Discussion on Refrigeration Cycle for Regenerative Cryocoolers

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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, respectively, 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,

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. 3. in the Carnot cycle. For a cycle, the first law of thermodynamics must be Fig. 3 shows various Lorenz cycles with adiabatic satisfied, so the work done in a cycle is compression and expansion processes and two poly- n À k w ¼ q À q ¼ C ½ðT À T ÞÀðT À T Þ: ð5Þ tropic processes. The cycle 1–2–3–4 expresses the Carnot h c n À 1 v 2 3 1 4 cycle, the 100–200–3–4 represents the modified Stirling The thermodynamic efficiency e for the Lorenz cycle cycle (two isentropic and two isochoric) and the 10–20–3– then becomes 4 is the Brayton cycle (two isentropic and two isobaric). The thermodynamic efficiency expressed by equivalent T2 À T3 e ¼ 1 À 1 : ð6Þ T1 À T4 For the isentropic processes 1 ! 2 and 3 ! 4, we have (see Fig. 3) T P ðkÀ1Þ=k T P ðkÀ1Þ=k 2 ¼ 2 and 4 ¼ 4 : ð7Þ T1 P1 T3 P3 Thus T Á T P Á P ðkÀ1Þ=k 4 2 ¼ 4 2 : ð8Þ T3 Á T1 P3 Á P1 For the polytropic processes 2 (20 or 200Þ!3and 4 ! 1ð10 or 100), we get T P ðnÀ1Þ=n T P ðnÀ1Þ=n 2 ¼ 2 and 4 ¼ 4 : ð9Þ T3 P3 T1 P1 Thus T Á T P Á P ðnÀ1Þ=n 4 2 ¼ 4 2 : ð10Þ T3 Á T1 P3 Á P1 Combining Eqs. (8) and (10), the following condition Fig. 3. Two adiabatic and two polytropic regenerative processes. must be met: 中国科技论文在线 http://www.paper.edu.cn

136 G. Chen et al. / Cryogenics 42 (2002) 133–139 P2 Á P4 T ¼ 1: ð11Þ e ¼ c P Á P 1 3 T h À T c q q q Thus ¼ c a À c s1À s4 s2 À s3 s1 À s4 P2 P3 q ¼ : ¼ 1 a À 1 P1 P4 qc From Eq. (7), we obtain the following expression: T2 À T3 ¼ 1 À 1 T1 À T4 T2 T3 T2 þ T3 T2 À T3 0 ¼ ¼ or ¼ : ð11 Þ Tc T1 T4 T1 þ T4 T1 À T4 ¼ : ð15Þ Ta À Tc Substituting the above relation into Eq. (6), expression This means that we can use T and T to express the of the thermodynamic efficiency e of the Lorenz cycle a c thermodynamic efficiency e for the Lorenz cycle. Com- can now be obtained as paring Eq. (12) with Eq. (1), the conclusion of the same T T thermodynamic efficiency e value for both the Carnot e ¼ 4 ¼ c ; ð12Þ T3 À T4 Ta À Tc cycle (at temperature limits Ta and Tc) and the Lorenz cycle (at equivalent temperature limits Ta and Tc) holds T T T T using a and c for 3 and 4, respectively. true. Here, the authors define Ta and Tc of polytropic This expression is derived also by use of the integral processes in Fig. 3 as the equivalent temperature limits mean temperature T . The efficiency of the Lorenz cycle corresponding to those of the Carnot cycle. shown in Fig. 2 could be expressed as follows: Normally the polytropic index n is expressed by R 1 6 n 6 k, but definition of the generalized polytropic q 1 T ds T c R 4 0iR c process becomes 0 6 n 6 1. In the Lorenz cycle, there e ¼ ¼ 2 1 ¼ : ð13Þ qh À qc T h À T c 3 Ti ds À 4 T0i ds are two special cases. When n ¼1, i.e. v ¼ const:, Eq. (3) becomes qc ¼ CvðT1 À T4Þ. This is the case of the Here, Ti and T0i represent instantaneous temperatures at Stirling cycle at the condition of two adiabatic processes the heat rejected and heat absorbed processes, respec- and two isochoric processes. When n ¼ 0, i.e. p ¼ const:, tively, and T and T represent the integral mean tem- h c Eq. (3) will become qc ¼ CpðT1 À T4Þ. This is the case of peratures of these two processes defined by Eq. (14) the Brayton cycle at the conditions of two adiabatic Q processes and two isobaric processes. Of course, in the T ¼ : ð14Þ special case of constant temperature, it is the Carnot DS cycle itself. In this meaning, all the above cycles are When the values of T3 and T4 in Fig. 2 are equal to Ta named the Lorenz cycle in the following analysis. and Tc, respectively, as shown in Fig. 3, using Eqs. (3), So far, we could draw a conclusion that all regener- (4), (110) and (14), we can obtain ative refrigeration cycles might be divided into two

Fig. 4. Branch chart of Carnot cycle: Reitlinger cycle and Lorenz cycle. 中国科技论文在线 http://www.paper.edu.cn

G. Chen et al. / Cryogenics 42 (2002) 133–139 137 groups, which were characterized by two isothermal and In an ideal refrigeration cycle, the specific heat of the two polytropic processes as well as by two adiabatic and working fluid is constant in the whole system. So, using two polytropic processes. As a summary of the above Ta and Tc for T4 and T1, respectively, we have discussion, a branch chart of the Carnot cycle is shown in Fig. 4. CpðT1 À T6Þ e ¼ C T T C T T pð3 À 4ÞÀ pð 1 À 6Þ ðT À T Þ ¼ 1 3 a À 1 5. Ideal cycles based on isentropic compression and ðT À T Þ expansion c 6 T ðT =T ÞÀ1 ¼ 1 a Á 3 a À 1 Tc 1 ÀðT6=TcÞ In the following we will discuss idealized refrigeration , ! ðkÀ1Þ=k cycles, in which ideal gas of the working fluid and 100% Ta Ph efficiency of heat transfer in the heat exchanger or in the ¼ 1 Á À 1 ð19Þ Tc Pl regenerator are assumed and fluid flow-resistance and other mechanical losses are to be neglected as well. because Ta and Tc are also equal to T2 and T5, respectively. 5.1. Brayton cycle

The thermodynamic efficiency e of the Brayton cycle 5.2. Modified stirling cycle (see Fig. 5) can be expressed as follows. The heat rejected to ambient at the constant pressure The modified Stirling cycle consists of two isentropic ph, and two isochoric processes. Comparing with the Brayton cycle, the two isobaric processes are replaced by qh ¼ h3 À h4: ð16Þ two isochorics in the modified Stirling cycle. Therefore, The heat removed from the cold end heat exchanger at the thermodynamic efficiency can be written as the constant pressure pl, namely cooling power of the system, CvðT1 À T6Þ e ¼ CvðT3 À T4ÞÀCvðT1 À T6Þ qc ¼ h1 À h6: ð17Þ ðT À T Þ The thermodynamic efficiency of the system then can be ¼ 1 3 a À 1 expressed as follows: ðTc À T6Þ Ta ðT3=TaÞÀ1 qc h1 À h6 ¼ 1 Á À 1 e ¼ ¼ : ð18Þ Tc 1 ÀðT6=TcÞ qh À qc ðh3 À h4ÞÀðh1 À h6Þ , ! T V kÀ1 ¼ 1 a Á l À 1 : ð20Þ Tc Vh

5.3. Giﬀord–McMahon cycle

For the ideal G–M cycle, the refrigeration produced per cycle is equal to its work diagram area: I

qc ¼ p dV ¼ V ðph À plÞ: ð21Þ

The input power of the cycle equals the work required to compress the working mass, which ﬁlls the expander: Z ph W ¼ðmc À maÞ v dp; ð22Þ pl

where mc À ma ¼ phV =RTc À plV =RTa. This is the quan- tity of the high pressure gas which ﬁlls the maximum cold space, V, minus that of the low pressure gas left in the maximum warm space, V, in a cycle. Thus, the input Fig. 5. Modiﬁed Brayton cycle with a regenerator. power can be expressed as follows: 中国科技论文在线 http://www.paper.edu.cn

138 G. Chen et al. / Cryogenics 42 (2002) 133–139

Fig. 6. Thermodynamic eﬃciency vs. pressure ratio (a) and vs. cooling temperature (b) of ideal cycles.

" # ðkÀ1Þ=k gest to use the modiﬁed Brayton cycle. However, in the cpTRV ph pl ph W ¼ À À 1 pulse tube refrigerator, the turbine expander of the R Tc Ta pl " # Brayton cycle is substituted by the expansion space of ðkÀ1Þ=k kVpl ph TR ph the pulse tube, and the recuperator is replaced by the À 1 À 1 ; ð23Þ regenerator. We are thus dealing with a variant of the k À 1 pl Tc pl Brayton cycle, denoted as the modiﬁed Brayton cycle

where TR is the inlet temperature of the compressor, [9]. In an ideal pulse tube refrigeration cycle, Eq. (19) which nearly equals the room temperature Ta. Finally, may be rewritten as the following expression (see Fig. 5), we have the thermodynamic eﬃciency of the G–M ma- since there is no expansion work recovery for the pulse chine per cycle [7]: tube refrigerator:

qc q h À h e ¼ e ¼ 2 ¼ 1 6 W w h3 À h2 k À 1 ph T À T ¼ À 1 ¼ c 0 k pl T À T ,( " #) ,h ()a ðkÀ1Þ=k ðkÀ1Þ=k ph TR ph T P À 1 À 1 : ð24Þ ¼ 1 a Á h : ð26Þ pl Tc pl Tc Pl

For the purpose of comparison, the thermodynamic 5.4. Solvay cycle efficiencies of various ideal cycles derived above are plotted in Fig. 6 for helium as the working fluid, as a One may regard the Solvay cycle as a variant of the function of pressure ratio (Fig. 6(a)) and of refrigeration G–M cycle. The expression of the thermodynamic effi- temperature (Fig. 6(b)). Here are assumed the ambient ciency for the Solvay cycle has been discussed in detail temperature of 300 K and the refrigeration temperature by Longsworth [8]: of 80K (Fig. 6(a)), or the pressure ratio of 2 (Fig. 6(b)). q The curve of the pulse tube refrigeration cycle in Fig 6 e ¼ c W has lower efficiency compared with that of the Braton ( ) 1=k cycle due to no expansion work recovery. Worthy of T p 1 p p k À 1 ¼ c Á i À i À l note, the irreversible losses of the orifice and the double T p k p p k R h h h inlet of the pulse tube refrigerator here are excluded. ,( ) p ðkÀ1Þ=k h À 1 : ð25Þ pl 6. Conclusion

5.5. Pulse tube refrigeration cycle The high-performance regenerative cycle postulates that which contains either two isothermal or two adia- In order to predict the thermodynamic eﬃciency of batic processes of compression and expansion in the the pulse tube refrigeration cycle [10], the authors sug- cycle. The former is the Reitlinger cycle with two iso- 中国科技论文在线 http://www.paper.edu.cn

G. Chen et al. / Cryogenics 42 (2002) 133–139 139 thermal and two polytropic processes. The latter is the References Lorenz cycle with two isentropic and two polytropic processes. Introducing the equivalent temperature lim- [1] Reitlinger J. Uber Kreisprozesse zwischen zwei isothermen. Z Ost its, the efficiency of this cycle containing heat exchangers Ing Arch Ver 1876. [2] Walker G. Cryocoolers, Part 1. New York: Plenum Press; 1983. or regenerators with temperature gradient can be rea- [3] Timmerhaus KD, Flynn TM. Cryogenic process engineering. New sonably compared with that of the Carnot cycle. York: Plenum Press; 1989. For simple analysis, the pulse tube refrigeration could [4] Lorenz H. Beitrage zur Beurteilung von Kuhlmaschincn. Z Ver be considered as a variant of the Brayton cycle, and Dtsch Ing 1894;38(3). Eq. (26) may be used for estimating the thermody- [5] Ravex A. Recent developments in cryocoolers. In: 20th Interna- tional Congress of Refrigeration, IIR, 1999; Sydney. namic efficiency of an ideal cycle of the pulse tube re- [6] Finkelstein T. Generalized thermodynamic analysis of Stirling frigeration. engines S.A.E., Paper number 118B, January 1960. [7] Gifford WE. The Gifford–McMahon cycle. Adv Cryo Eng 1961;11:152. [8] Longsworth RC. A modified Solvay-cycle cryogenic refrigerator. Acknowledgements Adv Cryo Eng 1970;16:195–204. [9] McCormick JA et al. Design and test of low capacity reverse This work is financially supported by the National Brayton cryocooler for refrigeration at 35 K and 60K. Cryoco- olers 1999;10:421–9. Natural Sciences Foundation of China. The authors [10] Chen GB, Gan ZH, Thummes G, Heiden C. Thermodynamic would like to thank late Prof. C. Heiden and Prof. G. performance prediction of pulse tube refrigeration with mixture Thummes for helpful discussions. fluids. Cryogenics 2000;40:262–7.