### Abstract

The Kirchhoff transformation is the classical method of solution to the nonlinear thermal diffusion problem with temperature dependent properties. It essentially converts the nonlinear problem into a linear one if the thermal diffusivity is approximately constant. Unfortunately, with the only exception of an exponential dependence of the thermal conductivity on temperature, all other thermal conductivity functions produce an inconvenient form for the inverse transform. This paper shows that the Kirchhoff transformation is a particular consequence of the more general Cole-Hopf transformation. However, the classical presentation of the Kirchhoff transformation in terms of a definite integral is more restrictive than the result obtained from the Cole-Hopf transformation and it is this restrictiveness that causes the practical inconvenience in the form of the inverse transform. It is shown that a more compact and practically convenient form of the inverse transform can be obtained by using directly the result from the Cole-Hopf transformation, hence, making its application more attractive.

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

Article number | 121302 |

Journal | Journal of Heat Transfer |

Volume | 132 |

Issue number | 12 |

DOIs | |

State | Published - 2010 |

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### Keywords

- Cole-Hopf transformation
- Kirchhoff transformation
- nonlinear conduction
- nonlinear thermal diffusion
- temperature dependent properties

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Materials Science(all)
- Condensed Matter Physics

### Cite this

**Analytical solution to nonlinear thermal diffusion : Kirchhoff versus Cole-Hopf transformations.** / Vadasz, Peter.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Analytical solution to nonlinear thermal diffusion

T2 - Kirchhoff versus Cole-Hopf transformations

AU - Vadasz, Peter

PY - 2010

Y1 - 2010

N2 - The Kirchhoff transformation is the classical method of solution to the nonlinear thermal diffusion problem with temperature dependent properties. It essentially converts the nonlinear problem into a linear one if the thermal diffusivity is approximately constant. Unfortunately, with the only exception of an exponential dependence of the thermal conductivity on temperature, all other thermal conductivity functions produce an inconvenient form for the inverse transform. This paper shows that the Kirchhoff transformation is a particular consequence of the more general Cole-Hopf transformation. However, the classical presentation of the Kirchhoff transformation in terms of a definite integral is more restrictive than the result obtained from the Cole-Hopf transformation and it is this restrictiveness that causes the practical inconvenience in the form of the inverse transform. It is shown that a more compact and practically convenient form of the inverse transform can be obtained by using directly the result from the Cole-Hopf transformation, hence, making its application more attractive.

AB - The Kirchhoff transformation is the classical method of solution to the nonlinear thermal diffusion problem with temperature dependent properties. It essentially converts the nonlinear problem into a linear one if the thermal diffusivity is approximately constant. Unfortunately, with the only exception of an exponential dependence of the thermal conductivity on temperature, all other thermal conductivity functions produce an inconvenient form for the inverse transform. This paper shows that the Kirchhoff transformation is a particular consequence of the more general Cole-Hopf transformation. However, the classical presentation of the Kirchhoff transformation in terms of a definite integral is more restrictive than the result obtained from the Cole-Hopf transformation and it is this restrictiveness that causes the practical inconvenience in the form of the inverse transform. It is shown that a more compact and practically convenient form of the inverse transform can be obtained by using directly the result from the Cole-Hopf transformation, hence, making its application more attractive.

KW - Cole-Hopf transformation

KW - Kirchhoff transformation

KW - nonlinear conduction

KW - nonlinear thermal diffusion

KW - temperature dependent properties

UR - http://www.scopus.com/inward/record.url?scp=79952094976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952094976&partnerID=8YFLogxK

U2 - 10.1115/1.4002325

DO - 10.1115/1.4002325

M3 - Article

VL - 132

JO - Journal of Heat Transfer

JF - Journal of Heat Transfer

SN - 0022-1481

IS - 12

M1 - 121302

ER -