Flotation Kinetics Beyond the First-Order Paradigm: A Multi-Scale, Heterogeneity-Aware Framework for Coal and Complex Minerals

For nearly a century, flotation kinetics has relied on deterministic first-order rate equations treating the cell as a homogeneous reactor, a paradigm that inherently fails for heterogeneous ores, especially coal, whose organic macerals, porosity, and oxidation susceptibility defy a single rate constant. Breaking from chronological cataloguing, this review proposes a three-dimensional taxonomy based on physical scale, inherent material heterogeneity, and epistemic certainty. We demonstrate that critical industrial prediction failures arise from structural mismatches between model physics and particle surface chemistry, notably time-dependent oxidation deactivation and selective maceral recovery. Six fundamental failure modes are identified, from neglected time-dependence of rate constants to the absence of a thermodynamic deactivation term, corroborated by experimental evidence from coal and base-metal flotation. Advanced microfluidic, automated mineralogical, surface-sensitive spectromicroscopic, CFD-DEM, and physics-informed machine learning tools are dismantling the black box of the flotation rate constant “k”. We introduce the Distributed Reactive Surface Kinetics (DRSK) framework, which embeds particle-scale heterogeneity into a population balance via an adaptive surface-sensitive selection function and treats kinetic uncertainty through stochastic differential equations. A comprehensive comparison table facilitates the transition from conventional models to the DRSK paradigm. We conclude with a roadmap for flotation kinetics 4.0, where digital twins, real-time froth analytics, and self-calibrating hybrid models transform this empirical discipline into a truly predictive engineering science. The framework is elaborated for coal and conventional minerals, underscoring why coal demands its own dedicated kinetic theory and how these lessons can revolutionize the processing of increasingly complex, low-grade ores and secondary resources.