On Propeller Efficiency and the Axial Interference Factor

There is an important variable in momentum theory called the axial interference factor. I have referenced it from page 79 of "Hydrodynamics of Resistance and Propulsion" and decided to add it to the issue of propeller efficiency.
(The same formula for propeller propulsive efficiency is derived as Eq. 1-3 in Volume 6 of the JAEA New Aeronautical Engineering Course, which I am citing here.)
The conventional definition of propeller efficiency only describes the efficiency of a specific propeller in a specific case. This is not only a restricted definition with conditions, but it also raised concerns for the future as propeller performance remains unknown, preventing performance improvements or breakthroughs.

When the drag D, which balances with the thrust T, is at power P, and the flight speed (cruising speed) is Vs:
η = TVs / P = DVs / P
In the conventional definition of propeller efficiency, as drag D decreases, both thrust and power requirements are reduced without any effort to improve propeller performance, leading to a higher T/P ratio and higher efficiency.
(The term "propeller efficiency" is also questionable; in essence, this definition refers to propulsive efficiency.)
There were some who doubted this in terms of achieving higher efficiency. However, referring to the axial interference factor 'a' (where 'a' is the coefficient of the ideal state in a lossless propeller where the slipstream velocity is twice the induced velocity), it can be seen that as flight speed increases, 'a' decreases regardless of the level of propeller performance, and 'η', which was defined as propeller efficiency, converges to 1.0 for all propeller aircraft.
η = T*Vs / P = 1 / (1 + a)
(*Although there is an explanation in Fig. 3.2 on page 79 of "Hydrodynamics of Resistance and Propulsion", as a→0, lim(η) = lim(1 / 1+a) = 1.0.)

The following chart shows not only the contradictions in the conventional definition of propeller efficiency, but also the misidentification problem where propeller efficiency converges to η = 1.0 at the limit where D→0, Vs→∞, and power P→0, which balances with thrust T.

(It is the propulsive efficiency, not the propeller efficiency, that converges to 1.0)

This means that the conventional definition (index) of propeller efficiency cannot represent propeller performance, and the calculation of propulsive efficiency, which is affected by aerodynamic design, is being confused with propeller efficiency.
Incidentally, if we compare the conventional propeller efficiency formula, ηp = 1 / {1 + (v/2Vs)} ... (1), with the propulsive efficiency formula derived from modern momentum theory (Introduction to Helicopters, Eq. 3.6), ηd = 1 / {1 + (u/Vs)} ... (2), the 'v' in formula (1) is the slipstream velocity, but the important induced velocity 'u' is missing.
In the propulsive efficiency formula of (2), the slipstream velocity 'v' is considered as the acceleration by which the propeller increases the induced velocity 'u', so 'v' exists, but it does not affect the formula.
Depending on Vs, the propeller efficiency formula (1) becomes the same as the propulsive efficiency, causing confusion.
Furthermore, if we assume that half of the slipstream velocity 'v' is equal to the induced velocity 'u' (*), the propeller efficiency formula (1) is exactly the same as the propulsive efficiency formula (2).
Formula (1) cannot be called propeller efficiency to indicate propeller performance, and this is an error.
When considering a superior propeller, I believe we must rely on different ideas from the past, such as the height of the T/P ratio (including the optimal number of blades problem for horsepower absorption) and the "grasping" of air.

(*) According to Eq. 3.8 of "Hydrodynamics of Resistance and Propulsion", the axial interference factor 'a' is defined as:
a = u/Vs = v/2Vs

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