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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 9139135, 11 pages
Research Article

An Iterative Method for Solving of Coupled Equations for Conductive-Radiative Heat Transfer in Dielectric Layers

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Academy of Sciences of Ukraine, 3-b Naukova Str., Lviv 79060, Ukraine

Correspondence should be addressed to Vasyl Chekurin; moc.liamg@nirukehc.v

Received 17 March 2017; Revised 31 May 2017; Accepted 6 June 2017; Published 16 August 2017

Academic Editor: Mariano Torrisi

Copyright © 2017 Vasyl Chekurin and Yurij Boychuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The mathematical model for describing combined conductive-radiative heat transfer in a dielectric layer, which emits, absorbs, and scatters IR radiation both in its volume and on the boundary, has been considered. A nonlinear stationary boundary-value problem for coupled heat and radiation transfer equations for the layer, which exchanges by energy with external medium by convection and radiation, has been formulated. In the case of optically thick layer, when its thickness is much more of photon-free path, the problem becomes a singularly perturbed one. In the inverse case of optically thin layer, the problem is regularly perturbed, and it becomes a regular (unperturbed) one, when the layer’s thickness is of order of several photon-free paths. An iterative method for solving of the unperturbed problem has been developed and its convergence has been tested numerically. With the use of the method, the temperature field and radiation fluxes have been studied. The model and method can be used for development of noncontact methods for temperature testing in dielectrics and for nondestructive determination of its radiation properties on the base of the data obtained by remote measuring of IR radiation emitted by the layer.