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

Abstract

There are many applications for problems involving thermal conduction in twodimensional (2D) cylindrical objects. Experiments involving thermal parameter estima tion 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 iter atively in order to obtain convergence with the measured temperature history by chang ing 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 accu racy. The present work involves the generation of an exact solution using Green's func tions where the principle of superposition is employed in combining a one-dimensional (ID) cylindrical case with a ID 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 gener ated 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 convec tive boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundaiy condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions. In troductio nVerification is an important step in the use of numerical codes for conductive heat transfer experiments. In order to find the ther mal properties of materials in heat transfer experiments, a direct solution for temperature as a function of time must be generated in an attempt to match the temperature histories obtained empiri cally. Developing these solutions using Green's functions can simplify the process of verification and the exact solutions gener ated can sometimes be used directly, thus eliminating the need for a numerical solution. The exact solutions sometimes converge in only a few terms and yield insight that numerical solutions may not provide. Cole et al.[1] provide a convenient method for gener ating these exact solutions using Green's functions. The specific problem addressed here involves a nonhomogeneous convective boundary condition on one of the planar surfaces of a solid cylin der and homogeneous convective boundary conditions elsewhere. Materials being tested using the laser flash method for the deter mination of thermal diffusivity are typically configured in cylin drical shapes...

A novel Hamiltonian-based method for two-dimensional transient heat conduction in a rectangle with specific mixed boundary conditions

“…Analytical studies were usually carried out by the method of separation of variables (Ozisik, 1990;Belghazi et al, 2010;Norouzi et al, 2012;Delouei and Norouzi, 2015) and Green's function approach (Cole et al, 2010;McMasters and Beck, 2014). Making use of the obtained exact solutions, some semi-analytical method was developed to solve the transient heat conduction (Gordeliy et al, 2008).…”