# Advanced Scale-Propeller Design Using a MATLAB Optimization Code
## 2 Materials and Methods
### Propeller/Rotor Efficiency
#### 2.3.2 Propulsive Efficiency  
The propulsive efficiency, \(\eta_{P}\), of a propeller driven propulsion system is a more useful expression of propeller performance, since in their operation, all propulsion systems will  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Propeller** & **Rotor** \\ \hline \(CTp\) & Propeller Thrust Coefficient & \(C_{T}\) & Rotor Thrust Coefficient \\ \hline \(CPp\) & Propeller Power Coefficient & \(C_{P}\) & Rotor Power Coefficient \\ \hline \(CQp\) & Propeller Torque Coefficient & \(C_{Q}\) & Rotor Torque Coefficient \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definitions of non-dimensional propeller and rotor coefficients.  
have axial flow with a non-zero velocity. Firstly, the advance ratio, J which is needed to calculate the propulsive efficiency, is defined in Equation (11).  
\[J=\frac{Axial\;Flow\;Velocity}{Tip\;Rotational\;Velocity}=\frac{V}{nD} \tag{11}\]  
Typically, in the range of 0-2, the advance ratio is the ratio of axial velocity to rotational velocity at the tip of the propeller. As can be seen in Equation (12), the propulsive efficiency is directly proportional to the advance ratio, \(J\).  
\[\eta_{P}=J\frac{C_{T}}{C_{P}}\;\equiv\;\frac{Tc}{Pc} \tag{12}\]  
As can be seen in Equation (12), the propulsive efficiency of a propeller is directly affected by the coefficients of thrust and power. Looking back to Equations (4) and (6), it can be seen that both the angular velocity and the propeller diameter will affect the propulsive efficiency. The fluid density, which is in the denominator of both coefficients, cancels out and therefore does not affect the efficiency. However, the propulsive efficiency is determined by other factors not explicitly seen in Equations (4) and (6). The propeller pitch/diameter ratio, airfoil geometry, spanwise chord, taper profile and material choice all affect the thrust and power terms seen in Equations (4) and (6); thus, they all affect the propulsive efficiency. The efficiency curve of an APC 10\({}^{\prime\prime}\)\(\times\) 7\({}^{\prime\prime}\) propeller is shown in Figure 5, with data from the UIUC propeller database [2].  
Well-designed propellers typically have maximum efficiencies at advance ratios of between 0.5 and 0.7. However, the maximum efficiency for a propeller only occurs at one advance ratio (J), meaning that as the advance ratio increases, the efficiency will quickly reduce to a minimum. This sudden drop means that designing an efficient propeller that will be operated over a range of advance ratios is very challenging, as one can select an advance ratio at which the propeller will operate at maximum efficiency, but any variation in advance ratio can lead to a drastic drop in efficiency. Therefore, the maximum efficiency of a propeller is limited to a single airspeed at a given angular velocity. Propellers on larger aircraft suffer from compressibility losses, whereby the tips of the blades approach the speed of sound, resulting in shock wave formation and the losses associated with these. Transonic flow begins at about Mach 0.8.  
Figure 5: Example efficiency curve of an APC 10\({}^{\prime\prime}\)\(\times\) 7\({}^{\prime\prime}\) Thin Electric propeller [2].