Microalgae growth determination using modified breakage equation model

Numerical solutions for multidimensional fragmentation problems using finite volume methods

“…In this section, we validate the efficiency of Scheme−1a (17), Scheme−2a (19) and Scheme−1b (22), Scheme−2b (24) by considering eight test problems defined by two particle properties, and two test problems with three particle properties. As mentioned in Section 1, three dimensional extension of the schemes of [21] and [5] are very difficult task and can be treated as a scope of future research.…”

“…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).…”

“…Introduction. Fragmentation or breakage, comprising all processes in which two or more objects are created from one initial object, occurs in many natural and industrial processes, for example: cell division, bubble break-up, milling and grinding of particulate materials [9,19,22,33].…”

“…Modeling of these processes, for instance for process design and optimisation, is inherently difficult, as they are also of a multiscale nature: fragmentation at single-object level and fragmentation at the level of the whole population of objects, for example all bubbles in a bubble column or particles in a grinding process. Due to its industrial importance, several studies have been performed to describe fragmentation at single-object level, for instance [7,8,12,30], using Discrete Element methods (DEM) or from experimental data [22,35,37,38]. Additionally, fragmentation has been studied at the level of populations, utilising the population balance approach (PBM) [2,14,25,40], allowing the estimation of product properties, e.g.…”

“…Theoretically, Scheme−2a estimates the evolution of total number of particles with high accuracy. In the second set, two models Scheme−1b [eqs (22), (23)] and Scheme−2b [eqs (24), (25)] are designed to conserve the particle hypervolume (21) during a breakage event. In this case also, the Scheme−2b is expected to estimate the zeroth moment with good accuracy.…”