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On the Use of a Complex Frequency for the Description Of ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 98 (2012) 232 –241 DOI 10.3813/AAA.918508 On the Use of aComplexFrequencyfor the Description of Thermoacoustic Engines M. Guedra, G. Penelet Laboratoire d’Acoustique de l’Université du Maine, UMR CNRS 6613, Avenue Olivier Messiaen, 72085 Le Mans Cedex9,France. [email protected] Summary In this paper,aformulation is proposed to describe the process of thermoacoustic amplification in thermoacoustic engines. This formulation is based on the introduction of acomplexfrequencywhich is calculated from the transfer matrices of the thermoacoustic core and its surrounding components. The real part of this complex frequencyrepresents the frequencyofself-sustained acoustic oscillations, while its imaginary part characterizes the amplification/attenuation of the wave due to the thermoacoustic process. This formalism can be applied to anytype of thermoacoustic engine including stack-based or regenerator-based systems as well as straight, closed loop or coaxial duct geometries. It can be applied to the calculation of the threshold of thermoacoustic instability, butitisalso well-suited for the description of the transient regime of wave amplitude growth and saturation due to non linear processes. All of the above mentioned aspects are described in this paper. PACS no. 43.35.Ud 1. Introduction Manyresearchers use the freely available software package called DeltaEC developed at Los Alamos Na- Thermoacoustic engines belong to atype of heat engines tional Laboratory [4] for the design of thermoacoustic sys- in which acoustic work is produced by exploiting the tem- tems. This computer code is avery powerful tool which is perature gradient between ahot source and acold sink mainly based on the linear (and weakly nonlinear)ther- [1, 2]. Typical arrangements of thermoacoustic engines moacoustic theory in the frequencydomain. Besides the are shown in Figure 1. It consists basically of an acous- limitations of this computer code for large acoustic am- tic resonator partially filled with apiece of an open-cell plitudes requiring proper account of nonlinear effects, an- porous material, often referred to as astack or aregener- other of its characteristics is that it is expressed in the ator.Animportant temperature gradient is imposed along Fourier domain, so that it describes steady state condi- this stack/regenerator,sothat above acritical temperature tions: the steady-state acoustic pressure amplitude is ob- gradient, acoustic modes of the resonator can become un- tained from atemperature field which itself is controlled stable and the thermoacoustic process results in the on- by the acoustic field due to acoustically induced heat trans- set of self-sustained, large amplitude acoustic oscillations. port. The multi-parameter shooting method which is em- Such kind of engines have been extensively studied in the ployed in this computer code is well suited for the predic- past decades [3], leading to adeeper understanding of their tion of an equilibrium state above the threshold of ther- operation and to the building of afew devices such as moacoustic instability.However,itisnot primarily de- thermoacoustically driventhermoacoustic refrigerators or voted to the determination of the threshold condition it- thermoelectric generators. These engines have interesting self (i.e. the required external thermal action above which features inherent to the absence of moving parts (pistons thermoacoustic oscillations begin to growupwith time). and crankshafts)which can be advantageously used for Moreover, under some circumstances, the transient pro- industrial applications at moderate power densities (typ- cess leading to the steady state sound should be consid- ically up to afew kilowatts). It is howeverworth noting ered, and the approach used in DeltaEC then becomes un- that the design of thermoacoustic engines is atedious task suitable. This is notably the case when the engine is oper- which comprises an important part of uncertainties, be- ated slightly above the threshold of thermoacoustic insta- cause the operation of these engines is by nature nonlinear, bility,where complicated effects may be observed. Forex- and because the existing efficient prototypes include vari- ample, the existence of ahysteretic loop [5, 6] in the onset ous elements likeflow straighteners, tapered tubes, mem- and damping of the engine, or the periodic switch on/off branes or jet pumps which are difficult to model accurately. of thermoacoustic instability [7, 8] have been reported for both standing and travelling wave engines. In such situa- tions, the fixed external thermal action on the system does Received16May 2011, accepted 12 December 2011. 232 ©S.Hirzel Verlag · EAA Guedra, Penelet: Description of thermoacoustic engines ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 98 (2012) T-junctions etc ...The works presented in this paper basi- Thermoacoustic core cally consist of ageneralization of previous works [10, 17] (a) T and its main originality is thus primarily to propose to the T h c reader arather simple modelling of anykind of thermoa- x coustic engine in order to determine its onset conditions or to describe the transient regime leading to steady state x x x x 0 l h r L sound in the frame of weakly nonlinear theory. In section 2the multiport network modelling of ther- (b) Th Tc moacoustic engines is presented, which leads to the analyt- ical expression of the complexfrequencyfrom the transfer x matrices of the thermoacoustic core and its surrounding xl xh xr components. In section 3, this formalism is applied to the determination of the conditions corresponding to the on- 0(L) set of thermoacoustic instability in the cases of astanding wave engine and of aclosed-loop travelling wave engine. Section 4isdevoted to the description of basic concepts to secondary acoustic load concerning the use of this approach to study the transient regime leading to steady state sound (ortomore compli- Figure 1. Simplified drawings of astanding wave engine (a) and cated dynamic behaviors of the thermoacoustic oscillator) atravelling wave loop engine (b),possibly coupled with asec- in thermoacoustic systems. ondary acoustic load. 2. Theory not correspond to aunique steady state solution for the Thermoacoustic engines are generally made up of aduct acoustic pressure amplitude. network inside which the thermoacoustic core is inserted. Though useful design tools are nowadays available, The term “thermoacoustic core” refers here to the hetero- an accurate description of thermoacoustic engines is still geneously heated part of the device in which the ampli- needed, and an important research effort has been devoted fication of acoustic wavesoperates: it is basically com- to the description of the onset of thermoacoustic instabil- posed of an open cell porous material -referred to as the ity and to its saturation by nonlinear effects. Various ana- stack (δκ ∼ r)orthe regenerator (δκ r)depending on lytical [9, 10, 11, 12] and numerical models [13, 14] have the value of the average radius r of its pores relative to been proposed in the literature, which are yet limited to the thickness δκ of the acoustic thermal boundary layer - the description of simple thermoacoustic devices of par- along which atemperature gradient is imposed using heat ticular geometry.Inthis context, the aim of this paper is exchangers. As illustrated in Figure 1, the great variety to propose ageneral modelling approach which is mainly of thermoacoustic engines can be schematically separated based on the transfer matrices formalism. As in previous into twodifferent classes. The first class of engines (Fig- analytical works [9, 12] the model presented in this pa- ure 1a)refers to some conventional waveguide arrange- per takes advantage of the significant difference between ment ensuring the resonance of agas column. Among this the instability time scale and the period of acoustic oscil- class of engines are the stack-based standing wave engines lations, which is exploited here by the introduction of a which were extensively studied during the past decades, complexfrequency, sometimes used for the treatment of butalso the cryogenic devices where Taconis oscillations transient oscillatory motions (note that the introduction of may occur [1]. The second class of engines (Figure 1b) complexfrequencyhas already been proposed in acon- refers to some waveguide arrangements where aclosed- ference by J. E. Parker et al. [15] to treat thermoacous- loop path exists, allowing the development of travelling tic oscillations, and also in asimilar network approach by acoustic wavesrunning along the loop. Among this class Q. Tu et al. [16]). Depending on its sign, the imaginary of engines are the stack-based travelling wave engine first part of this complexfrequencyrepresents an amplification studied by Yazaki et al. [18], the thermoacoustic-Stirling or an attenuation coefficient, which is calculated from the heat engine [19] first successfully designed by Backhaus et linear thermoacoustic theory applied to the thermoacous- al. [20], the regenerator-based co-axial engines [21] where tic system under consideration. The analytical treatment the feedback loop is formed by locating asmall diameter presented here is necessarily based on substantial approx- thermoacoustic core into alarger diameter waveguide, and imations but, as will be discussed in this paper,itiswell also as amatter of interest some kinds of free-piston Stir- suited to carry out extensive parametric studies of both the ling engines. transient and steady
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