Combination of External and Internal Diffusion Limitations

Complex reaction mechanisms and catalytic surface effects can also lead to multiplicity. Only multiplicity resulting from the combined effect of diffusion, adsorption, and reaction in an isothermal particle will be briefly dealt with here. Luss [1968] derived the following necessary and sufficient condition for uniqueness  

For an nth-order reaction, this condition requires that 

Interfacial concentration and temperature gradients as well as temperature gradients inside the particle widen the range of operating conditions which may lead to the occurrence of multiple steady states. For further discussion of this pathological behavior, the reader is referred to Lee and Luss [1969, 1970, 1971], Luss [1971,1976], and Varma and Aris [1977], 

The addition of fluid-phase resistance to internal resistance is relatively easy for a first-order reaction. It has to be accounted for that is not known and must be related to the fluid field value Cby 

Equation (3.11-2), when used in defining the effectiveness factor based on the bulk fluid concentration C, gives 

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