Dimension reduction of bivariate population balances using the quadrature method of moments

Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.