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In this paper, we focus on providing a discrete formulation for a reduced aggregation population balance equation. The new formulation is simpler, easier to code, and adaptable to any type of grid. The presented method is extended to address a mixed-suspension mixed-product removal (MSMPR) system where aggregation and nucleation are the primary mechanisms that affect particle characteristics (or distributions). The performance of the proposed formulation is checked and verified against the cell average technique using both gelling and non gelling kernels. The testing is carried out on two benchmarking applications, namely batch and MSMPR systems. The new technique is shown to be computationally less expensive (approximately 40%) and predict numerical results with higher precision even on a coarser grid. Even with a revised grid, the new approach tends to outperform the cell average technique while requiring less computational effort. Thus the new approach can be easily adapted to model the crystallization process arising in pharmaceutical sciences and chemical engineering.

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The application of multi-dimensional population balance equations (PBEs) for the simulation of granulation processes is recommended due to the multi-component system. Irrespective of the application area, numerical scheme selection for solving multi-dimensional PBEs is driven by the accuracy in (size) number density prediction alone. However, mixing the components, i.e., the particles (excipients and API) and the binding liquid, plays a crucial role in predicting the granule compositional distribution during the pharmaceutical granulation. A numerical scheme should, therefore, be able to predict this accurately. Here, we compare the cell average technique (CAT) and finite volume scheme (FVS) in terms of their accuracy and applicability in predicting the mixing state. To quantify the degree of mixing in the system, the sum-square χ2 parameter is studied to observe the deviation in the amount binder from its average. It has been illustrated that the accurate prediction of integral moments computed by the FVS leads to an inaccurate prediction of the χ2 parameter for a bicomponent population balance equation. Moreover, the cell average technique (CAT) predicts the moments with moderate accuracy; however, it computes the mixing of components χ2 parameter with higher precision than the finite volume scheme. The numerical testing is performed for some benchmarking kernels corresponding to which the analytical solutions are available in the literature. It will be also shown that both numerical methods equally well predict the average size of the particles formed in the system; however, the finite volume scheme takes less time to compute these results.

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In this study, a complete two dimensional (internal coordinates) population balance model (2D-PBM) is developed, calibrated and validated as a predictive tool for predicting the particle size and the liquid content distribution of the granules produced from twin screw granulation (TSG). The model is calibrated and validated using experimental distributions for the two internal coordinates that are captured using image processing. Granulation runs are conducted at multiple liquid to solid (L/S) ratios and liquid binder viscosities, and then used to calibrate and validate the 2D-PBM. The mathematical model accounts for aggregation and breakage of the particles occurring in three zones of the TSG with inhomogeneous screw configurations (2 conveying zones and 1 kneading zone). A Madec aggregation kernel, and a linear breakage selection function are used in the 2D-PBM and finite volume numerical approximation is used for solving the model. The calibrated model shows that the aggregation rate in the conveying elements is higher than in the kneading elements while the breakage rate in the kneading elements is much higher than in the conveying elements. Also, the increase in L/S ratio and liquid viscosity leads to higher aggregation rates and lower breakage rates.

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In this study, a compartmental population balance model (CPBM) is developed as a predictive tool of particle size distribution (PSD) for wet granulation in co-rotating twin-screw granulator (TSG). This model is derived in terms of liquid to solid ratio (L/S) and screw speed representing the main process parameters of the TSG. The mathematical model accounts for aggregation and breakage of the particles occurring in five compartments of the TSG with inhomogeneous screw configurations (3 conveying zones and 2 kneading zones). Kapur's aggregation kernel is implemented in granulation and finite volume numerical method is adapted for solving the mathematical model. The results show a dramatic improvement in solution accuracy compared to the cell average numerical method. Moreover, Kriging interpolation is used to interpolate for new values of empirical parameters at different L/S and screw speeds. Finally, the CPBM model is calibrated and validated using the experimental data.

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In this paper, a new mass-based numerical method is developed using the notion of Forestier-Coste & Mancini (Forestier-Coste & Mancini 2012, SIAM J. Sci. Comput. 34, B840-B860. (doi:10.1137/110847998)) for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method.

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Two-dimensional population balance model (PBM) is developed in order to model pharmaceutical granules formation in a twin-screw wet granulator. Granule size and liquid content are considered as internal coordinates, while axial length of granulator is considered as external coordinate. Two types of initial liquid distribution are considered for the model development, i.e. constant and linear distributions. The main focus is on modeling and validation of liquid content distribution of granules. Regime-separated approach was used in order to capture the non-homogeneity of the granulator. The plug flow regime is considered for the conveying zone, while well-mixed regime is assumed for the kneading zone of twin-screw granulator. Aggregation and breakage are considered as the main mechanisms for granule formation and size control. Cell average method is used for solution of the PBM based on lumped parameter approach. In order to determine experimentally the distribution of liquid, liquid binder by dye addition was used in the process. The model findings are calibrated and validated by comparing with measured liquid content in each size fraction. The measured data is collected on a 12 mm twin-screw wet granulator using microcrystalline cellulose (MCC) and water soluble dye plus water as binder. The model indicated to be valid for MCC and needs to be validated with further excipients. The results revealed that increasing screw speed led to more uniform liquid distribution. Finally, the model findings indicated that 2D PBM is capable of predicting liquid distribution, and can be used as predictive tool in pharmaceutical continuous granulation.

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