Concepts of statistical entropy for the evaluation of comminution and separation processes

Statistical entropy, corresponding to Shannon entropy in information theory or Gibbs entropy in physical thermodynamics, can be used to evaluate the additional disorder in a system induced by its subdivision in components or subsystems. Several variations of this concept have been introduced in the last 20 years to quantify the efficiency of a separation process (or a chain of it) within the context of minerals processing both in mining and recycling. This contribution presents a coherent frame connecting these concepts, and extends them to evaluate comminution processes, in such a way that joint thermoeconomic zoptimization of whole beneficiation plants becomes possible.

The main idea is to split the system simultaneously into three dimensions: (1) mineral or chemical components, (2) particles or particle classes, and (3) output streams. In each of these dimensions, a different ratio can be defined to describe the way the mass (or volume or matter) of the system is split into the subsystems: (1) a composition; (2) a particle-wise distribution; a (3) set of partition coefficients. Entropy contributions can then be defined for each of these dimensions at different levels of integration, e.g. (1) component-wise or for a total, (2) particle-wise or bulk-wise, (3) per stream or per stage feed.

By systematically considering all possible entropy decompositions, the following results were found. Total stage entropy does not depend on the ordering of integration through the three dimensions. Comminution cannot decrease the stage entropy; a comminution that would generate perfectly liberated particles without overgrinding would keep the stage entropy constant. A separation process cannot decrease stage entropy, and perfect separation would keep it constant. However, once the masses of the output streams are measured, the output entropy does indeed decrease for any moderately good separation process. The difference between the stage entropy with and without controlling output masses can then be understood as the information value of the output mass flow measurements.