### Abstract

Analytical solutions to transient heat conduction problems are often obtained by superposition of a particular solution (often the steady-state solution) and an eigenfunction series, representing the terms that decay exponentially with time. Here, a finite element realization of this method is presented in which conventional finite element discretization is used for the spatial distribution of temperature and analytical methods for the time dependence. This leads to a linear eigenvalue problem whose solution then enables a general numerical model of the transient system to be created. The method is an attractive alternative to conventional time-marching schemes, particularly in cases where it is desired to explore the effect of a wide range of operating parameters. The method can be applied to any transient heat conduction problem, but particular attention is paid to the case where the Biot number is small compared with unity and where the evolution of the system is very close to that with zero heat loss from the exposed surfaces. This situation arises commonly in machines such as brakes and clutches which experience occasional short periods of intense heating. Numerical examples show that with typical parameter values, the simpler zero heat loss solution provides very good accuracy. One also shows that good approximations can be achieved using a relatively small subset of the eigenvectors of the problem.

Original language | English |
---|---|

Pages (from-to) | 1579-1588 |

Number of pages | 10 |

Journal | Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science |

Volume | 226 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2012 |

### Fingerprint

### Keywords

- brakes
- clutches
- eigenfunction series
- finite element method
- modal analysis

### ASJC Scopus subject areas

- Mechanical Engineering

### Cite this

*Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science*,

*226*(6), 1579-1588. https://doi.org/10.1177/0954406211423615

**Finite element implementation of the eigenfunction series solution for transient heat conduction problems with low Biot number.** / Al-Shabibi, A.; Barber, J. R.

Research output: Contribution to journal › Article

*Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science*, vol. 226, no. 6, pp. 1579-1588. https://doi.org/10.1177/0954406211423615

}

TY - JOUR

T1 - Finite element implementation of the eigenfunction series solution for transient heat conduction problems with low Biot number

AU - Al-Shabibi, A.

AU - Barber, J. R.

PY - 2012/6

Y1 - 2012/6

N2 - Analytical solutions to transient heat conduction problems are often obtained by superposition of a particular solution (often the steady-state solution) and an eigenfunction series, representing the terms that decay exponentially with time. Here, a finite element realization of this method is presented in which conventional finite element discretization is used for the spatial distribution of temperature and analytical methods for the time dependence. This leads to a linear eigenvalue problem whose solution then enables a general numerical model of the transient system to be created. The method is an attractive alternative to conventional time-marching schemes, particularly in cases where it is desired to explore the effect of a wide range of operating parameters. The method can be applied to any transient heat conduction problem, but particular attention is paid to the case where the Biot number is small compared with unity and where the evolution of the system is very close to that with zero heat loss from the exposed surfaces. This situation arises commonly in machines such as brakes and clutches which experience occasional short periods of intense heating. Numerical examples show that with typical parameter values, the simpler zero heat loss solution provides very good accuracy. One also shows that good approximations can be achieved using a relatively small subset of the eigenvectors of the problem.

AB - Analytical solutions to transient heat conduction problems are often obtained by superposition of a particular solution (often the steady-state solution) and an eigenfunction series, representing the terms that decay exponentially with time. Here, a finite element realization of this method is presented in which conventional finite element discretization is used for the spatial distribution of temperature and analytical methods for the time dependence. This leads to a linear eigenvalue problem whose solution then enables a general numerical model of the transient system to be created. The method is an attractive alternative to conventional time-marching schemes, particularly in cases where it is desired to explore the effect of a wide range of operating parameters. The method can be applied to any transient heat conduction problem, but particular attention is paid to the case where the Biot number is small compared with unity and where the evolution of the system is very close to that with zero heat loss from the exposed surfaces. This situation arises commonly in machines such as brakes and clutches which experience occasional short periods of intense heating. Numerical examples show that with typical parameter values, the simpler zero heat loss solution provides very good accuracy. One also shows that good approximations can be achieved using a relatively small subset of the eigenvectors of the problem.

KW - brakes

KW - clutches

KW - eigenfunction series

KW - finite element method

KW - modal analysis

UR - http://www.scopus.com/inward/record.url?scp=84864426891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864426891&partnerID=8YFLogxK

U2 - 10.1177/0954406211423615

DO - 10.1177/0954406211423615

M3 - Article

VL - 226

SP - 1579

EP - 1588

JO - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

JF - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

SN - 0954-4062

IS - 6

ER -