A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

Abstract

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.