When the barrier has two or more independent diffusional pathways present in parallel, the total permeability coefficient Ptotal is the sum of P for each individual pathway as

On the other hand, when the barrier consists of two or more physically distinct layers in series, the reciprocal of Ptotal is the sum of reciprocal P for each individual layer as

With regard to Eq. (4.2), the general diffusion equation describing three standard geometries (slab sheet, cylinder, and sphere) is

where a is the half-thickness or radius (x = 0 at the center and x = a at the surface), and a = 0, 1, and 2 for planar, cylindrical, and spherical geometry, respectively. The initial and boundary conditions needed to solve Eq. (4.7) vary depending on the type of system under consideration.

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