Compartment-model for the simulation of the separation performance of stirred liquid – liquid-extraction columns

For detailed simulation and evaluation of stirred extraction columns a CFD based compartment-model was developed. Instead of simulating all effects in a computational expensive PBE-CFD-model, the velocity field calculation of the continuous phase is decoupled from the calculation of the dispersed phase (one-way coupling). In CFD only the continuous phase is simulated and the resulting velocity profile is used in the compartment-model to simulate the drop movement, coalescence, breakage and mass transfer for a representative number of drops (Monte-Carlo Method). This decoupling has a major impact on the calculated fluid-dynamics. Thus, the velocity profile of the CFD results is modified in the model to account for phase interaction. The compartment-model is applied for the simulation of a Kühni extraction column with the system toluene/water/acetone. The simulation results, namely holdup, drop size and concentration profiles over the column height, are in good agreement with experiments for different loads and different stirrer speeds.


| INTRODUCTION
For a model based design of liquid-liquid-extraction columns the movement of the drops as well as drop coalescence, breakage and mass transfer should be calculated. Especially models, which incorporate population balance equations (PBE), are capable of tracking the drop-property-distribution along the column height. PBE-based simulation tools for liquid-liquid-extraction columns can be classified in three approaches: • One-dimensional simulation tools, which only describe the drop properties along the column height, such as ReDrop or PBELab. [1][2][3][4] • CFD-models (computational fluid dynamics) with PBE, which calculate the fluid dynamics and mass transfer in an either two-or three-dimensional simulation domain. [5][6][7][8][9][10][11][12][13][14] • Compartment-models, which decouple the fluid dynamic simulation from the calculation of the drop-property-distribution of the PBE. 15,16 The advantage of one-dimensional models is the fast simulation of extraction columns. The model requires sub-models, which take the influence of the column geometry and energy input on drop movement, coalescence, breakage and mass transfer into account. Especially in stirred extraction-columns the retardation/slowing of drops caused by the stirrer-vortexes requires appropriate sub-models. 17,18 However, such sub-models are only applicable for -column-designs, for which the correlations were developed.
The influence of novel column designs on the performance of extraction columns can be determined with CFD-models in combination with PBE. The coupled interaction of all phenomena can be modeled in detail. Nevertheless, each independent property, which has to be considered in the PBE, increases the dimensionality of the PBE and therefore the computational effort. 19 For example Attarakih et al. 3 simulated an extraction column type RDC (rotating disc contactor) with a two-dimensional CFD-PBE model and required 4.63 days (without mesh refinement) to 27 days (with mesh refinement) to simulate the steady state operation point. The simulation with a three-dimensional CFD-PBE model would require even more simulation time. When simulating multiple mass-transfer components or reactions, which have to be included into the population balance, the simulation will become even more expensive.
Therefore, a hybrid model-approach was developed by Weber et al 16  with novel designs can be predicted, without the need to develop new sub-models for example, for the slowing factor which describes the retardation of the drops' rise velocity due to stirrer induced vortexes. Furthermore, the local energy input is calculated by the CFDsimulation. Consequently, drop breakage can be calculated according to the local turbulent dissipation rate, which is highest at the stirrer tip. The local turbulent dissipation rate is typically not used in onedimensional simulation models, which apply an averaged energy input for the breakage models. 10 Similar compartment-models were developed, for example, for homogeneous reactors, 20,21 loop-reactors, 15,22 crystallizers, 23 and bubble columns. 24,25 Weber et al 16 simulated an extraction column type Kühni with a compartment-model. The model is capable to simulate the fluid dynamic in the column. However, mass transfer was not considered and the applicability for different energy inputs was not shown. Furthermore, fitting parameters were required to calculate the coalescence in the dense packed layer under stators. Therefore, the model was further improved by including sub-models for mass transfer and simplifying the modeling of the coalescence below the stators.
In the following, the improved compartment-model is presented.
With the model the fluid dynamics and mass transfer of the extraction system toluene/water/acetone is simulated in a Kühni extraction column. The model is used to calculate various column loads and energy inputs. At the beginning of the article, we will present the simulated extraction column and the properties of the extraction system. The CFD-model is explained briefly and following the compartmentmodel, the implemented sub-models and the simulation setup are

| Adaption of the continuous phase velocity
Since the CFD-simulation neglects the disperse phase, the coupling between the continuous and disperse phase is principally one-way.
However, it is necessary to modify the continuous phase velocity Equation (2) is applied only in the areas under the stators, the so called coalescence regimes (see Figure 3). The height coordinate of the drops which enter these areas is randomly distributed so that a packed layer is formed with a holdup of approximately 74% (gray area in Figure 3). This represents the holdup value for the case of the clos- The reduced area A red is calculated according to the local holdup.
When applying Equation (4) in Equation (3) and with v rel = v c − v d the modified continuous phase velocity can be calculated.
For Equation (5)  Coalescence is determined with random numbers between 0 and 1.
When the random number is smaller than the coalescence probability, For τ c the model of Henschke 1 is applied, which was developed for liquid-liquid extraction columns.
Here ξ represents the coalescence parameter, which needs to be

| Modeling of mass transfer
The mass transfer from continuous into disperse phase is calculated according to Equation (8) where Δt is the time step, ρ d and ρ c the disperse and continuous with D d as the diffusion coefficient of the dispersed phase, C IP the instability parameter, μ d the dispersed phase viscosity and v rel the relative velocity between drop and continuous phase. The instability parameter was determined by Henschke 1 with single drop experiments to 9,445.

| Simulation plan
In this study, all column experiments of Garthe 17 are simulated with the extraction system toluene/water/acetone, which were performed in a Kühni extractor. Therefore, the compartment-model is applied for a different extraction system compared to the previous publication of Weber et al. 16 Furthermore, the accuracy of the compartment-model to simulate operation points with two different energy inputs (stirrer speeds) is determined. Finally, the measured concentration profiles are compared to the simulated concentration profiles. In Table 2 the simulated operation points are summarized. Garthe 17 explained that his measured operation points with the highest load were close to the flooding point. Therefore, in Table 2  Experimental and simulated operation points for the DN80 extraction column type Kühni for the extraction system toluene/water/acetone

| RESULTS AND DISCUSSION
In the following sections, the simulation results are compared with the experimentally measured holdup, sauter drop-diameter and concentration profiles. In the end, the compartment model is compared to one-dimensional simulation results of the same data set, which were published by Weber et al. 4

| Fluid dynamic in DN80 extraction column
In the compartment-model

| MASS TRANSFER
After having discussed the simulated fluid dynamic behavior, the concentration profiles are exemplarily compared to the measured profiles in Figure 9 for

| CONCLUSION
We developed a CFD based compartment-model, including PBE (Monte-Carlo approach) and sub-models for the drop movement, coalescence, breakage and mass transfer. The compartment-model requires the velocity profile of the continuous phase, which is calculated in CFD.
The dispersed phase is neglected in the CFD-simulation. However, the