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.
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Abbreviations
- a :
-
Parameter in thermal-conductivity expression
- A :
-
Constant
- Ai :
-
Airy function of the first kind
- Bi :
-
Airy function of the second kind
- C 1 ,C 2 :
-
Constants
- d :
-
Constant
- D :
-
Denominator
- f 1 ,f 2 :
-
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
- \({\varepsilon}\) :
-
Fin effectiveness
- η :
-
Fin efficiency
- θ :
-
Excess temperature
- a:
-
Ambient
- b:
-
Fin base
- t:
-
Fin tip
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Aziz, A., Rahman, M.M. Thermal Performance of a Functionally Graded Radial Fin. Int J Thermophys 30, 1637–1648 (2009). https://doi.org/10.1007/s10765-009-0627-x
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DOI: https://doi.org/10.1007/s10765-009-0627-x