# 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
However, on the back side, the speed of the fluid interferes with the surface. The ratio between the Grashoff number (\(Gr\)) and the second power of the Reynolds number (\(Re^{2}\)) is calculated and, according to \(Gr/Re^{2}\), evaluates the nature of convection: natural, forced or mixed [14]: \(Gr/Re^{2}\sim 1\) means mixed convection; \(Gr/Re^{2}>>1\) natural convection; \(Gr/Re^{2}<<1\) forced convection. In the case that convection is mixed and the flows of natural and forced convection are in opposition, the largest \(Nu\) is considered to be the proper value. In the case that the flows in mixed convection are not in opposition, \(Nu\) is considered as:  
\[Nu\ =\sqrt[3]{\left(Nu_{forced}\right)^{3}+\left(Nu_{natural}\right)^{3}} \tag{9}\]  
In order to estimate the PV cell temperature, it is important to correctly state the physical problem of the heat exchanges between the two plates, and in particular the convection exchanges. The temperature of the air standing around the planes is taken into account in this paper, and for this reason natural convection is considered. Figure 4 shows the geometric arrangement used to calculate the natural convection between two plane plates (i.e., backsheet and battery), where \(T_{b}\) is the temperature of the backsheet, and \(T_{batt-b}\) is the temperature of the battery, \(L_{1}\) and \(L_{2}\) are the dimensions of the first and the second plane, respectively. According to [14], the \(Ra\) number is given by:  
\[Ra_{planes}=\frac{g\cdot\beta\cdot(T_{w}{}^{*}-T_{a})}{\upsilon\cdot\alpha}\cdot \left(\frac{d}{L_{1}}\right) \tag{10}\]  
where \(T_{w}{}^{*}\) (K) is the average temperature of the plates. The \(Nu\) number can be evaluated considering the case of two opened planes surrounded by air, as follows [14]:  
\[Nu=\left[(Nu_{fd})^{-1.9}+(0.62\cdot Ra^{1/4})^{-1.9}\right]^{-1/1.9} \tag{11}\]  
\[Nu_{fd}=\frac{\left(4\cdot T^{*2}-7\cdot T^{*}+4\right)\cdot Ra_{planes}}{90 \cdot\left(1+T^{*}\right)^{2}},\text{ being }T^{*}=\frac{(T_{b}-T_{a})}{(T_{ batt-b}-T_{a})} \tag{12}\]  
By these assumptions, air temperature \(T_{a}\), backsheet temperature \(T_{b}\), and battery temperature \(T_{batt-b}\) are taken into account. It should be considered that the critical value of the \(Ra\) number has never reached (\(Ra<Ra_{cr}=10^{5}\)), which justifies the validity of this approach. The radiative heat exchange (\(Q_{md}\)) is simplified by considering parallel planar bodies with the same area (A), and the radiative heat coefficient (\(h_{rad}\)) is calculated with reference to the unity view factor (i.e., \(F_{12}=1\)), leading to:  
\[Q_{rad}=h_{rad}\cdot A\ \cdot\left(T_{1}^{4}-T_{2}{}^{4}\right),\text{ being }h_{rad}=\frac{\sigma}{\frac{1-\varepsilon_{1}}{ \varepsilon_{1}}+\frac{1}{\varepsilon_{12}}+\frac{1-\varepsilon_{2}}{ \varepsilon_{2}}}=\frac{\sigma}{\frac{1-\varepsilon_{1}}{\varepsilon_{1}}+1+ \frac{1-\varepsilon_{2}}{\varepsilon_{2}}} \tag{13}\]  
where \(T_{1}\) and \(T_{2}\) are the surface temperatures (K), \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the corresponding emissivity coefficients, and \(\sigma\) is the Stephan-Boltzmann constant [\(5.67\times 10^{-8}\) W/(m\({}^{2}\) K\({}^{4}\))]. Equation (13) is used to evaluate all the radiative exchanges, i.e., sky-PV front side, backsheet-ground, backsheet-aluminum plate, battery-ground, considering the corresponding areas. \[\text{Pr}=\frac{v}{\alpha},Gr=\frac{g\cdot\beta\cdot\Delta T\cdot L}{v},\,Ra= Gr\cdot Pr=\frac{g\cdot\beta\cdot\Delta T\cdot L^{3}}{v\cdot\alpha},\text{Re}= \frac{u\cdot L}{v} \tag{6}\]0 Natural convection between plane plates (i.e., backsheet and battery). \(d\) is the distance between the PV unit and the battery; \(T_{b}\) is the temperature of backsheet; and \(T_{batt-b}\) is the temperature of the battery; \(Ta\) is the temperature of the air; \(L_{1}\) and \(L_{2}\) are the dimensions of the first and the second plane, respectively.