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

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.

[1]  J. Szmelter.  Incompressible flow and the finite element method , 2001 .

[2]  Wolfgang Marquardt,et al.  Modular dynamic simulation for integrated particulate processes by means of tool integration , 2005 .

[3]  Wolfgang Hackbusch,et al.  Approximation of coalescence integrals in population balance models with local mass conservation , 2007, Numerische Mathematik.

[4]  Wolfgang Hackbusch.  Convolution of hp-functions on locally refined grids - extended version , 2007 .

[5]  Stefan Turek,et al.  Efficient Solvers for Incompressible Flow Problems - An Algorithmic and Computational Approach , 1999, Lecture Notes in Computational Science and Engineering.

[6]  Volker John,et al.  On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems , 2010 .

[7]  Onur Baysal,et al.  Stabilized finite element methods for time dependent convection-diffusion equations , 2012 .

[8]  K. Sundmacher,et al.  Crystal Aggregation in a Flow Tube: Image-Based Observation , 2011 .

[9]  Analysis and measurement of crystallization utilizing the population balance , 2002 .

[10]  Michael Manhart,et al.  Precipitation of nanoparticles in a T-mixer: Coupling the particle population dynamics with hydrodynamics through direct numerical simulation , 2006 .

[11]  M. Smoluchowski.  Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen , 1918 .

[12]  Dmitri Kuzmin,et al.  Explicit and implicit FEM-FCT algorithms with flux linearization , 2009, J. Comput. Phys..

[13]  Volker John,et al.  Techniques for the reconstruction of a distribution from a finite number of its moments , 2007 .

[14]  Wolfgang Marquardt,et al.  A framework for the simulation of mass crystallization considering the effect of fluid dynamics , 2006 .

[15]  Volker John,et al.  Reconstruction of a distribution from a finite number of moments with an adaptive spline-based algorithm , 2010 .

[16]  K. Sundmacher,et al.  Simulations of Population Balance Systems with One Internal Coordinate using Finite Element Methods , 2009 .

[17]  Michael Manhart,et al.  Predictive simulation of nanoparticle precipitation based on the population balance equation , 2006 .

[18]  Daniele Marchisio,et al.  Solution of population balance equations using the direct quadrature method of moments , 2005 .

[19]  Gunar Matthies,et al.  MooNMD – a program package based on mapped finite element methods , 2004 .

[20]  Robert McGraw,et al.  Description of Aerosol Dynamics by the Quadrature Method of Moments , 1997 .

[21]  Volker John,et al.  On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations , 2012, J. Comput. Phys..

[22]  Volker John,et al.  Finite element methods for time-dependent convection – diffusion – reaction equations with small diffusion , 2008 .

[23]  M. Mazzotti,et al.  L-Glutamic Acid Precipitation : Agglomeration Effects , 2008 .

[24]  Gunar Matthies,et al.  International Journal for Numerical Methods in Fluids Higher-order Finite Element Discretizations in a Benchmark Problem for Incompressible Flows , 2022 .

[25]  Wolfgang Hackbusch,et al.  On the Efficient Evaluation of Coalescence Integrals in Population Balance Models , 2006, Computing.