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TECHNICAL BRIEFS

# A Temperature Fourier Series Solution for a Hollow Sphere

[+] Author and Article Information
Gholamali Atefi

Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846-13114, Tehran, Iranatefi@iust.ac.ir

Mahdi Moghimi

Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846-13114, Tehran, Iran

J. Heat Transfer 128(9), 963-968 (Apr 21, 2006) (6 pages) doi:10.1115/1.2241914 History: Received October 05, 2005; Revised April 21, 2006

## Abstract

In this paper, we derive an analytical solution of a two-dimensional temperature field in a hollow sphere subjected to periodic boundary condition. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel’s theorem is used to solve the problem for a periodic boundary condition. The boundary condition is decomposed by Fourier series. In order to check the validity of the results, the technique was also applied to a solid sphere under harmonic boundary condition for which theoretical results were available in the literature. The agreement between the results of the proposed method and those reported by others for this particular geometry under harmonic boundary condition was realized to be very good, confirming the applicability of the technique utilized in the present work.

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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 1

Comparison between the result of the amplitude of a one-dimensional temperature field of a solid sphere (8) and the results for a solid sphere presented in (4) under the same boundary condition

Figure 2

Comparison between the result of the phase difference of a one-dimensional temperature field of a solid sphere (8) and the results for a solid sphere presented in (4) under the same boundary condition

Figure 3

Dimensionless amplitude, A, when r¯i=0.2, ψ=60

Figure 4

Dimensionless phase difference, ϕ, when r¯i=0.2, ψ=60

Figure 5

Dimensionless amplitude, A, when r¯i=0.2, ψ=30

Figure 6

Dimensionless phase difference, ϕ, when r¯i=0.2, ψ=30

Figure 7

Dimensionless amplitude, A, when r¯i=0.7, ψ=30

Figure 8

Dimensionless phase difference, ϕ, when r¯i=0.7, ψ=30

Figure 9

Dimensionless amplitude, A, when M=2, ψ=30

Figure 10

Dimensionless phase difference, ϕ, when M=2, ψ=30

Figure 11

Q(t)∕Q(p) when M=2, Bi=1 under periodic boundary condition

Figure 12

Q(t)∕Q(p) when M=2 under periodic boundary condition

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