Simultaneous precipitation and diffusion

An example of inverse calculation from dissolved Ni concentrations in the Eastern Pacific measured by Bruland (1980) is discussed in Chapter 5. Particularly important in inverting the data is to make sure that e must be larger than unity, since the rate constant is a positive parameter, o 

Let us assume a solid infinite matrix with C and CJ being the concentration of two conservative species i and j. i and j may react to form a compound, e.g., a local precipitate, an oxide. with fixed concentrations C0 and C0j. pp and pm, the densities of the compound and matrix, respectively, are assumed to be constant. The compound is finely dispersed, and we call p its volume fraction. Mass balance of element i requires 

Let us assume no advective flux and the diffusive flux to be proportional to the fraction 1 — p of matrix material 

At the onset of the precipitation, the product of gradient terms on the right-hand side can be neglected and we obtain 

The rate of precipitation depends on the rate of change in concentrations 

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