An Exact Expression for Shuttle Heat Transfer
A new and general analytical expression is obtained for the shuttle heat transfer rate in displacer-cylinder systems. The shuttle heat transfer occurs when the displacer has a reciprocating motion over the cylinder with an axial temperature gradient. The heat transfer might be important in small cryogenic refrigerators because it represents a loss of refrigeration in addition to the wall conduction. The oscillating temperature distributions of both the displacer wall and the cylinder wall are exactly obtained as functions of time and space from the conduction equations by introducing the complex temperatures. From the cyclic steady state solution of the temperatures, a simple mathematical expression is derived to calculate the shuttle heat transfer. The expression includes the axial temperature gradient, the stroke and the angular speed of the reciprocating motion, the heat transfer coefficient between two walls, and the thermal properties (such as thermal conductivity, density, or specific heat) of two walls. The usefulness of the results is justified by the approximate solutions of previous works and the physical interpretations are presented.
KeywordsEnthalpy Helium Biot Refrigeration
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