Liquid-liquid chromatography techniques [1], [2], [3], [4], [5], [6], [7], commonly known as countercurrent chromatography (CCC) [8], [9], [10], [11], [12], [13] and / or centrifugal partition chromatography (CPC) [14], [15], [16], [17], are based on centrifugal devices and primarily used for the analytical and preparative separations of natural products and pharmaceutical substances. [17], [18], [19], [20], [21], [22], [23], [24]. The performance of centrifugal chromatography instruments, due to their relatively small size (up to 25 liters capacity), is too low compared to extraction apparatuses, and they cannot be used in large-scale industrial processes, in particular in hydrometallurgy, where thousands of liters of metal solutions must be processed. Liquid–liquid extraction technologies based on a cascade of mixer–settler extractors are widely used in hydrometallurgy in separation of metals, in particular in separation of rare-earth metals [25]. The performance (productivity) of mixer–settler extractors is several orders of magnitude higher than that of CCC devices. Although preparative CCC devices are available on the market, the increasing complexity of large chromatographs limits the size of CCC devices, and a significant increase in the size of commercial CCC equipment is not possible. Thus, the disadvantage of CCC separation methods for applications in hydrometallurgy is their low productivity (performance). However, countercurrent chromatography methods have significant advantages over countercurrent extraction methods: separation of multicomponent mixtures in one technological stage, ensuring high purity of the products obtained, and low consumption of reagents and organic solvents. In addition, they are more environmentally friendly. Therefore, combining the advantages of both methods to create industrial-scale extraction-chromatographic separations based on a cascade of mixer-settler extractors provides new opportunities for CCC applications and represents a challenging and promising task, especially for hydrometallurgy. For simulation, optimal design and practical implementation of different operating modes of such separations, preliminary mathematical modeling is necessary. To determine the optimal design and/or operating mode to solve a given separation problem by experimental trial and error approach is time-consuming. Due to the absence of packing materials, the mathematical modeling of CCC separations is less complicated than that of liquid-solid chromatography, which makes it possible to find analytical expressions for describing linear chromatographic separations. To develop mathematical descriptions of the considered extraction-chromatographic separations, two approaches can be used: precisely description of solutes distributions in the cascade of mixer-settler extractors based on the equilibrium cell model [12]; approximate solution of the model equations based on the Gaussian distribution [26,27].

The cell model replicates the actual physical picture of the chromatographic separation processes in the cascade of mixer-settler extractors, since due to the intense contact of the mobile and stationary phases in the mixing chambers and the recirculation of the stationary phase between the mixing and separation chambers (to retain it in the extractors), each extractor in the cascade can be considered as a theoretical plate (equilibrium cell). The discrepancy between the above descriptions of separation processes depends on the value of two operating parameters: the number of the extractors in the cascade (the number of equilibrium cells, N) and the loaded volume of the solution of the components to be separated (sample volume). The discrepancy between the results of exact and approximate calculations increases with an increase in the sample size and a decrease in the number of equilibrium cells. As applied to analytical and preparative separations, when relatively small sample volumes are loaded and the efficiency of chromatographic columns N > 200, both approaches give identical results, and much simpler equations of approximate solutions can be used to simulate different versions of CCC separation. In the case of industrial extraction-chromatographic separations, large volumes of a solution of components should be loaded into the cascade to ensure the required performance, and the number of extractors in the cascade, as a rule, is N ≤ 50 - 100. As will be shown below, the cell model equations should be used to simulate these separations.

An active study of extraction processes is associated with growing demand for valuable metals of high purity. The main way to search for effective processing strategies is the study of new extractants. In this regard, the topics of deep eutectic solvents are being actively developed. They have already found their application of both the processing of samarium-cobalt magnets [28] and other important areas [29], allowing to realize the complete separation schemes [30]. However, at the moment, there are not so many works devoted to hardware techniques that make it possible to effectively realize the potential of modern extraction systems. In particular, resource-saving solutions are especially important to scale the extraction process. One of these solutions is a highly effective liquid-liquid chromatography.

Countercurrent chromatography has a variety of operating modes that can significantly improve chromatographic separations compared to the conventional isocratic batch injection mode [31]. In this study, we will compare accurate and approximate modeling of analytical, preparative and industrial CCC separations for two of the simpler modes of operation: conventional elution and closed-loop recycling modes.

Processes of analytical, preparative and industrial CCC separations differ in the conditions and volumes of the loaded solution of components and hardware design. Analytical and preparative CCC separations, as a rule, are based on centrifugal devices; relatively small sample volumes are on impulse injected into the chromatographic column using sample loops. Industrial CCC separations for hydrometallurgy could be based on a cascade of mixer-settler extractors with a semi-continuous loading of a solution of metals to be separated.