Abstract
In multiphase chemical reactor analysis the dispersed phase distribution plays a major role in obtaining reliable predictions. The population balance equation is a well established equation for describing the evolution of the dispersed phase. However, the numerical solution of this type of equations is computationally intensive. In this work, a time-property least squares spectral method is presented for solving the population balance equation including breakage and coalescence processes. In this problem, both property and time are coupled in the least squares minimization procedure. Spectral convergence of the L 2 least squares functional and L 2 error norms in time-property is verified using a smooth solution to the population balance equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Bochev, Finite element methods based on least-squares and modified variational principles, Technical report, POSTECH (2001)
Chen M., Hwang C., Shih Y.: A wavelet-galerkin method for solving population balance equations. Comp. Chem. Eng. 20(2), 131–145 (1996)
Delves L.M., Mohamed J.L.: Computational methods for integral equations. Cambridge University Press, London (1985)
Deville M.O., Fischer P.F., Mund E.H.: High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge (2002)
Dorao C.A., Jakobsen H.A.: Application of the least square method to population balance problems. Comp. Chem. Eng. 30(3), 535–547 (2005)
C.A. Dorao, H.A. Jakobsen, An evaluation of selected numerical methods for solving the population balance equation. Fourth International Conference on CFD in the Oil and Gas, Metallurgical & Process Industries. (SINTEF/NTNU Trondheim, Norway, 2005), 6–8 June 2005
Dorao C.A., Jakobsen H.A.: Application of the least squares method for solving population balance problems in \({\mathbb{R}^{d+1}}\) . Chem. Eng. Sci. 61(15), 5070–5081 (2005)
Dorao C.A., Jakobsen H.A.: A least squares method for solving advective population balance problems. J. Comp. Appl. Math. 201(1), 247–257 (2005)
Hackbusch W.: Integral equation: theory and numerical treatment. International series of numerical mathematics. Birkhauser Verlag, Basel (1995)
Jakobsen H.A., Lindborg H., Dorao C.A.: Modeling of bubble column reactors: progress and limitations. Ind. Eng. Chem. Res. 44(14), 5107–5151 (2005)
Jiang B.: The least-square finite element method: theory and applications in computational fluid dynamics and electromagnetics. Springer, New York (1998)
Liu Y., Cameron T.: A new wavelet-based method for the solution of the population balance equation. Chem. Eng. Sci. 56, 5283–5294 (2001)
De Maerschalck B. (2003) Space-Time least-squares spectral element method for unsteady flows - application and evaluation for linear and non-linear hyperbolic scalar equations, Master Thesis Report, Delft University of Technology, Dept. of Aerospace Engineering, The Netherlands, 2003
Mantzaris N.V., Daoutidis P., Srienc F.: Numerical solution of multi-variable cell population balance models. II. Spectral methods. Comp. Chem. Eng. 25, 1441–1462 (2001)
Post D., Kendall R.: Software project management and quality engineering practices for comples, coupled multi-physics, massively parallel computational simulations: lessons learned from ASCI. Int. J. High Perform. Comp. Appl. 18(4), 399–416 (2003)
Proot M.M.J., Gerritsma M.I.: A least-squares spectral element formulation for stokes problem. J. Sci. Comp. 17(1–4), 285–296 (2002)
J.P. Pontaza, J.N. Reddy (2004) Spectral/hp least-squares finite element formulation for the incompressible Navier–Stokes equation. J. Comput. Phys. 190(2), 523–549 (2003)
Pontaza J.P., Reddy J.N.: Space-time coupled spectral/hp least squares finite element formulation for the incompressible Navier–Stokes equation. J. Comput. Phys. 190(2), 418–459 (2004)
Ramkrishna D.: Population balances, theory and applications to particulate systems in engineering. Academic Press, San Diego (2000)
Roy C.J.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205, 131–156 (2005)
Subramain G., Ramkrishna D.: On the solution of statistical models of cell populations. Math. Biosc. 10, 1–23 (1971)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dorao, C.A., Jakobsen, H.A. Time-property least-squares spectral method for population balance equations. J Math Chem 46, 770–780 (2009). https://doi.org/10.1007/s10910-009-9546-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-009-9546-0