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RESEARCH PAPERS: Conduction

An Exact Solution to Steady Heat Conduction in a Two-Dimensional Annulus on a One-Dimensional Fin: Application to Frosted Heat Exchangers With Round Tubes

[+] Author and Article Information
A. D. Sommers

Department of Mechanical and Industrial Engineering, University of Illinois, 1206 West Green Street, Urbana, IL 61801asommers@uiuc.edu

A. M. Jacobi

Department of Mechanical and Industrial Engineering, University of Illinois, 1206 West Green Street, Urbana, IL 61801a-jacobi@uiuc.edu

J. Heat Transfer 128(4), 397-404 (Aug 31, 2005) (8 pages) doi:10.1115/1.2165210 History: Received May 19, 2005; Revised August 31, 2005

The fin efficiency of a high-thermal-conductivity substrate coated with a low-thermal-conductivity layer is considered, and an analytical solution is presented and compared to alternative approaches for calculating fin efficiency. This model is appropriate for frost formation on a round-tube-and-fin metallic heat exchanger, and the problem can be cast as conduction in a composite two-dimensional circular cylinder on a one-dimensional radial fin. The analytical solution gives rise to an eigenvalue problem with an unusual orthogonality condition. A one-term approximation to this new analytical solution provides fin efficiency calculations of engineering accuracy for a range of conditions, including most frosted-coated metal fins. The series solution and the one-term approximation are of sufficient generality to be useful for other cases of a low-thermal-conductivity coating on a high-thermal-conductivity substrate.

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Copyright © 2006 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Two different examples of the hexagonal pattern that emerges when using the sector method (9)

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Figure 2

Schematic of the composite slab, with the one-dimensional fin (material 1) and the two-dimensional frost layer (material 2)

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Figure 3

Using the conditions of Table 1, example fin efficiencies are shown for two values of the convective heat transfer coefficient over a range of frost thicknesses. The effect of neglecting conduction through the frost layer is clearly seen.

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Figure 4

Using the data from Table 2, the fin efficiency is calculated and shown for different methods and compared to the analytical solution. Note that the convective heat transfer coefficient is not constant; it decreases with frost thickness.

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Figure 5

The difference between the series solution and the one-term approximation is shown using the conditions of Table 1

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Figure 6

The difference between using the sensible heat transfer coefficient and the modified convective coefficient in the calculations is shown using the data from Table 2. Note that the convective heat transfer coefficient is not constant; it decreases with frost thickness.

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