A variational approach for estimating the distribution of body temperature in the form of a rectangular parallelepiped under the influence of temperature, heat exchange and heat isolation of some facets

Abstract

The article describes the use of a variational method involving the finite element method to estimate the law of temperature distribution in a body in the form of a rectangular parallelepiped. The case is considered when a certain temperature is maintained on one of the faces of a rectangular parallelepiped, and heat exchange with the environment occurs on the opposite side. In accordance with the proposed approach, an approximating function in the form of a polynomial of the third degree is proposed. To determine the law of temperature distribution in a solid in the form of a rectangular parallelepiped, a functional is compiled, which consists of terms that take into account temperature, heat exchange with the environment, isolation of the faces of a rectangular parallelepiped, as well as natural boundary conditions. Minimizing this functional, the temperature values at these nodes are determined by the nodal points of a rectangular parallelepiped. Further, substituting these values into the approximating function, we obtain the temperature distribution law. At the same time, variants are investigated when the remaining faces of a rectangular parallelepiped are thermally insulated or vice versa. The temperature distribution law is estimated for different amounts of partitioning of the sides of a rectangular parallelepiped. In addition, a comparison of the temperature distribution law for a rectangular parallelepiped and a rod close in size, other things being equal, was made. Their minor differences are shown.