Particle Shape and the Simulation of Crystallization Processes

Particle shape is typically described by face normals and the distances $h_i$ of each face from the center of the crystal. The population balance equation tracks the temporal development of the particle size and shape:

$$ \frac{\delta n}{\delta t}+\sum_i{\frac{\delta\left( g_i, n\right)}{\delta h_t}}=0 $$

The face-specific growth rates  are supersaturation dependent. In order to close the mass balance, the total crystal volume must be computed:

$$ I_{volume}=\int{\mu_{volume}\left(h_C\right) n\left(h_C\right)d h_C} $$

The computational complexity of the problem increases with the increase of the crystals’ geometrical complexity. The resulting numerical problem was solved by a combination of the method of characteristics and Monte Carlo integration [3].

 For aggregation processes, it is infeasible to fully describe the shape of growing crystal aggregates during a crystallization process by such a deterministic population balance equation. The usual approach to solving this problem is to assume that two crystals form a body of equivalent volume upon aggregation. However, the area of the volume-equivalent body is always too small. This can alternatively be solved by using Monte Carlo simulations where each particle in a small representative volume is simulated. Only these types of simulations lead to the correct computation of particle area and therefore the correct mass balance [6].

Alternatively to the shape representations described above, one can use convex geometry to describe a convex body  $B$ as a weighted sum of structuring elements $S_iS :

$$B=\sum_i{\lambda_i S_i} $$

The theoretical background was the foundation of this project. One major advantage of such shape representation is the possibility of describing rounded particles.

Combined with suitable image processing steps, this method has been used for describing particle attrition [2].

 Monte Carlo simulations of crystal aggregation and growth enable the correct description of the aggregate shape and the resulting evolution of surface area over time. However, such simulations are computationally intensive and therefore not suited for process control, optimization and large parameter studies. Furthermore, they cannot be easily coupled with other process steps that use a simpler shape description. The goal is to combine these methods with a less computationally expensive classical population balance approach. To this end, the temporal evolution of the area of single aggregates will be determined from several simulations. This area growth rate $G_A$  will then be used in a population balance model of crystal growth and aggregation that preserves both the volume $V$  V and the area  $A$ upon aggregation:

$$\begin{align}
\frac{\delta f}{\delta t} + \frac{\delta \left(G_V f\right)}{\delta_V} & + \frac{\delta \left(G_A f\right)}{\delta_A} = \\
& \frac{1}{2}\int_{V=0}^V \int_{A=0}^{A} {\beta \left(V', V-V', A', A-A'\right) f\left(V-V', A-A'\right) f\left(V',A'\right))dV' dA'} \\
& - f\left(V,A\right)\int_{V=0}^\infty \int_{A=0}^{\infty} {\beta \left(V', V, A', A\right)f\left(V',A'\right)dV' dA}\end{align}$$

Here, $f$ is the particle size distribution over the volume and area, $G_V=A G_h$ is the volumetric growth rate based on the face-specific growth rate $G_h$ and $\beta$ the aggregation kernel. The validation will be performed through comparison with full Monte Carlo simulations.

The image processing algorithms developed in the previous project stages were focused on the proof of concept for the chosen crystal geometries. For a generalized applicability, the algorithms will be expanded and adjusted for further crystal morphologies (see Image).