## 3 Mathematical (CFD) Model of the Vertilated Solar Wall
### Results
#### 3.4.2 Convective Heat Transfer Coefficients  
In Figure 6, we represent the local convective heat transfer coefficients calculated for the two surfaces of the internal and external air gap.  
The convective heat exchange between the air cavity surface and the air is defined as:  
\[q_{cv}=h_{cv}\left(T_{s}-T_{f}\right) \tag{22}\]  
By definition, \(T_{f}\) is the temperature of the fluid sufficiently far from the surface, but since we are working in a parallel-plate vertical channel, \(T_{f}\) was calculated as the air bulk temperature (see Equation (20)). The bulk temperature was used for flows of internal forced convection, such as pipe flows and heat exchangers [33; 34]. The local convection heat transfer coefficients \(h_{cv}\) between the air cavity surfaces and the airflow inside the channels were calculated using the following equation:  
\[h_{cv}=\frac{q_{cv}}{\left(T_{s}-T_{f}\right)} \tag{23}\]  
These coefficients are shown in Figure 6. The local convection heat transfer coefficients between the left and right surfaces and the air of the outer air gap are shown in Figure 6a and those of the inner air gap obtained by the CFD simulations are shown in Figure 6b.  
It is noted that, in the upper section of the outer air gap, the local convective heat transfer coefficients between the channel's left surface and the air were negative. This does not make physical sense. In fact, we did not follow the basic definition of the convective heat transfer coefficient equation where the fluid temperature was that of a fluid sufficiently far from the surface, but we used the definition of the bulk air temperature. The convective heat flux relative to this surface was positive, while the temperature difference was negative.  
[PERSON] identified this phenomenon while investigating the thermal behavior of parietodynamic windows [35]. It is well explained when observing the local temperature profile in the air gap (Figure 7): the temperature gradient was negative at the left wall of the air gap, while the flux \(q_{S}\) was positive. However, since the temperature difference (\(T_{sg}-T_{m}\)) was negative, we obtained a negative coefficient. [PERSON] concluded that the temperature difference (\(T_{sg}-T_{m}\)) under asymmetrical heating conditions is not representative of the direction of heat flow.  
Figure 6: Local convective heat transfer coefficients in the (**a**) outer gap and (**b**) inner gap.  
In fact, the air bulk temperature uses the temperature and velocity profiles cannot meaningfully be used to evaluate these coefficients. In the second air gap, extremely high values of local convective heat transfer coefficients were obtained between the left surface of the inner glazing and the air, where the air bulk temperature coincided with the surface temperature. For the above reasons, we chose to calculate the average convective heat transfer coefficients for the two air gaps.  
Figure 8 shows the evolution of the heat transfer coefficients as a function of mass flow rate. For the outer air gap, as the mass flow rate increased, the convective heat transfer coefficient on the left (l) side decreased, while that on the right (r) side increased. For the second air gap, when the mass flow rate increased, the convective heat transfer coefficient towards the left side decreased, while for the right side it was almost stable. The same appearance of the curves was found as for the Paziaud window [35].  
It should be noted that similar results were obtained for the other simulations carried out with different imposed mass flow rates.