Dissipation of boundary effects in multilayer heat conduction problems

Multilayer thermal conduction occurs commonly in engineering problems as varied as electronics cooling, nuclear engineering and bioheat transfer. A commonly found configuration is that of applying a thermal signal on a finite-thickness layer with a semi-infinite body underneath. In such a problem, the interest is in understanding the conditions in which the impact of the imposed thermal signal remains completely within the finite-thickness layer. Such information may be useful, for example, for designing nuclear plate-type fuel assemblies and thermoluminescence dosimeters, and guiding thermal-based therapies on skin to minimize collateral thermal damage to the deep tissue. For the special case of a single-layer body, this problem is governed by the well-known St. Venant's principle. The present work generalizes the thermal St. Venant's problem by accounting for the impact of the finite-thickness layer and interfacial thermal contact resistance. Temperature distribution in the two-layer body is derived as a function of parameters such as layer thickness, orthotropic thermal properties, interfacial thermal contact resistance and the mismatch in thermal conductivities of the two layers. Conditions in which the finite-thickness layer fully shields the semi-infinite body from the imposed thermal signal are derived. The impact of various non-dimensional parameters that represent the extent of orthotropy in the layers, interfacial thermal contact resistance and the relative thermal conductivities on the thermal shielding effect is investigated. Conditions under which thermal shielding is not particularly sensitive to some of these parameters are identified. Multiple representative problems related to bioheat transfer, nuclear reactors and thermoluminescence dosimeters are solved. In addition to generalizing the well-known St. Venant principle to a multilayer body, the present work may also aid in the thermal analysis and optimization of several practical multilayer problems.