Exact solution of steady nonlinear heat conduction in exponentially graded cylindrical and spherical shells with temperature-dependent properties

In this study, an exact analytical solution of steady heat conduction problem in exponentially graded inhomogeneous materials with temperature-dependent heat conductivity is presented. The FGM properties are assumed to depend exponentially on spatial coordinates whereas the temperature dependency is taken to be a linear function. The proposed method is based on an integral transformation of the temperature field, which transforms the nonlinear heat equation into a linear one for the transformed temperature. The boundary conditions are to be transformed as well. Once the linear equation is solved and the transformed temperature is obtained, the inverse transform is used to calculate the physical temperature field. The boundary conditions are enforced on the transformed temperature. Finally, in order to demonstrate the application of the proposed method, two numerical examples are worked out, i.e. nonlinear heat conduction in the radial direction of cylindrical and spherical thick-walled shells. In order to check the validity of the proposed solution scheme, a numerical solution of the problems has been performed and an excellent agreement has been established