Efficient Numerical Evaluation of Exact Solution for 1D, 2D and 3D Infinite Cylindrical Heat Conduction Problem

Abstract

Estimation of thermal properties or diffusion properties from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Bodies of infinite extent are a particular challenge from this perspective. Even for exact analytical solutions, because the solution often has the form of an improper integral that must be evaluated numerically, lengthy computer-evaluation time is a challenge. The subject of this thesis is improving the computer evaluation time for exact solutions for infinite and semi-infinite bodies in the cylindrical coordinate system. The motivating applications for the present work include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. In this thesis the computer evaluation time is improved by replacing the integral-containing solution by a suitable finite-body series solution. Although the series solution is approximate, the precision of the series solution may be controlled to a high level and the required computer time may be minimized, by a suitable choice of the extent of the finite body. An easy-to-use relationship is developed for the finite-body size needed as a function of desired precision and as a function of time. The method is demonstrated for the one-dimensional case of large body with a cylindrical hole and is extended to two-dimensional and three-dimensional geometries of practical interest. The computer-evaluation time for the finite-body solutions are shown to be hundreds or thousands of time faster than the infinite-body solutions, depending on the geometry. Future work is considered to apply the same method from the diffusion problem to other geometries problem.