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 eciency 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 coecient 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 eect 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 eciency 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 dierent 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 coecient of eciency 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 eciency 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 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 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 Tc) enters the regenerator and is warmed up q h h at constant pressure P to state 2 (P ; T T ambi- COP 2 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 T2 Ta). COP 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 In an ideal refrigeration cycle, where the speci®c heat T1 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 p 1 6 Cp T3 T4Cp T1 T6 constant pressure P1 in the cold-end heat exchanger (temperature T T ) and returns to state 1, ®nishing a T T T T 1 c 1 6 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 5 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 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 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 T1 and T2 T4 in Fig. 1, Eq. (5) can be rewritten as follows: T T T T T T T T 3 3 2 4 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 1 7 T3=T11 T3 T1 or 1 T T COP 6 6 ; 8 T4=T61 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 T ; 9 Dqr qrh qrc h4 h5 h2 h1: 12 6 1 COP 2 The enthalpy dierence 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 1 6 r h h h h Dq 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. 2 6 4 5 : 13 h3 h4 h2 h6 h4 h5 qrh 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 h2 h1: 11 In an ideal case, qrh 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.