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Research Papers: Thermal Systems

Optimizing a Functionally Graded Metal-Matrix Heat Sink Through Growth of a Constructal Tree of Convective Fins

[+] Author and Article Information
Jacob Kephart

Mem. ASME
Naval Surface Warfare Center,
Philadelphia Division,
Philadelphia, PA 19112
e-mail: jacob.kephart@navy.mil

G. F. Jones

Fellow ASME
Department of Mechanical Engineering,
Villanova University,
Villanova, PA 19085
e-mail: gerard.jones@villanova.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received February 26, 2015; final manuscript received February 18, 2016; published online April 26, 2016. Assoc. Editor: Oronzio Manca.This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Heat Transfer 138(8), 082802 (Apr 26, 2016) (12 pages) Paper No: HT-15-1147; doi: 10.1115/1.4033209 History: Received February 26, 2015; Revised February 18, 2016

Optimal material utilization in metal-matrix heat sink is investigated using constructal design (CD) in combination with fin theory to develop a constructal tree of optimally shaped convective fins. The structure is developed through systematic growth of constructs, consisting initially of a single convective fin enveloped in a convective medium. Increasingly complex convective fin structures are created and optimized at each level of complexity to determine optimal fin shapes, aspect ratios, and fin-base thickness ratios. One result of the optimized structures is a functional grading of porosity. The porosity increases as a function of distance from the heated surface in a manner ranging from linear to a power function of distance with exponent of about 2. The degree of nonlinearity in this distribution varies depending on the volume of the heat sink and average packing density and approaches a parabolic shape for large volume. For small volume, porosity approaches a linear function of distance. Thus, a parabolic (or least-material) fin shape at each construct level would not necessarily be optimal. Significant improvements in total heat transfer, up to 55% for the cases considered in this work, were observed when the fin shape is part of the optimization in a constructal tree of convective fins. The results of this work will lead to better understanding of the role played by the porosity distribution in a metal-matrix heat sink and may be applied to the analysis, optimization, and design of more effective heat sinks for electronics cooling and related areas.

Copyright © 2016 by ASME
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Figures

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Fig. 1

Assembly process of Bejan and Dan's [7] constructal tree of constant cross-sectional area fins

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Fig. 2

Elemental construct of the convective tree

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Fig. 3

First-construct convective tree shown as a single tree (a). First construct arranged to form flow channels (b).

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Fig. 4

Nusselt number correlation for fully developed flow in a rectangular channel of aspect ratio α*

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Fig. 5

First-construct convective tree shown with actual and rectangularized flow channels

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Fig. 6

Material overlap in first construct. Shaded area represents region of conductive material accounted for twice.

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Fig. 8

Second-construct convective trees show as a single tree (a). Second construct divided along an alternate line of symmetry (b).

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Fig. 7

D0/H0<1  and, therefore, a valid geometry (a). D0/H0>1  and, therefore, an invalid geometry (b).

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Fig. 9

First construct's optimized heat transfer rate is monotonically increasing with both Ã1 and ϕ1

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Fig. 10

Trend of optimized heat transfer as a function of area for a constructal tree of constant cross-sectional area fins observed by Bejan and Dan [7]

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Fig. 11

Nonlinear inequality constraint map of optimization for the first construct

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Fig. 12

Convection enhancement on the main stem of first construct from the conductive contribution of the stems of the elemental construct

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Fig. 13

Optimized exponent, m1, for first construct

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Fig. 14

Optimized exponent, m0, for first construct

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Fig. 15

Fraction of conductive material allocated to main stem of first construct

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Fig. 16

Number of elemental constructs in the first construct

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Fig. 17

Relative performance increase in a first construct with shaped fins (this work) compared with cross-sectional area fins

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Fig. 18

Porosity ratio versus dimensionless position from base for first construct

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Fig. 19

Plot of normalized porosity distributions (this study) and the allometric scaling investigated by Ortega et al. [30]. Ortega's distribution in the current notation is γ(x̃)/γ1,avg=(γbase/γ1,avg)γφx̃, where γφ is a scaling parameter.

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