Molecular Diffusion of Solids, Liquid and Gases

Is there diffusion in solids, like leaching, drying, etc. involving contact of fluids with solids and involving diffusion in the solid phase?

Fick's law for the steady-state diffusion can be written as 

NA = - DA dCA/dz

where NA is the rate of diffusion of A per unit cross-section of solid. DA is the diffusion of A through the solid, dCA/dz is the concentration gradient in the direction of diffusion.

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Molecular Diffusion of liquid

Study-state equimolar counter diffusion:

We Have NA = - NB

The flux equation for steady-state equimolar counter diffusion is given by

NA = DAB/z(CA1 - CA2) = DAB/z(p/M) avg. (xA1 - xA2) 

Where xA1-xA2 is the concentration difference in terms of mole fraction. 

(p/M) avg. = C avg. = [p1/M1 - p2/M2]/2

C avg. is the total average concentration of A and B in kmol/m3, p1 and p2 are the average densities of the solution in kg/m3 at locations 1 and 2 respectively, and M1 and M2 are the average molecular weight of the solution at locations 1 and 2 respectively in kg/kmol.

Steady-state diffusion of A through non-diffusing B:

We know that NA = Constant, NB = 0

The flux is given by 

NA = DAB/z.xAM(p/M) avg. (xA1 - zA2) 

Where   xB, M = (xB2 - xB1) / ln (xB2/xB1).

Molecular Diffusion of Gas

For the diffusion in the z-direction and for NA and NB both constant (steady-state) we will derive equations for the flux. 

For an ideal gas, we have

CA/C = PA/P = yA

Where PA is the partial pressure of component A and P is the total pressure. 

Diffusivity

Diffusivity is defined as the ratio of flux to the corresponding concentration gradient. The dimensions of the diffusivity are L2/θ and its SI units are m2/s. The diffusivity of any component is a measure of its diffusive mobility and is a function of the temperature, pressure, nature, and concentration of the other components.

Diffusivity mobility is a function of the number of collisions. Therefore, diffusivity increases with a decrease in pressure because the number of collisions is less at lower pressure. However, the effects of pressure are negligible in the case of liquids. Also as the random motion of the molecules increases with an increase in temperature, diffusivity increases with an increase in temperature. 

NA = JA + xA (NA + NB) 

Steady-state Equimolar counter Diffusion

The diffusion of two components in opposite directions each other in a stationary mixture is of importance in the distillation of a binary system. If the two components are at the same temperature and the same total pressure, then no net movement of the mixture will take place and it will be simply the replacement of the molecules of one component by the molecules of other components. In the language of diffusion, two components diffuse at the same molar flow rate in opposite directions each other. This overall transfer process is called equimolar counter diffusion. 

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Take these Notes is, Orginal Sources: Unit Operations-II, KA Gavhane