A numerical method for the simulation of an aggregation‐driven population balance system

Abstract

SUMMARY A population balance system that models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three‐dimensional domain, while the equation for the particle size distribution is given in a four‐dimensional domain. This problem is convection‐dominated and aggregation‐driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system, which is based on finite element schemes, a finite difference method and a modern method to evaluate convolutIon integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel. Copyright © 2011 John Wiley & Sons, Ltd.

A comparative study of a direct discretization and an operator-splitting solver for population balance systems

“…It is in principal the same model as considered in Hackbusch et al (2012). Only, the flow domain has a different form, hence also the flow field, such that a 3D-axisymmetric form of this model can be used in the numerical simulations.…”

“…Only, the flow domain has a different form, hence also the flow field, such that a 3D-axisymmetric form of this model can be used in the numerical simulations. It will be explained in Section 4 that the different flow field will also change the impact of growth and aggregation on the urea population compared with Hackbusch et al (2012).…”

“…The PSD at in is given by experimental data, see Hackbusch et al (2012). In summary, the considered PBS consists of the two convection-diffusion equations (3) and (5) for the molar concentration and the temperature, respectively, and the integro partial different equation of transport type (7) for the PSD.…”

“…A detailed description of the general approach can be found in Hackbusch et al (2012). Consider the time t n .…”

“…To this end, a model of urea synthesis from Hackbusch et al (2012), with a uni-variate population, will be used where some experimental data from Borchert and Sundmacher (2011) are available. This model will be considered in a comparable simple situation: the flow domain is a cylindrical pipe, the flow fields are stationary, the velocity field is given by a Hagen-Poiseuille profile, and the setup is such that the solution, in cylindrical coordinates, can be assumed to be independent of the angle.…”