## 5 Results and Discussion
\(dC_{p}\cong 0.073\) and \(\lambda(r)=5.6\) followed by a drop to a trough at \(dC_{p}\cong 0.052\) and \(\lambda(r)=6.0\) and rises thereafter. It is worth noting that the values obtained from both the ADT and the VRT are low relative to those of the RAST, so much so that the concavity of their curves is adversely affected when plotted together with the curve of the RAST, a consequence of which is the flatness of their curves for \(0.3\leq\lambda(r)\leq 5.6\). Also worthy of note is the fact that \(dC_{p}\) values for the VRT are higher than those estimated from the ADT, while the values estimated using the RAST are higher than those of the former two theories. This disparity in the magnitude of \(dC_{p}\) results from the fact that the RAST considers the local torque to be generated by the wake-induced angular speed on the rotor as stated in Equation (16), whereas the local torque estimated using the VRT results from the pressure difference across the WT rotor as expressed in Equation (64), while the power estimated from the ADT is the result of the product of the axial thrust expressed in Equation (2) and the wind speed at the rotor. The actuator disk and rotating annular stream-tube theories also consider the rotor as a disk. Figure 7: Local power coefficient as a function of the local tip-speed ratio: (**a**) plots of the ADT, the RAST and the VRT; and (**b**) those of the ADT and the VRT. The best fit curves for all the three theories are third-degree polynomials. The best fit curves are obtained with a 95% confidence interval using the MATLAB plotting tool. The best fit third-degree polynomial obtained for the ADT with a coefficient of determination of 0.9732, resulting in a correlation coefficient of 98.65%, is  
\[dC_{P}=-5.842\times 10^{-4}\lambda(r)^{3}+4.424\times 10^{-3}\lambda(r)^{2}+4.424 \times 10^{-3}\lambda(r)+8.685\times 10^{-4}. \tag{85}\]  
After integrating Equation (85) with respect to \(\lambda(r)\) within the limits \(0\leq\lambda(r)\leq\lambda\), the resulting expression is  
\[C_{P}=-1.4605\times 10^{-4}\lambda^{4}+1.4747\times 10^{-3}\lambda^{3}+2.212 \times 10^{-3}\lambda^{2}+8.685\times 10^{-4}\lambda, \tag{86}\]  
subject to the following constraints: \(0.1\leq\lambda\leq 11.4\) and \(0.0001104<C_{p}\leq 0.3102\). The optimum \(C_{p}\) for the HAWT in this study using the ADT is \(C_{p}=0.3102\) at \(\lambda=8.377\). The best fit third-degree polynomial obtained for the RAST with a coefficient of determination of 0.9755, resulting in a correlation coefficient of 98.76% is given by  
\[dC_{P}=-5.037\times 10^{-2}\lambda(r)^{3}+7.577\times 10^{-1}\lambda(r)^{2}-1  
\(dC_{T}/dr\cong 0.0016\) at \(r=61.633\) m. It is worth noting that despite the apparent flatness of the VRT curve, it has an upward concavity for local radii between 2.8667 m and 59 m. The ADT and RAST curves merge at all local radii. This merger is the upshot of the equality in the numerical values of \(dC_{T}/dr\) obtained from respective expressions for these theories. Despite the apparent cubic dependence of \(dC_{T}/dr\) in the case of the RAST on \(r\) and the linear dependence of \(dC_{T}/dr\) on \(r\) in the case of the ADT, both curves are linear and coincident, thus signifying a complete agreement of these two theories within this range of local radii. Equations (12) and (18) show the dependence of \(dC_{T}/dr\) on \(r\) for these two theories. The coincident curves rise linearly from \(dC_{T}/dr\cong 0.0046\) at \(r=2.8667\) m to \(dC_{T}/dr\cong 0.1097\) at \(r=61.633\) m. It is discernible from Figure 8 that the thrust coefficients estimated from both the actuator disk and rotating annular stream-tube theories are higher than those obtained from the VRT at all the local radii considered in this paper. This difference between the VRT and the other two theories is evident because the latter two theories use the entire cross-sectional area of the path swept by the rotor blades in estimating \(dC_{T}/dr\), whereas the VRT uses the planform area of each rotor blade to estimate \(dC_{T}/dr\).