## Preprint 2019-09

## Raimund Bürger, Stefan Diehl, María Carmen Martí:

### A system of conservation laws with discontinuous flux modelling flotation with sedimentation

### Abstract:

The continuous unit operation of flotation is extensively used in mineral processing, wastewater treatment, and other applications for selectively separating hydrophobic particles (or droplets) from hydrophilic ones, where both are suspended in a viscous fluid. Within a flotation column, the hydrophobic particles are attached to gas bubbles that are injected and float as aggregates forming a foam or froth at the top that is skimmed. The hydrophilic particles sediment and are discharged at the bottom. The hydrodynamics of a flotation column is described in simplified form by studying three phases, namely the fluid, the aggregates, and solid particles, in one space dimension. The relative movements between the phases are given by constitutive drift flux functions. The resulting model is a system of two scalar conservation laws with a multiply discontinuous flux for the aggregates and solids volume fractions as functions of height and time. The model is of triangular nature since one equation can be solved independently of the other. Based on the theory of conservation laws with discontinuous flux, steady-state solutions that satisfy all jump and entropy conditions are constructed. For the existence of the industrially relevant steady states, conditions on feed flows and concentrations are established and mapped as “operating charts”. A numerical method that exploits the triangular structure, is formulated on a pair of staggered grids, and is employed for the simulation of the fill-up and transitions between steady states of the flotation column.