What Is the Optimal Shape of a Fin for One-Dimensional Heat Conduction?

Abstract

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 fin. The temperature along a fin in 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, by imposing either its volume or its surface, 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.

An extremal eigenvalue problem arising in heat conduction

“…The statement of this lemma is close to [13,Lemma 3.4]. Nevertheless, a notable difference lies in the fact that we have to deal with perturbations of b that are the sums of characteristic functions of measurable sets, instead of open sets.…”

“…This section is devoted to the investigation of Problem (4). As highlighted in [13,Lemma 3.1], the class S a0, ,S0 does not share nice compactness properties. In particular, it is not closed nor bounded in W 1,∞ (0, ) (endowed with the strong topology), whereas it is bounded in L ∞ (0, ).…”

“…In [13], the authors dealt with a simplified one-dimensional stationary model of axisymmetric fin taking into account the lateral heat transfers of the fin. They analyzed the optimal design problem and in particular the existence issues as well as the determination of maximizing sequences.…”

Discrete variable topology optimization for simplified convective heat transfer via sequential approximate integer programming with trust‐region

“…Further, the checkerboard pattern is the inherent convection oscillatory feature that has also been observed in the studies for one-dimensional convection problems where the oscillating fins are preferred for the optimal shape. [45][46][47] Illustrated by this numerical study and results in Table 13, the proposed algorithms can balance the conduction and convection effects to automatically search the dissipated thermal pattern. Yan et al 48 have proven that the optimal configuration for the pure volume-to-point conduction is the lamellar needle structure because the most efficient shape from the heat sink to the heat source is the straight line.…”

Strong Optimal Controls in a Steady-State Problem of Complex Heat Transfer

“…Notice that the optimization of energy in/out flow, which improves heating/cooling systems, is a quite popular problem in many engineering applications. In [15][16][17][18], similar problems are considered in the context of shape optimization.…”