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International Journal of Thermophysics

, Volume 33, Issue 5, pp 924–941 | Cite as

Analytical Solution for Convective–Radiative Continuously Moving Fin with Temperature-Dependent Thermal Conductivity

  • Mohsen Torabi
  • Hessameddin Yaghoobi
  • A. Aziz
Article

Abstract

In this article, heat transfer in a moving fin with variable thermal conductivity, which is losing heat by simultaneous convection and radiation to its surroundings is analyzed. The calculations are carried out by using the differential transformation method (DTM) which is an analytical solution technique that can be applied to various types of differential equations. The effects of parameters such as the Peclet number, Pe, thermal conductivity parameter, a, convection–conduction parameter, N c, radiation–conduction parameter, N r, dimensionless convection sink temperature, θ a, and dimensionless radiation sink temperature, θ s, on the temperature distribution are illustrated and explained. The analytical solution is found to be in good agreement with the direct numerical solution. Moreover, the results demonstrate that the DTM is very effective in generating analytical solutions for even highly nonlinear problems.

Keywords

Convection–radiation heat transfer Differential transformation method Moving fin Thermal analysis Variable thermal conductivity 

List of Symbols

Variables

A

Fin cross-sectional area (m2)

a

Dimensionless thermal conductivity

cp

Specific heat of the material (J · kg−1 · K−1)

D

Domain

H

Constant

h

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

k(T)

Temperature-dependent thermal conductivity (W · m−1 · K−1)

ka

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

L

Fin length (m)

Nc

Dimensionless convection–conduction parameter

Nr

Dimensionless radiation–conduction parameter

P

Fin perimeter (m)

Pe

Peclet number

T

Temperature (K)

Ta

Sink temperature for convection (K)

Tb

Fin’s base temperature (K)

Ts

Sink temperature for radiation (K)

U

Speed of moving fin (m · s−1)

X

Dimensionless axial distance measured from the tip of the fin

X(k)

Transformed analytical function

x

Axial distance measured from the tip of the fin (m)

x(t)

Original analytical function

Greek Symbols

α

Thermal diffusivity of the material (m2 · s−1)

β

Slope of the thermal conductivity–temperature curve (K−1)

\({\varepsilon}\)

Emissivity

θ

Dimensionless temperature

θa

Dimensionless convection sink temperature

θb

Dimensionless fin’s base temperature

θs

Dimensionless radiation sink temperature

ρ

Density of material (kg · m−3)

σ

Stefan–Boltzmann constant

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Young Researchers Club, Central Tehran BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Mechanical Engineering, School of Engineering and Applied ScienceGonzaga UniversitySpokaneUSA

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