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Thermal Performance of a Functionally Graded Radial Fin

  • A. AzizEmail author
  • M. M. Rahman
Article

Abstract

This paper investigates the steady-state thermal performance of a radial fin of rectangular profile made of a functionally graded material. The thermal conductivity of the fin varies continuously in the radial direction following a power law. The boundary conditions of a constant base temperature and an insulated tip are assumed. Analytical solutions for the temperature distribution, heat transfer rate, fin efficiency, and fin effectiveness are found in terms of Airy wave functions, modified Bessel functions, hyperbolic functions, or power functions depending on the exponent of the power law. Numerical results illustrating the effect of the radial dependence of the thermal conductivity on the performance of the fin are presented and discussed. It is found that the heat transfer rate, the fin efficiency, and the fin effectiveness are highest when the thermal conductivity of the fin varies inversely with the square of the radius. These quantities, however, decrease as the exponent of the power law increases. The results of the exact solutions are compared with a solution derived by using a spatially averaged thermal conductivity. Because large errors can occur in some cases, the use of a spatially averaged thermal-conductivity model is not recommended.

Keywords

Fin effectiveness Fin efficiency Functionally graded radial fin Heat transfer rate Spatially averaged thermal-conductivity model 

List of Symbols

a

Parameter in thermal-conductivity expression

A

Constant

Ai

Airy function of the first kind

Bi

Airy function of the second kind

C1 ,C2

Constants

d

Constant

D

Denominator

f1 ,f2

Constants

F

Function of r b, r t, and n

h

Convective heat transfer coefficient

I

Modified Bessel function of first kind

J

Bessel function of the first kind

k

Fin thermal conductivity

K

Modified Bessel function of second kind

m

Fin parameter

n

Exponent

q

Fin heat transfer rate

r

Radial coordinate

T

Temperature

w

Fin thickness

Y

Bessel function of the second kind

Greek Symbols

\({\varepsilon}\)

Fin effectiveness

η

Fin efficiency

θ

Excess temperature

Subscripts

a

Ambient

b

Fin base

t

Fin tip

References

  1. 1.
    Incropera F.P., DeWitt D.P.: Fundamentals of Heat and Mass Transfer. Wiley & Sons, New York (2002)Google Scholar
  2. 2.
    Kraus A.D., Aziz A., Welty J.R.: Extended Surface Heat Transfer. Wiley, New York (2001)Google Scholar
  3. 3.
    Babaei M.H., Chen Z.T.: Int. J. Thermophys. 29, 1457 (2008)CrossRefGoogle Scholar
  4. 4.
    Noda N.: J. Therm. Stresses 22, 477 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Eslami M.R., Babaei M.H., Poultangari R.: Int. J. Pres. Ves. Pip. 82, 522 (2005)CrossRefGoogle Scholar
  6. 6.
    Hosseini S.M., Akhlaghi M., Shakeri M.: Heat Mass Transfer 43, 669 (2006)CrossRefADSGoogle Scholar
  7. 7.
    Lau W., Tan C.W.: J. Heat Transfer T. ASME 95, 549 (1973)Google Scholar
  8. 8.
    Yovanovich M.M., Culham J.R., Lemczyk T.F.: J. Thermophys. Heat Transfer 2, 152 (1988)CrossRefADSGoogle Scholar
  9. 9.
    Aparacido J.B., Cotta R.M.: Heat Transfer Eng. 11, 49 (1990)CrossRefADSGoogle Scholar
  10. 10.
    Bejan A.: Heat Transfer. Wiley, New York (1993)Google Scholar
  11. 11.
    Sahin A.Z.: J. Thermophys. Heat Transfer 11, 153 (1997)CrossRefGoogle Scholar
  12. 12.
    Abramowitz M., Stegun I.G.: Handbook of Mathematical Functions. Dover Publications, New York (1972)zbMATHGoogle Scholar
  13. 13.
    Aziz A.: Heat Conduction with Maple. R.T. Edwards, Inc., Philadelphia, PA (2006)Google Scholar
  14. 14.
    Jiji L.M.: Heat Conduction. Begell House, Inc., New York (2000)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringGonzaga UniversitySpokaneUSA
  2. 2.Department of Mathematics and Statistics, College of ScienceSultan Qaboos UniversityMuscatOman

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