H.‐J. Bart

Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

Abstract The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.

Mass transfer into droplets undergoing reactive extraction

Abstract The mass transfer into droplets in metal extraction is investigated in the system ZnSO 4 –H 2 SO 4 /di(2ethylhexyl)phosphoric acid in isododecane. Results of equilibrium measurements and stirred cell experiments are described in a mathematical model based on gradients of chemical potential. Concentration profiles measured with Laser Induced Fluorescence technology were calculated with the Maxwell–Stefan equations. Diffusion coefficients were estimated using the Wilke and Chang equation. Additional eddy diffusion coefficients measured in single droplet experiments in a venturi tube are described with different models known from literature. Whereas equilibria and mass transfer into droplets in stagnant conditions can be calculated with good agreement with the experimental results, none of the known models describes eddy diffusion in reactive extraction occurring in free moving droplets sufficiently well over a broad concentration range.

Predicting diffusivities in liquids by the group contribution method

Abstract The group contribution method to predict diffusivities in binary liquid systems is compared with other well-known semi-empirical correlations, such as Wilke–Chang, Hayduk–Minhas and Tyn–Calus, by means of literature data including also systems of commercial interest, e.g. sulfolane–aromatics systems. This comparison indicates that the group contribution method is superior to the other correlations investigated. Furthermore, the prediction of diffusivities with the group contribution method is explained step-by-step by means of several examples.

ON A HIGH-RESOLUTION GODUNOV METHOD FOR A CFD-PBM COUPLED MODEL OF TWO-PHASE FLOW IN LIQUID-LIQUID EXTRACTION COLUMNS

This paper is about the numerical solutions for a computational fluid dynamics-population balance model (CFD-PBM) coupled model of two-phase flow in a liquid-liquid extraction column. The model accounts for a complete description between both the dispersed and continuous phases, and constitutes a hyperbolic system of equations having a linearly degenerate nature. A numerical algorithm based on operator splitting approach for the numerical solution of the model is used. The homogeneous part is solved using the TVD MUSCL-Hancock scheme. Numerical results are presented, demonstrating the accuracy of the proposed methods and in particular, the accurate numerical description of the flow in the vicinity of the contact discontinuities.