Comparison of discretization methods to solve a population balance model of activated sludge flocculation including aggregation and breakage

Population balance models (PBMs) can be used to describe the evolution with time of distributions of properties of individuals. In this study a PBM for activated sludge flocculation including aggregation and breakage processes was investigated. The PBM is an integro-differential equation and does not have an analytical solution. A possibe method of solving the equation at relatively low computational cost is to use discretization. Two different discretization techniques, the fixed pivot and the moving pivot, were compared using geometric grids of different coarseness. Simulations were performed for three different processes: pure aggregation, pure breakage and combined aggregation -- breakage. The results for pure aggregation showed that the fixed pivot overpredicts the large particle sizes when using coarse grids since grid refinement results in a clear downward trend. The predictions of the moving pivot technique show even lower predictions for the large particle sizes, with a slight upward trend for finer grids. This suggests that these predictions are closer to the pseudo-analytical solution (i.e. at infinitely fine grid). For the pure breakage case it was found that the moving pivot predictions collapsed onto one curve. Since a binary breakage case was studied, a fixed pivot with a grid with geometric factor 2 also collapsed onto that curve. Grid refinement for the fixed pivot case resulted in overestimations. Similar conclusions could be drawn for the combined aggregation -- breakage case. Overall, the moving pivot is found to be superior since it produces more accurate predictions, even for much coarser grids. Despite the computational burden, the latter implies a lower computational load.