×

zbMATH — the first resource for mathematics

Parameter estimation and accuracy matching strategies for 2-D reactor models. (English) Zbl 1081.80008
The paper deals with a description of a numerical scheme applied to the solution of the PDE used in mathematical modeling of chemical reactions. The authors consider a 2-d time-dependent model in a cylindrical reactor module. The governing eqs. result from the balance equations for mass and energy, but the development is not described. The model also includes some uncertain parameters, which are determined in the numerical procedure combined with measurements. After spatial discretization a very large system of differential-algebraic equations is integrated by a one-step method using the linearly implicit extrapolation code LIMEX. The parameter identification procedure requires supplementing the governing equations with the sensitivity analysis equations. Then, the parameters can be obtained minimizing the nonlinear least-squares functional created from the simulation solutions and their measurement counterparts. For this purpose, the damped Gauss-Newton algorithm is employed. To increase the computational efficiency, the master/slave mode is used to control the Gauss-Newton iterations. Finally, some numerical experiments are presented.

MSC:
80A32 Chemically reacting flows
65N40 Method of lines for boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
92E20 Classical flows, reactions, etc. in chemistry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adler, R., Stand der simulation von heterogen-gaskatalytischen reaktionsabläufen in festbettreaktoren-teil 1, Chemie ing. tech., 72, 555-564, (2000)
[2] Adler, R., Stand der simulation von heterogen-gaskatalytischen reaktionsabläufen in festbettreaktoren-teil 2, Chemie ing. tech., 72, 688-699, (2000)
[3] Adler, R.; Schmoz, G.; Helmer, A.; Heilmann, U.; Nelles, J., Experimentelle und auswertungsmethoden zur reaktionstechnischen untersuchung heterogen-gaskatalytischen prozesse, teil xviiaufbau von versuchsraktoren mit dem baukastensystem gradientenbehafteter reaktoren, Chem. techn., 40, 69-73, (1988)
[4] M. Bauer, Theoretische und experimentelle Untersuchungen zum Wärmetransport in gasdurchströmten Festbettreaktoren, dissertation at the Martin-Luther-University of Halle-Wittenberg, 2001.
[5] Bock, H.G., Numerical treatment of inverse problems in chemical reaction kinetics, (), 102-125
[6] Deuflhard, P., Order and stepsize control in extrapolation methods, Numer. math., 41, 399-422, (1983) · Zbl 0543.65049
[7] P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Series Computational Mathematics 35, Springer, Berlin, 2004. · Zbl 1056.65051
[8] P. Deuflhard, V. Apostolescu, A study of the Gauss-Newton method for the solution of nonlinear least squares problems, in: Frehse/Pallaschke/Trottenberg (Eds.), Special Topics of Applied Mathematics, North-Holland, Amsterdam, 1980, pp.129-150. · Zbl 0443.65049
[9] Deuflhard, P.; Nowak, U., Extrapolation integrators for quasilinear implicit odes, (), 37-50 · Zbl 0617.65078
[10] R. Ehrig, U. Nowak, L. Oeverdieck, P. Deuflhard, Advanced extrapolation methods for large scale differential algebraic problems, in: High Performance Scientific and Engineering Computing, LNCSE Springer 8, Springer, Berlin, 1999, pp. 233-244.
[11] U. Fiand, Vergleichende Untersuchungen zum Wärmetransport in Schüttungen mit und ohne chemische Reaktion, dissertation at the University of Erlangen-Nuremberg, 1978.
[12] Fritsch, F.N.; Carlson, R.E., Monotone piecewise cubic interpolation, SIAM J. numer. anal., 17, 238-246, (1980) · Zbl 0423.65011
[13] A. Grah, Entwicklung und Anwendung modularer Software zur Simulation und Parameterschätzung in gaskatalytischen Festbettreaktoren, dissertation at the Martin-Luther-University of Halle-Wittenberg, 2004.
[14] E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Series in Computational Mathematics 8, 2nd rev. ed., Springer, Berlin, 1993. · Zbl 0789.65048
[15] Maly, T.; Petzold, L., Numerical methods and software for sensitivity analysis of differential-algebraic systems, Appl. numer. math., 20, 57-79, (1996) · Zbl 0854.65056
[16] Nowak, U.; Deuflhard, P., Numerical identification of selected rate constants in large chemical reaction systems, Appl. numer. math., 1, 59-75, (1985) · Zbl 0551.65051
[17] U. Nowak, A. Grah, M. Schreier, Numerical software for simulation, sensitivity analysis and parameter estimation in catalytic gas reactors, in: ACOMEN 2002: Proceedings of the Second International Conference on Advanced Computational Methods in Engineering (CD-ROM) 2002.
[18] Schittkowski, K., Numerical data Fitting in dynamical systems—A practical introduction with applications and software, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1018.65077
[19] Schlegel, M.; Marquard, W.; Ehrig, R.; Nowak, U., Sensitivity analysis of linearly-implicit differential-algebraic systems by one-step extrapolation, Appl. numer. math., 48, 83-102, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.