Overall parabolic diffusion equation

Fig. 6. Plotting of Cd desorption by chloride or citrate from Fe oxides formed at initial citrate/Fe(II) MRs of (a) 0 and (b) 0.1 based on overall parabolic diffusion equation.

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

Incorporation of a diffusion term in nucleation and growth reaction models has been proposed by Hulbert [68]. Interface advance is assiuned to fit the parabolic law and is proportional to but the nucleation step is uninhibited. The overall rate expressions have the same form as the Avrami-Erofeev equation ... 

The authors analysis (38) uses film theory on the gas-side and penetration theory on the liquid side. The penetration theory was adopted as offering a more realistic basis to describe the diffusion and reaction. The set of parabolic partial differential equations which describe this diffusion and reaction were solved in conjunction with the customary boundary conditions, plus a number of subsidiary relationships which ensure electrical neutrality and are compatible with the initial loadings of reagent amine. Equilibrium was assumed to be established in the bulk, but otherwise the fluxes were not jointed to the overall material balances on the gas and liquid phases. Even so, the computation appears quite formidable. 

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