### Project

# Extraction processes in pump-mixers – Development of a coupled CFD and PBM simulation method to investigate the effects of various input conditions on the dispersion generation

The focus of this project period was the flow simulation of pump-mixers which are apparatuses used for combined dispersion generation and pumping in technical liquid-liquid extraction processes. Since the mass transfer rate between the two phases is related to the surface area of the dispersed droplets, the control of the Droplet Size Distribution (DSD) is of utmost interest for process optimization. Hence, the aim of this project period was the enhancement of simulation methods for an a priori prediction of the DSD. Special emphasis was put on the influence of the turbulence model and the droplet-droplet interaction. A parameter study in terms of various integral volume fractions of the dispersed phase as well as the rotational speed of the stirrer was performed.

## Project Details

### Project term

March 1, 2022โApril 30, 2023

### Affiliations

Ruhr University Bochum

### Institute

Chair of Hydraulic Fluid Machinery

### Principal Investigator

### Researchers

### Methods

A tank stirred by a Rushton turbine, which is a simplified version of a pump-mixer as shown in Figure 1, was considered. The fluid was a liquid-liquid dispersion consisting of paraffin oil droplets dispersed in a continuous water phase. A homogeneous Euler-Euler mixture approach was applied, which means that both phases are approximated as interpenetrating Eulerian phases. Simulations were performed with a pressure-based flow solver employing the Pressure-Implicit with Splitting of Operators (PISO) [1] scheme, which is a modified pressure-correction predictor-corrector step algorithm, whereby corrector steps are omitted, facilitated by small time steps and an additional deferred correction loop. Details can be found, e.g., in the publications by Issa [1] and Jasak [2] as well as in our recent publications by Hundshagen et al. [3] and Rave et al. [4,5]. 3D Computational Fluid Dynamics (CFD) is coupled to a Population Balance Model (PBM) for a prediction of the DSD. Transient simulations were conducted in the full 360ยฐ domain. A computational grid with 3.1 million cells was employed. Scale-Adaptive Simulations (SAS) with the model by Egorov & Menter [6] were performed to assess the effect of turbulence scale resolution on the DSD. Results were compared to the statistical Shear Stress Transport (SST) model by Menter [7]. Coalescence and breakup of the oil droplets were approximated on the sub-grid scale by the simple kernels by Coulaloglou & Tavlarides [8] and Hsia & Tavlarides [9]. Both kernels require a test-case specific model parameter calibration on experimental data. Experiments had been conducted by our project partners at the Chair of Separation Science and Technology and at the Laboratory of Reaction and Fluid Process Engineering, which are both located at the Technical University Kaiserslautern.

### Results

Our latest results have recently been published by Rave et al. [5] and the reader is kindly referred to the publication for further details beyond this report. In Figure 2, time-averaged characteristic diameters of the DSDs simulated with both the SAS and SST models are shown and compared to experimental measurement data. The mean droplet size is expressed in terms of the Sauter mean diameter ๐ฬ
_{32} The width of the distribution is expressed in terms of the interquantile range (IQR) being the difference between the percentiles ๐ฬ
_{90,3} and ๐ฬ
_{10,3}, which is represented by a vertical bar bounded by two horizontal caps. The corresponding DSDs belong to the two distinct probe locations ML-I and ML-T which were positioned near the impeller blade and the liquid surface as depicted in Figure 1b), respectively. In the experiments, both the ๐ฬ
_{32} and IQR increased with decreasing rotational speed *N* and increasing integral volume fraction *A*_{dis} at both measurement locations. The trends are captured by both the SST and SAS turbulence models. However, with an increasing distance to the calibration point of the kernel parameter set at *N* = 400 [rpm] and *A*_{dis} = 0.05, the difference to the experimental data becomes more pronounced. At all operating points, the characteristic diameters as well as the IQRs predicted by the SAS model are larger compared to the SST model. Both effects can be attributed to the notably lower turbulent energy dissipation rate (TEDR) predicted by the SAS model, which is a decisive input parameter in the breakup and coalescence kernels and causes an increase of the characteristic time scale for droplet breakup ๐_{b} for droplets of all sizes. An increase of ๐_{b} causes the droplets to remain larger than for the SST model, increasing the ๐ฬ
_{32} ย Additionally, an increased ๐_{b} causes a larger size range of droplets to be unaffected by droplet breakup, leading to a broader DSD in terms of a larger IQR compared to the SST model. Moreover, the DSDs evaluated at both measurement locations are closer to each other for the SAS model in comparison to the SST model which is in agreement with experimental data. This observation is traced back to an enhanced turbulent mixing by turbulent dispersion being present for the SAS that contributes to a more homogeneous field of characteristic droplet sizes in the tank.

In Figure 3, the ๐ฬ
_{32} field for both turbulence models is depicted. For the SST results, large blurred structures are identifiable in Figure 3a). In contrast, smaller scale turbulent fluctuations resolved by the SAS model are clearly visible in Figure 3b). Hence, on the one hand, the spatial variation of the ๐ฬ
_{32} field is significantly narrower for the SAS, leading to a more homogeneous field of droplet sizes throughout the tank and especially at the measurement locations ML-I and

ML-T, which is in line with the experimental observations. On the other hand, there is some evidence from experiments that the TEDR might be underestimated by the SAS model. Although this effect may be compensated by a recalibration of kernel parameters, further studies are planned to enhance the prediction of the TEDR.

### Discussion

๐ด_{๐๐๐ } = 0.05.

**Refrences**

[1] Issa, R. I. (1986). Solution of the implicitly discretised fluid flow equations by operator-splitting. Journal of Computational Physics, 62(1), 40โ65. DOI: 10.1016/0021-9991(86)90099-9.

[2] Jasak, H. (1996). Error Analysis and Estimation for the Finite Volume Method With Applications to Fluid Flows. Ph.D. thesis, Imperial College of Science, Technology and Medicine, London.

[3] Hundshagen, M., Rave, K., Nguyen, B., Popp, S., Hasse, C., Mansour, M., Thevenin, D., Skoda, R. (2022). Two-Phase Flow Simulations of Liquid/Gas Transport in Radial Centrifugal Pumps with Special Emphasis On the Transition From Bubbles to Adherent Gas Accumulations. ASME Journal of Fluids Engineering, 144(10), 101202. DOI: 10.1115/1.4054264.

[4] Rave, K., Hermes, M., Wirz, D., Hundshagen, M., Friebel, A., von Harbou, E., Bart, H.-J., Skoda, R. (2022). Experiments and fully transient coupled CFD-PBM 3D flow simulations of disperse liquid-liquid flow in a baffled stirred tank. Chemical Engineering Science, 253, 117518. DOI: 10.1016/j.ces.2022.117518.

[5] Rave, K., Hermes, M., Hundshagen, M., Skoda, R. (2023). Assessment of scale-adaptive turbulence modeling in coupled CFD-PBM 3D flow simulations of disperse liquid-liquid flow in a baffled stirred tank with particular emphasis on the dissipation rate. Chemical Engineering Science, 270, 118509. DOI: 10.1016/j.ces.2023.118509.

[6] Egorov, Y., Menter, F. R. (2008). Development and Application of SST-SAS Turbulence Model in the DESIDER Project. In: Peng, S.-H., Haase, W. (Eds.), Advances in Hybrid RANS-LES Modelling. Springer, Berlin, Heidelberg, 261โ270. DOI: 10.1007/978-3-540-77815-8_27.

[7] Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8), 1598โ1605. DOI: 10.2514/3.12149.

[8] Coulaloglou, C., Tavlarides, L. (1977). Description of interaction processes in agitated liquid-liquid dispersions. Chemical Engineering Science, 32(11), 1289โ1297. DOI: 10.1016/0009-2509(77)85023-9.

[9] Hsia, M., Tavlarides, L. (1983). Simulation analysis of drop breakage, coalescence and micromixing in liquid-liquid stirred tanks. Chemical Engineering Journal, 26(3), 189โ199. DOI: 10.1016/0300-9467(83)80014-8.

## Additional Project Information

DFG classification: 404-03 Fluid Mechanics

Software: OpenFOAM

Cluster: CLAIX

### Publications

Rave, K., Hermes, M., Hundshagen, M., Skoda, R. (2023). *Assessment of scale-adaptive turbulence modeling in coupled CFD-PBM 3D flow simulations of disperse liquid-liquid flow in a baffled stirred tank with particular emphasis on the dissipation rate.* Chemical Engineering Science, 270, 118509. DOI: 10.1016/j.ces.2023.118509.