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.

A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

“…Here, the function is a smooth function and γ α for = , are two non-zero constants. Conservation laws with discontinuous flux function in appear in various physical model problems in science and engineering, for example in the modeling of two-phase flow in porous media [1], in sedimentation problems [2][3][4] and in traffic flow on roads with varying conditions [5,6]. We also note that conservation laws with discontinuous flux can be modeled in different forms.…”

Modelling and boundary optimal control design of hybrid column flotation

“…The model extends the modelling contributions in the literature. [14,24,26,39] The novelty of this contribution is that it has been tailored for a hybrid flotation column, accounts for a CSTR added to the column geometry, includes transport-reaction conservation laws, and accounts for macro-and micro-scale processes, including bubble-particle interactions and mass transfer between up-flow and down-flow. As illustrated in Figure 2, this process is modelled as an interconnection of three different subsystems, including a CSTR, which represents a well-mixed zone, and two plug flow transport reactors, representing pulp and froth zones.…”

“…Along similar lines, other works have considered conservative hyperbolic PDEs in a physically meaningful laboratory setting. [26,27] Flotation is generally effective over a size range of approximately 15-150 μm of mineral particles. [28] Mechanical flotation cells provide enhanced particle-bubble collision through agitation, while flotation columns have a large quiescent zone that enables fractionation.…”