Application of Gaussian cubature to model two-dimensional population balances

Abstract

In many systems of engineering interest the moment transformation of population balance is applied. One of the methods to solve the transformed population balance equations is the quadrature method of moments. It is based on the approximation of the density function in the source term by the Gaussian quadrature so that it preserves the moments of the original distribution. In this work we propose another method to be applied to the multivariate population problem in chemical engineering, namely a Gaussian cubature (GC) technique that applies linear programming for the approximation of the multivariate distribution. Examples of the application of the Gaussian cubature (GC) are presented for four processes typical for chemical engineering applications. The first and second ones are devoted to crystallization modeling with direction-dependent two-dimensional and three-dimensional growth rates, the third one represents drop dispersion accompanied by mass transfer in liquid-liquid dispersions and finally the fourth case regards the aggregation and sintering of particle populations.

Application of Polymers as a Tool in Crystallization—A Review

“…The best example is the production of sodium chloride in ancient times by sea water evaporation using the Sun’s energy. A decade ago, more than 70% of solids manufactured by industry were obtained by crystallization or precipitation [ 1 , 2 ]. Despite this, the phenomenon of nucleation and crystal growth is not fully understood, resulting in some serious issues which need to be managed.…”

Chemical and Process Engineering

“…The numerical analysis of dispersed systems through Population Balance Modelling (PBM) described in Bałdyga et al (2018) in its general formulation, consists of a nonlinear-integro-partial-differential equation, solved for instance by using the quadrature method of moments (QMOM) as in Jasińska et al (2014). Among the possible approaches, the 𝑆 − 𝛾 method of moments, as implemented by Lo and Zhang (2009) is adopted here.…”

Quadrature-based moment methods for the population balance equation: An algorithm review

“…Yoon and McGraw [141] developed the Tensor-product QMOM in the context of aerosol modeling. Baldyga et al [142] developed the Gaussian Cubature (GC) technique for crystallization, aggregation and sintering. Another method is the direct quadrature method of moments (DQMOM) which was developed by Marchisio and Fox [127].…”