Steady state population balance modelling of precipitation processes: Nucleation, growth and size-dependent agglomeration

In this work a numerical methodology to solve the steady state Population Balance Equation (PBE) is developed. Three crystallisation mechanisms are included, namely: nucleation, size-independent growth and size-dependent loose agglomeration. The numerical method is based on the discretisation of the crystal size as distributed variable. In order to describe the loose agglomeration, the numerical methodology solves two PBE: one including the nucleation and growth mechanisms and one accounting for the agglomeration process. From the first PBE, liquid phase composition, supersaturation, developed crystal surface and Crystallite Size Distribution (CSD) are obtained. Similarly, the second PBE leads to the Agglomerate Size Distribution (ASD). The study of the size -dependant agglomeration kernel induces an additional numerical difficulty due to the dependency of both PBE and agglomeration kernel on the particle size. An accelerated fixed point algorithm based on the crossed secant method is adapted to overcome the difficulty and accurately solve the agglomeration PBE. The oxalic precipitation of uranium is simulated using this numerical methodology. First, the experimental results of a reference case are compared with the numerical predictions in terms of particle size distribution, mean size, mass fraction and moments. Then, the operating conditions are varied in order to test the algorithm robustness and performances. In all cases, the crossed secant method ensures the size-dependent agglomeration PBE solution and properly predicts the ASD. The developed numerical methodology predicts the mean particle size under the experimental uncertainty in a reasonable computation time and number of iterations.