Population balance modeling of flotation pulp: The route from process frequency functions to spatially distributed models

The increased sophistication of mathematical models depicting industrial processes requires advancement both in terms of expressing the state of participating materials and of describing their spatial distribu-tions. In case of dispersed systems, the appropriate mathematical tools for such an advanced represen-tation are the population balance equations and the computational fluid dynamics algorithms. In the presence of turbulence the situation is even more difficult because turbulence affects all size scales of the problem. Since the direct simulation of turbulence is impracticable with the present computational means, a large effort in modeling turbulent interactions is required. The present work is the follow up of a series of two recent works on the development of expressions for turbulent collisions frequency be-tween bubbles and particles. Here, the whole route for transferring the localized collision information to the real-world device scale is presented, accounting for the dynamics of bubbles and particles popula-tions. A particular procedure to (i) select the state and independent variables, (ii) select the occurring process, and (iii) mathematically reduce the multivariate population balance system after careful inspec-tion of the problem in each stage of development, is proposed. The procedure is applied to a relatively complicated specific flotation process with the aim of demonstrating its details. A spatially homogeneous system of ordinary differential equations is derived and several results are illustrated in order to expose its usefulness. Finally, possible ways to transfer the spatially homogeneous model to a computational fluid dynamics environment are discussed. (c) 2021 Elsevier Ltd. All rights reserved.