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RESEARCH PAPERS

Heat Conduction in Multilayered Rectangular Domains

[+] Author and Article Information
James Geer, Anand Desai, Bahgat Sammakia

Department of Mechanical Engineering, Binghamton University, Binghamton, NY 13902-6000

J. Electron. Packag 129(4), 440-451 (Apr 07, 2007) (12 pages) doi:10.1115/1.2804094 History: Received October 16, 2006; Revised April 07, 2007

This paper presents the results of an analytical study of steady state heat conduction in multiple rectangular domains. Any finite number of domains that are equally sized (in plane) may be considered in the current analysis. The thermal conductivity and thickness of these domains may be different. The entire geometry composed of these connected domains is considered as adiabatic on the lateral surfaces and can be subjected to a wide range of thermal boundary conditions at the top and bottom. For example, the bottom of the stack may be adiabatic, while the top of the stack may be exposed to a uniform heat transfer coefficient. Spatially varying heat generation rates can be applied in each of the domains. The solutions are found to be in agreement with known solutions for simpler geometries. The analytical solution presented here is very general in that it takes into account the interface resistances between the layers. One application of this analytical study relates to the thermal management of three-dimensional stacks of computer devices and interconnect layers. The devices would have spatially nonuniform power dissipation within them, and the interconnect layers would have a significantly lower thermal conductivity than the devices. Interfacial defects, such as delamination or air voids, between the devices and the interconnect layers may be included in the model. Another possible application is to the study of hot spots in a chip stack with nonuniform heat generation. Many other potential applications may also be simulated.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic view of a rectangular stack with N layers and an indication of the coordinate system

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Figure 2

The temperature along the z axis (x=y=0) for Example 1 plotted as a function of the normalized distance z∕L5. The locations of the second and fourth layers are indicated by dashed lines.

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Figure 3

Some isotherms in the plane y=0 for Example 1 plotted as functions of the normalized distances z∕L5 and x∕a

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Figure 4

Schematic view of the rectangular stack considered in Examples 2–4. The stack consists of five layers, with three cooling tubes embedded in the middle layer.

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Figure 5

The temperature for Example 2 at y=0 and x=±a, x=±5a∕8, and x=0, plotted as a function of the normalized distance z∕L5. The locations of the second and fourth layers are indicated by dashed lines.

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Figure 6

Some isotherms in the plane y=0 for Example 2 plotted as functions of the normalized distances z∕L5 and x∕a. The locations of the cooling tubes (in layer 3) are indicated by blue shaded areas.

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Figure 7

The temperature for Example 3 at y=0 and x=−a, x=−5a∕8, and x=0, plotted as a function of the normalized distance z∕L5. The locations of the second and fourth layers are indicated by dashed lines. Note the jumps in the temperature along x=−5a∕8 near the fourth layer.

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Figure 8

The temperature for Example 3 at y=0 and x=0, x=5a∕8, and x=a, plotted as a function of the normalized distance z∕L5. The locations of the second and fourth layers are indicated by dashed lines. Note the jumps in the temperature along x=5a∕8 near the second layer.

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Figure 11

Some isotherms in the plane y=0 for Example 4 plotted as functions of the normalized distances z∕L5 and x∕a. The locations of the cooling tubes (in layer three) and the “hot spot” (in layer 5) are indicated by blue and red shaded areas, respectively.

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Figure 10

, The temperature for Example 4 at y=0 and x=±a, x=±a∕2, and x=0, plotted as a function of the normalized distance z∕L5. The locations of the second and fourth layers are indicated by dashed lines.

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Figure 9

Some isotherms in the plane y=0 for Example 3 plotted as functions of the normalized distances z∕L5 and x∕a. The locations of the cooling tubes (in layer 3) are indicated by blue shaded areas, while the regions of interfacial resistance on the surfaces of the second and fourth layers are indicated by heavy black lines.

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