In this paper, in closed form, the problems of determining stationary temperature fields in multilayer (flat, cylindrical and spherical) structures in the presence of discrete-continuous internal and point heat sources are solved. The one-dimensional differential equation of thermal conductivity in different coordinate systems is given through one parametric family of quasi-differential equations. It is assumed that the coefficients of the differential equation of thermal conductivity are piecewise constant functions. A system of two linearly independent boundary conditions is added to the equation, which in the general case are nonlocal. The solutions of such problems are constructive and are expressed exclusively through their initial data. The basic provisions of the concept of quasi-derivatives, the provisions of the theory of heat transfer, the theory of generalized systems of linear differential equations, elements of the theory of generalized functions are used. For the mathematical model of stationary thermal conductivity, the practical use of the concept of quasi-derivatives is illustrated, for the efficient construction, in a closed form, of solutions of boundary value problems with the most general boundary conditions. As an example, the problem of finding the critical radii of thermal insulation of multilayer hollow cylinders and spheres, taking into account the internal heat sources in the layers. Boundary conditions of the first and third kind. It is established that the value of the critical radius does not depend on the number of layers and the intensity of internal heat sources, but only on the thermal conductivity of the outer layer of the structure and the heat transfer coefficient between the structure and the environment. The formula for determining the critical radius of thermal insulation for a multilayer cylindrical and spherical structure is derived. The methods developed in this work have the prospect of further development and can be used in engineering calculations.
Discussion: “Dynamic Response of Heat Exchangers Having Internal Heat Sources—Part IV” (Yang, W. J., Clark, J. A., and Arpaci, V. S., 1961, ASME J. Heat Transfer, 83, pp. 321–336)
