Stirling and Pulse-tube Cryo-coolers

Chapter 3

Ideal Stirling cycle — real gas

3.1 Background

The internal processes of the Stirling cycle machine do not lend themselves to intuitive prediction; neither are the intricate details readily probed experimentally. Computer simulation is widely advocated as a route to the insights sought, but a lack of experimental values with which to compare computed temperatures, flowrates, etc. places an onus on demonstration ofself-consistency.

Measures that have provided useful yardsticks to date include setting heat transfer coefficients to infinity (e.g. by setting Stanton number N_st = ∝), flow losses to zero (e.g. by setting friction factor C_f = 0), and checking for coincidence with a closed-form solution such as that provided by the Schmidt analysis (1871).

These measures, however, presuppose ideal gas behaviour. When attention turns to coolers operating on the reversed cycle, and when it thus becomes desirable to take account of real gas behaviour, no comparable resource is to hand. At first sight, the problem is merely that of re-formulating the Schmidt treatment around a suitable equation of state, such as van der Waals'. This turns out to be feasible algebraically although by no means trivial and not, unfortunately, the end of the matter: the Schmidt (or ‘isothermal’) cycle is a statement of mass conservation; it does not set out explicitly to conserve energy. Back-substituting an energy balance into the ideal gas treatment nevertheless yields the Carnot coefficient of performance ...