×

What Is the optimal shape of a fin for one-dimensional heat conduction? (English) Zbl 1302.49062

Summary: This article is concerned with the shape of small devices used to control the heat flowing between a solid and a fluid phase, usually called fins. The temperature along a fin in the stationary regime is modeled by a one-dimensional Sturm-Liouville equation whose coefficients strongly depend on its geometrical features. We are interested in the following issue: is there any optimal shape maximizing the heat flux at the inlet of the fin? Two relevant constraints are examined, imposed either on the volume or the surface of the fin, and analytical nonexistence results are proved for both problems. Furthermore, using specific perturbations, we explicitly compute the optimal values and construct maximizing sequences. We show in particular that the optimal heat flux at the inlet is infinite in the first case and finite in the second one. Finally, we provide several extensions of these results for more general models of heat conduction, as well as several numerical illustrations.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
80A20 Heat and mass transfer, heat flow (MSC2010)

Software:

AMPL
PDFBibTeX XMLCite
Full Text: DOI arXiv