Abstract
Both 1-D and 2-D analytic methods are used for a rectangular fin optimization. Optimum heat loss is taken as 98% of the maximum heat loss. Temperature profile using 2-D analytic method and relative error of temperature along the fin length between 1-D and 2-D analytic methods are presented. Increasing rate of the optimum heat loss with the variation of Biot number and decreasing rate of that with the variation of the fin base length are listed. Optimum fin tip length using 2-D analytic method and relative error of that between 1-D and 2-D analytic methods are presented as a function of Biot numbers ratio.
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Abbreviations
- Bi :
-
Fin top and bottom Biot number, (h l’) / k
- Bi e :
-
Fin tip Biot number, (h e l’) /k
- h :
-
Fin top and bottom heat transfer coefficient [W/m2°C]
- h e :
-
Fin tip heat transfer coefficient [W/m2 °C]
- k :
-
Thermal conductivity of fin material [W/ m°C]
- l’:
-
One half fin base height [m]
- L’ b :
-
Fin base length [m]
- Lb :
-
Dimensionless fin base length,L’ b /l’
- L’ e :
-
Fin tip length [m]
- L e :
-
Dimensionless fin tip length,L’ e /l’
- q :
-
Heat loss per unit width [W/m]
- Q :
-
Dimensionless heat loss,q/(kϕ i )
- T :
-
Fin temperature [°C]
- T b :
-
Fin base temperature [°C]
- T i :
-
Temperature of inside wall [°C]
- T ∞ :
-
Ambient temperature [°C]
- x’:
-
Length directional variable [m]
- x :
-
Dimensionless length directional variable,x’/l’
- y’:
-
Height directional variable [m]
- y :
-
Dimensionless height directional variable
- β:
-
Ratio of Biot numbers,Bi e /Bi
- θ:
-
Dimensionless temperature, (T-T ∞ )/(T i -T ∞ )
- λn :
-
Eigenvalues (n = 1,2,3, …)
- ϕi :
-
Adjusted temperature of inside wall [°C], (T i -T ∞ )
- 1:
-
One-dimensional analysis
- 2:
-
Two-dimensional analysis
- b:
-
Fin base
- e:
-
Fin tip
- i:
-
Inside wall
- ∞:
-
Surrounding
- ’:
-
Dimensional quantity
- *:
-
Optimum
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Kang, HS. A rectangular fin optimization including comparison between 1-D and 2-D analyses. J Mech Sci Technol 20, 2203–2208 (2006). https://doi.org/10.1007/BF02916337
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DOI: https://doi.org/10.1007/BF02916337