Practically all particle separation processes depend on more than one particulate property. In the case of the industrially important froth flotation separation, these properties concern wettability, composition, size and shape. Therefore, it is useful to analyze different particle descriptors when studying the influence of particle wettability and morphology on the separation behavior of particle systems. A common tool for classifying particle separation processes are Tromp functions. 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. In the present paper, an alternative parametric approach based on copulas is proposed in order to compute multivariate Tromp functions and, in this way, to characterize the separation behavior of particle systems. In particular, bivariate Tromp functions for the area-equivalent diameter and aspect ratio of glass particles with different morphologies and surface modification have been computed, based on image characterization by means of mineral liberation analysis (MLA). Comparing the obtained Tromp functions with one another reveals the combined influence of multiple factors, in this case particle wettability, morphology and size, on the separation behavior and introduces an innovative approach for evaluating multidimensional separation. In addition, we extend the parametric copula-based method for the computation of multivariate Tromp functions, in order to characterize separation processes, also in the case when image measurements are not available for all separated fractions.

Froth flotation predominantly separates particles according to their differences in wettability. However, other particle properties such as size, shape or density significantly influence the separation outcome as well. Froth flotation is most efficient for particles within a size range of about 20–200 μm, but challenges arise for very fine or coarse particles that are accompanied by low recoveries and poor selectivity. While the impact of particle size on the separation behavior in flotation is well known by now, the effect of particle shape is less studied and varies based on the investigated zone (suspension or froth) and separation apparatus used. Beyond these complexities, many particle properties are correlated, making it challenging to analyze the isolated impact of individual properties on the separation behavior. Therefore, a multidimensional perspective on the separation process, considering multiple particle properties, enhances the understanding of their collective influence. In this paper, the two-dimensional case is studied; i.e., a parametric modeling approach is applied to determine bivariate Tromp functions from scanning electron microscopy-based image data of the feed and the separated fractions. With these functions it is possible to characterize the separation behavior of particle systems. Using a model system of ultrafine (<10 μm) particles, consisting of either glass spheres or glass fragments with different wettability states as the floatable fraction and magnetite as the non-floatable fraction, allows for the investigation of the influence of descriptor vectors consisting of size, shape and wettability, on the separation. In this way, the present paper contributes to a better understanding of the complex interplay between certain descriptor vectors for the case of ultrafine particles. Furthermore, it demonstrates the benefits of using multivariate Tromp functions for evaluating separation processes and points out the limitations of SEM-based image measurements by means of mineral liberation analysis (MLA) for the studied particle size fraction.

Froth flotation predominantly separates particles according to their differences in wettability. However, other particle properties such as size, shape or density significantly influence the separation outcome as well. Froth flotation is most efficient for particles within a size range of about $20-200μm$, but challenges arise for very fine or coarse particles that are accompanied by low recoveries and poor selectivity. While the impact of particle size on the separation behavior in flotation is well-known by now, the effect of particle shape is less studied and varies based on the investigated zone (suspension or froth) and the separation apparatus used. A multidimensional perspective on the separation process, considering multiple particle properties, enhances the understanding of their collective influence. In this paper the two-dimensional case is studied, i.e., a parametric modeling approach is applied to determine bivariate Tromp functions from scanning electron microscopy-based image data of the feed and the separated fractions. With these functions it is possible to characterize the separation behavior of particle systems. Using a model system of ultrafine ($<10μm$) particles, consisting of differently shaped glass particles with different wettability states as the floatable and magnetite as the non-floatable fraction, allows for investigating the influence of particle size, shape and wettability, on the separation. In this way, the present paper contributes to a better understanding of the complex interplay between certain property vectors for the case of ultrafine particles. Furthermore, it demonstrates the benefits of using multivariate Tromp functions for evaluating separation processes, and points out the limitations of SEM based image measurements by means of mineral liberation analysis (MLA) for the studied particle size fraction.

This paper introduces a novel method for characterizing fracture mechanisms in composite materials using 3D image data gained by computed tomography (CT) measurements. In mineral liberation, the understanding of these mechanisms is crucial, particularly whether fractures occur along the boundaries of mineral phases (intergranular fracture) and/or within mineral phases (transgranular fracture). Conventional techniques for analyzing fracture mechanisms are focused on globally comparing the surface exposure of mineral phases extracted from image measurements before and after fracture. Instead, we present a virtual reassembling algorithm based on image registration techniques, which is applied to 3D data of composite materials before and after fracture in order to determine and characterize the individual fracture surfaces. This enables us to conduct a local quantitative analysis of fracture mechanisms by voxelwise comparing adjacent regions at fracture surfaces. A quantitative analysis of fracture mechanisms is especially important in the context of geometallurgical recycling processes. As primary deposits are decreasing worldwide, the focus is shifting to secondary raw materials containing low concentrations of valuable elements such as lithium. To extract these elements, they can be enriched as engineered artificial minerals in the slag phase of appropriately designed cooling processes. The subsequent liberation through comminution processes, such as crushing, is essential for the extraction of valuable minerals. A better understanding of crushing processes, especially fracture mechanisms in slags, is crucial for the success of recycling. The reassembling algorithm presented in this paper is evaluated through a simulation study, followed by an application to a naturally occurring ore and a slag resulting from a recycling process.