Got this basic propeller formula from a friend that does his phd on hydrofoils/hydrodynamics, its very simplified, but it takes into account that the propeller is moving through a medium.
There are several important trends that are apparent upon consideration of these equations. We see that the propulsive efficiency is zero when the flight velocity is zero (no useful work, just a force), and tends towards one when the flight velocity increases. In practice, the propulsive efficiency typically peaks at a level of around 0.8 for a propeller before various aerodynamic effects act to decay its performance as will be shown in the following section.
Assumptions:
Flight velocity: 20km/h = 5.6m/s
Thrust: 300N (might be a bit high)
Propeller diameter: 140mm, 70mm and 50mm ( We only play with this size in this example)
Propulsive efficiency with prop-diameter 140mm= 0.80 (140mm looks like a good size for our use)
Propulsive efficiency with prop-diameter 70mm= 0.58
Propulsive efficiency with prop-diameter 50mm= 0.47
So a prop diameter of 140mm compared to a prop diameter of 70mm theoretically gives 38% higher efficiency, which is a LOT. The practical efficiency gain is probably lower due to a larger duct drag, but still…
Another interesting formula, which gives us the Power needed for a certain speed/thrust (given an ideal propeller), from the same variables. This values are a bit easier to comprehend.
This is the ideal (minimum) power required to drive the propeller. In general, the actual power required would be about 15% greater than this.
Assumptions power consumption with “ideal propeller”:
Flight velocity: 20km/h = 5.6m/s
Thrust: 300N (might be a bit high)
Propeller diameter: 140mm, 70mm and 50mm ( We only play with this size in this example)
Power needed to run at 20km/h with an optimal prop, diameter 140mm= 2098W
Power needed to run at 20km/h with an optimal prop, diameter 70mm= 2892W
Power needed to run at 20km/h with an optimal prop, diameter 50mm= 3593W
These values are similar to the values we have measured on our system (We do not have a optimal propeller) but we can see that the correlations are there!