A NUMERICAL MODEL OF A 4K TWO-STAGE PULSE TUBE COOLER WITH MIXED EULERIAN-LAGRANGIAN METHOD

In order to simulate and visualize the internal processes and performance of twostage pulse tube coolers operating in the 4K-temperatures, a mixed Eulerian-Lagrangian numerical model is proposed to follow the exact tracks of gas particles as they move with pressure oscillation in the pulse tube and to simulate the detailed variations of dynamic parameters in the regenerators. A variety of physical factors, including real thermal properties of helium gas, multi-layered magnetic regenerative materials, pressure drop and heat transfer in the regenerator, are taken into account in this model. Predicted results of the time-variations of gas temperature, pressure, mass flow rate, enthalpy flow, in the 1and 2-stage regenerators are presented. Particular attention is given to the effects of different regenerative materials on the performance of 4K two-stage pulse tube coolers. INTRODUCTION Due to several advances in design and the development of magnetic materials, pulse tube coolers have developed rapidly in recent years. In contrast to the GM coolers pulse tube coolers operate without cold moving parts, thereby eliminating the potential problems (mechanical vibration, lifetime, reliability) inherent to the GM coolers. Since 1994 when Matsubara and Gao first obtained a temperature of 3.6K using a three-stage pulse tube cooler [1], the lowest temperature of pulse tube coolers [2-4] is now well below the inversion temperature of He and He. As such, it can serve as a precooling stage for further reducing the temperatures to the millikelvin range using He-He dilution refrigerators [5]. 4K-pulse tube coolers, which provide 0.5W cooling capacity at 4.2K and with COP up to 10" [6], are now commercially available. This cooling power is already sufficient for several applications, such as for the liquefaction of helium [7] and for cooling small-sized superconducting magnets [8]. However, the COP of 4K-pulse tube coolers is still much lower than that of GM coolers. Most of this is due to the absence of solid displacer in the pulse tube, the phase CP613, Advances in Cryogenic Engineering: Proceedings of the Cryogenic Engineering Conference, Vol. 47, edited by Susan Breon et al. © 2002 American Institute of Physics 0-7354-0059-8/027$ 19.00 715 shifter located at room temperature does not lead to an efficient phase shifting (by the socalled "gas piston") of the moving liquid helium at the cold end. The other reasons are due to lack of fundamental knowledge of the effects of the salient characteristics of real helium properties, the unique features of thermal properties of magnetic material, and the existence of helium in the void space of the regenerator on the performance of the coolers. Since the internal processes in 4K-pulse tube coolers are not well understood at present, a mixed Eulerian-Lagrangian model [9] recently developed by the present author is used to simulate and visualize the internal processes and performance of a 4K two-stage pulse tube cooler. Here, we briefly summarize the model and present some predicted results of the time-variations of gas temperature, pressure, mass flow rate, enthalpy flow, in the 1and 2 -stage regenerators. Particular attention is given to the effects of different regenerative materials on the performance of the 4K two-stage pulse tube cooler. FORMULATION The schematic diagram of a 4K two-stage pulse tube cooler is shown in Fig. 1. The analysis assumes 1-D, compressible gas flow and neglects any geometric complexity in the regenerator, in order to reduce CPU time and avoid any difficulty involved with the twoor three-dimensional models. Other basic assumptions were as follows: 1. The gas flow is laminar, no turbulence 2. Constant wall temperature at the cold and hot end heat exchangers 3. The regenerative material in the regenerator is incompressible, uniform porosity 4. Boundary and variable permeability effects are neglected Under these assumptions, the governing equations are given by: dp dpu = dt dx (1) pM^-W(^ (2) dx dx dt ax I dx) dp a ox (3) (4) The energy equation for the solid regenerative materials in the regenerator is: