Numerical simulation of a three-stage Stirling-type pulse-tube refrigerator

Abstract

The pulse-tube refrigerator (PTR) is a rather new device for cooling down to extremely low temperatures, i.e. below 4 K. The PTR works by the cyclic compression and expansion of helium that flows through a regenerator made of porous material, a cold heat exchanger, a tube, a hot heat exchanger and an orifice, in series. In a Stirling-type PTR compression and expansion are generated by a piston. The compression increases the temperature of the helium in the tube and makes it flow towards the orifice; the expansion decreases the temperature and makes the helium flow backwards to the regenerator. The net effect of warmer helium flowing in one direction and colder helium in the opposite direction is that of cooling power at the cold heat exchanger. Three PTRs are inter-connected aiming to obtain the desired 4 K lowest temperature. The conservation laws of mass, momentum and energy, and an equation of state, are simplified using asymptotic analysis based on low Mach-numbers. The regenerator is modelled one-dimensionally with Darcy’s law for flow resistance. The tube is modelled either one-dimensionally without resistance or two-dimensionally with axisymmetric laminar viscous flow. The heat transfer in the porous medium of the regenerator and in the solid tube wall is taken into account. The gas can be either ideal or real. All the material properties, including viscosity and conductivity, are taken temperature and pressure dependent. Three single-stage PTRs are connected with the regenerators in series and the tubes in parallel and six flow possibilities at the junctions are considered. Three by-passes (double-inlets) are used to enhance and tune the performance. The governing equations are numerically solved with a finite-difference method of nominally second-order accuracy in space and time. Pressure correction, flux limiter, 1D-2D connections and domain decomposition are the keywords here. Special attention is paid to suitable initial conditions, high resolution in the boundary layers and to the correct calculation of the three-way junctions in multi-stage PTRs. The model describes the fluid dynamics and thermodynamics of the pulse tubes and regenerators. The heat exchangers are assumed to be ideal in the whole analysis. The equation of state for real gas and other real properties of gas and regenerator material, which are temperature and/or pressure dependent, are applied in the three-stage PTR which works with extreme low temperatures, where the ideal gas law does not hold. The numerical methods require special attention. Typically for flow problems we deal with various length scales. Straight forward discretisation will result in unnecessary fine grids and therefore unacceptable computational time. We developed robust and efficient algorithms to deal with boundary layer problems. The employed domain decomposition technique allows for using coarse grids in areas where the solution does not change significantly. It also decouples a larger system into smaller ones, leading to smaller complexities. The objective of computing accurately and efficiently the steady oscillatory flow and heat transfer in a PTR has been achieved. In particular the tiny viscous and thermal boundary layers of a PTR operating at high frequency (20 Hz) could be resolved. The simulated three-stage PTR was able to cool down close to 5 K. The developed software is intended for use in design and optimisation of multi-stage PTRs.