# Effect of Wind Direction and Velocity on PV Panels Cooling with Perforated Heat Sinks
## 3 Materials and Methods
### Sensitivity Analysis of Grid Spacing  
The spatial discretization error analysis was also performed. The Grid Convergence Index (_GCI_) method was used, according to [35; 36]. The _GCI_ represents a very accurate method used to evaluate how far the numerical computed values are compared to the asymptotic one [36]. The analysis was implemented on both base and heat sink cases. The following equations present the calculations used and the detailed values obtained. Three grid refinement levels were considered and the following _GCI_ values were calculated:  
\[GCI_{1-2}=\frac{F_{s}|\varepsilon_{1-2}|}{\left(r_{1-2}^{p}-1\right)} \tag{1}\]  
and  
\[GCI_{2-3}=\frac{F_{s}|\varepsilon_{2-3}|}{\left(r_{2-3}^{p}-1\right)} \tag{2}\]  
where:  
_GCI\({}_{1-2}\)_--Grid Convergence Index for fine and medium grids;  
_GCI\({}_{2-3}\)_--Grid Convergence Index for coarse and medium grids;  
_F\({}_{s}\)_--Factor of safety (_F\({}_{s}\)_ = 1.25 in the present study);  
_\(\varepsilon\)_--Relative error:  
\[\varepsilon_{n-n+1}=\frac{f_{n+1}-f_{n}}{f_{n}} \tag{3}\]  
_p_--Order of convergence:  
\[p=\frac{ln\frac{f_{n+2}-f_{n+1}}{f_{n+1}-f_{n}}}{\ln r} \tag{4}\]  
_f_--Significant parameter chosen for the comparison (_f\({}_{1}\)_--the temperature calculated with the finest grid,_f\({}_{2}\)_--with the medium grid,_f\({}_{3}\)_--with coarsest one);  
_r_--Constant refinement ratio (_r_ = 2 in the present study):  
\[r_{n-n+1}=\frac{h_{n+1}}{h_{n}} \tag{5}\]  
_h_--Grid spacing, with:  
\[h_{1}(=1\;\text{mm})<h_{2}(=2\;\text{mm})<h_{3}(=3\;\text{mm}) \tag{6}\]  
For the present study, the_f_ parameter consists in the average operating temperature of the PV panel,_t\({}_{PV}\)_.  
Figure 2: Mesh independence study for the numerical simulation. In order to achieve precise results from numerical simulations by using the lowest number of cells, the following auditing ratio should be as close as possible to 1:  
\[\frac{GCI_{2-3}}{r^{p}GCI_{1-2}}\to 1 \tag{7}\]  
In the case of using the finest grid, the predicted value at \(h=0\) is:  
\[f=f_{1}+\frac{f_{1}-f_{2}}{r_{1-2}^{p}-1} \tag{8}\]  
For the specific values of the base case analysis (\(r=2,f_{1}=47.28,f_{2}=47.40,f_{3}=47.38\), \(F_{s}=1.25\)), the following values were obtained: the order of convergence \(p=1.68\) and the predicted value \(f=47.31\). Also, \(\varepsilon_{1-2}=0.053\%\) and \(\varepsilon_{2-3}=0.169\%\), while the \(GCI_{1-2}=0.030\%\) and \(GCI_{2-3}=0.096\%\). The calculated auditing ratio is:  
\[\frac{GCI_{2-3}}{r^{p}GCI_{1-2}}=0.9977 \tag{9}\]  
For the specific values of the heat sink case analysis (\(r=2,f_{1}=39.39,f_{2}=39.44\), \(f_{3}=39.49\), \(F_{s}=1.25\)), the following values were obtained: The order of convergence \(p=0.17\) and the predicted value \(f=39.80\). Also, \(\varepsilon_{1-2}=0.116\%\) and \(\varepsilon_{2-3}=0.130\%\), while the \(GCI_{1-2}=1.139\%\) and \(GCI_{2-3}=1.281\%\). The calculated auditing ratio is:  
\[GCI_{2-3}=\frac{F_{s}|\varepsilon_{2-3}|}{\left(r_{2-3}^{p}-1\right)} \tag{2}\]0  
Therefore, the main verification of the grid quality confirms that the results are in the asymptotic range of convergence and the numerical model was further conducted by using the lowest cell number (coarse grid). A large number of research studies from the literature are analyzing the photovoltaic panels either in the horizontal [19] or vertical [8; 23] position. In addition, the surrounding air is most often simulated as a channel behind the photovoltaic panel [18]. Moreover, studies on cooling inclined photovoltaic panels are present in the literature, but in reduced number [6; 11; 27]. During this study, the numerical simulations were done considering the air circulation in the vicinity of an inclined PV panel cooled with heat sinks. It was possible due to the high accuracy mesh, which assured a realistic simulation of the airflow and heat transfer from the PV panel to the heat sink and to the surrounding air. In the present study, in order to simplify the numerical model, the influence of the relative humidity of the air was neglected.