Unified Multilayer Diffusion Model and Application to Diffusion Experiment in Porous Media by Method of Chambers

Abstract

Diffusion coefficient is an important parameter for examining contaminant transport in the environment. Chamber methods (with or without external mixing devices) are the most popular methods for measuring effective diffusion coefficients in porous media (Deff) through air or water. The objectives of this paper were to apply simplified and unified analytical methods for both perfectly mixed and nonmixed (one- or two-) chamber systems and to examine how mixing affects the estimation of Deff. An analytical solution for a multilayer transient diffusion model was applied to the chamber methods without external mixing. By increasing the diffusion coefficient in reservoirs (D1 and D3) more than 10 times from the value for air or water (D0), the model was sufficient to approximate the well-mixed condition and, consequently, can be used to model transient diffusion in chamber systems with external mixing devices. We demonstrated that at long time Deff was related to the first eigenvalue (beta1) of a transcendental equation, which provided a quick method for determining Deff accurately from experimental data. The error caused by using the well-mixed approximation can be significant for a single-chamber system when there are no external mixing devices. This error increased rapidly with decreases in the experimental duration. A good fit for the concentration versus time curve could not be obtained forthe well-mixed solution, especially when sampling ports were near the boundary (x=0) and interface (x = l1). The proposed solutions are useful when the reservoir or chamber methods are used for measuring Deff and have wide applications in predicting contaminates transport in porous media and groundwater.

A physicist's guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

“…4 to investigate spectral sums in these domains that are often appear in modeling heat transfer in composite materials, drug release from multilayered capsules and drug-eluting stents, radioactive contamination of soils and waste disposal, etc. [48,49,50,51,52,53,54,55]. One can also consider diffusion in wedges, angular sectors and solid angles [3,20], as well as in the presence of confining potentials [56].…”

Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

“…with T and F defined in Eqs. (20), (22), respectively, in which λ is replaced by s. To obtain the propagator in time domain, one needs to perform an inverse Laplace transform. This is done by looking for the poles s = λ n ofG and the above formula shows that they are given by the zeros of F (s), as expected.…”

“…We provide here a rough analysis of Eq. (22) in order to study this phenomenon. We discard the elementary case of a single interval (m = 1) where the roots of F are explicitly known [1,2].…”

“…2.1) and obtain the equation of the spectrum as a transcendental equation F (λ) = 0 (Eq. (22)). Three following subsections are more technical and may be omitted in a first reading.…”

Diffusion dynamics of small molecules from mesoporous silicon films by real-time optical interferometry

“…This phenomenological model is employed for its simplicity, and its applicability is well supported by the data-fitting results presented herein. It is noted that the general approach presented in this work can be extended to different physical scenarios, such as diffusion of molecules into a closed-ended porous film [27,28], diffusion through open-ended porous media/films [25,[35][36][37], and diffusion of solute through multilayer porous media with spatially varying diffusion coefficient [22,23], so long as proper boundary conditions are enforced and a valid concentration profile is employed. Refinements such as inclusion of concentration dependence or spatially varying diffusion coefficient may also be studied with proper analytical solutions of the concentration function.…”