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Research Papers: Conduction

Thermal Analysis and Optimization of Orthotropic Pin Fins: A Closed-Form Analytical Solution

[+] Author and Article Information
Syed M. Zubair, A. F. M. Arif

Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Mostafa H. Sharqawy

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

J. Heat Transfer 132(3), 031301 (Dec 30, 2009) (8 pages) doi:10.1115/1.4000059 History: Received June 02, 2009; Revised August 16, 2009; Published December 30, 2009; Online December 30, 2009

Analytical solutions for temperature distribution, heat transfer rate, and fin efficiency and fin effectiveness are derived and presented for orthotropic two-dimensional pin fins subject to convective-tip boundary condition. The generalized results are presented and discussed in terms of dimensionless variables such as radial and axial Biot numbers (Bir,Biz), fin aspect ratio, L/R, and radial-to-axial conductivity ratio k. Several special cases are derived from the general solution, which includes the insulated-tip boundary condition. It is also demonstrated that the classical temperature distribution and heat transfer rate from the two-dimensional isotropic pin fin introduced earlier in literature can easily be recovered from the general solutions presented in this paper. Furthermore, dimensionless optimization results are presented for orthotropic pin fins that can help to solve many natural and forced convection pin fin problems.

FIGURES IN THIS ARTICLE
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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Pin fin schematic

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Figure 2

Orthotropic pin fin heat transfer rate versus thermal conductivity ratio; comparison of results with Bahadur and Bar-Cohen (14)

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Figure 3

Dimensionless temperature contour plots for a two-dimensional orthotropic pin fin having aspect ratio L/R=10: (a) radial Biot number Bir=0.1 and longitudinal Biot number Biz=0.1, (b) Bir=2.0 and Biz=0.1, (c) Bir=0.1 and Biz=1.0, and (d) Bir=2.0 and Biz=1.0

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Figure 4

Dimensionless heat transfer rate from the pin fin as a function of radial Biot number Bir and thermal conductivity ratio k∗ for fin aspect ratio L/R=10

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Figure 5

Orthotropic pin fin efficiency and effectiveness as a function of radial Biot number Bir and thermal conductivity ratio k∗ for fin aspect ratio L/R=10

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Figure 6

Dimensionless heat transfer rate from the pin fin as a function of axial Biot number Biz and thermal conductivity ratio k∗ for fin aspect ratio L/R=10; comparison with one-dimensional solution

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Figure 7

Orthotropic pin fin efficiency and effectiveness as a function of axial Biot number Biz and thermal conductivity ratio k∗ for fin aspect ratio L/R=10; comparison with one-dimensional solution: (a) low values of Biz and (b) high values of Biz

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Figure 8

Percentage error in approximate (one-term) solution for dimensionless heat transfer and efficiency as a function of radial and axial Biot numbers (Bir,Biz) for a fin aspect ratio L/R=10

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Figure 9

Dimensionless heat transfer rate from the pin fin as a function of aspect ratio L/R and radial-to-axial thermal conductivity ratio k∗ for a dimensionless fin volume V∗=1×10−6.

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Figure 10

Dimensionless heat transfer rate from the pin fin as a function of aspect ratio L/R and dimensionless fin volume V∗ for a radial-to-axial thermal conductivity ratio k∗=0.05

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Figure 11

Optimal fin aspect ratio, (L/R)opt as a function of dimensionless fin volume V∗ for different values of radial-to-axial thermal conductivity ratio k∗; comparison with one-dimensional optimal solution

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