A degenerating convection-diffusion system modelling froth flotation with drainage

TL;DR

This paper models the formation and stability of froth layers in mineral flotation using a degenerating convection-diffusion PDE system, incorporating drainage effects, and provides numerical simulations and feasibility conditions.

Contribution

It introduces a hyperbolic PDE model with drainage effects for froth flotation, constructs steady states, and develops a monotone numerical scheme with stability analysis.

Findings

Feasibility conditions for steady froth layers visualized in operating charts

A monotone numerical scheme successfully simulates flotation dynamics

Boundedness of volume fractions proven under CFL condition

Abstract

Froth flotation is a common unit operation used in mineral processing. It serves to separate valuable mineral particles from worthless gangue particles in finely ground ores. The valuable mineral particles are hydrophobic and attach to bubbles of air injected into the pulp. This creates bubble-particle aggregates that rise to the top of the flotation column where they accumulate to a froth or foam layer that is removed through a launder for further processing. At the same time, the hydrophilic gangue particles settle and are removed continuously. The drainage of liquid due to capillarity is essential for the formation of a stable froth layer. This effect is included into a previously formulated hyperbolic system of partial differential equations that models the volume fractions of floating aggregates and settling hydrophilic solids [R. B\"{u}rger, S. Diehl and M.C. Mart\'i, {\it IMA J. Appl. Math.} {\bf 84} (2019) 930--973]. The construction of desired steady-state solutions with a froth layer is detailed and feasibility conditions on the feed volume fractions and the volumetric flows of feed, underflow and wash water are visualized in so-called operating charts. A monotone numerical scheme is derived and employed to simulate the dynamic behaviour of a flotation column. It is also proven that, under a suitable Courant-Friedrichs-Lewy (CFL) condition, the approximate volume fractions are bounded between zero and one when the initial data are.