One-Group Reduced Population Balance Model for CFD Simulation of a Pilot-Plant Extraction Column Christian Drumm, † Menwer Attarakih, †,‡ Mark W. Hlawitschka, † and Hans-Jo ¨rg Bart* ,† Lehrstuhl fu ¨r Thermische Verfahrenstechnik and Center for Mathematical and Computational Modeling, TU Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany, and Chemical Engineering Department, Faculty of Engineering Technology, Al-Balqa Applied UniVersity, P.O. Box 15008, 11134-Amman, Jordan In this work, a one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM) developed recently by Attarakih et al. (In Proceedings of the 19th European Symposium on Computer Aided Process Engineering, ESCAPE-19, Cracow, Poland, June 14-17, 2009; Jezowski, J., Thullie, J., Eds.; Elsevier: New York, 2009; ISBN-13: 978-0-444-53433-0) is implemented in the commercial computational fluid dynamics (CFD) package FLUENT 6.3 for solving the population balance equation in a combined CFD-population balance model (PBM). The one-group reduced population balance conserves the total number (N) and volume (R) concentrations of the population by directly solving two transport equations for N and R and provides a one-quadrature point for closing the unclosed integrals in the population balance equation. Unlike the published two-equation models, the present method offers accuracy improvement and internal consistency (with respect to the continuous population balance equation) by increasing the number of primary particles (sections). The one-group reduced population balance provides the possibility of a one-equation model for the solution of the PBM in CFD based on the mathematically consistent d 30 instead of the classical d 32 mean droplet diameter. Droplet breakage and coalescence are considered in the PBM, which is coupled to the fluid dynamics in order to describe real droplet behavior in a stirred liquid-liquid extraction column. As a case study, a full pilot-plant extraction column of a rotating disk contactor (RDC) type consisting of 50 compartments was simulated with the new model. The predicted results for the mean droplet diameter and the dispersed-phase volume fraction (holdup) agree well with literature data. The results show that the new CFD-PBM model is very efficient from a computational point of view (a factor of 2 less than the QMOM and a factor of 5 less than the method of classes). This is because the one-group reduced population balance requires the solution of only one equation (the total number concentration) when coupled to the CFD solver. It is therefore suitable for fast and efficient simulations of small-scale devices and even large-scale industrial processes. 1. Introduction Computational fluid dynamics (CFD) has been used with great success in multiphase flow applications arising in chemical engineering such as bubble and liquid-liquid extraction columns. 1-8 This allows for the prediction of detailed hydro- dynamics and turbulence characteristics using real equipment geometry. One widely used multiphase model for CFD simula- tions is the Eulerian multiphase model, which can describe dispersed multiphase flow and accounts for interactions between the dispersed continuous phases. 1,7,8 In Euler-Euler multiphase flow, a two-fluid model is usually applied with a constant size for the bubbles, droplets, or particles in the dispersed phase. In reality, a wide particle size distribution can exist in the apparatus due to particle growth, aggregation, or breakage resulting from either mass transfer (particle growth) or interactions between the moving particles and the turbulent continuous phase or/and internal equipment geometry. 9-11 Accordingly, this distribution of sizes results in different particle velocities with respect to size. Therefore, the constant-particle- size approach cannot describe the hydrodynamic behavior and, often, the coupled hydrodynamics and mass transfer of the dispersed phase. In fact, when coupling CFD and population balance models, many authors have shown the strong depen- dence of the bubble and droplet relative velocities on their diameters (e.g., Bhole et al. 1 and Drumm et al. 3 ). Moreover, in reality, the basic purpose of bubble or liquid extraction columns is to carry out mass-transfer operations. Because the local mass- transfer coefficients (in both the continuous and dispersed phases) depend on the droplet or bubble diameters, 11,12 there is no a priori justification for the assumption of equal rise velocities. 1 As a possible solution, multifluid models 6-8 divide the dispersed phase into classes of different sizes where each class represents one fluid. In this way, the model can predict size-dependent velocities because each class size moves with its own velocity field. Such an approach is more realistic, but unfortunately, it demands a high computational cost and hence a long CPU time. Moreover, it ignores the basic natural phenomena leading to particle size distributions such as breakage and coalescence. On the other hand, population balance modeling (PBM) takes into account particle growth, aggregation, breakage and nucle- ation and accommodates the dependence of velocity and mass transfer on particle size. This actually gives a full description of the dispersed phase with a result of an infinite number of partial differential equations due to the continuous variation of the particle internal properties (for example, the particles size ranges mathematically from 0 to ∞). Because of this dramatic increase in the computational load imposed by coupling the population balance equation to commercial CFD packages, there exist many numerical methods in the literature as attempts to reduce and solve the PBE in a reasonable computational time. * To whom correspondence should be addressed. Tel.: +496312052- 414. E-mail: bart@mv.uni-kl.de. † TU Kaiserslautern. ‡ Al-Balqa Applied University. Ind. Eng. Chem. Res. 2010, 49, 3442–3451 3442 10.1021/ie901411e  2010 American Chemical Society Published on Web 02/23/2010