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Small Wind Energy Systems 173

where
n c is the number of cylinders
ω t is the angular speed of the turbine
3
ρ= 1 225 kg/m is the air density at sea level at 15.5 °C, according to the International Standard
.
Atmosphere
A is the area swept by the spinning cylinders (m 2 )
V is the wind velocity (m/s )
As the terms T L and T D of the Magnus turbine torques are usually very complicated mathematical
expressions, it is better to obtain them experimentally by a curve fitting procedure, that is, as a func-
tion of the rotor and the cylinders TSR, respectively, λ t and λ c . These coefficients are related to the
radius of the area swept by the cylinder r t , to the angular speed of the cylinders ω c , to the cylinder
radius r c , and to the wind speed V as
ω tt r
λ t = (7.20)
V
ω cc r
λ c = (7.21)
V

Given the mechanical power, it is necessary to subtract the cylinder friction losses during its rota-
tion in a laminar air flow and the motor losses in the cylinder drivers:

P t = P mec − P losses (7.22)
The electromechanical power losses due to air friction can be expressed by
ρπ d ω c ( r t − )
3
4
r o
P losses = 1 328 n c ⋅ (7.23)
.
−
16η elect mech Re d
where
r o is the radius of the rotor hub
Re d = ρω π 2 c r 2 µ is the Reynolds number
c ( ) 2/
µ is the air viscosity coefficient
η elect mech− is the electromechanical efficiency
These equations are quite challenging, and a practical engineer would prefer to consider experi-
mental results or other forms of approximation to establish a polynomial equation whose form and
coefficients are obtained by curve fitting. The optimal ratio dimensions of the cylinder have been
established in the literature by (r t − r o )/16. Figure 7.17 illustrates the performance for a Magnus
turbine specified in Table 7.7, where the different effects on the dimensionless power coefficients λ c
and λ t are related, respectively, to the rotation of the cylinder around its shaft and the turbine rotor
shaft. The reduced rotation of the Magnus cylinder–rotor is about two to three times lower when
compared to the blades in the conventional turbines, ensuring much less air turbulence with higher
operational safety and durability.
This is a major advantage compared to the low efficiency of other turbine types for most usual
wind velocities (5–15 m/s) due to the small lift coefficient of an ordinary blade turbine under such
conditions. It is clear that the Magnus wind turbine must be explored in a wider range of rotor and
wind velocities, varying from 2 to 35 m/s compared to a traditional blade turbine, typically limited
in a range from 3 to 25 m/s. This variable speed is well suited for the PMSG and IG characteristics
and, certainly, when associated with a hill-climbing control (HCC) for the optimal speed of the
turbine rotor and cylinders can be achieved for maximum generator power production. The power
coefficient of the ordinary wind turbines drops rapidly to zero at about wind velocity on the order of