# Performance of a Closed Cycle Power Takeoff for a Shore-Based Wave Energy Device
## 2 Materials and Methods
### Preliminary 1D Design of the Turbine
Based on the turbine requirements obtained above for the breakwater at Mutriku, the turbine geometry can be determined. An important aspect of it are the rotor and stator blade angles. The blade angles are defined in Figure 7 and can be expressed in terms of the degree of reaction \(R\) and the velocity ratio \(C_{x}/U\) (where \(C_{x}\) is the axial velocity through the turbine), as follows  
\[\alpha_{2}=\tan^{-1}\left(\frac{2(1-R)}{C_{x}/U}+\tan\ \alpha_{1}\right) \tag{22}\]  
\[\beta_{2}=\tan^{-1}\left(\tan\ \alpha_{2}-\frac{1}{C_{x}/U}\right) \tag{23}\]Figure 6:0  
In a single-stage turbine, the inlet angle \(\alpha_{1}=0\). The stagger angles define the position of the blade chord relative to the axial direction. It was calculated assuming a circular arc camber line:  
Figure 6:1  
Figure 6:2  
This leads to the angles presented in Figure 6:3 As the angles depend on the choices of non-dimensional parameters only, they are the same for both cases considered. The blade height \(H\) and mean radius \(r_{m}\) together define the turbine flow area \(A=2\pi r_{m}H\). The area is determined by the air mass flow rate required to pass through the turbine. Indeed, seeing that the axial velocity \(C_{x}\) is constant through the turbine, the mass flow rate is \(\dot{m}=\rho AC_{x}\) where \(\rho\) is the air density. Since the axial flow velocity is proportionally linked to the blade speed \(U\) through the design parameter \(C_{x}/U\), the problem reduces to the estimation of the blade speed, which is completed by considering the total-to-total efficiency [20]:  
Figure 6:4 \begin{table}
\begin{tabular}{l l l} \hline \hline Stator inlet metal angle & \(\alpha_{1}\) & \(0^{\circ}\) \\ Stator outlet metal angle & \(\alpha_{2}\) & 75.96\({}^{\circ}\) \\ Stator stagger angle & \(\xi_{stator}\) & 37.98\({}^{\circ}\) \\ \hline \hline Rotor inlet metal angle & \(\beta_{2}\) & 33.69\({}^{\circ}\) \\ Rotor outlet metal angle & \(\beta_{3}\) & 73.30\({}^{\circ}\) \\ Rotor stagger angle & \(\zeta_{rotor}\) & 19.81\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Blade angles. Figure 7: Blade geometric parameters definition. Numbers indicating the location in the stage correspond to the subscripts on pressure, temperature and enthalpy. where \(\Delta W=h_{01}-h_{03}\) is the work produced by the turbine and \(\Delta h_{is}=h_{01}-h_{38}\) is the isentropic enthalpy drop across the turbine, which was already calculated in Equation (18) based on the pressures obtained by the lumped model of the complete system. Using the relations linking the blade angles and the design parameters for \(\alpha_{1}=0\), the work output can be written as  
\[\Delta W=UC_{x}\tan\alpha_{2}=2(1-R)U^{2} \tag{28}\]  
Thus, Equation (27) can be used to estimate the blade speed \(U\) by assuming the nominal total-to-total efficiency. Choosing \(\eta_{tt}=0.9\), the nominal blade speeds obtained for the higher and the lower energy cases are, respectively, 41.6 and 28.9 m/s. The mean radius is then simply derived as \(r_{m}=U/\Omega\). The resulting blade dimensions are presented in Table 3. Note that the geometry does not differ much between the two cases; the relative difference between the two blade heights or the two blade mean radii is 1.5%. Indeed, the different mass flow rates are accounted for by different nominal speeds rather than different geometries. The rest of this study will, therefore, focus on one turbine design only, whose geometric parameters are taken as the average between those two cases. The last design aspect to be determined is the number of stator and rotor blades. If there are many blades, the friction losses will be large; conversely, if there are very few blades, the fluid will lack guidance. An optimum can be found between these two extremes. This is usually completed using [PERSON]'s criterion.