**3.1. The ideal bare wind turbine model**

The one-dimensional bare wind turbine model without losses has been treated extensively and is well documented in textbooks [15]. The velocity of the air crossing the wind turbine velocity is assumed to be a fraction of the upstream air velocity. This fraction is introduced as a parameter, *a*, which expresses the ratio between the latter and the former. Parameter *a* is assigned a value in the range of numbers between zero and one. It is important to note that the physical range of this parameter is limited, for example, 0<a<0.5, otherwise the velocity at the downstream becomes negative. To help clarify this point, calculations were performed covering the full range of parameter *a*. Equations (6) and (8) give the power coefficient and the thrust coefficient, respectively. Figure 5 shows these coefficients as functions of parame‐ ter *a* (a very well-known result in the professional literature in the field).

**Figure 5.** Power and thrust coefficients for the ideal bare wind turbine as a function of the parameter a (the ratio between the air velocity crossing the turbine blades and the upstream velocity of the air). The plot is reproduced simi‐ lar to what is known in the literature, but highlighting the physical region (0<*a*<0.5) with thicker black color and the non-physical region (0.5<a<1) with thinner red color.

For explicit presentation, the physical range of parameter *a* (up to the value of 1/2) was drawn in a thick black color, while the rest of the plot was prepared using a thinner red col‐ or. It is clear from the figure that the coefficients vanish at the zero and one values of the parameter. In between, the maximum thrust coefficient (with a value of unity) occurs at the value of *a*=1/2. On the other hand, the maximum value of the power coefficient occurs at the value of *a*=1/3, for which the Betz limit is given (*CP* = *B* = 16/27). If we plot the power coeffi‐ cient as a function of the thrust coefficient, as Figure 6 shows, a loop shape would be pro‐ duced. Again, the physical range was highlighted using a thick black color.

**Figure 6.** Power coefficient for the ideal bare wind turbine as a function of the thrust coefficient. The plot has been extended to include the non-physical region as was done in Figure 5 for reasons of consistency.

Two important points must be noted on such a plot: the maximum power coefficient (the Betz limit) for which the thrust coefficient receives a value of 8/9; and the maximum thrust (with the value of unity), for which the power coefficient gets a value of 1/2. These relations can be checked using equation (9), or more explicitly by using equation (17).

#### **3.2. The bare wind turbine with losses**

**3. Numerical considerations**

18 Advances in Wind Power

**3.1. The ideal bare wind turbine model**

In this section sample, plots of the results are considered.

ter *a* (a very well-known result in the professional literature in the field).

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

**Power and thrust coefficients.**

non-physical region (0.5<a<1) with thinner red color.

The one-dimensional bare wind turbine model without losses has been treated extensively and is well documented in textbooks [15]. The velocity of the air crossing the wind turbine velocity is assumed to be a fraction of the upstream air velocity. This fraction is introduced as a parameter, *a*, which expresses the ratio between the latter and the former. Parameter *a* is assigned a value in the range of numbers between zero and one. It is important to note that the physical range of this parameter is limited, for example, 0<a<0.5, otherwise the velocity at the downstream becomes negative. To help clarify this point, calculations were performed covering the full range of parameter *a*. Equations (6) and (8) give the power coefficient and the thrust coefficient, respectively. Figure 5 shows these coefficients as functions of parame‐

> **Power and thrust coefficients for the bare wind turbine with zero loss coefficeint vs. the parameter a**

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

**Figure 5.** Power and thrust coefficients for the ideal bare wind turbine as a function of the parameter a (the ratio between the air velocity crossing the turbine blades and the upstream velocity of the air). The plot is reproduced simi‐ lar to what is known in the literature, but highlighting the physical region (0<*a*<0.5) with thicker black color and the

For explicit presentation, the physical range of parameter *a* (up to the value of 1/2) was drawn in a thick black color, while the rest of the plot was prepared using a thinner red col‐

**0<a <0.5 0.5<a<1**

> In this section, sample plots are given to demonstrate the effect of the losses as modeled in section 2.2. The losses are due to friction and are modeled as proportional to the velocity of the square of the velocity of the air flowing through the wind turbine. The plots are pre‐ pared for three different values of the non-dimensional loss coefficient C\* loss: 0, 0.05, and 0.1. Figure 7 shows a plot of the power coefficient and of the thrust coefficient as a function of parameter *a,* covering the physical range while accounting for losses.

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

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

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