How to calculate in the right way the efficiency of a propeller?

If we know the engine power, speed of the plane and the thrust of its propeller, what is the correct method, (1) or (2), for calculating the efficiency of the propeller?

Efficiency of a propeller

Assuming that Method 1 is the correct one, it appears that the efficiency of a propeller must satisfy the inequality:

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It looks like Method 1 is correct and 2 is wrong as long as the page Performance of Propellers, MIT calculates the efficiency on an ideal propeller and gets the same inequality. If a propeller of diameter, d, delivers the thrust, T, while the plane travels at the speed, V, it always has a maximum possible efficiency that can be calculated, is below 1 and can not be improved. Not even an ideal propeller of diameter, d, has, in general, an efficiency that reaches 100%. The absolute minimum, reference power, is always $TV$ and not something else and the efficiency is always: $$TV/Power_{absorbed}$$ where the minimum absorbed power is calculated with Froude's Propeller Theory.

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    $\begingroup$ Whilst we have contributors here who should be able to answer this, I feel this belongs in Physics.SE $\endgroup$ – Ben Jan 21 '16 at 21:32
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    $\begingroup$ Physicists are not familiar with Froude's theory and other theories related to propellers. Only naval or aerodynamic engineers can give qualified answers. $\endgroup$ – Robert Werner Jan 22 '16 at 2:29
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    $\begingroup$ I second @Ben, this seems better suited for Physics S.E. $\endgroup$ – Firee Jan 22 '16 at 10:59
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    $\begingroup$ I do not understand the title, the definition of Power as Velocity x Force is a physical one, is always valid and does not depend on the system considered. $\endgroup$ – Federico Jan 22 '16 at 11:32
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    $\begingroup$ To those voting this "off-topic", just because this question could also be a valid question on another SE does not mean that it cannot be valid here as well. $\endgroup$ – J Walters Jan 22 '16 at 14:27

Propellers are designed for an optimal tip speed being a certain multiple of the air speed they are flying inside. The angle of the propellers aerofoil shape to the propellers axel changes as you move from the Center to the tip. The more pronounced the blade rotation the higher the intended tip speed.

From memory having the tip speed running at 1-3 times the the air flow speed is not efficient, but at least the propeller works over a wide range of air speeds.

Highly tuned propeller tips are moving at 6x the air speed. However, their effectiveness at moving air disappears rapidly if the air flow is too slow. (The propeller blades are stalled)

The last thing you want on a propeller plane is an efficient propeller.

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