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

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#### 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

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

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

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

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

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

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

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

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

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