## 2 Materials and Methods  
### Specification of the Experimental Designs  
When reviewing the literature, information to serve as a starting point for this research regarding the implementation of multi-blade profiles in Savonius type rotors was not obtained. Therefore, seeking the greatest constructive ease, an arc-circular geometry was defined as the profile of the secondary element and the fundamental dimensions with which its construction is completely defined were identified. For this, three factors were established; one of them corresponds to the diameter that constitutes the leading edge of the secondary element (\(O\)) and the remaining ones comprise the distances between the leading (\(P\)) and trailing (\(Q\)) edges of the blade studied with respect to the edge outside the main element, as shown in Figure 1. These factors are dimensioned according to the percentage represented respecting the rotor diameter (\(D\)). Three levels were defined for each factor constituting an experimental design \(3^{3}\) (i.e., three levels for the three factors). Table 1 shows the factors and levels used in the experimental designs. The radius of the secondary element is obtained by construction, so that the angle at the trailing edge of the secondary element is ensured to be coincident with the angle of the outer edge of the main element. The wind velocity was defined as a constant, with a value of 4 m/s. A FCCD does not require all the level combinations that a FFD requires, but only the central combinations between each pair of factors. In this regard, for the case of a design with 3 factors evaluated at 3 levels, a cubic structure is obtained with points of analysis at each vertex, at the center of the faces, and at the center of the body of the experimental domain, which determines a total of 15 treatments [29]. By constructing the geometries according to the values provided by the FFD and the FCCD, it was possible to obtain a total of 27 and 15 models, corresponding respectively to each treatment (i.e., combination of levels). All geometries are bounded by a circumference of 200 mm in diameter (\(D\)) and built considering a thickness of 1 mm. The performance of each geometry is evaluated in terms of the torque coefficient (\(C_{T}\)) from which the power coefficient (\(C_{P}\)) is determined as the response variable. The \(C_{T}\) is estimated as the relationship between the torque generated by the turbine on its shaft (\(T\)) and the torque that is possible to generate under the given conditions. This can be seen expressed in Equation (1), where \(\rho\) is the density of the air, \(v\) is the wind velocity in free flow and \(A=DH\) is the cross-sectional area of the turbine, being \(H\) its height (unitary in two-dimensional analysis). \[C_{T}=\frac{T_{turbine}}{T_{available}}=\frac{T}{\frac{1}{4}\,\rho\,\,Av^{2}D} \tag{1}\]  
Likewise, the \(C_{P}\) consists of the relationship between the power generated by the turbine and the energy flow carried by the fluid. This can be estimated by means of Equation (2), where \(\omega\) is the angular velocity of the turbine [30]. \[C_{P}=\frac{P_{turbine}}{P_{available}}=\frac{T\,\omega}{\frac{1}{2}\,\rho\,\, Av^{3}}=\frac{T}{\frac{1}{4}\,\rho\,\,Av^{2}D}\,\frac{\omega D}{2v} \tag{2}\]  
The speed ratio at the tip of the blade (\(TSR\)) gives a proportion of the angular velocity of the turbine in dimensionless terms according to Equation (3). \[TSR=\frac{\omega D}{2v} \tag{3}\]  
In this way, Equation (2) can be rewritten as Equation (4). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Factor** & **Symbol** & **Level** & **Unit** \\ \hline Tip diameter & O & 96, 98, 100 & \% \\ Inlet gap & P & 12, 16, 20 & \% \\ Outlet gap & Q & 6, 8, 10 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the experimental designs (3\({}^{3}\)). Figure 1: Geometric parameters used in experimental designs.