## Theory of Heat Conduction for Two Region Problems Using Green's Functions

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#### Abstract

This paper derives equations which describe transient temperature distributions in adjacent regions which share a common boundary. These regions consist of materials with distinct, constant physical properties. The theory is developed for two types of boundary contact. The first formula is developed for perfect contact where there is continuity of both temperature and flux. The second formula allows for a thermal resistance at the boundary which retains continuity of flux, but causes a temperature drop across the boundary. As an example, the two-region theory is applied to quadrants 1 and 2 which are separated by an infinitely thin resistive layer on the y-axis and are heated along the x-axis. This problem was solved by a direct approach in a previous paper and the two-region Green’s function approach gives the same results.

Keywords Two Regions Heat Conduction Unsteady State Laplace Transform

#### References

References

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

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

[3] Amos, DE (2011), Transient Heat Conduction in Adjacent Materials Heated on Part of the Common Boundary, http://nanohub.org/resources/12390.

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

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

[6] Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B (2010) Heat Conduction Using Green's Functions, 2nd Ed., CRC Press, Boca Raton, 643 pp

[7] Morse PM, Feshbach H (1953) Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 997 pp

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

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