Green's Function for Heat Conduction in a Slab X55 with Type 5 Boundary Condtions

By Donald E. Amos

Sandia National Laboratories, Retired



Published on


The Green's function is the principal tool in construction of the general solution to the classical heat conduction problem. The solution is presented in terms of the internal heat generation, initial temperature and integrals which reflect the physical influence of the boundary. In the current literature ( ) the common boundary conditions are presented as Types 1,2,3,4, and 5 ranging from a specified temperature (Type 1) to the most general form (Type 5) where input energy (flux), heat loss to the surroundings, heat storage on a boundary layer and conduction into the material are considered. Since the driving energy for the Green's function is internal, the homogeneous form of the boundary condition is used to define the Green's function. The thrust of this work is to derive the Green's function, labeled X55, for a slab with Type 5 boundary conditions on both faces.

In the general solution with a Type 5 boundary, the usual integrals emerge, but an extra term which accounts for the release (or absorption) of energy stored in the boundary layer also appears. The results are used to construct the solution to an X55T0 slab problem where fluxes are the energy sources at the Type 5 boundaries x=0 and x=L. Another problem, describing the cooling of a boundary layer, is constructed to utilize only the extra term where the energy source is the heat stored in a boundary layer and the slab acts as a heat sink. Both solutions agree with the direct Laplace transform solutions. Finally, a closed system where there is no heat loss or gain is considered. The initial temperature differences of the slab and boundary layers provide the driving force for a redistribution of energy. The transient temperature distribution and equilibrium temperatures are calculated and agree with known results.


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) Theory of Heat Conduction with a Type 5 Boundary Condition,
  3. Amos, DE, (2014) Heat Conduction in a Slab X55T0 and Sub-cases,

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