Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 181, pp. 1-13.
Title: Optimal design of a bar with an attached mass
for maximizing the heat transfer
Authors: Boris P. Belinskiy (Univ. of Tennessee, Chattanooga, TN, USA)
James W. Hiestand (Univ. of Tennessee, Chattanooga, TN, USA)
Maeve L. McCarthy (Murray State Univ., Murray, KY, USA)
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.