Analytical modeling of one-dimensional diffusion in layered systems with position-dependent diffusion coefficients

Diffusion in stratified porous media is common in the natural environment. The objective of this study is to develop analytical solutions for describing the diffusion in layered porous media with a position-dependent diffusion coefficient within each layer. The orthogonal expansion technique was used to solve a one-dimensional multi-layer diffusion equation in which the diffusion coefficient is expressed as a segmented linear function of positions in the porous media. The behavior of the solutions is illustrated using several examples of a three-layer system, with constant diffusion coefficient α1 in layer 1 (0 < x < l1), α3 in layer 3(l2 < x < l3), and a linearly position-dependent diffusion coefficient α1(1 + Δ(x − l1)/(l2 − l1)) in the center layer (Δ = α3/α1 − 1). Because of the asymmetry of the layered system, the diffusion and related concentration distributions are also asymmetrical. For a given Δ value, the smaller the value of (l2 − l1)/l3, the more significant the accumulation of concentration in the middle transition zone (l1 < x < l2), the sharper the change in the concentration profile of spatial distribution. Therefore, transition between two layers has significant effects on diffusion.