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Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperature-dependent thermal conductivity

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Abstract

Here, the optimal homotopy asymptotic method (OHAM), one of the newest analytical methods which is powerful and easy-to-use, is applied to solve heat transfer problems with high nonlinearity order. In this research, the OHAM is used to investigate two problems: the temperature distribution equation in a convective straight fin with temperature-dependent thermal conductivity and the convective–radiative cooling of a lumped system with variable specific heat. The validity of our results is verified by numerical results. The OHAM provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. This realizes using a number of auxiliary constants which are optimally determined. Thus, unlike perturbation methods, the OHAM does not depend on any small physical parameters and it is valid for both weakly and strongly nonlinear problems. It has been attempted to show the capabilities and wide-range applications of the OHAM in comparison with the previous methods in solving heat transfer problems.

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Abbreviations

A :

Area (m2)

A c :

Cross-sectional area of fin (m2)

b :

Fin length (m)

c :

Specific heat (J kg−1 K−1)

c a :

Specific heat at ambient temperature (J kg−1 K−1)

E :

Surface emissivity (W)

h :

Convection heat transfer coefficient (W m−2 K−1)

k :

Thermal conductivity of fin material (W m−1 K−1)

k a :

Thermal conductivity at ambient temperature (W m−1 K−1)

k b :

Thermal conductivity at base temperature (W m−1 K−1)

P :

Fin perimeter (m)

T :

Temperature (K)

T a :

Ambient temperature (K)

T b :

Base temperature of fin (K)

Ti :

Initial temperature (K)

T s :

Effective sink temperature (K)

V :

Volume (m3)

x :

Distance measured from fin tip (m)

β :

Volumetric thermal expansion coefficient (K−1)

θ :

Dimensionless temperature (–)

λ :

Slope of thermal conductivity-temperature curve (K−1)

ξ :

Dimensionless length (–)

ρ :

Density (kg m−3)

σ :

Stefan–Boltzman constant (–)

τ :

Dimensionless time (–)

ψ :

Thermo-geometric fin parameter (–)

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Authors and Affiliations

  1. Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, 15875-4413, Tehran, Iran

    Saeed Dinarvand & Reza Hosseini

Authors
  1. Saeed Dinarvand
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  2. Reza Hosseini
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Correspondence to Saeed Dinarvand.

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Dinarvand, S., Hosseini, R. Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperature-dependent thermal conductivity. Afr. Mat. 24, 103–116 (2013). https://doi.org/10.1007/s13370-011-0043-9

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  • DOI: https://doi.org/10.1007/s13370-011-0043-9

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