Polytropic gas dynamics – and other potential resources
The methods of wave mechanics are so frequently discussed in terms of homentropic (uniform entropy) flow as to give the impression of being confined to that special case. The recent application of wave methods to the Stirling engine may be the first instance of re-working to cope with the alternative prescribed temperature case.
Wave mechanics formulations yield insights not obtainable from other approaches. The effects of flow friction are readily accommodated. Heat transfer is a degree more difficult. In homentropic flow, heat transfer is zero by definition, while to maintain a prescribed temperature distribution, infinite Stanton number is called for. Under either extreme the pulse-tube would not function as a cooler!
Adapting the wave solutions to cope with heat transfer means incorporating the energy equation – in one form or another – into the working equations for the physical and state planes. If the assumptions of one-dimensional flow (slab flow) are acceptable, Shapiro's derivation (but not the implementation he illustrates – see below) offers rigour. The price of that rigour, however, is intricacy in coding for mechanized solution.
An alternative, believed to be novel, calls for modest adjustment to either the ‘homentropic’ or the ‘temperature-determined’ construction. Relative to the full path-line method there is some loss of fidelity to the underlying physical processes.
The simplicity ...