Research Papers: Conduction

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

[+] Author and Article Information
Suneet Singh

Department of Energy Science and Engineering,
Indian Institute of Technology (IIT) Bombay,
Powai, Mumbai 400076, India
e-mail: suneet.singh@iitb.ac.in

Prashant K. Jain

Reactor and Nuclear Systems Division,
Oak Ridge National Laboratory,
Oak Ridge, TN 37831


Department of Nuclear, Plasma
and Radiological Engineering,
University of Illinois at Urbana—Champaign,
216 Talbot Lab,
104 S. Wright Street,
Urbana, IL 61801

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 21, 2015; final manuscript received April 25, 2016; published online June 7, 2016. Assoc. Editor: William P. Klinzing.This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The U.S. government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. government purposes.

J. Heat Transfer 138(10), 101301 (Jun 07, 2016) (11 pages) Paper No: HT-15-1488; doi: 10.1115/1.4033536 History: Received July 21, 2015; Revised April 25, 2016

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

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Fig. 1

(a) Coordinate system for the problem and (b) 2D cross-sectional view of n concentric spherical layers at θ  = π/2 plane

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Fig. 2

The plot of heat flux applied at the surface of the sphere for the illustrative example given by Eq. (45)

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Fig. 3

Plot of eigencondition f[λ] against λ for (a) m = 1 and (b) m = 2. Roots of the transcendental eigencondition are obtained by using Mathematica [27] and are graphically verified by plotting f[λ].

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Fig. 4

Transient temperature distribution in the radial direction along θ = 0. The change of azimuthal coordinate will not have any change in the values at θ = 0.

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Fig. 5

Transient temperature distribution in the radial direction along θ=π/4 for different azimuthal (ϕ) locations: (a) ϕ=0, (b) ϕ=π/2, and (c) ϕ=3π/2

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Fig. 6

Transient temperature distribution in the radial direction along θ=π/2 for different azimuthal (ϕ) locations: (a) ϕ=0, (b) ϕ=π/2, and (c) ϕ=3π/2

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Fig. 7

Temperature contours in the θ=π/2 midplane of the three-layer sphere at different times. Coordinate system is same as in Fig. 1(b).




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