# Numerical and Parametric Analysis for Enhancing Performances of Water Photovoltaic/Thermal System
## 2 Methodology
### Heat Balance Governing Equations
#### 2.2.1 Solid Layers  
The heat flux for the PV/T solid layers can be obtained using Equation (1). It is a function of the density \(\rho_{S}\), specific heat capacity \(c_{p,S}\) and the thickness \(\delta_{s}\) of the selected layer as well the thermal heat exchange coefficient \(h_{s}\):  
\[\rho_{S}\ \delta_{s}c_{p,s}\ \frac{\mathrm{dT_{s}}}{\mathrm{dt}}=\Sigma\ h_{s}\ \Delta\mathrm{T_{s}}+\alpha\ E_{sun}-\theta E_{e} \tag{1}\]  
where \(\Delta\mathrm{T_{s}}\) is the temperature gradient and \(E_{sun}\) is the absorbed solar irradiance for each solid layer. In fact, a significant part of the incident solar irradiance is absorbed by the PV cells and converted to electrical energy \(E_{e}\) while the remainder is lost with environment. Thus, the absorption and electricity production terms for PV cell are \(\alpha\
eq 0\); \(\theta=1\) and for the other solid layers are \(\alpha=0\); \(\theta=0\). The expression of heat exchange coefficient \(h_{s}\) is directly dependent on the heat transfer mechanisms between layers. In case of solid layers _i,j_ where heat is transferred by conduction, \(h_{s}\) is given by:  
\[h_{s}\ =\ hc_{i,j}=\frac{1}{\left(\frac{\delta_{s}}{\lambda_{i}}+\frac{\delta_{ j}}{\lambda_{j}}\right)A_{i,j}} \tag{2}\]  
where: \(\lambda_{i}\) and \(\delta_{i}\) are the thermal conductivity and thickness of the layer \(i\); \(\lambda_{j}\) and \(\delta_{j}\) are the thermal conductivity and thickness of the layer \(j\) and \(A_{i,j}\) is the exchange area between the layers _i.j_. For the front of the PV/T system, three expressions of the heat exchange coefficient \(h_{s1}\), \(h_{s2}\), and \(h_{s3}\) are used, due to the heats transfer by radiation between the glass-sky, glass-ground and by convection between the glass-ambient:  
\[h_{s1}\ =\ hr_{g,sky}\ =\ \sigma\varepsilon_{g}F_{g,sky}\ A\left(T_{sky}+T_{g} \right)\cdot\left(T_{sky}^{2}+T_{g}^{2}\right) \tag{3}\]  
\[h_{s2}\ =\ hr_{g,gr}\ =\ \sigma\varepsilon_{g}F_{g,gr}\ A\left(T_{gr}+T_{g} \right)\cdot\left(T_{gr}^{2}+T_{g}^{2}\right) \tag{4}\]  
\[h_{s3}\ =\ hv_{g,a}=\ \left(\hv_{forced}^{3}+\hv_{free}\right)^{1/3} \tag{5}\]  
where \(\varepsilon_{g}\), is the glass emissivity; \(F_{g,sky}\) and \(F_{g,gr}\) are the view factors [24]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dimensions** & & **1665 \(\times\) 996 (mm \(\times\) mm)** & \\ \hline
**Properties** & **Glazing** & **PV Cells** & **Tedlar** & **Aluminum Absorber** \\ \hline Density (kg/m\({}^{3}\)) & 2300 & 2330 & 1500 & 2700 \\ Thermal conductivity (W/m\(\cdot\)K) & 1.0 & 168.0 & 0.2 & 160 \\ Specific heat (kJ/kgK) & 500 & 757 & 1200 & 900 \\ Thickness (m) & 0.003 & 0.00035 & 0.0015 & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PV/T investigated parameters. The glass-ambient heat transfer coefficient \(h_{s3}\) is determined as the sum of the forced and natural convective heat transfer coefficients \(hv_{forced}\) and \(hv_{free}\). If Gr/Re\({}^{2}<<1\), the forced convection regime has been evaluated. Thus, the following correlation [27] based on the wind speed velocity \(w\) is used:  
\[hv_{forced}=2.8w+3 \tag{6}\]  
Otherwise, for Gr/Re\({}^{2}>>1\), the correlation of [PERSON] et al. [28] depending on the air thermal conductivity \(\lambda a\) and Nusselt number \(Nu_{free}\) is adopted:  
\[hv_{free}=\frac{Nu_{free}\ \lambda a}{L} \tag{7}\]  
\[Nu_{free}=0.14\Big{[}(Gr\ Pr)^{\frac{1}{3}}-(Gr_{cr}\ Pr)^{\frac{1}{3}}\Big{]}+ 0.56(Gr_{cr}Pr\cos\theta) \tag{8}\]  
At the back side of the PV/T system, the heat losses by radiation between the absorber-sky, absorber -ground, and by convection between the absorber-ambient were considered.