Multiphase systems are important in minerals processing, and usually include solid–solid and solid–fluid systems, such as in wet grinding, flotation, dewatering, and magnetic separation, among several other unit operations. In this paper, the current trends in the process system engineering tasks of modeling, design, and optimization in multiphase systems, are analyzed. Different scales of size and time are included, and therefore, the analysis includes modeling at the molecular level (molecular dynamic modeling) and unit operation level (e.g., computational fluid dynamic, CFD), and the application of optimization for the design of a plant. New strategies for the modeling, design, and optimization of multiphase systems are also included, with a strong focus on the application of artificial intelligence (AI) and the combination of experimentation and modeling with response surface methodology (RSM). The integration of different modeling techniques such as CFD with discrete element simulation (DEM) and response surface methodology (RSM) with artificial neural networks (ANN) is included. The paper finishes with tools to study the uncertainty, both epistemic and stochastic, based on uncertainty and global sensitivity analyses, which is present in all mineral processing operations. It is shown that all of these areas are very active and can help in the understanding, operation, design, and optimization of mineral processing that involves multiphase systems. Future needs, such as meso-scale modeling, are highlighted.

Modeling the global markets is complicated due to the existence of uncertainty in the information available. In addition, the lithium supply chain presents a complex network due to interconnections that it presents and the interdependencies among its elements. This complex supply chain has one large market, electric vehicles (EVs). EV production is increasing the global demand for lithium; in terms of the lithium supply chain, an EV requires lithium-ion batteries, and lithium-ion batteries require lithium carbonate and lithium hydroxide. Realistically, the mass balance in the global lithium supply chain involves more elements and more markets, and together with the assortment of databases in the literature, make the modeling through deterministic models difficult. Modeling the global supply chain under uncertainty could facilitate an assessment of the lithium supply chain between production and demand, and therefore could help to determine the distribution of materials for identifying the variables with the highest importance in an undersupply scenario. In the literature, deterministic models are commonly used to model the lithium supply chain but do not simultaneously consider the variation of data among databases for the lithium supply chain. This study performs stochastic modeling of the lithium supply chain by combining a material flow analysis with an uncertainty analysis and global sensitivity analysis. The combination of these methods evaluates an undersupply scenario. The stochastic model simulations allow a comparison between the known demand and the supply calculated under uncertainty, in order to identify the most important variables affecting lithium distribution. The dynamic simulations show that the most probable scenario is one where supply does not cover the increasing demand, and the stochastic modeling classifies the variables by their importance and sensibility. In conclusion, the most important variables in a scenario of EV undersupply are the lithium hydroxide produced from lithium carbonate, the lithium hydroxide produced from solid rock, and the production of traditional batteries. The global sensitivity analysis indicates that the critical variables which affect the uncertainty in EV production change with time.

The heap leaching of minerals is one of the more commonly used processes in the mining industry. This process has been modeled by many authors. However, the validation, verification, and implementation of these models are difficult since there is uncertainty about the operating conditions and the leaching model parameters. This work uses the uncertainty quantification, based on uncertainty and sensitivity analysis, for studying the model strength against uncertainties in heap leaching. The uncertainty analysis (UA) is used to quantify the effect of the magnitude of the uncertainties of the input variables on the recovery of heap leaching. Global sensitivity analysis (GSA) is used to study the nature of connections between the recovery and input variables of the leaching model. In addition, GSA facilitates the detection of whether a leaching model is over-parameterized. The information obtained allows studying some applications of the kinetic model. The Mellado et al. kinetic model is used as an example. The UA results indicate that the kinetic model can estimate the recovery behavior considering the full range of uncertainties of input variables. The GSA indicates that the kinetic model is over-parameterized on the uncertainties range considered; this conclusion contradicts the results when the local sensitivity analysis is used. However, the model shows a good correlation between the results of GSA and the kinetic behavior of heap leaching. In addition, the kinetic model presents versatility because it allows the determination of operating regions for heap leaching.

Response surface methodology has been applied in numerous studies using polynomial models despite some of them exhibit poor correlation coefficient $R^2$, which implies an incorrect optimization. In this work, a methodology for obtaining response surfaces when a small data set is available and its behavior is complex is presented. The methodology consists of four steps. First, the classic experimental design is used for obtaining a data set. Second, using the experimental results from a design of experiment, the classical kriging is employed for estimating properties at unsampled locations. Third, a response surface is obtained by training an artificial neural network using the kriging-estimated data set, and a hybrid algorithm based on differential evolution and backpropagation algorithms. The verification of the model is accomplished with the experimental data set. Fourth, the uncertainty quantification is utilized for studying the behavior of the response against uncertainties, which guarantee the robustness of the model developed. The methodology was applied to three cases, considering verification, validation, and uncertainty quantification. The results indicate that the proposed methodology is robust, and it provides more stable response surfaces than the approaches commonly used for polynomials and artificial neural networks. As a result, better optimal conditions are attained.

The design of a flotation circuit based on optimization techniques requires a superstructure for representing a set of alternatives, a mathematical model for modeling the alternatives, and an optimization technique for solving the problem. The optimization techniques are classified into exact and approximate methods. The first has been widely used. However, the probability of finding an optimal solution decreases when the problem size increases. Genetic algorithms have been the approximate method used for designing flotation circuits when the studied problems were small. The Tabu-search algorithm (TSA) is an approximate method used for solving combinatorial optimization problems. This algorithm is an adaptive procedure that has the ability to employ many other methods. The TSA uses short-term memory to prevent the algorithm from being trapped in cycles. The TSA has many practical advantages but has not been used for designing flotation circuits. We propose using the TSA for solving the flotation circuit design problem. The TSA implemented in this work applies diversification and intensification strategies: diversification is used for exploring new regions, and intensification for exploring regions close to a good solution. Four cases were analyzed to demonstrate the applicability of the algorithm: different objective function, different mathematical models, and a benchmarking between TSA and Baron solver. The results indicate that the developed algorithm presents the ability to converge to a solution optimal or near optimal for a complex combination of requirements and constraints, whereas other methods do not. TSA and the Baron solver provide similar designs, but TSA is faster. We conclude that the developed TSA could be useful in the design of full-scale concentration circuits.