## 3 Validations of the TLBM Approach
### Evaporating Mass Flux  
Evaporating mass flux indicates the capability of evaporation heat transfer. In this part, the mass flux in a thin-film evaporating pore is investigated using TLBM, and is then compared with the theoretical predictions. [PERSON] proposed that the interfacial mass flux can be estimated by the following equation [46; 47],  
\[m^{\prime\prime}=\frac{2\hat{\sigma}}{2-\hat{\sigma}}\left(\frac{M}{2\pi R} \right)\left(\frac{P_{v_{0}}(T_{lv})}{T_{lv}^{1/2}}-\frac{P_{v}}{T_{v}^{1/2} }\right), \tag{36}\]  
where \(\hat{\sigma}\) is an accommodation ratio, \(M\) is the molecular weight, \(R\) is the universal gas constant, \(T_{lv}\) is the interface temperature, \(P_{v_{0}}\) is the equilibrium vapor pressure associating with the temperature \(T_{lv}\), \(P_{v}\) and \(T_{v}\) are the pressure and temperature of vapor, respectively. The above equation was derived from the kinetic theory. Denote \(J\) as the following expression  
\[J=\frac{P_{v_{0}}(T_{lv})}{T_{lv}^{1/2}}-\frac{P_{v}}{T_{v}^{1/2}}. \tag{37}\]  
Then mass flux is proportional to \(J\)[31], i.e.,  
\[m^{\prime\prime}\propto J. \tag{38}\]  
In the TLBM simulations investigating the evaporating mass flux, the schematic and boundary conditions introduced in the previous part are still employed. The heat flux applied to the side walls changes from \(1\times 10^{-4}\) to \(8\times 10^{-4}\). Hence, different evaporation mass fluxes are achieved. The temperature and pressure in the centre line of the channel are evaluated in order to obtain \(J\) as well as the theoretical prediction of \(m^{\prime\prime}\). On the other hand, the mass flux of vapor exiting the pore can also be calculated via  
\[m^{\prime\prime}=\sum_{\mathbf{x}\in\mathbf{x}_{\text{out}}}\sum_{i=0}^{8}e_{y,i}f_{i}(\mathbf{x})/W, \tag{39}\]  
where \(W\) is the channel width, \(e_{y,i}\) is the component of \(\mathbf{e}_{i}\) at the y direction, and \(\mathbf{x}_{\text{out}}\) is the set of nodes locating at the outlet layer. The above equation evaluates the evaporating mass flux directly from the numerical simulations. The comparisons of simulations and theoretical predictions are illustrated in Figure 9, which yields good agreement.  
Figure 8: The wall temperature distribution under constant flux: black crosses are experimental data from [5]; red line is obtained by the thermal lattice Boltzmann method (TLBM) simulations; \(X\) and \(T^{*}\) are normalized location and temperature, respectively; density profile of the half channel is presented in the mini plot.  
The mass flux discussed above corresponds to an ensemble property of the evaporating pore. It is of interest to look into the details of the liquid-vapor interface, which is an advantage of TLBM. Denote the unit normal vector of the interface as \(\mathbf{\hat{n}_{i}}\). The local mass flux across the interface is estimated by  
\[m_{i}^{\prime\prime}(\mathbf{x})=\sum_{i=0}^{8}e_{y,i}f_{i}(\mathbf{x}) \mathbf{\hat{n}_{i}}(\mathbf{x}). \tag{40}\]  
The simulations for interfacial mass flux is shown in Figure 10, where the red solid line represents the contact line throughout adhesion, evaporating and intrinsic meniscus regions. Maximum interfacial mass flux is observed in the evaporating region compared with the adhesion region and the intrinsic meniscus region. It coincides with the classic understanding of TFE.