Forced oscillations of chemical reactors. (English) Zbl 0722.92021
Spatial inhomogeneities and transient behaviour in chemical kinetics, Proc. Int. Conf., Brussels/Belgium 1987, Proc. Nonlinear Sci., 13-30 (1990).
[For the entire collection see Zbl 0691.00017.]
For understanding physical systems, mathematical model building and its validation by experimental tests is necessary. Mathematical models involve simplifications and serve as purpose even when they are no longer capable of reflecting the behavior of the physical system with quantitative accuracy. A model is good even though its ability to reproduce data may be limited.
This paper presents five model reactions that have played important roles in the study of chemical oscillations, describes some features of their autonomous behaviour and discusses some characteristics of their response to periodic excitations. The five models are:
1) Two first-order reactions \(A\to X\to B\) taking place in sequence, with A concentration constant; 2) the isothermal operation of a continuous flow stirred tank reaction; 3) the autocatalytic reactions \(A+2B\to 3B\), \(B\to C\), taking place in a stirred tank; 4) the Brusselator, which is a mechanism for \(A+B\to D+E\) in four steps: \(A\to X\), \(B+X\to Y+D\), \(Y+2X\to 3X\), \(X\to E\), A and B concentrations being constants; 5) the catalytic oxidations \(A+B\to C\) in which two absorbed molecules require two neighbouring vacant sites to react: \(A+S\to A^*\), \(B+S\to B^*\), \(A^*+B^*+2S\to C+4S.\)
The two next sections present the autonomous behaviour of the models and the behaviour in response to forcing. The above models take the form of a pair of ordinary differential equations with constant parameters. The solution, presented in the so-called phase plane, appears as a curve from a starting point to the boundary or to an invariant set. In order to characterize qualitatively and quantitatively the autonomous behaviour of the system a bifurcation diagram is exemplified when the topological structure of the parameter space is established.
If one of the operating parameters is chosen and subjected to forcing by a periodic function, the response of the system may be periodic with the same period or some integral multiple of it, or the response may be quasiperiodic or chaotic. In the case of forcing, a stroboscopic phase portrait plays the same role as the phase portrait in the autonomous case. It could describe the forced response of a system by dividing the parameter space into regions of qualitatively identical stroboscopic phase planes. The transition implicated in continuation methods is important in dividing the space of the parameters with equivalent domains for the autonomous systems and in drawing the excitation diagram.
For understanding physical systems, mathematical model building and its validation by experimental tests is necessary. Mathematical models involve simplifications and serve as purpose even when they are no longer capable of reflecting the behavior of the physical system with quantitative accuracy. A model is good even though its ability to reproduce data may be limited.
This paper presents five model reactions that have played important roles in the study of chemical oscillations, describes some features of their autonomous behaviour and discusses some characteristics of their response to periodic excitations. The five models are:
1) Two first-order reactions \(A\to X\to B\) taking place in sequence, with A concentration constant; 2) the isothermal operation of a continuous flow stirred tank reaction; 3) the autocatalytic reactions \(A+2B\to 3B\), \(B\to C\), taking place in a stirred tank; 4) the Brusselator, which is a mechanism for \(A+B\to D+E\) in four steps: \(A\to X\), \(B+X\to Y+D\), \(Y+2X\to 3X\), \(X\to E\), A and B concentrations being constants; 5) the catalytic oxidations \(A+B\to C\) in which two absorbed molecules require two neighbouring vacant sites to react: \(A+S\to A^*\), \(B+S\to B^*\), \(A^*+B^*+2S\to C+4S.\)
The two next sections present the autonomous behaviour of the models and the behaviour in response to forcing. The above models take the form of a pair of ordinary differential equations with constant parameters. The solution, presented in the so-called phase plane, appears as a curve from a starting point to the boundary or to an invariant set. In order to characterize qualitatively and quantitatively the autonomous behaviour of the system a bifurcation diagram is exemplified when the topological structure of the parameter space is established.
If one of the operating parameters is chosen and subjected to forcing by a periodic function, the response of the system may be periodic with the same period or some integral multiple of it, or the response may be quasiperiodic or chaotic. In the case of forcing, a stroboscopic phase portrait plays the same role as the phase portrait in the autonomous case. It could describe the forced response of a system by dividing the parameter space into regions of qualitatively identical stroboscopic phase planes. The transition implicated in continuation methods is important in dividing the space of the parameters with equivalent domains for the autonomous systems and in drawing the excitation diagram.
Reviewer: S.Curteanu (Iaşi)
MSC:
| 92E20 | Classical flows, reactions, etc. in chemistry |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 80A30 | Chemical kinetics in thermodynamics and heat transfer |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 37-XX | Dynamical systems and ergodic theory |
