Discussion of Numerical Methods used in Positive Displacement Comprehensive Mechanistic Models: Case Study using the Z-Compressor

Comprehensive mechanistic models have been widely used for simulating positive displacement (PD) compressors or expanders. Such models are based on a thermodynamic analysis of generic control volumes that describe the working chambers. The thermodynamic analysis results in a set of non-linear ordinary differential equations (ODEs) that describe conservation of mass and energy. The derived ODEs are used to predict the evolution of the thermodynamic properties of the working fluid with respect to crank angle or time. The solution scheme of a mechanistic model involves different layers of solvers that handle the step-by-step integration, the cycle-to-cycle continuity as well as the overall energy balance. Since most of the PD machines feature multiple control volumes that merge or split (e.g. scroll-type) or wrap-around a rotor (e.g. screw-type) the integration scheme faces challenges due to the stiffness of the equations or discontinuities in the control volume definition. In this paper, a discussion of different integration schemes and numerical considerations are proposed that are applied to a novel compressor called the z-compressor which presents a high-degree of intrinsic stiffness. A simplified comprehensive model is developed which is validated against experimental mass flow rate data for a prototype compressor with an average model-predicted error of less than 2%. This model is used to explore the differences in two ODE solvers from the ODE suite in MATLAB, (ODE15s, and ODE113). These results are further compared against an existing, simplified, Z-compressor model developed in PDSim which uses an explicit, lower order, solver the Adaptive Runge-Kutta (RKF). The results suggest that the semi-implicit solver (ODE15s) provides an efficient means for stepping through the compression process ODEs compared to the explicit solvers.