Solid-state diffusion continuity equation

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

Continuum Theory. Solid-state diffusion is described in terms of a continuity equation known as Fick s second law ... 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

The steady-state continuity equation for A in the pore, accounting for diffusion in the pore axial direction and effective diffusion inside the solid at the pore wall, is... 

The diffusion theory states that matter is deposited in a continuous way on the surface of a crystal at a rate proportional to the difference in concentration between the bulk and the surface of the crystal. The mathematical analysis is then the same as for other diffusion and mass transfer processes and makes use of the film concept. Sometimes, the film theory is considered to be an oversimplification for crystallization and is replaced by a random surface removal theory (20-23). For both theories the rate of crystal growth (dm/dt) is given by equation XVII, where m, is the mass of solid deposited in time t k, the mass transfer coefficient by diffusion. A, the surface area of the crystal, c, the concentration in the supersaturated solution and Cj, the concentration at the crystal-solution interface (3). For the stagnant film and random surface removal model, equations XVIII and XIX can be used, respectively (3,4) D is the diffusion coefficient, x, the film thickness and f, the fractionai rate of surface renewal. 

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