Analytical solution of 1-d multiple layer dual phase lag heat conduction problem with generalized time dependent boundary condition

Abstract

The Fourier law of heat conduction is unable to describe the phenomena such as self-heating of micro-electronics, situations involving very low temperature near to absolute zero, heat transport in living tissues and heat sources like laser heating. The dual phase lag (DPL) heat conduction model is found to be more useful to describe these phenomena. Several analytical solutions of 1D single layer to multiple layer (composite material) DPL heat conduction problems for mixed boundary conditions (BCs) are obtained by representing the BCs with Newton’s law of cooling combined with Fourier law of heat conduction. However, this is contradictory to the assumption of non-applicability of Fourier law of heat conduction in the DPL heat conduction model. In the present work it is shown that several approaches such as separation of variables (SOV), finite integral transform (FIT) and orthogonal eigenfunction expansion method (OEEM) are not applicable if BCs are consistent with DPL heat conduction model assumptions. Moreover, such methods are also not applicable in cases of heat conduction in multiple layer composite material. Since, only in the case of Dirichlet type BCs and single layer material such discrepancy is not present hence, the above mentioned approaches can be applied to obtain analytical solutions. It is also shown that the Laplace transform (LT) can be successfully used to obtain analytical solutions of single as well as multiple layer DPL heat conduction problems with generalized BCs when Taylor series expansion of the phase lag operator is taken into consideration.