# Small Magnus Wind Turbine: Modeling Approaches
## 2 Magnus Effect-Based Wind Turbine Simulation Methods
This function uses TSR and CSR as inputs, similar to the previous one, and introduces cylinder aspect ratio \(\gamma\), which can be calculated as shown in (12), where \(2r\) is the diameter of the cylinder. Researchers conclude that a two-cylinder configuration is optimal for the MWT, and it allows the generator to reach the maximum power coefficient, which is equal to 0.2 in that particular case [19]. Figure 2.2  
Figure 2.3  
Another method proposed (referred to as correlation model in text) by the same research group is presented in [25]. Similar to the previously mentioned studies, a mathematical model of the horizontal-axis MWT is derived, however, instead of the direct method, high correlation models are used to achieve optimal \(C_{p}\) functions for different configurations of wind turbines. The proposed mathematical model can be adapted to different cases according to the number of cylinder blades. In the case of a two-bladed cylinder, the function of \(C_{p}\) is presented in (13). Research confirms that a two-cylinder configuration is preferable, however, the maximum value of \(C_{p}\) reached during the simulation was not higher than 0.11. A decrease in drag coefficient, however, can lead to a higher \(C_{p}\) of up to 0.24. In both correlation and regressive models, the validity of obtained results is limited to a Reynolds number of \(10^{5}<Re<10^{6}\). Figure 2.4  
Three models of the power coefficient were built using the presented algorithms. Parameters of the experimental turbines are presented in Figure 2.5  
Concerning these parameters, \(C_{p}\) against tip speed ratio \(\lambda_{1}\) and cylinder speed ratio \(\lambda_{2}\) was plotted for three formulas. Results for the analytical model are presented in Figure 3a. The plot shows steep dependency for both TSR and CSR. The maximum value of \(C_{p}\) reached during this simulation is 0.569, with multiple values of \(\lambda_{1}\) and \(\lambda_{2}\) corresponding to a maximum value of wind power coefficient. In the case of the regression model, as shown in Figure 3b the maximum value \(C_{p}\) obtained by this model is 0.247 at \(\lambda_{1}=5\) and \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Radius of the Cylinder, \(r\) & 6.5 cm \\ Rotor radius, \(R\) & 0.6 m \\ Number of cylinders, \(N\) & 2 \\ Torsional damping, \(B\) & 1.4 N\(\cdot\)m\(\cdot\)s/rad \\ Rotational inertia, \(J\) & 2 kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the experimental MWT. \(\lambda_{2}=1.7\). Finally, a high correlation model is presented in Figure 3c. Here the maximum value reached is 0.047 at \(\lambda_{1}=1\) and \(\lambda_{2}=1.7\). The direct comparison shows significant discrepancies between different approaches to simulate the power coefficient. The primary explanation for such discrepancy is a different approach to the deduction of \(C_{p}\) function. In the case of regression and correlation models, experimental data obtained by previous researchers was used to find dependencies between \(C_{p}\) and different parameters of the windwheel. In the case of the analytical model, however, the formula for \(C_{p}\) is derived from the Blade Element Momentum theory. Therefore, the first two models depend on the mechanical characteristics of the generator, which were used to obtain experimental data. Some of these characteristics can be exclusive to the wind turbine and were not taken into account when deriving \(C_{p}\). Meanwhile, the analytical model is more universal and does not depend on the unique mechanical parameters of the wind turbine. Thus, the difference between different models can be explained by the unique structural properties of the experimental generator. Another explanation is that correlation and regression models function in a specified Reynolds number area, and the \(Re\) number of simulated turbine exceeds this range.