(4f) Solution of Population Balance Equations for Wet Granulation and Combination with the FLUENT Software

Purpose: To compare different methodologies for solution of monovariate and bivariate population balance equation (PBE) models of the wet granulation processes. To combine PBE models with the computational fluid dynamics (CFD) codes. To simulate impact of the physically based kernels and non-ideal mixing on the product non-uniformity. Methods: A fundamental model of the wet granulation should capture and combine the following three key features of this complicated process: 1.Population balancing of growth and breakage of different granules; 2. Hydrodynamic modeling of the gas ? solid mixture flow; 3. Modeling of contact mechanics and granule formation. For pharmaceutical application, the models should predict both, the granule size and the granule composition and the multivariate PBE models must be solved. Rigorous models [7] for both, the monovariate and bivariate PBE models were developed and solved first [1]. The methodology was validated by comparison of linear models solutions (for constant kernels) with the analytical solutions. Then the rigorous models were combined with the FLUENT Granular Model [1, 2] to simulate impact of more realistic flow conditions in different granulators. The combined (rigorous PBE + CFD) models are computationally very expensive [1]. On the other hand, the direct quadrature method of moments (DQMOM) was recently formulated, validated and tested for several monovariate and multivariate cases [3,4] allowing drastic reduction of CPU time. Again, we first tested and validated the DQMOM methodology for solution of the wet granulation models with constant kernels. Then the physically based agglomeration kernels were employed [5,6]. Results: In our work the rigorous PBE models for homogeneous systems were developed and integrated using the VODE solver from the IMSL library. The number of adopted classes was 2000 for monovariate models and 7500 for bivariate models. Theoretical results for the homogeneous systems showed that non-uniform granule composition can be caused by the granule ? formation mechanism (kernel). The models using the physically based kernels also successfully predict some interesting phenomena observed experimentally, for example the induction (coating, wetting) period without granule growth for the binder solutions with low concentration of the active component. Solutions of the combined (FLUENT + rigorous PBE) models showed, that the non-uniform granule composition can be caused also by the non-ideal mixing in the granulator. However, only a limited number (15) of classes could be employed for the combined models. As already shown before [3,4], the DQMOM was excellent for the linear cases for both, monovariate (size) and bivarite (size and composition) wet granulation processes. Reasonably accurate results were obtained with only N = 2 nodal points, i.e. by solving 2N = 4 ordinary differential equations (ODE) for the monovariate case and 3N = 6 ODE for the bivariate case. The accuracy was not significantly worse even for strongly nonlinear and discontinuous (cut-off) kernels based on the kinetic theory of granular flow (KTGF) [1,5,6]. However, several numerical problems were observed during the testing simulations with DQMOM models. The main problem was to find acceptable initial conditions for the integration of the ODE systems and to avoid physically meaningless solutions (e.g. negative size or composition). For the discontinuos (cut-off) kernels there are additional numerical and theoretical problems related to unrealizable solutions for such cases and certain choice of moments. Conclusions: The implementation of the PBE models in CFD codes can be important for several practical cases. DQMOM is a very promising method for solution of multivariate PBE models of the wet granulation, particularly in combination with the CFD codes, when the employment of the rigorous or classes methods is still computationally intractable. The main challenge is the optimal choice of the moments, solution for the discontinuous (cut-off) kernels which are typical for the wet granulation processes and the moments interpretation when comparing with experiments.

References: [1] Rajniak, P., & Chern, R.: Mathematical modeling of wet granulation: Combination of population balances and FLUENT granular model. Poster No.: T3160, AAPS Annual Meeting, Baltimore, 2004. [2] FLUENT 6.1. User's Manual, 2003. [3] Fan, R., Marchisio, D.L., & Fox, R.O.: Application of the direct quadrature method of moments to polydispers gas-solid fluidized beds. Powder Technology 139 (2004) 7-20. [4] Marchisio, D.L., & Fox, R.O.: Solution of PBE using the DQMOM. Aerosol Science 36 (2005) 43 - 73. [5] Stepanek, F., Rajniak, P., Chern, R., & Mancinelli, C.: Modeling of multi-component granule formation in a wet granulation process. WCPT 2006, Orlando. [6] Liu, L.X., Litster, J.D., Iveson, S.M., & Ennis, B.J.: Coalescence of deformable granules in wet granulation processes. AIChE J. 46 (2000) 529-539. [7] Vanni, M.: Approximate PBE for aggregation-breakage processes. J. Colloid Interface Sci. 221 (2000) 143 ? 160.