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Research Papers

Geometric Optimization of C-Shaped Cavities According to Bejan's Theory: General Review and Comparative Study

[+] Author and Article Information
G. Lorenzini

Dipartimento di Ingegneria Industriale,
Università degli Sudi di Parma,
Parco Area delle Scienze 181/A,
43124 Parma, Italy

C. Biserni

Dipartimento di Ingegneria Industriale,
Università degli Studi di Bologna,
Viale Risorgimento 2,
40136 Bologna, Italy

L. A. O. Rocha

Universidade Federal do Rio Grande do Sul,
Departamento de Engenharia Mecânica,
Rua Sarmento Leite,
425, Porto Alegre,
Rio Grande do Sul 90.050-170, Brazil

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received February 18, 2013; final manuscript received March 22, 2013; published online June 4, 2013. Assoc. Editor: Gary Miller.

J. Electron. Packag 135(3), 031007 (Jun 04, 2013) (7 pages) Paper No: EP-13-1014; doi: 10.1115/1.4024113 History: Received February 18, 2013; Revised March 22, 2013

The aim of this paper is to consider, by means of the numerical investigation, the geometric optimization of a cavity that intrudes into a solid with internal heat generation. The objective is to minimize the maximal dimensionless excess of temperature between the solid and the cavity. The cavity is rectangular, with fixed volume and variable aspect ratio. The cavity shape is optimized for two sets of boundary conditions: isothermal cavity and cavity cooled by convection heat transfer. The optimal cavity is the one that penetrates almost completely the conducting wall and proved to be practically independent of the boundary thermal conditions, for the external ratio of the solid wall smaller than 2. As for the convective cavity, it is worthy to know that for values of H/L greater than 2, the best shape is no longer the one that penetrates completely into the solid wall, but the one that presents the largest cavity aspect ratio H0/L0. Finally, when compared with the optimal cavity ratio calculated for the isothermal C-shaped square cavity, the cavities cooled by convection highlight almost the same optimal shape for values of the dimensionless group λ ≤ 0.01. Both cavities, isothermal and cooled by convection, also present similar optimal shapes for ϕ0 < 0.3 and ϕ0 > 0.7. However, in the range 0.3 ≤ ϕ0 ≤ 0.7, the ratio (H0/L0)opt calculated for the cavities cooled by convection is greater than the one presented by isothermal cavities. This difference is approximately 17% when λ = 0.1 and ϕ0 = 0.7, and 20% for λ = 1 and ϕ0 = 0.5.

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References

Figures

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Fig. 2

The minimization of the global thermal resistance when the external shape of the heat generating body is fixed

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Fig. 3

The optimized geometry and performance when the external shape is square

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Fig. 4

The effect of the external shape H/L on the global thermal resistance minimized as in Fig. 2

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Fig. 5

The optimal shape of the lateral intrusion, as a function of H/L and ϕ

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Fig. 6

Correlation of the optimal shape results of Fig. 5

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Fig. 7

The optimized geometry and performance when the internal shape is square

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Fig. 8

The effect of the external shape H/L on the global thermal resistance minimized for constant H/L

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Fig. 9

The optimal shape obtained by holding H/L fixed, versus the optimal shape obtained by holding H0/L0 fixed

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Fig. 10

Minimization of the dimensionless global thermal resistance as function of H0/L0 for several values of the fraction of the area of the solid inserted into the cavity

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Fig. 11

The behavior of the best shapes and the minimal dimensionless global thermal resistance calculated in Fig. 10 as function of the volume fraction ϕ0

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Fig. 12

Illustrations of some optimal shapes from Fig. 11

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Fig. 13

The behavior of the dimensionless global thermal resistance θmax,min as function of the dimensionless group λ and the volume fraction ϕ0

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Fig. 14

The effect of the dimensionless group λ on the optimal shape of the cavity for several values of the volume fraction ϕ0

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Fig. 15

Comparison between the optimal ratio (H0/L0)opt with reference to convective C-shaped cavities and isothermal square cavities

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Fig. 16

The behavior of the minimal dimensionless global thermal resistance as function of the ratio H/L and the volume fraction ϕ0

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Fig. 17

The behavior of the optimized ratio H0/L0 as function of the ratio H/L and the volume fraction ϕ0

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Fig. 1

C-shaped lateral intrusion into a two-dimensional conducting body with uniform heat generation

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Fig. 18

Examples of the best shapes calculated in Fig. 17 for the volume fraction ϕ0 = 0.1

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