Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T16:49:25.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor. II Diabatic operation

Published online by Cambridge University Press:  17 February 2009

M. I. Nelson  and
H. S. Sidhu
M. I. Nelson
Affiliation:
School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, NSW 2522, Australia; e-mail: mnelson@uow.edu.au.
H. S. Sidhu
Affiliation:
School of Physical, Environmental and Mathematical Sciences, University College, University of New South Wales, Australian Defence Force Academy, Canberra 2600, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We extend an investigation into the bifurcation phenomena exhibited by an oxidation reaction in an adiabatic reactor to the case of a diabatic reactor. The primary bifurcation parameter is the fuel fraction; the inflow pressure and inflow temperature are the secondary bifurcation parameters. The inclusion of heat loss in the model does not change the static steady-state bifurcation diagram; the organising centre is a pitchfork singularity for both the adiabatic and diabatic reactors. However, unlike the adiabatic reactor, Hopf bifurcations may occur in the diabatic reactor. We construct the degenerate Hopf bifurcation curve by determining the double-Hopf locus. When the steady-state and degenerate Hopf bifurcation diagrams are combined it is found that there are 23 generic steady-state diagrams over the parameter region of interest. The implications of these structures from the perspective of flammability in the CSTR are discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 3 , January 2004 , pp. 303 - 326
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Doedel, E. J., Fairgrieve, T. F., Sandstede, B., Champneys, A. R., Kuznetsov, Y. A. and Wang, X., AUTO 97: Continuation and bifurcation software for Ordinary Differentia1 Equations (with Hom-Cont), March 1998. Available by anonymous ftp from ftp.cs.concordia.ca/pub/doedel/auto.Google Scholar
[2]Forbes, L. K., “Limit-cycle behaviour in a model chemical reaction: the Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. London Ser A 430 (1990) 641–651.Google Scholar
[3]Forbes, L. K., Myerscough, M. R. and Gray, B. F., “On the presence of limit-cycles in a model exothermic chemical reaction: Sal'nikov's oscillator with two temperature-dependent reaction rates”, Proc. Roy. Soc. London Ser. A 435 (1991) 591–604.Google Scholar
[4]Gray, B. F. and Roberts, M. J., “A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters”, Proc. Roy. Soc. London Ser. A 416 (1988) 361–389.Google Scholar
[5]Gray, P. and Scott, S. K., “Experimental systems 2: Gas-phase reactions”, in Chemical oscillations and instabilities: non-linear chemical kinetics, Volume 21 of International Series of Monographs on Chemistry, 1st ed., (Clarendon Press, Oxford, 1990) Ch. 15.CrossRefGoogle Scholar
[6]Kuznetsov, Y. A., Elements of applied bifurcation theory, Applied Mathematical Sciences 112, 1st ed. (Springer, New York, 1995).CrossRefGoogle Scholar
[7]Nelson, M. I. and Sidhu, H. S., “Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor. I Adiabatic operation.”, J. Math. Chem. 31 (2) (2002) 155–186.Google Scholar
[8]Sal'nikov, I. Ye., “A thermokinetic model of homogeneous periodic reactions”, Dokl. Akad. Nauk SSSR 60 (3) (1948) 405–408, in Russian.Google Scholar
[9]Sal'nikov, I. Ye., “Contribution to the theory of the periodic homogeneous chemical reactions: II A thermokinetic self-excited oscillating model”, Zh. Fiz. Khimii 23 (1949) 258–272, in Russian.Google Scholar
[10]Sexton, M. J. and Forbes, L. K., “An exothermic chemical reaction with linear feedback control”, Dynam. Stability Systems 11 (3) (1996) 219–238.Google Scholar