Analytical Solution for Steady-State Heat Conduction in an Annular Sector Plate with Uniform Internal Heat Generation and Isothermal Boundary Conditions

This paper presents an exact analytical solution for steady-state heat conduction in a solid plate with annular sector geometry. The two-dimensional domain, defined by concentric circular arcs and radial lines, is subjected to uniform volumetric internal heat generation. Such configurations are critical in the thermal management of rotary machinery, turbine blades, and compact electronic components. The governing mathematical model is the Poisson partial differential equation in polar coordinates. The problem is solved under general Dirichlet boundary conditions, allowing for distinct constant temperatures along the four edges of the sector. The analytical derivation employs the principle of superposition to decouple the governing equation into a particular solution, representing the source term, and a homogeneous solution satisfying the Laplace equation. The method of separation of variables is utilized to resolve the homogeneous field, resulting in a rapidly convergent Fourier series expansion. To validate the proposed model, the analytical results are compared against Finite Element Method (FEM) simulations, showing excellent agreement. Furthermore, the study conducts a parametric investigation across three distinct thermal states by varying the internal heat generation rate (G) and the boundary temperatures (T ۱ through T ۴). The results demonstrate the sensitivity of the temperature field to boundary constraints and provide a rigorous benchmark for verifying numerical algorithms in polar coordinate systems.

کلیدواژه ها:

Heat Conduction ، Annular Sector ، Poisson Equation ، Analytical Solution ، Internal Heat Generation