Optimum shape of radiation-cooled ring fins

Grodzovskii, G. L.; Pasechnik, Z. V.

doi:10.1007/bf00913229

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…Many engineering works focused on modeling the direct problem in order to assess the efficiency of different fin shapes [3]. Nevertheless, very few theoretical results and mathematical proofs do exist (see for instance [2,10,16]). The study proposed in the present article tackles this shape optimization problem, taking into account several relevant constraints on the admissible class of designs.…”

Section: Introductionmentioning

confidence: 99%

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

Marck¹,

Nadin²,

Privat³

2014

SIAM J. Appl. Math.

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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.

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Section: Introductionmentioning

confidence: 99%

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

Marck¹,

Nadin²,

Privat³

2014

SIAM J. Appl. Math.

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“…Many engineering works focused on modeling the direct problem in order to assess the efficiency of different fin shapes. Notice that these studies are mainly numerical and no mathematical approach is used to determine the optimal profiles of fins (see for instance [3,4,9,15]).…”

Section: Motivations In Convection-conduction Theorymentioning

confidence: 99%

An extremal eigenvalue problem arising in heat conduction

Nadin

Privat

2016

Journal de Mathématiques Pures et Appliquées

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International audienceThis article is devoted to the study of two extremal problems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric fins and model this problem as the issue of (i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjoint operator having a compact inverse. We impose a pointwise lower bound on the radius of the fin, as well as a lateral surface constraint. Using particular perturbations and under a smallness assumption on the pointwise lower bound, one shows that the only solution is the cylinder in the first case whereas there is no solution in the second case. We moreover construct a maximizing sequence and provide the optimal value of the eigenvalue in this case. As a byproduct of this result, and to propose a remedy to the non-existence in the second case, we also investigate the well-posedness character of another optimal design problem set in a class enjoying good compactness properties

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Optimum shape of radiation-cooled ring fins

Cited by 2 publications

References 6 publications

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

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

An extremal eigenvalue problem arising in heat conduction

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