## 5 Lattice Boltzmann Methods (LBMs)
### Application to Fluid Flows by Advection/Diffusion with Phase Change in Porous Media
In Equations (14)-(19), \(\stackrel{{\rightarrow}}{{u}}\), \(p\), \(T_{f}\), \(T_{s}\), \(\epsilon\), \(\rho\), \(\
u_{f}\), \(C_{p}\), \(\lambda_{eff}\) and \(\Gamma\) represent the velocity vector filed, pressure, fluid and porous medium temperatures, structure porosity, density, kinematic viscosity of PCM, thermal capacity, the equivalent thermal conductivity and the liquid fraction, respectively. Subscripts \(f\) and \(s\) point out the fluid and solid phases, respectively. Using the enthalpy method, the liquid fraction \(\Gamma\) can be computed as [118; 119]:  
\[\Gamma(T)=\left\{ \begin{array}{ll}0&T_{f}\leq T_{m}-\Delta T\\ \left(T_{f}-T_{m}+\Delta T\right)/2\Delta T&if\quad T_{m}-\Delta T\prec T_{f} \prec T_{m}+\Delta T\\ 1&T_{f}\geq T_{m}+\Delta T\end{array} \right. \tag{20}\]  
\(\Delta T\) being the PCM's melting temperature range. The permeability and the Forchheimer's form coefficient can be expressed, respectively, as:  
\[F_{\epsilon}=2.12\times 10^{-3}(1-\epsilon)^{-0.132}(\frac{d_{p}}{d_{f}})^{1.63} \tag{21}\]  
\[K=\epsilon^{3}d_{p}^{2}\Big{(}150(1-\epsilon)^{2}\Big{)}^{-1} \tag{22}\]  
\(d_{f}\) and \(d_{p}\) being, respectively, the average diameters of ligaments and of pores, which can be written:  
\[d_{f}=1.18((1-\epsilon)/3\pi)^{1/2}d_{p}\text{ and }d_{p}=22.4\times 10^{-3}/\omega \tag{23}\]  
In Equations (16) and (18), the specific surface area of the porous structure (\(a_{sf}\)) and the interfacial heat transfer coefficient between porous medium and PCM (\(h_{sf}\)) can be estimated via the following correlations:  
\[a_{sf}=3\pi d_{f}\Big{(}1-e^{-(1-\epsilon)/0.004}\Big{)}/0.59/d_{p}^{2} \tag{24}\]  
and  
\[h_{sf}=\left\{ \begin{array}{ll}0.76\cdot\text{Re}_{d}^{0.37}\frac{\lambda_ {f}}{d_{f}}\\ 0.52\cdot\text{Re}_{d}^{0.5}\text{Pr}^{0.37}\frac{\lambda_{f}}{d_{f}}\\ 0.26\cdot\text{Re}_{d}^{0.6}\text{Pr}^{0.37}\frac{\lambda_{f}}{d_{f}}\end{array}  \right.\quad for\quad\left\{ \begin{array}{ll}1\leq\text{Re}_{d}\leq 40\\ 40\leq\text{Re}_{d}\leq 10^{3}\\ 10^{3}\leq\text{Re}_{d}\leq 2.10^{5}\end{array} \right. \tag{25}\]  
More details can be found in Refs. [114; 118; 119; 48]. Generally speaking, natural convection is considered in certain circumstances and is sometimes negligible to simplify the mathematical model. However, [PERSON] et al. [164] showed that for solar organic Rankine cycle systems, natural convection strongly affects the temperature field and the energy stored in the PCM, when the solar radiation decreases. On \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **REVScale** & **PoreScale** \\ \hline Evolution of solid/liquid phase interface [163; 124] & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._the other hand, [PERSON] et al. [165] conducted a numerical investigation of the PCM melting process using two approaches, viz., pore-scale and averaged-volume models. Their findings showed that natural convection reduced the total melting time by 28%. In addition, these authors demonstrated the ability of the pore-scale model to capture local melting behavior and indicated that this approach could be used as a benchmark to handle the application of the volume-averaged method with the single temperature model.