## 4 Mathematical Model
### Thermal Balance in the ventilated-Air Layer  
The thermal balance in the ventilated air layer is considered to be a heat transfer by convection between the wall surfaces and the circulating air, as well as a heat exchange by radiation between the inner and outer surfaces of the storage wall and insulating panel:  
\[\phi_{w}|_{x=e_{g}+\delta_{1}+e_{m\_pcm}}=h_{\prime 2}\left(T_{ins,ext}-T_{w, int}\right)+h_{\prime 2}\left(T_{f2}-T_{w,int}\right) \tag{22}\]  
\[\phi_{ins}|_{x=e_{g}+\delta_{1}+e_{m\_pcm}+\delta_{2}}=h_{\prime 2}\left(T_{w, int}-T_{ins,ext}\right)+h_{\prime 2}\left(T_{f2}-T_{ins,ext}\right) \tag{23}\]  
where:  
\[h_{\prime 2}=\frac{\sigma\left(T_{w,int}^{2}+T_{ins,ext}^{2}\right)\left(T_{w,int}+T_{ins,ext}\right)}{1/\varepsilon_{w,int}+1/\varepsilon_{ins,ext}-1} \tag{24}\]  
The convective heat transfer coefficient of the fluid movement is calculated using Churchill correlations [67]:  
\[Nu=\frac{h_{\prime 2}H}{\lambda_{f2}}=\begin{cases}0.68+\frac{0.67 Ra^{1/4}}{ \left(1+\left(\frac{0.492 Ra}{Pr}\right)^{9/16}\right)^{4/9}}&Ra<10^{9}\\ \\ 0.825+\frac{0.387 Ra^{1/6}}{\left(1+\left(\frac{0.492 Ra^{1/6}}{Pr}\right)^{9/16 }\right)^{8/27}}\end{cases}\end{cases} \tag{25}\]  
Figure 9: (**a**) Enthalpy curve vs. temperature, (**b**) Derivative of the enthalpy vs. temperature T.  
where:  
\[Ra=Gr_{f2}.Pr_{f2}=\left(\frac{\beta_{f2}g\rho_{f2}^{2}\delta_{2}^{3}\left|T_{p}-T _{f2}\right|}{\mu_{f2}^{2}}\right)\left(\frac{\mu_{f2}c_{f2}}{\lambda_{f2}}\right) \tag{26}\]  
Panel temperature: \(T_{p}=T_{w}\) for storage wall, and \(T_{p}=T_{ins}\) for insulating wall. In this study, considering the dimension and weather conditions, the Rayleigh number remains below the limits of (10\({}^{9}\)). Therefore, the correlation considered for all calculations is the first one during the measurement period.  
Temperature \(T_{f2}\) of the air-fluid circulating in the ventilated air layer depends on the heat fluctuations released from the two wall surfaces (storage and insulating walls) and the inlet air temperature at the lower (or upper) vent of the ventilated air layer. The numerical method used to calculate this average air-fluid temperature is given in [57]:  
\[T_{f2}=\left(\frac{1}{a_{0}H}-\frac{e^{-a_{0}H}}{\left(1-e^{-a_{0}H}\right)} \right)T_{inlet}+\left(-\frac{1}{a_{0}H}+\frac{1}{\left(1-e^{-a_{0}H}\right)} \right)T_{outlet} \tag{27}\]  
\[T_{outlet}=T_{inlet}.e^{-a_{0}H}+\frac{b_{0}}{a_{0}}\left(1-e^{-a_{0}H}\right) \tag{28}\]  
where:  
\[a_{0}=\frac{h_{c2,w}W+h_{c2,ins}W}{\hat{m}c_{f2}}\ ;\ b_{0}=\frac{h_{c2,w}WT_{w}+h_{c2, ins}WT_{ins}}{\hat{m}c_{f2}} \tag{29}\]  
The airflow rate \(\hat{m}\) results from the fluctuations in temperatures \(T_{inlet}\) and \(T_{outlet}\), and the total head loss coefficient of fluid during its natural thermocirculation in the ventilated layer. This entire analysis was detailed in [23; 57]. The airflow rate \(\hat{m}\) can be calculated using the following equations:  
\[\hat{m}=\rho_{f2}A\left(\frac{2 gH}{\xi_{total}}\left(\frac{T_{outlet}-T_{inlet} }{T_{outlet}+T_{inlet}}\right)\right)^{1/2} \tag{30}\]  
\[\zeta_{total}=\zeta_{entry}.\rho_{f,bottom}\left(\frac{\rho_{f2}A_{air,gap}}{ \rho_{f,bottom}A_{\rm IV}}\right)^{2}+\zeta_{lv,bottom}.\rho_{f,bottom}\left( \frac{\rho_{f2}A_{air,gap}}{\rho_{f,bottom}A_{\rm IV}}\right)^{2} \tag{31}\]  
\[+\zeta_{elbow\,bottom}.\rho_{f,bottom}\left(\frac{\rho_{f2}A_{air,gap}}{\rho_{ f,bottom}A_{\rm IV}}\right)^{2}+\zeta_{\rm linear}.\rho_{f2}+\zeta_{elbow\,top}. \rho_{f2}\]  
\[+\zeta_{uv}.\rho_{f,top}\left(\frac{\rho_{f2}A_{air,gap}}{\rho_{f,bottom}A_{ \rm uv}}\right)^{2}+\zeta_{out}.\rho_{f,top}\left(\frac{\rho_{f2}A_{air,gap} }{\rho_{f,top}A_{\rm uv}}\right)^{2}\]  
It then becomes possible to validate the accuracy of correlations used in the calculation and summation of heat fluxes measured from the fluxmeter:  
\[P_{cal}=\hat{m}c_{f2}\left|T_{outlet}-T_{inlet}\right| \tag{32}\]  
\[P_{flux}=\sum\phi A_{exch} \tag{33}\]