ANALYTICAL SOLUTION OF THE PROBLEM OF CONVECTIVE HEAT EXCHANGE IN TWO-LAYER MEDIUM - Pyshnograev, Y. N.; Shtanko, A. I.; Pyshnograev, E. Y. (Computer Science and Applied Mathematics, 2017)
When building blocks are designed, there is an important problem to determine the temperature distribution in contacting materials. It is well known that obtaining a general analytical solution in such cases is difficult. For this reason, numerical methods are used to calculate the parameters of temperature fields. However there is a problem of errors accumulation when numerical methods are used. In some cases this can lead to unreliable results. For this reason, there is a need for analytical research. In this research a medium with two layers is is considered. Heat is distributed according to the laws of heat conduction and convection. The boundary conditions of the first kind are given on the outer boundaries. Layers contact each other with the conditions for an ideal thermal contact. Temperature distribution in the layers at the initial instant is constant. Current research determines the temperature function, which depends on the spatial variable and time. The mathematical model consists of a one-dimensional nonstationary heat conduction equation with a convective component, boundary and initial conditions. The solution of the problem is carried out by the method of finite integral transformations. In order to achieve this, the method of the separation of variables is used. This helps to reduce the original problem to the spectral problem with respect to the spatial variable. Then, using the proper replacement, the problem is transformed to the Sturm-Liouville problem. To solve this problem, the three intervals of the arrangement of eigenvalues are considered. At each of the intervals the set of eigenvalues is determined from the respective transcendental equations. This values correspond to eigenfunctions, which are a linear combination of trigonometric or hyperbolic functions. It is shown that the found eigenfunctions satisfies the orthogonality property and can be used as kernels of integral transformations. Conducting an integral transformation makes possible reducing the original partial differential equation to an ordinary differential equation with respect to time. Using the inversion formula, the final solution of the problem is written in the form of a functional series. Given solution can be extended to solve more complex two-dimensional and three-dimensional problems. It is also possible to solve the problem when the number of contacting materials is increased.