## 2 System Description  
A vertical section (2D) of the ventilated solar wall studied numerically is shown in Figure 2. This comprises (from outside to inside) an outer glazing in contact with the exterior environment, an outer air gap, a second glazing with a low-emissivity coating on the outer side, an inner air gap, a massive wall, and insulation. The two air gaps form a U-shaped channel. Fresh air enters the southern-facing solar wall through the inlet opening located at the top of the outer pane at a temperature equal to that of the outdoor air. It flows down into the first air gap delimited by the two glazings and then flows upward into the second air gap delimited by the glazing and the storage wall. The solar energy absorbed by the two panes and mainly by the concrete massive wall is transferred to the fresh air passing through the ventilated cavities. The air is also preheated by recovering part of the heat loss from the interior. The thermal energy stored in the form of sensible heat in the concrete massive wall is used to meet energy needs at night or during cloudy daytime periods. Preheated air enters the building through the outlet opening at the top of the opaque layer with an unknown temperature. The flow of air at a low speed in the solar wall is induced by the depressurization of interior spaces by mechanical means. The glazing with a low-e coating prevents heat from re-radiating to the outside, thus improving the performance of the VSW by taking advantage of the greenhouse effect during the day and reducing losses to the outside at night. A black heat-absorbing material is installed on the exterior surface of the massive wall to maximize the solar energy absorbed by the latter. Both the upper and the lower horizontal surfaces of the VSW are treated as adiabatic. Figure 2 shows the different heat transfer mechanisms involved in the model: thermal conduction within each solid layer, convection, long-wave radiation (LWR) in the two air gaps as well as in the surrounding environments of the VSW, and solar radiation absorption. The thermophysical properties of the outer and inner glass panes as well as the opaque layers of the VSW are given in Tables 2 and 3 respectively. Figure 2: The physical model of the VSW (_not to scale_). * Properties of materials
* Absorption of solar radiation in the two glazings and the massive wall considering multiple reflections  
When the incident solar radiation (direct and diffuse radiation) reaches the external surface of the first glazing, it is partly absorbed by the latter and partly transmitted to the second glazing, and the remaining part is reflected back to the outside environment. Likewise, the part which reaches the second glazing is divided into reflected, absorbed, and transmitted radiation to the massive wall. The fraction which is transferred to the massive wall is then absorbed or reflected by it. The part reflected towards the first glazing is again reflected, absorbed, or transmitted by it. The determination of the absorbed heat at each layer of the VSW is relatively complicated due to the multi-reflection between them. The solar energy absorption factors for the two panes as well as the massive wall, which depend on the optical characteristics of the panes as well as the absorption factor of the storage wall, were calculated:  
\[\xi_{v1}=\alpha_{v1}+\frac{\alpha_{v1}(1-\alpha_{m})\tau_{v1}}{1-\varrho_{v1} (1-\alpha_{m})} \tag{4}\]  
\[\xi_{v2}=\frac{\tau_{1}\alpha_{2}}{1-\varrho_{2}\varrho_{1}}+\frac{\tau_{1} \tau_{2}\varrho_{m}\alpha_{2}}{1-\varrho_{m}\varrho_{2}}(1+\frac{\varrho_{2} \varrho_{1}}{1-\varrho_{2}\varrho_{1}}) \tag{5}\]  
\[\xi_{m}=\alpha_{m}\tau_{1}\tau_{2}(1+\frac{\varrho_{1}(\varrho_{2}+\tau_{2} \varrho_{m})}{1-\varrho_{1}\varrho_{2}}+\frac{\varrho_{m}\varrho_{2}}{1- \varrho_{m}\varrho_{2}}) \tag{6}\]  
The results obtained from these relations were compared with the absorption coefficients calculated with the WINDOW 7 software [29], that is based on an exact analytical calculation. Errors were estimated at less than 4.5% (Table 4).