Thermoacoustics: Underlying Principles and Research Challenges

An Introduction to Thermoacoustics: Underlying Principles and Research Challenges

Anthony A. Atchley

Graduate Program in Acoustics, The Pennsylvania State University, State College, PA 16804, USA

The purpose of this paper is to very briefly introduce thermoacoustic heat transport and acoustic heat engines to those unfamiliar with the field. After a review of some relevant properties of conventional heat engines, it is explained that a parcel of fluid undergoing acoustic oscillations near a solid boundary can exhibit essential features common to all heat engines. Following this discussion, a few of the important design details are discussed briefly to indicate some of the parameters involved. The nature of current research challenges is outlined.

INTRODUCTION sum of the work and rejected energy must equal the input energy. The internal combustion engine is a

common example of a prime mover. The term "heat engine" often conjures up images of 19th century steam engines. However, heat engine technology continues to evolve and it would be 1 difficult to understate the importance of their role in Tcold Q society. Moreover, because their use is so widespread, P out = 3 W ∫ PdV even modest improvements in fuel efficiency, 2 reliability, and acquisition and maintenance costs of 5 2 T heat engines may lead to significant economic and Qin hot environmental benefits. Thermoacoustic engines are a Tcold 4 class of heat engine that use the mechanical power in 1, 5 Qin the acoustic field to accomplish the engine's intended V function. The underlying physics of the heat transfer 4 3 process offers the potential for mechanically simple Tcold Qout heat engines that have competitive thermal efficiencies T and do not use ozone-depleting or global warming hot chemicals. In contrast to the power levels usually encountered FIGURE 1. Illustration of a Carnot Heat Pump. in acoustics, in thermoacoustics it is not uncommon to Although there are many different types of heat consider kilowatts of acoustic power. These large engine, they all have three common features: a power levels lead to many design challenges and working fluid (in the case of thermoacoustics), some fundamental research opportunities. means of doing work on or extracting work from the working fluid, and thermal contact between the HEAT ENGINE FUNDAMENTALS working fluid and the sources and sinks of heat at different temperatures. For illustration purposes, the Heat engines fall into one of two categories: heat operation of an idealized Carnot heat pump is depicted pumps and prime movers. A heat pump is a device that in Fig. (1). The working fluid is inside a container uses mechanical work supplied to it to transport heat fitted with a frictionless piston. In State 1, the fluid is from a region at one temperature to another region at a thermally isolated in the container at temperature Thot. different temperature. Refrigerators and air In going from State 1 to State 2, the fluid is conditioners are common examples. A prime mover is compressed, decreasing its volume and increasing its a device that converts thermal energy stored at a high pressure and temperature. The fluid is then brought temperature into mechanical work. According to the into thermal contact with a sink of heat and compressed second law of thermodynamics, this energy conversion further, rejecting heat and arriving at State 3. The fluid process cannot be 100% efficient. The portion of the is then isolated from the heat sink and expanded, high-temperature thermal energy that cannot be leaving the fluid at Tcold, in State 4. Next, the fluid is converted to work is rejected as heat at a lower brought into thermal contact with a source of heat at temperature. Conservation of energy requires that the Tcold, expanded further, extracting heat from the source and arriving at State 5. Finally, the fluid is isolated amplitude. Thermoacoustic devices usually operate at from the heat source, bring the system back to its initial one of the normal modes of a low-loss, acoustic state, completing the cycle. A graph of pressure P in resonator to achieve high pressure amplitudes. Typical the fluid versus its volume V indicates that the net amplitudes are 1 – 10% of the mean gas pressure in the result of the cycle is the transfer of heat from Tcold to resonator. Thot. An amount of work W is done on the fluid to Recall that the essential features of a heat engine are accomplish this heat transport. The efficiency of a heat that the working fluid must experience a temperature engine, and whether it operates as a heat pump or a change and be brought into thermal contact with prime mover, depends on the phasing of the volume (or sources and sinks of heat at different temperatures. density) changes relative to the pressure changes. This means that the stack or regenerator must be A fluid parcel undergoing acoustic oscillations near located in a region of the acoustic field where both a boundary can exhibit the essential heat engine pressure and displacement (or velocity) change. If the features. Again using a heat pump as an example, the acoustic field is a standing wave, then the stack or fluid parcel is compressed during the positive pressure regenerator cannot be located at either a pressure or a phase of the acoustic cycle, and displaced under the velocity node. To reduce viscous dissipation in the action of the net force exerted on it by the acoustic stack, arising from the requisite fluid velocity, the stack pressure gradient. If there is a thermal gradient along is placed nearer the pressure antinode (velocity node) the boundary in the direction of the displacement, the and the pore size is relatively large. This is a common parcel is brought into thermal contact with regions of characteristic of standing waves acoustic engines. the boundary having different temperatures as it Restrictions on stack location and pore size lead to oscillates back and forth. These different-temperature an inherent inefficiency in standing wave engines. regions can act as sources or sinks of heat. Under the Engines employing traveling wave acoustic phasing proper phasing conditions, net work can be done on the allows the use of stacks (called regenerators in this fluid, and heat can be transported between regions of application) with much smaller pore sizes, resulting in different temperature, just as in the Carnot heat pump higher thermal efficiencies than in standing wave example cited above. This is the essence of acoustic devices [1,2]. Traveling wave engines have heat engines. demonstrated thermal efficiencies of 30%, which is comparable to common internal combustion engines. DETAILS Much of the current research is devoted to the developing theoretical, computational and experimental Whereas conventional heat engines use mechanical understanding of heat transport in high amplitude, means to achieve the proper phasing, in acoustic heat oscillatory, compressible flows. This two-page paper engines this is accomplished by the inherent phasing of can only scratch the surface of thermoacoutics. Many the acoustic oscillations and the degree of thermal papers have been published on the subject. However, contact between the fluid parcel and the boundary. A space limitations allow room for citing only a few consequence of this "natural" phasing is that acoustic references. The reader is urged to consult these for heat engines are mechanically much simpler than more complete discussions. Additional citations can conventional engines, requiring fewer moving parts. also be found there. To achieve the largest possible temperature change for a given pressure change, the working fluid should ACKNOWLEDGMENT have as high a ratio of specific heat capacities γ as possible. Therefore, chemically-inert, noble gases, This work is supported by the Office of Naval such as helium, argon, and xenon, or mixtures of them, Research. are usually used as the working fluid. As a result, the working fluid is environmentally benign. REFERENCES To maximize the amount of gas in thermal contact with the solid boundary, a poorly-thermally- 1. de Blok, C. M., "Thermoacoustic system", Dutch Patent. conducting, porous structure having a large surface International Applications Number PCT/NL98/00515 area to volume ratio, and a large heat capacity relative (1998). See http://www.aster-thermoacoustics.com . to that of the gas, is used as the heat exchange medium. 2. Backhaus, S. and Swift, G. W., Nature, 399, 335-338 (1999). This structure is referred to either as a stack or a 3. Swift, G. W., Journal of the Acoustical Society of regenerator, depending on the size of the pores and on America, 84, 1145-1180 (1988). particular type of heat engine under consideration. 4. Swift, G. W., Thermoacoustics: A unifying perspective The amount of heat pumped by an acoustic engine for some engines and refrigerators, available for (or the power output, in the case of a prime mover) is download at http://www.lanl.gov/thermoacoustics, to be proportional to the square of the acoustic pressure published by the Acoustical Society of America (2001). A Miniature Thermoacoustic Refrigerator for ICs

Thomas J. Hofler and Jay A. Adeff

Department of Physics, Naval Postgraduate School, Monterey, CA 93943, USA

A miniature thermoacoustic refrigerator is being developed for the purpose of cooling integrated circuits below their failure temperature in hot environments. Work has been done on a piezoelectric acoustic driver operating at 4 kHz. A simple refrigerator has been built and tested that uses one atmosphere of air as a working medium, is 5 cm long, and has produced 12° C of cooling. A more advanced refrigerator is under development, which will use pressurized noble gas mixtures, and span 30 degrees centigrade with good efficiency.

MOTIVATION AND GOALS PRELIMINARY REFRIGERATOR

The motivation behind this project is to cool a An inexpensive audio tweeter unit consists of a medium power integrated circuit in a hot environment piezoceramic disk bonded to a paper cone in a below its failure or malfunction temperature using a compression horn-driver configuration. The horn was miniature refrigeration device. The target environment removed and a plastic tube was coupled to the paper is a military aircraft avionics package where the heat cone volume. The internal length of the tube is ¼ rejection temperature may approach 90° C. While an wavelength for the working gas, which is air at one obvious technology choice is the thermoelectric (TE) atmosphere for this device. The plastic tube contains cooler, we hope to produce better efficiency with a the thermoacoustic “stack” and an ambient heat thermoacoustic cooler. Also, a fundamental problem rejection exchanger, as shown in Fig. 1. with TE coolers is poor longevity when rejecting heat at temperatures in excess of 80° C. Mic port

End cap & ambient heat reservoir (Cu) Specifications Amb. heat exchanger

A short list of specifications for the device are Stack given below: Cold Table 1. Refrigerator Specifications. Thermocouple Specification Value Overall length 6 cm (approx.) Velocity Paper cone Acoustic frequency 4 kHz Antinode Driver transduction PZT piezoceramic flex disk PZT Disk Gas pressure & type 15 bar He-Xe mix Rejection temperature 90° C FIGURE 1. A section drawing of the preliminary “Cold” temperature 60° C refrigerator device. The driver housing details are not Useful cooling power 1 W (minimum) shown.

The ambient heat exchanger is a disk of copper A commercially available piezoceramic flexural screen material connected to a 21 g copper mass, disk was chosen as the acoustic power transduction which is also the rigid end cap for the tube. An element, which has a fundamental mechanical attachment point for a thermocouple is provided at the resonance of 4 kHz. Because the issues of impedance cold end of the stack, in lieu of a cold heat exchanger. matching and acoustic power delivery were not Thus, no quantitative cooling powers were measured. certain, it was decided to first build a simple Stack structures consisting of random polyester preliminary refrigerator as a test of the piezoceramic fiberfill and a spiral polyester film roll were tested. disk. A temperature difference of 1° to 2° C was measured with the fiberfill stack and a 12° C temperature difference was measured with the spiral roll stack. Reassuringly, a large peak pressure amplitude of 6% to 8% of the mean pressure was also measured.

PRESSURIZED REFRIGERATOR

A more advanced pressurized refrigerator was optimized using new thermoacoustic analysis software called DSTAR[1,2]. It is designed to meet the specification in Table 1. The resonator design FIGURE 3. Single PZT disk driver output in a test- topology is the same as previous work by Hofler[3,4]. resonator. The dynamic pressure transducer output to drive Without elaborating on the quantitative design, a voltage ratio is plotted for 4 different drive voltages. schematic section of the device is shown is Fig. 2. In Fig. 3, the 4.6 kHz mechanical resonance peak occurs at a drive voltage of 1 V rms and a gas resonance is also seen on this curve at 4.23 kHz. Bent tuning curves and hysteresis are apparent for high drive levels when frequencies are swept up and down. At a drive voltage of 14 V rms, a merging of the mechanical resonance peak with the gas resonance peak at 4.23 kHz occurs, which yields the largest peak. While the acoustic power output of the single disk driver is low, a two-disk driver has delivered 0.5 W to the test resonator at a dynamic pressure of 0.46 bar pk, which is adequate for refrigeration purposes. We hope to have cooling data with the pressurized refrigerator by the time of ICA meeting in September.

ACKNOWLEDGMENTS

FIGURE 2. A section drawing of the pressurized We acknowledge the laboratory work of students refrigerator. The driver is shown with 2 piezoceramic disks Omer Livvarcin and Seyhmus Direk of the Turkish and a pressure transducer installed. Navy and funding from DARPA through the Rockwell Science Center. One useable driver has been completed, as is the machining of the resonator parts. We are currently assembling the heat exchangers and resonator. REFERENCES

Piezoceramic Driver Development 1. Thomas J. Hofler and J. A. Adeff, ARLO 2, 37 (2001).

The driver design couples one or two PZT flexural 2. DSTAR thermoacoustic analysis software is available at: disks to a push-rod, which is connected to a smaller http://phserver.physics.nps.navy.mil/hofler/ diameter stainless steel foil diaphragm. The 3. J. C. Wheatley, T. Hofler, G.W. Swift, A. Migliori, Am. diaphragm radiates acoustic power into the J. Phys. 53, 147 (1985) thermoacoustic resonator. Early driver testing was performed with a special 4. T. Hofler, John Wheatley, G. W. Swift, and A. Migliori, pressurized test-resonator having a tunable resonance U.S. Patent 4,722,201 (1988). frequency. Substantial nonlinearities were discovered during testing and are attributable to the foil diaphragm, as seen in Fig. 3. Thermoacoustic Spontaneous Oscillation and its Relation to Thermodynamics A.Tominaga

Institute of Physics, University of Tsukuba,Tsukuba, Japan 305-0006 E-mail: [email protected]

Thermoacoustic spontaneous oscillation is one of fascinating phenomena and has attracted scientists from the early part of the 19th century. The fluid-mechanical discussion on the stability limit revealed that essential to thermoacoustics was oscillating entropy of fluid element contrary to the adiabatic oscillation in a free space. The work flow and the heat flow (or the entropy flow) appeared in thermodynamical discussions on thermoacoustic phenomena are revival of classical concepts, which formulated the first and the second laws of thermodynamics. In addition to these fundamental laws the law of minimum increment of entropy flow is important for understanding heat-engines and thermoacoustic spontaneous oscillations. Deeper insight into the second law leads us a concept of the local entropy production rate.

INTRODUCTION

Thermoacoustic spontaneous oscillations attracted . (1) many scientists in the 19th century, for example, Sond- hauss, de Rijke, Tyndal[1], and Rayleigh[2]. The put- put toy boat was also invented in the end of the Thus, the for given is a decreasing function of century[3]. In the middle of the 20th century, the fluid- the cooling power . If we accept the law of mini- mechanical discussion on the stability limit from mum increment of entropy flow, we can expect that the Kramers[4] to Rott[5] gradually revealed that essential becomes as large as possible. According to the sec- to thermoacoustics was oscillating entropy of fluid ele- ond law (i.e. the Clausius' inequality ) the is ment contrary to the adiabatic oscillation in free space. smaller than , the cooling power of Thermodaynamical discussion[6] on thermoacoustic phenomena started in the end of 20th century revived the ideal engine . the work flow , the heat flow , and the entropy flow Let us consider next a prime mover which absorbs , which were classical and hidden concepts ap- heat at the hot end of temperature and emits peared in discussion of heat engines from Carnot to heat at the cold end of temperature . According Clausius. The energy flow is summation of the and to the first law, the prime mover emits the output work the , and the second law known as the Clausius' ine- . The increment of entropy flow in quality is expressed by the . These flow quantities play the prime mover is given by an important role in nonequilibrium systems. Thermoa- coustic theory[6] relates these flow quantities to famil- iar state quantities appeared in thermodynamics of the . (2) equilibrium state and the fluid mechanics. The theorem of minimizing the entropy production rate proved by Prigogine[7] is one of most important discovery in the 20th century. The theorem is however Thus, the for given is a decreasing function of restricted to systems of homogeneous temperature. the . If we accept the law of minimum increment This paper proposes, as one of natural laws, a law of of entropy flow, we can expect that the becomes minimizing the entropy production rate, which is an ex- as large as possible. According to the second law (i.e. pansion of the theorem to systems of inhomogeneous the Clausius' inequality ) the is smaller temperature. than , the output work of the ideal en- HEAT ENGINE gine.

Let us discuss the increment of entropy flow in heat THERMOACOUSTIC SPONTANEOUS engines. Let us consider at first a heat pump which ab- OSCILLATION sorbs the input work and emits heat at the hot end of temperature . According to the first law, the Let us imagine an experimental device containing a heat pump absorbs heat at the cold stack; hot end of the stack is heated by an electric heater end of temperature . The increment of entropy flow and temperature of the cold end is kept constant by cir- in the heat pump is given by culating water of temperature . Temperature of the heat conduction

thermoacoustic oscillation

FIGURE 1. Typical results of experiment[7] on thermoacoustic spontaneous oscillation of fluid. The oscillation is ob- FIGURE 2. The increment of the entropy flow, , as a func- served only beyond the . Both the and the tion of the heater power, , for possible two branches. The show a kink at the . branch of thermoacoustic oscillation is lower than that of heat conduction. hot end is possibly an increasing function of the heater power . Many experiments shows that there is is the total of the time dependent entropy flow through some critical heater power : below the outer surface of the volume. Since the for an there is no oscillation of fluid in the device, but beyond isolated system is absent, we have the fluid oscillates spontaneously and the stack emits the output work . The thermoacoustic , (7) spontaneous oscillation is therefore one of dissipative for the isolated system. The law of increase of entropy structures. requires According to experiment[8] both the and the and at the equilibrium state. For a steady state( ), such as steady heat con- show a kink at the (see Fig.1); duction in a solid material, we have and has a jump. The increment of entropy flow expressed by Eq.(2) is also an . (8) increasing function of the and For cyclic phenomena, such as thermoacoustic phenomena, time average of the (3) vanishes so we have , (9) has a jump at the , since and changes discontinuously. where indicates time average. As a matter of course The as a function of the is schematically the is a quantity expressed by up to this point. shown in Fig.2. In this figure there are two thermodyna- Eqs.(8) and (9) with lead to the Clausius' inequali- mic branches; a heat conduction branch and a thermoa- ty; the requirement of includes the Clausius' ine- coustic oscillation branch. The latter branch is lower quality automatically. than the former branch. If we accept the law of mini- Therefore the law of minimum increment of entropy mizing the increment of entropy flow, we can expect flow is equivalent to the law of minimizing the entropy that beyond the the latter branch is realized and production rate. the fluid begins to oscillate spontaneously. Therefore the law of minimum increment of entropy REFERENCES flow is, in addition to the first and the second laws, essential for us to understand heat engines and thermoa- 1. Thomas,J.M., Michael Faraday and the Royal Institution- coustic spontaneous oscillation. The Genius of Man and Place, IOP, Bristol, 1991, p.156. 2. Rayleigh,J.W.S., The Theory of Sound, Dover, New York, 1945, Chap.16. LOCAL ENTROPY PRODUCTION RATE 3. Piot,D.T., Great Britain Patent No.26,823 (1897). 4. Kramers,H.A., Physica 15, 971-984(1949). Let us consider local entropy production rate 5. Rott.N., Z.Angew Math.Phys.20, 230-243(1969); ibid. 24, 54-72(1973); ibid.26, 43-49(1975). , (4) 6. Tominaga,A., Cryogenics 35, 427-440(1995); Fundamen- tal Thermoacoustics, Uchida Rokakuho, Tokyo, 1998, where is the entropy in a unit volume and the Chap.4-9. time dependent entropy flow through a unit area. Total 7. Nicolis,G. and Prigogine,I, Self-Organization in Nonequili- of the in a volume, , is brium Systems, John Wiley & Sons, New York, 1977, Chap.3. (5) 8. Biwa,T., et al, "Transition from Standing to Traveling Waves in Thermoacoustic Gas Oscillations" in Proceed- where ings of The 50th Nat.Cong.of Theoretical & Applied Me- chanics 2001, Japan Society of Civil Engineers, Tokyo, (6) 2001, pp.351-352. Oscillatory Minor Losses in Jet Pumps

A. Petculescu and L. A. Wilen

Department of Physics and Astronomy, Ohio University, Athens, OH 45701 USA

Empirical studies of steady-flow minor losses at transition regions in ducts date back to the beginning of the first millennium when curators of the Roman aqueducts specified the use of special nozzles to restrict the quantity of water tapped from storage cisterns[1]. It is surmised that enterprising citizens soon discovered that greater (but not entirely legal) quantities of water could be extracted if reverse nozzles (or diffusers) were used to transition back to larger size pipes[2]. Detailed studies of minor losses for oscillatory flow have been performed only very recently, and mostly for orifice-like geometries and pipe bends[3-5]. We have made extensive impedance measurements of jet pumps subjected to oscillatory flow. At high velocity amplitudes, it is found that the resistance of the sample increases linearly with velocity, and that the DC pressure across the sample increases as the square of the velocity. From these quantities, the minor loss parameters Kin and Kout for a variety of jet pumps are determined using an analysis described by Thurston et al[5] and also Swift et al[6]. We compare the oscillatory results to those obtained for the same samples using steady flow as a test of the so-called “Iguchi hypothesis.”

INTRODUCTION studies in general, we undertook a systematic study of oscillatory minor losses in jet pumps. Consider steady, inviscid, laminar flow along a pipe of gradually decreasing cross-section. Conservation of mass requires the velocity to increase as the pipe narrows, and conservation of energy, or Bernoulli’s principle, dictates that the pressure will decrease correspondingly. If the pipe cross section now gradually opens back up, the pressure will recover to the original value as the extra kinetic energy (velocity) is converted back into potential energy. If, on the other hand, the cross section of the pipe abruptly increases, processes may occur (jetting, turbulence, vortex FIGURE 1. Experimental setup. formation[7]) which lead to a conversion of some, or all, of the kinetic energy into heat, in which case the pressure will not completely recover to its original value. The resulting pressure drop is known as a minor EXPERIMENT loss. Minor losses occur at any kind of sharp transition or bend in a pipe and have been studied extensively for The experimental setup is shown in Fig. 1. A steady flow[8]. Since the pressure drop is related to the loudspeaker attached to a bellows acts as an oscillatory Bernoulli effect, it is often characterized by a volume velocity source driving a tube leading to a jet dimensionless minor loss parameter K, where pump. The tube compliance is acoustically in parallel with the impedance of the jet pump. The volume 1 2 p K v (1) velocity, as well as the pressure in the compliance, is drop 2 monitored with a lock-in amplifier. From these measurements, the total lumped-element impedance For an asymmetric transition (referred to here as a jet can be determined. Typical measurements are pump), the minor losses may be different for flow in performed at 96 Hz; hence the wavelength is much opposite directions. When a jet pump is subjected to larger than the system. Since the tube compliance can oscillatory (AC) flow, a time averaged pressure drop be measured separately, the impedance of the jet pump will develop across the transition. Swift et al have alone is easily determined. The time-averaged pressure taken advantage of such an effect to counter drop across the jet pump is also measured. If one undesirable steady flow in toroidal traveling-wave makes the assumption that AC minor losses can be thermoacoustic devices. Motivated by this application described by a formulism similar to that for DC flow, in particular, and the dearth of oscillatory minor loss function of opening angle for samples that have a fixed 1 2 namely that pdrop (t) Kin/out v(t) , where Kin opening curvature. Also displayed are the results 2 obtained from steady flow measurements. Kin and and Kout refer to flow in the positive and negative Kout were also determined as a function of opening directions respectively, measurements of the resistance curvature for samples with fixed angle (not shown). R and time-averaged pressure pave (versus velocity amplitude |v1|) can be used to determine Kin and Kout. Thurston’s calculations show[5]:

1 2 CONCLUSIONS p v K K ave 8 1 in out (2) 2 The assumption that high amplitude oscillatory flow R v K K 3 1 in out may be described at any instant using the equations for steady flow was stated and tested by Iguchi and co- We have measured the impedance and time-averaged workers and is sometimes referred to as Iguchi’s pressure drop for jet pumps with a variety of angles hypothesis[3]. The results of the current work are and opening curvatures (refer to Fig. 1). A separate consistent with this assumption. In future work, we setup was used to measure steady-flow minor losses. plan to extend our measurements to look at AC minor losses in pipe bends and other unusual geometries.

7.00E+06 90 6.00E+06 80 70 5.00E+06 60 4.00E+06 50 ACKNOWLEDGMENTS 3.00E+06 40 30 2.00E+06 20 1.00E+06 10 The authors gratefully acknowledge support from the 0.00E+00 0 0 1020304050 0 500 1000 1500 2000 Office of Naval Research. The authors thank B. Smith 2 |v1| |v1| and A. Atchley for many useful discussions.

FIGURE 2. R vs. |v | and p vs. |v |2 1 ave 1 REFERENCES

1.2 1. Sextus Julius Frontinus, De Aquaeductibus Urbis Romae, 1 ca. 97 AD. 0.8 Kout (AC) Kin (AC) 0.6 Kout (DC) 2. R.W. Fox and A.T. McDonald, Introduction to Fluid 0.4 Kin (DC) Mechanics, New York: John Wiley and Sons, Inc., 1992,

0.2 pp. 369-372.

0 0 5 10 15 3. M. Iguchi and M. Ohmi, Bull. JSME, 25, 1537-1543, Angle (degrees) (1982).

4. J.R. Olson and G.W. Swift, J. Acoust. Soc. Am. 100, FIGURE 3. Kin and Kout versus opening angle. 2123-2131, (1996).

5. G.B. Thurston, J. Acoust. Soc. Am. 30, 452-455 (1958).

RESULTS 6. G.W. Swift, D. L. Gardner and S. Backhaus, J. Acoust. Soc. Am. 105, 711-724 (1999).

Fig. 2 shows the resistance of one jet pump plotted 7. B.L. Smith and A. Glazer, Phys. Fluids 10, 2281-2297 against the velocity amplitude at the narrow end of the (1998). sample and also the time-averaged pressure drop plotted against the velocity amplitude squared. Both 8. I.E. Idelchik, Handbook of Hydraulic Resistance, New plots exhibit linear behavior at high amplitudes. The York: Begell House, 1994. slopes of these curves, along with Eq. 2, determine Kin and Kout[9]. Similar curves were obtained for all the 9. A correction is required because the jet pump is not driven directly with a current source. A full discussion of samples. Fig. 3 shows the results for Kin and Kout as a this point is beyond the scope of this brief article. The benefit of linear approach and its experimental validation for a thermoacoustic system efficient analysis

M. X. François, E. Bretagne

LIMSI/CNRS, B.P.133, F-91403-Orsay-France

Analysis of acoustic and temperature fields in a thermoacoustic (TA) system is rather difficult. It is nevertheless of great interest, especially for new complex thermally driven thermoacoustic or pulse tube refrigerator. In this paper combining different linear approaches performs the analysis of such a thermoacoustic machine. These models give some important information, directly applicable to the design of the device, mostly for the production of acoustic power and its transmission to the load. It is shown furthermore that a deterministic reading of the system dynamic may improve the analysis and help to interpret the deviation from the existing theory. The prime mover standing states are indeed shown to be very sensitive to the reference state.

INTRODUCTION Furthermore, acoustic and temperature fields are closely related to complex non-linear processes and A pulse tube or thermoacoustic refrigerator driven energy balance. The interpretation of their deviation by a TA prime mover (TADR) is a very attractive but from existing theory and models has thus to be complex machine. It is also difficult to compare considered very carefully. different systems and analyse their limitations only In the second part of this presentation we give some from a performance point of view. evidences that the standing states are not intrinsic and In the design of such system, a first specificity is that must be associated with some states of reference the behaviour and performance of a TADR can not be consistent with a linear analysis. deduced from the investigation of the two separated The most important remark is the following one: the systems. A TA prime mover indeed operates by onset of the thermoacoustic instability in a prime resonance and then a new introduced element as a mover is actually the bifurcation of a deterministic Pulse Tube may induce large perturbations in its system, and probably in most cases a supercritical dynamics. bifurcation. Thus, if the design of a specific prime mover or a The theory of deterministic system provides us a link refrigerator is still a very complex work, the between linear results near onset of self-oscillation computations become more and more intuitive if the and amplitudes in a weakly non-linear domain. two devices are associated. Note that a parametric investigation by a numerical A gradual methodology simulation can be rapidly time consuming due to numerous geometrical data. In a thermoacoustic device, the harmonic content is By combining different approaches it is however generally weak even with drive ratio p1/ p0 as high as possible to improve the design of an efficient system. 10%. A rough estimate of the acoustic fields in the In the first part of this presentation we focus on the whole resonator is so obtained from linear models. analysis of the power production and transmission in The use of a lumped parameter model allows a a TADR. preliminary selection of interesting topology for the We then show that a linear and thus crude approach resonator. of such thermoacoustic systems can be very useful to With simplified scheme it is thus possible to get have a quick and pretty good insight of the dynamics. easily a good idea of first, the power transmission For this, using Rott’s model [1] with different level of efficiency from a TA prime mover to its load, second, approximation allows to deal with the acoustical dimensional ratio which leads to the regenerator impedance conditions and power transmission. proper phasing, that is a travelling wave phasing. For An experimental validation for two typical structures example in the case B figure 1, it can be is presented. l ω2 A TA prime mover is also different from a demonstrated that the condition loop ≈ is the 2 0.1 mechanical driver because the phenomenon attached 2c to the control of the wave amplitude is an instability appropriate condition to obtain an acoustic power mechanism.

Inﬂuence of Resonator Shape on Sound Fields in Thermoacoustic Engines M. F. Hamilton, Yu. A. Ilinskii and E. A. Zabolotskaya Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712–1063, USA

The influence of resonator shape and boundary layers in the stack on sound fields in thermoacoustic engines is investigated theoreti- cally. Analytical results for the natural and resonance frequency shifts are presented for an empty resonator, without a stack. Numerical results are presented for the dependence of a finite-amplitude waveform on the shape of a resonator containing a stack.

INTRODUCTION by Eq. (2) determines the entire spectrum of axial modes. Comparison with numerical solutions of the Webster horn Both variation in the cross-sectional area of a thermoa- equation indicates that Eq. (1) is accurate for the range of coustic engine, and boundary layers within the stack, con- area perturbations considered below. tribute to changes in the frequency response. The natural frequency shift is a linear effect of these factors and it determines two nonlinear effects, the resonance frequency RESONANCE FREQUENCY SHIFT shift, and changes in the ﬁnite-amplitude waveforms. Our objective here is to summarize brieﬂy a theoretical inves- Now consider an empty resonator that is driven near ω ω tigation of these phenomena. To obtain analytical ex- its fundamental natural frequency, 1. The area

pressions for the natural and resonance frequency shifts variation is chosen to be Eq. (2) with n ¢ 1, such that

we consider a resonator without a stack [1]. Waveform only the fundamental natural frequency is shifted and thus

¦ ω ω ¢

distortion is calculated with a numerical solution that in- ω ω ¢

¥ ¥ 1 ¥ 01 1 a1 2, with n 01 n for n 2. In this

cludes the effect of the stack [2]. case, the dependence of the resonance frequency ωres on

the particle velocity amplitude u1 max at x l ¥ 2 is [1]

2 2

ω γ ¡

NATURAL FREQUENCY SHIFT res ¤ 1 u1 max

1 ¤ ω 2

1 64 c0

We begin by considering an empty resonator for which 6 4

ω ω ¡ γ ω

¥ ¤

01 1 4 01

¡£¢

¤ the cross-sectional area may be expressed as S x S0 ¤ (3)

¦ ω ω ¡ γ ω ¤ 1 01 ¥ 1 1 1 δS x ¡ , where x is the axial coordinate of the resonator, and δS is a small, slowly varying perturbation. We refer to the

where γ is the ratio of speciﬁc heats for the gas. This ¡

case of constant cross section δS ¢ 0 as a cylindrical ω ¢

result is based on the assumption of low losses and 1

geometry, even though S0 may describe a cross section

ω ω ¢ ω

(more generally, 2 ). that is not necessarily circular. All walls of the resonator 01 1 2 The ﬁrst term in the square brackets in Eq. (3) is as- are assumed to be rigid. The nth natural frequency for the

sociated with second-harmonic generation, the second

ω ¢ ω axial modes may be expressed in the form ¤

n 0n with static displacement and cubic nonlinearity. The ﬁrst

δω ω ¢ π n, where 0n n c0 ¥ l is the natural frequency of a ω term is the dominant contribution for ω1 01 and in-

cylindrical resonator, c0 the small-signal sound speed, l ω ω ω ω dicates res 1 (hardening behavior) for 2 1 2,

the length of the resonator, and ω ω ω ω and res 1 (softening behavior) for 2 1 2. Since

¨ l

ω ω

δωn 1 1, the hardening or softening behavior corresponds

¢§¦ ¡ π ¡ ¥ ln S ¥ S0 cos 2n x l dx (1) ω ω0n l 0 to whether the second harmonic 2 of the drive frequency is greater or less, respectively, than the second natural fre- the relative natural frequency shift due to the variation in quency ω2. The phase of the second harmonic differs by cross section [1]. π in the two cases, which determines the sign of its inﬂu-

It is convenient to choose ence back on the drive frequency component. Amplitude-

frequency response curves predicted analytically for n ¢

¢ © π ¡ ¥ S ¥ S0 exp an cos 2n x l (2)

1 in Eq. (2) are shown in Fig. 1, where (a) corresponds to

ω ω ¢ ¢§¦ ¡ δω ω ¢ ¦

¥ ¥

because Eq. (1) yields n ¥ 0n an 2. One may thus a convex resonator for which 1 01 1 1 a1 0 2 ,

ω ω ¢

shift the nth natural frequency independently of all oth- and (b) to a concave resonator for which 1 ¥ 01 0 9

¡ ers, and a superposition of spatial modulations deﬁned a1 ¢ 0 2 [1]. 0.15 1.4 (a) (a) 0 1.2 0.1 p/p | 0 1 /c 1 |u 0.05 0 0.5 1 1.5 2

1.4 (b) 0

1.095 1.1 1.105 1.11 1.115 0 1.2

ω /ω01 p/p 0.15 1 (b) (a) 0 0.5 1 1.5 2 0.1 | 0

/c 1.4 (c) 1 |u 0 0.05 1.2 p/p 1 0 0.89 0.895 0.9 0.905 0.91 0 0.5 1 1.5 2 ω /ω01 time (periods)

FIGURE 1. Analytical predictions of amplitude-frequency re- FIGURE 2. Numerical predictions of pressure waveforms in a

ω ω ω ¢ ω

¡ ¤

sponse curves for an empty resonator with 2 01 2 0 and two-dimensional thermoacoustic prime mover for (a) 2 1 2,

ω ω ¢ ω ω ω ω ω ¢ ω

¡ ¡ ¥ ¦

(a) 1 01 1 1, (b) 1 01 0 9. (b) 2 1 2, (c) 2 1 2.

φ ¢ ¦ π ω ε ω

£ ¤ and thus p ¤ p cos t cos2 t, which

WAVEFORM DEPENDENCE 1 2 1 1 ω ω φ ¢

is U-shaped as in (a). For 2 1 2, 0, and thus

ω ¦ ε ω £ p1 ¤ p2 cos 1t cos2 1t, which is V-shaped as in

We conclude by considering waveform dependence on (c). The effects of area variation and stack dispersion

ω ω

the shape of the resonator and presence of the stack com- combine for a1 ¢§¦ 0 10 such that 2 1 2. In this case

π ω ¦ ε ω

bined. An algorithm was developed recently for model- φ ¢¦

£ ¤ ¥ 2 and p1 p2 cos 1t sin2 1t. Because ω ing the thermal and acoustical ﬁelds in a two-dimensional 2ω1 2 provides resonance, harmonic generation is ef- resonator containing a stack with negligible plate thick- ﬁcient, and the shock formation observed in (b) occurs.

ness [2]. The area variation is taken again to be Eq. (2) ¢

with n 1, the stack has length l ¥ 8, and it is centered

¢ ¢ at x l ¥ 4. Helium is the gas, with 1 67 and Prandtl ACKNOWLEDGMENT

δ number Pr ¢ 0 67. The plate spacing in the stack is 4 κ, δ where κ is the thermal penetration depth, and the ratio This work was supported by the U.S. Ofﬁce of Naval of the heat capacity of the stack to that of the gas is 50. Research. In this example the engine is operated as a prime mover, which drives the resonator principally at its fundamental ω ω natural frequency 1. Here n are natural frequencies for REFERENCES a resonator including a stack.

Pressure waveforms at x ¢ 0 are shown in Fig. 2 for

1. M. F. Hamilton, Yu. A. Ilinskii, and E. A. Zabolotskaya,

ω ω ω ω

¢§¦ ¡ ¢§¦ ¡ (a) 2 1 2 a1 0 25 , (b) 2 1 2 a1 0 10 , “Linear and nonlinear frequency shifts in acoustical res-

ω ω

¢ ¡ ¥ and (c) 2 1 2 a1 0 25 , where p p0 is the ra- onators with varying cross sections,” J. Acoust. Soc. Am. tio of total to ambient pressure. The waveforms may 110, July (2001). be interpreted with oscillator theory. At second order, 2. M. F. Hamilton, Yu. A. Ilinskii, and E. A. Zabolotskaya, the second harmonic p2 satisﬁes an oscillator equation “Nonlinear two-dimensional model for a thermoacoustic

∝ 2 ω ¦ with forcing function F2 p1, where p1 £ cos 1t is engine,” 1st International Workshop on Thermoacoustics, the fundamental. Let φ be the phase of p2 relative to ’s-Hertogenbosch, the Netherlands, 23–25 April 2001. ω ω ω

F2. For 2 1 2 (drive frequency above 2) we have Direct Simulations of Thermoacoustic Stacks and Heat Exchangers O.M. Knio & E. Besnoin Department of Mechanical Engineering The Johns Hopkins University Baltimore, MD 21218-2686 USA

The velocity and temperature fields in an idealized thermoacoustic refrigerator are analyzed computationally using an unsteady, two- dimensional, low-Mach-number, flow model. The model simulates the the flow behavior in the neighborhood of a parallel-plate thermoacoustic stack / heat exchanger configuration. A parametric study is conducted of the role of heat exchanger length, the length of gap between the stack and heat exchangers, and the drive ratio. In particular, the simulations indicate that the cooling load peaks at a well-defined heat exchanger length and gap width between the heat exchangers and the stack. The optimal gap width and heat exchanger length vary with drive ratio and should be optimized simultaneously.

INTRODUCTION these methods enable the use of efficient FFT-based Pois- son solvers. A convergence acceleration scheme is also The flow field around thermoacoustic stacks and heat used that is based on a sucessive mesh refinement strat- exchangers is dominated by the periodic shedding of con- egy [2]. Specifically, computations are first performed on centrated vortices, which may lead to complex veloc- a coarse computational mesh. Once steady state condi- ity and temperature patterns, and significantly affect heat tions are reached, the solution is interpolated on a finer transfer correlations. This paper discusses recent results grid, and the process is repeated until steady state condi- of direct numerical simulation effort that aims at investi- tions are reached on a grid that is fine enough for quan- gating these phenomena. tities to be accurately determined. As discussed in [2], As reflected below, attention is focused on simplified this approach leads to dramatic reduction in the CPU time parallel-plate thermoacoustic stack and heat exchanger needed to reach steady conditions, and at the same time configuration. The model is based on a simplified rep- provides useful information on the accuracy of the pre- resentation of resonance tube acoustics and on detailed dictions. (2D) representation of the unsteady flow in a neighborhood of the stack and heat exchangers. The present model accounts for finite plate thickness and gap effects and thus RESULTS AND DISCUSSION extends earlier constructions in [1] –which assumed that the stack and heat exchanger plates are in perfect thermal

A parametric study is performed of the effects of drive

contact, and in [2] – where plates of vanishingly small

datio (2% Dr 8%), temperature difference between

thickness were considered. As discussed below, the com- the hot and cold heat exchangers (6 K ∆T˜ 18 K), putations are used, in particular, to explore the depen- gap width g˜ between the heat exchangers and the stack dence of the thermoacoustic cooling load on the length plates, and heat exchanger length L˜ on the cooling load of the heat exchangers and on the size of the gap between c (Q˜). Analysis of the computations reveal that the cool- the heat exchangers and the stack. ing load Q˜ peaks at a well-deﬁned values of the gap

width and heat exchanger length. The optimal combi-

¡ £ nation Lc ¢ g opt varies with drive ratio (Dr) and temper- NUMERICAL MODEL ature difference (∆T). However, for all cases considered, the optimal heat exchanger length Lopt only varies with Direct numerical simulations are performed of the un- the gap width (Fig. 1); it is insensitive to Dr and ∆T. The

steady mass, momentum and energy conservation equa- optimal gap width varies with ∆T, but not with Dr; it is

¥ £¨§

¡ δ ¦ tions for a perfect gas in the low-Mach-number limit. insensitive to Lc in the range Lc ¤ 2 Rp k 1 (Fig. 2). These conservation equations are coupled with the un- The presence of a finite gap between the stack plates steady heat conduction equation in the stack plates. A and heat exchangers can lead to a significant increase in second-order discretization of the vorticity form of the the cooling load. For the operating conditions indicated governing equation is used. The flow solver incorporates in figure 3, optimization of the gap size results in sig- domain decomposition / boundary Green’s function tech- nificant increase in the cooling load over arrangements niques for the reconstruction of the velocity field [3], and having either small or large gap width. 1.8 2 5

dr=5%; ∆Τ=12 Κ 9

dr=5%; ∆Τ= 9 Κ

1.6 ∆Τ= 6 Κ r

dr=5%; e

dr=2%; ∆Τ= 6 Κ g

dr=8%; ∆Τ=18 Κ a

)

∆Τ=12 Κ c

κ dr=8%;

1.4 x

δ ∆Τ= 6 Κ E + dr=8%;

t a e H

/ (2.R 1.2 d l 3 opt o 9 L C

1 0 8

0.8

0© 0.2 0.4 0.6 0.8 δ ) g / (2.R p+ κ

FIGURE 1. Variation of the optimal heat exchanger length with 1 9 gap size. Lopt is deﬁned as the length for which the ﬂux peaks

for a given (ﬁxed) gap width. ) ) m m m ( m (

0.5 y y 8 7

0.4 9 8

dr=8%; ∆Τ=18 Κ

) κ 0.3 ∆Τ=18 Κ

δ dr=5%; +

p dr=5%; ∆Τ=15 Κ dr=8%; ∆Τ=12 Κ dr=5%, ∆Τ=12 Κ

/ (2.R 0.2 ∆Τ= 6 Κ

opt dr=8%; g dr=5%, ∆Τ= 6 Κ dr=2%; ∆Τ= 6 Κ 0.1 7 8 e t a 6 l 7 P 0

0 0.5 1 1.5 2 c

δ ) a L h/c / (2.R p+ κ t S FIGURE 2. Variation of the optimal gap width with heat exchanger length. gopt is deﬁned as the gap width for which the

1 0 5 5 5

ﬂux peaks for a given (ﬁxed) heat exchanger length. . .

1 0 8

) m m ( x 5 5 1 0

. .

1 0 ) m m ( x 12 NO WIDE GAP GAP 10 FIGURE 4. Instantaneous velocity vectors around the ther-

moacoustic stack and the cold heat exchanger; left: Dr 5%,

∆

8 ∆T 15 K; right: Dr 6 5%, T 18 K.

(W/m) 6 δ ) = 0.45 ~ Q L c /(2.R p+ κ δ ) = 0.67 L c /(2.R p+ κ δ ) = 1.12 L c /(2.R p+ κ

ACKNOWLEDGMENTS 4 δ ) = 1.79 L c /(2.R p+ κ

2 This work is supported by the Ofﬁce of Naval Re-

0 0.2 0.4 0.6 g / (2.R +δ ) p κ search. Computations were performed at the National Center for Supercomputing Applications.

FIGURE 3. Variation of the cooling with the gap width for a stack operating at Dr 8% and ∆T˜ 18 K. Curves are plotted for different values of the heat exchanger length. REFERENCES

1. A.S. Worlikar and O.M. Knio, Numerical Heat Transfer A, The results also indicate that for the present set of con- 35, 49–65 (1999). ditions, a loss of symmetry occurs in the ﬂow ﬁeld for 2. E. Besnoin and O.M. Knio, Numerical Heat Transfer A, to appear (2001). Dr 5% (Fig. 4). This phenomenon is accompanied by the appearance of large-amplitude wavy motion within 3. A.S. Worlikar, O.M. Knio and R. Klein, J. Comput. Phys. the gap and between the plates. 144, 299–324 (1998). Separation of Gas Mixtures by Thermoacoustic Waves

D. A. Geller, P. S. Spoora, and G. W. Swift

Condensed Matter and Thermal Physics Group, Los Alamos National Laboratory, Los Alamos NM 87545 USA a Current address: CFIC Inc., 302 Tenth Street, Troy NY 12180 USA

Imposing sound on a binary gas mixture in a duct separates the two gases along the acoustic-propagation axis. Mole-fraction differences as large as 10% and separation fluxes as high as 0.001 M-squared c, where M is Mach number and c is sound speed, are easily observed. We describe the accidental discovery of this phenomenon in a helium-xenon mixture, subsequent experiments with a helium-argon mixture, and theoretical developments. The phenomenon occurs because a thin layer of the gas adjacent to the wall is immobilized by viscosity while the rest of the gas moves back and forth with the wave, and the heat capacity of the wall holds this thin layer of the gas at constant temperature while the rest of the gas experiences temperature oscillations due to the wave's oscillating pressure. The oscillating temperature gradient causes the light and heavy atoms in the gas to take turns diffusing into and out of the immobilized layer, so that the oscillating motion of the wave outside the immobilized layer tends to carry light-enriched gas in one direction and heavy-enriched gas in the opposite direction. Experiment and theory are in very good agreement for the initial separation fluxes and the saturation mole-fraction differences.

INTRODUCTION standing-wave phasing in a channel whose diameter is much larger than these penetration depths, we might A few years ago, we were experimenting [1] with think of the wave as consisting of four sequential steps acoustically coupled acoustic resonators, using two equally spaced in time, as illustrated in Fig. 1. In the identical half-wavelength resonators, each of which first step, while the pressure is high, the time- was driven at its natural resonance frequency by a dependent part of the temperature has a steep gradient thermoacoustic engine. Under some circumstances, within a thermal penetration depth of the wall due to these experiments demonstrated mode locking when the adiabatic temperature rise in the gas far from the the two resonators were connected through a smaller- wall and the large solid heat capacity of the wall itself. diameter half-wavelength coupling tube. When those During this time, thermal diffusion drives the heavy measurements were made with a helium-xenon component down the temperature gradient toward the mixture, we sometimes noticed a totally unexpected near-perfect equality in the resonance frequencies of the two resonators while we were trying to force unequal resonance frequencies by imposing a temperature difference (as large as 15oC) between the resonators. This behavior could not be explained by any obvious candidates such as temperature uncertainty or geometry mismatch between the resonators. We concluded that the sound wave in the small-diameter acoustic coupling tube was somehow separating the helium and xenon, thereby enriching one resonator with helium and the other with xenon.

We realized that this mass separation could be due to a combination of three effects in the boundary FIGURE 1. Illustration of the separation process occurring layers adjacent to the inside surface of a tube: near a solid wall (hatched) in a standing wave in a gas mixture with Prandtl number near ¼. Solid arrows show oscillating temperature gradients in the thermal motion of the heavy component, and light arrows show boundary layer, thermal diffusion, and oscillating motion of the light component. The arrow lengths represent velocity gradients in the viscous boundary layer. In a velocity, and widths represent local concentrations. (a) typical mixture of helium and xenon, the Prandtl through (d) show processes occurring at time intervals number is about 1/4, so the viscous penetration depth separated by ¼ of the period of the wave, as described in the is about half of the thermal penetration depth. For text. wall and the light component up the temperature one end and argon toward the other end. The gradient away from the wall. Hence, at the end of this concentrations at the ends, which reach extremes of time the gas near the solid wall is enriched in the 45% and 55%, are detected as functions of time by heavy component and depleted of the light component, exciting acoustic resonance (near 3 kHz) in small while the gas approximately a thermal penetration cavities at each end and inferring the concentrations depth from the wall is enriched in the light component there via the dependence of sound speed on average and depleted of the heavy component. In the second molar mass [5]. step, the gas moves upward, with a steep gradient of velocity within a viscous penetration depth of the wall Measurements include the initial rates of mole flux due to viscosity. toward each end, before a significant concentration gradient has had time to develop, and the saturation During this time, the heavy-enriched gas is concentration difference reached in steady state. relatively immobilized in the viscous boundary layer, These are measured as functions of pressure while the light-enriched gas, just outside of the viscous amplitude, volume-velocity amplitude, and phase boundary layer, moves easily upward. In the third between pressure and volume velocity. step, low pressure reverses the sign of the temperature gradient, so the thermal diffusion reverses direction, In these experiments, tube length and frequency have forcing the heavy component away from the boundary been chosen so that the tubes are “short,” much shorter and the light component toward the boundary. Thus, in than a wavelength and also short enough that the the fourth step, light-enriched gas is relatively resulting concentration differences from one end to the immobilized while heavy-enriched gas moves easily other do not cause large differences in thermophysical downward. The net, time-averaged effect of these four properties. This eases comparison of the experimental steps is that some of the heavy component moves results with the theory. We are confident that longer downward while some of the light component moves tubes and/or higher frequencies would result in larger upward. separations. The larger tube diameter is chosen to be ten times the largest of the three penetration depths, so RESULTS TO DATE that the boundary-layer approximation is useable in the theory. The smaller of the two tubes is chosen to test Thus far, the agreement between our theory [2] of this the more challenging Bessel-function aspects of the process and our measurements [3,4] of it is very good. theory.

The theory is based on a monofrequency, steady-state acoustic approximation to the equations of momentum, continuity, heat transfer, and mass diffusion, assuming ACKNOWLEDGMENTS that the viscous penetration depth, thermal penetration depth, mass-diffusion penetration depth and the tube This work is supported by the Office of Basic Energy radius are all small compared to the tube length. The Sciences in the US Department of Energy. three penetration depths themselves are of comparable magnitudes. Results of the theory include the time- averaged mole fluxes of each component along the tube axis. This result is expressed as a function of REFERENCES pressure amplitude, volume-velocity amplitude, phase between pressure and volume velocity, axial 1. P. S. Spoor and G. W. Swift, J. Acoust. Soc. Am. 106, concentration gradient, and gas properties (including 1353-1362 (1999). mean pressure and temperature, thermal diffusion ratio, viscosity, thermal conductivity, and mass 2. G. W. Swift and P. S. Spoor, J. Acoust. Soc. Am. 106, diffusivity). Setting the time-averaged mole fluxes 1794-1800 (1999). equal to zero yields the axial concentration gradient at which the effect saturates. 3. P. S. Spoor and G. W. Swift, Phys. Rev. Lett. 85, 1646- 1649 (2000). The experiments begin with a mixture of 50% helium 4. D. A. Geller and G. W. Swift, “Saturation of boundary- (mole fraction) and 50% argon near atmospheric layer thermoacoustic mixture separation,” submitted to pressure. The separation tubes are 1 meter long and J. Acoust. Soc. Am. (2000). either 5 mm or 15 mm in diameter, filled with the mixture and driven by bellows-sealed pistons. When a 5. E. Polturak, S. L. Garrett, and S. G. Lipson, Rev. Sci. tube is insonified at 10 Hz, helium separates toward Instrum. 57, 2837-2841 (1986). Integration of Resonant Electric Power Conversion with Thermoacoustic Engines and Coolers

J. A. Corey & P. S. Spoor CFIC, Inc. 302 Tenth St. Troy, NY USA 12180

Thermoacoustic conversion of energy (and power) has been proven effective, reliable, and inexpensive in the laboratory. Thermal power is both available as a source and useful as an output, but because acoustic power is limited in applicability, a second transduction seems essential to broad application of thermoacoustics. This paper describes high-efficiency, mechanically-resonant acoustic-electric transducers that can be incorporated directly into sealed thermoacoustic devices to make thermal-to-electric convertors (engine-generators, coolers, and heat pumps) that retain all of the qualities that make thermoacoustics attractive but couple directly to the ubiquitous electric power infrastructure. Fundamentals of such transducers are explained, with examples of past and present art. System and impedance-matching issues are presented with methods for resolution. Design and performance implications are presented for incorporating resonant electro-acoustic transducers into thermoacoustic machines (for both 'standing' and 'travelling' wave phasings), with examples from current CFIC practice and resulting products.

INTRODUCTION magnet systems combine the structural robustness of The authors are developing compact thermoacoustic the switched reluctance with the fully-utilized iron of energy convertors, both engine-generators and heat the moving coil, providing lower mass and scalability pumps (including refrigeration). We are using to multi-kilowatt sizes. Most moving magnet designs resonant linear motor/alternators in place of a long are axisymmetric (Redlich, Bhate)2. This may derive gas column or Helmholtz cavity resonator. We use from historical use of gas bearings or familiarity with an efficient, flexure-supported, moving permanent rotary equipment. One moving magnet type (Yarr)3, magnet design for the electroacoustic transducer. called STAR motors, uses a non-rotating flexure These transducers have proven successful in bearing and a non-circular array of salient magnets. capacities from 100 to10,000 watts. This compacts the unit and allows further increases in scale without structural deficiency. This type is the TRANSDUCER TYPES focus of the systems work reported here. Electroacoustic power transduction is commonly done with moving coil or moving armature drives1. IMPEDANCE, DYNAMICS, AND In smaller capacities, piezoelectric elements have CAPACITY MATCHING also been successful. Broadband transducers of PARAMETERS either type are rarely more than a few percent It is through tuned, resonant operation at a single efficient, with theoretical limits below 70%. frequency that dynamic reciprocating transducers can Meanwhile, apart from the larger acoustics achieve high efficiency and power-handling capacity. community, free-piston Stirling engines and coolers The velocity of the magnet-carrying piston must be were developed over the last 20 years, using moving- very nearly in phase with the sum of forces acting on magnet and switched reluctance alternators and it, such that only real power flows through the motors. These can be considered as electroacoustic system, while the motion is sustained. The drives, too, though not broadband devices. These transducer typically has an electrical circuit electrodynamic transducers, like moving coil resonance, too, but that is designed far above machines, convert between current and force, through operating frequency. the well known relation F BI . For loads with high impedance, where the transducer Moving magnet systems have the greatest variety of must provide high force at low flow, piezoelectric sub-types. Moving coil designs are restricted by their transducers are well suited. In applications at typical requirement for thin coils to minimize the airgap that power systems frequencies (e.g., 60 or 400 Hz), the magnets must overcome, so the coils are almost powers at kilowatt levels (and above) require either always cylindrical for low mass and rigidity. excessive area or stroke in piezo units. Switched reluctance systems use their iron in a single Electrodynamic transducers are essential. Large -sided manner, requiring twice as much as iron as stroke-handling capability is especially important designed for fully-reversed flux loading. Moving when the transducer replaces the bulk of an acoustic resonator, as the moving piston must provide the transducer drives for 50 and 60 Hz pulse tube same reactive volume flux as the resonating gas cryocoolers, delivering from 150 to 16,000 acoustic column it replaces. watts. In TARGET, there is a 6.5 bar pressure wave In properly tuned resonant operation, the inherent on a 40 bar mean, while a 5-inch piston delivers 0.07 stiffness of the motor plus the reactive pressure m 3 /s at 180 Hz and a stroke of 5 mm peak; the forces on the piston create a spring which resonates phase between pressure and velocity is 88.4 degrees. with the moving mass at the drive frequency. In that A typical pulse tube drive (midrange, at about 2 case, the current to the transducer creates just that kilowatts) has a 3 bar wave on an 18 bar mean, with 3 force IBF necessary to deliver Vp power to twin 4.5-inch pistons delivering 0.06 m /s at 50 Hz the load and overcome mechanical drag; and a stroke of 9 mm peak; the phase angle between pressure and velocity here is 76 degrees. This drive mathematically, cos RApIB MX v. The has demonstrated over 80% efficiency (electric phase angle between the pressure p X across the power in to acoustic power out). piston and its velocity v can vary from near 0 degrees for a low power, close coupled transducer CONCLUSIONS driving a Stirling or pulse-tube cryocooler load, to Integral electroacoustic transducers of the moving near 90 degrees for standing-wave thermoacoustic magnet electrodynamic type, in particular STAR devices and high-power pulse-tube drivers. A phase motor/alternators have been applied to both standing angle near 90 degrees implies that most of the and travelling wave devices, with overall power pressure force on the piston is reactive, not work- efficiencies of over 80%. Matched combination of producing or friction-overcoming. The inherent thermoacoustic and electroacoustic convertors stiffness of our motors is primarily magnetic, with the provides a new class of compact, reliable, and suspension straps contributing slightly (<10%). As practical cooling machines and engine-generators the motors grow in size and power, the moving ranging from 10's of watts to 10's of kilowatts. magnet mass grows faster than the magnetic stiffness, so more reactive pressure force is needed to maintain resonance at a given frequency. In addition, some ACKNOWLEDGMENTS loads look more reactive, i.e. “springy,” than others; Work reported here has been supported in part by standing-wave acoustic engines and refrigerators in the New York State Energy Research & particular are very reactive by nature. Since many Development Authority, and by commercial partners. systems large and small have charge pressures and oscillating pressures that are limited by factors such as pressure-vessel safety, the reactive pressure, or “gas spring,” is usually increased by increasing the REFERENCES piston area A. This in turn affects other design parameters; for instance, a large part of the velocity- 1. Hunt, Frederick V., Electroacoustics, the Analysis of transduction and its Historical Background, 1982 dependent loss factor RM represents gas leaking past Acoustical Society of America the piston seals. As the seal perimeter grows, so does 2. S Patent numbers 4602174(Redlich), 4349757 (Bhate) the seal-leakage loss, which may affect the desired 3. Yarr, George A. et al, US Patent 5146123 stroke. The maximum motor efficiency occurs when 2 electrical loss (e.g. RI E ) matches the “mechanical 2 loss” v RM , which implies an optimum stroke for a given amount of output power desired. Thus, designing an efficient resonant power system requires careful consideration of the load impedance, the load power, the motor parameters, and the various loss factors, in order to assure success.

TEST EXAMPLES Examples of systems using integral, tuned electrodynamic transducers at CFIC include TARGET, a compact 200 Hz thermoacoustic electric generator of 500 watts capacity, and a range of twin- Sound Synthesis of Combustion Noise for Perceptive Evaluation. G.Guyadera,b, B. Saint-Loubrya, N.Hamzaouib, C. Boissonb

a Direction de la Recherche Renault, 78288 Guyancourt, France, mail: [email protected] b Laboratoire Acoustique et Vibration, INSA Lyon, 69621 Villeurbanne, France

The document presents a method to synthesise the combustion noise radiated by the cylinder head. The first stage is the calculation of the excitation, the cylinders pressures, based on the Wiebe’s combustion energy release model. The interest and the originality of the method lie in the estimate of the transfers of the energy of combustion. The transfers through the cylinder head are estimated starting from simple temporal recordings of the pressure in cylinder and the vibrations of cylinder head in some points. Measurements are taken on engine under operation and a post treatment makes it possible to retain only the useful part of the signal. From the treated signal one calculates the transfer functions between the pressure and the point of the cylinder head considered. This method makes it possible to obtain a transfer for each cylinder representative of the transfer of the energy of combustion (by the cylinder head only). The noise of combustion is thus generated in temporal space by convolution product of the pressures cylinders by the transfers.

INTRODUCTION COMBUSTION ENERGY WITH WIEBE’S MODEL The objective of the method is to create a sound synthesis for internal combustion engine. The harmonic Burned fraction and heat supply approach of the issue has been replaced by a typical view of combustion events, at a “microtemporal” scale, so Wiebe’s model is based on the burned fraction and as to reach a realistic result of combustion noise. We take it is shown according to the crankshaft angle. into consideration the increase of the pressure induced by ! . m 11 combustion like short temporal events which are fa() 1 exp" /0 2 (1) recurrent at regular intervals. The structure of the # 03 synthesis is based on the physical or signal analogy with : beginning of the combustion which is described in Figure 1. 0 variation of an angle during the The main idea is to get back the contribution of the combustion energy of combustion on a few points on the cylinder m and a coefficients influencing the head of the GMP and to transform them in acoustic combustion speed. pressure. The obtained sound synthesis allows us to assess the contribution of the main characteristics of the The amount of heat brought by combustion for each considered sources in the radiated noise by a GMP. We increment of the crankshaft angle gives : have introduced in this paper a part which characterizes RQdf 11fm() a. m the transfer between the combustion and the vibration of QQ /o the cylinder head.. R injd inj 0 (2)

Qinj est la quantité de chaleur globale

Energy Source signal Cylinder pressure generation generation Classical laws of thermodynamics allow us to determine the gradient of pressure with some heat Vibratory filtering transfer supply and a light variation of : dP dQ C dV R ( (1 v )P ) (3)

d d R d CvV Acoustical Mixing radiation

Figure 1: Physics/signal analogy TRANSFER ESTIMATION AND convolution product is done for each cycle SYNTHESIS. (compression, combustion, relaxation) with different pression cylindre pression transferée à 6 x 10 6 l'extérieur du moteur The studied path is the transfer of combustion noise 5 140 4 C through the cylinder. Modeling has been done from 3 120 transfer functions and audio filters have been 2 O 1 N 100 assessed from those transfer functions. 0 V 80 -1 0 500 1000 1500 2000 2500 O 60

-4 L x 10 2.5 Procedure of treatment U 40 2

1.5 T 20 1 I 0 The original data base consists of a succession of 0.5 O 0 measures with three cylinder pressures and three N -20 -0.5 0 500 1000 1500 2000 2500

-1 accelerometers which have been simultaneously 0 50 100 150 200 measured. Measurement files have been taken into filtre audio (type reponse account so as to only keep temporal areas which are impulsionelle) useful for computation. Figure 2 describes the process showing the chosen temporal areas for the Figure 3: synthesis principle computation of filters connected with combustion noise. The example consists of two curves filters to transfer energy. Then, signals have been representing two measurement points on a cylinder gathered one after another to obtain the signal on and a third curve represents the pressure into a each considered cylinder point. We can thus build a sound synthesis (an accelerometric one for the zones utilisées pour les calculs de filtres audio moment) at the level of one or several points of the cylinder, from parametric variations (Wiebe’s 50 model, RPM, ……). 0

-50 Zone non utile CONCLUSION 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 The obtained results with the studied structure in this Zone non utile 50 paper are encouraging and allow us to contemplate a 0 generalization of the method for other sources of the -50 engine. Nevertheless, the estimation of the transfers 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 is rather complicated and may be problematic for 6 x 10 6 some sources which could be less energetic or

4 concealed. It will be then preferable to establish a

2 Zone non utile measurement with a calibrated excitation or to use 0 results from a structure computation. The suggested -2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 method also remains to be completed by the calculation of the acoustic radiation of the structure; an orientation to simplified integral methods seems Figure 2: cylinder pressure and vibration of a cylinder to be adequate concerning this issue. head and areas of interest. cylinder. The measurement file which has been used, ACKNOWLEDGEMENTS not only contains the researched events, but also all the other phenomena like valve impacts... The precision of areas allowing to extract the adequate This text is a part of a thesis of a doctorate initiated data, is an important criterion for the quality of the by the direction of research of Renault with the obtained transfer functions. We compute the scientific monitoring of the vibroacoustic laboratory of INSA Lyon. averaged FRF(/P) between a vibratory measurement point and a measured cylinder pressure from extracted temporal samples. Once this FRF is REFERENCES obtained, it will allow us to compute the audio filter. The computed cylinder pressure (eq3) has been 1. M.Ishihama & Y.Sakai, Automotive engine sound creation convoluted with the impulsional response of the specifying a small number of parameters., in Proceedings of internoise 2000 Nice France, edited by D.Cassereau audio filter using Wiebe’s model (eq1&2), so as to 2. T.Priede & &K. Dutkiewicz, The effect of normal combustion generate a vibratory acceleration on one point of the and knock on gasoline engine noise., SAE 891126. cylinder, as it is described in figure 3. In order to use 3. H. Kamp & J. Spermann, New methods of evaluating and different transfer functions for each cylinder the improving piston related noise in internal combustion engines, SAE 951238. Thermodynamical Mode Selection Rule Observed in Spontaneous gas oscillations T. Biwa, Y. Ueda, T. Yazaki, and U. Mizutani (1) Department of Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan (2) Department of Physics, Aichi University of Education, Kariya, 448, Japan e-mail: [email protected]

We have built a thermoacoustic engine that shows a transition from the standing to the traveling wave mode through quasiperiodic states with increasing heat supplied to the engine. We found that an increase in entropy flow determines the oscillating mode. We experimentally propose a thermodynamical mode selection rule, which acts as a guiding principle for oscillating modes appearing in a thermoacoustic engine.

Work flow I˜ and heat flow Q˜ are proposed as the mode with a frequency of 100 Hz. The energy conver- ˜ basic concepts in thermoacoustic phenomena from the sion is illustrated in the inset to Fig. 1. Heat flow QH view point of heat engines[1-3]. Experimental studies entering from the hot end of the stack is partly converted on thermoacoustic engines so far concentrated on stand- to the output power DI˜ within the stack, and the rest, ˜ ing wave engines [4] using a resonator, where a stand- QC , is removed from the cold end of the stack. The ing wave contributes to the energy conversion through energy conservation law assures the relation Q˜ = ˜ ˜ H the irreversible thermodynamical process. More re- QC +DI . cently, traveling wave engines [5, 6] have been con- We focus on entropy flow S˜ and its increase DS˜ structed by using a looped tube, where a traveling wave through the stack [7]. Entropy flow S˜ is defined as is capable of converting the energy through the revers- S˜ = Q˜ / T by using heat flow Q˜ passing through the ible thermodynamical process. In this experiment, we cross section at temperature T. Therefore, the increase have built a thermoacoustic engine by combining both the looped tube and the resonator, so that both modes 150 can be induced. It is of great importance to examine what determines the oscillating mode in this engine. We ex- 100 perimentally propose a thermodynamical mode selec- 50 Q*˜ tion rule, which serves as an important guiding principle (a) in thermoacoustic engines. 0 Figure 1 shows the schematic illustration of the 600 no oscillation present engine. Total lengths of the loop and the reso- oscillation nator are 1.23 and 1.00 m, respectively. A ceramic stack, 500 having square channels with their sides of 0.93 mm, is 400 located in the loop together with two heat exchangers. ˜ (b) When heat flow QH supplied by an electrical heater ex- 300 0.25 ceeds a critical value Q˜ * , the gas in the present engine no oscillation spontaneously begins to oscillate in the standing wave 0.20 oscillation

TH 0.15 ˜ QH heat stack DI˜ 0.10 ˜ exchangers QC T 0.05 stack H (c) TC TC 0.00 0 20 40 60 80 100 120 ˜ QH (W) looped tube resonator ˜ ˜ FIGURE 2. The variation of DI (a), T H (b) and DS (c) as ˜ FIGURE 1. Schematic diagram of the present thermoacoustic a function of Q H , when the gas oscillations are absent (solid engine and an illustration as a heat engine (inset). An electri- circles) and present(open circles). Open diamonds in (b) are cal heater was wound around the hot heat exchanger (upper) measured when the oscillations are suppressed and those in (c) represents the corresponding variation of S˜ . The verti- and the cold heat exchanger (lower) was kept at TC (room D termperature) by a cooling water pipe. cal dashed line represents the critical heat flow Q˜ * = 60W.. ˜ ˜ ˜ ˜ in the entropy flow DS is given by DS =QC /TC-QH /TH. ing non-oscillating state. In other words, since the os- When the oscillation is self-sustained, the output power cillating branch of DS˜ deviates downward from the non- DI˜ must be finite, and hence, DS˜ is rewritten as oscillating one above Q˜ * =60 W, the present engine selects the oscillating one, being a lower branch. ˜ ˜ ˜ DS = (1/ TC - 1/ T H ) ×QH - DI / TC , (1) Figure 3 shows the variations of TH with and without the gas oscillations up to 250 W. As shown by the ˜ and therefore, DS is a mesurable quantity through mea- circles in Fig. 3, TH in the present engine increases lin- ˜ ˜ Q˜ * surements of DI and TH as a fucntion of QH . early above , but the slope of T H becomes small, when ˜ ˜ ˜ Figures 2(a) and (b) show the QH -dependences of DI QH exceeds 160 W. This is brought about by a change and TH , respectively. The variation of TH , measured in the oscillating mode from the standing to the when the oscillations are suppressed, is also shown in quasiperiodic states involving both the standing and trav- Fig. 2(b) by the dotted curve. The increase in entropy eling wave modes [8]. For comparison, we also plotted ˜ flow DS in the absence of oscillations is simply deduced in Fig. 3 the variation of TH , by the dashed line, mea- ˜ ˜ ˜ from eq. (1) as DS = (1/ TC - 1/ T H ) ×QH , since DI is sured for the looped tube engine generating the same zero. The increase in entropy flow DS˜ without oscilla- traveling wave oscillations. The dashed line for the trav- tions calculated by inserting above data into eq. (1) is eling wave mode lies above the solid line for the stand- shown by the dotted curve in Fig. 2(c). In the region ing wave mode above Q˜ * , but crosses with it at about Q˜ Q˜ * ˜ when H > , the self-sustained oscillations are in- 200 W. The TH versus QH should directly reflect the ˜ ˜ ˜ ˜ duced. Consequently, DI starts to increase linearly DS versus QH relation, since the contribution of DI to ˜ * ˜ ˜ * ˜ above Q , while TH versus QH reduces its slope at Q . DS is so small that we can safely ignore it. Therefore, The branch of DS˜ in the presence of oscillations is also we see that the standing wave mode is induced instead plotted in Fig. 2(c) by a solid curve. Apparently, it lies of the traveling wave mode immediately above Q˜ * be- below the non-oscillating one marked with dotted line cause of the possession of DS˜ of the former lower than * in the region above Q˜ . This is because both the pro- that of the latter. This validates the applicability of the ˜ duction of DI and the reduction in TH contribute to the thermodynamical mode selection rule that we propose. reduction in DS˜ given by eq. (1). Now we propose the thermodynamical mode selection rule such that nature REFERENCES selects a mode having the lowest increase in entropy flow among the thermodynamically permissible modes includ- [1] J. Wheatley, T. Hofler, G. W. Swift, and A. Migliori, Phys. standing and Rev. Lett. 50, 499 (1983); J. Acoust. Soc. Am. 74, 153(1983); no oscillation standing wave traveling waves Am. J. Phys. 53, 147, (1985). [2] G. W. Swift, Phys. Today 48, No. 7, 22 (1995); J. Acoust. 700 Soc. Am. 84, 1145, (1988). [3] A. Tominaga, Cryogenics 35, 427, (1995). [4] A. A. Atchley, J. Acoust. Soc. Am. 95, 1661 (1994); A. 600 A. Atchley and F. Kuo, J. Acoust. Soc. Am. 95,1401 (1994); T. Yazaki, A. Tominaga, and Y. Naramara, J. LowTemp. Phys.

41, 45 (1980). 500 [5] T. Yazaki, and A. Tominaga, Proc.R. Soc. Lond.A 454, 2113, (1998). [6] S. Backhaus, and G. W. Swift, Nature 399, 335 (1999). 400 [7] A. Tominaga, Butsuri 55, 326, (2000) (in Japanese);A. Tominaga, “Thermoacoustic Phenomena: Interdisciplinary

300 Phenomena of Fluid mechanics and Thermodynamics” in Pro- ceedings of The 50th Nat. Cong. of Theoretical & Applied 0 50 100 150 200 250 ˜ Mechanics 2001, Japan Society of Civil Engineering, Tokyo. QH (W) [8] The traveling wave mode refers to the acoustic traveling FIGURE 3. The variation of TH (solid and open circles) measured in the present engine. Vertical dotted lines repre- wave with a fundamental frequency of 273 Hz running around sent boundaries among the no-oscillation, standing wave the looped tube. mode, quasiperiodic states. The dashed line with open diamonds represents the variation TH when the gas oscillations are absent, and the dashed line is measured in a looped tube engine without a resonator in Fig. 1. Study on basic characteristics of thermal compressor for pulse tube cryocooler H. Sugitaa,S.Toyamaˆ a, S. Zhoub and Y. Matsubarab a Ofﬁce of Research and Development, National Space Development Agency of Japan, Tsukuba, Ibaraki, Japan bAtomic Energy Research Institute, Nihon University, Funabashi, Chiba, Japan

Standing-wave-type thermal compressors can generate oscillatory pressure waves with the thermoacoustic effect. Because this type of compressor without mechanical moving parts is driven only by the thermal input, reliability of a compressor system can be greatly enhanced. In this study, a tentative thermal compressor for a pulse tube cryocooler is designed and investigated. That compressor consists of heat exchangers, a stack of SUS, a buffer, a reservoir and a resonator tube of 7 m at the maximum. The hot heat exchanger is heated up by electrical heaters, while the cold heat exchanger is cooled down by water of 300 K. First, characteristics of the resonator connected to a mechanical compressor is examined under various driving conditions. The energy ﬂow in the whole compressor system and characteristics of each component will be analyzed and discussed to clarify the loss mechanism.

RESEARCH BACKGROUND allowing for efficiency of a connected pulse tube cryocooler. Therefore, the Helium gas or a mixing gas of In recent years, the number of space missions with the Helium and the Argon is used as a working fluid. mechanical cryocoolers continues to grow, owing to im- Principal specifications of the oscillator are shown in provement in sensors with high sensitivity for observa- Table 1. The hot heat exchanger is given the maximum tions of the earth and the universe. Practical use of cry- thermal input of 300 W by heaters, while the cold heat ocoolers enables the long mission period, and reduces the exchanger is cooled by water with room temperature. The weight and the volume which have been occupied by con- resonator tube is of the inner diameter of 39.4 mm and is ventional cryostats and cryogens so far. of the maximum length of 7 m, which is changeable by 1 Pulse tube cryocoolers are of small vibration and of m. A removable reservoir of 20 l can be attached to the high reliability because of no moving part in the cold end of the tube. head. It is, thus, expected that pulse tube cryocoolers Figures 1 and 2 illustrate experimental apparatus used would be in great demand not only for the ground use in this study. but also for space. The authors have begun a research on space cooling systems for the next generation by re- placing conventional mechanical compressors with ther- EXPERIMENTAL RESULTS mal compressors, in order to enhance reliability and to diversify energy sources as space applications. Currently, Here, the resonator is connected to a mechanical com- a feasibility of thermal compressors for pulse tube cry- pressor, in order to investigate characteristics of the res- ocoolers of 1-2 W @ 60-80 K has been investigated. onator under the condition of the mean pressure of 1.0 Thermoacoustic engines such as a thermal compressor MPa. Pressure amplitudes for various driving frequen- have a possibility that the thermal energy (e.g. the solar cies are depicted in Figures 3 and 4. Figure 3 shows that energy) could be effectively utilized to control temper- resonance frequencies are located around 40 Hz. Figures atures of components or instruments in spacecrafts and 4 and 5 are experimental results when the 100% Helium space stations in the wide range. gas is used without the reservoir of 20 l. Figure 4 gives the In this paper, we designed and produced a ground test resonance frequency of 71 Hz, roughly matching the cal- unit (GTU) of a standing-wave-type thermal compressor culated data. Then, the consumed work in the resonator in order to obtain basic characteristics of a thermoacous- reaches about the maximum of 15 W. Next, the swept vol- tic oscillator and a resonator. ume of the mechanical compressor is changed, fixing the frequency of 71 Hz. Figure 5 means that the square of the pressure amplitude is proportional to the consumed work EXPERIMENTAL APPARATUS in the resonator. In this congress, the performance of the GTU will be reported based on results obtained in further experiments In this study, the standing-wave-type of a thermoacus- and analyses. tic driver is chosen to study the loss mechanism in each component. Its driving frequency is aimed below 50 Hz, Table 1. Specifications of the thermoacoustic oscillator

Type Material Size Cold heat exchanger Radial multi-slit Copper Heat exchange area 980 cm2 Stack Mesh Stainless steel Mesh size No.10, Wire φ : 0.505 mm Hot heat exchanger Radial multi-slit Stainless steel Heat exchange area 980 cm2 Buffer Cylinder Stainless steel Inner volume : 0.660 l

FIGURE 1. Experimental apparatus of a standing-wave-type thermal compressor

FIGURE 2. Scheme of the thermoacoustic oscillator FIGURE 4. Relation between the frequency and the pressure amplitude, when the dead volume and the swept volume of the mechanical compressor are of 45 cc and 4 cc, respectively.

FIGURE 3. Relation between the frequency and the pressure amplitude : For A-D, a mixed gas (He:Ar=3:1) is used with the FIGURE 5. Relation between the square of the pressure ampli- reservoir. For E, the 100% He is used without the reservoir. The tude and the consumed work in the resonator when the swept swept volume and the dead volume of the mechanical compres- volume of the mechanical compressor is changed and the fre- sor for A-D are (4cc, 45cc), (10cc, 45cc), (10cc, 10cc), (12cc, quency is ﬁxed at 71 Hz. 45cc) and (10cc, 45cc), respectively. Influence of Acoustic Streaming on Temperature Distribution in Annular Thermoacoustic Prime-movers.

S. Joba, V. Gusevb, P. Lottona, M. Bruneaua

a Laboratoire d’Acoustique, UMR-CNRS 6613, IAM, Faculté des Sciences, Université du Maine, av. O. Messiaen, 72085 Le Mans Cedex 09, France. b Laboratoire de Physique de l’Etat Condensé, UMR - CNRS 6087, Faculté des Sciences - ENSIM, Université du Maine, 72085 Le Mans Cedex 9, France.

Experiments have been done with an annular travelling wave thermoacoustic prime-mover, to measure the velocity of spontaneously generated acoustic streaming, when the thermoacoustic instability is reached. Results of temperature measurements, below and above this threshold, at some characteristic points of the annular engine clearly indicate the presence of a closed-loop mass flow, generated by the travelling acoustic wave. The experimental estimation of the fluid velocity lies in the range of 1 cm/s to 10 cm/s (while the acoustic velocity lies in the range of 0.3 m/s to 3 m/s.) and is found to be proportional to the square of the pressure wave amplitude, in accordance with our recent theoretical predictions [1].

INTRODUCTION Our experimental set-up (Figure 1), constructed from a gas-filled annular wave-guide of length L ≈ 2 m , Since few decades, a lot of research works, initiated by employs a quasi-adiabatic interaction between a non- << N. Rott [2], dealing with thermoacoustic (TA), render uniformly heated stack of length H L (a ceramic the growing interest in studying interaction phenomena with square pores) and the gas oscillations. The left between thermal and acoustical waves. The idea of a side of the stack is maintained at cold temperature TC traveling-wave TA engine was put forward more than (heat flux QC ), while the other side is heated ( QH ) at 20 years ago [3]. However the first annular TA prime- temperature T . A second heat flux Q is applied, at mover, which employs a quasi-adiabatic interaction H C between a non-uniformly heated porous material and the right side of TH , to control the spatial extent of the the gas oscillations for the transformation of the decreasing temperature. The fluid mean temperature is thermal energy into mechanical energy, was built only maintained at cold temperature TC outside the TA in 1997 [4]. In 1999, a closed-loop TA engine that stack. Above the threshold condition (in terms of the employed quasi-isothermal sound propagation through ratio T T ) of the TA instability, a travelling the stack (through the regenerator) was reported [5]. H C This experiment demonstrated that, in a TA Stirling acoustic wave and an acoustically induced mass flow engine, circulation of the gas is induced by the are observed in the system. This closed-loop mass traveling wave. flow contributes to the enthalpy transport in an annular device, and because the thermoacoustic amplification Q effect is a function of the temperature distribution in H Q QC C the stack, this acoustic steaming flow must be taken (a) into account to predict the behavior of annular thermoacoustic prime-movers. 0 H (b) VELOCITY MEASUREMENT

In order to determine the mass flow and the streaming velocity, the following simplified model of the FIGURE 1. Annular thermoacoustic prime-mover. temperature distribution is used. The temperature (a : thermoacoustic stack, b : annular wave-guide). distribution inside the cylindrical tube is considered to be independent of the radial coordinate. The enthalpy υ In Figure(3), xm is the streaming velocity averaged flux J is over the cross-section of the wave-guide , and p + is = − ∂ ∂ + J A Tm / x c p MTm , (1) the measured amplitude of the generated pressure wave travelling in the positive axial direction in the where x is the axial coordinate, T (x) is the mean wave-guide. Note that the more heating is supplied, the m bigger is the acoustic wave amplitude. temperature of the fluid (see Figure 2), c p is the air heat capacity and M is the mass flux. 0.06

TH υ xm (m/s) ( ) Tm x 0.00 + 2 0.0 p (kPa)2 3.0

TC 0 x H FIGURE 3. Measurement (dots with error bars), and predicted value (line) of the streaming velocity. FIGURE 2. Temperature distribution in the non uniformly heated thermoacoustic core, below (dashed line) and above (solid line) the TA threshold. CONCLUSION

The first term in Eq. (1) describes the thermal It clearly appears that acoustically-induced mass flow induces large heat convection, leading to strong conduction, where A ≡ (kS) is the equivalent thermal // modification of the temperature profile. As a resistance ( k is the thermal conductivity and S is the consequence, this streaming should be included into cross-section) of all materials (in parallel) subjected to analytical formulation to predict the behavior of the heat flux (wall of the tube, ceramic, air…). Then, thermoacoustic amplification. by modeling the heat exchange between the external surface of the steel tube and the surroundings using Newton’s law of cooling, a closed equation for the ACKNOWLEDGMENTS temperature distribution is obtained : ∂J / ∂x + h(T − T ) = 0 . Defining the normalized m C This work was supported by the Délégation Générale θ ( )= [ ( )− ] [ − ] temperature x Tm x TC TH TC (where TH is de l’Armement under contract N° 99.34.072/DSP and the hottest temperature in the non-uniformly heated through the DGA-CNRS Ph.D. fellowship of the first stack) and substituting Eq. (1), we get author.

∂ 2θ ∂ 2 − ∂θ ∂ − θ = A / x c p M / x h 0 , (2) REFERENCES where h is a characteristic coefficient of heat transfer exchange with surrounding (principally due to natural 1. V. Gusev, S. Job, H. Bailliet, P. Lotton and M. Bruneau, J. Acoust. Soc. Am., 108(3), 934-945, (2000) convection and radiation). It is straightforward to find the solution (depending on position x , and on 2. N. Rott, Z. Angew. Math. Phys., 20, 230-243, (1969). parameter h and M ) of the linear differential equation with constant coefficients (2) satisfying 3. P. H. Ceperley, J. Acoust. Soc. Am. 66, 1508-1513 appropriate boundary conditions. Finally, by mapping (1979). the solution of Eq. (2) to the temperature measurement at five characteristic point in the TA stack, the mass 4. T. Yazaki, A. Iwata, T. Maekawa and A. Tominaga, Phys. Rev. Lett. 81, 3128-3132 (1998). flux M can be experimentally determined. The velocity associated to this mass flux is presented in 5. B. Backhaus and G. W. Swift, J. Acoust. Soc. Am. 107, Figure (3), and is compared to its predicted value [1]. 3148-3166 (2000). Thermoacoustic Simulation with Lattice Gas Automata

Y. Chen, X. L. Chongxu Yang, Z. Li and Y. Ding Tsinghua University, Beijing 100084, P.R.Chian A two-dimensional 9-bits lattice gas model with 3 speeds was used to simulate a thermoacoustic engine. The model consists of a tube closed at two ends. The temperature gradient is applied to the boundary near one end of the tube. A thermal boundary condition is used in the heating range. In the simulation, the sound vibration is excited and propagated inside the tube. The optimized design of a thermoacoustic engine is calculated.

INTRODUCTION to be suitable to describe waves so that must be smaller than a quarter of a wavelength. The macroscopic In daily life, thermal phenomenon in sound waves density of the particle number n, the momentum nu can be neglected, however it intrinsically exists. The and the internal energy are defined as follows: thermoacoustics is dealt with both sound and heat f n a a effects so that its mathematics is rather complicated and furthermore it is usually a nonlinear phenomenon f e nu a a a in a thermoacoustic application. 1 A lattice gas model with temperature is a simple f (e u) (e u) n a a a a numerical method suitable for studying thermoacoustic 2 problems since the effects of sound and heat can be where ea is a particle momentum in a direction. The considered simultaneous and it is also valid in weak temperature of the lattice gas T is determined by nonlinear situation. The lattice gas method has been developed since the eighties of the last century, which k T was proposed to simulate hydrodynamic problems [1]. b In a lattice gas model, time, space and a fluid all are where kb is the Boltzmann constant. To display the discrete. The fluid consists of particles, which reside on features of sound waves, we need the pressure p, which a finite region of a regular lattice. These particles move depends on both the density and the temperature. From at regular time intervals from a lattice node to another statistical physics we know that the pressure is given along the lattice links. They collide at a lattice node by obeying the basic conservation laws. Lattice gas p n . models satisfy some macroscopic equations such as Navier-Stokes equations. Statistics of particles gives macroscopic features of a fluid motion. THERMOACOUSTIC ENGINE SIMULATION LATTICE GAS MODEL WITH TEMPERATURE The model of thermoacoustic engine consists of a two-dimensional tube closed at two ends filled with a In this paper a two-dimensional 9-bits lattice gas fluid and the temperature gradient is applied to the boundary near one end of the tube. A thermal boundary model with 3 speeds, 0, 1, and 2 [2] was used to condition is used in the heating range. When the simulate a thermoacoustic engine. With use of multi- particle impinge on a boundary if the local temperature speed model the temperature can be considered. In of fluid is greater than that of the boundary the particle collision, the particle number, momentum and energy would loss speed. Otherwise a particle gains a speed. are conserved. The two-body collision rules are the At the low temperature range, it is an elastic boundary. same as in Ref.[2], we also consider more body The simulation domain is 1000×50 lattice nodes. Once collision in the computing. The particle distribution function at node x and time t the calculation starts, the density at the range with temperature gradient changes and this disturbance averaged in direction α is denoted by f (x,t) . In a state a propagate towards the equilibrium range. When the of non-equilibrium, the range of average is considered disturbance reaches a closed end, it reflected. At last a standing wave of half wavelength is formed as shown in figure 1. first half period - - - - -Second half period (a) Stack near a closed end sound pressure sound sound pressure sound 0 200 400 600 800 1000 0 4000 8000 12000 16000 X time step

FIGURE 1 sound pressure in the tube (b) Stack near

We have studied the sound generation with different locations of the stack, which is the range with the temperature gradient. The calculation results after 20000 time steps are shown in figure 2. When the stack is near a closed end, the noise and the vibration are almost in the same order as shown in figure 2a. A sound pressure growing mode can be seen in figure 2b which is useful 0 4000 8000 12000 16000 time step of forming sound resonate. With increasing the time the amplitude of sound pressure increases for the stack near /8. When the stack is near a quarter wavelength the amplitude is saturated after a half of the period time as shown in figure 2c. It was found that the growing (c) Stack near mode only takes place for stack in a small range located less than /8. This result is consistent with that of experiments. On comparing figure2b and 2c, it can be seen that sound speed with longer temperature range is faster than that of shorter one. The calculation was not convergent because of the small scale model and pressure sound the small temperature gradient. The average pressure 0 4000 8000 12000 16000 time step grows obviously e.g. as shown in figure 2c. This method has being used in the further study of thermoacoustic problems. FIGURE 2. Vibration with time at the hot end

ACKNOWLEDGMENT

Thanks to the Tongfang Company for financial support

REFERENCES

1. J.P.Rivet, J.P.Boon, Lattice gas hydrodynamics, Cambridge university press, Cambridge, 2001.

2. S.Chen, K.H.Zhao & G.D.Doolen, Physica D 37 (1/3) 42-59, 1989. Modeling Sound Propagation Through a Combustion Zone

R. I. Sujith Department of Aerospace Engineering, Indian Institute of Technology Madras, India 600036 Currently at Deutsches Zentrum für Luft und Raumfahrt, Bunsenstraße 10, 37073 Göttingen, Germany

It is shown using analytical solutions that when an acoustic wave encounters a steep rise in mean temperature, it changes its nature of propagation. Though the acoustic waves are "geometrically" plane waves, they are evanescent waves in an environment of changing mean temperature. The fluid velocity is in phase with the pressure when the ratio of the length scale of the flame zone to the wavelength is large, and out of phase when this ratio is small. Exact analytical solutions for one-dimensional sound propagation through a combustion zone, taking the effects of mean temperature gradient and oscillatory heat release into account, are presented. Using appropriate transformations, solutions are derived for the cases of linear and exponential mean temperature gradients in terms of confluent hypergeometric functions and Bessel functions respectively.

INTRODUCTION 1 P¢ = [Ae-is + BAeis ] (4) T 1 / 4 The occurance of combustion instabilities has been a The first term in the bracket represents a wave that is plaguing problem in the development of combustors propagating in the right direction and the second term for rockets, jet engines and power generating gas denotes a wave that is propagating in the left direction. turbines. Predicting and controlling combustion Considering first only a wave that is propagating in the instability requires an understanding of the interactions Ae-is right direction, i.e., P¢ = 1 / 4 between the combustion process and the acoustic T waves. In this context, it is important to model sound 1 dP¢ 1 dP¢ dT propagation through an unsteady combustion zone, U = - = - = iwr dx iwr dT dx including the effects of mean temperature gradient and oscillatory heat release. e-is é 1 1 a gR ù P¢ é i ù A ê + iú = ê1- ú (5) T 1/ 4 ê r c 3 wT 1/ 4 ú r c ë s û WAVE PROPAGATION THROUGH A ë û ZONE OF TEMPERATURE GRADIENT where c = gRT . Also noting that dx dx 3 T 1/ 4 s The 1-D Helmholtz equation for a constant area duct ò = ò = = (6) with a mean temperature gradient, for a perfect, c(x) gR(ax + b)2 / 3 gR a w inviscid and non heat conducting gas, in the absence of the relation between acoustic pressure and velocity can mean flow is [1] be recast as 2 2 d P' 1 dT dP' w P¢ é dx ù + + P' = 0 (1) U' = 1- i / w (7) dx2 T dx dx gRT ê ò ú rc ë c( x )û Although the phenomena investigated is independent This relationship is of the same form as the relationship of the mean temperature profile chosen, a temperature between the acoustic pressure and velocity for an 4 / 3 profile of the form T = (ax + b) leads to solutions outward propagating spherical wave [3]. in terms of half order Hankel functions which can be é ù P¢ i expressed as simple exponential functions which are U' = ê1- ú (8) more intutive. For this temperature profile, the r c ê w r ú ë c û expression for acoustic pressure is Performing the same analysis for a wave propagating 1 ~ 1 ~ 2 ¢ ( ) ( ) 1 +is P = [AH 1/ 2 (s)+ BH 1/ 2 (s)] (2) to the left, i.e; P¢ = 1 / 4 Be yields the following T 1/ 8 T relationship between acoustic pressure and velocity: where s = 3w T 1/ 4 . Using the following expression a gR P¢ é i ù U ¢ = ê-1- ú (9) for Hankel functions [2], r c ë s û 2 eiz 2 e-iz H (1) (z) = -i ; H (2) (z) = i (3) This relationship is of the same form as that between 1/ 2 p z 1/ 2 p z the acoustic pressure and velocity for an inward Eq. (2) can be rewritten as propagating spherical wave [3]. Though the acoustic waves are "geometrically" plane For constant values of Â and S, Eq. (14) can be waves, they are evanescent waves as a result of the reduced to the Bessel's differential equation. Solution variation in mean temperature. The pressure amplitude to Eq. (14) can therefore be obtained in terms of Bessel is inversely proportional to the one-fourth power of the functions [2] as mean temperature. The fluid velocity is in phase with é ù (B / d -1) 2 é d ù é d ù the pressure when the length scale of the flame zone is P = T êc1Jn ê ú + c2 J-n ê úú large compared to the wavelength and out of phase ëê ë T û ë T ûûú with the pressure when the length scale of the flame if n is not an integer (15a) zone is small when compared to the wavelength. When é ù (B / d -1) 2 é d ù é d ù an acoustic wave encounters a temperature rise, a part P = T êc1Jn ê ú + c2Yn ê úú of it would be reflected and a part of it would be ëê ë T û ë T ûûú transmitted such that the acoustic pressure and the if n is an integer (15b) acoustic velocity are continuous across the interface. W The magnitude of the reflection and transmission where d = depends on the frequency of the wave and the gRd 2 temperature gradient. Linear Mean Temperature Profile SOLUTIONS IN THE PRESENCE OF OSCILLATORY HEAT RELEASE For a mean temperature profile of the formT = ax + b , Eq. (13) can be reduced to the The linearized acoustic momentum and energy following confluent hypergeometric equation, for equations in harmonic domain are [4]: constant values or Â and S: ¶P iwr U + = 0 (10) d 2P¢ dP¢ ¶x s + (1- s) + e P¢ = 0 (16) ds2 ds ¶U iw P + g P = (g -1)Q (11) B W2a2 x where s = T e = ¶ 2 The unsteady combustion process can be modeled as a a g RB function of the acoustic quantities in the combustion The solution to Eq. (16) can be expressed in the form region, i.e., Q = Â P + S U, where Â and S are the of confluent hypergeometric series [2] pressure and velocity coupled responses [4]. F(s) = C1 1F1(e ;1;s)+ C2 U (e;1;s) (17) The wave eqn. can be derived from Eq. (10) & (11) as d 2P é 1 dT (g -1)S ù dP éw 2 + iwÂ(g -1)ù + - + P = 0 ACKNOWLEDGMENTS 2 ê ú ê ú dx ëT dx g P û dT ëê g RT ûú (12) This work was supported by the Alexander von For obtaining solutions, Eq. (12) is rewritten with the Humboldt foundation through a Humboldt fellowship mean temperature as the independent variable to the author. 2 édT ù d 2 P é 1 d é dT ù dT ù dP W2 + T - B + P = 0 REFERENCES ê ú 2 ê ê ú ú ë dx û dT ëêT dx ë dx û dx ûú dT g RT (13) 1. A. Cummings, A., J. Sound and Vibration 51, pp. 55-67 (1977). (g -1)S where B = and W2 = w 2 + iwÂ(g -1) 2. M. Humi, and W. Miller, Second Course in g P Ordinary Differential Equations for Scientists and Engineers, New York: Springer-Verlag, 1988. Exponential Mean Temperature Profile 3. P. M. Morse and K. U. Ingard Theoretical Acoustics, McGraw-Hill New York, 1968. For a mean temperature profile of the form T = Cedx 4. F. E. C. Culick, Astronautica ACTA, 12, 113-126, 1966. where C and d are constants, Eq. (13) can be reduced to the following form: d 2 P é B ù dP W2 P + ê2 - ú + = 0 (14) dT 2 ë d û dT g Rd 2 T 3 Experimental Nonlinear Analysis of the Acoustic Field in a Combustion Chamber

G. Cammarata1, A. Fichera1, C. Losenno2, A. Pagano1

1Dip. Ingegneria Industriale e Meccanica – Università di Catania, Viale A. Doria 6, 95125 Catania, Italy Tel. +39 095 7382450; Fax +39 095 337994, [email protected] [email protected] 2School of Mechanical Engineering, University of Edinburgh, King’s Buildings Mayfield Road, Edinburgh EH9 3JL, Scotland UK Tel. +44 (0)131 6505685; Fax +39 (0)131 6673677, [email protected]

Thermo-acoustic instabilities present a serious threat to combustion systems, which can lead to system performance and structure degradations. Due to many non-linear interactions, a number of processes are generated, which are often very complex. Many linear approaches have been used to provide a deeper understanding of the underlying phenomena. Unluckily, linear analysis may be affected by an oversimplified view of the problem and a more detailed non-linear analysis may be required. This paper presents a non-linear analysis of experimental measurements observed in a methane-fuelled laboratory combustor, aiming to investigate the dynamical behaviour of the burner by means of the deterministic chaos theory. Lyapunov exponents of the internal and external acoustic field have been calculated allowing for the dynamic behaviour of the system to be described. Results of the analyses show the existence of a chaotic source of the dynamics in the combustion system in study is demonstrated.

pressure INTRODUCTION transducer Thermo-acoustic instabilities in combustion chambers take place as large amplitude and low frequency pressure oscillations [1]. Combustion mixing instabilities represent a serious threat to zone 70 cm combustion process because of many harmful primary effects. Owing to the potential harm of combustion zone microphone instabilities, many studies have been performed to provide a deep understanding of the underlying Figure 1 Experimental apparatus phenomena. The development of an analytical Experimental testes were performed by varying the approach capable of taking into account the non- equivalent ratio l (ratio of air rate over linearities of the phenomenon is still far from being stochiometric air rate) and the pilot fuel percentage full understood. This paper aims to achieve a PFP (percentage of methane fuelling the diffusive deeper insight into complex combustion behaviours stabilising flames). Two different values of l (1.3- by means of a non-linear approach to the analysis 1.5) and a single value of PFP=0 were considered. of experimental data generated by unstable combustion phenomena. NON-LINEAR ANALYSIS EXPERIMENTAL APPARATUS The application of chaos theory to investigate non-linear processes may reveal important aspects The experimental apparatus used for the testing is of the system dynamics, such as the existence of showed in Fig. 1. The experiments were performed complex behaviours. The dynamic behaviour of on a lean premixed methane-fuelled laboratory- any system must be studied in an appropriate scale combustor, which is described in [2]. The representation space, called phase or state space burner was provided with 12 gas injectors for the [3]. The representation of experimental time series premixed flame and 12 injectors for the pilot of a bounded deterministic system generates a diffusive flame. The pilot flame percentage varied stable structure of long-term trajectories, referred to from 0% up to 15%. A wall-mounted pressure as attractors of the system [4]. The existence of a transducer (KISTLER 6061B) measured pressure regular structure is a primary aspect, which is oscillations inside the duct, whereas a microphone usually exploited through the characterisation of (Larson & Davis 2560), located 70 cm far from the the attractor by means of quantities not depending combustor axis, detected pressure oscillations on the system representation and on the initial outside the duct. conditions, namely dynamical invariants or simply Signals coming from the sensors were collected invariants. One of the most important invariant at a frequency of 10000 Hz over a sampling characteristics are the Lyapunov Exponents [3-5]. interval of 12,8 s for each operating condition. Their knowledge allows assessing whether the globally stable behaviour of a deterministic system c) one of the Lyapunov exponents is close to zero, (i.e. not stochastic) is chaotic or not. This is the confirming that the system is autonomous [3], i.e. reason why they have been herein considered. the system dynamics is governed by an internal Figures 2 show the 2-D attractors of the time source and is not determined by an external series detected during the experimental tests. In forcing; these plots tr(t) and mic(t) indicate the time series d) the sum of the whole set of Lyapunov detected respectively by the pressure transducer exponents is negative, i.e. a net contraction effect and the microphone, and tr(t+tau) and mic(t+tau) dominate the system dynamics and ensure that the

1 1 system is globally stable. 0.8 0.6 0.5 0.4 h1 h2 h3 h4 h5 h6 Sum 0.2 0 0 Microphone Signal -0.2 t+tau) t+tau)

-0.4 -0.5 C130 0.252 0.139 -0.022 -0.218 -0.659 -0.508 -0.6 -0.8 C150 0.221 0.161 0.063 -0.032 -0.164 -0.516 -0.267 -1 -1 -1 -0.5 tr(t)0 0.5 1 -1 -0.5 0 0.5 1 tr(t) Pressure Transducer Signal 1 1 C130 0.243 0.142 0.041 -0.075 -0.241 -0.725 -0.615 0.5 0.5 C150 0.341 0.189 0.020 -0.202 -0.784 -0.436

0 0 t+tau) t+tau) Table 1 - Lyapunov exponents for the external and -0.5 -0.5 internal pressure field

-1 -1 -1 -0.5 0 0.5 1 mic(t) -1 -0.5 mic(t) 0 0.5 1 the corresponding delayed copies. CONCLUSIONS

This paper has presented a study of thermo- Fig. 2. Attractor of the time series for the pressure transducer (first row) and the microphone (second row). acoustic combustion instabilities performed with Operating conditions l=1.3 - PFP=0 (first column) and experimental non-linear data analysis. The use of l=1.5 - PFP=0 (second column). non-linear analysis tools allowed characterising the dynamical behaviour of the system. The main As the plots show, the attractors describing result of this work consisted in evidencing the different operating conditions of the combustor existence of chaos in the system in study. This was present sensible shape variations. This indicates a achieved looking at first at the attractors of the strong phase space non-uniformity of the system herein examined. Afterwards, chaos combustion process at different values of the air existence was analytically proved by means of the excess ratio. Moreover, the attractors depicted in calculation of Lyapunov spectra for the Fig. 2 are morphologically very similar to those experimental time series. The results point out the shown by other experimental systems. In particular, opportunity to define a model of the system that they are characterised by a phase space distribution takes into account the non-linearities affecting the similar to that of the “air noise system” described process, which would represent an important step by Mackenzie and Sandler [6]. The existence of towards the definition of an effective control similarities between the attractors of physically system. A non-linear model-based control system different systems represents a very important might be able to ensure a reduction of combustion feature of chaos analysis. Such similarities may in instabilities stronger than the one obtained by fact reflect the existence of an analogous (if not controllers based on linear approaches. identical) mathematical model for the similarly behaving systems, though it is clear that the variable of the model itself refers to physically REFERENCES different quantities. In such cases, it may be [1] Rayleigh J. W., Royal Institute Proceeding, 3: 536- possible to define the model of a unknown system 542 (1878). simply on the base of the morphological analogy [2] Fichera, A., Losenno, C., Pagano, A., Applied between its attractors and those of a system for Energy, vol. 69/2 pp. 101-117, 2001. which the mathematical model is known. The [3] Thompson J. M. T. and Stewart H. B., Nonlinear values of Lyapunov exponents for the external and Dynamics and Chaos, J. Wiley & Sons, New York, 1986. [4] Abarbanel H. D. I., Analysis of Observed Chaotic internal pressure field are reported in Tab. 1. The Data, Institute for Non-Linear Science, San Diego, 1995. Lyapunov spectra of both variables give clear [5] Rasband S. N., Chaotic Dynamics of Nonlinear evidence of the existence of chaos. In fact: Systems, J. Wiley & Sons, New York, 1990. a) there are at least two positive exponents, i.e. [6] Mackenzie, J., Sandler, M., Modelling Sound with two directions along which attractor expansion Chaos, M., Proc. IEEE ISCAS 94, London, pp. 6.93-6.96, occurs; hence, the system is hyperchaotic [7]; (1994) b) there are at least two negative Lyapunov [7] Hudson, J.L., Rõssler, O.E., Killory, H.C., IEEE exponents, which correspond to two directions Trans. On Circuits and Systems, 35, 7, 902-908, (1998). along which contraction of the attractor occurs;

Work Flow Measurement on Thermoacoustic Stirling Engine

Y. Ue d a a, T. Biwa a, T. Yazaki b, U. Mizutani a

a Department of Crystalline Materials Science, Nagoya Umiversity Furo-cho, Chikusaku, Nagoya 464, Japan b Department of Natural Science, Aichi University of Education, Kariya 448, Japan E-mail: [email protected]

We measured simultaneously the pressure and the velocity of the oscillating gas to determine the work flow and the phase difference between the pressure and velocity in a thermoacoustic traveling wave engine. We revealed that traveling waves can produce the work flow in the regenerator. Moreover, we could determine a position where only the traveling wave is formed. An installation of a second regenerator at this particular position led us to construct a thermoacoustic Stirling cooler with low viscous losses.

In thermoacoustic phenomena brought about by their engine. It is found that the present engine changes thermal interactions between acoustic waves and solid energy through a Stirling cycle. Moreover, by using walls, the work flow I is converted into the heat flow the experimental data derived from the present engine, Q (heat engine) or Q into I (cooler) [1-3]. Particularly we could succeed in building a thermoacoustic Stirling when traveling waves isothermally exchange heat with cooler with low viscous losses. solid walls, these energy flows I and Q can be The experimental apparatus is schematically converted from the one to the other through a Stirling illustrated in Fig. 1. Both a looped tube and a resonator cycle [4]. A thermoacoustic Stirling engine was first are made of Pyrex glass with its internal diameter of built in 1998 by using a looped tube. Work flow 40 mm. Total lengths of the loop tube and the measurement on it revealed that the work flow was resonator are 118 and 104 cm, respectively. The amplified in the regenerator by traveling acoustic wave, internal volume of the tank is 20 l. A 35 mm long running around the tube [5]. regenerator consists of a stack of 60-mesh stainless- Recently, Backhaus and Swift developed a steel screens to make a good thermal contact with thermoacoustic engine consisting of a looped tube and acoustic waves. Hot and cold heat exchangers are a resonator. They reported that their engine could placed at both ends of the regenerator. An electrical convert Q into I in a Stirling cycle and that its heater is wound around the hot heat exchanger to efficiency reached 42% of the theoretical maximum control its temperature TH, and a cooling water pipe Carnot efficiency [6]. However, details of the around the cold heat exchanger to keep its temperature thermodynamic cycle and acoustic waves in their at TR close to room temperature of 18ºC. This engine is engine have not been experimentally clarified. filled with one atmospheric air. In this work, we performed work flow measurements on a thermoacoustic engine similar to

FIGURE 1. The present thermoacoustic engine (a) consists of a looped tube, a resonator and a tank. The cooling stack (b) is made of ceramic and has numerous small square holes. A joint position connecting the looped tube with the resonator is set to X=0 as an origin and the direction of X is taken anticlockwise in the looped tube and towards the right in the resonator, respectively.

The acoustic wave with the frequency of 41Hz is conclude that the present engine converts Q into I spontaneously generated, when TH exceeds 210ºC. almost through a Stirling cycle. As a result, the work Experiments were performed while keeping TH to be at flow I is amplified in the regenerator, which is shown 278ºC. We simultaneously measured both the pressure in Fig. 2(c). The difference of I between the cold and P=p cos ω t and the cross-sectional mean velocity hot ends of regenerator represents the output power

Um=umcos(ω t +Φ)=um cosΦcosω t + um sinΦcos(ω t generated by the energy conversion. The present work + π/2) by using a pressure sensor and laser Doppler revealed that the phase delay Φ crossed 0 at the low velocimetry, respectively and determined the work temperature end of the regenerator. We consider that flow I as (1/2)·p·um·cosΦ [7]. Here we call um this is critically important to extract most effectively the output power from the thermoacoustic Stirling cosΦcosω t the traveling wave component of Um, engine. which is in phase with P, and um sinΦcos(ω t + π/2) the standing wave component, which is π/2 out of Finally we propose a themoacoustic Stirling cooler phase with P. Only the traveling wave component as a useful application of the present engine. It is Φ contributes to the work flow I. important to note that at X=85, and um takes zero and minimum, respectively. We can naturally assume that heat pumping must be achieved through a Stirling cycle with low viscous losses in the stack, if a cooling stack acting as a heat pump is placed at X=85. Based on the discussion above, we built a thermoacoustic Stirling cooler by inserting the cooling stack (Fig. 1(b)) at X=85. We could successfully lower the temperature of the cooling exchanger by 16 ºC. We proved that the present engine operates as a traveling wave heat engine by analyzing the generated acoustic wave. Moreover, we succeeded in building a thermoacoustic Stirling cooler, which makes both the energy conversion of Q into I and I into Q through a Stirling cycle.

ACKNOWLEDGMENTS

We are grateful to T. Yamamoto and Y. Ichikawa of NGK Insulators, LTD. for supplying us with cooling stacks.

REFERENCES

FIGURE 2. The axial distribution of (a) p(o), um(•), Φ 1. Tominaga, A., Cryogenics 35, 427 (1995) (b) and (c) I in the resonator (-104