## 2 Materials and Methods
\[\eta_{el\_TEG}=\ \frac{T_{h}-T_{c}}{Th}*\frac{\sqrt{1+Z*\overline{T}}}{+\frac{T_{c}}{T_{h }}} \tag{6}\]  
where \(T_{h}\) is the temperature of the hot side of the TEG, \(T_{c}\) Is the temperature of the cold side of the \(TEG\), \(\overline{T}\) is the average temperature of the TEG and \(Z\) is the figure of merit of the module. In Equation (6) temperatures are given in K. According to the manufacturer, the product between the figure of merit and the average temperature under nominal conditions has a value of 0.73. As well as thermal performance, the electrical performance of a PVT panel ( \(\eta_{el\_PVT}\)) is related electrical production with the incoming energy from the sun  
\[\eta_{el\_PVT}=\ \frac{W_{PVT}}{E_{s}} \tag{7}\]  
and performance of TEG modules ( \(\eta_{el\_TEG}\)) is related with the heat flux that go throght the thermoelectrical generators (\(E_{TEG}\))  
\[\eta_{el\_TEG}=\ \frac{W_{TEG}}{E_{TEG}} \tag{8}\]  
In such a way, the electrical performance of PVT + TEG panel is calculated as the addition between the electrical performance of a conventional PVT panel and the electrical performance of TEGs modules considering gradient temperature in the hybrid panel. \[\eta_{el\_PVT+TEG}=\ \eta_{el\_PVT}+\ \eta_{elTEG} \tag{9}\]  
In this paper, a three-dimensional numerical simulation of the PVT panel and the PVT panel with TEGs has been performed in the software ANSYS [45]. The 3D analysis is selected due to the serial configuration of the TEGs integrated along the tube. This application works under the finite element method for structural analysis and the finite volume method in the case of fluid studies. This software includes different work modules that allow the study of structural, thermal, electrical, and fluid dynamics aspects. In this case study, module Fluent is employed, which is a computation fluid-dynamics (CFD) simulator. Definition of the three-D model of the PVT and PVT + TEG panels is the first step that must be done. For that purpose, technical specifications of the different layers that compose the PVT panel has been provided by the fabricant and are shown in \[E_{s}=I\;*\;A_{C} \tag{1}\]0 Due to the high complexity of the study, some assumptions have been set in order to simplify the PVT model:  
* The thermal behavior of the small pipes of the water collector is assumed identical inside all tubes. For this reason, only one of the 10 pipes is represented;
* Both models of the panel (PVT and PVT + TEG) have been simulated under steady state conditions;
* Water mass flow inside the simulated pipe is considered to be incompressible and laminar;
* The thermophysical properties of the different materials remain constant during the study period. The employed governing equations in the performed numerical simulation of conventional PVT and PVT + TEGs panels are:  
Continuity equation:  
\[E_{s}=I\;*\;A_{C} \tag{1}\]1  
Momentum equation:  
\[E_{s}=I\;*\;A_{C} \tag{1}\]2  
Energy equation:  
\[E_{s}=I\;*\;A_{C} \tag{1}\]3  
where \(\overrightarrow{\mathrm{v}}\) is the velocity vector, \(\rho\) is the density of the fluid, \(P\) is the pressure vector, \(\mu\) is the viscosity of the fluid, \(c_{p}\) is the heat capacity and \(k\) represents the conductivity. Considered boundary conditions are shown below:  
* External surfaces of the PVT panels are adiabatic. \(\
abla(k\
abla T)\ =\ 0\). * Velocity of the fluid in the solid walls is null. \(\overrightarrow{\mathrm{v}}\ =\ 0\). * In the wall of a solid, velocity and temperature of the fluid are that of the solid element. \(-k\frac{\partial T}{\partial n}\Big{|}_{f}=\ -k_{s}\frac{\partial T_{s}}{ \partial n}\Big{|}_{s}\). * Inlet conditions are: \(T=T_{in}\); \(\mathrm{u}=0\); \(\mathrm{v}=0\) and \(\mathrm{w}=-V_{in}\). * Outlet condition is \(P=0\). Table 3 includes the main dimensions of the different bodies that compound the PVT panel simulated in the employed numerical software. Once the physical 3D model (Figure 1) that is going to be simulated is presented, a mesh must be done.