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International Journal of Thermophysics

, Volume 33, Issue 1, pp 143–159 | Cite as

Analysis of Transient Heat Conduction in a Hollow Sphere Using Duhamel Theorem

  • Mohammad A. Abdous
  • Hasan Barzegar Avval
  • Pouria Ahmadi
  • Nima Moallemi
  • Ibrahim Dincer
Article

Abstract

In this article, analytical modeling of two-dimensional heat conduction in a hollow sphere is presented. The hollow sphere is subjected to time-dependent periodic boundary conditions at the inner and outer surfaces. The Duhamel theorem is employed to solve the problem where the periodic and time-dependent terms in the boundary conditions are considered. In the analysis, the thermophysical properties of the material are assumed to be isotropic and homogenous. Moreover, the effects of the temperature oscillation frequency, the thickness variation of the hollow sphere, and thermophysical properties of the sphere are studied. The temperature distribution obtained here contains two characteristics, the dimensionless amplitude (A) and the dimensionless phase difference (\({\varphi}\)). Moreover, the obtained results are shown with respect to Biot and Fourier numbers. Comparison between the present results and the findings from a previous study for a hollow sphere subjected to the reference harmonic state show good agreement.

Keywords

Fourier series Hollow sphere Periodic boundary condition Transient heat conduction 

List of Symbols

a2

Inverse of thermal diffusivity (s · m−2)

Aknm

Dimensionless amplitude of temperature

Bi

Biot number

Cp, c

Specific heat (J · kg−1 · K−1)

f(ψ)

Spherical space function

Fo

Fourier number

g(t)

Time function of boundary condition

h

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

k

Thermal conductivity (W · m−1 · K−1)

M

Defined in Eq. 33

P

Time period

Q

Stored thermal energy

r

Radius (m)

\({\overline r}\)

Dimensionless radius, defined by Eq. 28

t

Time (s)

\({\overline t}\)

Dimensionless time, defined by Eq. 28

V

Volume (m3)

x

Dimensionless thickness, defined by Eq. 28

Xc

Electrical reactance

R

Electrical resistance

C

Capacitor constant

Greek Symbols

θ

Temperature field

ζ

Defined by Eq. 13

ωkn

Eigenvalue

ω

Time frequency

\({\Phi ,\eta}\)

Eigenfunction

\({C_n^{\left( {\rm i} \right)} ,C_n^{\left( {\rm o} \right)}}\)

Defined by Eq. 13

Pn (ζ)

Legendre function

Jn (ωknr)

Bessel function

Δ (n )

Defined by Eq. 14

\({\alpha _n^{\left( {\rm i} \right)} ,\alpha _n^{\left( {\rm o} \right)}}\)

Defined by Eq. 15

\({\beta _n^{\left( {\rm o} \right)} ,\beta _n^{\left( {\rm i} \right)}}\)

Defined by Eq. 15

ψ

Spherical angle

δkn

Defined by Eq. 17

Dkn

Defined by Eq. 25

Tkn

Defined by Eq. 25

\({\varphi _{kn} ,\varphi}\)

Phase difference

ρ

Density (kg · m−3)

γkni

Defined by Eq. 30

τ

Time (s)

Subscripts

0

steady

1

unsteady

i

inner

o

outer

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Mohammad A. Abdous
    • 1
  • Hasan Barzegar Avval
    • 2
  • Pouria Ahmadi
    • 1
  • Nima Moallemi
    • 1
  • Ibrahim Dincer
    • 3
  1. 1.Islamic Azad University, Roudan BranchRoudanIran
  2. 2.Energy Optimization Research and Development GroupTehranIran
  3. 3.Department of Mechanical Engineering, Faculty of Engineering and Applied ScienceUniversity of Ontario Institute of Technology (UOIT)OshawaCanada

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