# Advanced Scale-Propeller Design Using a MATLAB Optimization Code
## 3 Results
### Reynolds Number Distribution  
Using Equation (13), the Reynolds number distribution along the blade can be calculated, using the optimal chord distribution output from the program. The re-arrangement of Equation (13) to an equation suitable for the computer program is found below.  
\[Re(r)=\frac{\rho\cdot W(r)\cdot\ c(r)}{\mu}=\frac{\rho\cdot\ \Omega\ r\cdot c(r)}{\mu} \tag{37}\]  
As already discussed in this paper, a Reynolds number of 100,000 represents a transition point at which the lift-to-drag ratio of a smooth airfoil increases drastically. In propeller design, it is therefore highly desirable for the Reynolds number to be above 100,000 at most radial co-ordinates. Whilst this could be achieved at all radial co-ordinates by simply increasing the chord length close to the propeller hub, structural requirements dictate that this chord profile is not commonly seen in small (less than 21-inch diameter) propellers. There would also be a drag penalty associated with increasing the chord length, which would reduce the lift-to-drag ratio of the blade sections close to the hub.  
Figure 8: Computation time for MATLAB program with varying number of blade elements. The processor of the system used for this computation is an Intel (R) Core (TM) i3-6006U CPU @ 2.00 GHz, 1992 MHz.  
Figure 7: Optimal chord distribution for a propeller with design inputs of: 20-inch diameter, a design advance ratio of 0.59, two-blades, a shaft power of 3 kW, zero angle of attack, standard density of 1.225 kg/m\({}^{3}\) and a lift-to-drag ratio of 20.  
Increasing of Aircraft Propeller Efficiency by Using Variable Twist Propeller Blades', a paper by [PERSON] (2008) [PERSON] (2008) (2018), details a method which, given that the above radial variables have been calculated at every radial co-ordinate, can be easily used to calculate the radial distribution of thrust generated by the propeller. After derivation, the equation used in the computer program is shown in Equation (38).  
\[dT(r)=2\pi\rho V\cdot(1+a)\cdot(2 VaF)\cdot dr \tag{38}\]  
An example thrust distribution, plotted by calculating \(dT\) at every radial co-ordinate, is shown in Figure 9.  
As can be seen in Figure 9, the majority of the thrust developed by a propeller is between the non-dimensional radii of 0.4 and 0.95. Using numerical integration, it is found that for the example in Figure 9, 85% of the thrust is developed between 0.4 and 0.95 of the propeller radii, but other propellers will yield very similar results. Therefore, in the computer program created for this paper, the program ensures that all blade stations between non-dimensional radii of 0.4 and 0.95 experience Reynolds numbers above the 100,000 threshold.  
The unmodified Reynolds number distributions of two example propellers are found in Figure 10.  
Figure 10: Radial Reynolds number distribution along a 10-inch and 20-inch propeller blade with design inputs of: velocity of 10 m/s, 2000 RPM, two blades and a hub radius of 0.15.  
Figure 9: Radial thrust distribution along a propeller blade with design inputs of: 20-inch diameter, a design advance ratio of 0.59, 2-blades, a shaft power of 3 kW, zero angle of attack, standard density of 1.225 kg/m\({}^{3}\) and a lift-to-drag ratio of 20.  
For clarification, a flow chart outlining the basic architecture of the MATLAB program which outputs optimal propeller geometry and subsequently modifies this to minimise the effect of low Reynolds number on propeller efficiency is shown in Figure 11.  
Figure 11: Flow chart of MATLAB program which outputs optimal propeller geometry.