# 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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## Details

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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