1-D Green's Functions For Heat Conduction Between Semi-infinite Slabs With Perfect and Imperfect Boundary Contact

Sandia National Laboratories, Retired

Abstract

This document presents two derivations for 1-D Green's functions for semi-infinite slabs in contact along the boundary x=0. The case of imperfect contact with a heat transfer coefficient h is derived and the case of perfect contact is obtained by taking h to infinity. The two dimensional case with source point (x',y') is reduced to the one dimensional case by applying a constant source in the y' direction. Because the two-dimensional source solutions have complex representations, we get 1-D complex representations also. However, these complex, 1-D forms can also be reduced to all real, closed forms which agree with a direct attack using the 1-D equations.

The case of perfect insulation on x=0 is also computed by taking h to zero. The result is 'the method of images' solution in the source region and zero in the other region.

The convolution for a continuous (constant) source in time is carried out to produce a point source solution with continuous heat generation.

References

References

 Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover Publications Inc., New York, 1046pp

 Amos DE (2012) Handbook of Integrals Related to Heat Conduction and Diffusion, http://nanohub.org/resources/13874

 Amos DE, Beck JV, de Monte F (2011) Transient Heat Conduction in Adjacent Quadrants Separated by a Thermal Resistance, http://nanohub.org/resources/12465

 Amos DE (2012) Transient Heat Conduction in Adjacent Materials Heated on Part of the Common Boundary, http://nanohub.org/resources/12390

 Amos, DE (2012), Green's Functions For Heat Conduction in Adjacent Materials, http://nanohub.org/resources/12856

 Amos, DE (2012), Theory of Heat Conduction for Two-region Problems Using Green's Functions, http://nanohub.org/resources/13671

 Cole DC, Beck JV, Haji-Sheikh A, Litkouhi B (2010) Heat Conduction Using Green's Functions, 2nd Ed., CRC Press, 643p.

 Carslaw HS, Jaeger JC (1948) Conduction of Heat in Solids, Oxford Univ Press, London, 386pp

Cite this work

Researchers should cite this work as follows:

• Donald E. Amos (2013), "1-D Green's Functions For Heat Conduction Between Semi-infinite Slabs With Perfect and Imperfect Boundary Contact," https://nanohub.org/resources/15237.