Linear reaction diffusion system, stationary solution

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

For d 4 singular term (d — l)(d — 3)/(4 j2) does not allow to find the solution at r/ = 0. It has simple interpretation in systems with so large space dimensionalities no variable rj = r/fo exists there. Similar to d = 3 in the linear approximation, for d 4 we can find the stationary solutions, Y (r, oo) = y0(r). For them the reaction rate K(oo) = Kq — const and the classical asymptotics n(t) oc Ya, ao = 1 hold. Therefore, for a set of kinetic equations derived in the superposition approximation the critical space dimension could be established for the diffusion-controlled reactions. 

Reinmuth has examined chronopotentiometric potential-time curves and proposed diagnostic criteria for their interpretation. His treatment applies to the very limited cases with conditions of semi-infinite linear diffusion to a plane electrode, where only one electrode process is possible and where both oxidized and reduced forms of the electroactive species are soluble in solution. This approach is further restricted in application, in many cases, to electrode processes whose rates are mass-transport controlled. Nicholson and Shain have examined in some detail the theory of stationary electrode polarography for single-scan and cyclic methods applied to reversible and irreversible systems. However, since in kinetic studies it is preferable to avoid diffusion control which obscures the reaction kinetics, such methods are not well suited for the general study of the mechanism of electrochemical organic oxidation. The relatively few studies which have attempted to analyze the mechanisms of electrochemical organic oxidation reactions will be discussed in detail in a following section. 

The simplest approximation for treating a complex flame system corresponds to assuming the pseudo-stationary state concentration for each of the chemical intermediates in the expressions for the net rate of production of the fuel and product molecules, and setting the mole fractions and fractions of the mass rate of flow for each of these intermediates equal to zero in the diffusion, energy balance, and motion equations. The resulting equations correspond to a simple one step chemical reaction system with only a single linearly independent and the solution of these equations presents no difficult mathematical problems. 

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