http://www.paper.edu.cn Cryogenics 42 (2002) 133–139 Discussion on refrigeration cycle for regenerative cryocoolers Guobang Chen *, Zhihua Gan, Yanlong Jiang Cryogenics Laboratory, Zhejiang University, Hangzhou 310027, PR China Received 15 February 2001; accepted 15 January 2002 Abstract Based on review and analysis of thermodynamic efficiency e of the Carnot cycle and the cycle with two isothermal and two polytropic processes, another thermodynamic cycle with two isentropic and two polytropic processes, which can achieve the Carnot value of thermodynamic efficiency, is testified theoretically. Thermodynamic efficiency expressions of a number of ideal regenerative refrigeration cycles are derived, including the ideal pulse tube refrigeration cycle. A classified branch chart and a plot of ideal thermodynamic efficiency of regenerative refrigeration cycles are given for the purpose of comparison. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermodynamics; Refrigeration cycle; Carnot cycle; Brayton cycle; Pulse tube 1. Introduction same thermodynamic efficiency value too. The Lorenz cycle is such a cycle consisting of two adiabatic processes Thermodynamic efficiency e is an important thermo- of compression and expansion, and two arbitrary poly- dynamic measure of quality for cryogenic refrigerators. tropic processes [4]. This cycle can also attain the Carnot The Carnot engine is the most efficient unit and has the value of thermodynamic efficiency e at the same equiv- maximum value of the thermodynamic efficiency e.In alent temperature limits. this cycle, the adiabatic compression and expansion of the working fluid are assumed to be reversible. And in the isothermal processes, heat transfer between the 2. Carnot cycle working fluid and the heat sinks is also supposed to be reversible. The Carnot value of thermodynamic effi- The Carnot cycle consists of two reversible isentropic ciency e is the reference index for all other refrigeration (compression and expansion) processes and two revers- cycles. In fact, many other reversible thermodynamic ible isothermal (heat rejection and absorption) pro- cycles, which are based on the two isothermal processes, cesses. If the working fluid is an ideal gas, one can can also attain the same performance at the same tem- readily show that the thermodynamic efficiency e is perature limits. The most general form of idealized simply given by thermodynamic cycle analyzed by Reitlinger [1] and Q T 1 Walker [2] consists of two isothermal processes of com- e ¼ c ¼ c ¼ i Q À Q T À T ðT =T ÞÀ1 pression and expansion and two polytropic regenerative a c a c a c 1 processes. The Stirling cycle containing two isothermal ; 1 ¼ ðkÀ1Þ=k ð Þ and two isochoric processes is a popular example, which ðÞP2=P1 À 1 can attain the same thermodynamic efficiency e as that where Q is the heat removed at the refrigeration tem- of the Carnot cycle [3]. c perature T , Q is the heat rejected at the ambient tem- Besides, there is another thermodynamic cycle, which c a perature T , P and P refer to the higher and lower is based on two adiabatic processes and can achieve the a 2 1 pressures at both the ends of an adiabatic process, re- spectively, and k ¼ Cp=Cv is the adiabatic index. * Corresponding author. Tel.: +86-571-87951771; fax: +86-571- Typical values of e for the Carnot cycle operating 87952464. between the ambient temperature (Ta ¼ 300 K) and E-mail address: [email protected] (G. Chen). various refrigeration temperatures (100, 80, 20, 4.2 K, 0011-2275/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0011-2275(02)00017-6 转载 中国科技论文在线 http://www.paper.edu.cn 134 G. Chen et al. / Cryogenics 42 (2002) 133–139 Table 1 tropic heat transfer processes in place of the adiabatic Typical thermodynamic efficiency e and required pressure ratio for the compression and expansion of the Carnot cycle. Heat Carnot refrigerator (Ta ¼ 300 K) rejection and reception during the polytropic processes Tc must be provided by thermal storage in a regenerator. 100K 80K 20K 4.2 K All these cycles are characterized by the same thermo- ei 0.50 0.36 0.071 0.014 dynamic efficiency e as expressed in Eq. (1). P2=P1 15.6 27.2 871 43 100 However, it must be emphasized that the compression and expansion processes of both Stirling and Ericsson refrigerators are practically much closer to adiabatic respectively) are given in Table 1. Substituting k ¼ 1:67 processes rather than isothermal [2–6]. This means that for helium in Eq. (1), the corresponding pressure ratios the adiabatic model is suitable for most design and op- required for the cycle are also shown in Table 1. timization of the Stirling cryocoolers and is a good deal Obviously, it is practically impossible for a cryogenic better than the isothermal one. Obviously, the modified refrigerator to be operated at the refrigeration temper- adiabatic model of the Stirling cycle is concerned in the atures following the Carnot cycle. The required pressure Lorenz cycle rather than the Reitlinger cycle. ratio would greatly exceed the limitation of compressor technology. Fortunately, in practical application, the high-pressure ratio has been brought down to a rea- sonable value by means of regenerative heat transfer 4. Deduction and testification of Lorenz cycle processes in the cycle. Based on the above analysis, the authors make the following deduction. The high-performance regenerative cycle postulates that containing either two isothermal or 3. Reversible refrigeration cycle two adiabatic processes of compression and expansion in the cycle. The former is the Reitlinger cycle described The thermodynamic efficiency e for the Carnot engine above. The latter is based on two isentropic processes in is the maximum value that can be achieved by a refrig- the Carnot cycle, but the rest two isothermal processes erator operating between temperature limits Ta and Tc. of heat reception and rejection are replaced by two poly- However, it is well known that other reversible ther- tropic processes. The above two reversible cycles can modynamic cycles based on isothermal compression and attain the thermodynamic efficiency e corresponding to expansion can attain the same performance [2]. The the Carnot value between the same or equivalent tem- various forms of the idealized thermodynamic cycle perature limits. called the Reitlinger cycle are shown in Fig. 1. The cycle According to the thermodynamics, even though the 1–2–3–4 represents the Carnot cycle, which consists of heat sinks have temperature gradient, the cycle consist- two adiabatic and two isothermal processes. The 1–20– ing of two no temperature-difference heat transfer pro- 30–4 is the Stirling cycle with two isothermal processes cesses and two isentropic processes has the maximum and two isochoric processes. And the 1–200–300–4 is the thermodynamic efficiency. This cycle is called the Lorenz Ericsson cycle with two isothermal and two isobaric cycle [4]. Fig. 2 shows the T–S plane of the Lorenz cycle processes. The 2nd and 3rd cycles incorporate poly- with adiabatic compression and expansion processes and two polytropic processes. In the Lorenz cycle, all Fig. 1. Two isothermal and two polytropic regenerative processes. Fig. 2. T–S plane of Lorenz cycle. 中国科技论文在线 http://www.paper.edu.cn G. Chen et al. / Cryogenics 42 (2002) 133–139 135 processes are reversible, so it has the same thermody- temperature limits for the polytropic cycle is testified as namic efficiency value as that of an equivalent Carnot follows. cycle. As a matter of fact, the Lorenz cycle could be The state equation of a polytropic process is expressed considered to consist of many differential Carnot cycles. as PV n ¼ const:, where n is the polytropic index. Rela- Its thermodynamic efficiency could be obtained through tions of state parameters for a polytropic process of the integral of the rejected or received heat of differential ideal gas can be expressed as Carnot cycles. We could use the integral mean temper- P V n T V nÀ1 T P ðnÀ1Þ=n ature of heat transfer processes to express its thermo- 2 ¼ 1 ; 2 ¼ 1 and 2 ¼ 2 : dynamic efficiency. It is less well known that the Lorenz P1 V2 T1 V2 T1 P1 cycle is stem of several cryogenic cycles. The Brayton ð2Þ cycle widely used in cryogenics [9] is basically a variant of the Lorenz cycle. Besides, there are a number of Let us assume that polytropic index n is neither equal to thermodynamic cycles, which incorporate the polytropic adiabatic index k nor 1, since the conditions of n ¼ k heat transfer processes in place of the isothermal pro- and n ¼ 1 correspond to the adiabatic and the isother- cesses of heat rejection and addition in the Carnot cycle mal processes, respectively, which constitute the Carnot and can achieve the Carnot value of e; such as the cycle itself. Then, we can testify that the thermodynamic modified Stirling cycle (constant volume), the modified efficiency e of the Lorenz cycle is in accordance with that Ericsson cycle (constant pressure) and, as the special of the Carnot cycle. Referring to Fig. 3, the second law case, the Carnot cycle itself (constant temperature). All of thermodynamics yields the expressions above cycles are referred to the Lorenz cycle in the n À k the heat absorbed q ¼ C ðT À T Þ; ð3Þ following analysis so as to avoid confusion with the c n À 1 v 1 4 Reitlinger cycle. However, it is difficult for the cycle with n À k polytropic processes to be compared with the Carnot the heat rejected q ¼ C ðT À T Þ: ð4Þ h n À 1 v 2 3 cycle at the same constant temperature limits. So, we 0 00 0 introduce the equivalent temperature limits into the Here, the suffix 1 stands for 1 or 1 and the suffix 2 for 2 00 Lorenz cycle instead of the constant temperature limits or 2 in Fig.