Piecewise uniform optimal design of a bar with an attached mass

Abstract

We minimize, with respect to the cross sectional area, the mass of a bar given the rate of heat transfer. The bar enhances the heat transfer surface of a larger known mass to which the bar is attached. This article is an extension of a previous publication by two coauthors, where heat transfer from the sides of the bar was neglected and only conduction through its length was considered. The rate of cooling is defined by the first eigenvalue of the corresponding Sturm-Liouville problem. We compare the mass of the computed variable cross-section bar with the mass of a bar with constant cross-sectional area and the same rate of heat transfer, and conclude that a fin design with constant, or near constant, cross-sectional area is best.

Keywords

• Optimal design
• heat transfer
• heat equation
• least eigenvalue
• Sturm-Liouville problem
• calculus of variations
• transcendental equation
• computer algebra