http://www.paper.edu.cn

Thermodynamic performance prediction of pulse tube refrigeration with mixture ¯uids q Guobang Chen a,*, Zhihua Gan a, G. Thummes b, C. Heiden b

a Cryogenics Lab, Zhejiang University, Hangzhou 310027, People's Republic of China b Institute of Applied Physics, University of Giessen, Giessen, Germany Received 11 January 2000; accepted 17 May 2000

Abstract A refrigeration cycle with two isentropic and two isobaric processes, referred to as the modi®ed Brayton cycle, is introduced for predicting the thermodynamic performance of pulse tube refrigeration with a binary mixture refrigerant. The corresponding the- oretical expressions of cooling power, thermodynamic eciency and required work of a refrigeration cycle are established. Based on the prediction calculations for a number of cryogenic ¯uids, promising mixture pairs of working refrigerants are recommended. The computed results show that 9.5% of the coecient of performance (COP) and 6.7% of the cooling power at 80 K could be gained if a mixture pair of 10% nitrogen and 90% helium is used instead of pure helium for the pulse tube refrigeration. The minor mixture pairs which have a positive e€ect on pulse tube refrigeration at temperatures near 80 K, including hydrogen±helium, argon±helium and neon±helium, are also discussed. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Gas mixture; Thermodynamics; Brayton cycle; Pulse tube refrigeration

1. Introduction calculations, it can be expected that an increase in thermodynamic eciency can be obtained at 80 K Pulse tube refrigerators bear the promise of high re- without any penalty in size, weight or cost of the re- liability and economic operation and can ± at least in frigerator if a proper binary working ¯uid is used. For a principle ± be adapted to many di€erent particular ap- mixture of 10% nitrogen and 90% helium, an increase of plications. In order to increase their thermodynamic cooling power of 6.7% and an increase of coecient of eciency such that they become competitive with other performance (COP) of 9.5% were obtained for temper- cooler types, we focused on the use of mixed working atures near 80 K. ¯uids [1±4]. For this purpose, the refrigeration cycle in the pulse tube refrigerator is modeled by a cycle con- sisting of two isentropic and two isobaric processes, 2. Refrigeration cycle similar to the Brayton cycle. For the calculations, a number of cryogenic ¯uids To predict the refrigeration performance of pulse tube have been considered in order to select an appropriate refrigerators, a proper thermodynamic cycle model is component to work with helium as a pair of mixed re- indispensable. The authors adopt a practical method, frigerants. Mixtures of hydrogen±helium (possibly single which can meet the requirement of thermodynamic phase) and two-phase mixtures of nitrogen±helium, ar- analysis. In the case of pulse tube refrigeration, the gon±helium and neon±helium are considered and com- following assumptions are made: pared with pure helium as refrigerants. From the model 1. the processes of ambient compression of the working refrigerant and expansion for refrigeration are much closer to adiabatic processes than isothermal ones; q The work was carried out at the Institute of Applied Physics, 2. the heat rejection of compression to a coolant in the University of Giessen. after cooler is an isobaric process; * Corresponding author. Tel.: +86-571-7951771; fax: +86-571- 7951358. 3. the cooling power of the pulse tube refrigerator is ab- E-mail address: [email protected] (G. Chen). sorbed over a whole range of temperatures rather

0011-2275/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 0 ) 0 0 034-5 转载 中国科技论文在线 http://www.paper.edu.cn

262 G. Chen et al. / Cryogenics 40 (2000) 261±267 than at a constant temperature. This is modeled by a 3. Thermodynamic eciency constant pressure process from minimum to maxi- mum refrigeration temperature; The COP of the modi®ed Brayton cycle, which con- 4. heat transfer processes in a regenerator are carried sists of two isobaric and two adiabatic processes, can be out under constant pressure. The enthalpy unbal- expressed as follows: ance of the compressed, warmer stream and expand- The heat rejected to ambient is: 3±4±10±20±3. ed, cooler stream in the regenerator must be taken q ˆ h h : 1† into consideration. In particular, the enthalpy de®- 1 3 4 ciency cannot be ignored if a mixed working ¯uid The heat removed from the cold-end heat exchanger at

is used. the constant pressure P1, namely cooling power of the This model then leads to a cycle consisting of two system, is 6±1±10±60±6. adiabatic and two isobaric branches. Fig. 1 shows the q ˆ h h : 2† corresponding T±S diagram. The working refrigerant 2 1 6 leaving the cold-end heat exchanger in state 1 The COP of the system then can be expressed as follows: (P1; T1 ˆ Tc) enters the regenerator and is warmed up q h h at constant pressure P to state 2 (P ; T ˆ T  ambi- COP ˆ 2 ˆ 1 6 : 3† 1 1 2 a q q h h † h h † ent temperature). It is compressed adiabatically, in a 1 2 3 4 1 6

compressor, from state 2 to 3 (P2, T3) and passes In practical situations, Eq. (3) may be replaced by through the after cooler to state 4 (P2; T4 ˆ T2 ˆ Ta). COP ˆ q2=q1 since, for a pulse tube, there is no expan- The working ¯uid then enters the regenerator and is sion work recovery.

cooled at a constant pressure P2 to state 5 (P2; T5 ˆ In an ideal refrigeration cycle, where the speci®c heat T1 ˆ Tc). It is then expanded adiabatically at the cold of the working refrigerant is constant in the whole sys- end of the pulse tube to its lowest refrigerating tem- tem, we have perature in state 6 (P , T ), which is lower than T . 1 6 c C T T † Finally, the working refrigerant absorbs heat at a COP ˆ p 1 6 Cp T3 T4†Cp T1 T6† constant pressure P1 in the cold-end heat exchanger   (temperature T ˆ T ) and returns to state 1, ®nishing a T T † T T 1 c ˆ 1 6 ˆ 1 3 4 1 : 4† complete cycle. T3 T4† T1 T6† T1 T6 The cycle shown in Fig. 1 is essentially a Brayton Both processes 2±3 and 5±6 are isentropic. Therefore cycle. However, in the conventional Brayton cycle, a  recuperative heat exchanger is used. Furthermore, tur- T T P K1=K 3 ˆ 5 ˆ 2 ; 5† bines are usually employed as compressors and ex- T T P panders in the cycle [5]. However, in the case of the pulse 2 6 1 tube refrigerator, the turbine expander is substituted by where K ˆ Cp=Cv is the adiabatic index. For an ideal the expansion space of the pulse tube and the recuper- ¯uid mixture, the total speci®c heat is the weighted av- ator by the regenerator. We are thus dealing with a erage of each component partP in the mixture at the same variant of the Brayton cycle, which we will denote as the temperature, namely K ˆ XiKi, where Xi is the mole modi®ed Brayton cycle in this paper. fraction of the ith component in the mixture and Ki is the adiabatic index of the ith component.

Since T5 ˆ T1 and T2 ˆ T4 in Fig. 1, Eq. (5) can be rewritten as follows: T T T T T ‡ T T T 3 ˆ 3 ˆ 2 ˆ 4 ˆ 3 4 or 3 4 : 6† T5 T1 T6 T6 T1 ‡ T6 T1 T6 Finally, substituting Eq. (6) in Eq. (4), the COP is simply expressed by the temperature ratio 1 T COP ˆ ˆ 1 7† T3=T1†1 T3 T1 or 1 T T COP ˆ ˆ 6 ˆ 6 ; 8† T4=T6†1 T4 T6 T2 T6

where temperatures T3 or T6 can be computed by Eqs. Fig. 1. T±S diagram of pulse tube refrigeration cycle with two isen- (5) and (6). From Eq. (8), the lowest refrigeration tem- tropic and two isobaric processes. perature of the system can be calculated as 中国科技论文在线 http://www.paper.edu.cn

G. Chen et al. / Cryogenics 40 (2000) 261±267 263 COP T ˆ T ; 9† Dqr ˆ qrh qrc ˆ h4 h5† h2 h1†: 12† 6 1 ‡ COP 2 The enthalpy di€erence can be considered as an addi- where T2 is the room temperature. tional heat load (if it is a positive value) or an additional The above analysis was based on the assumption of refrigeration (if it is a negative value) in the cycle; then balanced enthalpy ¯ow in the regenerator. It must be expression (3) is rewritten as stressed that the phenomenon of unbalanced enthalpy h h †Dq ¯ow in the regenerator cannot be neglected in the COP ˆ 1 6 r h h †‰ h h †Dq Š computation process. The heat rejected by compressed 3 4 1 6 r 0 0 h h † h h † working refrigerant in the regenerator is 4±5±6 ±1 ±4. ˆ 2 6 4 5 : 13† h3 h4†‰ h2 h6† h4 h5†Š qrh ˆ h4 h5: 10† The calculations have shown that the enthalpy de®cit The heat absorbed by the expanded working refrigerant cannot be eliminated if mixed ¯uids are used as working in the regenerator is 1±2±20±10±1. refrigerants. qrc ˆ h2 h1: 11†

In an ideal case, qrh ˆ qrc. This is mostly true when pure 4. Computed results helium is used as the working refrigerant. However, for mixed ¯uids, the enthalpy ¯ow of the warmer refrigerant In order to ®nd out an appropriate pair of mixed could be greater or less than that of the cooler one in the ¯uids which might be suitable for pulse tube refrigera- regenerator. In this case, an enthalpy de®cit occurs. The tors at temperatures near 80 K, the ¯uids to be selected enthalpy de®cit may be positive or negative depending can be divided into three groups: single phase ¯uids, on the ¯uids and operating parameters and can be ex- vapor±liquid two-phase ¯uids and noble gas mixtures. pressed as In the analytical calculations, a room temperature of

Fig. 2. Theoretical thermodynamic eciency and cooling power of helium and nitrogen mixture. 中国科技论文在线 http://www.paper.edu.cn

264 G. Chen et al. / Cryogenics 40 (2000) 261±267

Ta ˆ 300 K, a ®lling pressure of the working refrigerant a number of ¯uid pairs to be selected such as helium of 1.7 MPa and a pressure ratio of P2=P1 ˆ 2.0 MPa/1.0 with nitrogen, hydrogen, neon and argon are computed MPa ˆ 2.0 are assumed for ease of comparison with according to Eqs. (2) and (13). The calculation results

experimental results. The cooling power qc and COP of are summarized in Figs. 2±5 and Table 1.

Fig. 3. (a) Speci®c thermodynamic eciency and (b) cooling power of helium and hydrogen mixture.

Fig. 4. (a) Speci®c thermodynamic eciency and (b) cooling power of helium and neon mixture.

Fig. 5. (a) Speci®c thermodynamic eciency and (b) cooling power of helium and argon mixture at 100 K. 中国科技论文在线 http://www.paper.edu.cn

G. Chen et al. / Cryogenics 40 (2000) 261±267 265

Table 1 Calculation results for mixed refrigerants

Mixture He + N2 He + H2 He + Ne He + Ar 80 K 90 K 100 80 K 90 K 100 80 K 90 K 100 80 K 90 K 100 K K K K He fraction (%) 90 76 62 97 60 82 97 97 97 79

COPmix/COPHe (%) 1.095 1.078 1.042 1.0175 1.023 1.018 1.014 1.013 1.01 1.078 He fraction (%) 90 76 62 98 98 98 98 98 98 79

qc;mix/qc;He (%) 1.067 1.046 1.015 1.008 1.006 1.0041 1.0078 1.0056 1.004 1.055

Nitrogen is the most promising ¯uid to compose a two-phase working refrigerant with helium near 80 K. Fig. 2 and Table 1 show that it has a positive speci®c cooling power q =q and speci®c coecient of c;He‡N2 c;He performance COP /COP in the range of nitrogen He‡N2 He fraction up to 18%. The peak values of speci®c COP of 0.095, 0.078 and 0.042 and the speci®c cooling powers of 6.7%, 4.6% and 1.5% appear at nitrogen fractions of 10%, 24% and 48% and temperatures of 80, 90 and 100 K, respectively. These peak values occur at tem- peratures near the corresponding liquid±vapor phase change. In addition, the second phase change may occur with additional refrigeration when cooling to tempera- Fig. 6. Enthalpy de®cit of helium and hydrogen mixture. tures below the triple point (63.15 K) of nitrogen. However, the latent heat of the liquid±solid phase change is excluded in the calculation results due to lack view of either thermodynamic properties or transport of data in the PROMIX program [6]. Obviously, nitro- characteristics, seems to be worthless as a mixed ¯uid gen is the most promising ¯uid with helium as a vapor± for increasing the cooling power or thermodynamic liquid two-phase mixed refrigerant for pulse tube eciency of pulse tube refrigerators near 80 K. The refrigeration near 80 K. mixture of argon and helium, however, exhibits in- Hydrogen is also expected to be one of the major spiring results as a two-phase mixture at the 100 K components forming a single-phase mixture with helium range. due to its excellent transport properties. Fig. 3 and Ta- ble 1 show that it has positive speci®c cooling power and speci®c COP in the range of hydrogen fractions up to 5. Discussion 35% at 80 K compared with those of pure helium, though the values are not as high as expected. In fact, the The above analysis and calculation were based on the enthalpy di€erence of isentropic expansion of hydrogen conditions of ®xed thermal-boundary of the refrigera- is higher than that of helium at the same condition. tion cycle and ideal heat transfer in the regenerator. It However, most of the cooling power of hydrogen is o€set means that the temperatures at both hot and cold ends by its enthalpy de®cit in the regenerative heat-exchange of the regenerator have constant values and the positive process. Fig. 6 gives the enthalpy de®cit of hydrogen in enthalpy di€erence (Dqr > 0) in the regenerator can be the regenerator as a function of the helium fraction. It o€set by the cooling power of the refrigerator via heat shows that a negative value of enthalpy di€erence can, transfer process. However, if the positive enthalpy dif- nevertheless, lead to a refrigeration gain for fractions of ference is so great or the performance of the heat ex- hydrogen up to 14%. Attempts at reducing the enthalpy changer is not high enough that the additional heat load de®cit in the regenerator will probably be limited by the cannot be o€set by its cooling power, the refrigeration operation parameters of the system. Besides, the higher process will take place at a higher temperature. thermal conductivity and lower viscosity of hydrogen Considering an unbalanced enthalpy ¯ow in a regen- can, perhaps, lead to an improvement in heat transfer erator, if Dqr > 0, then the refrigeration cycle is 1±2±3± and ¯uid ¯ow. These possible bene®ts, however, are not 4±5a±6a±1 in Fig. 7, where h4 h5a ˆ h2 h1. In this included in the thermodynamic analysis. case, It is disappointing that the noble gas mixture of helium±neon (see Fig. 4 and Table 1), from a point of Dqr ˆ h4 h5† h2 h1†ˆh5a h5: 14† 中国科技论文在线 http://www.paper.edu.cn

266 G. Chen et al. / Cryogenics 40 (2000) 261±267

Fig. 7. T±S Diagram for Dqr > 0. Fig. 8. T±S Diagram for Dqr < 0.

Substituting Eq. (14) into Eq. (13), The calculation results are listed in Table 2. The val- ues of COP and q are slightly di€erent from those in h h † h h † c COP ˆ 1 6 5a 5 : 15† Table 1. h3 h4†‰ h1 h6† h5a h5†Š On the other hand, a COP of the refrigeration cycle (1± 2±3±4±5a±6a±1) in Fig. 7 can be expressed as follows: 6. Conclusions h h † COP ˆ 1 6a The proposed modi®ed Brayton cycle, which consists h3 h4† h1 h6a† h h † h h † of two isentropic and two isobaric processes, can be ˆ 1 6 6a 6 : 16† e€ectively used for describing the thermodynamic per- h h †‰ h h † h h †Š 3 4 1 6 6a 6 formance of pulse tube refrigeration. The COP and the

The cooling power qc could be expressed as qc ˆ h6a h1 cooling power of pulse tube refrigerators with various in Fig. 7. binary mixture ¯uids have been predicted. The most

If Dqr < 0, then the refrigerator cycle is 1±2a±3a±4±5± promising ¯uid in the 80 K range is the two-phase 6±1 in Fig. 8, where h4 h5 ˆ h2a h1. In this case, mixture of helium and nitrogen. The computed results show that an improvement of 6.7% for cooling power Dq ˆ h h † h h †ˆ h h †; 17† r 4 5 2 1 2 2a and 9.5% for COP can be obtained in comparison with then data for pure helium if a mixed refrigerant of 10% ni- h h †‡ h h † trogen with 90% helium is used. COP ˆ 1 6 2 2a : 18† h3 h4†‰ h1 h6† h2 h2a†Š According to the de®nition of COP, Acknowledgements h h † COP ˆ 1 6 h3a h4† h1 h6† The project is ®nancially supported by the DAAD of h h † Germany and the National Natural Sciences Founda- ˆ 1 6 : 19† tion of China. The authors would like to thank the h h †‰ h h † h h †Š 3 4 3 3a 1 6 referee of this paper who provided the analysis and

In this case, the cooling power is qc ˆ h6 h1 in Fig. 8. calculation results of the discussion section.

Table 2 Results calculated by PROMIX

Mixture Refrigeration temperature (K) COP COPmix/COPHe qc; mix (kJ/gmol) qc; mix/qc; He 100% He 80 0.2531 1 0.4021 1

90% He + 10% N2 80 0.2798 1.1056 0.4303 1.0702 97% He + 3% H2 80 0.2546 1.0107 0.4004 1.0042 中国科技论文在线 http://www.paper.edu.cn

G. Chen et al. / Cryogenics 40 (2000) 261±267 267

References [4] Patwardhan KP, Bapat SL. Cyclic simulation of Stirling cryogen- erator with two-component two-phase ¯uid, Cryocooler 9. New [1] Walker G. Stirling-cycle cooling engine with two-phase, two- York:Plenum Press, 1997. p. 157±62. components working ¯uid. Cryogenics 1974;8:459. [5] McCormick J, et al. Design and test of low capacity reverse [2] Chen GB, Gan ZH, Yu JP. Study on two-component gas mixtures Brayton cryocooler for refrigeration at 35 K and 60 K, Cryocoolers in regenerative refrigerators. Proc of ICEC 1998;17:117±22. 10. New York:Kluwer Academic/Plenum, 1999. p. 421±30. [3] Yu JP, Chen GB, Gan ZH, et al. Discussion on regenerator [6] Cryodata, The PROMIX program, 1995. performance improvement with binary gas mixture. Proc of ICEC 1998;17:117±22.