High dimensional population balances for the growth of faceted crystals: Combining Monte Carlo integral estimates and the method of characteristics

Numerical analysis of crystallization of high aspect ratio crystals with breakage

“…Assuming that the quadrature moments (22) are continuous in time and possess continuous derivates, we can substitute the approximating forms (21) and (22) …”

“…High resolution algorithms were introduced by Ma et al [1] for nucleation and growth problems, Puel et al [2,3] and Ma et al [14] applied the method of classes, Alexopoulos and Kiparissides [15] applied an improved sectional grid technique, Chakraborty and Kumar [16] used discretization, Kumar et al [17] used an extension of the cell average technique while Singh et al [18] applied for that purpose unstructured triangular grids. The Monte Carlo method has proved to be an effective solution method for bivariate cases [19][20][21][22] but this method can be applied even for solution of higher dimensional population balance models [23,24].…”

Simulations of crystal aggregation and growth: Towards correct crystal area

“…The size and shape of the obtained crystals influence the further processing steps such as filtration, but also play an important role in the final application by, for example, affecting the dissolution rate. Apart from performing experiments, the evolution of these properties can also be examined by modeling the crystallization process using the population balance approach . This method, as well as the involved crystallization phenomena and challenges related to shape control have been reviewed in the literature .…”

“…If crystal growth is the only considered phenomenon, crystal shape can be modeled in a quite straightforward way in its full complexity. Crystals are then described by the distance of each face group from the crystal center, leading to a multidimensional population balance equation . In these simulations, special care must be taken to appropriately handle crystal faces that disappear and reappear during the process .…”

“…Crystals are then described by the distance of each face group from the crystal center, leading to a multidimensional population balance equation . In these simulations, special care must be taken to appropriately handle crystal faces that disappear and reappear during the process . In classical aggregation models, particles are described by one size dimension, such as length or volume, leading to a one‐dimensional (1D) population balance equation (see reference [] for a recent review).…”