@tallakt , I am not sure to understand these two comments. So, as I can, I will quote something I understood before forget it, then you can indicate if that matches your calculation and definition.
https://www.researchgate.net/publication/324565013_Innovation_in_Wind_Turbine_Design (author Peter Jamieson), pages 127-129:
The secondary rotor concept (Figure 6.16) also has potential for very large wind turbine
systems where the conventional drive train solutions will be extremely heavy and expensive
on account of low shaft speed and high rated torque. This concept involves rotors which are carried by the wind turbine blades of the main rotor. The secondary rotors thereby
experience a much higher apparent wind speed than the ambient wind speed and can be of
comparatively small diameter, high speed, low torque and low weight. The idea is quite old,
possibly preceding patent applications of the 1980s and 1990s (Watson [19], St-Germain
[20] and Jack [21]). It has sparked a few misconceptions. One is that the Betz limit will
apply twice to the overall power conversion but, more curiously, another recent one is that
the secondary rotor can achieve a power coefficient of unity [22]. Discounting any ground
related effect, the power produced by a rotor must be exactly the same whether a wind
turbine system is stationary in an ambient wind field of velocity, V , or is mounted on
a vehicle (such as a moving wind turbine blade) and transported at velocity, V , through
still air. In either case the relative axial fluid velocity local to the rotor disc is not V but
V (1 − a) where a is the axial induction factor at the rotor plane.
A brief analysis of the secondary rotor concept follows. It is clearly the thrust (and
not power) of the secondary rotor that provides reaction torque to extract power from the
primary rotor. It appears that extraction of power from the primary rotor is most efficient
when the axial induction of the secondary rotor is small. An interesting trade-off then arises
between having larger and therefore more expensive, lightly loaded secondary rotors to
improve efficiency and hence reduce cost of the major primary rotor system.
Notation:
Air density ρ
Primary rotor radius R
Primary rotor angular speed ω
Primary rotor blade number N
Secondary rotor radius r
Secondary rotor power coefficient Cp
Secondary rotor power coefficient Ct
Secondary rotor axial induction aThe primary rotor produces power P which is as usual subject to the Betz limit. In terms of
the thrust reaction T of each of the N secondary rotors and assuming for present convenience
that they are mounted at the tip of each blade,
P = NTRω
T = 0.5ρ (ωR)² π r² Ct neglecting ambient wind speed compared to tip speed
P = N0.5ρ (ωR)² π r²Ct Rω
P = N0.5ρω³ R³ π r² CtThe power extracted by the secondary rotors is
Pe = 0.5ρN (ωR)³ π r² Cp
Thus Pe / P = Cp / Ct
Considering the ideal Betz model:
If
Cp / Ct = (4a (1 − a)²) / (4a (1 − a)) = (1 − a).
If the secondary rotor is optimised in its own right, then the usual choice of a = 1/3
applies and the overall limit is 16 / 27 (1 − 1/3) = 0.395. This exceeds Betz squared by a little as
(16 / 27)² = 0.351.
However it is much better to trade reduced specific loading, Ct , on the secondary rotors
at the cost of making them a little bigger. In a specific design study a = 0.2 was about
optimum. Hence the ratio Pe/P is (1 − 0.2) = 0.8 and the overall limit is 16/27× 0.8 =
0.474. The power coefficient of the secondary rotors is reduced to a theoretical limit of 4 ×
0.2 (1 − 0.2)² = 0.512 and the secondary rotors are somewhat larger and more expensive
but this can be a very worthwhile trade off.
I quote again the main part of his analysis:
It is clearly the thrust (and not power) of the secondary rotor that provides reaction torque to extract power from the primary rotor. It appears that extraction of power from the primary rotor is most efficient when the axial induction of the secondary rotor is small.
Apart from that, a fly-gen wing (Makani-like) does not undergo the structural issue of tip blade loading as for a wind turbine with tip rotors.