Analytical modeling of one-dimensional diffusion in layered systems with position-dependent diffusion coefficients |
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| Authors: | Gang Liu Bing C Si |
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| Institution: | 1. Department of Soil Science, University of Saskatchewan, Saskatoon, SK, Canada S7N5A8;2. Laboratory for Plant–Soil Interaction Processes, Ministry of Education, College of Resources and Environment, China Agricultural University, No. 2 Yuanmingyuan Xi Lu, Beijing 100094, PR China |
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| Abstract: | 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. |
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| Keywords: | Multi-layer porous media Diffusion Non-homogenous boundary conditions Orthogonal expansion Eigenfunction Eigenvalue |
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