### Abstract

The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions - a problem solved using the SOV method - it is also "free" from imaginary eigenvalues.

Language | English (US) |
---|---|

Pages | 144-154 |

Number of pages | 11 |

Journal | Nuclear Engineering and Design |

Volume | 241 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

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### ASJC Scopus subject areas

- Nuclear Energy and Engineering
- Mechanical Engineering
- Safety, Risk, Reliability and Quality
- Materials Science(all)
- Nuclear and High Energy Physics
- Waste Management and Disposal

### Cite this

*Nuclear Engineering and Design*,

*241*(1), 144-154. DOI: 10.1016/j.nucengdes.2010.10.010

**Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions.** / Singh, Suneet; Jain, Prashant K.; Rizwan-Uddin.

Research output: Research - peer-review › Article

*Nuclear Engineering and Design*, vol 241, no. 1, pp. 144-154. DOI: 10.1016/j.nucengdes.2010.10.010

}

TY - JOUR

T1 - Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

AU - Singh,Suneet

AU - Jain,Prashant K.

AU - Rizwan-Uddin,

PY - 2011/1

Y1 - 2011/1

N2 - The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions - a problem solved using the SOV method - it is also "free" from imaginary eigenvalues.

AB - The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions - a problem solved using the SOV method - it is also "free" from imaginary eigenvalues.

UR - http://www.scopus.com/inward/record.url?scp=78650698255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650698255&partnerID=8YFLogxK

U2 - 10.1016/j.nucengdes.2010.10.010

DO - 10.1016/j.nucengdes.2010.10.010

M3 - Article

VL - 241

SP - 144

EP - 154

JO - Nuclear Engineering and Design

T2 - Nuclear Engineering and Design

JF - Nuclear Engineering and Design

SN - 0029-5493

IS - 1

ER -