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Afrika Matematika

, Volume 24, Issue 1, pp 103–116 | Cite as

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

  • Saeed DinarvandEmail author
  • Reza Hosseini
Article

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.

Keywords

Nonlinear heat transfer equations Extended surface Convective–radiative cooling Temperature-dependent thermal conductivity Optimal homotopy asymptotic method (OHAM) Homotopy perturbation method (HPM) 

List of symbols

A

Area (m2)

Ac

Cross-sectional area of fin (m2)

b

Fin length (m)

c

Specific heat (J kg−1 K−1)

ca

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)

ka

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

kb

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

P

Fin perimeter (m)

T

Temperature (K)

Ta

Ambient temperature (K)

Tb

Base temperature of fin (K)

Ti

Initial temperature (K)

Ts

Effective sink temperature (K)

V

Volume (m3)

x

Distance measured from fin tip (m)

Greek symbols

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 (–)

Mathematics Subject Classification (2000)

35A24 (Methods of ordinary differential equations) 49K15 (Problems involving ordinary differential equations) 

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Copyright information

© African Mathematical Union and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran

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