Evolution of Mathematical Models of Non-stationary Heat (Mass) Conductivity Processes in Bodies of Canonical Form

Abstract

Currently, there are a large number of materials that are subject to thermal effects during their production; from the point of view of the principles of geometry, their shape can be reduced to classical bodies of canonical shape: plate, cylinder, or sphere. During thermal treatment of solid materials (heat and moisture treatment, drying, firing), the transfer potentials (temperature, mass content) change critically with respect to the process time. When solving boundary value problems of heat and mass (moisture) conductivity in similar cases, it is proposed to use the “zonal” method and the “microprocesses” method. The main positions of the “microprocesses” method, as applied to the modeling of boundary value problems of heat and mass transfer for bodies of canonical shape under boundary conditions of the first kind (Dirichlet conditions), are presented in the previous works of the authors [1–3]. The current paper presents a methodology based on the “microprocess” method for solving boundary value problems of heat and moisture conductivity under more general boundary conditions, conditions of the third kind (Riemann–Newton). The high adaptability of these conditions lies in the fact that, depending on the values of the Biot number (Bi), they are transformed into a condition of the first (Bi → 0) or second (Bi →∞) kind. The paper shows that for mathematical modeling of heat- and mass-transfer processes in systems with a solid phase based on the “microprocesses” method, it is promising to search for solutions in the region of small values of the Fourier numbers (Fo < 0.1). Mathematical calculations for solving the corresponding boundary value problems are presented and examples of the results of their numerical implementation are shown. The solution to the problems of heat conduction and diffusion for bodies, including those of canonical form, is obtained in the form of Fourier series, which is typical for conditions with an uneven initial distribution of the heat- and mass-transfer potentials of matter, but solutions for small values of the Fourier numbers are not given in the sources. At the same time, with a decrease in the process time, the numerical values of the Fourier criteria also decrease and thus the number of members of the infinite series increases, which entails an increase in the error in the subsequent calculations. The paper presents solutions for bodies of canonical shape—plate, cylinder, and sphere—and also presents nomograms of the dimensionless temperature of the body surface depending on the values of the Biot and Fourier numbers for specific values of the number Bi.