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

Published online by Cambridge University Press:  31 May 2012

Sashikumaar Ganesan
Sashikumaar Ganesan*
Affiliation:
Numerical Mathematics and Scientific Computing, Supercomputer Education and Research Centre, Indian Institute of Science, 560012 Bangalore, India. [email protected]
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Abstract

We present a heterogeneous finite element method for the solution of a high-dimensionalpopulation balance equation, which depends both the physical and the internal propertycoordinates. The proposed scheme tackles the two main difficulties in the finite elementsolution of population balance equation: (i) spatial discretization with the standardfinite elements, when the dimension of the equation is more than three, (ii) spuriousoscillations in the solution induced by standard Galerkin approximation due to pureadvection in the internal property coordinates. The key idea is to split thehigh-dimensional population balance equation into two low-dimensional equations, anddiscretize the low-dimensional equations separately. In the proposed splitting scheme, theshape of the physical domain can be arbitrary, and different discretizations can beapplied to the low-dimensional equations. In particular, we discretize the physical andinternal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG)finite elements, respectively. The stability and error estimates of the Galerkin/SUPGfinite element discretization of the population balance equation are derived. It is shownthat a slightly more regularity, i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for theoptimal order of convergence. Numerical results are presented to support the analysis.

Keywords

Population balance equationsoperator-splitting methoderror analysisstreamline Upwind Petrov Galerkin finite element methodsbackward Euler scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1447 - 1465
Copyright
© EDP Sciences, SMAI, 2012

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References

S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3th edition. Springer (2008).
Burman, E., Consistent SUPG-method for transient transport problems : Stability and convergence. Comput. Methods Appl. Mech. Eng. 199 (2010) 1114–1123. Google Scholar
Cameron, I.T., Wang, F.Y., Immanuel, C.D. and Stepanek, F., Process systems modelling and applications in granulation : a review. Chem. Eng. Sci. 60 (2005) 3723–375. Google Scholar
Campos, F.B. and Lage, P.L.C., A numerical method for solving the transient multidimensional population balance equation using an Euler-Lagrange formulation. Chem. Eng. Sci. 58 (2003) 2725–2744. Google Scholar
Chen, P., Sanyal, J. and Dudukovic, M.P., CFD modeling of bubble columns flows : implementation of population balance. Chem. Eng. Sci. 59 (2004) 5201–5207. Google Scholar
Eriksson, K. and Johnson, C., Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput. 60 (1993) 167–188. Google Scholar
S. Ganesan and L. Tobiska, Implementation of an operator-splitting finite element method for high-dimensional parabolic problems. Faculty of Mathematics, University of Magdeburg, Preprint No. 11-04 (2011).
Ganesan, S. and Tobiska, L., An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems. Chem. Eng. Sci. 69 (2012) 59–68. Google Scholar
Glowinski, R., Dean, E.J., Guidoboni, G., Peaceman, D.H. and Rachford, H.H., Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Japan J. Ind. Appl. Math. 25 (2008) 1–63. Google Scholar
Gunawan, R., Fusman, I. and Braatz, R.D., High resolution algorithms for multidimensional population balance equations. AIChE J. 50 (2004) 2738–2749. Google Scholar
Gunawan, R., Fusman, I. and Braatz, R.D., Parallel high-resolution finite volume simulation of particulate processes. AIChE J. 54 (2008) 1449–1458. Google Scholar
T.J.R. Hughes and A.N. Brooks, A multi-dimensional upwind scheme with no cross-wind diffusion, in Finite element methods for convection dominated flows, edited by T.J.R. Hughes. ASME, New York (1979) 19–35.
Hulburt, H.M. and Katz, S., Some problems in particle technology : A statistical mechanical formulation. Chem. Eng. Sci. 19 (1964) 555–574. Google Scholar
John, V. and Novo, J., Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal. 49 (2011) 1149–1176. Google Scholar
John, V., Roland, M., Mitkova, T., Sundmacher, K., Tobiska, L. and Voigt, A., Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci. 64 (2009) 733–741. Google Scholar
Kulikov, V., Briesen, H., Grosch, R., Yang, A., von Wedel, L. and Marquardt, W., Modular dynamic simulation for integrated particulate processes by means of tool integration. Chem. Eng. Sci. 60 (2005) 2069–2083. Google Scholar
Kulikov, V., Briesen, H. and Marquardt, W., A framework for the simulation of mass crystallization considering the effect of fluid dynamics. Chem. Eng. Sci. 45 (2006) 886–899. Google Scholar
Lian, G., Moore, S. and Heeney, L., Population balance and computational fluid dynamics modelling of ice crystallisation in a scraped surface freezer. Chem. Eng. Sci. 61 (2006) 7819–7826. Google Scholar
Ma, D.L., Tafti, D.K. and Braatz, R.D., High-resolution simulation of multidimensional crystal growth. Ind. Eng. Chem. Res. 41 (2002) 6217–6223. Google Scholar
Ma, D.L., Tafti, D.K. and Braatz, R.D., Optimal control and simulation of multidimensional crystallization processes. Comput. Chem. Eng. 26 (2002) 1103–1116. Google Scholar
Majumder, A., Kariwala, V., Ansumali, S. and Rajendran, A., Fast high-resolution method for solving multidimensional population balances in crystallization. Ind. Eng. Chem. Res. 49 (2010) 3862–3872. Google Scholar
Marchisio, D. and Fox, R., Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci. 36 (2005) 43–73. Google Scholar
Marchisio, D. L. and Fox, R.O., Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci. 36 (2005) 43–73. Google Scholar
Nandanwara, M. N. and Kumar, S., A new discretization of space for the solution of multi-dimensional population balance equations : Simultaneous breakup and aggregation of particles. Chem. Eng. Sci. 63 (2008) 3988–3997. Google Scholar
D. Ramkrishna, Population Balances, Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego (2000).
Ramkrishna, D. and Mahoney, A.W., Population balance modeling : Promise for the future. Chem. Eng. Sci. 57 (2002) 595–606. Google Scholar
V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 3th edition. Springer (2008).