Numerical simulation of pulse-tube refrigerators

A new numerical model has been developed for simulating oscillating gas ??ow and heat transfer in the tube section of a pulse-tube refrigerator. Pulse-tube refrigerators are among the newest types of cryocoolers. They work by the cyclic compression and expansion of gas, usually helium. Introduced in 1963, pulse-tube refrigerators typically reached temperatures of about 120 K. By the end of the 1990s temperatures below 2 K had been reached. The practical use of pulse-tube cryocoolers is still at an early stage. However, they are beginning to replace the older types of cryocoolers in a wide variety of applications: military, aerospace and medical industries. Advantages such as simplicity, low cost and reliability, combined with high performance, have resulted in an extensive study of pulse tubes in recent years. The ??rst and second laws of thermodynamics have been major tools to investigate pulse-tube refrigerators theoretically. However, a clearer understanding of the ??uid dynamical properties is necessary if one wishes to make quantitative improvements in pulse tube performance. In this study we concentrate solely on the tube section of the pulse-tube refrigerator to identify undesired effects that occur in the tube and reduce the ef??ciency of the coolers. The developed mathematical model is based on the conservation of mass, momentum and energy, and the equation of state. The conservation equations for compressible viscous unsteady ??ow are written in differential form using primitive variables. One-dimensional and two-dimensional cylindrical axisymmetrical cases are considered. According to dimensional analysis, the tube conveys a low-Mach-number compressible ??ow. Therefore, we expanded all relevant variables in terms of powers of M2, a parameter related to the Mach number. This asymptotic consideration reveals several key features of pulse tube ??ow. Two physically distinct roles of pressure are to be distinguished: one as thermodynamic variable and one as hydrodynamic variable. The thermodynamical pressure appears in the energy equation and in the equation of state. It is spatially uniform, thus a function of time only, and is responsible for the global compression and expansion. The hydrodynamical pressure appears in the momentum equations and is induced by inertia and viscous forces. The acoustic pressure does not play a role in pulse tubes. Due to the non-linearity of the resulting system of equations, general analytical solutions are not available. Therefore numerical modelling has been applied. For the numerical solution of the resulting system of equations ??nite difference methods are used. The energy equation for the temperature is a convection-diffusion equation, mostly of a convective nature. It is solved with state-of-the-art ??ux-limiter schemes in an attempt to preserve the steep temperature gradients in a pulse tube. When large gradients are present, either internally or adjacent to a boundary, more accurate solutions can be obtained by grid re??nement. Re??ning a grid throughout the entire computational domain can be expensive, particularly in multi-dimensions. Instead of applying non-uniform locally re??ned grids, we use several uniform grids with different mesh sizes that cover different parts of the domain. One coarse grid covers the entire domain. The mesh size of this global grid is chosen according to the smoothness of the solution outside the high-activity regions. Besides the global grid, ??ne local grids are used which are also uniform. They cover only parts of the domain and contain the high-activity regions. The mesh size of each of these grids follows the activity of the solution. The solution is approximated on the composite grid which is the union of the uniform subgrids. This re??nement strategy is known as local uniform grid re??nement (LUGR). To deal with the problem of pressure-velocity coupling in the ??ow computation, we employ a pressure correction method. It is specially designed for low-Mach-number compressible ??ows. Combining the continuity equation and the energy equation, we derive an expansion equation or velocity divergence constraint. Our pressure correction scheme is based on this expansion equation and not on the continuity equation, which is different from the common approach in the simulation of compressible ??ows. The simulation tool, based on the proposed model, is constructed and tested on classic problems with known analytical solutions. Finally, the model was applied to a typical pulse-tube refrigerator. Results of one-dimensional and two-dimensional axisymmetrical simulations are presented and interpreted. The proposed model is more accurate and versatile than the widely used harmonic analysis and computationally less expensive than a full three-dimensional simulation with commercially available codes. It can be used for practical simulations, for calculating optimal values of the real system design parameters and for investigating different physical effects in the pulse tube.