The article proposed a general research scheme of heat transfer process in multilayer constructions for three basic geometric forms in order to simulate fire distribution. The scheme is based on linear differential equations, the Fourier method and the modified method of Eigen functions. The work considers five different layers' design and does not take into account internal heat sources. In this regard, a one-parameter family of boundary problems were solved. The authors simulated heat transfer for the Cartesian, cylindrical and spherical coordinates. Structures comprised several materials each having thermal properties varying with temperature.

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The article is devoted to the application of the direct method to the study of heat transfer processes in the "continuous cylinder inside a multilayer cylindrical shell" system. To solve the initial problem, an auxiliary problem is posed with a “remote” cylinder of sufficiently small radius. The solution is based on the reduction method, the concept of quasiderivatives, the Fourier scheme using the modified eigenfunctions method. The solution to the original problem was obtained by following the radius of the remote cylinder to zero.

Purpose. Characteristics of heat transfer processes in multilayer bodies of basic geometric shapes simultaneously under conditions of convective heat transfer on its surfaces and taking into account imperfect thermal contact between the layers. Methodology. A direct method was applied to solve a one-parameter family of boundary value problems in the theory of heat conduction. This method is based on the reduction method, the concept of quasiderivatives, a system of differential equations with impulse action, the method of separation of variables, and the modified method of eigenfunctions of Fourier. It is worth noting that the application of the concept of quasiderivatives allows you to circumvent the well-known problem of multiplication of generalized functions, which arises when using the differentiation procedure of the coefficients of a differential equation. Such a procedure, in our opinion, casts doubt on the equivalence of the transition to the differential equation obtained in this way with generalized coefficients. Findings. The solution to the problem is obtained in a closed form. The proposed algorithm does not contain a solution to volume conjugation problems. It includes only: a) finding the roots of the corresponding characteristic equations; b) the multiplication of a finite number of known (2 2) matrices; c) the calculation of certain integrals; d) summing the required number of members of the series to obtain the specified accuracy. As an illustration, we consider model examples of heating eight-layer structures in a fire. Originality. For the first time, the direct method has been applied to solving the problem of the distribution of an unsteady temperature field over the thickness of multilayer structures of basic geometric shapes simultaneously, in the presence of imperfect thermal contact between the layers. Practical value. The implementation of the research results allows us to effectively study the heat transfer processes in multilayer structures, which are found in a number of applied problems.

Purpose. To investigate the peculiarities of distribution of a nonstationary temperature field over the thickness of a multilayer hollow cylinder under convective heat exchange conditions on its surfaces, taking into account the presence of internal (distrib uted) heat sources. Methodology. In order to achieve this goal, a direct method of solving boundary value problems of the theory of thermal con ductivity was applied, which includes the application of the method of reduction, the concept of quasi derivatives, the method of separation of variables, and the modified method of Fourier eigenfunctions. findings. The solution of the boundary value problem was obtained in a closed form, which allowed us to create an algorithm for calculating the propagation of a nonstationary temperature field in multilayer hollow cylindrical structures under convective heat exchange on its surfaces and the presence of internal heat sources. It should be noted that such algorithms include only: a) finding the roots of the characteristic equation; b) multiplication of finite number of known (2 × 2) matrices; c) calculation of defined integrals; d) summing the required number of members of the series to obtain the specified accuracy. Changing the third order boundary conditions to any other boundary conditions does not cause any significant difficulty in solving the problem. originality. A closed solution is obtained for the propagation of a nonstationary temperature field in a multilayer hollow cyl inder in the presence of internal sources of heat and convective heat exchange on its surfaces. Practical value. Implementation of the research results allows us to investigate the processes of heating or cooling multilayer hollow structures, taking into account the internal heat sources encountered in several applied problems. These are tasks that can be related to the processes of cooling of thermal elements of nuclear power plants, changes in the temperature field during microar ray oxidation, heating of electronic components during the passage of electric current.