Використання методу джерел для розв’язання практичних задач теплопровідності

The subject of the study is analytical solutions of the non-stationary heat equation for practical application in engineering calculations. The aim of the work is to obtain and analyze analytical solutions of the non-stationary heat equation under type I and type II boundary conditions of a semi-infinite rod using the method of instantaneous sources and the fundamental Kelvin solution. The research methodology is based on potential theory, the method of instantaneous sources and the Gauss error functions. The fundamental Kelvin solution, or the temperature effect function of the instantaneous source of the three-dimensional non-stationary heat equation in the case of the initial point source of heat is analyzed. Variants of the fundamental Kelvin solution for one-dimensional and two-dimensional heat conduction problems, which are examples of Green's functions and have wide applications in potential theory, are also considered. Analytical solutions of the non-stationary heat equation under type I and type II boundary conditions of a semi-infinite rod for cooling/heating regimes are obtained and analyzed, which is convenient for practical application in engineering calculations. It is shown that the functional dependencies for determining the temperature and heat flux density during heating under type I and type II boundary conditions can be obtained from the solution of the heat equation for the cooling mode under type I boundary conditions using a simple change of variable. It is shown that the derivatives dΘ(Fox)/dFox and dΘ(x)/dτ have an extremum at Fox = 1/6 and τ = x2/6a respectively, the magnitude and time of occurrence of the extremum depends on x, where Θ – dimensionless temperature, Fox – Fourier number, τ – time, s; x – coordinate, m; a – temperature conductivity, m2/s. Using the obtained extremum values, it is possible to track the coordinate and time of the maximum heating rate in a semi-infinite rod, or the maximum cooling rate during cooling. The obtained solutions of the non-stationary heat equation of a semi-infinite rod can be used for the operational determination of rational thermal parameters during laser hardening of materials, the study of temperature dependences of thermophysical properties of materials, in particular composite ones, etc.