A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles

Numerical solutions for multidimensional fragmentation problems using finite volume methods

“…It can be easily proved that both the Scheme−1b (22) and Scheme−2b (24) follows hypervolume conservation law (21). Additionally, Scheme−2b (24) follows the discrete number formulation given by (15).…”

“…Primarily in most of the articles, authors have adopted different methodologies to design numerical schemes approximating the aggregation problems. In the literature, the sectional methods by [17,18] (cell average technique), [20,21,36] (fixed pivot technique), the method of higher-order moment-conserving classes by [5,6], method of moments by [24,27,39], by Monte-Carlo simulations [15,27,28,34], finite volume methods by [11,23] are well recognized because of their efficiency to predict different moments with good accuracy. However, the consideration of multidimensional fragmentation is limited to the works of [5,6,21], where the authors have designed their respective schemes to approximate the two-dimensional coupled aggregation-fragmentation equations.…”

“…However, the consideration of multidimensional fragmentation is limited to the works of [5,6,21], where the authors have designed their respective schemes to approximate the two-dimensional coupled aggregation-fragmentation equations. The work of [21] solves the fragmentation problems for uniform daughter distribution function. It is very difficult to design their schemes for daughter distribution functions which are nonuniform in nature.…”

“…It is very difficult to design their schemes for daughter distribution functions which are nonuniform in nature. Moreover, extension of the [21] scheme for three or higher dimensions can be treated as a new research problem. On the other hand, the works [5] and [6] are quite similar.…”

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

“…Several numerical methods, method of moments and its variants, discretization methods, finite difference, least square method, spectral and finite element methods, have been proposed and used to solve PBEs by several authors, see for example [16,17,[22][23][24][25][26] and the references therein. However, most of these methods are restricted to one-dimensional (spatially) PBEs or applied to a system of one-dimensional (1D) PBEs which are obtained by splitting the high-dimensional PBE.…”