Pool chemical model with diffusion

We may also briefly consider the behaviour of the simple autocatalytic model of chapters 2 and 3 under reaction-diffusion conditions. In a thermodynamically closed system this model has no multiplicity of (pseudo-) stationary states. We now consider a reaction zone surrounded by a reservoir of pure precursor P. Inside the zone, the following reactions occur  

Using the dimensionless concentrations and rate constants introduced in chapter 3, the governing rate equations for this scheme can be written in the form 

At low p0, the system has a high stationary-state concentration of A relative to that of the autocatalyst. Typically, both profiles have a maximum at the centre of the reaction zone, p = 0, as shown in Fig. 9.11 (a). High reactant concentrations favour larger concentrations of the autocatalyst B and lower 

Stationary-state loci (a) ass(0)-/io and (b) / s,(0)-/io for diffusive model of autocatalysis with decaying precursor, showing hysteresis with D = 0.05 and = 0.01. The extinction and ignition points occur at = 0.636 and 0.72 and there is a Hopf bifurcation at fi0 = 1.105. 

These features become even more accentuated if the dimensionless diffusion coefficient D is made smaller. In the limit D - 0, as may be expected, the profiles tend to be virtually uniform across most of the region, with ass and j3ss given by eqn (9.33). At the edges of the reaction zone the system develops thin boundary layers, with thickness of order D 12. 

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