From Columns to Heaps: Dimensionless Similarity with PSD-Distributed Damk\"ohler Numbers and Dual-Porosity Flow

Abstract

This work develops a unified, dimensionless framework for comparing geometrically similar reacting porous-flow systems across scale, with emphasis on hydrometallurgical heap leaching, when particle size distribution (PSD) and intraparticle pore structure differ. Under dynamic similarity, the dimensionless liquid residence-time distribution (RTD) is identical, but differences in PSD and internal porosity break microscopic similarity. Using the shrinking-core model (SCM), the analysis shows how a PSD in particle diameter maps to a distribution of particle-scale Damk\"ohler numbers that governs heap-averaged conversion. Explicit PSD to Damk\"ohler transformations are derived for (i) external film control, (ii) intraparticle diffusion control, and (iii) mixed control via additive rates. Dual-porosity hydrology relevant to sedimentary or strongly stratified ores is then incorporated by coupling SCM kinetics to mobile and immobile liquid domains, introducing additional dimensionless groups that describe interporosity exchange. Two numerical examples map a lognormal PSD into film- and diffusion-controlled Damk\"ohler distributions and compare column/heap conversion for different PSDs. A practical workflow is outlined for hydrometallurgical column-test interpretation, combining tracer RTD calibration with PSD-aware kinetic fitting in dimensionless time. The framework clarifies why diffusion-controlled leaching is far more sensitive to PSD tails and dual-porosity structure than film-controlled leaching, and it identifies the compact set of dimensionless groups that must be matched to ensure similarity between laboratory columns and industrial heaps.