From Residence-Time Distribution and Population-Balance Modeling to Predictive Digital Twins of Industrial Ball Mills: A Comprehensive Critical Review and Evidence-Derived Research Framework

Ball mills remain among the most energy-intensive and operationally complex units in mineral-processing circuits, while many existing monitoring and modeling approaches address only isolated aspects of their behavior. This critical review examines the scientific foundations required for the development of predictive digital twins of industrial ball mills, with particular emphasis on residence-time distribution (RTD), population-balance modeling (PBM), particle-size distribution (PSD), grinding-media and liner wear, process sensing, condition monitoring, and hybrid physics–data modeling. Industrial and laboratory studies are critically compared to assess how material transport, mixing, hold-up, breakage kinetics, classification, wear evolution, and hidden operating states have been measured and modeled. Particular attention is given to the coupling between RTD and PBM, the identification of effective mixing structures from tracer data, the distinction between liquid- and solids-based residence-time measurements, and the additional challenges introduced by hydrocyclone recycle in closed grinding circuits. The review shows that no single fixed transport topology can be assumed universally for industrial ball mills; instead, effective residence-time structure and model parameters should be identified from process-specific measurements. Based on the reviewed evidence, an integrated research framework is proposed in which an identified multi-zone RTD model is coupled with a dynamic PBM for full PSD prediction, followed by explicit sump and hydrocyclone models, grinding-media and liner-wear states, sensor-based hidden-state estimation, and data-driven parameter adaptation or residual compensation. The proposed architecture is intended to support multi-horizon prediction, uncertainty-aware monitoring, what-if simulation, and eventual deployment as an industrial predictive digital twin. Major limitations in the literature include restricted access to industrial datasets, limited long-term validation, weak integration of wear with grinding kinetics, and insufficient treatment of uncertainty and model transferability.