Multivariate stochastic modeling of particle descriptors for studying the influence of wettability on flotation-based separation behavior

In the context of mineral processing, the separation of particles of some feed material into a valuable and a waste fraction is commonly achieved by processes like magnetic (1) or flotation separation (2). The outcome of such processes often depends on particle properties like wettability, but it can also depend on the composition, size and shape of particles. Therefore, with regard to the goal of optimizing separation processes, it is useful to analyze different particle descriptors when studying the influence of wettability on the particle separation behavior. A common tool for classifying particle separation processes are Tromp functions (3). Recently, multivariate Tromp functions, computed by means of non-parametric kernel density estimation, have emerged which characterize the separation behavior with respect to multidimensional vectors of particle descriptors (4). However, estimating multivariate probability densities of particle descriptors using methods of kernel density estimation requires a relatively large sample size (5). Therefore, we propose an alternative parametric approach based on copulas (6) in order to compute multivariate Tromp functions and, in this way, to characterize the separation behavior of particle systems, see Figure 1. Moreover, the parametric modeling approach is extended by an optimization routine to handle separation processes when measurements are not available for all separated fractions. A potential application of the optimization routine is to reduce the measurement effort in a series of separation experiments for a given feed material and various separated fractions. A simulation study has been performed in order to quantitatively compare the parametric with the non-parametric modeling approach. This comparison focuses especially on scenarios for which only a relatively small number of particles is observed in measurements. Such scenarios occur, for example, when image data of particle systems are considered, from which descriptors of individual particles can be determined, but
the number of observed particles tends to be smaller than the number of particles obtained with other measurement techniques. However, some particle properties like wettability cannot be directly deduced from image data, although many separation processes, such as flotation-based particle separation processes, rely on differences in wettabilities. Building on recent studies regarding flotation separation (2), bivariate Tromp functions for the area-equivalent diameter and aspect ratio of glass particles with differently modified wettabilities have been computed. Comparing the Tromp functions obtained in this way reveals the influence of particle wettability on the separation behavior.