Stirling and Pulse-tube Cryo-coolers

Chapter 8

Classic regenerator problem - real gas

8.1 Introduction

The otherwise daunting ‘regenerator problem’ becomes trivial on separating the calculation of particle trajectories from that of the temperature histories. Under the traditional assumption of uniform density, the separation process causes no compromise of solution accuracy.

Applying the separation technique to the ideal gas (p/ρ = RT) or to a ‘real’ gas as modelled by van der Waals’ equation, (p + aρ²)(1/ρ — b)= RT, threatens compromise: particle motion now depends on local, instantaneous density, and density is not precisely known prior to obtaining the temperature solutions.

The potential problem is of little consequence unless one insists on analysing regenerators of low-temperature recovery ratio or of generally poor thermal performance. For present purposes only, high thermal recovery is of interest, and:

• cyclic temperature swing of the matrix is slight (high thermal capacity ratio, N_TCR);
• fluid and matrix temperatures differ only slightly over a cycle (high NTU).

In the case of the ideal gas, these factors in combination ensure that fluid temperature is never far from a linear distribution between nominal expansion temperature, T_E, and compression space temperature, T_C. The accuracy with which the particle trajectories are specified is thus not a limitation of the method.

At this stage it is not known whether the matrix temperature distribution under the assumption of van der Waals gas behaviour and of variable ...