Simulation of crystal size and shape by means of a reduced two-dimensional population balance model

Commonly, population balance modeling for crystallization processes only considers one inner variable. Usually, a variable characterizing particle size, like a sphere-equivalent diameter is employed. However, crystal structures are obviously not spherical but exhibit a complex habit. Often, the habit is even crucial for the quality of the product or for the operability of downstream filter units. To describe the transient behavior of crystals in a batch process considering two-dimensional growth and nucleation, a multidimensional population balance needs to be employed. Considering two characteristic lengths of a crystal, the standard discretization of such a system (by, e.g. finite differences) leads to quite a large model size, which may be unsuited for model-based control and parameter identification purposes. In this contribution a model reduction which is based on two steps is proposed. First, a coordinate transformation is performed to model the system not in terms of two characteristic lengths but by means of the crystal volume and a shape factor. In a second step the actual model reduction is performed by generating cross-moments for the two-dimensional representation. An ansatz for the two-dimensional crystal size distribution, which gives full flexibility in the volume coordinate but restricts the dependence in the shape factor κ, allows the closure of the system of moments. The reduced system consists of three coupled population balance equations in which all three are structurally similar to a single one-dimensional population balance equation, where growth and nucleation only are considered. Solving this reduced system allows the detailed simulation of the easily measurable volume-based number density distribution and preserves average and dispersity information on the crystal shape. The resulting model size, however, scales only linearly with the number of discretization grid points instead of the quadratic scaling for standard discretizations. Numerical results for the crystallization of potassium dihydrogen phosphate (KH₂PO₄) in a batch process are presented for illustration.