Regenerator dynamics in the harmonic approximation

In this paper we derive expressions describing the DC and first-harmonic AC behaviour of the regenerator in Stirling type refrigerators, based on energy conservation, mass conservation, and the flow resistance of the material. We show that the temperature profile is determined by thermal conduction, that the average temperatures of the gas and the matrix are equal, and that the pressure is constant. For the AC components of the various quantities first-order differential equations are derived which can be solved numerically. In an important special case the relations can be solved analytically in terms of Bessel functions for the mass flow and the pressure variation in the regenerator