Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, half-space, slab and layered media in the frequency domain

Abstract

Analytical Green's functions in the frequency domain are presented for the three-dimensional diffusion equation in an unbounded, halfspace, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the three-dimensional time convection-diffusion equation for a heat point source to be obtained as a summation of one-dimensional responses. The problem is recast in the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high frequency contribution to the response is small.The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the Boundary Element Method solutions for the case of the half-space, slab and layered media. q

A method of recursive images extended to solve diffusion in multilayer materials under convective boundary conditions

“…This diffusion problem raises continuing and considerable interest and various approaches have been proposed for its solution [6][7][8][9][10][11][12][13][14]. Recently, we have proposed an extension to the method of images which could give exact solutions for the diffusion equation in a material with any number of layers, in a way that is conceptually simple [15].…”

Dynamic simulation of heat conduction using a BEM model in the frequency domain: an experimental validation

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“…Null fluxes and prescribed temperatures 0 ( ) T t are imposed along the boundary sections 1 S and 2 S , respectively, while continuity of temperatures and heat fluxes are assumed along the remaining part of the boundary, 3 S , (  …”

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A method of recursive images to solve transient heat diffusion in multilayer materials

“…Nevertheless, diffusion in multilayer materials has been solved analytically using the method of separation of variables, [5][6][7][8] the Laplace and Fourier transforms [9][10][11] and also numerically through the method of fundamental solutions [12] or using proprietary or commercial software packages employing finite elements, finite differences [1] or boundary element algorithms. For a thorough account of the state of the art in this subject one can take a look at the papers of de Monte [5,6].…”