# Performance of a Closed Cycle Power Takeoff for a Shore-Based Wave Energy Device
## 2 Materials and Methods
### Design Refinement  
The preliminary design bases all calculations on a point at the mid height of the blade. This overlooks the variation in velocities across the blade in the spanwise direction. According to [24], if the hub-tip ratio \(r_{h}/r_{s}\) is greater than 0.8, which means that the blade height is relatively small compared to the rotor radius, the two-dimensional flow assumption is reasonable. In this study, the hub-tip ratio was 0.73. Two approaches have been taken to refine the design. First approach is leaning the blade, which consists of stacking the same blade shape developed in the preliminary design but with an angle in the radial direction. The aim is to counteract the radial pressure gradient created by the variation in centripetal force along the blade. The second approach is varying the blade angles radially, i.e., the free vortex design. Figure 13 presents a visualisation of these two approaches, which can be compared to the straight blade previously shown in Figure 8.  
The free vortex condition states that \(C_{w}r\) remains constant for all radii \(r\), where \(C_{w}\) is the swirl (or whirl) velocity. Combined with the conditions of constant axial velocity and constant specific work through the turbine, the free vortex condition ensures that the flow is in radial equilibrium. In practice, this means varying the blade angles so that  
\[\tan\alpha_{i}=\frac{r_{m}}{r}\,\tan\alpha_{im} \tag{33}\]  
Figure 12: Variation in pressure drop with flow coefficient for different rotational speeds. Figure 13: Leaned rotor blade (**a**) and free vortex design blades (**b**). \[\tan\beta_{i}=\tan\alpha_{i}-\frac{U}{C_{x}} \tag{34}\]  
The resulting angles at the hub and the shroud are presented in Table 5. With varying angles, the degree of reaction will also vary, from 0.29 at the hub to 0.48 at the shroud. Figure 14 shows the turbine's efficiency for three blades: the straight blade obtained from the preliminary design, the leaned blade and the free vortex design blade. The leaned blade results collapse quite well with the straight blade ones. The free vortex blade improves the peak efficiency by 1 percentage point, from 91.2% to 92.2%. The efficiencies are also improved by the free vortex design at low flow coefficients, with a wider plateau of high efficiencies. However, they are slightly worsened at very high flow coefficients. Figure 15 shows a visualisation of the flow by plotting velocity streamlines around the straight blade and the free vortex design blade at \(N=2000\) rpm and \(\Delta P=3000\) Pa, which is the point of peak efficiency in Figure 14. There is a recirculation zone on the pressure side of the straight rotor blade, while the flow closely follows the shape of the free vortex blade. Moreover, the velocities on the suction side of the straight blade are higher compared to the free vortex blade. This explains the improvement in efficiency brought about by the free vortex design. The straight blade is, however, expected to be easier to manufacture than the free vortex blade. In both cases, variations from the ideal design are expected during the manufacturing process. The variation in efficiencies remains at most a couple of percentage points when altering the 3D design of the blades. This shows that the turbine design is overall quite robust and can be expected to withstand manufacturing irregularities. Figure 14: Variation in efficiency with flow coefficient for different blade geometries.