Method for calculating the theoretical performance of a gerotor compressor based on modeling the geometry of a cycloidal gearing

The development of screw technologies for compressing various media is associated with the development of internally geared screw compressors and requires the creation of engineering calculation methods for their performance at the early design stages. This is due to the fact that the determination of volumetric flow rate is complicated by the complex curvilinear geometry of the conjugate profiles of the working elements. The aim of the research is to develop a methodology for calculating the theoretical capacity of an internally geared screw compressor considering the specific features of the cycloidal gearing of the working elements. The article demonstrates that the capacity of an internally geared screw compressor is determined by the volume of the working cavity on the suction side and the rotational speed of the driving rotor. The key challenge in this calculation is determining the net cross-sectional area of the end-face profile, which is defined as the difference between the areas bounded by the initial and conjugate contours of the working elements. To solve this problem the authors propose the following methods: a numerical integration method based on Green's formula and the trapezoidal rule, applicable to arbitrary profile geometries; an analytical expression for the specific case of an ideal hypocycloidal gear engagement; approximate engineering correlations derived from the post-processing of extensive numerical solution datasets; a geometric analogy method based on a model of the annular clearance between equivalent circles. As a result, the authors derive correlations for calculating the working volume of the suction-side cavity based on the net cross-sectional area and the geometric parameters of the rotors. The proposed set of methods, offering varying levels of accuracy and computational effort, enables the calculation of the theoretical capacity of an internally geared screw compressor at various design stages without resorting to numerical simulation.