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Performance analysis and optimization of radiating fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method

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Abstract

Although tapered fins transfer more rate of heat per unit volume, they are not found in every practical application because of the difficulty in manufacturing and fabrications. Therefore, there is a scope to modify the geometry of a constant thickness fin in view of the less difficulty in manufacturing and fabrication as well as betterment of heat transfer rate per unit volume of the fin material. For the better utilization of fin material, it is proposed a modified geometry of new fin with a step change in thickness (SF) in the literature. In the present paper, the homotopy perturbation method has been used to evaluate the temperature distribution within the straight radiating fins with a step change in thickness and variable thermal conductivity. The temperature profile has an abrupt change in the temperature gradient where the step change in thickness occurs and thermal conductivity parameter describing the variation of thermal conductivity has an important role on the temperature profile and the heat transfer rate. The optimum geometry which maximizes the heat transfer rate for a given fin volume has been found. The derived condition of optimality gives an open choice to the designer.

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Abbreviations

a:

Dimensionless cross-sectional area defined in Eq. (27)

A:

General differential operator

Ac :

Cross-sectional area of the fin, m2

B:

Boundary operator

C1,2,3 :

Integral constants

k:

Thermal conductivity W/(mK)

ℓ:

Length of the thin section of the fin, m

L:

Length of the entire fin, m

N:

Nonlinear operator

Nc :

Conduction-radiation parameter

q:

Dimensionless heat transfer rate

Q:

Heat transfer rate per unit fin depth, W/m

t:

Unreduced semi-thickness of the fin, m

T1 :

Temperature within the thin section of the fin, K

T2 :

Temperature within the thick section of the fin, K

Tb :

Base temperature, K

x:

Axial coordinate for entire fin, m

x1 :

Axial coordinate for the thin section of the fin, m

x2 :

Axial coordinate for the thick section of the fin, m

α:

Thickness parameter

β:

Thermal conductivity parameter

δ:

Dimensionless fin semi thickness

ε:

Emissivity of the fin material

η:

Fin efficiency

κ:

A constant for variable thermal conductivity, K−1

λ:

Length ratio

σ:

The Stefan-Boltzmann constant, W/(m2 k4)

θ:

Dimensionless temperature within the thin section of the fin

ϕ:

Dimensionless temperature within the thick section of the fin

ξ:

Dimensionless axial coordinate of the thin section of the fin

τ:

Dimensionless axial coordinate of the thick section of the fin

ζ:

Dimensionless axial coordinate for the entire fin

ADM:

Adomian decomposition method

HPM:

Homotopy perturbation method

SF:

Step fin

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Authors and Affiliations

  1. Department of Mechanical Engineering, Ataturk University, 25240, Erzurum, Turkey

    Cihat Arslanturk

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  1. Cihat Arslanturk
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Correspondence to Cihat Arslanturk.

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Arslanturk, C. Performance analysis and optimization of radiating fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method. Heat Mass Transfer 47, 131–138 (2011). https://doi.org/10.1007/s00231-010-0673-8

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  • DOI: https://doi.org/10.1007/s00231-010-0673-8

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