Xa1 Xa2

■mln where Xm]n is the logarithmic mean of the molecular fractions. For a small difference between XA1 and XA2, the arithmetic mean (Xm) can be used. For a dilute solution, 1 - XA1 ~ 1 - XA2 and Xm = 1.0, giving


Diffusion in Solids

Diffusion of gases and liquids in solids are phenomena occurring in many processes such as drying, packaging, catalytic reactions, leaching, and membrane separations. The two classifications of diffusion in solids are structure independent and structure dependent. In the former, the diffusing substance is dissolved in the solid to form a homogeneous solution, and thus Fick's law can be directly applied. In the latter, pores, capillaries, and other interconnected voids in the solid allow some flow of the diffusing substance. Because this porous solid is anisotropic, the diffusion path of molecules will be different than that described by Fick's law and an overall, apparent, or effective diffusivity coefficient may be used.

Structure-Independent Diffusion in Solids. A homogeneous solution in the solid is assumed here; therefore, diffusion is independent of the actual structure of the solid. The solute concentration gradient is the actual driving force, and the general equation 20 for binary diffusion can be applied directly. Usually the concentration CA is small compared to the total concentration C, and CA/C can be neglected. By further assuming that DAR is constant, we can integrate to get

If we set Cj^i *** Cj^Xj^i and C^2 ^ Qa-^A2> equation 22 becomes na = ^f(ca ca2)

where XA1 and XA2 are the molecular fractions of A at point 1 and point 2, respectively. Similarly, for liquid B, we have

When dealing with diffusion of gases in a solid, the solubility S of a gas solute A is usually expressed as (vol of A at 0°C and 1 atm)/(vol of solid • pressure of A). To convert this into concentration CA (kmol of A/vol of solid), the following equation can be used.

spa 22.414

Because the diffusivity D¿g is usually concentration dependent, an average concentration CA should be used when calculating the value of .Dab-

Diffusion of Liquid A through a Stagnant Layer of Liquid B.

In this case, NB = 0, and equation 17 can be rewritten where 22.414 is the volume (in m3) of 1 kmol of an ideal gas at 0°C and 1 atm.

In the case of diffusion through a solid cylinder in the radial direction, the total diffusing area is not constant. For a cylinder with an inner radius rt, outer radius r2, and length I, we have

dCA dr

Because A = 2nrl for the cylinder, substituting this expression into equation 29, rearranging, and integrating gives na d

2 ni

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