S.J. Neethling

Modelling flotation froths

Abstract This paper gives a brief description of a fundamentally based model for the behaviour of a flotation froth. It includes descriptions for all three phases present in a flotation froth. The model for the gas motion uses either the simpler Laplace equation for describing the gas motion, or a more complex model for froths in very complex geometries or with very high flow rates. A methodology for predicting coalescence in a flowing foam or froth is included together with some theoretical forms for the coalescence frequency function. The liquid motion is predicted as the result of a force balance that includes gravity, viscosity and surface tension, acting by means of gradients in the curvature of the Plateau border interface. The solid motion model include the effects of liquid motion, particle settling and two types of dispersion, Plateau border and geometric. Lastly, the model is used to predict the performance of three different froth handling designs. The simulations compare the performance of each of these designs as the air rate into the cell is varied, and gives some experimental verification.

The entrainment of gangue into a flotation froth

Abstract The flotation froth structure and motion determine the amount of entrained gangue that is collected to the concentrate. Despite this important role in the overall performance, the behaviour of the froth is still ill understood. To date, predominantly empirical models have been developed to describe entrainment. A fundamentally based model is described and used here in an attempt to improve the understanding of entrainment. This general froth simulator (UMIST FrothSim) allows the modelling of a wide range of flotation conditions, as it takes account of a large number of the physical phenomena that occur within flotation froths. The model firstly describes gas motion and bubble coalescence, and the water motion based on gravitational, viscous and capillary effects. The motion of the solids distinguishes between various solids classes based on hydrophobicity, particle size and density and models them by including the effects of Geometric and Plateau border dispersion, particle settling and the motion of the water. The aim of this paper is to show the applicability of this model to explain the observed entrainment and collection of gangue. This is done by comparing model predictions with the experimentally observed relationship between the gangue and water recoveries. The model predicts the identical trends, which are explained in terms of the interaction between the linear effects of water motion and hindered settling and the non-linear effect of particle dispersion.

A foam drainage equation generalized for all liquid contents

Recently the study of liquid drainage through foams has generated considerable experimental and theoretical research. This research has usually made the assumptions that the liquid content is very low and that the viscous resistance is limited to occurring within either the Plateau borders or vertices. This paper presents a foam drainage model that not only describes the low-liquid-content extreme, but also takes account of liquid content effects. These include the increase in the foam volume with increasing liquid content and the increase in the liquid flowrate in the Plateau borders over the average flowrate brought about by the presence of vertices. The relative importance of the Plateau borders and vertices to viscous loss within the foam is also made dependent on the liquid content of the foam. This model is verified experimentally for two surfactant types and various bubble sizes using a standard forced drainage system.

Solids motion in flowing froths

Abstract Flotation is a widely used process within the minerals processing industry, as well as being used for water treatment and de-inking of recycled paper. The froth phase and its role in the separation achieved is as yet ill understood. A fundamentally based model of the behaviour of solids within a flowing froth allows for a fuller understanding of the froth phase of flotation vessels and process optimisation. This paper outlines a model for the motion of solids within a flowing froth. It builds on earlier work on the modelling of bubble and liquid motion within a flowing froth and includes all the effects of same phenomena that effect liquid motion, as well as including the effect of solids concentrations on liquid motion. The solids are divided into two classes for the purposes of modelling, namely the attached material, which follows the bubbles, and the unattached material, which mainly follows the liquid, but can move relative to the water by means of hindered settling and geometric and Plateau border dispersion. The attached material consists of hydrophobic particles, while the unattached material can consist of both hydrophobic and hydrophilic particles. Attached particles can become unattached due to coalescence or bursting. Results from simulations are shown to illustrate the movement and concentration of the solids from the pulp–froth interface to the upper, bursting surface and overflowing the weir.

Prediction of the water distribution in a flowing foam

Abstract A two-dimensional mathematical model is presented which describes the time-averaged steady-state water distribution in a coalescing, flowing foam. The model uses previous work that predicts foam flow velocity and bubble coalescence. Drainage is described using the transient model of Verbist et al., extended into two-dimensions. The effect of coalescence is accommodated by introducing the concept of Plateau border length for each bubble. The formulation and solution technique of the highly non-linear partial differential equation is discussed in some detail. Simulations are shown that compare the water distribution for non-coalescing and coalescing foams. The model will have application for determining the liquid content of flowing foams, the entrainment of solids in flotation systems and for designing foam process equipment.

Simulation of the effect of froth washing on flotation performance

Abstract The froth phase is extremely important in the operation of a flotation cell, seeing that it is critical in determining the amount of unwanted gangue collected to the concentrate and thus the purity of the product. This paper uses a fundamentally based model of flowing froths to simulate the performance of a flotation cell. The study concentrates specifically on the effect of wash water addition on the overall performance, i.e. the grade and the recovery. The froth model that is used within this work includes a large number of the effects seen within a flotation froth and approaches their description from a fundamental point of view. Some of the phenomena that are included are bubble coalescence, liquid drainage including the effects of gravity, surface tension and viscous dissipation, particle settling and particle dispersion. The results show the advantages and disadvantages of different water addition strategies on the performance of flotation vessels. Since most recent flotation work has been concerned with improving recoveries, rather than grade, the addition of water into flotation froths has been largely limited to column cells. This work demonstrates how water addition can be optimised in terms of both water addition point and quantity in order to produce the desired performance.

The effect of particle hydrophobicity, separation distance and packing patterns on the stability of a thin film

Hydrophobic particles attached to bubble films in foams can increase the capillary pressure required to cause coalescence or bursting. Previous studies have considered the effects of changing particle spacing and contact angle in 2 dimensions (2D), but there are limitations to this approach; in 2D when the separation distance is zero and the particles are touching, the critical capillary pressure tends to infinity as there is no exposed film. In 3 dimensions (3D) spherical packing ensures that there are always exposed sections of film between particles even when they are close packed. Using Surface Evolver, the effects of contact angle and particle separation on the stability of a particle laden film were investigated in 2D and 3D. The 2D model was compared and validated with an analytical approach developed by Ali et al. [Ind. Eng. Chem. Res. 39 (2000) 2742-2745] and a 3D model was used to investigate the critical capillary pressures of square and hexagonal packing of monodisperse particles. It was found that when the stability of the film was compared with the area of film per particle both packing patterns have the same stability.

The recovery of liquid from flowing foams

This paper examines the recovery of liquid from stable overflowing foams. The foams are formed by bubbling gas into the bottom of a column and the liquid is collected from the foam that flows over the lip at the top. The paper demonstrates, using a foam drainage equation, that the recovery of liquid will rapidly decrease towards an asymptotic value as the foam height is increased. An expression for this liquid recovery is then developed. That the liquid recovery becomes constant as foam height is increased in a stable foam is also demonstrated experimentally. The mathematical analysis of the problem suggests that the amount of liquid collected is proportional to the gas rate squared. This relationship is verified experimentally for an aqueous foam.

Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media

Numerical simulation of multiphase flow in porous media is of great importance in a wide range of applications in science and engineering. The governing equations are the continuity equation and Darcy’s law. A novel control volume finite element (CVFE) approach is developed to discretize the governing equations in which a node-centered control volume approach is applied for the saturation equation, while a CVFE method is used for discretization of the pressure equation. We embed the discrete continuity equation into the pressure equation and ensure that the continuity equation is exactly enforced. Furthermore, the scheme is equipped with dynamic anisotropic mesh adaptivity which uses a metric tensor field approach, based on the curvature of fields of interest, to control the size and shape of elements in the metric space. This improves the resolution of the mesh in the zones of dynamic interest. Moreover, the mesh adaptivity algorithm employs multi-constraints on element size in different regions of the porous medium to resolve multi-scale transport phenomena. The advantages of mesh adaptivity and the capability of the scheme are demonstrated for simulation of flow in several challenging computational domains. The scheme captures the key features of flow while preserving the initial geometry and can be applied for efficient simulation of flow in heterogeneous porous media and geological formations.

Quantifying and minimising systematic and random errors in X-ray micro-tomography based volume measurements

X-ray micro-tomography (XMT) is increasingly used for the quantitative analysis of the volumes of features within the 3D images. As with any measurement, there will be error and uncertainty associated with these measurements. In this paper a method for quantifying both the systematic and random components of this error in the measured volume is presented. The systematic error is the offset between the actual and measured volume which is consistent between different measurements and can therefore be eliminated by appropriate calibration. In XMT measurements this is often caused by an inappropriate threshold value. The random error is not associated with any systematic offset in the measured volume and could be caused, for instance, by variations in the location of the specific object relative to the voxel grid. It can be eliminated by repeated measurements. It was found that both the systematic and random components of the error are a strong function of the size of the object measured relative to the voxel size. The relative error in the volume was found to follow approximately a power law relationship with the volume of the object, but with an exponent that implied, unexpectedly, that the relative error was proportional to the radius of the object for small objects, though the exponent did imply that the relative error was approximately proportional to the surface area of the object for larger objects. In an example application involving the size of mineral grains in an ore sample, the uncertainty associated with the random error in the volume is larger than the object itself for objects smaller than about 8 voxels and is greater than 10% for any object smaller than about 260 voxels. A methodology is presented for reducing the random error by combining the results from either multiple scans of the same object or scans of multiple similar objects, with an uncertainty of less than 5% requiring 12 objects of 100 voxels or 600 objects of 4 voxels. As the systematic error in a measurement cannot be eliminated by combining the results from multiple measurements, this paper introduces a procedure for using volume standards to reduce the systematic error, especially for smaller objects where the relative error is larger. Methodology for quantifying random and systematic errors in microCT images presented.Unexpected power law scaling for error in small particle volume as threshold changes.Random component of error in volume insensitive to threshold value.