Comprehensive analytic solutions for the geometry of symmetric constant-wall-thickness scroll machines

Abstract

In this work, a comprehensive framework and geometric solutions are presented for the most common type of scroll machine, the symmetric constant-wall-thickness scroll compressor based on the involute of a circle. This analytic treatment comprises solutions for the volume, derivative of volume, centroid, normalized force and moment parameters, and leakage areas for all chambers in the machine. The representations for the centroid and area are based on Green's Theorem analysis which allows for general solutions that can then also be applied to other machines. Analytic solutions are also presented for the discharge geometry corresponding to arc-line-arc, perfect-meshing-profile, one-arc, and two-arc discharge geometries. Each of the derived parameters is validated against numerical solutions based on polygons. Source code in the Python programming language is provided for the analytic derivations as well as the implementation of each of the analytic outputs.