# Numerical and Parametric Analysis for Enhancing Performances of Water Photovoltaic/Thermal System
## 2 Methodology
### Heat Balance Governing Equations
#### 2.2.2 Fluid Layers  
For the cooling fluid flows in the duct of the PV/T system, the energy balance equation is given as:  
\[\left(\rho_{\rm f}\delta_{\rm f}\ C_{f}\right)\frac{dT_{f}}{\rm dt}=hc_{f,abs1} \left(T_{abs1}-T_{f}\right)+hc_{abs2,f}\left(T_{abs2}-T_{f}\right)-\hat{m}C_{f} \left(T_{out}-T_{int}\right) \tag{20}\]  
where \(\hat{m}\), \(C_{f}\), and \(T_{int}\) and \(T_{out}\) are respectively the fluid mass flow rate, the fluid specific heat capacity, and the temperature at the inlet and outlet of the PV/T system.  
The heat transfer coefficients \(hc_{f,abs1}\) and \(hc_{f,abs2}\) characterize the convection between the two absorbers and fluid and can be evaluated as a function of the fluid thermal conductivity \(\lambda_{f}\), duct hydraulique diameter \(D_{h}\), and Nusselt number \(Nu_{f}\):  
\[hc_{f,abs1}=hc_{f,abs2}=\frac{Nu_{f}\ \lambda_{f}}{D_{h}} \tag{21}\]  
The Nusselt number for water cooling flow between absorber plates is deduced from the correlations of [PERSON] et al. (2018):  
\[Nu_{f} =1.86\left[R_{e}\ P_{r}\left(\frac{D_{h}}{L}\right)\right]\ ^{\frac{1}{2}}\ \left(\frac{\mu_{f}}{\mu_{f} }\right)^{0.14}\text{, case of laminar flow} \tag{22}\] \[Nu_{f} =0.023\ [\ R_{e}\ ^{0.8}P_{r}\ ^{0.33}]\text{,}\qquad\qquad\qquad \text{case of turbulent flow} \tag{23}\]  
where \(R_{e}\), \(P_{r}\), and \(\mu_{f}\) are respectively the Reynolds number, Prandlt number, and fluid dynamic viscosity; \(\mu_{f}^{\prime}=\mu_{f}\big{(}T_{p}\big{)}\)([PERSON] et al., 2018).  
By rearranging the governing Equations (15)-(20), a set of energy differential equations was built. Thereafter, a MATLAB solution program was developed where the Runge-Kutta (RK4) numerical method was used to solve the energy balance system. Finally, the basic simulation allowed to determine the temperature profile as well as the thermal, electrical, and global performances. The calculation steps in MATLAB software are summarized in flow chart form as shown in Figure 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Layers** & **Equations** \\ \hline Glass & \(\left(\rho_{\rm g}\delta_{\rm g}C_{\rm g}\right)\frac{dT_{f}}{\rm dt}=\alpha_{ \rm g}E_{sun}+hr_{g,\rm sky}\left(T_{sky}-T_{g}\right)+hr_{g,gr}(T_{gr}-T_{g})\) \\  & \(+\hbar v_{g,g}(T_{a}-T_{g})+hc_{PV,g}PF\left(T_{PV}-T_{g}\right)+\) \\  & \(+\hbar c_{T_{end,g}}(1-PF)\big{(}T_{T_{\rm{end}}}-T_{g}\big{)}\) \\  & \(+\hbar c_{T_{end,g}}(1-PF)\big{(}T_{T_{\rm{end}}}-T_{g}\big{)}\) \\ \hline PV cells & \(PF(\rho_{\rm PV}\delta_{\rm PV}\rm{CPV})\frac{dT_{\rm{PD}}}{\rm dt}\) \\  & \(=\left[\left(\tau_{\rm g}\alpha_{\rm PV}-\mu_{e}\right)\ E_{sun}+hc_{PV,g} \big{(}T_{g}-T_{PV}\big{)}\right.\) \\  & \(+hc_{T_{end,PV}}(T_{T_{\rm{end}}}-T_{PV})\big{]}\) \\ \hline Tedlar & \(\left(\rho_{\rm Ted}\delta_{\rm Ted}C_{\rm Ted}\right)\frac{dT_{\rm{PD}}}{ \rm dt}\) \\  & \(=\left(1-PF\right)T_{g}\alpha_{\rm T}+\left(1-PF\right)hc_{T_{end,g}}\left(T_{g}-T_{ T_{\rm{end}}}\) \\  & \(+\hbar c_{abs1,T_{end}}(T_{abs1}-T_{\rm{end}})\) \\ \hline Absorber 1 & \(\left(\rho_{\rm abs}\ \delta_{\rm abs}\ C_{\rm abs}\right)\frac{dT_{\rm{PD}}}{ \rm dt}\) \\  & \(=hc_{abs1,T_{end}}(T_{T_{\rm{end}}}-T_{abs})\) \\  & \(+hr_{abs1,abs2}(T_{abs1}-T_{\rm{end}})\) \\ \hline Absorber 2 & \(\left(\rho_{\rm abs}\delta_{\rm abs}C_{\rm abs}\right)\frac{dT_{\rm{PD}}}{ \rm dt}\) \\  & \(=hc_{abs2,f}\left(T_{f}-T_{abs2}\right)+hr_{abs1,abs2}(T_{abs2}-T_{abs1})\) \\  & \(+hr_{abs2,sky}\left(T_{sky}-T_{abs2}\right)+hr_{abs2,gr}\left(T_{gr}-T_{abs2}\right)\) \\  & \(+hr_{abs1}(T_{a}-T_{abs2})\) \\ \hline \hline \end{tabular} (19)  
\end{table}
Table 3: Governing equations for the PV/T solid layers.