**3.3. The shrouded wind turbine**

**Power and thrust coefficients of a bare wind turbine for different values of loss coefficient versus a**

**0 0.1 0.2 0.3 0.4 0.5 a**

**Figure 7.** Power and thrust coefficients for the bare wind turbine as a function of parameter *a*, accounting for friction‐

It is clear from the figure that there is degradation in both coefficients in the order of a few percentage points. The two points (maximum power coefficient and maximum thrust coeffi‐

> **Power coefficient of a bare wind turbine for different values of loss coefficient versus thrust coefficient**

> > **Loss coefficient**

**0 0.05 0.1**

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Thrust coefficient**

**Figure 8.** Power coefficient for the bare wind turbine as a function of the thrust coefficient, accounting for frictional

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1**

cient) can be better visualized as illustrated by Figure 8.

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7**

**Power coefficient**

**Power and thrust coefficients.**

al losses, with different values.

20 Advances in Wind Power

losses, with different values.

The shrouded wind turbine was analyzed based on the extended Bernoulli equation, while accounting for frictional losses in the same manner as was done for the bare wind turbine. The increased air mass flow due to the larger drop in pressure was modeled as proportional to the kinetic energy difference, using the coefficient *CF* (pressure drop coefficient—see equation (17)). As can be seen from Figure 9 and Figure 10 (equations (18) and (19) respec‐ tively), the power coefficient and the thrust coefficient are increased proportionally to the pressure coefficient, which is in agreement with the findings in the literature.

**Figure 9.** Power coefficient for the shrouded wind turbine as a function of parameter *a*, accounting for frictional loss‐ es and for augmentation coefficient *CF*.

The maximum power coefficient and the maximum thrust points are illustrated in Figure 11.

By consulting Figure 11, one can observe that both maximum points are degraded by an in‐ creasing loss coefficient.

#### **3.4. Efficiency of the wind turbine**

As was considered by Betz, the power coefficient as originally defined agrees with the defi‐ nition of efficiency for a device that extracts work from a given amount of energy. Thus, for the bare wind turbine, the maximum efficiency that could be extracted is actually given by the Betz limit. The effect of friction on wind turbine efficiency, as was expressed through the power coefficient, decreases with friction. A similar observation could be stated for the shrouded wind turbine if we use the definition as given by equation (22). Accordingly, the Betz limit is exceeded, that is, the shrouded wind turbine produces more power, but the amount of energy extracted per unit of volume with a shroud is the same as for an ordinary bare wind turbine. These results were found to be in agreement with results observed in [Van Bussel, 2007]. The efficiency of maximum power output that was observed by the finite time analysis was approximately 36%, which is comparable to experimental findings [14]. When compared to heat engines, the efficiency of the wind turbine could be expressed in terms of the Betz number by using equation (23). If the Betz number is substituted, the effi‐ ciency could be approximated as 47%, but if the other factors are taken into account, the practical efficiency could reach much lower values.

**Figure 10.** Thrust coefficient for the shrouded wind turbine as a function of parameter *a,* accounting for frictional losses and for augmentation coefficient *CF,* with different values.
