### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 261-273 |

Number of pages | 13 |

Journal | International Journal of Thermal Sciences |

Volume | 47 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Mechanical Engineering

### Cite this

*International Journal of Thermal Sciences*,

*47*(3), 261-273. DOI: 10.1016/j.ijthermalsci.2007.01.031

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

Research output: Contribution to journal › Article

*International Journal of Thermal Sciences*, vol 47, no. 3, pp. 261-273. DOI: 10.1016/j.ijthermalsci.2007.01.031

}

TY - JOUR

T1 - Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction

AU - Singh,Suneet

AU - Jain,Prashant K.

AU - Rizwan-uddin,

PY - 2008/3

Y1 - 2008/3

N2 - 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.

AB - 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.

KW - Analytical solution

KW - Multi-layer

KW - Polar coordinates

KW - Transient heat conduction

UR - http://www.scopus.com/inward/record.url?scp=37349011827&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37349011827&partnerID=8YFLogxK

U2 - 10.1016/j.ijthermalsci.2007.01.031

DO - 10.1016/j.ijthermalsci.2007.01.031

M3 - Article

VL - 47

SP - 261

EP - 273

JO - International Journal of Thermal Sciences

T2 - International Journal of Thermal Sciences

JF - International Journal of Thermal Sciences

SN - 1290-0729

IS - 3

ER -