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Suppose you had a system consisting of a large propeller and engine. In this scenario, the total weight of the apparatus is $160\hspace{1mm}\text{kg}$ and the length of a single propeller blade is $80\hspace{1mm}\text{cm}$ (diameter is $160\hspace{1mm}\text{cm}$).

The force of gravity acting on the system is thus $1568\hspace{1mm}\text{N}$, and the control surface (in this case, the area of the circle which the propellers make upon a revolution, equal to $\pi \cdot 0.8^2$) is approximately $2.01\hspace{1mm}\text{m}^2$

Im trying to calculate the ideal wattage (no losses or otherwise) required by the engine which will produce just enough thrust to counteract the force of gravity (meaning a thrust of $1568\hspace{1mm}\text{N}$).

During my research, I stumbled across this website and its 12th equation:

$$P=\frac{T^{\frac{3}{2}}}{\sqrt{2 \rho A}}$$ where:

  • $P$ is power (in Watts)

  • $T$ is thrust force (in Newtons)

  • $\rho$ is the fluid density of the surrounding medium (the unit is kg/m3, and since the medium is air, its around $1.204 \hspace{1mm} \text{kg}/\text{m}^3$)

  • and $A$ is the control surface area (I believe it to be in units of $\text{meters}^2$)

*$A$ is not explicitly described in the article as being the area of a propeller revolution, so i could be misinterpreting it, but i believe it refers to the control surfaces used in the Reynolds Transport Theorem, being a surface which fluid flows through, which would indeed be the propellers.

By setting the thrust force to be the force of gravity, and performing all other relevant substitutions, I calculated a value of around $28218\hspace{1mm}\text{Watts}$, or nearly a $38\hspace{1mm}\text{hp}$ motor.

This leaves me with my final question, is any of this correct? I did dimensional analysis on the equation, and the units check out, but am I using this formula as intended? Is this proper application? Is this formula even practical? (as in it makes too many simplifying assuumptions to be reliable)

I am by no means an aeronautical engineer, so it would come as no surprise if I misinterpreted several things across this process. All help is appreciated, thank you.

Jacob
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A is not explicitly described in the article as being the area of a propeller revolution

Per this article, yes, $A$ is the area of the rotor disk.

This leaves me with my final question, is any of this correct? ... I am by no means an aeronautical engineer,

I'm not an aeronautical engineer either, but I'm a very interested amateur. It sounds correct as an absolute upper limit on the required power. It's basically about as far as you can push the calculations without having to accurately model the fluid flow, and that modeling is difficult. You should probably plan on needing at least twice as much power at the rotor.

Note, too, that we're talking about fluid dynamics, which is very much not amenable to exact solutions -- the flow is both three-dimensional and turbulent; even computational fluid dynamics, as a useful thing, only really started making their way out of academic hands and into big industry in the early 2000's.

I've never tried it, but there are rules of thumb out there for how to start with a scale model, measure its performance, and have an idea of how the full-size prototype will work. It still won't get you to exact numbers (spend any time with your nose in the history of aircraft development, and you'll see that the first article prototype always gets modified, sometimes extensively, after the first set of flight tests). What it might get you is a prototype that will get off the ground, and will refrain from crashing for long enough that you can determine what modifications need to be done for the next step.

* Or spend years becoming a good enough aeronautical engineer to put together a valid model and do simulation on it.

TimWescott
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