A numerical scheme for a model of a flotation column including the transport of liquid components

Abstract. Froth flotation in a column is a widely used unit operation in mineral processing, wastewater treatment, and other applications. The flotation process selectively separates finely divided hydrophobic materials (valuable minerals or ores; repelled by water) from hydrophilic (slimes or gangue; attracted to water), where both are suspended in a viscous fluid. A flotation column roughly functions as follows: gas is introduced close to the bottom and generates bubbles that rise through the continuously injected pulp that contains the solid particles. The hydrophobic particles attach to the bubbles, forming foam or froth (the concentrate) that is removed through a launder. The hydrophilic particles do not attach to bubbles, but normally settle to the bottom, and are removed continuously. Additional wash water, injected close to the top, may assist with the rejection of entrained impurities and increase froth stability. A recently formulated partial differential equation model [R. Bürger, S. Diehl, M.C. Martí, Y. Vásquez, IMA J. Appl. Math. 87 (2022) 1151–1190] describes the process by a pair of degenerate parabolic PDEs with discontinuous flux for the volume fractions of bubbles and hydrophilic solid particles as functions of height and time. An extension of that model is presented, which includes the effect of wash water to be injected into the froth as well as the transport of an arbitrary number of components (such as slimes or chemical reagents) with the liquid. The numerical scheme for bubble and particle volume fractions is extended to simulate percentages representing the liquid components, which are proven to remain nonnegative and sum up to one. In addition, a theory of desired steady states of the flotation column is outlined. It is proven that the condition of “positive bias” in a determined zone of the flotation column (i.e., net downward flow of water) coincides with the mathematically derived condition for the existence of a stationary bubble concentration profile, including a stable froth layer. It is demonstrated how steady-state solutions to the governing model can be constructed and conditions for their existence can be conveniently mapped through so-called “operating charts.” Numerical simulations are presented.