Manchester Applied Mathematics and Numerical Analysis Seminars Spring 2000

March 15, 2000, 4.00 pm

Room OF/B9, Oddfellows Hall, Grosvenor Street

A Mathematical Model of Flotation Froths

Dr. Jan J. Cilliers, Froth and Foam Research Group, Dept. of Chemical Engineering, UMIST

Flotation is a long established and widely used separation method in minerals processing and paper de-inking, with vast research having been devoted to it. Despite this, there has been virtually no fundamental modelling of the froth phase, which is a prerequisite for flotation. This work is an attempt to redress this shortfall, and to produce a model framework that can be used to accommodate research in various aspects of the froth phase.

The model describes a vertical cross-section through a froth flowing upwards in a column, until it either bursts at the top surface or overflows the sides. This requires four inter-related models that describe the gas motion, the bubble behaviour, liquid motion and solid motion. Bubble shapes are also visualised.

Laplace's equation is used to model the gas motion, and individual bubble shapes determined by carrying out force balances between the neighbouring lamellae. A criterion is included to accommodate coalescence, based on deformation and lifetime. The liquid motion is determined from a force balance over a unit volume. Three forces are considered; gravity, capillary and viscous dissipation. This is combined with a continuity equation that includes the effect of bubble coalescence on the behaviour of the liquid. The solids are divided into two classes, the particles that are attached to the bubbles (hydrophobic) and the particles that are unattached (hydrophobic or hydrophilic). Attached particles can become unattached through bursting or coalescence. Unattached particles can move relative to the liquid by means of three mechanisms; hindered settling, geometric dispersion and Plateau border dispersion. These mechanisms will be described.

This takes the form of a boundary value problem, and is solved using a finite difference approximation. Four types of boundaries are considered: impermeable walls, the bursting surface, the pulp-froth interface and the weir overflow. The model makes use of the physical dimensions of the system and can thus be used for equipment design and optimisation.

Examples of results from typical simulations of a foam flowing vertically and overflowing to the right are shown below. Results showing the motion of liquid and entrained solids will also be shown, as well as the effect of bubble coalescence and weir angle.

Figure 1: Single frame from bubble visualisation	Figure 2: Motion of unattached, hydrophobic solids

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).