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