Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.