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I'm confused about the relation between IAS, TAS, thrust, and shaft power, for an electric application. Yes, higher air is thinner, and thrust goes down due to higher airspeed. What I've researched uses only air breathing engines for calculations.

Gas turbines either are limited to lower altitudes or higher altitudes with high speed, to intake enough air to maintain combustion. The other limiting factor is they have very strict maximum rpm. Modern turbofans do not have a variable transmission to counteract this. The same concept applies to electric motors, to a lesser degree.

For example, an electric driven ducted fan has a tip speed of Mach 0.85 and was set to maintain the same power output of say 50kw no matter the altitude.

  • What would happen to the TAS, IAS, thrust and the shaft rpm of the motor as well as tip speed of the ducted fan as the altitude increases?

  • Is the power necessary for an increase in TAS and a constant IAS the same up to the point where the increase in rpm (less load) due to thinner air causes the fan blades to see supersonic tip speeds?

  • If an aircraft can fly at 60 knots with 50 kw at sea level, does that mean at 50,000 ft it can realistically fly at 185 knots TAS for the same given power? (I used this calculator).

Also, how does thinner air affect static thrust? Is the same thrust maintained as long as the same amount of power is given to spin the fan?

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The Helios was an electric powered prop driven aircraft that achieved an altitude of over 96,000 feet and maintained it in level flight for more than 40 minutes.

as altitude increases

You maintain wing lift with constant IAS until the wing airflow approaches the speed of sound (TAS increases). Shaft power output must increase to spin prop faster to maintain thrust or... changing prop pitch is helpful instead of increasing rpm. This helps avoid the "sound barrier" for the prop.

as motor rpm increases

Even when not limited by supersonic prop tips, increased engine bearing friction from higher rpm requires a greater power draw. Again, the ability to vary prop pitch is enormously helpful to maintain optimal prop AOA as TAS increases.

can I fly with the same amount of power at 50,000 feet as I can at sea level

You can get a lot closer by designing your aircraft to fly at 50,000 feet by:

 1.  Reducing wing loading by removing weight
 2.  Increasing wing aspect ratio and/or increasing wing area
 3.  Increasing the **area** of your prop so you can push more air at the same rpm.

With props, you must compensate for thinner air, but higher rpm is not the only way.

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First let's look at the difference between IAS and TAS

TAS is your true airspeed and is independent of altitude, it's the speed at which you're moving through air. If there is no wind it is equal to your airspeed. On the other hand IAS is the airspeed you read on your flight instruments as indicated airspeed and is a function of the dynamic pressure. Your pitot tube is measuring the stagnation pressure and deducing the static pressure you get the dynamic pressure related to the airspeed. $$ P_{stagn}=P_{static}+\frac{1}{2}\rho V^2$$ As your instruments are calibrated for a given $\rho$ the indicated airspeed IAS will be different than the TAS with variation of the air density. IAS is used to provide a good idea of the aircraft performances as in drag and lift equation the term $\frac{1}{2}\rho V^2$ is always there. At same IAS, the aircraft aerodynamic performances will be the same whether you are at see level or up a mountain.

Now moving to the engine part of the question. Modern turbofans actually work with fairly constant airflow speed inside fixed by their design. Thrust is produced by accelerating air through the engine. The more air you can accelerate and the more speed increase you can provide the more thrust you get.

As the exhaust leaves the engine nozzle at a fairly fixed velocity, the acceleration provided is directly linked to the incoming airspeed, and here it's the true airspeed that matters. The faster the aircraft speed, the less the intake air is being accelerated. Therefore thrust decreases with increasing airspeed. Furthermore the higher you fly the less air density you have. Then for the same volume of intake air passing through your engine, less mass is accelerated and therefore thrust is also reduced while flying at high altitude. Both effects are mitigated by the fact that flying faster increases compression in the intake and lets the whole engine operate at a higher pressure level, thus increasing the output thrust. As airspeed increase, more volume of air is directed in the inlet where it needs to slow down before entering the compressor, thus increasing the inlet pressure your engine is seeing.

Modern turbofans are highly oversized, as for twin engines airliners they must be able to fly level for more than 180min with one engine shut down (ETOPS 180 certification). Therefore they never use the maximal power output they can generate even at take-off. Then even if with the altitude and airspeed thrust decrease there is still plenty enough to cruise at transsonic speed. Finally the shaft power is a function of the torque and RPM of the engine. The torque is directly related to the thrust and the RPM are maintained at quite a constant value in new engines. Therefore if thrust decreases, the aerodynamic torque on the fan decreases and the shaft power required to drive it will decrease. Maintaining the same IAS means your drag will remain constant in level flight whatsoever your flying altitude. And thrust requirement remains constant. On ducted fan the inlet compressive effect at high speed mitigate the need of higher RPM. On propeller on the other hand, if you don't adapt your blade pitch, at constant power output, the RPM will increase which is not really good for the engine and the blade tip going supersonic. If you change your pitch your blade can remain at constant speed and the RPM will remain constant.

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