Thermodynamic analysis of GM-type pulse tube coolers q Y.L. Ju * ,1 Cryogenic Laboratory, Chinese Academy of Sciences, P.O. Box 2711, Beijing 100080, People's Republic of China Received 29 January 2001; accepted 28 June 2001 Abstract The thermodynamic loss of rotary valve and the coecient of performance COP) of GM-type pulse tube coolers PTCs) are discussed and explained by using the ®rst and second laws of thermodynamics in this paper. The COP of GM-type PTC, based on two types of pressure pro®les, the sinusoidal wave inside the pulse tube and the step wave at the compressor side, has been derived and compared with that of Stirling-type PTC. Result shows that additional compressor work is needed due to the irreversible entropy productions in the rotary valve thereby decreasing the COP of GM-type PTC. The eect of double-inlet mode on the COP of PTC has distinct improvement at lower temperature region larger T H =T L ). It is also shown that the COP of GM-type PTC is independent of the shape of the pressure pro®les in the ideal case of no ¯ow resistance in the regenerator. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Pulse tube cooler; GM-type; Thermodynamic analysis 1. Introduction In general pulse tube cooler PTC) requires a phase shifter system, located at the hot end of the pulse tube, to achieve an optimum phase angle h between the gas ¯ow rate and the pressure oscillation inside the pulse tube to increase the cooler performance. The so-called ori®ce [1] and double-inlet [2] modes are two of the most well-known con®gurations. Other innovations are multi- bypass [3], two-piston [4], four-valve [5], inertance tube [6], active-buer [7], inter-phasing [8], and double-ori®ce [9]. In recent years multi-stage 4 K-PTCs [9±11] have been reported with multi-layered hybrid magnetic materials in the coldest regenerator region. Two-stage PTCs can provide more than 0.5 W cooling power at 4.2 K and meet the cooling requirements of superconducting de- vices operating at 4 K [12]. By using 3 He as the working ¯uid the lowest temperature below 1.8 K has been achieved [13]. All these machines use a compressor and a valve system to produce pressure oscillation in the cooler system. They are called GM-type PTCs. A GM-type PTC, shown in Fig. 1, only diers from the Stirling-type at the compressor side of the cooler. Instead of a piston compressor the GM-type uses a ro- tary valve or several electromagnetic valves to generate the pressure oscillations in the cooler. In the ideal case the compressor is isothermal and reversible. The com- pressor heat is removed in the aftercooler. The rotary valve connects the PTC system alternatively to a con- stant high pressure p H and a low pressure p L . This paper is a continuation of our previous papers on the thermodynamic aspects of pulse tubes [14±17]. We will discuss the thermodynamic losses of the rotary valve and the coecient of performance COP) of GM-type PTC in this paper. General expressions for the COP of GM-type PTC, based on two types of pressure pro®les, the sinusoidal wave inside the pulse tube and the step wave at the compressor side, have been derived and compared with those of Stirling-type PTCs. 2. Thermodynamic analysis Fig. 2 shows the compressor and the rotary valve parts of a GM-type PTC with enthalpy ¯ows and en- tropy ¯ows in it. Consider the control volume at left Cryogenics 41 2001) 513±520 www.elsevier.com/locate/cryogenics q This work was partly carried out at Faculty of Applied Physics, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands. * Tel.: +86-10-6262-7302; fax: +86-10-6256-4049. E-mail address: yonglin@cl.cryo.ac.cn Y.L. Ju). 1 On leave from Faculty of Applied Physics, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Nether- lands. 0011-2275/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0011-227501)00123-0