Citation: Betancourt, F.; Bürger, R.; Diehl, S.; Gutiérrez, L.; Martí, M.C.; Vásquez, Y. A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments. Minerals 2023, 13, 344. https:// doi.org/10.3390/min13030344 Academic Editor: Saeed Chehreh Chelgani Received: 31 January 2023 Revised: 22 February 2023 Accepted: 25 February 2023 Published: 28 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). minerals Article A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments Fernando Betancourt 1 , Raimund Bürger 2, * , Stefan Diehl 3 , Leopoldo Gutiérrez 4 , M. Carmen Martí 5 and Yolanda Vásquez 2 1 Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería Metalúrgica, Facultad de Ingeniería, Universidad de Concepción, Casilla 160-C, Concepción 4030000, Chile 2 Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción 4030000, Chilel 3 Centre for Mathematical Sciences, Lund University, P.O. Box 118, S-221 00 Lund, Sweden 4 Departamento de Ingeniería Metalúrgica, Facultad de Ingeniería, Universidad de Concepción, Casilla 160-C, Concepción 4030000, Chile 5 Departament de Matemàtiques, Universitat de València, Avda. Vicent Andrés Estellés s/n, E-46100 Burjassot, Spain * Correspondence: rburger@ing-mat.udec.cl; Tel. +56-41-2661317 Abstract: The operation of a froth flotation column can be described by a nonlinear convection–diffusion partial differential equation that incorporates the solids–flux and drift–flux theories as well as a model of foam drainage. The resulting model predicts the bubble and (gangue) particle volume fractions as functions of height and time. The steady-state (time-independent) ver- sion of the model defines so-called operating charts that map conditions on the gas and pulp feed rates that allow for operation with a stationary froth layer. Operating charts for a suitably adapted version of the model are compared with experimental results obtained with a laboratory flotation column. Experiments were conducted with a two-phase liquid–bubble flow. The results indicate good agreement between the predicted and measured conditions for steady states. Numerical simulations for transient operation, in part for the addition of solid particles, are presented. Keywords: froth flotation; drainage; drift flux; mathematical model; partial differential equation; steady state; numerical simulation 1. Introduction Froth flotation is the most important concentration operations in mineral processing and is widely used for the recovery of valuable minerals from low-grade ores (cf. [1], ([2], Chapter 12) or ([3], Part 7)). This unit operation is an important stage particularly for copper mining in Chile. The flotation process selectively separates hydrophobic materials (that are repelled by water) from hydrophilic (that would be attracted to water), where both are suspended in a viscous fluid. It is well known that a flotation column works as follows: gas is introduced close to the bottom and generates bubbles that rise through the continuously injected pulp that contains the solid particles. The hydrophobic particles (the valuable mineral particles) attach to the rising bubbles, forming froth that is removed through a launder. The hydrophilic particles (slimes or gangue) do not attach to bubbles but settle to the bottom (unless they are trapped in the bulk upflow) and are removed continuously as flotation tailings. Close to the top, additional wash water can be injected to assist with the rejection of entrained impurities and to increase the froth stability [1,4,5]. This unit operation is particularly suitable for processing low-grade ores, such as copper ores in Chilean deposits; however, this requires huge amounts of process water. Minerals 2023, 13, 344. https://doi.org/10.3390/min13030344 https://www.mdpi.com/journal/minerals