# Modeling and Analysis of Contactless Solar Evaporation for Scalable Application
## 2 Methodology
### Modelling Framework  
Based on the working principles of two devices shown above, the modelling framework considered the heat transfer, fluid flow and moisture transport, resulting in a complex multiple-physical process. Initially, as shown in Figure 1, solar energy is absorbed by the absorber-emitter as the input energy (\(Q_{\text{in}}\)), which heats up the water near the air-water interface through radiation (\(Q_{\text{rad}}\)) and conduction (\(Q_{\text{cond}}\)). Meanwhile, there exists the top heat loss (\(Q_{\text{L\_top}}\)) through the absorber surface to the environment. The latent heat (\(Q_{\text{v}}\)) of evaporation carries the energy, which is delivered to the vapor through phase change at the air-water interface. The mass flux (\(m\)) associated with the heat flux enters the air layer simultaneously, leading to a continuous vapor generation, which then escapes from the holes on the absorber-emitter. There also exists the bottom heat loss (\(Q_{\text{L\_bottom}}\)) through the bulk water to the environment. The following assumptions were applied in this study:  
1. The moist air is in saturated state (RH = 100%) on the water surface;
2. In the scalable model, the view factor is set to 1 between the infrared emitter surface and the water surface;
3. Both in laboratory and scalable models, the bulk water is set as solid domain. To capture the heat transfer in both solid and fluid domains, governing equations shown in Equations (1) and (2) were applied for water, air, thermally insulated foam and container walls. \[\rho C_{\text{p}}\mathbf{u}\cdot\
abla T+\
abla\mathbf{q}=Q_{\text{h}} \tag{1}\]  
\[\mathbf{q}=-k\
abla T \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parts** & \multicolumn{2}{c}{**Dimension**} \\  & **Laboratory Device** & **Scalable Device** \\ \hline Absorber–emitter [A \(\times\) B \(\times\) C] & 35 mm \(\times\) 35 mm \(\times\) 1.5 mm & 11.67 mm \(\times\) 11.67 mm \(\times\) 1.5 mm \\ Bulk water [A \(\times\) B \(\times\) C] & 35 mm \(\times\) 35 mm \(\times\) 50 mm & 11.67 mm \(\times\) 11.67 mm \(\times\) 50 mm \\ Container side wall thickness & 20 mm & None \\ Container bottom wall thickness & 20 mm & 20 mm \\ Air gap thickness & 2 mm & 2 mm \\ Number of straight holes & 9 & 1 \\ Thickness of the insulation foam & 3.5 mm & None \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the laboratory and scalable devices. Figure 2: Schematic diagrams of (**a**) the laboratory SCE device, (**b**) the scalable SCE device and (**c**) the detailed layout of absorber–emitter of both devices. where, \(\rho\), \(C_{p}\) and \(\mathbf{u}\) denote the density, heat capacity at constant pressure and velocity vector, respectively. \(\mathbf{q}\), and \(Q_{\mathrm{h}}\) represent the conductive heat flux vector and the heat source, respectively. \(k\) and \(T\) are thermal conductivity and temperature. To describe the flow in air domain, governing equations for laminar flow were adopted as shown in Equations (3) and (4), considering the slow movement. \[\rho(\mathbf{u}\cdot\
abla)\mathbf{u}=\
abla\cdot[-p\mathbf{I}+\mathbf{K}]+ \mathbf{F}+\rho\mathbf{g} \tag{3}\]  
\[\
abla\cdot(\rho\mathbf{u})=0 \tag{4}\]  
where, \(\mathbf{I}\), \(\mathbf{K}\) and \(\mathbf{F}\) represent identity matrix, the shear tensor, and the volume force vector, respectively. \(\mathbf{g}\) is the gravity acceleration. To capture the vapor transport in air domain, governing equations shown in Equations (5)-(7) were adopted for moisture transport in air. \[M_{v}\mathbf{u}\cdot\
abla c_{v}+\
abla\cdot\mathbf{g}=G \tag{5}\]  
\[\mathbf{g}=-M_{v}D\
abla c_{v} \tag{6}\]  
\[c_{v}=\varphi c_{\mathrm{sat}} \tag{7}\]  
where, \(M_{v}\), \(c_{v}\), \(\mathbf{g}\), and \(G\) represent the molar mass of water vapor, water vapor concentration, diffusion flux of vapor and the moisture source, respectively. \(D\) is the vapor diffusion coefficient in air. \(\varphi\) and \(c_{\mathrm{sat}}\) are the relative humidity (RH) and the vapor saturation concentration, respectively.