Mathematical Modelling of Stationary State and Oscillatory Regimes in Oregonator
DOI:
https://doi.org/10.14529/cmse180101Keywords:
Belousov–Zhabotinsky’s reaction, oregonator, steady states, oscillatory regimesAbstract
Oscillatory systems are widely used in various areas of research (chemical, biological, environmental, oscillators). The article presents the five-stage Field-Noyes-Köros model of the Belousov-Zhabotinsky reaction and the corresponding mathematical model of the oregonator. A system of equations for the stationary states oregonator is derived. Stationary states of oregonator are calculated depending on the speed of direct reactions for various values of stoichiometric coefficient. Simulation of homogeneous stationary state of the system was conducted according to the experimental data of the authors of the model. The stationary solutions corresponded to the physical meaning of the model. In framework of the system of ordinary differential equations of the kinetics reactive systems of oscillatory regimes are calculated. The time for oscillatory regime is determined. The amplitudes of the oscillations are corresponded to the experimental data of the authors of the model. The instability of the stationary state of the oregonator to perturbations is investigated.
References
Belousov B.P. Periodicheski deystvuyushchaya reaktsiya i eye mekhanizmy [Bath Reaction and Its Mechanism. The Collection of Abstracts on Radiation Medicine]. М.: Мedgiz Publ., 1959. pp. 145–148.
Volter B.V., Salnikov I.E. Ustoychivost rezhimov raboty khimicheskikh reaktivov [Stability of Operation Modes of Chemical Reactors]. M.: Khimiya, 1972. (in Russian).
Volterra V. Matematicheskaya teoriya borby za sushchestvovaniye [Mathematical Theory of the Struggle for Existence. Russian Translation]. М.: Nauka, 1976. 287 p.
Zhabotinsky А. М. Kontsentratsionnyye kolebaniya [Concentration Auto-Oscillations]. M.: Nauka, 1974. (in Russian)
Zhabotinskiy A. M. Periodicheskiye reaktsii okisleniya v zhidkoy faze [Periodic Liquid Phase Reactions]. Proc. Ac. Sci. USSR, 1964. vol. 157. pp. 392–395. (in Russian)
Zhabotinskiy A. M. Periodicheskiy protsess okisleniya malonovoy kisloty v rastvore (issledovaniye kinetiki reaktsii Belousova) [Periodic Process of the Oxidation of Malonic Acid in Solution (Study of Kinetics of Belousov’s Reaction)]. Biofizika. vol. 9. 1964. pp. 306–311. (in Russian)
Kholpanov L.P., Prokudina L.A. Mathematical Modeling of Unstable Mass Transfer Complicated by Chemical Reactions. Theoretical Foundations of Chemical Engineering, 2005. vol. 39, no 1. pp. 39-49.
Yaparova N.M. Numerical Method for Solving an Inverse Problem for Nonlinear Para-
bolic Equation with Unknown Initial Conditions. Vestnik Yuzho-Uralskogo gosudarstvennogo universiteta. Seriya " Vychislitel'naya matematika i informatika " [Bulletin of South Ural State University. Series: Computational Mathematics and Software Engineering]. 2016. vol. 5, no. 2. pp. 43–58. (in Russian) DOI: 10.14529/cmse160204.
Field R.J., Koros E., Noyes R. Oscillations in Chemical Systems. II. Thorough Analysis of Temporal Oscillation in the Bromate-Cerium-Malonic Acid System. Journal of the American Chemical Society. vol. 94, no. 25. 1972. pp. 8649–8664. DOI: 10.1021/ja00780a001.
Field R.J., Noyes R.M. Oscillations in Chemical Systems. IV. Limit Cycle Behavior in a Model of a Real Chemical Reaction. The Journal of Chemical Physics. vol. 60, no. 5. 1974. pp. 1877–1884. DOI: 10.1063/1.1681288.
Edelson D., Field R.J., Noyes R.M. Mechanistic Details of the Belousov-Zhabotinskii reaction. International Journal of Chemical Kinetics. vol. 7. 1975. pp. 417–432. DOI: 10.1021/j100381a039.
Lotka A.J. Contribution to the Theory of Periodic Reactions. Journal of the Chemical Society. no. 14(3). 1909. pp. 271–274. DOI: .10.1021/j150111a004.
Lotka A.J. Undamped Oscillations Derived from the Law of Mass Action. Journal of the American Chemical Society. no. 42(8). 1920. pp. 1595–1599. DOI: 10.1021/ja01453a010.
Prokudina L.A., Kholpanov L.P. Nonlinear Development of Perturbations of Diffusion-Complicated Autocatalytic Reaction. Theoretical Foundations of Chemical Engineering. vol. 38(6). 2004. pp. 636–643. DOI: 10.1007/s11236-005-0037-0.
