Quantification of Solid-Liquid Mixing Uniformity via Three-Dimensional Ripley’s L Function in DEM-VOF Simulations

Uniformity in solid-liquid mixing is a critical aspect for mass and heat transfer efficiency in multiphase reactors. This highlights the necessity for rigorous quantitative approaches capable of resolving spatial heterogeneity across multiple scales. In this work, a coupled discrete element method-volume of fluid (DEM-VOF) framework is employed to simulate the suspension dynamics of 20,000 particles, each 2 mm in diameter, within a liquid medium. To achieve a quantitative and multiscale characterization of three-dimensional particle distributions, Ripley’s L function, rooted in spatial statistics, is introduced and systematically applied. Its validity and robustness are further corroborated through Monte Carlo-based numerical experiments. The findings demonstrate that this methodology not only discriminates effectively between distinct spatial distribution regimes, but also captures with high fidelity the temporal evolution of particle organization during agitation. Interestingly, comparative analysis under identical operating conditions reveals that 45° pitched-blade impellers consistently outperform their 90° straight-blade counterparts. Among the configurations investigated, the 45° turbine impeller delivers the most rapid homogenization, attaining a uniform state within 2.4 s, followed by the 45° four-blade wide hydrofoil design. By contrast, the conventional four-blade impeller exhibits the lowest efficiency, requiring 3.7 s to reach a comparable level of uniformity.