A New Thermal Lattice Boltzmann Formulation for Modeling Thermal Transport in Complex Heterogeneous Media

Abstract:

Modeling heat transfer in porous media has numerous industrial and biological applications. Natural porous structures which can be found in many geological and biological systems are complex and generally heterogeneous over a wide range of length scales. The ability of multicomponent media to transfer heat at the continuum scale depends directly on the transport of heat through interfaces between the different constituents. Therefore constraining heat and also mass balance at a macroscopic level depends on the development of quantitative models that account for the processes occurring at smaller scales. Consequently, one needs to deal with several temporal and spatial scales which makes modeling of transport phenomena a complicated task.

In the present study, we first investigate thermal transport in natural heterogeneous structures at the discrete scale. We introduce a new and simple lattice Boltzmann formulation which handles conjugate thermal boundary conditions at interfaces between two phases/components. Verification of the present interface treatment on benchmark problems confirms the accuracy and simplicity of the proposed approach. The model’s implementation is independent of the interface geometry and provides a powerful method to model thermal transport in heterogeneous media with random microstructures. Because we are ultimately interested in developing macroscale (homogenized) conservation laws for heterogeneous media, we introduce a macroscopic thermal model based on variable-order (VO) time and space derivatives. The proposed thermal model maps the heterogeneities in temporal and spatial scales into the order of the fractional derivative, which allows us to steer away from a classical diffusion equation for complex heterogeneous media. We then verify the VO thermal model for benchmark problems and discuss the possible links between values of VO derivatives in the new conservation equation and microstructure through spatial correlation functions.