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.
The main topic of this study is the mathematical modelling of bubble size distributions in an aerated stirred tank using the population balance method. The air-water system consisted of a fully baffled vessel with a diameter of 0.29 m, which was equipped with a six-bladed Rushton turbine. The secondary phase was introduced through a ring sparger situated under the impeller. Calculations were performed with the CFD software CFX 14.5. The turbulent quantities were predicted using the standard k-ε turbulence model. Coalescence and breakup of bubbles were modelled using the MUSIG method with 24 bubble size groups. For the bubble size distribution modelling, the breakup model by Luo and Svendsen (1996) typically has been used in the past. However, this breakup model was thoroughly reviewed and its practical applicability was questioned. Therefore, three different breakup models by Martínez-Bazán et al. (1999a, b), Lehr et al. (2002) and Alopaeus et al. (2002) were implemented in the CFD solver and applied to the system. The resulting Sauter mean diameters and local bubble size distributions were compared with experimental data.