Data Availability Statement: 
The data that support the findings of this study are available from the corresponding author, P.Q, upon reasonable request.Paulina Quintanilla would like to acknowledge the National Agency for Research and Development (ANID) for funding this research with a scholarship from “Becas Chile”. The Society of Chemical Industry is also greatly acknowledged for the support granted by the SCI Messel Scholarship 2020

Brito-Parada, PR

Neethling, SJ

Quintanilla, P

Materials Proceedings

Brunel University Research Archive

AbstractDevelopment and Validation of a Dynamic Model for FlotationPredictive Control Incorporating Froth Physics †Paulina Quintanilla * , Stephen J. Neethling and Pablo R. Brito-ParadaCitation: Quintanilla, P.; Neethling,S.J.; Brito-Parada, P.R. Developmentand Validation of a Dynamic Modelfor Flotation Predictive ControlIncorporating Froth Physics. Mater.Proc. 2021, 5, 13. https://doi.org/10.3390/materproc2021005013Academic Editor:Konstantinos SimeonidisPublished: 9 November 2021Publisher’s Note: MDPI stays neutralwith regard to jurisdictional claims inpublished maps and institutional affil-iations.Copyright: © 2021 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;s.neethling@imperial.ac.uk (S.J.N.); p.brito-parada@imperial.ac.uk (P.R.B.-P.)* Correspondence: p.quintanilla18@imperial.ac.uk† Presented at International Conference on Raw Materials and Circular Economy, Athens, Greece,5–9 September 2021.Keywords: froth flotation; flotation modelling; flotation control; model calibration; model validation;model predictive controlIn mining, froth flotation is the largest tonnage separation process used to separatevaluable minerals from waste rock. The froth flotation process is carried out in stirred tanksin which chemical reagents and air are added. Reagents make the valuable mineral particleshydrophobic, so they repel water. As a consequence, the valuable mineral particles attach toair bubbles, covering them and generating bubble–particle aggregates. The bubble–particleaggregates rise to the top of the tank, forming a froth that overflows as a mineral-richconcentrate. The waste rock particles do not attach to the bubbles and leave from thebottom of the tank.As froth flotation is a large-scale process, even small improvements in the separationefficiency would translate into important increments in production [1,2]. Optimisingthe process to increase metal recovery also increases the sustainability of the operation.One of the most efficient ways to optimise the process is to implement advanced controltechniques [3]. Model predictive control (MPC), in particular, is widely considered as one ofthe most efficient strategies to optimise a multivariable process, using an explicit, dynamicmodel of the process to predict the future behaviour of the most important variables [4].However, implementing MPC for froth flotation is very challenging, as modelling thisprocess is a difficult task due to the very complex dynamics involved and since it is amultiphase (gas-liquid-solid) system with inherent instability [2,3,5–8]. While complexmodels to describe the phenomena of the flotation process can be found in the literature,models for control purposes must be simple enough—yet robust—to solve the controlproblem in real time. Both characteristics conflict with each other; thus, it is necessary tofind a trade-off between simplicity and robustness.In this study, we propose a novel dynamic flotation model for MPC purposes. Thefull model development can be found in [9]. Our model differs from the others foundin previous studies since, for the first time, the physics of the froth phase is included forcontrol. The importance of including froth physics lies in the fact that the phenomenaoccurring in the froth—such as air recovery, bubble coalescence, and bubble bursting—ultimately determine the separation efficiency and thus the recovery of minerals [10]. Themodel proposed can be classified as a differential and algebraic equation (DAE) problem,which has 31 + 12i + 5k equations, where i is the number of mineralogical classes, and k isthe number of bubble size classes in the pulp phase. An analysis of degrees of freedomfor control revealed that two variables need to be fixed to have a completely determinedmodel. Since the model is quite extensive, here, we only discuss some of the equationsused to model each of the phases of a flotation system.Experiments were carried out in an 87 L laboratory flotation tank at Imperial CollegeLondon in order to calibrate and validate the model. All experiments were performed in aMater. Proc. 2021, 5, 13. https://doi.org/10.3390/materproc2021005013 https://www.mdpi.com/journal/materprocMater. Proc. 2021, 5, 13 2 of 3three-phase system (solid–liquid–gas). The solids system consisted of silica with an averageparticle size of 75 µm. In the experiments, we measured important flotation variables,such as pulp level (dynamic), air recovery (dynamic), and pulp bubble size distribution(steady state) while fixing the two degrees of freedom: air and tailings flowrates (throughtheir respective control valves). These flowrates are the typical manipulated variables inflotation control. Three values of air flowrates and tailings flowrates were fixed during theexperiments, having a total of nine different operating conditions. The experiments in eachoperating condition were carried out in triplicate for further statistical analysis.The experimental data obtained were used to calibrate and validate the proposedmodel, as shown in [11]. In the pulp phase, one of the most important variables for flotationcontrol is the pulp level. Results for the validation of the dynamic model for pulp levelprediction, based on the model proposed by [12], revealed high accuracy in terms of theprediction and trends for this variable. In the froth phase, air recovery plays a crucial role,as it is strongly linked to froth stability and, in turn, with the overall flotation performance.However, little work has been conducted in terms of including this variable in controlstrategies. Therefore, one of our main aims was to include this variable in the model inorder to obtain a complete flotation model capable of capturing the effect of changes in frothstability. The experimental data analysed here are in exceptionally good agreement withthe model predictions. The standard error of estimate calculated for air recovery suggestedthat the average difference between the observed values and the model prediction will fallwithin the range from 1.29% to 6.77%, depending on the operating condition. What is evenmore important here, though, is that the model can follow the trends in air recovery. Thisability is crucial for the implementation of our model into predictive control strategies forenhanced metallurgical performance.In summary, we validated a new dynamic flotation model, which is suitable for MPCstrategies and incorporates froth physics. Having models of the froth phase for MPC willallow detailed predictions and, therefore, better control strategies to maximise the amountof valuable metal recovered. This enhancement in the recovery of valuable metals will helpin taking a step towards better utilisation of the resources, contributing to the sustainabilityof the process.Data Availability Statement: The data that support the findings of this study are available from thecorresponding author, P.Q, upon reasonable request.Acknowledgments: Paulina Quintanilla would like to acknowledge the National Agency for Re-search and Development (ANID) for funding this research with a scholarship from “Becas Chile”.The Society of Chemical Industry is also greatly acknowledged for the support granted by the SCIMessel Scholarship 2020.Conflicts of Interest: The authors declare no conflict of interest.References1. Ferreira, J.; Loveday, B. An improved model for simulation of flotation circuits. Miner. Eng. 2000, 13, 1441–1453. [CrossRef]2. Quintanilla, P.; Neethling, S.J.; Brito-Parada, P.R. Modelling for froth flotation control: A review. Miner. Eng. 2020, 162, 106718.[CrossRef]3. Bergh, L.; Yianatos, J. The long way toward multivariate predictive control of flotation processes. J. Process. Control. 2011, 21,226–234. [CrossRef]4. Camacho, E.F.; Bordons, C. Model Predictive Control; Springer: London, UK, 2007. [CrossRef]5. Desbiens, A.; Hodouin, D.; Najim, K.; Flament, F. Long-range predictive control of a rougher flotation unit. Miner. Eng. 1994, 7,21–37. [CrossRef]6. Desbiens, A.; Hodouin, D.; Mailloux, M. Nonlinear Predictive Control of a Rougher Flotation Unit Using Local Models. IFACProc. Vol. 1998, 31, 287–292. [CrossRef]7. Sbarbaro, D.; del Villar, R. Advanced control and supervision for mineral processing. In Advances in Industrial Control; Springer:Berlin/Heidelberg, Germany, 2010. [CrossRef]8. Shean, B.; Cilliers, J. A review of froth flotation control. Int. J. Miner. Process. 2011, 100, 57–71. [CrossRef]9. Quintanilla, P.; Neethling, S.J.; Navia, D.; Brito-Parada, P.R. A dynamic flotation model for predictive control incorporating frothphysics. Part I: Model development. Miner. Eng. 2021, 173, 107192. [CrossRef]Mater. Proc. 2021, 5, 13 3 of 310. Neethling, S.; Brito-Parada, P. Predicting flotation behaviour—The interaction between froth stability and performance. Miner.Eng. 2018, 120, 60–65. [CrossRef]11. Quintanilla, P.; Neethling, S.J.; Navia, D.; Brito-Parada, P.R. A dynamic flotation model for predictive control incorporating frothphysics. Part II: Model calibration and validation. Miner. Eng. 2021, 173, 107190. [CrossRef]12. Shean, B.; Hadler, K.; Neethling, S.; Cilliers, J. A dynamic model for level prediction in aerated tanks. Miner. Eng. 2018, 125,140–149. [CrossRef]

Development and Validation of a Dynamic Model for Flotation Predictive Control Incorporating Froth Physics

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Development and Validation of a Dynamic Model for Flotation Predictive Control Incorporating Froth Physics

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