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Design of Thermoacoustic Refrigerators M.E.H Cryogenics 42 (2002) 49–57 www.elsevier.com/locate/cryogenics Design of thermoacoustic refrigerators M.E.H. Tijani, J.C.H. Zeegers, A.T.A.M. de Waele * Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 12 November 2001; accepted 5 December 2001 Abstract In this paper the design of thermoacoustic refrigerators, using the linear thermoacoustic theory, is described. Due to the large number of parameters, a choice of some parameters along with dimensionless independent variables will be introduced. The design strategy described in this paper is a guide for the design and development of thermoacoustic coolers. The optimization of the different parts of the refrigerator will be discussed, and criteria will be given to obtain an optimal system. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermoacoustic; Refrigeration 1. Introduction temperature of À65 °C. The measurement results can be found elsewhere [3,4]. The theory of thermoacoustics is well established, but quantitative engineering approach to design thermoa- coustic refrigerators is still lacking in the literature. 2. Design strategy Thermoacoustic refrigerators are systems which use sound to generate cooling power. They consist mainly of We start by considering the design and optimization of a loudspeaker attached to an acoustic resonator (tube) the stack which forms the heart of the cooler. The coef- filled with a gas. In the resonator, a stack consisting of a ficient of performance of the stack, defined as the ratio of number of parallel plates and two heat exchangers, are the heat pumped by the stack to the acoustic power used installed, as shown in Fig. 1. The loudspeaker sustains by the stack, is to be maximized. The exact theoretical an acoustic standing wave in the gas at the fundamental expressions of the acoustic power and cooling power in resonance frequency of the resonator. The acoustic the stack are complicated, so one can try to use the standing wave displaces the gas in the channels of the simplified expressions deduced from the short stack, and stack while compressing and expanding. The thermal boundary-layer approximations [1,3]. These expressions interaction between the oscillating gas and the surface of still look complicated and they contain a large number of the stack generates an acoustic heat pumping. The heat parameters of the working gas, material and geometrical exchangers exchange heat with the surroundings, at the parameters of the stack. It is difficult to deal in engi- cold and hot sides of the stack. A detailed explanation of neering with so many parameters. However, one can the way thermoacoustic coolers work is given by Swift reduce the number of parameters by choosing a group of [1] and Wheatly et al. [2]. In this paper the design and dimensionless independent variables. Olson and Swift [5] development procedure of a thermoacoustic refrigerator wrote a paper about similitude and dimensionless pa- is reported. The thermoacoustic refrigerator, con- rameters for thermoacoustic devices. Some dimension- structed on basis of the design procedure given in this less parameters can be deduced directly. Others can be paper, has operated properly and it has reached a low defined from the boundary-layer and short-stack as- sumptions [1,3]. The parameters, of importance in ther- moacoustics, which are contained in the work flow and heat flow expressions are given in Table 1 [3]. * Corresponding author. Tel.: +31-40-247-4215; fax: +31-40-243- The goal of the design of the thermoacoustic refrig- 8272. erator is to meet the requirements of a given cool- _ E-mail address: [email protected] (A.T.A.M. de Waele). ing power Qc and a given low temperature TC. This 0011-2275/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0011-2275(01)00179-5 50 M.E.H. Tijani et al. / Cryogenics 42 (2002) 49–57 plify the number of parameters. Olson and Swift [5] proposed to normalize the acoustic power W_ and the _ cooling power Qc by the product of the mean pressure pm; the sound velocity a, and the cross-sectional area of the stack A: pmaA. The amplitude of the dynamic pres- sure can be normalized by the mean pressure. The ratio p0=pm is called the drive ratio D. In practice the stack Fig. 1. A simple illustration of a thermoacoustic refrigerator. An material can be chosen so that the thermal conductive acoustically resonant tube containing a gas, a stack of parallel plates term in the heat flow expression can be neglected [1]. In and two heat exchangers. A loudspeaker is attached to one end of the this case the parameters of the stack material do not tube and the other end is closed. Heat Q_ is pumped up the stack so that the cold heat exchanger becomes colder and the hot heat exchanger have to be considered in the performance calculations. hotter. The porosity of the stack, sometimes called blockage ratio and defined as y0 Table 1 B ¼ ð1Þ Operation-, working gas-, and stack parameters y0 þ l Operation parameters Working gas parameters is also used as a dimensionless parameter for the ge- ometry of the stack. The thermal and viscous penetra- Operating frequency: f Dynamic viscosity: l tion depths are given by Average pressure: pm Thermal conductivity: K sffiffiffiffiffiffiffiffiffiffiffi Dynamic pressure amplitude: Sound velocity: a 2K p0 dk ¼ ð2Þ Mean temperature: Tm Ratio of isobaric to isochoric qcpx specific heats: c and Stack sffiffiffiffiffiffiffiffiffi Material Geometry 2l dm ¼ ; ð3Þ Length: Ls qx Thermal conductivity: Ks Stack center position: xs where K is the thermal conductivity, l is the viscosity, q Density: qs Plate thickness: 2l Specific heat: cs Plate spacing: 2y0 is the density, cp is the isobaric specific heat of the gas, Cross-section: A and x is the angular frequency of the sound wave. The resultant normalized parameters are given an extra index n and are shown in Table 2. The number of pa- requirement can be added to the operation parameters rameters can once more be reduced, by making a choice shown in Table 1. of some operation parameters, and the working gas. The boundary-layer and short-stack approximations assume the following [1,3]: • The reduced acoustic wavelength is larger than the 3. Design choices stack length: k=2p Ls; so that the pressure and ve- locity can be considered as constant over the stack For this investigation we choose to design a refriger- and that the acoustic field is not significantly dis- ator for a temperature difference of DTm ¼ 75 K and a turbed by the presence of the stack. cooling power of 4 W. In the following, we will discuss • The thermal and viscous penetration depths are smal- ler than the spacing in the stack: dk; dm y0. This as- Table 2 sumption leads to the simplification of Rott’s Normalized operation-, working gas-, and stack parameters functions, where the complex hyperbolic tangents Operation parameters can be set equal to one [1,3]. Drive ratio: D ¼ p0=pm Q Q_ =p aA • The temperature difference is smaller than the average Normalized cooling power: cn ¼ c m Normalized acoustic power: W ¼ W_ =p aA temperature: DT T , so that the thermophysical n m m m Normalized temperature difference: DTmn ¼ DTm=Tm properties of the gas can be considered as constant within the stack. Gas parameters Prandtl number: r The length and position of the stack can be normalized Normalized thermal penetration depth: dkn ¼ dk =y0 by k=2p. The thermal and viscous penetration depths can be normalized by the half spacing in the stack y0. Stack geometry parameters The cold temperature or the temperature difference can Normalized stack length: Lsn ¼ kLs Normalized stack position: x ¼ kx be normalized by Tm. Since dk and dm (see below) are n related by the Prandtl number r, this will further sim- Blockage ratio or porosity: B ¼ y0=ðy0 þ lÞ M.E.H. Tijani et al. / Cryogenics 42 (2002) 49–57 51 the selection of some operation parameters, the gas and thermal conductivity of all inert gases. Furthermore, stack material. helium is cheap in comparison with the other noble gases. A high thermal conductivity is wise since dk is 3.1. Average pressure proportional to the square root of the thermal conduc- tivity coefficient K. The effect of using other gases is Since the power density in a thermoacoustic device is discussed elsewhere [10]. proportional to the average pressure pm [1], it is favor- able to choose pm as large as possible. This is determined 3.5. Stack material by the mechanical strength of the resonator. On the other hand, dk is inversely proportional to square root of The heat conduction through the stack material and pm, so a high pressure results in a small dk and a small gas in the stack region has a negative effect on the per- stack plate spacing. This makes the construction diffi- formance of the refrigerator [1,3]. The stack material cult. Taking into account these effects and also making must have a low thermal conductivity Ks and a heat the preliminary choice for helium as the working gas (see capacity cs larger than the heat capacity of the working below), the maximal pressure is 12 bar. We choose to gas, in order that the temperature of the stack plates is use 10 bar. To minimize the heat conduction from the steady. The material Mylar is chosen, as it has a low hot side of the stack to the cold side, we used a holder heat conductivity (0.16 W/m K) and is produced in made of a material with low thermal conductivity (e.g.
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