Models of Steady Heat Conduction in Multiple Cylindrical Domains

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

Department of Mechanical Engineering, State University of New York, Binghamton, NewYork 13902

J. Electron. Packag 128(1), 10-17 (May 18, 2005) (8 pages) doi:10.1115/1.2159003 History: Received July 29, 2004; Revised May 18, 2005

An analytical model of steady state heat conduction in multiple cylindrical domains is presented and discussed. The domains are axisymmetric, contiguous, and coaxial. Three domains are considered in the current study. The thermal conductivities, thicknesses, and radii of the domains may be different. The entire geometry composed of the three connected domains is considered as adiabatic on its lateral surfaces and is subjected to uniform convective cooling at one end. The other end of the geometry is subjected to a constant heat flux, while uniform heat generation is imposed in one of the domains. The analytical solution of the model is found and special cases of it are shown to be in agreement with known solutions for simpler geometries. One application of this model relates to the thermal management of computer chips that are attached to a heat sink or a heat spreader. The three layers could simulate the chip, the thermal adhesive and the heat sink. Another application is the simulation of a nanotube or nanocylinder connecting a region of the chip to a region of the heat sink. Many other potential applications may be simulated using the different possible configurations for the solution presented.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Schematic view of the geometry of the model, along with an indication of the geometrical and thermal parameters

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Figure 2

The four cases considered determined by the relative magnitudes of the radii a,b, and c

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Figure 3

(a) Case I application—a thermal management system in which vertically aligned nanotubes are added to the thermal interface material, to form a low resistance path through the thermal interface material. (b) Isotherms near the silicon, nanotube interface for the example for case I. The isotherms are nearly parallel away from the interface (as expected). The bending of the isotherms increases near the interface with the silicon, and then they become nearly parallel again away from the interface.

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Figure 4

(a) Case II application—a thermal management system consisting of a silicon layer, thermal interface material, and a heat sink. This particular arrangement is used in air-cooled systems, such as desktop computers, workstations, and low level servers. (b) Isotherms for an example of case II. The thick lines represent the boundaries of the model. The upper most surface has convective cooling and bottom most surface is subjected to a specified heat flux. All the other surfaces are adiabatic. The isotherms are nearly parallel away from the interface and bend as they approach the interface. (c) A magnified view of the temperature contours near the interface for the example of case II. The magnified view shows the silicon and thermal interface region. The temperature contour in the thermal interface region is a band owing to its low thermal conductivity.

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Figure 5

(a) Case III application—a chip, thermal interface material, and heat pipe assembly. This system is commonly used for temperature regulation in laptops. The bottom most layer is silicon, the thin layer is thermal interface material, and the top layer is the heat pipe. The heat pipe then extends out to a heat sink, which is cooled by a fan. (b) Isotherms for the example for case III. The isotherms bend near the interfaces of the different materials, and are nearly parallel away from them. The band in the upper layer is due to geometry and relatively high thermal conductivity of the heat pipe, where the isotherms are nearly parallel throughout this region. (c) A magnified view of the temperature contours near the thermal interface material for the example of case III. The isotherms bend due to the different thermal conductivities of the different layers, and also due to changes in geometry of the different materials.

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Figure 6

(a) Case IV application—a printed circuit board with components on both sides cooled using a heat sink. This example also can be thought of as a system with a heat source in the middle, which is being cooled by heat sinks. (b) Isotherms for the example for case IV. The direction of bending of the isotherms changes from the bottom cylinder towards the top cylinder, due mainly to the geometry of the example.



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