Research Papers: Conduction

Analytical Solution to Nonlinear Thermal Diffusion: Kirchhoff Versus Cole–Hopf Transformations

[+] Author and Article Information
Peter Vadasz

Department of Mechanical Engineering, Northern Arizona University, P.O. Box 15600, Flagstaff, AZ 86011; Faculty of Engineering, University of Kwa-Zulu Natal, Durban 4041, South Africa

J. Heat Transfer 132(12), 121302 (Sep 22, 2010) (6 pages) doi:10.1115/1.4002325 History: Received November 01, 2009; Revised July 23, 2010; Published September 22, 2010; Online September 22, 2010

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.

Copyright © 2010 by American Society of Mechanical Engineers
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