*2.2. Ideal Blade Shape*

A tentative blade design could be determined using BEM theory. A detailed explanation of the equations and their derivation can be found in [9], as only the aspects relative to

blade design are briefly presented herein. By manipulating BEM equations to express the power coefficient for each radial section without considering drag [9], one gets:

$$\mathcal{C}\_P(r) = \frac{8}{\lambda^2} \lambda\_r^3 a'(1-a) \tag{1}$$

The equation can be rearranged and written in terms of the flow angle *ϕ*:

$$\mathbb{C}p(r,\lambda,\varphi) = \frac{8}{\lambda^2} \sin^2 \varphi (\cos \varphi - \lambda\_I \sin \varphi)(\sin \varphi + \lambda\_I \cos \varphi) \lambda\_r^2 \tag{2}$$

The flow angle distribution along the span that maximizes the power coefficient (*Cp*) can be found by setting the partial derivative of Equation (2) equal to zero:

$$\frac{\partial}{\partial \boldsymbol{\varphi}} \left( \mathbb{C}\_{\mathcal{P}} (\boldsymbol{r}, \boldsymbol{\lambda}, \boldsymbol{\varphi}) \right) = 0 \tag{3}$$

Solving Equation (3), one then gets:

$$
\varphi = \frac{2}{3} \tan^{-1} \left( \frac{r}{R} \lambda \right) \tag{4}
$$

The local blade twist can be calculated based on the flow angle as:

$$
\gamma = \left. \varrho - \theta - \mathfrak{a}\_{des} \right. \tag{5}
$$

where *θ* is the blade pitch angle and *αdes* is the local design angle of attack. The local blade chord can also be expressed as [9]:

$$z = \frac{8\pi r}{BC\_l} (1 - \cos\varphi) \tag{6}$$

The twist and chord distributions obtained from Equations (5) and (6) do not account for drag and tip losses, and so the design angle of attack *αdes* should be selected as the angle of attack that maximizes the glide ratio of the airfoil employed at the selected local radius. As shown in [10], when the airfoil glide ratio exceeds 40, the assumption of neglecting drag can be reasonably assumed. Moreover, the proposed design method determines the ideal blade shape in design conditions, with set tip–speed ratio (TSR) and pitch angle. It is then apparent that the design conditions should be chosen carefully. Rather than choosing the rotor and wind speed at rated conditions, a sounder choice would be to choose operating conditions based on the design wind speed distribution. In the present test case, a design wind speed of 8.5 m/s was chosen as the mean wind speed of a class IIA wind speed distribution. Another good choice could be the mode of the wind speed distribution. When designing a fixed-speed wind turbine, the mode of the wind speed distribution (i.e., the most frequent wind speed) should be chosen as a design point in order to ensure the turbine is operating at its design TSR most of the time. A variable speed wind turbine, on the other hand, can vary rotor speed to maintain a nominal TSR as wind speed varies, and the mean wind speed is therefore also a good choice because it ensures that the rotor speed is closer to the nominal value at the design point. The design TSR must also be chosen carefully, as this will contribute to determining rotor speed. Modern rotors generally operate between TSRs of 4 and 10 [25]. Higher tip–speed ratios decrease blade solidity and increase aerodynamic noise [26,27]. Therefore, based on these considerations and similar existing turbine designs [14,28–30], a medium-low TSR of 5.7 was selected here.

#### *2.3. Airfoil Families*

In order to obtain smooth chord and twist distributions, airfoils from the same family must be used along the entire blade. Several airfoil families have been designed specifically for wind turbines over the years by laboratories, scientists, and technical institutions such

as NREL (USA), Risø (Denmark), and Delft (The Netherlands). [18,31–33]. The selection of the required airfoils plays a crucial role in the aerodynamic design process. The shape of the selected airfoils is a compromise between performance, regulation characteristics (especially important in stall-regulated wind turbines), and structural stiffness. The mid and outer sections of a wind turbine blade are typically optimized for high aerodynamic performance, while the inner sections are designed to provide the required structural integrity and stiffness for the blade. Suggesting a family of airfoils is definitely not an easy task, since any of them have specific benefits and drawbacks that may be relevant to each different application; also, companies sometimes are willing to design proprietary airfoils tailored for their machine. However, the scope of the present work was to show how, even in case where one selects a very well-known family of "standard" airfoils, effective turbine designs can be achieved. In detail, research into airfoil families that would be suitable for a 50 kW wind turbine with a rotor diameter of 16 m led to the selection of two different airfoil families belonging to the S800 group developed and tested by NREL [18] for medium-size turbines rated at 20–150 kW with blades from 5 to 10 m in length, which was the size category of our interest.

The first family considered (Figure 1a) [18,34], with thin tip airfoils, was designed in 1987 and includes the S805A, S806A, S807, and S808 airfoils. This airfoil family was designed to have a low tip maximum lift coefficient (*Clmax*) (1.0) for a Reynolds number just over 1\*106, and it is suitable for stall-regulated blades. The "A" designation stands for an improved version of an airfoil, based on wind-tunnel test results for a similar airfoil.

**Figure 1.** (**a**) Thin-airfoil family for medium blades (low tip *Clmax*), (**b**) thick-airfoil family for medium blades (low tip *Clmax*), (**c**) thick-airfoil family for large blades (low tip *Clmax*).

The second family (Figure 1b) [18,35], having thick tip airfoils, was designed in 1993 and consists of the S819, S820, and S821 airfoils. This family was designed to have performance characteristics similar to the previous family. The greater tip-region thickness helps accommodate overspeed-control mechanisms for stall-regulated rotors at the expense of a slightly higher drag [36,37]. Though these mechanisms are not used in modern turbines and were thus not included in this case study, the increased thickness is structurally beneficial. The S821 blade-root airfoil was designed to have restrained maximum lift coefficients, and have low profile-drag coefficients, and to be as insensitive as possible to roughness.

The low design lift coefficient of these airfoil families is indeed a design trait [33,36,38]. Specifically, on an SWT, the operating Reynolds number must be as high as possible to achieve the best aerodynamic performance. Based on Equation (6), decreasing the design lift coefficient implies an increase of the chord size required to reach a certain performance level. In turn, this increases the operating Reynolds number, which helps to lower drag and increase the glide ratio of the blade [10].

Finally, the S812, S813, and S814 (Figure 1c) [18,39–41] airfoil family was designed for large rotors rated at 100–400 kW with blades 10–15 m in length. Though this family of airfoils did not seem to fit the specification of the test case, it has been used successfully on the Atlantic Orient AOC 15/50 three-bladed wind turbine. The designation 15/50 refers to the 15 m diameter rotor and its rated output of 50 kW [18,28,42]. This rated output is achieved at 12 m/s by the 50 Hz version and at 11.3 m/s by the 60 Hz version. This airfoil family was therefore also taken into consideration.

#### *2.4. Preliminary Performance Curves*

Following the steps to determine the blade design presented in Section 2.1, the first steady analyses on OpenFAST for the three airfoil families were carried out. The focus in this phase was put on the stall-regulated turbine, as in this case, the aerodynamic design also influenced the regulation characteristics. As discussed previously, the design point was chosen to be 8.5 m/s, which is the mean wind speed of a class IIA (see Table 1).


**Table 1.** Turbine specifications. IEC: International Electrotechnical Commission.

Figure 2 shows the aerodynamic power produced as a function of the wind speed. These power curves were obtained without stall delay correction, and—only afterward once the most promising design was chosen—were the polars 3D-corrected and further refinements done (see Section 2.5). The regulation method for this preliminary design was variable speed stall regulation. In particular, the powers produced at 12 m/s were as follows: 50.4 kW for the S819–21 family, 52.6 kW for the S812–14 family, and 52.1 for the S805–8 family.

**Figure 2.** Turbine curves of generator power from the steady simulations in OpenFAST. Maximum rotor speed was set to 60 rpm.

Upon examination of the performance comparison in terms of power output, one could notice that the S821, S819, and S820 family was preferable to the others. In fact, it allowed us to reach the set power target of 50 kW at 12 m/s with good stall regulation, and it generated more power for below-rated wind speeds than the S805, S806, S807, and S808 family. The main preliminary characteristics of the turbine after the first preliminary design phase are shown in Table 1.
