Reaction-Diffusion Equations for an Autocatalytic Reaction

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This Demonstration shows the behavior of a reaction-diffusion system in which an autocatalytic reaction takes place.

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Consider the reaction scheme with rate of reaction , where and are the reactant concentrations and is the reaction rate constant. The reaction takes place in a capillary tube of length filled with a fluid separated by an impermeable membrane. Half of the tube's length contains reactant and the other half contains reactant . The membrane is removed at time and the reaction-diffusion process begins. The plots of functions and are shown for user-selected values of time , diffusivity , and reaction rate constant .

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Contributed by: Clay Gruesbeck (March 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The governing equations are

and

.

With the membrane at , the initial conditions are

,

and , with boundary conditions

at and . Here and are the diffusion coefficients of and , respectively, is distance, and is time.

The system can be simplified by making the transformations

and

to obtain the system

,

,

which has the following analytical solution when [1]:

,

,

with

, where is a free parameter taken as .

Reference

[1] A. H. Salas, L. J. Martinez H., and O. Fernandez S., "Reaction-Diffusion Equations: A Chemical Application," Scientia et Technica, 17(46), 2010 pp. 134–137. www.redalyc.org/pdf/849/84920977041.pdf.



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