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An Analytical Study on a Model Describing Heat Conduction in Rectangular Radial Fin with Temperature-Dependent Thermal Conductivity

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Abstract

The coupling of the homotopy perturbation method (HPM) and the variational iteration method (VIM) is a strong technique for solving higher dimensional initial boundary value problems. In this article, after a brief explanation of the mentioned method, the coupled techniques are applied to one-dimensional heat transfer in a rectangular radial fin with a temperature-dependent thermal conductivity to show the effectiveness and accuracy of the method in comparison with other methods. The graphical results show the best agreement of the three methods; however, the amount of calculations of each iteration for the combination of HPM and VIM was reduced markedly for multiple iterations. It was found that the variation of the dimensionless temperature strongly depends on the dimensionless small parameter \({\varepsilon_1}\). Moreover, as the dimensionless length increases, the thermal conductivity of the fin decreases along the fin.

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Abbreviations

A s :

Fin surface area (m2)

A c :

Cross-sectional area of the fin (m2)

h :

Coefficient of natural convection (W · m−1 · K−1)

r :

Radius (m)

R :

Dimensionless length

T :

Temperature (K)

T a :

Ambient temperature (K)

T b :

Temperature at the base (K)

k(T):

Thermal conductivity (W · m−1 · K−1)

k a :

Thermal conductivity at T =  T a (W · m−1 · K−1)

L :

Fin length (m)

L(u):

Linear term

N(u):

Nonlinear term

HPM:

Homotopy perturbation method

VIM:

Variational iteration method

\({\varepsilon}\) :

Small parameter

β :

Coefficient of linear conductivity (K −1)

θ :

Dimensionless temperature

a:

Ambient

s:

Surface

b:

Base

n:

Number of iteration

i:

Inner

o:

Outer

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Authors and Affiliations

  1. Department of Mechanical Engineering, Islamic Azad University, Sari Branch, Mazandaran, P.O. Box 48164-194, Sari, Iran

    F. Hedayati & S. M. Hamidi

  2. Department of Mechanical Engineering, Babol University of Technology, P.O. Box 484, Babol, Iran

    D. D. Ganji

  3. Mechanical Engineering Department, AmirKabir University of Technology, 424 Hafez Ave, P.O. Box 15875-4413, Tehran, Iran

    A. Malvandi

Authors
  1. F. Hedayati
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  2. D. D. Ganji
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  3. S. M. Hamidi
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  4. A. Malvandi
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Correspondence to D. D. Ganji.

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Hedayati, F., Ganji, D.D., Hamidi, S.M. et al. An Analytical Study on a Model Describing Heat Conduction in Rectangular Radial Fin with Temperature-Dependent Thermal Conductivity. Int J Thermophys 33, 1042–1054 (2012). https://doi.org/10.1007/s10765-012-1222-0

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  • DOI: https://doi.org/10.1007/s10765-012-1222-0

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