Electron. J. Diff. Equ., Vol. 2012 (2012), No. 181, pp. 1-13.

Optimal design of a bar with an attached mass for maximizing the heat transfer

Boris P. Belinskiy, James W. Hiestand, Maeve L. McCarthy

Abstract:
We maximize, with respect to the cross sectional area, the rate of heat transfer through a bar of given mass. The bar serves as an extended surface to enhance the heat transfer surface of a larger heated known mass to which the bar is attached. In this paper we neglect heat transfer from the sides of the bar and consider only conduction through its length. The rate of cooling is defined by the first eigenvalue of the corresponding Sturm-Liouville problem. We establish existence of an optimal design via rearrangement techniques. The necessary conditions of optimality admit a unique optimal design. We compare the rate of heat transfer for that bar with the rate for the bar of the same mass but of a constant cross-section area.

Submitted August 15, 2012. Published October 19, 2012.
Math Subject Classifications: 74A15, 74P10, 49K15, 34B24.
Key Words: Optimal design; heat transfer; heat equation; least eigenvalue; Sturm-Liouville problem; Helly's principle; calculus of variations.

Show me the PDF file (266 KB), TEX file, and other files for this article.

Boris P. Belinskiy
Department of Mathematics
University of Tennessee at Chattanooga
615 McCallie Avenue
Chattanooga, TN 37403-2598, USA
email: Boris-Belinskiy@utc.edu
James W. Hiestand
College of Engineering
University of Tennessee at Chattanooga
615 McCallie Avenue
Chattanooga, TN 37403-2598, USA
email: James-Hiestand@utc.edu
Maeve L. McCarthy
Department of Mathematics and Statistics
Murray State University, 6C Faculty Hall
Murray, KY 42071-334, USA
email: mmccarthy@murraystate.edu

Return to the EJDE web page