Calculation of Thermal Conductivity in Normal Conical Coordinates

Modeling fields of various structures using analytical methods is widely used in previously studied coordinate systems, such as cylindrical, spherical, and others. However, if the fields and their sources have a more complex structure, other approaches are required for their study. A normal conical coordinate system was previously considered in 11; 12. This article examines the application of the normal conical coordinate method to modeling physical fields in conical bodies. The theoretical section substantiates the transition from graphical methods for studying simple fields to the use of differential field theory for objects with complex geometries. A system of normal conical coordinates (t, u, v) is described, the dependences for the transition to rectangular Cartesian coordinates are established, and expressions for partial derivatives and Lamé coefficients are obtained. Based on these relationships, the Laplace operator (Laplacian) of a scalar field in a curvilinear coordinate system is derived. The main part of the paper formulates assumptions that allow the problem of steady-state heat conduction for a conical wall to be reduced to a spatially one-dimensional problem. Using the obtained parameters, a partial differential equation for heat conduction is derived, which is reduced to an ordinary differential equation through transformations. An analytical solution is obtained for the temperature distribution T(v), depending on the coordinate normal to the cone-determinant. It is proven that the isothermal surfaces in this formulation are cones, equidistant surfaces of the system's determinant. Integration constants are determined based on first-order boundary conditions, a law for temperature change is established, and a formula for calculating heat flux using Fourier's law is derived. The final section presents a numerical example of calculating the temperature field and heat flux for the thermal insulation layer of a conical shell, confirming the viability of the proposed mathematical framework.