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Theory of Heat Conduction with Type 5 Boundary Condition

By Donald E. Amos

Sandia National Laboratories, Retired


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In the classical theory, the general solution of the heat conduction problem is expressed in terms of the Green's function. Terms which take into account volumetric heat generation, an initial temperature distribution and boundary conditions can be identified. In the current literature ( http://Exact.unl.edu ) a numbering system is used to describe a large set of possible problems arising from a variety of physical conditions and geometries. In this system, the classical boundary Types 1 through 3 define problems with known temperature, flux and heat loss on the boundary. These are extended to create Types 4 and 5 from Types 2 and 3 by adding a surface film or boundary layer with only heat capacity. This makes the boundary layer a source or sink for heat flow into or out of the material, thereby distinguishing Types 4 and 5 from classical types by the presence of a time derivative. The thrust of this development is to present the form of the boundary integral in the general solution for the classical types and, by analogy, extend the results to Types 4 and 5.


This work was supported by
NSF Award 1250625, Exact Analytical Conduction Toolbox,
administered by the University of Nebraska,
Kevin Cole, Director



[1] Cole, KD, Beck, JV, et. al. (2010), Heat Conduction Using Green's Functions, 2nd Ed., CRC Press
Boca Raton, 643pp

[2] Amos, DE, (2014) Green's Function for Heat Conduction in a Slab X55 with Type 5 Boundary
Conditions, http://nanohub.org/resources/20575

[3] Amos, DE, (2014) Heat Conduction in a Slab X55T0 and Sub-cases,

[4] Amos, DE, (2014) Green's Function for Heat Conduction in a Hollow Cylinder R55 with Type 5
Boundary Conditions, http://nanohub.org/resources/20661

[5] Amos, DE, (2014) Heat Conduction in a Hollow Cylinder R55T0 and Sub-cases
Conditions, http://nanohub.org/resources/20535

Cite this work

Researchers should cite this work as follows:

  • Donald E. Amos (2014), "Theory of Heat Conduction with Type 5 Boundary Condition," http://nanohub.org/resources/20365.

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