Shock Waves (1996) 6:301-308 SlheCx 9 Springer Verlag 1996 Numerical study of wave processes in a pressure-wave refrigerator A. Galyukov 1 , E. Timofeev 2, P. Voinovich 1 1 Advanced Technology Center, P.O. Box 29, St. Petersburg 194156, Russia 2 Ioffe Physical-Technical Institute, Russian Academy of Sciences, Polytekhnicheskaya str. 26, St. Petersburg 194021, Russia Received 14 February 1996/Accepted 1 June 1996 Abstract. The paper provides some results concerning the nu- merical study of the strongly transient gasdynamic processes in a pressure-wave refrigerator (PWR). A hierarchical set of nu- merical models from the simplest one-dimensional to a fully three-dimensional formulation is introduced. The computa- tions show that one-dimensional solutions give a reasonable foundation for the understanding of PWR operational prin- ciples but cannot satisfactorily predict the refrigeration effi- ciency. Good agreement with experiment is achieved by con- sidering two- and three-dimensional effects of gas mixing by overlap of rotating nozzles and expansion tubes. Key words: Pressure-wave refrigerator, Numerical simula- tion, Wave processes 1 Introduction The pressure-wave refrigerator (PWR), called also a "thermal separator," represents a simple facility (see Fig. 1) for gas cool- ing by decompression which has many applications in energy transformation and chemical processes (Marchal et al. 1985, Yu et al. 1989, Fang et al. 1991). A rotating gas distributor con- tains nozzles from which the high-pressure gas enters radially mounted expansion tubes. The opposite ends of the tubes are closed or, alternatively, are connected to tanks for shock damp- ing. The gas motion in the tubes and nozzles, as their openings coincide with each other, is similar to that in a shock tube. After a nozzle has passed a tube opening, the expanded gas comes out of the tube into the receiver in which the distributor rotates and then into the low pressure pipe. The process repeats periodically in each nozzle and tube. The operational cycle of the PWR is conceptually close to that of the pressure wave supercharger (PWS). The latter was studied extensively both experimentally and numerically (Zehnder 1971, Zhang & So 1990, Saito et al. 1993). Some ex- perimental results concerning the PWR can be found in Mar- chalet al. (1985), Yu et al. (1989) and Fang et al. (1991). Prior to the present study, the authors could find no reports of numer- ical study of a PWR in the literature. Our numerical approach and results were first published in Galyukov et al. (1994, 1995). Correspondence to: E. Timofeev The present paper contains an extended summary of the above publications and includes up-to-date data. Progressive (not instant) overlapping of high- and low- pressure channels is a universal feature of pressure-wave ma- chines. The phenomenon influences essentially the wave pro- cesses and efficiency of the devices (Zhang & So 1990) and requires two-dimensional (in the rotation plane) or three- dimensional modelling to adequately simulate the processes of gas mixing. The respective codes are typically CPU time and memory consuming. Therefore, investigation of an actual PWR with long expansion tubes over a number of cycles is practicable today only on the basis of one-dimensional mod- els. The alternative employed in the present paper is to analyze the results of one-, two- and three-dimensional simulations in combination. 2 Numerical models A hierarchical set of one-dimensional (l-D), quasi-one- dimensional, two-dimensional (2-D) and three-dimensional (3-D) models has been introduced to study gasdynamic pro- cesses in a thermal separator. It will be shown below, only a conjoint usage of the above models successfully predicts both the wave processes in the PWR and its efficiency. It has been demonstrated that the role of viscous effects on the gas motion in the PWS is negligibly small (Zhang & So 1990). Unlike PWS channels, the expansion tubes of the PWR have a rather high length-to-diameter ratio which might lead to attenuation of shock waves. Based on the experiments by Fang et al. (1991), the computations reported here were carried out for ratios for which the above phenomenon could be neglected. Centrifugal and Coriolis forces acting on the gas inside the rotating distributor were also disregarded because the respective terms are negligible at rotational speeds typical for PWRs. Consequently, we proceeded from the unsteady l- D, 2-D or 3-D Euler equations written in an integral form and supplied with appropriate initial and boundary conditions. For instance, the impermeability condition was provided for the nozzle and tube walls. Several shock-capturing finite-volume numerical tech- niques based on both structured and unstructured grids were