(339h) Numerical Simulation of Spatially Distributed Population Balances

Population balance models have widely been used to model various crystallization systems, for instance reactive crystallization in emulsion droplets [1,2,3]. However, these models do not consider spatial variations of solute concentrations and spatial particle distributions. Recently, we have proposed a model for emulsion-assisted precipitation which overcomes this drawback [4]. Besides mass balances, a (1+1)-dimensional population balance equation (PBE) is used. The first coordinate is particle size, the second is position along the radial direction of a spherical emulsion droplet.

For the solution of the (1+1)-dimensional PBE the Finite Volume Method has been employed. Higher order schemes were implemented to evaluate the fluxes at the cell boundaries. It is well known that such schemes produce spurious oscillations - especially when the solution shows steep gradients. With a Koren flux limiter function these oscillations were suppressed. Then a total variation diminishing (TVD) numerical scheme is obtained. From the literature it is known that many flux limiter functions are available guaranteeing TVD. This contribution compares different flux limiter methods for selected test cases.

Instead of utilizing flux limiter functions, oscillations can also be suppressed with Essentially Non-Oscillating (ENO) schemes. Instead of approximating the flux at the cell boundary with a fixed set of neighboring grid points (stencil), the flux at a given cell boundary is evaluated on different stencils. From these stencils the one is selected on which the fluctuation of the state function is minimal. A further development of this method are Weighted Essentially Non-Oscillating (WENO) schemes. With this method, the full set of stencils is used but with different weights according to their magnitude of fluctuation. Our poster compares the WENO schemes against flux limiter methods with respect to accuracy and computational effort for some test cases and the 2-D emulsion crystallization PBE model.

[1] Niemann, B., Rauscher, F., Adityawarman, D. Voigt, A. Sundmacher, K., ?Microemulsion-assisted precipitation of particles: Experimental and model-based process analysis?, Chemical Engineering and Processing 45 (2006), 917-935.

[2] Weiss, C. Hennig. T. Kümmel, R., Tschernjaew, J., ?Modeling Mass Transfer and Crystallization in Disperse Systems?, Chemical Engineering Technology 23 (2006), 485-488.

[3] Feltham, D. L., Garside, J., ?A mathematical model of crystallization in an emulsion?, The Journal of Chemical Physics 122 (2005), 174910.

[4] Fricke, M., Sundmacher, K., ?Model-Based Analysis of an Emulsion-Assisted Precipitation Process?, Industrial and Engineering Chemistry Research, submitted.