Evolution of Mathematical Models of Processes of Non-Stationary Heat (Mass) Conduction in Bodies of Canonical Form

Currently, there are a large number of materials that are thermally affected during their production. From the point of view of the principles of geometry, their shape can be reduced to classical bodies of canonical form: a plate, a cylinder, a ball. In the heat 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 method of micro-processes. The main positions of the method of micro-processes, as applied to modeling boundary value problems of heat and mass transfer for canonical bodies under boundary conditions of the first kind (Dirichlet conditions), were outlined in previous articles by the authors. In this paper, a technique based on the method of micro-processes for solving boundary value problems of heat and moisture conduction under more general boundary conditions, conditions of the third kind (Riemann-Newton) is presented. 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 kind ((Bi→0) or the second (Bi→∞). The paper shows that for mathematical modeling of heat and mass transfer processes in systems with a solid phase based on the method of micro-processes, it is promising to search for solutions in the field of small Fourier numbers (Fo <0,1). Mathematical calculations for solving the corresponding boundary value problems are given and examples of the results of their numerical implementation are shown. The solution to the problems of heat conduction and diffusion for bodies, including the canonical form, is obtained in the form of Fourier series, which is typical for conditions with an uneven initial distribution of heat and mass transfer potentials, but solutions for small Fourier numbers are not given in the sources. At the same time, as the process time decreases, the numerical values of the Fourier criteria also decrease, and thus there are more members of the infinite series, which entails an increase in the error in further calculations. The paper presents solutions for canonical bodies – plates, cylinders and spheres, also presents nomograms of the dimensionless temperature of the body surface depending on the values of the Biot and Fourier numbers at specific values of the Bi number.

S.V. FEDOSOV¹, Academician Russian Academy of Architecture and Construction Sciences (RAACS), Doctor of Sciences (Engineering), Professor (This email address is being protected from spambots. You need JavaScript enabled to view it.),
M.O. BAKANOV², Adviser of RAACS, Doctor of Sciences (Engineering), Head of the Educational and Scientific Complex "Fire Fighting"(This email address is being protected from spambots. You need JavaScript enabled to view it.)

¹ National Research Moscow State University of Civil Engineering (26, Yaroslavskoe Highway, Moscow, 129337, Russian Federation)
² Ivanovo Fire and Rescue Academy of the State Fire Service of the Ministry of Emergency Situations of Russia (33, Stroiteley Prospect, Ivanovo, 153011, Russian Federation)

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