Fixed point convergence and acceleration for steady state population balance modelling of precipitation processes: Application to neodymium oxalate

The present work focuses on the development of a performing numerical methodology to solve the steady state Population Balance Equation (PBE) including nucleation, independent size growth and loose agglomeration as crystallization mechanisms. The methodology is based on the solution of two PBEs: one for the isolated crystallites and one describing the loose agglomerates formation. Both are solved by a discretization method and only the last one is reformulated as a fixed point problem. The algorithm solving PBE for agglomeration includes the crossed-secant algorithm as a fixed point acceleration method. The numerical PBE solution method is first validated by comparison to analytical solutions and then applied to the neodymium oxalate precipitation in order to compare to experimental results in a wide range of operating conditions. The methodology is tested under highly restrictive numerical conditions: narrow tolerances, a large amount of points in the discretization scheme and a zero vector as initial condition. The crossed-secant method demonstrates to improve the robustness of the standard fixed point iterations by ensuring the convergence of the agglomerates PBE when penalizing conditions are applied and by reducing the number of iterations otherwise. In all cases, the developed methodology predicts accurately the crystal size distribution under the experimental uncertainty in a reasonable computation time and number of iterations. (authors)