An efficient multiscale bi-directional PBM-DEM coupling framework to simulate one-dimensional aggregation mechanisms

The mesoscale population balance modelling (PBM) technique is widely used in predicting aggregation processes. The accuracy and efficiency of PBM depend on the formulation of its kernels. A model of the volume- and time-dependent one-dimensional aggregation kernel is developed for predicting the temporal evolution of the considered particulate system. To make the developed model physically relevant, the PBM model needs three unknown parameters as input: volume-dependency in collisions, collision frequency per particle and aggregation probability. For this, the microscale discrete element model (DEM) is used. The system’s collision frequency is extracted periodically using a novel collision detection algorithm that detects and ignores duplicate collisions.

Finally, a multiscale bi-directional PBM–DEM coupling framework is presented to simulate the aggregation mechanism. PBM and DEM simulations take place periodically to update the particle size distribution (PSD) and extract the collision-frequency, respectively. The coupling framework successfully explains the dependence between the PSD and the collision frequency. Additionally, computational cost of the algorithm is optimized while maintaining the accuracy of the results. Lastly, the accuracy and efficiency of the developed framework are verified using two different test cases. In one of the examples, a simple aggregation is simulated directly inside the DEM for the first time.

Conservative Finite Volume Schemes for Multidimensional Fragmentation Problems

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems.

Mass-based finite volume scheme for aggregation, growth and nucleation population balance equation

In this paper, a new mass-based numerical method is developed using the notion of Forestier-Coste & Mancini (Forestier-Coste & Mancini 2012, SIAM J. Sci. Comput. 34, B840-B860. (doi:10.1137/110847998)) for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method.