Numerical Investigation of the Actual Volumetric Flow Rate and Volumetric Efficiency and Optimization of the Geometric Parameters of a Three-Rotor Pump with Lantern Meshing—Part II

Part II of this study builds upon the mathematical framework developed and validated in Part I for describing the geometry and volumetric performance indicators of an innovative three-rotor hydraulic pump with bilateral lantern meshing. This part focuses on the numerical investigation and multi-objective optimization of these indicators through the proper selection of geometric parameters. The aim of the study is to establish the isolated and combined influence of the dimensionless geometric parameters—number of teeth z, relative lantern radius r^*_c,  and cycloid shortening coefficient λ—on the actual flow rate Q and the volumetric efficiency η_v under various operating conditions, while maintaining the overall dimensions of the pump element in the radial and axial directions. Through detailed numerical analysis and subsequent rigorous analytical proof, it has been established that the optimal values of the geometric coefficients r^*_c,opt and λ_opt are strictly determined and provide a simultaneous global maximization of both indicators (Q, η_v), regardless of the operating pressure p, rotational speed n,  or the viscosity of the working fluid. However, an analytically irresolvable conflict regarding the number of teeth z has been identified: a small number maximizes the flow rate, whereas a large number increases the volumetric efficiency. To overcome this contradiction, the problem is formulated within the class of mixed-integer nonlinear programming (MINLP), and multicriteria Pareto optimization is applied, combined with the PSIMS method for the selection of optimal compromise solutions. An empirical relationship (with a coefficient of determination of R² = 0.9603) has been derived, which defines the optimal number of teeth z_opt as a function of the operating pressure and rotational speed. The proposed methodology provides a reliable and applicable tool for designing highly efficient three-rotor pumps tailored to specific operational requirements.