Dissolving the Ore
Leaching is the foundational step in hydrometallurgy: a solid ore particle is immersed in an acidic (or alkaline) solution that selectively dissolves the target metal. The shrinking-core model, developed by Yagi and Kunii in 1955 and refined by Levenspiel, provides an elegant mathematical framework for predicting how fast this dissolution proceeds. As the acid penetrates the particle surface, it reacts with the mineral, leaving behind a porous shell of gangue through which fresh acid must diffuse to reach the retreating reaction front.
Shrinking-Core Mathematics
The model distinguishes three sequential resistances: external film diffusion, product-layer diffusion, and surface chemical reaction. In most industrial settings, product-layer diffusion dominates for particles larger than ~1 mm. The key equation relates fractional conversion X to dimensionless time: 1 - 3(1-X)^(2/3) + 2(1-X) = t/τ, where τ is the time for complete conversion. This τ depends on particle radius squared, making size reduction the single most powerful lever for accelerating extraction.
Temperature and Activation Energy
The Arrhenius equation governs how temperature accelerates leaching. Diffusion-controlled processes have low activation energies (10–25 kJ/mol), while reaction-controlled processes show higher values (40–80 kJ/mol). This distinction is diagnostic: if doubling temperature barely changes the rate, diffusion controls. If the rate surges, chemical reaction is limiting. Industrial heap leaches operate at ambient temperature for cost reasons, while autoclave processes push to 150–250°C for refractory ores.
Industrial Applications
Heap leaching of copper oxides with sulfuric acid is one of the largest hydrometallurgical operations on Earth, processing billions of tonnes of ore annually in Chile, Peru, and Arizona. The same shrinking-core principles apply to pressure oxidation of gold ores, alkaline leaching of bauxite (Bayer process), and the emerging field of direct lithium extraction from brines. This simulator lets you explore how particle size, acid strength, and temperature interact to determine extraction efficiency and processing time.