Originally Posted by TangoKilo
As you have previously said, there is a certain power required to move a body at a certain speed. Where does that power come from?
The only place it can come from is a fan moving air along a tube. So the power required to move a body through the air must be equal to the power required to move an amount of air through a tube. Where else can power come from?
Yes the thrust (and power) must come from moving the air. That does not mean that there must be equality between the rate of kinetic energy addition to the air and the power supplied to move the airplane.
Consider two airplanes moving at the same speed with the same drag. The propulsion systems of both airplanes impart horizontal momentum to the air at the same rate. In any given second, the propulsion system of Airplane A imparts a velocity change of V m/s to a volume of air with mass M kg. In the same second, the propulsion system of Airplane B imparts a velocity change of V/2 m/s to a volume of of air with mass 2M kg. The thrust, drag and momentum transfer rates are the same. Because the airplanes are moving at the same speed, the power supplied to keep them moving is the same (Drag*TAS). I think you will find that the rate at which the propulsion system of Airplane A changes the air's kinetic energy is different from the rate at which the propulsion system of Airplane B change's the air's kinetic energy.
The extra kinetic energy Airplane A "dumps" into the air has to come from somewhere, so the fuel flow (or battery drain rate) of Airplane A must be higher. So both propulsion systems provide the same power to move the airplane through the air, but one of the systems does it with less power consumption. You want to say that the net propulsive efficiencies of these systems are the same. That seems hard to justify.