ColHySE: An advanced column hydrodynamic-based model for solvent extraction

Highlights

• A novel PBM-based model is introduced to simulate extraction column hydrodynamics.

• Interfacial area and holdup are predicted from the turbulent energy distribution.

• The sensitivity of the results to the empirical parameters of the model is studied.

• The effect of geometry and operating parameters is well reflected by the model.

Abstract

An original one dimensional population balance model (PBM)-based model of liquid–liquid extraction columns is reported. Compared to existing simulators, ColHySE implements a more realistic description of the flow patterns in the contactor, and predicts its effect on the local droplet–droplet interactions (i.e. breakage and coalescence rates). Proper turbulent properties, extracted from single-phase flow CFD simulations, are used in the source terms of the PBM to evaluate locally the inhomogeneous breakage and coalescence rates, using the averaged Coulaloglou and Tavlarides kernels (Castellano et al., 2018). The sensitivity of the predicted droplets mean diameter, d_32, and the holdup, φ, to the parameters of the used empirical and phenomenological models, on the one hand, and to the operating conditions of the column, on the other hand, was studied. Although some model parts must be refined, and an experimental validation remains necessary, the results confirm that the 1D-PBM methodology used in ColHySE is relevant for predicting the interfacial area in the pulsed column as a function of the operating conditions and geometry, hence highlighting its relevance to study the hydrodynamic stability and tendency to flooding. The sensitivity analysis has moreover highlighted the needs for an improved slip velocity model.

Introduction

Various types of contactors can be used to implement solvent extractions: extraction columns, mixer-settlers and centrifugal contactors. In the case of columns, where the two liquid phases flow counter-currently, the extraction proceeds continuously along the axial direction. Agitated columns, as Rotating Discs Columns (e.g. Kühni or Batman) exhibit a relative flexibility towards the solvent properties or the feed concentration, and are preferred in hydrometallurgy, whereas pulsed column, packed with discs and doughnuts (DD) or with perforated plates internals, remain the most preferred contactors in nuclear applications, due to their robustness and low maintenance requirements (Godfrey and Slater, 1994). The typical geometries and operating conditions of these different contactors strongly influence the concentration of interface area between the two liquid phases and hence affect the extraction efficiency. While a large interfacial area favors the extraction, reducing too much the size of the droplets can lead to flooding and other stability and phase separation or phase entrainment issues, which are highly detrimental to the column performances. It is hence of great interest for process design, optimization purposes, and the selection of the appropriate technology for a given liquid–liquid extraction problem, to be able to predict the interfacial area and its evolutions across scales. In pulsed columns, the interfacial area between the two non-miscible liquids results from the joint effects of the packing geometry (e.g. the free area and the spacing between the packing elements) and the pulsation intensity.

Accurate prediction of the drop size distribution (DSD) and the dispersed phase volume fraction in the column (or the holdup, φ), as a function of its geometry and operating conditions is hence required. Empirical correlations are available in the literature to estimate these properties (Boyadzhiev and Spassov, 1982; Kumar and Hartland, 1996; Pacek et al., 2005) but they are only valid within the range of operating conditions in which they were established, and their extrapolation is hazardous. Better predictions can be obtained by implementing a Population Balance Equation model (PBE) that requires the computation of the droplet breakage and coalescence rates used as source terms. Since the prior work of Casamatta (1981), many works have been devoted to the modeling of liquid–liquid extraction columns using PBE, either based on class method (Al Khani et al., 1989; Attarakih et al., 2017), or on the method of moments (Attarakih et al., 2015). In each case, the column is considered as a 1D axial domain along which the PBE is solved, and in most of them, if not all, the turbulent breakage and coalescence kernels are based on the inertial subrange of the Kolmogorov theory (Batchelor, 1982), and are only dependent on the turbulent kinetic energy dissipation rate ε. They moreover suppose uniform source terms, which means that a uniform turbulent dissipation rate is assumed to prevail in the column. It is however well known that ε is far from being uniform in most liquid–liquid contactors and that this assumption, despite the obvious simplification it brings, can lead to significant uncertainties in the performance predictions across scales. It is now generally accepted that a significant improvement in the predictive character of solvent extraction models cannot be obtained without a better description of the flow in the apparatus, and of its couplings with the population of droplets.

A large number of fluid dynamic studies have therefore been dedicated in recent years to the simulation of flows in extraction columns. Similarly, thanks to increasing numerical resources, coupled CFD-PBE simulations are becoming more easily applicable. The latter are usually performed using a RANS approach, which is particularly convenient when dealing with droplets breakage and coalescence, as it directly gives the turbulent dissipation rate. In these coupled approaches, the CFD code, usually ANSYS- FLUENT(Amokrane et al., 2016) or OpenFOAM (Li et al., 2017), provides the local flow field variables and the energy dissipation rate, while the PBE is solved in each cell of the computational domain.

In a previous study Amokrane et al. (2016), predicted the discrete phase holdup and droplet mean diameter in a 25 mm inner diameter DD column. They solved the PBE using the Quadrature Method Of Moments (QMOM) algorithm (Marchisio et al., 2003), considering the Coulaloglou and Tavlarides kernels as source terms (Coulaloglou and Tavlarides, 1977). The four kernels parameters had been identified beforehand using a first series with in situ DSD and hold-up measurements. More recently (Alzyod et al. (2018), developed a coupled CFD-PBE model to predict the extraction performances of pulsed sieve plate columns, i.e. including the mass transfer problem. As in Amokrane et al. (2016), the breakage and coalescence model parameters had to be evaluated in advance, in this case by simulations. While being promising, CFD-PBE simulations require heavy multicore equipment and are therefore not suitable for simulating an entire extraction column. Advanced modeling approaches in chemical engineering are still needed to predict the sensitivity of extraction columns functioning towards geometrical and operating parameters (Hlawitschka et al., 2016). This is the aim of the original model ColHySE introduced in this work.

Previous work dedicated to emulsification in closed stirred tanks (Castellano et al., 2018) demonstrated that the Sauter mean diameter of the dispersed phase, and its evolution with the stirring rate, and nature of the dispersed phase, can be efficiently predicted by a 0D simulation. Indeed, using the steady-state probability density function f(ε) of the turbulent dissipation rate in the closed vessel, the prediction of the mean diameter can be conveniently decoupled from the 3D hydrodynamic description of the contactor. This methodology, based on the computation of volume-averaged coalescence and breakage kernels has enabled to identify a set of empirical parameters which, although not universal, are more robust to changes in the flow properties. It is here extended to the case of a 1D contactor, where (i) the flow is periodically varying with time, and (ii) along which the dispersed phase properties and holdup are not uniform, nor constant over time. By this way, relation with fluid dynamic simulation results is ensured but in a simplified 1-D hydrodynamic model, which ensures faster computation, while accounting for the main turbulence effects on the dispersed phase properties to be predicted, without systematic adjustment of the parameters.

The study is hydrodynamic only, i.e. considers the evolution of the size of droplets and the holdup, but not mass transfer. It is based on the 25 mm inner diameter pulsed liquid–liquid extraction column for which some experimental values of holdup and Sauter mean diameter have already been published (Amokrane et al., 2016). The founding principles and the structure of the model are described in Section 2, giving the details of the equations solved in the 1D domain. In Section 3, we discuss the impact of the time-step used when averaging the pdf(ε) in the periodical pulsed flow. An analysis of the sensitivity of the predicted interfacial area to the adjustable parameters of the closure models, including the 4 parameters of the breakage and coalescence kernels, is also given. The aim is to highlight the weight of the different interactions between the phases and to point out which closure models require special attention or additional developments. At last in Section 4, we perform a sensitivity analysis of the dispersed phase properties to the column operating parameters, to highlight the great flexibility of the proposed advanced hydrodynamic description for the prediction of the interfacial area, which is of prime importance for solvent extraction operations.