This need can be addressed in various ways. Currently, various efforts are taken in order to generalize solution methods for one-dimensional population balance models to two- and multidimensional problem formulations. Though conceptually there is no such limitation, population balance modeling will most likely be employed only up to a few internal dimensions due to computational effort and the increasing complexity of implementation. Alternatively, stochastic (or Monte-Carlo) methods gain increasing popularity for the simulation of particulate processes (Haseltine et al., 2005). Instead of the continuous, deterministic formulation of the population balance, stochastic methods track the evolution of an ensemble of discrete particles. Since the particles are individually characterized, stochastic methods are easily extensible to more complex particle characterizations. The statistical relevance of the stochastic simulation of course is mainly affected by the number of particles considered in the ensemble.
Considering only non-agglomerated particles, already more than one characterizing variable is often needed to adequately represent the particle's geometry. For most of the crystals appearing in practical applications, plates or needles are the predominant geometry. For their characterization two length dimensions, namely the length and width, are sufficient. If the geometry of agglomerates is to be described in detail, the number of necessary dimensions quickly increases, since the relative orientation of the primary particles to each other have to be taken care for.
This work uses a novel representation of agglomerate structures (Briesen, 2005) for the simulation of crystallization processes by stochastic methods. The basis of this representation is a hierarchical characterization, which uses three dimensions to characterize each primary particle as an arbitrary cuboids. Additionally, each primary particle is associated with a 4x4 matrix which specifies the relative position of each of the primary particles within the agglomerates. For evaluating aggregation events in a reasonable computation time the geometrical complexity of this characterization is too high. Therefore, an additional level of characterization is used for the overall agglomerates. By principal component analysis (PCA) the main geometry of the agglomerate structure is determined. The agglomerate is then characterized by a substitution system comprising a set of 7 point masses, which reflect the overall geometry and preserve the moments of inertia of the full agglomerate characterization. For this substitution system geometrical computation like an aggregation of two colliding particles can be performed much more efficient. Therefore, this approach allows to track the evolution of the explicit morphological structure of the agglomerates in a realistic way. Hence, the evaluation of the secondary information like degree of agglomeration, shape of the particles, or particle porosity is possible. Additionally, the approach provides the potential for introducing a modeling of the crystallization rate processes using a very detailed characterization of the particles geometry as it may be necessary for agglomeration or breakage. Note that the position of the particles in the space is not considered i.e. no Lagranian view is taken for the simulation. The proposed method is tested and validated by means of simulation studies considering simultaneous particle growth and aggregation.
Briesen, H.: Hierarchical characterization of agglomerated crystal structures for Monte-Carlo simulations, submitted to AIChE J., 2005.
Haseltine, E.; Patience, D. & Rawlings, J. On the stochastic simulation of particulate systems Chem. Eng. Sci., 2005, 60, 2627-2641.
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