Asymptotic Behavior in the Transport Limited Regime

In this section we shall generalize some results of Section 2.3 concerning the asymptotic behavior of distributed chemical systems. We shall show that as the size of the system increases, the rates of chemical reactions and transport processes are appreciable only in a surface layer of constant thickness while the interior of the system is very nearly at equilibrium. Using this result, we shall obtain some a priori bounds for the effectiveness factor. The related problem of uniqueness of the steady state at large values of Vq has not been solved rigorously. It is expected, however, that as in the case of a single reaction, the steady state is unique for sufficiently large Tq. 

An important property of distributed chemical systems is that in addition to the a priori bounds for the state variables, a priori bounds can be obtained for the derivatives of the state variables, i.e., the fluxes. The procedure of obtaining the a priori bounds for the fluxes is based on the following observation Regardless of the value of the radius Tq, the a priori bounds for the concentrations will be violated if the values of the fluxes become too high. 

The same exactly bounds hold in the case of planar regions. 

Effectiveness factors can be defined for each of the reactions or each bp Using the definition of Section 2.3, we can write the effec- 

Lower bounds for can be similarly obtained if these quantities 

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The remaining four sections will be concerned with problems of uniqueness, stability, and asymptotic behavior of the steady states. The results are in close analogy with those obtained in Section 1.9, especially in regard to the behavior of the system in the transport and reaction limited regimes. 

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