The generalized periodic boundary condition for microscopic simulations of heat transfer in heterogeneous materials

Abstract

Porous and composite materials have been increasingly used in many heat transfer applications due to their enhanced thermal performance, and extensive studies have been conducted for a better scientific understanding of the heat transfer processes in such materials especially at the microscopic level. For heterogeneous materials with periodic microscopic structures, analysis can be performed over one such periodic unit if appropriate boundary conditions are applied. However, assumptions with no mathematical or physical justifications are often involved in previous studies, and this casts doubt on the accuracy and reliability of these results. In this study, a new set of boundary conditions are established based on rigorous mathematical derivations. The classical periodic boundary condition is generalized to incorporate the temperature differences between each pair of opposite boundaries of the periodic unit, and the global thermal gradient can be applied in arbitrary directions relative the structural periodicity. Also the interface between constituents in composite materials (or the fluid-solid interface in porous materials) is modeled according to the conjugate condition. This explicit approach incorporates the thermophysical properties of individual constituent materials and the possible thermal resistance at constituent interface, and thus provides a more complete and realistic representation of the heterogeneous system. This method with the generalized periodic boundary condition (GPBC) is carefully validated by comparing the temperature fields and effective conductivity for a two-dimensional (2D) heterogeneous model obtained from a direct simulation of a large domain size without using our boundary method and from one-unit simulations but with various periodic unit selections. The excellent agreement observed in these results indicates that our GPBC method correctly and accurately represents the underlying relationships in the thermal fields among periodic units. We also compare the simulation results for a 2D heterogeneous model with inclined poor conductivity rods in the middle, using our GPBC method and the insulated unit method (IUM) in previous publications; and remarkable differences in temperature fields and heat flux patterns are observed. Analysis and discussions are provided to show the unrealistic features in the IUM results due to the artificial assumptions on the boundary conditions. On the other hand, the temperature field from our GPBC method is consistent with intuitive physical considerations. The effects of constituent composition, conductivity and interface conductance on the effective thermal conductivity for a simple 2D heterogeneous material model, as well as the thermal flows with different Reynolds and Prandtl numbers through a staggered array of square blocks, are also examined. Despite the model simplicity, several interesting features are noticed, such as the heat flux response to changes in constituent conductivity and interface resistance and the saturation states of effective conductivity. Such information is valuable for a better understanding of the complex relationships between the microscopic structure and properties and the macroscopic thermal performances of composite and porous materials. The GPBC method developed in this work can also be readily implemented in typical numerical techniques in computational heat transfer such as the finite volume method, finite element method, and lattice Boltzmann method.