Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations

Research output: Contribution to journalArticle

16 Citations (Scopus)

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 languageEnglish (US)
Article number121302
JournalJournal of Heat Transfer
Volume132
Issue number12
DOIs
StatePublished - 2010

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Inverse transforms
Thermal diffusion
thermal diffusion
Thermal conductivity
Thermal diffusivity
thermal conductivity
Temperature
thermal diffusivity
temperature
causes

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.

In: Journal of Heat Transfer, Vol. 132, No. 12, 121302, 2010.

Research output: Contribution to journalArticle

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