A Two-Dimensional Cylindrical Transient Conduction Solution Using Green's Functions

Abstract

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.