Unsteady Analytical Model for Appendix Gap Losses in Stirling Cycle Machines

For the design of a Stirling cycle machine a high-accuracy model is needed. Recent experimental results [1] indicate that the thermal losses associated with the cylinder system, particularly those caused by the annular gap around the displacer, have been underestimated in their influence on the engine performance. Therefore, existing models for these thermal losses, which are commonly referred to as appendix gap losses, have been reviewed [2]. The analytical approaches are of major importance, as they may allow the derivation of practically applicable design formulas for the gap geometry. The previously available models appear to be partly based on questionable assumptions. Therefore, a new analytical model will be developed. It has been shown that the flow in the gap is laminar but unsteady [2], and that analytical solutions for the flow pattern and the temperature profile can be directly derived from the conservation laws. Herewith, the new model will be able to consider inertia effects in the unsteady flow of the gas in the gap. In contrast to an existing solution [3], which is based on the assumption of a pure drag flow (Couette flow), the new model will also take into account the evidently prevailing pressure-driven flow (Poiseuille flow) component. Additionally, all energy transport mechanisms in the cylinder system will be investigated with respect to their dependency on the axial position. Hereby, the simplifying assumptions of constant wall and gas temperature gradients along the gap, which is made in all previous models, will be checked, since earlier temperature measurements at Stirling machine cylinders and numerical simulations suggest the expectation of a curved cylinder wall temperature profile instead.