## 5 Lattice Boltzmann Methods (LBMs)
### Application to Fluid Flows by Advection/Diffusion with Phase Change in Porous Media  
Due to its distinctive advantages, the LB methods have proven to be promising in simulating thermal flows in porous media with or without phase change. In this framework, the LBMs are generally split into two classes, viz representative elementary volume (REV) scale and pore scale approaches [157; 158; 159]. At REV-scale LB simulation, neither the detailed properties of the fields (_dynamic_, _thermal_, etc.) nor the geometry detail (e.g., _porous medium involved_) are covered, while pore-scale LB simulation targets the local details of flow and heat transport processes considering the real geometry. It requires prohibitive computing times and high computing resources and should be applied only to small regions where a verydetailed analysis is required. In terms of equations, only one averaged conservation equations set deals with the REV concept, whereas two sets of equations for the porous medium and the phase change material, respectively, are involved in the pore scale approach. Although many efforts have been deployed, the application of the latter concept remains tedious. On the other hand, much research has demonstrated that the REV-LBMs simulations can lead to sufficiently accurate solutions for many flow average properties in even complex geometries (e.g., _porous media embedding phase change materials_). In this approach, the porous structure and the fluid occupying the pores are assumed to form a single computational control volume. Only statistical (_averaged_) parameters, such as porosity, permeability, effective thermal conductivity, etc. are considered without any detail on the pores structure, which results in non-excessive computation times. The REV-LBM simulation method is based on a generalized Navier-Stokes equations model that describes the fluid flow in porous media with or without phase change [158; 159; 160] to cite a few. In this approach, the flow field LB-equation is supplemented with an additional term based on empirical or semi-empirical models (e.g., _Darcy model_, _[PERSON]'s extended Darcy model_, _and non-Darcy model generalized_), which represents the porous medium. Obviously, according to the model chosen, the REV scale method yields suitable results corroborating other studies (_experimental or numerical if there are any_). Thereby, the REV-scale LB method is increasingly being adopted to study fluid flows and heat transfer in large porous media systems. Several models based on the REV-scale method have been proposed to describe the interaction between the porous medium and the fluid involved. According to the key points and main application features, Table 9 compares the REV-scale and pore-scale LBM-approaches to simulate heat transfers in a porous structure (_either during phase change or during sensitive storage process_). \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Method**} & **REVScale** & **PoreScale** \\ \hline Simulation approach & Volume average simulation (_FVM_, _FEM_, _LBM_). & Direct numerical simulation (_LBM_, _DNS_). \\ \hline Advantages & Fluent implementation and programming; less computational requirements; large computational domain size. & Reflection of pores’ effect on mechanisms involved; interstitial heat transfer study; no need for empirical models. \\ \hline Disadvantages & Less reflection of pores’ transport mechanisms; need for semi-empirical models. & Important computing platform; small computational domain; high computational demands; tedious implementation and programming. \\ \hline \hline \end{tabular}
\end{table} 
Table 9: REV scale vs. pore scale (principle, advantage/disadvantage, geometry, governing equations).