Analytical Solution for One-Dimensional Transient Thermal Conduction in a Hollow Cylinder

Abstract

The primary use of analytical solutions in the area of thermal conduction problems is for verification purposes, comparing the calculated temperatures and heat flux values to the results from numerical codes. The contribution from the analytical solutions can be especially significant where large temperature gradients are found. This is because the temperature and heat flux results can be found very precisely: normally to eight or 10 significant figures using analytical solutions. Normally, numerical solutions require extremely fine grids in high-heat-flux locations, but analytical solutions can provide insight into the adequacy of grid densities in these types of situations. One area of particular interest is in nonrectangular analytical solutions because generalized numerical grids do not naturally lend themselves well to curved surfaces without a large number of nodes. In this current study, four solutions are offered in a geometry involving a hollow cylinder. Two of the solutions examine heating from the inside of the cylinder, and two involve heating from the outside. Development of the solutions is provided along with results that show the temperature responses to the various sets of boundary conditions. A mathematical identity is used as part of the solution, which eliminates the need to evaluate an infinite series, along with need to find roots of transcendental equations and evaluate Bessel functions. Intrinsic verification is also applied in order to provide assurance that the solutions are properly formulated and evaluated.