Diffusion of decaying admixtures in materials with random spherical inclusions
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DIFFUSION OF DECAYING ADMIXTURES IN MATERIALS WITH RANDOM SPHERICAL INCLUSIONS A. R. Tors’kyi,1 E. Ya. Chaplya,2 and O. Yu. Chernukha1
UDC 517.958:536.72
We study the process of diffusion of admixtures in a layer of an inhomogeneous material with randomly located spherical inclusions modeling, e.g., the lavalike fuel-containing masses formed at nuclear power plants. The contact boundary-value problem of diffusion is reduced to a boundary-value problem of mass transfer in the entire body by the methods of the theory of generalized functions. We propose an equivalent integro-differential equation and obtain its solution by the method of successive approximations in the form of a Neumann integral series. The convergence of this series and the theorem of existence of the solution are proved. We deduce an approximate formula for the evaluation of the concentration of radionuclides averaged over the ensemble of phase configurations and perform its numerical analysis. It is shown that the characteristics of the material affect the behavior and concentration of admixtures in a body with randomly inhomogeneous structure. If the diffusion coefficient in the inclusions is lower than in the matrix, then the average concentration in the layer increases, and vice versa. As a result of the process of decay of foreign particles, their average concentration may become several times lower.
In the mathematical description of the process of migration of admixtures, it is often necessary to take into account their natural degradation (decay) and the structure of the material of the body [1–3]. Thus, microdefects (whose influence on the intensity of diffusion is significant) are formed as a result of decay in radioactive fuelcontaining materials [2, 4]. At the same time, the problem of mathematical simulation of the transfer of admixtures is directly connected with certain schematic representations of porous media and the corresponding prognostic estimates, e.g., of the protection of underground waters, must take into account the process of decay [5, 6]. In what follows, for the description of the local structure of a body, we use the approximation of inclusions randomly distributed over the main phase of the material. It is assumed that the inclusions are spherical and their volume fraction is constant. The decay of foreign particles is taken into account in the key equations of transfer. The conditions of contact between the inclusions and the matrix are imperfect. By the methods of the theory of generalized functions, we determine the distributions of the field of concentrations of admixtures averaged over the ensemble of configurations of the phases. Under certain physical conditions, the process of diffusion can be described by using the continual models of heterodiffusion [7] or the solutions of boundary-value problems, in particular, for regular structures [8]. In the stationary case, the total amount of admixtures passing through a layer can be found by using the effective diffusion coefficient introduced according to the wel
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