Abstract
The ultimate purpose of this paper is to present a new analytical formulation to simulate diffusion with retention in a reactive medium under stable thermodynamic conditions. The analysis of diffusion with retention in a continuum medium is developed after the solution of an equivalent problem using a discrete approach. The new law may be interpreted as the reduction of all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect namely a circulatory motion. It is remarkable that the governing equation requires a fourth order differential term as suggested by the discrete approach. The relative fraction of diffusion particles β is introduced as a control parameter in the diffusion-retention law as suggested by the discrete approach. This control parameter is essential to avoid retention isolated from the diffusion process. Two matrices referring to material properties are introduced and related to the real phenomenon through the circulation hypothesis. The governing equation may be highly non-linear even if the material properties are constant, but the retention effect is a function of the concentration level, that is, β is a function of the concentration.
Highlights
Spreading of particles or microorganisms immersed in a given medium or deployed on a given substrata is frequently modeled as a diffusion process
The main purpose of this paper is to propose a model that can be used to simulate with good accuracy the retention effect in a diffusion process
The governing equations for diffusion with retention appearing in the literature so far, to the best of our knowledge, belong to the class of the second order partial differential equation as in the classical diffusion approach
Summary
Spreading of particles or microorganisms immersed in a given medium or deployed on a given substrata is frequently modeled as a diffusion process. Trying to solve the retention problem by imposing an artificial dependence of the diffusion coefficient on the particle concentration p(x, t) or introducing extra differential terms while keeping the second order rank of the governing equations, or both, disguises the real physical phenomenon occurring in the process. Since the corresponding governing equation includes for this case two new parameters B and k, it would be presumably possible to obtain a better representation of experimental results by adjusting properly the values of these two parameters This is a refinement of the theoretical framework, but the basic event remains the same, belonging to the class of diffusion problems without taking into account any other phenomenon as retention for instance. That is the components ci j of C refer to a circulation effect that adequately simulates the real phenomenon with respect to the dynamical and geometric characteristics of the circulatory motion
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