# Thermal and Performance Analysis of a Photovoltaic Module with an Integrated Energy Storage System
## 2 Thermal Model of a Single PV Module
### Convective and Radiative Coefficients  
Convective and radiative coefficients are calculated according to the geometry of the PV module and ambient conditions such as wind speed, air temperature, and ground temperature. The convective heat exchange is given by:  
\[Q_{conv}=h_{conv}\cdot A\ \cdot\Delta T,\text{ being }h_{conv}=\frac{Nu\cdot k}{L} \tag{5}\]  
being \(h_{conv}\) [W/(m\({}^{2}\) K)] the convective coefficient, \(A\) the exchange area (m\({}^{2}\)), \(\Delta T\) the difference between two surfaces at different temperatures, \(L\) the characteristic length of the geometry of the PV module (_area/perimeter_), \(k\) the thermal conductivity of the fluid for a reference temperature [W/(m K)], and \(Nu\) the dimensionless Nusselt number. In addition to the Nusselt number, the following numbers are helpful in order to determine the heat transfer:  
\[\text{Pr}=\frac{v}{\alpha},Gr=\frac{g\cdot\beta\cdot\Delta T\cdot L}{v},\,Ra= Gr\cdot Pr=\frac{g\cdot\beta\cdot\Delta T\cdot L^{3}}{v\cdot\alpha},\text{Re}= \frac{u\cdot L}{v} \tag{6}\]  
where the Prandtl number (_Pr_) gives the information about the type of fluid. It also provides information about the thickness of the thermal and hydrodynamic boundary layer. Reynolds number (_Re_) gives information about whether the flow is inertial or viscous force dominant, in order to determine if the flow is laminar or turbulent. The Grashoff number (_Gr_) is used in the correlation of heat and mass transfer due to the thermally induced natural convection of a solid surface immersed in a fluid. Figure 3: Battery and PV module layers. \(T\) is temperature; \(b\) is backsheet; \(e\) is ethylene vinyl acetate (EVA);\(f\) is front side; \(g\) is ground and batt is battery. The Rayleigh number (\(Ra\)) is defined as the product of the Grashof number and the Prandtl number, and it describes the relationship between momentum diffusivity and thermal diffusivity. Parameters of interest in (Equation (6)) are: dynamic viscosity \(v\) (m\({}^{2}\)/s); thermal diffusivity \(\alpha\) (m\({}^{2}\)/s); gravity acceleration \(g\) (m/s\({}^{2}\)); isobaric compression ratio \(\beta\) (1/K); temperature difference \(\Delta T\), and wind speed \(u\) (m/s). All these thermal properties are evaluated at the reference temperature \(T_{ref}=(T_{a}+T_{w})/2\), where \(T_{a}\) is the air temperature (i.e., the fluid that is surrounding the PV module) and \(T_{w}\) is temperature of the surface under consideration (K). The transition from laminar to turbulent flow was determined by the critical values (\(cr\)): \(Ra_{cr}=10^{9}\), \(Re_{cr}=10^{5}\), \(Gr_{cr}=10^{9}\). Measurements were taken when the wind speed was not too high, so Reynolds number was usually \(Re<Re_{cr}\). In this way Nusselt number for forced convection is empirically calculated as [14]:  
\[Nu_{forced}=0.664 Re^{0.5}Pr^{1/3} \tag{7}\]  
In natural convection, empirical [PERSON] and Chu formulas are used to calculate the convective coefficient for the PV module front and back surfaces. The Churchill-Bernstein equation is valid for a wide range of Reynolds and Prandtl numbers, and it can be used for any object of cylindrical geometry in which boundary layers develop freely, without constraints imposed by other surfaces [15], leading to:  
\[Nu_{natural}\ =\ \left(0.825\ +\ \frac{0.387 Ra^{1/6}}{\left[1\ +\ \left(\frac{0.492}{Pr} \right)^{9/16}\right]^{8/27}}\right)^{2} \tag{8}\]  
Note that the Nusselt number (Equation (8)) does not change in a forced convection field, whereas it does change from the front to back sides of the PV module in natural convection. For the back side, \(Nu\) has to hold on gravity force, so it is important to correct the Rayleigh number with a coefficient \(g\) cos\(\beta\)[13]. The \(g\) is the acceleration due to gravity, and the cos\(\beta\) is the tilt angle of PV module. The buoyancy force on the tilt plane has to be divided into parallel and perpendicular components. On the front side these components do not interfere with the fluid and both contribute to the natural heat exchange.