The cell average technique for solving multi-dimensional aggregation population balance equations

In this paper, the cell average discretization [Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., & Mörl, 2006a. Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique. Chemical Engineering Science, 61, 3327–3342] is extended to solving multi-dimensional population balance equations. Similar to the one-dimensional case, the scheme is based on an accurate prediction of certain moments of the population. The formulation is quite simple to implement, computationally not expensive and highly accurate. Numerical diffusion is a common problem with many numerical methods when applied on coarse grids. The presented technique nearly eliminates numerical diffusion and predicts four moments (zeroth, first, first cross and second) of the distribution function with high accuracy. The technique may be implemented on any type of grid. The accuracy of the scheme has been analyzed by comparing analytical and numerical solutions of three test problems. The numerical results are in excellent agreement with the analytical results and show the ability to predict higher moments very precisely. Additionally, an extension of the proposed technique to higher dimensional problems is discussed.