## 4 Methodology  
The design parameters of the 5 MW NREL reference OWT outlined in Reference [22] as shown below, are used to validate the VRT in conjunction with the existing theories discussed in this paper:  
WT rating: 5 MW,  
Rotor diameter: 126 m,  
Rotor orientation: upwind,  
Hub diameter: 3 m,  
Number of blades: 3,  
Hub height: 90 m,  
Cut-in wind speed: 3 m/s,  
Rated tip-speed: 80 m/s,  
Cut-out wind speed: 25 m/s,  
Overhang: 5 m,  
Cut-in RPM: 6.9,  
Shaft tilt angle: 5\({}^{\circ}\),  
Rated rotor RPM: 12.1,  
Pre-cone: 2.5\({}^{\circ}\),  
Rated wind speed: 11.4 m/s. Suffice it to emphasize that the present paper is solely a theoretical paper based on analytical methods as demonstrated in the previous section. Equations (7), (8), (12), (14), (15), (16), (18), (20), (21), (65), (73), (75), (79) and (80) are coded in MATLAB and graphs are plotted in Section 5. Based on the assertion that the axial induction factor of an optimum rotor lies within \(1/4\leq a\leq 1/3\)[5], a value of \(a=0.293\) is used in determining \(U_{R}\) at the hub height. A value of \(U_{R}=5.93\) is estimated. The value \(a=0.293\) is the average of the range of optimum values. It is worth noting that based on Equation (21), \(a^{\prime}=\infty\) at \(a=1/4\) and \(a^{\prime}=0\) at \(a=1/3\). To estimate the wind speed distribution across the rotor blade span, the wind shear equation is used. In this case, the WT rotor hub height \(h\) is the reference height. Based on 24-year data for the Maryland offshore wind farm, the average wind speed at a hub height of 90 m for the NREL OWT under investigation is estimated to be 8.37 m/s. The corresponding angular speed of the rotor is 0.93 rad/s. The wind speed distribution across the span of the rotor is given by  
\[U_{0}(r)=U_{h}(\frac{r+h}{h})^{\varepsilon}, \tag{81}\]  
where \(U_{h}\) is the average wind speed at hub height and \(\varepsilon\) is the wind shear exponent estimated using [PERSON]'s correlation formula:  
\[\varepsilon=0.096\log_{10}(z_{0})+0.016(\log_{10}(z_{0}))^{2}+0.24, \tag{82}\]  
in which \(z_{0}\) is the surface roughness. An average value of \(z_{0}=3.5\times 10^{-4}\)m is used in this paper, which is estimated from corresponding values for calm open and blown seas [3]. A value of \(\varepsilon=0.099\) is obtained from the above value of \(z_{0}\). Substituting for \(U_{h}=8.37\)m/s, \(h=90\) and \(\varepsilon\) in Equation (81) yields  
\[U_{0}(r)=8.37(\frac{r+90}{90})^{0.099}. \tag{83}\]  
Similarly, the wind speed at the WT rotor is estimated using the wind shear equation:  
\[U_{R}(r)=5.92(\frac{r+90}{90})^{0.099}. \tag{84}\]  
The local axial induction factor due to solidity in the case of the VRT or the local axial induction factor in the cases of the actuator disk and rotating annular stream-tube theories are obtained using Equation (8). Trial values of the half cone angle within the range \(0^{\circ}\leq\alpha<90^{\circ}\)are used in estimating \(a_{w}(r)\), \(a^{\prime}(r)\) and their corresponding pressure differences in the case of the VRT, while the value of \(a^{\prime}\) corresponding to \(a=0.293\) is estimated using Equation (21) in the case of the RAST, and their corresponding pressure differences are estimated. After iterating with trial values of \(\alpha\) between \(0^{\circ}\) and \(90^{\circ}\), an average half cone angle of \(45^{\circ}\) is adopted in this paper. The MATLAB curve fitting tool is used to obtain the best fit third-degree polynomials for the three theories; the resulting expressions for \(dC_{p}\) are integrated with respect to the local tip-speed ratio within the limits \(\lambda_{h}\leq\lambda(r)\leq\lambda\) to yield \(C_{p}\) for each theory. Trial values of the tip-speed ratio are used to ensure that the \(C_{p}\) values estimated from all the three theories do not exceed 0.593.