Application of the B-Spline Method to Solve Nonlinear Problem of Heat Conduction With Radiation Boundary Conditions Using Kirchhoff Transformation

Abstract

This paper concerns the joint application of the B-spline method and the Kirchhoff transformation to solve the nonlinear problem of thermal conduction with radiation type boundary conditions. The proposed method requires few iterations, sometimes none, for solids subjected to prescribed temperature boundary conditions. This method can be deployed by other numerical approaches (boundary element method, finite element method, finite element method, etc.) for the resolution of the heat conduction equation (linear or nonlinear), in terms of the Kirchhoff transformation θ. For numerical implementation, the steady-state finite element method is considered. The numerical validation was performed for a hollow aluminum cylinder whose outer surface is subjected to radiation. Three types of thermal conductivities are considered: (i) constant, (ii) linear, and (iii) nonlinear. As an application, we studied the thermal response of an aluminum reactor, in the form of an annular disk with cooling tubes, exposed to thermal radiation.