# Advanced Scale-Propeller Design Using a MATLAB Optimization Code
## 2 Materials and Methods
### Calculating Propeller Geometry  
The numerical method is based on the Betz criteria [24]. The Betz criterion for optimum loading for a lightly loaded actuator disk in axial flight states that the trailing edge vortex sheets, after an initial deformation close to the rotor disk, move axially downward as a rigid screw surface (i.e., the ratio of translational velocity to rotational velocity is constant). For every iteration of this method, the displacement velocity is assumed to be radially constant. If the wake helix angle is \(\varphi_{s}\), the axial and swirl velocity components are therefore given by Equations (14) and (15).  
\[w_{axial}=v^{\prime}Cos^{2}2\phi_{s} \tag{14}\]  
\[w_{swirl}=v^{\prime}\cos\phi_{s}\mathrm{sin}\,\phi_{s} \tag{15}\]  
In this method, the displacement velocity will be represented by the displacement velocity ratio, which is defined in Equation (16). This is the variable which will be optimised to result in optimum propeller geometry, but this will be discussed later.  
\[\zeta=\frac{v^{\prime}}{V} \tag{16}\]  
Both [PERSON] (1919) and [PERSON] (1929) derived momentum loss functions, which describe the energy lost in a propeller wake. Whilst [PERSON]'s function is more accurate, [PERSON]'s momentum loss function, F, will be used here for simplicity.  
\[F(r)=\frac{2}{\pi}\mathrm{arccos}e^{-f} \tag{17}\]  
\[f(r)=\frac{B}{2}\cdot\frac{1-\xi}{\sin\phi_{t}} \tag{18}\]  
\[\xi(r)=\frac{r}{R} \tag{19}\]  
\(\xi\) is the non-dimensional radius and is calculated by dividing the radial co-ordinate, \(r\), by the propeller tip radius, \(R\) as in Equation (19). \(\varphi_{t}\) is the flow angle at the propeller tip, and is a function of the displacement velocity, as seen in Equation (20). From the flow tip angle, the flow angle, \(\varphi\), of every radial point can be found using Equation (21).  
\[\tan\phi_{t}=\frac{V}{\Omega R}\cdot\left(1+\frac{\zeta}{2}\ \right) \tag{20}\]  
\[\phi(r)=\arctan\frac{\tan\phi_{t}}{\xi} \tag{21}\]  
From Equations (17)-(21), the Equations (22) and (23) can be derived, which describe the circulation, \(\Gamma\), at any radial point \(r\). The circulation function, \(G\), is more accurate for vortex sheets which are flat, parallel and closely spaced, however, it is an adequate approximation for most propellers.  
\[\Gamma(r)=\frac{2\pi V^{2}\zeta G}{B\Omega} \tag{22}\]  
\[G(r)=F\cdot\frac{\Omega r}{V}\cdot\cos\phi\mathrm{sin}\,\phi \tag{23}\]  
Applying the [PERSON] theorem for a blade element at radius \(r\), the lift per unit length of the blade can be equated as in Equation (24), where \(W\) is the local total velocity.  
\[\frac{\rho W^{2}cc_{L}}{2}=\rho W \tag{24}\]Applying the expression for \(\Gamma\) in Equation (22), results in Equation (25).  
\[Wc(r)=\frac{4\pi V^{2}G\zeta}{c_{L}B\Omega} \tag{25}\]  
The total velocity \(W\) can be expressed by Equation (26).  
\[W(r)=\frac{V(1+a)}{\sin\phi} \tag{26}\]  
\(a\) and \(a^{\prime}\) are variables which appear several times in this method. The derivation of them will not be included in this paper, but can be found in 'Increasing of Aircraft Propeller Efficiency by Using Variable Twist Propeller Blades', a paper by [PERSON] (2008) [26].  
\[a(r)=\frac{\zeta}{2}\cdot cos^{2}\phi\cdot(1-\epsilon\tan\phi) \tag{27}\]  
\[a^{\prime}(r)=\frac{\zeta}{2}\cdot\frac{V}{\Omega r}\cdot\cos\phi\sin\phi \left(1+\frac{\epsilon}{\tan\phi}\right) \tag{28}\]  
\(E\) is the section drag to lift ratio and is the only airfoil property that is required for this method. By dividing Equation (25) by Equation (26), the value of \(c\) can be calculated for any radial co-ordinate. Using Equation (29), the angle of twist (\(\beta\)) can be found at any radial co-ordinate.  
\[\beta(r)=\phi+\alpha \tag{29}\]