Modular dynamic simulation for integrated particulate processes by means of tool integration

Kulikov, V. D.; Briesen, Heiko; Grosch, R.; Yang, Aidong; Wedel, L. von; Marquardt, Wolfgang

doi:10.1016/j.ces.2004.11.037

Cited by 23 publications

(8 citation statements)

References 9 publications

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“…There are still only few approaches for the simulation of the equation for the PSD in the higher-dimensional domain [1][2][3][4]. Of course, the accurate and efficient simulation of such population balance systems poses a great challenge.…”

Section: Introductionmentioning

confidence: 99%

“…Of course, the accurate and efficient simulation of such population balance systems poses a great challenge. There are still only few approaches for the simulation of the equation for the PSD in the higher-dimensional domain [1][2][3][4]. Currently, more widely used are several proposals for model simplification, which replace the higher-dimensional equation for the PSD by a system of equations in three dimensions.…”

Section: Introductionmentioning

confidence: 99%

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

Hackbusch

John

Khachatryan

et al. 2011

Numerical Methods in Fluids

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Hackbusch

John

Khachatryan

et al. 2011

Numerical Methods in Fluids

View full text Add to dashboard Cite

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“…The matrix L does not posses positive off-diagonal entries and the diagonal matrix M L is called lumped mass matrix. Instead of (19), the equation…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…Functionals of the distribution of particles, in particular at the outlet of the chemical device, are of primary interest in applications. There are only initial attempts to simulate the population balance system describing a precipitation process [5,16,19,20,30]. The present paper will study the example from [16].…”

Section: Introductionmentioning

confidence: 99%

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

John

Roland

2010

Numerical Meth Engineering

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Population balance systems are models for processes in nature and industry which lead to a coupled system of equations (Navier-Stokes equations, transport equations, . . .) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three-dimensional equation of a precipitation process in a two-dimensional flow domain. The numerical schemes for the three-dimensional equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time-dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes.

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“…Several numerical methods, method of moments and its variants, discretization methods, finite difference, least square method, spectral and finite element methods, have been proposed and used to solve PBEs by several authors, see for example [16,17,[22][23][24][25][26] and the references therein. However, most of these methods are restricted to one-dimensional (spatially) PBEs or applied to a system of one-dimensional (1D) PBEs which are obtained by splitting the high-dimensional PBE.…”

Section: Introductionmentioning

confidence: 99%

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

Ganesan

2012

ESAIM: M2AN

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Abstract. We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.Mathematics Subject Classification. 35K20, 65M60, 65M12.

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Modular dynamic simulation for integrated particulate processes by means of tool integration

Cited by 23 publications

References 9 publications

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

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

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

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