(358a) A Mechanistic Kernel for the Aggregation Phenomenon in Population Balance Models of Granulation Processes

Granulation is the generic term for particle size enlargement processes. It is the process of agglomerating powders (primary particles) to form larger, semi-permanent aggregates by spraying a liquid (binder) onto the primary particles as they are agitated in a fluidized bed, high shear mixer or any other similar device. Despite its widespread use, economic importance and many years of research, granulation processes are highly inefficient and current industrial plants often operate with high recycle ratios (e.g. 5:1) and suffer from cyclic behaviour, surging, erratic product quality and unforeseen shutdowns [2]. This clearly shows that there is an economic incentive for better understanding and for effective operation and control of granulation processes.

A suitable framework for the mathematical modelling of the granulation process and its mechanisms is through population balance equations (PBE) which account for nucleation, consolidation (negative growth), aggregation and breakage. For an accurate depiction of the granulation process, the population balance model must account for distribution along granule size, binder content and granule porosity. These granule attributes can then be re-internalized in the PBE as volumes of solid, liquid and gas, thus necessitating three internal coordinates.

One major challenge in developing population balance models is the identification of suitable kernels (rate constants) for the sub-processes (e.g. aggregation). The aggregation kernel is a very important parameter in the PBE, as it provides the functional dependency of the growth rate on the process and material variables [3]. Due to the limited knowledge of the mechanisms that underlie the granulation process, most of the proposed aggregation kernels are empirical or semi-empirical in nature and have been largely formulated based on experimental observations. This involves fitting the coefficients in these kernels to plant or laboratory data. Such kernels have limited validity and utility in the broader perspective. Other studies have also considered physically based kernels and whilst these aggregation rate constants are more meaningful, the rate constants are still fitted from experimental data and therefore are not completely predictive [4]. Therefore, in order to further improve the predictive capabilities of the aggregation models and to extend their region of validity, one has to directly incorporate the mechanistic features of the process and develop a kernel strongly based on first principles. Immanuel and Doyle have presented a methodology for developing these first principle aggregation kernels (see [5] for details). Their mechanistic aggregation kernel is based on aggregation mechanisms reported [6, 7] and constitutes a novel approach. The advantage in the mechanistic formulation is its applicability to a wider range of conditions than its empirical/semi-empirical/physically based counterparts. It was also reported that preliminary results qualitatively captured experimental trends [5]. However, no quantitative model validation (incorporating the above three-dimensional mechanistic kernel) by means of suitable experimentation has been performed.

In this paper, experimental validation of the aggregation model incorporating the mechanistic kernel is studied and carried out. The experiments are conducted in a laboratory scale fluidized bed granulator (4M8-Fluidbed, ProCept). A material system consisting of glass ballotini and hydroxypropylcellulose (HPC) is used to investigate the effect of (1) binder spray rate, (2) binder droplet size, (3) binder position, (4) bed temperature and (5) fluidizing air velocity, on granule size, binder content and granule porosity. Model data will be compared with experimentally measured data from the ballotini-HPC system. Typically in experimental validation of a granulation process, only one granule attribute (e.g. granule size) is measured [4, 8]. In this study, granule size, granule porosity and granule binder content are measured. The inclusion of the latter two attributes adds to the authenticity of the model validation. Measurement techniques (e.g. light scattering, image analysis, porosimetry, gravimetry) used in the data acquisition of the granule attributes will also be presented. For the purpose of model validation, a bound constrained nonlinear optimization problem is also formulated and coded to optimize the adjustable constant in the mechanistic kernel.

A robust and efficient numerical solution technique is employed for the solution of the complex three-dimensional population balance model. The PBE is a hyperbolic nonlinear integro-partial differential equation whose solution involves complex finite difference/element techniques due to the presence of the integral terms [9]. The additional computational load for such an equation limits its utility for engineering problems where real time analysis is required for process control constraints. A fast and efficient solution technique (known as the hierarchical two-tier technique) was developed for one dimensional population balance models (see [10] for details) and extended to the three dimensional case (see [5] for details). In this study, the hierarchical two-tier technique is used to solve the three-dimensional population balance model. The main feature of the technique is that the offline analytical solution proposed for the aggregation quadratures leads to a substantial reduction in computational load. To further expedite solving the population balance model based on this technique, a parallel programming framework is implemented to the code. Results show that parallel computing is effective in further reducing execution time, enabling the complex population balance model to be potentially relevant for process control purposes.

References

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