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 states. Moreover, this model has some Whateverthe specificgeometry of the engine under interesting similarities with the computer code DeltaEC in consideration, all of these devices use the fact that when a the sense that it consists of amultiport network approach strong temperature gradient is imposed along the stack/re- which is well-suited for the description of complicated generator,part of the heat supplied is converted into acous- acoustic networks including thermoacoustic cores, ducts tic work inside the stack/regenerator.This thermoacoustic with constant or varying cross-sections, grids, membranes, amplification process results in the onset of self-sustained,
233 ACTA ACUSTICA UNITED WITH ACUSTICA Guedra, Penelet: Description of thermoacoustic engines Vol. 98 (2012) large amplitude acoustic wavesoscillating at the frequency instance, if astanding-wavedevice as the one depicted in of the most unstable acoustic mode of the complete device. Figure 1a is considered, writing the lossy propagation of In the following, the onset of this thermoacoustic insta- harmonic plane wavesatangular frequency ω in the ducts bility will be described by the introduction of acomplex of respective lengths xl and L−xr leads to the expressions frequency, the real part of which describes the frequency of the reflected impedances of acoustic oscillations and the imaginary part of which describes the wave amplitude growth or attenuation. The p (x ) Z0 + jZc tan kxl Z = l = , (3) analytical treatment presented here can be applied to any l −1 u(xl) 1 + jZ0Zc tan kxl kind of thermoacoustic engine (and also to free piston Stir- p (x ) ZL − jZc tan k(L − xr) ling engines), butitisconvenient here for clarity to sepa- Z = r = (4) r −1 rate the cases where there exists or not aclosed loop path u(xr) 1 − jZLZc tan k(L − xr) for the acoustic waves. Forthe sakeofsimplicity,the first class of engine will be referred to as “standing wave”en- where gine, while the second one will be referred to as “travelling ω 1 + (γ − 1)f wave”engine. The description of the acoustic field will be k = κ (5) operated in the frequencydomain in the frame of the linear c0 1 − fν approximation. Assuming that harmonic plane wavesare propagating along the centerline of the ducts, the acoustic and pressure p(x, t)and acoustic volume velocity u(x, t)are ρ0c0 1 Zc = (6) written as S (1 − f )(1 + (γ − 1)f ) ν κ −jωt ζ(x, t) = ζ(x)e , (1) are the complexwavenumber and the characteristic impe- dance of the duct, respectively.Inequations (5) and (6), ˜ where ζ may be either p or u, ζ denotes the complexampli- ρ0 is the fluid density at room temperature, c0 is the adia- tude of ζ, () denotes the real part of acomplexnumber, batic sound speed, γ is the specificheat ratio of the fluid, and x denotes the position along the duct axis (see Fig- S is the duct cross-section, and the functions fκ and fν ure 1). characterize the thermal and viscous coupling between the As shown in Figure 1, the apparatus consists of ather- oscillating fluid and the duct walls [2, 23]. In equations moacoustic core connected at both sides to straight (or (3) and (4), Z0 and ZL stand for the acoustic impedances curved)ducts. The propagation of acoustic wavesthrough at positions x = 0and x = L,respectively.Theycan be, the thermoacoustic core is described as an acoustical two- for instance, the radiation impedance of an open pipe [24], port relating the complexamplitudes of acoustic pressure the infinite impedance of arigid wall, or the acoustic im- and volume velocity at both sides, pedance of an electrodynamic alternator,depending on the configuration of the standing-waveengine. Finally,com- p (x ) T T p (x ) r = pp pu × l , bining equations (3) and (4) with equation (2) and solving u(xr) Tup Tuu u(xl) the associated system of twoequations leads to the equa- p (xl) tion = TTC × . (2) u(xl) ZlTpp + Tpu − ZlZrTup − ZrTuu = 0. (7) The transfer matrix of the thermoacoustic core, TTC,de- pends on the geometrical and thermophysical properties Equation (7) is the characteristic equation which accounts of its components. It also depends on the temperature dis- for both the processes operating through the thermoacous- tic core and the dissipative/reactive processes operating in tribution Tm(x)along the stack (x ∈ [xl,xh]) and the its surrounding components. This equation must be satis- thermal buffer tube (x ∈ [xh,xr]),and on the angular frequency ω.Ifthe imposed temperature distribution is fiedtodescribe the complete device properly. known, the transfer matrix TTC can be obtained theoret- 2.1.2. Travelling wave engines ically [1, 2, 17], butitcan also be obtained from experi- ments under various heating conditions [22]. If the case of atravelling wave engine is nowconsidered, and if the matrix TTC is known, it is also possible to derive 2.1. Derivation of the characteristic equation acharacteristic equation similar to equation (7).This im- plies to describe acoustic propagation at both sides of the 2.1.1. Standing wave engines thermoacoustic core. The basic idea is to makeone loop The case of astanding wave engine is first considered in the device -each of the individual components being here. If the matrix TTC is known, the theoretical mod- characterized by its transfer matrix -sothat after one loop, elling of the complete device requires knowledge of the the characteristic equation will ensure that one arrivesat acoustic propagation through the components surround- the same starting point. More precisely,there exists on ing the thermoacoustic core. This can be realized by de- the first hand the equation characterizing the propagation riving the expressions of the reflected impedances Zl,r = through the thermoacoustic core, equation (2),and on the p˜(xl,r)/u˜(xl,r)atboth sides of the thermoacoustic core. For other hand, it is possible to obtain an additional relation
234 Guedra, Penelet: Description of thermoacoustic engines ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 98 (2012) between the acoustic pressure and acoustic volume veloc- describe from equation (13) the wave amplitude growth ity at both sides of the thermoacoustic core by means of occuring after the onset of thermoacoustic instability un- the total transfer matrix Tsur of the components surround- der the quasi-steady state assumption. To do this, let the ing the thermoacoustic core, angular frequencybeallowed to have an imaginary part g, p (xl) p (xr) = Tsur × . (8) u(xl) u(xr) ω =Ω+j g, (14) Forinstance, if the particular geometry of Figure 1b is so that the acoustic pressure considered, and if the effects of the loop curvature are ne- p(x, t) = p (x)e−jωt = e gt p (x)e−jΩt . (15) glected, the matrix Tsur can be written as T = T × T × T , (9) is assumed to oscillate at frequency Ω= (ω), while the sur l load r attenuation/growth of the sound wave is characterized by where the matrices the thermoacoustic amplification coefficient g.However, g is assumed to be small compared to the real part Ω of cos(kdl,r)jZcsin(kdl,r) angular frequency, which means that the amplitude of the Tl,r = −1 (10) jZc sin(kdl,r)cos(kdl,r) wave varies slowly at the time scale of fewacoustic pe- riods, diminishing or growing depending on the sign of characterize lossy propagation through the ducts of respec- ,inasmuch as the temperature distribution T (x)stays tive lengths d = x and d = L − x (L is the unwrapped g m l l r r constant at the scale of afew acoustic periods. length of the closed-loop), and where the matrix Under this quasi-steady state assumption, Ω,and g 10 for aconstant temperature distribution Tm (atthe time- Tload = (11) scale of afew acoustic periods)itispossible to solve −Yload 1 equation (13) using conventional numerical methods, and accounts for the effect of the secondary acoustic load, to findacouple Ω, g which characterizes both the fre- by means of its reflected acoustic admittance Yload (this quencyofacoustic oscillations and the wave amplitude acoustic load can be asecondary acoustic resonator [20], growth/attenuation. The advantages of this formulation are an electrodynamic alternator [25] or anyother component that it is well-suited for the prediction of the onset of ther- characterized by its reflected admittance Yload). moacoustic instability (aswill be shown in the next sec- Finally,combining equation (8) with equation (2) leads tion)but also more generally,aswill be discussed in sec- to the characteristic equation tion 4, for the prediction of the engine’sefficiency, pro- vided that the nonlinear effects operating after the onset det TTC × Tsur − I2 = 0, (12) are properly described. where I2 stands for the identity matrix 2 × 2, and det() denotes the determinant of amatrix. 3. Determination of the threshold of ther- moacoustic instability 2.2. Determination of the complex frequency The theoretical modelling presented in section 2can be The proper description of the thermoacoustic device re- used at first to determine the marginal stability conditions quires to satisfy the corresponding characteristic equation of thermoacoustic devices, i.e. to findthe purely real an- gular frequency ω =Ωand the temperature distribution f (ω, Tm) = 0, (13) Tm for which equation (13) is satisfied. However, before where the function f corresponds to the left-hand-side of illustrating this with twoparticular examples, tworemarks equation (7) or equation (12),depending on the system un- need to be formulated. der consideration. It is important to point out that all of the Firstly,the model of section 2isactually incomplete above equations are derivedinthe frequencydomain, and since the heat transport within the thermoacoustic core due to this, it is implicitly assumed from equation (1) that needs to be described. Though it is well-established that the thermoacoustic system operates on steady state: this thermoacoustic amplification depends significantly on the means that the angular frequency ω is purely real. In fact, details of the temperature distribution, we will assume the only condition for which equation (13) can be satisfied here for simplicity that alinear temperature distribution is that the temperature distribution Tm(x)isfixedinsuch is imposed along the thermoacoustic core. In the absence away that there exists avalue of the angular frequency of an appropriate description of heat transfer through the ω which cancels the function f.Under such acondition thermoacoustic core, the effect of heating will thus be rep- the acoustic wavesare neither amplified nor attenuated, resented by the temperature ratio Th/Tc,where Th refers and since nonlinear effects saturating the wave amplitude to the hot temperature at position xh and Tc = 300K growth are not considered here, the solutions ω and Tm is the room temperature. In this paper,the transfer ma- correspond to the threshold of thermoacoustic instability. trix TTC (ω, Th/Tc) of the thermoacoustic core is obtained However, as it will be proposed in the following, one may from the transformation of the well-known differential
235 ACTA ACUSTICA UNITED WITH ACUSTICA Guedra, Penelet: Description of thermoacoustic engines Vol. 98 (2012) wave equation of thermoacoustics [1, 2] into an equiva- lent Volterra integral equation of the second kind (see refs. [10, 17] for more details). Secondly,inmost of thermoacoustic devices the self- sustained oscillations are generated at afrequencywhich corresponds to the lower order acoustic mode of the sys- Ω tem, butsince some of the components of the device con- Ωr sist of aduct of finite length, equation (13) has actually an infinite number of solutions under the plane wave as- sumption (under some circumstances, higher order acous- tic modes may even become more unstable than the first one, e.g. in refs [18, 26]). Since conventional numerical methods for the solving of equation (13) should converge to asingle solution, it is required to define an appropriate initial condition in the numerical scheme in order that the algorithm converges to the desired solution. In practice, the temperature ratio Th/Tc is fixed, act- ing as aparameter,and the characteristic equation (13) is
g solved using an iterative Newton-Raphson method, which Ω is suitable for finding the roots of acomplexfunction of a complexvariable [27]. The solution of (13) is found using the recurrence relation f(ω ) ω = ω − k , (16) k+1 k f (ωk)
where the first derivative f (ωk )iscalculated using asim- ple first order finite difference Tn df f(ωk +Δωk)−f(ωk) f (ωk) = = . (17) Figure 2. Normalized frequency Ω/Ωr (a) and amplification co- dω Δωk ω=ωk efficient g (b) as functions of the temperature ratio Tn = Th/Tc for three different locations of the stack in the resonator: x = Forour calculations, the step of finite difference is fixed h L/4(...), xh = L/2(−−)and xh = 3L/4(—).For xh = 3L/4, to Δω = 1.10−3ω .Inorder to avoid the divergence of k k the onset threshold corresponds to Tn 1.5. the method, the initial value ω0 is fixed to the angular frequencyofaresonant mode of the complete device. In the case of resonant modes resulting from acomplicated of respective lengths dl = xl and dr = L−xr are connected coupling between different elements (see for instance the to the thermoacoustic core. Assuming that the thermoa- loop engine presented in Sect. 3.2), this initial angular fre- coustic engine is closed at both ends, i.e. Z0 = ZL = ∞, quencyisdetermined graphically by plotting the modulus the expressions of the reflected impedances |f(Ω)| and by pointing at alocal minimum of the func- Zl = −jZc cot(kdl), (19) tion. The initial value for g is fixed to g = 0. The itera- tive computation is stopped when asufficient accuracy e is Zr = jZc cot(kdr), (20) obtained on the solution, i.e. when are introduced into equation (7) to compute the angular f(ω ) e ω − ω = k ≤ , (18) frequencyofacoustic oscillations Ω and the thermoacous- k+1 k f (ωk) 10 tic amplification coefficient g in functions of the hot tem- perature T .Inthis example, the total length of the engine −8 h with e = 10 in our case. With this computation process, is fixed to L = 1m,while the lengths of the stack and the it is then possible to calculate ω =Ω+j gas afunction of thermal buffer tube are fixed to ls = lw = 5cm. Air at the temperature Th.The threshold of thermoacoustic insta- atmospheric pressure is used as aworking fluid, and the bility then corresponds to the hot temperature Th for which stack is modelized as an assembly of 0.5mm-spaced par- g = 0: it is determined by means of azero-finding method allel plates (50 µminthickness). acting on the function g (Th). Figure 2presents the angular frequency Ω and the cor- responding amplification coefficient as functions of the 3.1. Standing wave engine g temperature ratio Tn = Th/Tc for three different locations As abasic illustration of the applicability of the model pre- of the stack in the resonator: xh = L/4(...), xh = L/2 sented in Sect. 2, the case of astanding-waveengine closed (−−)and xh = 3L/4(—).InFigure 2, we focus only on at both ends is studied. Aschematic drawing of this engine the first eigen mode of the half-wavelength resonator and is shown in Figure 1a: twostraight cylindrical waveguides Ω remains close to the angular frequency Ωr = πc0/L.For
236 Guedra, Penelet: Description of thermoacoustic engines ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 98 (2012)
Ω Ωr Ω Ωr
g Ω g Ω
onset mode 3
onset mode 2 onset mode 1
x h L Tn
Ω Figure 3. Normalized frequency Ω/Ωr (a) and amplification co- Figure 4. Normalized frequency (a) and amplification coeffi- Ωr efficient (b) as functions of the position of the stack x in the Th g h cient g (b) as functions of the temperature ratio Tn = for the Tc resonator for the three first modes of the half-wavelength res- three first modes of the half-wavelength resonator.The location onator: Ω = πc /L (—),2Ω (−−)and 3Ω (...). Here, the r 0 r r of the stack xh = 0.9L is chosen so that the onset of the three temperature ratio is fixed to Th/Tc = 4. modes is achievable.
the twopositions xh = L/4and xh = L/2, when Tn > 1, and the spatial distribution of the acoustic pressure fields the amplification coefficient g is negative and is contin- in ahalf-wavelength resonator is quite direct and confirms uously decreasing, meaning that the stack has adamping that these favourable positions are those for which the tem- 3L perature gradient has the same sign as the acoustic pres- effect for these positions. By contrast, for xh = 4 ,the onset condition g = 0isfound for atemperature ratio sure gradient. If the stack is near enough to the rigid wall on the right, T0 1.5. It can also be noticed that g < 0for Tn
237 ACTA ACUSTICA UNITED WITH ACUSTICA Guedra, Penelet: Description of thermoacoustic engines Vol. 98 (2012)
102
101 Ω 2π logf
100
10− 1
10− 2
log Ω 2π
Figure 5. Modulus of the function f(Ω,Tn = 1) computed for aclosed-loop engine (−−)and for the closed-loop engine cou- pled with the double alternator (—).The frequencies 80Hz and g 1020Hz correspond to the mechanical resonance of the alterna- Ω tor and the first mode of the loop, respectively.
In the previous section, it has been demonstrated that the choice of the frequencyinthe vicinity of which the on- set frequencyiscomputed is important. As it is illustrated in Figure 5, when the resonant modes result from acom- plicated coupling between different elements, this partic- T ular frequencycan be determined graphically by plotting n the modulus |f (Ω)| and by pointing at alocal minimum Figure 6. Frequency(a) and amplification coefficient (b) as func- of the function. It is then possible to study the onset con- Th tions of the temperature ratio Tn = for the first mode of the Tc ditions of the thermoacoustic instability close to the first loop without alternator (...)and for the twomodes resulting resonance of the loop, butalso for alower frequency, near from the coupling of the twoelements: the mode close to the the mechanical resonance of the alternator. resonance of the alternator (—) and the mode close to the first resonance of the loop (−−). In Figure 6, the amplification coefficient g is plotted as afunction of the temperature ratio Tn for these twodiffer- ent modes. When the loop engine is not coupled with asec- ondary acoustic load (see dotted lines in Figure 6),the on- surprisingly,close to the one of the second mode of the set of the thermoacoustic instability is found for atemper- loop (λ = L/2) butnot the first mode (λ = L). In ad- dition, theyalso investigated the same thermoacoustic de- ature ratio T0 2.4and the corresponding acoustic wave- length is close to the unwrapped length of the loop. Intro- vice acting as a“standing-wave” engine by blocking the ducing the alternator as asecondary acoustic load does not loop with arigid partition: theyobserved the threshold of impact significantly the onset conditions (see dashed lines the fourth standing-wavemode (for which λ = L/2). in Figure 6):asmall increase of the onset temperature is In order to evaluate the consistencyofour model, the observed, due to additionnal losses in the alternator.How- thermoacoustic engine built by Yazaki et al. in 1998 has ever,itappears that the onset temperature for the coupled been considered and the stability curves have been calcu- mode (continuous lines in Figure 6) is lower than the one lated by varying the static pressure P0 inside the engine. corresponding to the first mode of the loop. This behaviour The onset temperature ratio T0 is presented in Figure 7 is usually verified in practice, as this type of engine has as afunction of the square of the ratio r/δκ.The results generally an operating point close to the resonance of the obtained by Yazaki et al. [18] for both configurations (◦: alternator [25]. annular device, •:straight device)are compared with the theoretical ones (straight lines), when the oscillating fre- quencyissearched close to the frequencyofthe mode 3.3. Comparison with experimental results λ = L/2. This frequencycorresponds to the 2nd mode of In 1998, Yazaki et al. studied aclosed-loop thermoacous- the annular device (called “TW” in Figure 7) and to the tic device and measured the stability curves as functions 4th mode of the straight device (called “SW” in Figure 7). 2 of the ratio (r/δκ) ,together with the acoustic work flow The model reproduces quite well the left branches of the using LDV[18]. Theynotably observed that the frequency stability curves, and it predicts an optimal ratio r/δκ close of the sound wave amplified by thermoacoustic effect was, to the experimental one. Forlarger r/δκ,the onset tem-
238 Guedra, Penelet: Description of thermoacoustic engines ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 98 (2012)
Table I. General characteristics of the travelling wave engine Table II. Characteristics of the thermoacoustic engine designed computed with the model. by Yazaki et al. [18].
Loop [18] Model Total length L 1m Total length 2.58 m2.58 m Inner radius R 1cm w Inner radius R 20.1 mm 20.1 mm Location of the hot exchanger x 0.9 m 1 h Length of the glass pipe unknown 0.94 m Length of the regenerator l 4cm s Inner radius of the glass pipe R 18.5 mm 18.5 mm Length of the thermal buffer tube l 8cm 2 w Location of the center of the stack 0.5 m0.5 m Regenerator Length of the stack 4cm4cm Length of the thermal buffer tube unknown 20 cm Semi-width Rs 40 µm Porosity Φ 0.75 Length of the ambient heat unknown 2cm exchanger Fluid (Helium) Length of the hot heat unknown 2cm exchanger Room temperature Tc 300 K Semi-width of one stack pore 0.44mm 0.44 mm Static pressure P0 3MPa Porosity of the stack 0.72 0.72 Electrodynamic Alternators Semi-width of one exchanger unknown 2 × 0.4mm
Radius of the piston Ralt 2cm pore Moving mass Mm 0.2 kg Porosityofexchangers unknown 0.88 Mechanical losses Rm 2Ns/m Room temperature Tc unknown 293 K 4 Stiffness of the suspensions Km 5 · 10 N/m Bl product 20 N/A
Coil electrical resistance Re 6Ω Coil inductance Le 50 mH Back Cavity (cylinder)
Inner radius Rcav 4cm Length lcav 6cm 0 T perature ratio is less important when computed with the model. The differences between the model and the mea- surements realized by Yazaki et al. may be explained with the following statements. Firstly,the temperature distribu- tions are supposed to be linear along the stack and the ther- mal buffer tube. Secondly,some parameters in our model, 2 r such as those of the heat exchangers, the length of the ther- δκ mal buffer tube or the length of the glass pipe used for LDV, were fixed arbitrarily because theywere unknown Figure 7. Stability curves for the second mode of the loop (◦) (all the parameters used for the design are reported in Ta- and the fourth mode of the straight device (•)obtained by Yazaki et al. [18], compared with the theoretical ones (straight lines). In ble II). addition are plotted the stability curves for the first mode (dashed The model may also be used to investigate the typical line)and the second mode (dotted line)ofthe loop when no mode selection observed by Yazaki et al. In his experi- cross-sectionnal area constriction is introduced in the model. ments [18], Yazaki replaced one part of the loop with a glass pipe of smaller inner radius used for velocity mea- Yazaki is not favourable to the onset of the first mode of surements by LDV. The resulting cross-sectional area con- the loop, due to this cross-sectionnal area constriction. striction is not large (see Table II), butwestated the fact that this constriction would be responsible for the mode selection, by “killing” the first mode of the loop. Indeed, 4. About the applicability of the model for when no constriction is introduced in the model, the first transient regimes mode (λ = L,dashed line in Figure 7) naturally be- comes unstable for lower temperature ratios than the sec- As shown in section 3, the model presented in section 2 ond mode (dotted line). But when the cross-sectionnal area can be used for the determination of the onset conditions constriction is added, the first mode of the loop may the- of the thermoacoustic instability,but another advantage of oretically become unstable for much larger,physically in- this model is that it can also be used for the theoretical 2 consistent, temperature ratios (e.g. T0 20 for (r/δκ) study of the transient regime leading to steady-state sound. 2.6),and one can reasonably say that the onset is im- If the accurate account of various nonlinear effects and possible. To conclude, the thermoacoustic engine built by the proper description of heat transfer through the ther-
239 ACTA ACUSTICA UNITED WITH ACUSTICA Guedra, Penelet: Description of thermoacoustic engines Vol. 98 (2012) moacoustic core are discarded in the context of this study, processes, and also to perform comparisons with experi- it is actually quite direct to propose the basic formula- mental data. We will defer such studies to afuture publi- tion which is necessary to compute the transient process of cation. Our main goal here is to point out the advantages of wave amplitude growth and saturation. On the first hand, the formulation proposed in this paper,which can be used let’sassume that it is possible to describe heat transfer to study unsteady processes occuring in thermoacoustic through the thermoacoustic core by means of the follow- systems of complicated geometries. The transient regime ing differential equation: which precedes the stabilization of acoustic pressure am- plitude can exhibit complicated behaviours [7, 8, 34, 35], ∂tT (x, t) = g T (x, t),... , (21) and it provides much more information than the only value of asteady state acoustic pressure. If one is able to re- where ∂t stands for partial time derivative,and where produce such complicated dynamics using an appropriate T (x, t)refers to the time-dependent, cross-sectional av- simplified model, then one is able to get abetter physical erage temperature distribution along the thermoacoustic insight on the nonlinear processes which control the satu- core. In this equation the function g is supposed to account ration of the thermoacoustic instability. for the heat transfer processes (diffusion, convection, radi- ation)controlling the temperature distribution, while the dots in the argument of the function g refer to the geo- 5. Conclusion metrical and thermophysical parameters of the device un- der consideration. On the other hand, the thermoacoustic We presented aformulation for the description of the wave amplification operating in the thermoacoustic core is de- amplification in different kinds of thermoacoustic engines. scribed with the ordinary differential equation The model is suitable for the determination of the thresh- old of thermoacoustic instability,and it can be coupled dtprms(x0,t) = g T(x, t) prms(x0,t), (22) to the equations describing heat transfer through the ther- moacoustic core in order to describe unsteady processes where dt denotes time derivative,where prms(x0,t) = leading to steady-state sound in the frame of the weakly 2 p (x0,t) is the root mean square amplitude of acoustic nonlinear theory.Inthe frame of the linear theory,this pressure oscillations at some position x0 along the device model captures some interesting properties of thermoa- (< ···>denotes time averaging overanacoustic period), coustic engines, as the stack-location influence on the on- and where g is the imaginary part of the complexfre- set of the thermoacoustic instability or the selection of the quencyintroduced in Sect. 2. Note that this equation is di- unstable resonant mode of the device which can be am- rectly derivedfrom equation (1) because prms(x0,t+Δt) = plified. As acomplementary tool to the computer code Δt e g prms(x0,t), and because it is assumed that g Ω and DeltaEC or to direct numerical simulations, the proposed that during the time scale Δt of afew acoustic periods, the analytical model should prove useful for designing ther- variations of the temperature distribution are negligible, so moacoustic engines and for investigating the nonlinear that g stays constant. processes involved in these devices. In order to compute the initial start-up of thermoacous- tic oscillations up to the final stabilization of acoustic pres- Acknowledgements sure amplitude, one must solvethe set of coupled equa- tions (21) and (22) with appropriate boundary and ini- We would liketothank Pierrick Lotton for his useful ad- tial conditions, provided that the nonlinear effects satu- vices concerning the manuscript. rating the wave amplitude growth are included in these equations. Forinstance, the effect of thermoacoustic heat References pumping by the acoustic wave,which tends to reduce the temperature gradient externally imposed along the stack, [1] N. Rott: Thermoacoustics. Advances in Applied Mechanics can be included in equation (21) in the form of an acousti- 20 (1980)135–175. 2 [2] G. W. Swift: Thermoacoustics -Aunifying perspective cally (proportional to prms)enhanced thermal conductivity [12, 29] of the stack .More generally,asinthe case of for some engines and refrigerators. Acoustical Society of America, Melville, NY,2002. the DeltaEC computer code, it is possible to account (ina necessary simplified way) for some of the nonlinear effects [3] S. L. Garrett: Resource letter: TA-1: Thermoacoustic en- involved in the saturation process, such as minor losses at gines and refrigerators. Am. J. Phys. 72 (2004)11–17. the edges of the stack [30], higher harmonics generation [4] W. C. Ward, G. W. Swift, J. P. Clark: Interactive analysis, [9, 10, 12], or heat convection due to acoustic streaming design, and teaching for thermoacoustics using DeltaEC. J. Acoust. Soc. Am. 123 (2008)3546–3546. [31, 32, 33]. As mentioned above,nofurther derivations are pre- [5] G. B. Chen, T. Jin: Experimental investigation on the onset and damping behavior of the oscillation in athermoacoustic sented in this paper concerning the modeling of unsteady prime mover. Cryogenics 39 (1999)843–846. processes in thermoacoustic engines. This would indeed [6] T. Jin, C. S. Mao, K. 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