THE DETERMINATION AND MODELLING OF FLOATABILITY DISTRIBUTIONS OF MINERAL ORES

Abstract Froth flotation is the most widely used method of separation used in the mineral processing industry. It is the preferred method of mineral recovery for many of the most important minerals that are recovered, and large tonnages of ore are processed by flotation annually. The modelling of industrial flotation process is reasonably described by the floatability component method. In a perfectly mixed flotation cell the recovery for a group of similar particles is described by the standard Savassi – AMIRA (Australian Mineral Industry Research Association) model. This model includes the properties of the pulp, froth phase, and the hydrodynamic characterisation of the flotation cell. These properties are encapsulated in the variables froth recovery, bubble surface area flux, cell residence time, water recovery term, degree of entrainment and particle floatability. Most of these properties can be measured directly or easily derived from measured quantities except the particle floatability. The difficulty with regards to the particle floatability is because there is not one or two but many floatabilities corresponding to the many types of particles present. In fact, given the huge number of particles present, it is more realistic to assume that the particle floatability is a continuous distribution. The determination of the floatability distribution is an essential difficulty in the floatability component model. Calculating the flotation rate distribution from first principles has proved too complex to be implemented for plant simulations. This leaves us with an empirical method based on a mathematical regularization technique, or the model developed in this thesis by analogy with the methods of Statistical Mechanics. The most practical empirical method to estimate the floatability distribution of the ore to date is to use the batch cell test. Particle property based measurement such as liberation to correlate to flotation rate while in principle possible, is too time-consuming to be practical at the moment. Compounding the problem of estimating ore’s floatability distribution is that the estimation of floatability distribution from measured mass fraction remaining (complement of recovery) from batch cell test is mathematically an ill-posed problem. This is manifested in that there are infinite set of floatability distributions that can fit the measured mass fraction remaining to the accuracy of the data. Similarly, a small change in the data introduces a large variation in the calculated floatability distribution. Hence the main aims of this thesis are : To develop a method to determine the floatability distribution of particle – environment system with certain physico-chemical properties from batch cell test time resolved mass fraction remaining measurements. To propose a model based on a physical analogy for the floatability distribution. To address the first aim of the thesis the Generalized Singular Value Decomposition coupled with Tikhonov regularization method was used. The choice of Tikhonov regularization parameter is to find a Tikhonov solution such that the integral is as close to unity as possible. Being based on a linear method there is no need for an initial estimate and thus it is very robust. It was shown by numerical experiment that for a slowly varying floatability distribution the floatability distribution can be determined accurately while for a rapidly varying distribution a Monte-Carlo method gives a good estimate of the distribution. Applying the above method to accurate batch cell data for minerals Chalcopyrite Pentlandite and Pyrrhotite showed the richness of the method. For Pentlandite with large nonfloating fraction the distribution is bimodal. This seems to support the metallurgist’s commonly used conjecture that the flotation rate distribution composed of two components gives reasonable accuracy. There is also an almost exponential shaped flotation rate distribution shown by in Pyrrhotite with low nonfloating component. A study on the floatability distribution of three streams connected to the same continuous flotation cell was conducted. The traditional nodal analysis is a statement of mass conservation. Floatability is conserved if the cell feeds and products have the same mass of particles in every rate interval of the floatability rate distribution curves. This thesis showed that for this single continuous flotation cell to a good degree the method proposed produced floatability distributions such that mass in flotation rate interval k and k + dk is conserved within the uncertainty of the calculation. A novel method in determining the mass density functions in a flotation circuit was developed. This method uses the plant survey information, the batch cell data on circuit streams, and the Tikhonov regularization coupled with a non-negative least square technique. The method is based on simple and reasonable assumptions such as conservation of floatability, smoothness in the probability distribution of the flotation rates and observable plant properties such as the decrease of concentrates as the stream goes down the flotation cell bank. The estimated mass density functions are consistent with what can be observed, that is, the relevant assays, the batch cell test, and total flows. These estimates are extremely robust in that any physically meaningful initial estimate gives the same final estimate in the mass density function in all the streams. To address the second aim of the thesis a model based on an analogy with Statistical Mechanics was developed. The statistical mechanics model of floatability distribution was able to reproduce the quantitative features of the batch cell derived flotation rate distribution of Pentlandite, Chalcopyrite and Pyrrhotite. The statistical mechanics model seems to indicate that in the case of Pentlandite treating the nonfloating fraction to be as high as the last experimental point is not consistent with its prediction. This shows the importance of experimentally measuring the nonfloating fraction of the particle. This only involves doing the batch cell test for a longer period of time. The statistical mechanics model was able to quantitatively reproduce the flotation rate as a function of liberation. The model was able to explain the sharp increase in the flotation rate of galena particles in the liberation class 90 – 100% when compared to the galena particles in the 80-90% liberation class