will pump heat from both ends of the stack toward the middle. If we alter this idea by using a continuous stack that has, owing to dis- persive and nonlinear effects, a localized, or solitary, vibrational disturbance in the longitudinal direction, then heat is pumped from the wings to the center of the disturbance. Because the stack is continuous, the thermodynamic symmetry is not broken geometrically rather it is

28 Fall 1986 LOS ALAMOS SCIENCE The Natural Heat Engine

broken dynamically. We call such a device each tube linking resonators to measure itudinal field, to attend localization of a nonlinear natural engine. In principle. the direction of heat flow and also put a vibrational energy. The fundamental mo- such a localized disturbance could be a dynamic pressure sensor in each resonator lecular vibrational frequencies arc of the vibrational excitation of a mesoscale ob- to measure its level of vibration. The order of 1012 Hz or greater. but a time- ject. whole system was driven symmetrically dependent confirmational change in the Localized waves in lower-dimensional from the center by an acoustic driver. longitudinal field could beat a much lower vibrational systems have received a great When we drove the system at a frequency frequency-possibly low enough to create deal of theoretical attention because of less than the low-amplitude resonance fre- natural engine effects. These conjectures their potential application to biological quency. localization of energy did occur have motivated us to begin experimental processes. However, macroscopic model- above a certain threshold amplitude. work with a number of other collaborators ing experiments in a water wave trough at Further, heat was pumped toward the re- on the general question of the localization the University of California. Los Angeles. gion of high amplitude. But the localiza- of vibrational energy in materials. This is a have been very valuable in developing tion was stronger and occurred at a much case where we think we know what we are insight about solitary waves. (An lower drive amplitude than expected. This looking for, but we don’t know what we outstanding example is the Wu-ton, a non- localization was also attended by a low- will find. propagating soliton in water surface frequency modulation-typically at 1/11 Thus, whether shaking loudly in the waves. ) As a result, we decided to build an or 1/12 of the drive frequency but often laboratory or, perhaps, vibrating sound- acoustical model that might give insight with components a factor of 100 or more lessly in a molecule, natural engines may into how a coherently vibrating molecular times lower than the drive. be a widespread phenomenon of general system might behave. If such objects are What happened? Our resonators importance. Not only are natural engines indeed found to be real, we believe the performed as expected so far as alteration simple, they use a necessary thermody- field of potential applications will be much of the resonant frequency was concerned. namic evil—irreversibilities—as a broader than just lower-dimensional sys- However, we had unwittingly introduced a positive feature of the engine. We hope an tems. second set of vibrational systems into the understanding of these concepts will serve Our apparatus, which we call the experiment: plate-like vibrations on the mankind well in his quest for appropriate dodecagon, has been likened to a 12-ele- Kapton-oil system, We believe that under engines and will help us to comprehend ment benzene ring. It consists of a circle of suitable conditions the driver resonantly better the behavior of molecular vibra- twelve coupled acoustic Helmholtz re- excites the Kapton-oil system and induces tional systems. ■ sonators with a nonlinear element in- the film to make a hysteretic transition to cluded within each resonator (Fig. 22). We a different geometry that facilitates the introduce the nonlinearity by building the localization. resonator from two bulbs connected by a Our acoustical model experiments have neck with a thin Kapton plastic film that been helpful in inspiring thought on mo- Further Reading flexes with changes in pressure. To pre- lecular-scale or mesoscale systems. vent the neck from flapping at the acoustic Localization of energy and heat pumping frequency or its harmonics, we loaded the John Wheatley, T. Hofler, G. W. Swift, and A. did occur. More important, though, at- Migliori. 1985. Understanding some simple plastic film with oil. tending and preceding the localization, we phenomena in thermoacoustics with applica- According to one mathematical analy- observed behavior that changed on an en- tions to acoustical heat engines. American sis, localization of energy can occur if the tirely different time scale than the acoustic Journal of Physics 53:147. resonant frequency of any given resonator phenomenon. decreases as the amplitude increases. In We conjecture that in molecular and our resonators, as the dynamic pressure of mesoscale systems it is important to have J. C. Wheatley, T. Hofler. G. W. Swift, and A. Migliori, 1983. An intrinsically irreversible the acoustic wave increases, the velocity of two or more interacting. or coupled, thermoacoustic heat engine. Journal of the fluid through the neck increases. which “fields.” These coupled fields could be Acoustic Society of America74:153. means, from Bernoulli’s principle, that the some of the normal optical vibrational average pressure there decreases. The modes of a molecular system. In Kapton neck then flexes inward, reducing particular, torsional or Vibrational modes G. W. Swift, A. Migliori, T. Hofler, and John the cross-sectional area and, thus, the reso- of motion are almost certainly coupled Wheatley. 1985. Theory and calculations for an intrinsically irreversible acoustic prime mover nant frequency (which is proportional to nonlinearly with the longitudinal modes using liquid sodium as primary working fluid. the square root of the area). of motion. We expect a time-dependent Journal of the Acoustic Society of America We installed a thermoacoustic couple in confirmational change, say in the long- 78:767.

Fall 1986 LOS ALAMOS SCIENCE 29 The Natural Heat Engine

The Liquid Propylene Engine

n ideal use of geothermal energy is Fig. 1. In this propylene-to-water heat to warm buildings by extracting exchanger, made up of a stack of hun- A heat from ground water at dreds of stainless steel sheets copper- temperatures of only about 10°C. This ap- brazed together at Los Alamos, the plication involves the pumping of large propylene flows in at the top right of the amounts of heat across small temperature stack and across through the propylene differences (of the order of 30°C). An effi- manifolds and channels, then moves up cient way to effect such heat transfer is and out through the other propylene from one liquid to another. As a result, a duct. The arrow in the figure traces the heat pump that appears well suited for this path through just one of the sets of purpose is a conventional reciprocating channels and manifolds; similar flow oc- heat engine using a liquid for a working curs through the other, lower propylene substance. channels and manifolds. At the same We have been studying just such an time, water flows in and up through one engine—a Stirling engine that uses liquid water duct and across the stack (but propylene as its working fluid. Our dis- through alternate sets of plates and cussion of this device will both contrast across the plates in a direction perpen- the simplicity of natural engines with the dicular to the corresponding propylene complexity of more traditional engines flow) until it returns, exiting through the and, more important, will introduce the other water duct. Because of the in- use of a liquid as a thermodynamic work- timate thermal contact between fluid ing substance. (The section in the main and stainless steel, heat can be trans- article called “The Liquid Sodium ferred at a rate of 230 W/°C. ➤ Acoustic Engine” discusses a natural heat engine that uses a liquid as its primary thermodynamic medium.) It is a common misconception that ties. These facts were first appreciated by than that of other fluids with similar criti- liquids behave much like an idealized John Malone, who in the 1920s built sev- cal temperatures, hydraulic fluid, with density independent eral Stirling prime movers that used liquid A major advantage of a liquid working of temperature and pressure, In fact, water with pressures as high as 700 bars as substance is that liquids have a very large especially near the critical point (where the the working substance. We chose liquid heat capacity per unit volume compared liquid and gaseous phases become indis- propylene (C3H 6) for our work because its to gases, making it possible to build effi- tinguishable), a typical real liquid is some- critical temperature is just above room cient and compact heat exchangers and what compressible, has a large thermal temperature and its Prandtl number regenerators. This point is illustrated by expansion coefficient (comparable to or (which can be thought of as a measure of the compact propylene-to-water heat ex- larger than that of an ideal gas!), and has the material’s viscous losses in relation to changer we have developed for our engine other attractive thermophysical proper- its thermal transport capacity) is lower (Fig. 1). The exchanger is made of hun-

30 LOS ALAMOS SCIENCE Fall 1986 The Natural Heat Engine

Fig. 2. The heat engine shown here consists of four Stirling engines of the Rider form operating from a common crankshaft but phased 90 degrees apart. The working medium is liquid propylene, and heat exchange between water and the propylene takes place in the stainless-steel exchangers depicted in Fig. 1.4

photograph, when contrasted with photo- graphs of natural engines (See the main article) is nevertheless a dramatic rep- resentation of the complex of a more conventional reciprocating engine. In its heat-pump mode, our engine uses work supplied by an electric motor to transfer heat from a source at or below room temperature to a heat sink consisting of flowing water at or above mom temperature. For convenient measurement, the low-temperature source is an electric heater. Mean pressure, oscillating pressure amplitude, volumetric displacement, shaft rotation frequency f, and hot and cold temperatures are all independently con- trollable. We can measure both the rate at which heat is pumped away from the heat

In addition. our laboratory engine has valves that quickly change it from the ordinary heat-pump configuration to one dreds of chemically milled stainless-steel Because of this quality it is possible to in which there is no flow of propylene sheets copper brazed together (several of build a high-power engine that uses a short through the regenerators and heat ex- the individual plates are shown on the stroke, making the mechanical elements changers, even though crankshaft and cover). Although the exchanger (4 by 4 by very efficient without corn promising on piston motion, pressure amplitudes, 9 centimeters in size) entrains only a few the size and efficiency of the thermal ele- temperatures, and so forth remain the cubic centimeters of propylene, it transfers ments. same. This feature allow’s uS to accurately heat between the two fluid streams at a Our Laboratory-scale Iiquid-propylene rate of 230 watts per °C with only a few Stirling engine (Fig. 2) uses the same con- quired to pump the heat, with the back- watts of power required to pump the fluids figuration of parts shown in Fig. 3 of the ground torques due to bearing and seal through the exchanger. main article (the Rider form of the Stirling friction, piston blowby, and the like Another advantage of a liquid working engine), except that we have four such eliminated. substance is that liquids are typically assemblies, These assemblies operate Large amounts of heat can be pumped much less compressible than gases. Thus from a common crankshaft and are by the engine (Fig. 3a)-around 1300 the large pressure amplitudes needed to mechanically phased 90 degrees apart watts at a crankshaft rotation frequency of pump large amounts of heat can be so that the shaft torque oscillations are 4.5 Hz—and the data points match very achieved with only small displacements of minimized, eliminating the need for a big well curves predicted from theory for the a piston, even for a substantial volume of flywheel. Although much of the wiring in particular geometry of’ the engine and for entrained liquid in the thermal elements. Fig. 2 is for diagnostic purposes, the the use of propylene as the working fluid.

Fall 1986 LOS ALAMOS SCIENCE 31 The Natural Heat Engine

Fig. 3. (a) The rate at which the propylene engine pumps heat Q as a PROPYLENE ENGINE DATA function of crankshaft rotation frequency f at two different oscillating (a) pressure amplitudes agrees very well 14001- with theoretical curves predicted from the physical properties of propylene and the geometry of the engine. (b) The Oscillating Pressure Amplitude as a function of f, is just that part of the torque needed to pump the heat. In both graphs the blue data points represent no temperature difference across the regenerators, whereas the red data

The lines drawn on Fig. 3b represent the torque required by an engine with the Carnot efficiency to pump the observed amount of heat added to the torque associated with just the viscous losses of pushing the fluid through the regenerators and heat exchangers. Our measured torque differences agree well with these 0 1 2 3 4 theoretical curves. Crankshaft Rotation Frequency (Hz) Our laboratory engine is very far from a practical, economically useful device. Its scale and most of its design are appropriate for experimental measurements (b) and for the understanding of- principles. not for optimized efficicncy or low manu- facturing or operating costs in a specific application. But, as expected, we are learning that liquids are good heat engine working substances, Liquid engines may ultimately be of great technological importance. We arc also learning much about the practical details of the use of liquids in engines. For example, we suspect that the next logical step in the development of practical liquid engines is to abandon the reciprocating Stirling engine entirely. In- stead, we would use the liquid in, say, a Brayton engine with rotary compressors and expanders. Such a configuration Crankshaft Rotation Frequency (Hz) would reduce losses from such things as bearing and seal friction that, until now, we have regarded as quite uninter- esting. ■

LOS ALAMOS SCIENCE Fall 1986 The Natural Heat Engine

John C. Wheatley (1927-1986) joined Los Ala- mos in 1981. During his tenure here, he performed experiment; on novel heat engines and on the fundamentals of thermal and statistical physics. He received his B.S. in electrical engineering in 1947 from the University of Colorado and his Ph.D. in physics in 1952 from the University of Pittsburgh. He was elected a member of the National Academy of Sciences in 1975 and appointed to the Academy of Finland in 1980. His many honors include the two top awards given by the low-temperature physics community: the Simon Memorial Prize and the Fritz London Memorial Award. At the time of his death, he was the first joint Fellow of the University, of California, Los An- geles, and Los Alamos National Laboratory

Albert Migliori earned his B.S. in 1968 from Carnegie-htellon University and his Ph.D. in physics in 1973 from the University of Illinois. where he studied superconducting thin films. He then joined Los Alamos as a pos-doctoral fellow and studied high-field and self-field behavior of hard type II superconductors. In 1975 he was awarded a National Science Foun- dation Fellowship to study internal and surface magnetic fields in current-carrying superconductors with the Mossbauer effect. In 1976 he became a staff member of the Condensed Mat- ter and Thermal Physics Group.

Gregory W. Swift is a staff- member in the Condensed Matter and Thermal Physics Group, where he has been working on novel heat engines, acoustics, and superfluid helium-3 since 1981. He received his B.S. in physics and mathematics from the University, of Nebraska and his Ph.D. in physics from the University of California, Berkeley. From 1983 to 1985 he held an Oppenheimer Fellowship at Los Alamos.

LOS ALAMOS SCIENCE Fall 1986 33