Algorithms for Numerical Solution to the Problem of Diffusion in Nanoporous Media

Introduction. Mathematical modeling of mass transfer in heterogeneous media of microporous structure and construction of solutions to the corresponding problems of mass transfer was considered by many authors [1–9, etc.]. In [6, 7] authors proposed a methodology for modeling mass transfer systems and parameter identification in nanoporous particle media (diffusion, adsorption, competitive diffusion of gases, filtration consolidation), which are described by non-classical boundary and initial-boundary value problems taking into account the mutual influence of micro- and macro-transfer flows, heteroporosity, the structure of microporous particles, multicomponent and other factors. In [8, 9] for a mathematical model of nonstationary diffusion of a single substance in a nanoporous medium described in [2] in the form of a multi-scale differential mathematical problem, the classical problems in the weak formulation were obtained. In this paper, algorithms for solving the above mathematical problems are constructed by using the finite element method. The results of the numerical solution of the test problem are presented. The results confirm the efficiency of the developed algorithms. The purpose is to solve a problem of nonstationary diffusion of single substance in nanoporous medium by constructing discretization algorithms using FEM quadratic basis functions. Results. Algorithms for the numerical solution of the problem of nonstationary diffusion of single substance in a nanoporous medium are proposed. Peculiarities of discretization of the region and construction of the matrix of masses, stiffness, and vector of right-hand sides when solving the problem by using FEM are described. The efficiency of the developed algorithms is confirmed by the results of solving a model example. Keywords: mathematical modeling, numerical methods, nonstationary diffusion, nanoporous medium, finite element method.

TO THE SOLUTION OF ISSUES OF NONSTATIONARY DIFFUSION IN LAYERED ENVIRONMENTS

The issues of nonstationary diffusion in layered structures are considered. When designing the devices for implementing mass transfer processes, it is necessary to take into account the properties of the substance and the nature of the processes. Design time reduces significantly and the efficiency of the devices is higher if a good physical model is built and a mathematical analysis with kinetics of the processes is applied. The difficulties of theoretical analysis and calculation of mass transfer are determined by the complexity of the transfer mechanism to and from the phase boundary. Therefore, simplified models of mass transfer processes are used in which the mass transfer mechanism is characterized by a combination of molecular and convective mass transfer. Many important practical problems involve the calculation of nonstationary diffusion (Fick's second law of diffusion) for a certain volume of substance (substances). For qualitative evaluation of processes, in the case of symmetry, volumetric issues can be considered as one-dimensional tasks, i.e. dependent on one coordinate. The general solution of the non-stationary diffusion equation for layered environments is proposed. The case of non-stationary boundary conditions of the third kind on the external surface and boundary conditions of the fourth kind conjugation for contiguous layers has been considered. The solution is obtained by separating the Fourier variables by the eigenfunctions of the problem using the Duhamel integral. The proposed solution is explicit and due to the recurrent form of the basic relations can be useful in numerical calculations

Effect of Adsorbent Grain Size on the Pressure Swing Adsorption

An analytical solution is obtained for a system of differential equations consisting of the equation for diffusion and absorption of a component in adsorbent grains and the balance equation for this component moving in the adsorbent layer.The technique for calculation of steady-state periodic adsorption processes based on expansion of a concentration signal into Fourier series of eigenfrequencies of this adsorption process is proposed. After decomposition each of the eigenwaves independently of the others is passed through the adsorbent layer, and at the output from this layer all this solutions for individual eigenwaves are summed. The solution to a problem of passing an arbitrary periodic concentration signal through the adsorbent layer is obtained as a result of summing the solutions for individual eigenwaves.The proposed method takes into account nonstationary diffusion processes inside the adsorbent grains. It makes possible to analyze effect of the adsorbent grain sizes on a pressure swing adsorption processes.

Numerical Analysis of Inverse Extremum Problems for the Nonlinear Nonstationary Diffusion-Reaction Equation

The model of transfer of substance with Dirichlet boundary condition is considered. Inverse extremum problem of identification of the main coefficient in a nonstationary diffusion-reaction equation is formulated. The numerical algorithm based on the conjugate gradient method for solving this extremum problem is developed and is programmed on computer. The results of numerical experiments are discussed.