A Single Expression for the Solution of the One-Dimensional Transient Conduction Equation for the Simple Regular-Shaped Solids

Abstract

For transient conduction in certain simple regular-shaped solids, a dimensionless geometric parameter {omega} can be defined and used for writing the governing equations for each case as a single equation. Analytical solutions for the infinite slab, for a very long cylinder, and for a sphere are well known and can be found in several references including Carslaw and Jaeger (1947), Arpaci (1966), and Ozisik (1993). The solutions given are specific to each geometric shape. This article gives a single solution which can be applied to each of the three basic geometries by a mere substitution of the appropriate value of the geometric parameter {omega}. The Heisler charts (Heisler, 1947) provide solutions for each of the three basic geometries in a graphical form and can be found in several undergraduate heat transfer textbooks. A definite advantage of the single solution is that an appropriate algorithm based on the generalized solution can be written and incorporated into a computer program for transient conduction analysis involving any of the three basic geometries as an alternative to the use of separate equations.