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Transient Heat Conduction in Adjacent Quadrants Separated by a Thermal Resistance

By Donald E. Amos1, James Vere Beck2, Filippo de Monte3

1. Sandia National Laboratories, Retired 2. Michigan State University 3. University of L'Aquila, Italy



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Abstract Two linear, transient heat conduction problems set in quadrants 1 and 2 of the (x,y) plane are solved. In each problem, the quadrants have distinct, constant physical properties and are separated by an infinitely thin thermal resistance along the y-axis. Each region is initially at zero temperature. In Problem I, constant fluxes are specified along the x-axis boundaries to complete the problem definition; while in Problem II, constant temperatures are specified.

An attempt at a solution to Problem I by classical application of the Laplace transform results in an integral representation for the temperature in each quadrant. Unfortunately the integrals only converge in 45 degree wedges which are close to the x-axis. However, a modification in the path of integration into the complex plane leads to a complete solution in terms of integrals of co-error functions of a complex variable. Details on high accuracy numerical evaluation of error functions and quadratures are provided.

Problem II is solved by manipulating the solution of Problem I.

Explicit solutions for over 12 special cases, including some quarter plane problems, evolve in terms of other functions by taking limits or specializing parameters. Numerical experiments compare the general solution with a number of these cases.

Keywords Two regions Heat Conduction Unsteady State Laplace Transform



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

[2] Amos DE (2012), "Handbook of Integrals Related to Heat Conduction and Diffusion,"

[3] Amos DE, Burgmeier JW (1973) Computation with Three-Term Linear Recursion Relations, SIAM Review, Vol. 15, No. 2, April: 335-351

[4] Gautschi W (1967) Computational Aspects of Three-Term Recurrence Relations, SIAM Review, Vol. 9, No. 1, January: 24-82

[5] Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B (1992) Heat Conduction Using Green's Functions, Hemisphere Press, Washington D.C., 523 pp

[6] Amos DE (1990) Computation of Exponential Integrals of Complex Argument, ACM Trans. Math. Software, Vol. 16, No. 2, June: 169-177

[7] Amos DE (1990) Algorithm 683, A Portable Subroutine For Exponential Integrals of Complex Argument, ACM Trans. Math. Software, Vol. 16, No. 2, June: 178-182

[8] Amos DE (1978) Evaluation of Some Cumulative Distribution Functions by Numerical Quadrature, SIAM Review, Vol. 20, No. 4, October: 778-799

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

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