Simple Two-Dimensional Corrections for One-Dimensional Pulse Tube Models

Abstract

One-dimensional oscillating flow models are very useful for designing pulse tubes. They are simple to use, not computationally intensive, and the physical relationship between temperature, pressure and mass flow are easy to understand when described with phasor diagrams. They do not possess, however, the ability to directly calculate thermal and momentum diffusion in the direction transverse to the oscillating flow. To account for transverse diffusion effects, either parameter corrections must be obtained through experimentation, or solutions to the twodimensional differential fluid equations must be found.

A linearized two-dimensional solution to the fluid equations has been obtained. The solution provides lumped parameter corrections for one-dimensional models. The model accounts for heat transfer and shear flow between the gas and the tube wall. The complex Nusselt number and complex shear wall are useful in describing these corrections, with phase relations and amplitudes scaled with the Prandtl and Valensi numbers. The calculated enthalpy flow ratio, α, between a two-dimensional solution of the oscillating temperature and velocity and a one-dimensional solution for the same shows α scales linearly with Va for Va < 30. In this region α<0.5, that is, the enthalpy flow calculated with a two-dimensional model is 50% of a calculation using a one-dimensional model. For Va > 250, α=0.8, showing that diffusion is still important even when it is confined to near the tube wall.