Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.