### Abstract:

A computational model for a resonantly coupled alpha free-piston Stirling cooler
is presented. The cooler consists of two isothermal working spaces for compression and
expansion connected by a regenerator consisting of a stack of narrow parallel channels.
The regenerator is assumed to have a linear temperature distribution along its axial
direction and the working fluid is taken as an ideal gas. Control volume analysis is
adapted in this model, in which each of the components of the cooler is considered a
separate control volume. The compression piston is given a predetermined motion to
provide the work needed by the cooler. The expansion piston and the gas trapped
between the piston and the walls of the expansion cylinder are modeled as a mass,
spring, and damper system. The motion of the compression piston generates a pressure
difference across the cooler, and forces the working fluid to pass through the
regenerator. The expansion piston responds to the pressure in its space according to
Newton's second law of motion. The motion of the expansion piston is governed by the
forces originating from the pressure and the cold side gas spring and dash-pot. In this
way the dynamics of the moving pistons are coupled to the thermodynamics of the
cooler system.
A definition for the coefficient of performance (COP) that considers the heat
transfer by conduction through the material making up the regenerator is introduced.
This definition of the COP reflects the dependence of the cooler's performance on the
length of the regenerator. From a systematic variation of this regenerator length, an
optimal value can be found for a given set of operating parameters.
Conservation laws of mass, momentum and energy along with ideal gas
relations are used to form a set of equations fully describing the motion of the pistons
and the thermal state of the cooler. A marching-in-time technique with a Runge-Kutta
scheme of the fourth order is adapted to integrate the equation of motion. The plots of
the motion of the pistons, the pressure-volume diagrams of the workspaces and the COP
plots are provided to describe the cooler behavior.