Keith Williams:
So Power Required is proportional to TAS cubed.
The Indicated Airspeed (IAS) at any given altitude is determined by the TAS and the air density. As altitude increase the air density decreases, so more TAS is required to produce any given IAS.
At mean sea level in ISA conditions (ignoring instrument errors and pito system errors) the TAS is equal to IAS.
So at ISA MSL 150 KIAS = 150 Kts TAS.
But at 5000 ft in ISA conditions 150 KIAS = approximately 161 Kts TAS
And at FL300 in ISA conditions 150 KIAS = approximately 242 Kts TAS.
Looking at these last two figures, 242 kts is approximately 1.5 times greater than 161, So the power required at 150 KIAS at FL300 will be approximately 1.5 cubed = 3.375 times the power required at 150 KIAS at 5000 feet.
The power required is only proportional to TAS cubed if ρ is
constant, which of course
it won't be at two different altitudes. So at constant IAS the power at FL300 will be 1.5 times that at FL50,
not 3.375 times. Bloggs was right to call that error out.
Also:
The important thing to note in all of this is the fact that the thrust was not actually produced by the acceleration of the exhaust gas. It was produced by the rearward force which we exerted on the exhaust gas. The acceleration of the gas and the resulting loss of energy was just a side effect of the process. It happens because the air is not sufficiently stiff to resist the rearward force which we exerted upon it.
It is the mass of the jetwash/propwash/propellant that is the measure of its inertia, not “stiffness”.
You are driving down a motorway at 70 mph. The thrust required to maintain this speed is produced by your wheels exerting a rearward force on the surface of the road...because the road is stiff enough to resist the forces involved, it does not experience a rearward acceleration. The road surface meets your wheels at 70 mph and leaves your wheels at 70 mph. In this case all of the energy that your engine is providing to the wheels is being used to propel you forward, so your propulsive efficiency is 100%. But this is only possible because the material on which you are exerting your rearward force (the road) is stiff enough to prevent it form being accelerated rearwards.
In that case your propulsive efficiency is 100% because the chosen frame of reference is the same thing you are pushing against. Replace the car and Earth analogy with a hamster wheel and you find the wheel is spinning whilst the hamster is stationary. The hamster wheel is 'stiff' and yet it spins noticeably because it has a relatively small mass compared to the hamster. It is not about stiffness (except insofar as pushing against something stiff may transfer the force to something more massive).
Now it might be argued that the entire Earth is actually being accelerated backwards relative to your car, and because the mass of the Earth is enormous, the resulting acceleration is to small to detect.
Well, it would be very strange to argue that the Earth was accelerating relative to a car that was 'maintaining 70mph' relative to the Earth. But it would not be the least bit strange to point out that the force between tyre and Earth is a de facto torque on the Earth causing a – albeit infinitesimal – change of angular velocity. According to
NASA:
"Any worldly event that involves the movement of mass affects the Earth's rotation, from seasonal weather down to driving a car"
To test this argument let’s imagine that we have 2 identical cars standing back to back and tied together with a strong rope. The drivers start their engines and put the cars into first gear then slowly let out the clutches. The cars move forward until the rope becomes stretched and are then brought to a standstill. The cars are pointing in opposite directions so they cannot both be accelerating the road rearwards. But both cars are producing thrust and it is this which is exerting tension on the rope.
That would not test the argument anyway. If the cars are pointing in opposite directions the net force is zero so there is no torque on the Earth only compression of the surface and tension in the rope.