# Performance of a Closed Cycle Power Takeoff for a Shore-Based Wave Energy Device
## 2 Materials and Methods
### Preliminary 1D Design of the Turbine  
In this section, the process followed to obtain a suitable turbine design is presented. A complete description of such a work can be found in turbomachinery books (e.g., [20]). Out of the six cases presented in Table 1, two cases will be considered: the most energetic case (high tide, \([Z_{w},T_{w}]=[2\,\mathrm{m},10.5\,\mathrm{s}]\)) and the least energetic case (low tide, \([Z_{w},T_{w}]=[1.5\,\mathrm{m},10\,\mathrm{s}]\)). The best-suited type of turbine for this high flow rate/small pressure drop application is an axial flow turbine, where the flow direction is perpendicular to the axis of rotation. It is the most common configuration for gas turbines and has the advantage of being efficient on a wider range of speeds than radial or mixed flow turbines. It is thus expected to be better able to extract power from a wide range of sea states. The turbine can be one stage only, which limits the costs and is more compact. \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline
**Wave Height m** & **Wave Period s** & **Tide (-)** & **Pressure Drop Pa** & **Mass Flow Rate kg/s** \\ \hline \multirow{2}{*}{1.5} & \multirow{2}{*}{10} & low & 1390 & 0.55 \\  & & mean & 1650 & 0.60 \\  & & high & 2020 & 0.66 \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{10.5} & low & 2020 & 0.67 \\  & & mean & 2410 & 0.73 \\ \cline{1-1}  & & high & 2890 & 0.80 \\ \hline \end{tabular}
\end{table} 
Table 1: Turbine operating conditions for different sea states and tide levels. Figure 6: Distribution of mass flow (kg/s) (**a**) and pressure drop (**b**) for the two extreme cases considered in Table 1.  
The specific speed is defined by  
\[\Omega_{s}=\frac{\Omega\left(\frac{\dot{m}}{\rho}\right)^{0.5}}{\left(\Delta h_{is} \right)^{0.75}} \tag{17}\]  
where \(\Omega\) is the rotational speed in rad/s, \(\dot{m}\) is the mass flow rate in kg/s, \(\rho\) is the density in kg/m\({}^{3}\) and \(\Delta h_{is}\) is the isentropic enthalpy in J/kg. It represents the maximum available energy and can be calculated as follows  
\[\Delta h_{is}=C_{p}T_{01}\left(1-\left(\frac{P_{3}}{P_{01}}\right)^{\frac{2-1} {\gamma}}\right) \tag{18}\]  
where \(C_{p}\) is the specific heat capacity, \(T_{01}\) and \(P_{01}\) are, respectively, the inlet total temperature and pressure, \(P_{3}\) is the outlet static pressure and \(\gamma\) is the heat capacity ratio. For an axial flow turbine, the optimal specific speed is about \(\Omega_{s}=0.6\), as plotted by, for instance, [21]. This leads to an optimal rotational speed of \(N=2400\) rpm and \(1700\) rpm for, respectively, the most energetic and the least energetic of the considered cases. The turbine consists of a row of fixed blades called the stator, or the nozzle guide vanes, followed by a row of rotating blades called the rotor. In an impulse turbine, all the expansion occurs in the stator, while, in a \(100\%\) reaction turbine, all the expansion occurs in the rotor. The degree of reaction is defined by  
\[R=\frac{h_{2}-h_{3}}{h_{1}-h_{3}} \tag{19}\]  
and is chosen in between those two extremes at \(R=0.4\), where the best performances are expected for this application. Another important dimensionless design parameter is the velocity ratio  
\[\frac{C_{x}}{U} \tag{20}\]  
where \(C_{x}\) is the axial flow velocity and \(U\) is the rotor blade speed at the mean radius. A small velocity ratio is usually linked to higher efficiencies, but it implies a larger turbine. Here, a velocity ratio of 0.3 is taken, but that value could easily be modified depending on the physical constraints on the turbine size. The last dimensionless design parameter to choose is the blade aspect ratio  
\[\frac{H}{b} \tag{21}\]  
where \(H\) is the height of the blades and \(b\) is their axial chord length. In this work, it was taken to be 1.25. This choice results from a compromise between keeping the turbine compact and ensuring that the tip clearance remains proportionally small.