Abstract
Understanding and controlling the behavior of chemical processes are important issues, for safety as well as economical reasons. Some processes can have multiple steady states and even switch between them in a complex way, the reasons for the multiplicity not always being well understood. A singularity theory approach for investigating such behavior leads to nonlinear systems whose solutions correspond to specific singular states of the process. In order to exclude certain types of singularities, rigorous methods must be used to check the solvability of the matching systems. As these systems are highly structured, our solution method combines a symbolic preprocessing phase (term manipulation for utilizing the structure) with a branch-and-bound type rigorous interval-based solver. We report on our experience with this approach for small-to-medium sized example problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
G. Alefeld and J. Herzberger. Introduction to Interval Computations. Academic Press, New York, 1983.
V. Gehrke and W. M arquardt. A singularity theory approach to the study of reactive distillation. Comput. Chem. Engng, 21 (Suppl.) S1001 - S1006, 1997.
M. Golubitsky and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory, Volume I. Springer-Verlag, New York, 1985.
A. Griewank. Evaluating Derivatives — Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia, PA, 2000.
E. R. Hansen. Preconditioning linearized equations. Computing, 58: 187–196, 1997.
R. B. Kearfott. Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers, Dordrecht, 1996. For the current version of the software, see http://www.mscs.mu.edu/globsol/.
R. B. Kearfott and X. Shi. A preconditioner selection heuristic for efficient iteration with decomposition of arithmetic expressions for nonlinear systems. Interval Computations, 4 (1): 15–33, 1993.
R. Klatte, U. Kulisch, M. Neaga, C. Ullrich, and D. Ratz. Pascal-XSC-Language Description with Examples. Springer-Verlag, Berlin, 1992.
R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4: 187–201, 1969.
B. Lang. Verifizierte Lösung von Gleichungs-und Ungleichungssystemen. Z. angew. Math. Mech., 75 (S II): S541 - S542, 1995.
R. E. Moore and L. Qi. A successive interval test for nonlinear systems. SIAM J. Numer. Anal., 19 (4): 845–850, 1982.
A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 1990.
R. Seydel. Practical Bifurcation and Stability Analysis. Springer-Verlag, New York, 2nd edition, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bischof, C.H., Lang, B., Marquardt, W., Mönnigmann, M. (2001). Verified Determination of Singularities in Chemical Processes. In: Krämer, W., von Gudenberg, J.W. (eds) Scientific Computing, Validated Numerics, Interval Methods. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6484-0_25
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6484-0_25
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-3376-8
Online ISBN: 978-1-4757-6484-0
eBook Packages: Springer Book Archive