## Transient Heat Conduction in Adjacent Materials Heated on Part of the Common Boundary

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#### Abstract

This paper considers a classical linear, transient heat conduction problem set in Regions 1 and 2 defined by the half planes x>0 and x<0. These regions are in perfect contact along the boundary x=0, are initially at zero temperature and have distinct, constant physical properties. The energy source is an infinitely thin, semi-infinite uniform source of heat along the lower half of the boundary x=0 between the two materials. In a three dimensional setting we have a uniform half-plane heat source sandwiched between two semi-infinite slabs.

The solution for temperatures in each half plane is obtained by a classical application of the Laplace transform. The results are line integrals in the complex plane in terms of iterated co-error functions of a complex variable. Quadratures are required for numerical evaluation and computation of error functions of a complex variable are discussed. Temperatures along the boundary are of special interest, but because of slow convergence near the origin, special formulae which accelerate the convergence and display the asymptotic behavior for y to zero are developed.

A solution for a uniform source on a finite segment of the common boundary can be constructed from the semi-infinite source solution. By translating the endpoints of these source segments and using superposition, some interesting solutions for step source patterns distributed along the boundary x=0 are possible.

The special case where corresponding physical properties are the same in both half planes is known. Numerical experiments compare the general solution for the semi-infinite source with this special case.

Keywords Two regions Heat Conduction Unsteady State Laplace Transform

#### References

- Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover Publications Inc., New York, 1046 pp
- 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 (2006) Handbook of Integrals Related to Heat Conduction and Diffusion, http://nanohub.org/resources/13874
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- Amos DE (1990), ALGORITHM 683, A Portable Subroutine for Exponential Integrals of a Complex Argument, ACM Trans Math Software, Vol 16, No 2, pp 178-182

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#### Tags

- thermal
- nanoelectronics
- heat transfer
- heat conduction
- conduction heat transfer