Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction

Suneet Singh, Prashant K. Jain, Rizwan-uddin

Research output: Research - peer-reviewArticle

Abstract

Closed form analytical double-series solution is presented for the multi-dimensional unsteady heat conduction problem in polar coordinates (2-D cylindrical) with multiple layers in the radial direction. Spatially non-uniform, but time-independent, volumetric heat sources are assumed in each layer. Separation of variables method is used to obtain transient temperature distribution. In contrast to Cartesian or cylindrical (r, z) coordinates, eigenvalues in the direction perpendicular to the layers do not explicitly depend on those in the other direction. The implication of the above statement is that the imaginary eigenvalues are precluded from the solution of the problem. However, radial (transverse) eigenvalues are implicitly dependent on the angular eigenvalues through the order of the Bessel functions which constitute the radial eigenfunctions. Therefore, for each eigenvalue in the angular direction, corresponding radial eigenvalues must be obtained. Solution is valid for any combination of homogenous boundary condition of the first or second kind in the angular direction. However, inhomogeneous boundary conditions of the third kind are applied in the radial direction. Proposed solution is also applicable to multiple layers with zero inner radius. An illustrative example problem for the three-layer semi-circular annular region is solved. Results along with the isotherms are shown graphically and discussed.

LanguageEnglish (US)
Pages261-273
Number of pages13
JournalInternational Journal of Thermal Sciences
Volume47
Issue number3
DOIs
StatePublished - Mar 2008

Fingerprint

polar coordinates
conductive heat transfer
eigenvalues
Heat conduction
boundary conditions
Boundary conditions
Bessel functions
Isotherms
Temperature distribution
Hot Temperature
heat sources
eigenvectors
isotherms
temperature distribution
radii
Eigenvalues and eigenfunctions

Keywords

  • Analytical solution
  • Multi-layer
  • Polar coordinates
  • Transient heat conduction

ASJC Scopus subject areas

  • Fluid Flow and Transfer Processes
  • Mechanical Engineering

Cite this

Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction. / Singh, Suneet; Jain, Prashant K.; Rizwan-uddin.

In: International Journal of Thermal Sciences, Vol. 47, No. 3, 03.2008, p. 261-273.

Research output: Research - peer-reviewArticle

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