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This content will become publicly available on June 7, 2017

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

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 three types of boundary conditions in the outer and inner surface of the sphere. Lastly, the solution procedure is demonstrated on an illustrativemore » problem for which results are obtained.« less
 [1] ;  [2] ;  [3]
  1. Indian Inst. of Technology (IIT) Bombay, Mumbai (India). Dept. of Energy Science and Engineering
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Reactor and Nuclear Systems Division
  3. Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States). Dept. of Nuclear, Plasma and Radiological Engineering
Publication Date:
OSTI Identifier:
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Journal of Heat Transfer
Additional Journal Information:
Journal Volume: 138; Journal Issue: 10; Journal ID: ISSN 0022-1481
Research Org:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
Country of Publication:
United States
36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING Transient; Multilayer; Spherical; Analytical; Conduction; Heat Transfer; Steady state; Temperature; Eigenfunctions; Boundary-value problems