## 3 Development of Basic Aerodynamic Theories
### The VRT
#### 3.1.2. Blade Sectional Parameters and Local Blade Forces  
In a bid to calculate the forces on the rotor segment based on the static pressure determined using the VRT, the geometry and kinetics of the WT rotor blades must be taken into consideration. Some readers may be doubtful as to why the local planform area of the rotor is used in estimating the resultant force on the rotor blade; this is so because each rotor blade segment has a local twist angle such that the resultant static pressure does not act normally to the rotor blade but acts normally to the orthogonal projection of the rotor blade. Based on the local solidity ratio \(\sigma^{\prime}(r)\), at a radius \(r\) from the center of the rotor's hub, the local projected or planform area of a \(k\)-blade rotor \(A(r)\), where \(k\) is the number of rotor blades, is given by  
\[A(r)=\pi\sigma^{\prime}(r)r^{2}. \tag{66}\]  
However, \(\sigma^{\prime}(r)=kc(r)/2\pi r\), where \(c(r)\) is the local chord length of the rotor's airfoil cross-section at a radius \(r\) from the rotor hub [2; 3]. Substituting for \(\sigma^{\prime}(r)\) in Equation (62) and simplifying the resulting equation yield  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]  
Differentiating Equation (67) with respect to \(r\) leads to  
\[d(A(r))=\frac{kc(r)dr}{2}+\frac{krd(c(r))}{2}. \tag{68}\]  
For very long slender rotor blades, \(dc(r)\) is very small and \(dc(r)\to 0\), leading to  
\[\frac{dA(r)}{dr}=\frac{kc(r)}{2}. \tag{69}\]  
Let \(p_{1}(r)-p_{2}(r)\) be equal to \(\Delta p\). Then, the elemental resultant force per unit span on the WT rotor, \(F^{\prime}{}_{R}\) is given by \(F^{\prime}{}_{R}=\frac{dA(r)}{dr}\Delta p\) and after substituting for \(\frac{dA(r)}{dr}\) in the expression for \(F^{\prime}{}_{R}\), one has  
\[F^{\prime}{}_{R}=k\Delta pc(r)/2. \tag{70}\]  
Substituting the expression for \(\Delta p\) from Equation (65) in Equation (70) yields  
\[F^{\prime}{}_{R}=\frac{1}{4}k\rho c(r)[\delta(r)U_{0}^{2}(r)(2-\delta(r))+ \omega^{2}r^{2}(1+a^{\prime}(r))^{2}(1-(\frac{1-\delta(r)}{1-a_{s}}^{2}))]. \tag{71}\]  
The tangential force per unit span on the rotor segment is  
\[\frac{dF_{T}}{dr}=F^{\prime}{}_{R}\cos[\theta(r)], \tag{72}\]  
leading to  
\[\frac{dF_{T}}{dr}=\frac{1}{4}k\rho c(r)[\delta(r)U_{0}^{2}(r)(2-\delta(r))+ \omega^{2}r^{2}(1+a^{\prime}(r))^{2}(1-(\frac{1-\delta(r)}{1-a_{s}}^{2}))] \cos[\theta(r)]. \tag{73}\]  
The normal force per unit span on the rotor segment is given by  
\[\frac{dF_{N}}{dr}=F^{\prime}{}_{R}\sin[\theta(r)], \tag{74}\]  
leading to  
\[\frac{dF_{N}}{dr}=\frac{1}{4}k\rho c(r)[\delta(r)U_{0}^{2}(r)(2-\delta(r))+ \omega^{2}r^{2}(1+a^{\prime}(r))^{2}(1-(\frac{1-\delta(r)}{1-a_{s}}^{2}))]\sin [\theta(r)]. \tag{75}\]  
The local thrust coefficient per unit span is given by  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]0The local torque is given by  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]1  
The local power captured by the rotor segment is given by  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]2  
The local power coefficient is given by  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]3  
where \(\theta(r)\) is the angle of inclination of \(F_{R}^{\prime}\) to the tangential force on each rotor segment. It is estimated from the tangential speed on the rotor and the axial wind speed at the rotor by expressing its tangent as  
\[A(r)=\frac{kc(r)r}{2}. \tag{67}\]4  
It is evident from Equation (15) that the local tip-speed ratio is a function of \(r\), and to reduce the number of equations, \(a^{\prime}(r)\), \(a_{w}(r)\) and \(dC_{p}\) are expressed as functions of \(r\) instead of being expressed as functions of the local tip-speed ratio.