Simple model of equilibrium froth height for foams: an application for CNN image analysis

The design of a control system to monitor the washing of coal by a froth flotation mechanism is considered. The froth in a batch cell, due to steady sparging by air, reaches an equilibrium height h. This height is determined by the cumulative effects of several resistance mechanisms dissipating the air pressure gradient: viscous fiction of the rising air and of the falling liquid, the surface tension of bubbles, and the buoyancy forces. This control system is based upon a hydrodynamic model for the resistance and a feedback loop consisting of an image processing system that computes bubble density and size distribution needed by the model. The model hypothesis is that bubble flow is an air flow through a porous medium with an effective resistance coefficient K which depends on the dissipative mechanisms given above. The pressure gradient needed to estimate the froth height is found from Darcy´s law when the froth is idealized as a set of vertical tubes, with radius R chosen to be the average bubble size, which varies with vertical position, allowing the air to flow through with an average velocity V_m. The model equation is grad p=K V_m/R₀². The cellular neural network (CNN) paradigm was chosen for its ability to process images quickly for use as control system element to compute k and thus infer changes to h by changing the set point for air flow rate or by addition of more liquid or surfactant, which would change the drainage rate or the surface tension