In this paper, the mathematical and numerical models of heat transfer processes based on the dual-phase lag equation (DPLE; the energy equation with two delay times) are considered. This type of equation is, as a rule, applied for mathematical description of thermal processes taking place in the microdomains (microscale heat transfer) and also for the modeling of problems associated with the heat transfer in biological tissue domains, which follows from the specific tissue structure. The purpose of this paper is to show the correct form of boundary conditions supplementing the DPL equation (equations). The solutions of different DPLE variants discussed in the literature are obtained using the classical form of boundary conditions (as in the Fourier equation), which is not completely correct. In this paper, the proper mathematical form of the Neumann, Robin and continuity conditions is presented.
The second part of the paper is devoted to the numerical aspects of solving the problems basing on the mathematical description formulated in this way. At the stage of computations, the finite difference method in an implicit scheme is applied (1D and axially-symmetrical problems are considered). One of the examples was also solved using the generalized boundary element method. For numerical computations, an authorial computer program was developed, which performs simulations related to the modeling of thermal processes based on DPL as well as on the Cattaneo–Vernotte and Fourier equations.
The results of numerous simulations concerning both microscale heat transfer and bioheat transfer are shown, including a comparison of solutions using the classical approach to boundary-initial conditions and the approach presented in this paper.
Delay times values are not known for all materials, whereas the values presented in the literature sometimes differ from each other (especially in the case of biological tissue). In some works, it is emphasized that there are some limitations concerning the delay times of material considered, which assure the physical correctness of DPLE.
The correct formulations of boundary and initial conditions supplementing the dual-phase lag model are presented.
