×

zbMATH — the first resource for mathematics

Different scenarios in a controlled tubular reactor with a countercurrent coolant. (English) Zbl 0801.35140
Summary: A system of three partial differential equations, representing the dynamics of a tubular reactor with axial diffusion refrigerated by means of a countercurrent fluid, is studied. The PDE system is reduced to an ODE one applying the usual finite differences scheme. The use of a semi- implicit Runge-Kutta method, that has proved itself the most reliable integrator for highly oscillating systems, requires a careful analysis of the numerical integration procedure in order to save computational time. Simulation results show the possibility of chaotic behaviour in a specific parameters range. The variable temperature coolant system does not alter the features of the simplified model with a constant coolant temperature. However, beside the usual period doubling cascade, a mechanism of chaotic transition through type III intermittency and a hysteresis phenomenon are observed. After a preliminary characterization of the chaotic regime by means of the known methods, the paper focuses on the analysis of the new types of scenario.

MSC:
35Q80 Applications of PDE in areas other than physics (MSC2000)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
Software:
YSMP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lamba, P.; Hudson, J.L., Experimental evidence of multiple oscillatory states in a continuous reactor, Chem. engng commun., 32, 369-375, (1985)
[2] Turner, J.S., Complex periodic and nonperiodic behaviour in the Belousov-Zhabotinskii reaction, Adv. chem. phys., 55, 205-217, (1984)
[3] Retzloff, D.G.; Chan, P.C.H.; Chicone, C.; Offin, D.; Mohamed, R., Chaotic behaviour in the dynamical system of a continuous stirred tank reactor, Physica, 25D, 131-154, (1987) · Zbl 0623.58035
[4] Chan, P.C.H.; Retzloff, D.G.; Mohamed, R.; Berdouzi, B.; Chicone, C.; Offin, D., The dynamic behaviour and chaos for two parallel reactions in a continuous stirred tank reactor, Chem. engng commun., 57, 105-138, (1987)
[5] Caneba, G.T.; Crossey, M.J., Chaos in periodically perturbed CSTR, Chem. engng commun., 51, 1-18, (1987)
[6] Hasegawa, S.; Watanaba, N.; Matsubara, M., Bifurcations and chaos appearing in a periodically controlled CSTR, Chem. engng commun., 30, 35-53, (1984)
[7] Chang, H.-C.; Chen, L.-H., Bifurcation characteristics of nonlinear systems under conventional PID control, Chem. engng sci., 39, 1127-1142, (1984)
[8] Pellegrini, L.; Biardi, G., Chaotic behaviour of a controlled CSTR, Comp. chem. engng, 11, 1237-1247, (1990)
[9] Pellegrini, L.; Albertoni, S.; Biardi, G., The occurrence of the chaos in a tubular reactor with axial backmixing, Chem. engng sci., 47, 2463-2468, (1992)
[10] Villadsen, J.; Michelsen, M.L., Solution of differential equations models by polynomial approximation, (1978), Prentice-Hall Englewood Cliffs, NJ · Zbl 0464.34001
[11] Eisenstat, S.C.; Gursky, M.C.; Schulz, M.H.; Sherman, A.H., Yale sparse matrix package — II. the nonsymmetric codes, () · Zbl 0492.65012
[12] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Determining Lyapunov exponents from a time series, Physica, 16D, 285-317, (1985) · Zbl 0585.58037
[13] C. Tablino Possio and L. Pellegrini, Numerical algorithms for chaos detection, in preparation. · Zbl 0788.92027
[14] Bergé, P.; Pomeau, Y.; Vidal, Ch., Order within chaos, (1984), Wiley Paris
[15] Nandapurkar, P.J.; Hlavacěk, V., Chaotic behaviour of a diffusion reaction system, Chem. engng sci., 11, 2747-2760, (1986)
[16] C. Tablino Possio and L. Pellegrini, An example of type III intermittency in chemical engineering, Chem. Engng Sci., in press. · Zbl 0880.92039
[17] Tresser, C.; Coullet, P.; Arneodo, A., On the existence of hysteresis in a transition to chaos after a single bifurcation, J. phys. lett., 41, L243-246, (1980)
[18] Mercader, I.; Massaguer, J.M.; Net, M., Hysteresis in a transition to chaos, Phys. lett. A, 149, 195-199, (1990)
[19] Teymour, F.; Ray, W.H., Intermittency and hysteresis in the dynamic model of a polymerization reactor, Chaos, solitons & fractals, 1, 295-315, (1991) · Zbl 0744.58047
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.