Dynamic Behavior of a Nonisothermal Chemical System

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A nonisothermal chemical system displays interesting dynamic behavior ranging from period-one oscillations to period doubling and chaos, depending on the value of the bifurcation parameter, .
Contributed by: Housam Binous and Zakia Nasri (March 2011)
Open content licensed under CC BY-NC-SA
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The following nonisothermal reaction system is theoretical. The steps are as follows:
Here is a chemical precursor with constant concentration,
is the final product,
and
are intermediate chemical species,
,
and
are rate constants for the reactions, and
,
, and
are the concentrations of the hypothetical chemical species
,
, and
.
The autocatalytic reaction is the following step: , with
catalyzing its own formation. This step introduces the nonlinear term
in the governing equations.
The last reaction, B → C + Heat is exothermic. The rate constant of the first reaction,
P → A, follows the Arrhenius rate-law. Thus
depends on the temperature.
The governing equations for the two intermediate species and the temperature are usually written in the form:
,
,
.
The dimensionless governing equations are:
,
,
.
Here
,
, and
are dimensionless concentrations of
,
, and the dimensionless temperature, and the four parameters
,
,
, and
depend on the rate constants of the individual reactions
,
,
, and
, the concentration of the precursor
, the molar density
, the molar heat capacity
, the surface heat transfer coefficient
, the surface area
, the surrounding temperature
, the heat of reaction
for the reaction
, and the activation energy
of the reaction
.
The Demonstration illustrates the dynamics of the concentrations ,
, and the temperature
for various values of the bifurcation parameter
. Choose "time series" to get a plot of
versus time or "phase space" to get a three-dimensional parametric plot of
.
For = 0.5586, the
phase-space graph is that of a spiral attractor.
Reference: S. K. Scott and A. S. Tomlin, "Period Doubling and Other Complex Bifurcations in Non-isothermal Chemical Systems," Philosophical Transactions of the Royal Society of London, Series A: Mathematical and Physical Sciences, 332(1624), 1990 pp. 51–68.
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