Is this scheme for a single-fan electric airship with a rocket-like geometry workable?

Question

I am currently designing a vertical take-off and landing vehicle that uses a Tesla powertrain connected to an electric ducted fan. Please comment on if my thought process is sound on my feasibility analysis.

My initial mission goal is extremely modest to grease the wheels of my feasibility analysis and have a realistic utility in society (and for people in my sport, recreational/experimental skydiving).

The vehicle simply needs to lift itself and a 100kg payload (coincidentally the approximate mass of a skydiver with his parachute) to the altitude of 1 mile.

I think I need to calculate my total delta_v, but because I don't need a high speed (like an orbital escape velocity or something like that) I am not positive how to go about getting it without first understanding the maximum thrust from my system.

My current plan for determining thrust from the tesla-system: 1) identify maximum output of tesla battery and motor (lets say Model S P100D) 2) calculate maximum propeller size for #1 based on algorithms available in electric drone aircraft websites/forums 3) calculate potential thrust produced by #2 4) use #3 to determine if liftoff is possible carrying at least #1

My initial calculations are not intended to include all outside forces; I will incorporate these if it is in fact, theoretically feasible at all.

Based on information above as well as the weights of the battery, motor, and an estimate of a basic rocket-shaped frame, I can figure out if a basic propeller will work for this, or if I will need to incorporate an electric turbojet, which would blow up my budget for this.

It is also feasible I could have a combination of several motors / batteries attached to multiple propellers with something like a rocket sitting in the middle of an oversized quadcopter. Again, budget goes up considerably.

Answer 1

First, you need thrust-to-weight¹ ratio more than 1, so the device can maintain flight. You should be able to find data about how much thrust (lift in case of rotors; it's the same thing) they can produce and how much power from the engine it takes for some existing propellers and rotors.

The efficiency, that is thrust-to-power ratio, generally increases with the disc area. That is why helicopters have large rotors—they need less power to produce the desired lift.

So you'll also need enough area. You can chose a rotor like a normal helicopter, or you can choose multiple propellers in a style similar to

(via this answer)

or

(from this company)

You should be able to get (static) thrust and efficiency for some standard propellers to use in your estimates.

Keep in mind, that the maximum thrust decreases with air density and you'll need the thrust up to your desired altitude. At 5000 ft, the density is ~86% of that at sea level, further subject to weather as density also decreases with temperature.

Now when you have a set of propellers that can produce enough thrust with your available power, you'll still need some excess power. This excess power will roughly indicate your maximum climb rate: just divide the excess power by weight².

Note that the excess power will decrease with altitude as the efficiency will decline with density.

And last you'll need to make sure you have enough energy stored to maintain full power for the time it will take you to climb to that altitude.

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¹ Weight is a force acting on a body due to gravity, i.e. mass times gravity constant, so it can be directly compared to other forces (thrust).

² Since weight is force, dividing power by it yields (vertical) velocity.

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Note regarding delta_v. That is a characteristic quantity for rockets, where the limiting factor is the momentum they can impart to the exhaust gasses that are generated from the fuel only. And since momentum is mass times velocity, dividing the momentum by mass gives a useful characteristic value, the $\Delta V$, with dimension of velocity.

However you have propellers and they impart the momentum to surrounding air, which you have unlimited supply of. So the momentum is not limiting for you, the energy that you lose with it is. You minimize the energy by using a lot of the air, i.e. increasing the mass of the reactive mass, not its velocity. Rockets can't increase the mass, as the fuel is all they have, so they must maximize the exhaust velocity instead and are intentionally inefficient in the sense of the thrust-to-power ratio mentioned above.

Answer 2

The best propeller driven vertical take-off and landing craft would be a helicopter, actually. Yes a duct around a propeller increases thrust, but upsizing a propeller creates way more thrust from the available power.

Propeller thrust has scaling effects with size due to different Reynold numbers, data does not scale up well. Best way forward from where you are now would be to look at the weight, rotor dimensions and installed power of existing designs for small helicopters such as the R-22. Or better yet, design an electrical version of the Hiller Flying Platform Platform.