**3.2 Lift, drag, and dimensionless parameters**

The airflow over an airfoil produces a force distribution on the surface. The flow velocity increases over the convex surface resulting in lower average pressure on the "suction" side of the airfoil compared to its concave "pressure" side. Meanwhile, the viscous friction between the air and the surface of the airfoil slows the airflow to a certain point near the surface [6].

There are three forces of vital importance for aerodynamic analysis as seen in **Figure 2**, which are:


**Figure 2.**

*Forces and moments in an aerodynamic section, an angle of attack; c, chord. The direction of positive forces and moments is indicated by the direction of the arrow [5].*


$$\mathbf{C}\_{L} = \frac{\mathbf{L}/l}{\frac{1}{2}\rho U^2 c} \tag{3}$$

$$\mathbf{C}\_{D} = \frac{D/l}{\frac{1}{2}\rho U^{2}c} \tag{4}$$

where c is the chord of the blade. The lift and drag coefficients are expressed as a function of the angle of attack (γ). **Figure 3** shows the typical coefficients of wind turbine blades. Note that the CL coefficient grows approximately linearly with the angle of attack, while CD remains at a low value. For angles of attack greater than 13°, CL decreases while CD grows rapidly, and the blades go into loss.

The power output is produced through the lift force generated on the airfoil surface. As the turbine rotates, the airfoils encounter an incident wind velocity that is the vector summation of the surrounding flow velocity and the turbine rotation **Figure 4** [8].

**Figure 3.** *Coefficients of lift and drag of a blade [7].*

*Vertical Axis Wind Turbine Design and Installation at Chicamocha Canyon DOI: http://dx.doi.org/10.5772/intechopen.99374*

#### **Figure 4.**

*Vertical axis wind turbine principle of operation. α is the relative angle of attack of the incident flow velocity U incident, and e is the angle of rotation [8].*

#### **3.3 Reynolds number (Re)**

Defines the characteristics of flow conditions Eq. (5):

$$Re = \frac{\text{UL}}{v} = \frac{\rho \text{UL}}{\mu} = \frac{\text{Inercial Force}}{\text{Viscous Force}} \tag{5}$$

where μ is the fluid viscosity, *U* and *L* are the velocity and length that characterize the flow scale. These can be the inflow velocity of the flow, *Uwind* and the chord length of an airfoil. In addition to the Reynolds number of the flow conditions, the Reynolds number based on the chord *c* is also important.

Brusca et al. [9], defined the Reynolds number based on the chord ( *Rec*), and can be expressed as follow the Eq. (6)

$$Re\_{\varepsilon} = \frac{cw}{v} \tag{6}$$

where *c* is the chord, *w* is the air's relative velocity to the aerodynamic surface and *v* is the air's kinematic viscosity. The *c* can be expressed as a function of the solidity (σ) of the turbine Eq. (7):

$$
\sigma = \frac{\sigma C p\_{\text{max}}}{Nb} R \tag{7}
$$

where σ is the solidity, *Cp* is the Power Coefficient, *Nb* is the number of blades and *R* is the turbine's radio. Therefore, *Rec* is directly proportional to σ and *Cp* as follow Eq. (7):

$$Re\_{\varepsilon} = \frac{\sigma C p\_{\text{max}} R w}{N b v} \tag{8}$$

The Reynolds number strongly influences the power coefficient of a vertical-axis wind turbine. Furthermore, it changes as the main dimensions of the turbine rotor change. Increasing rotor diameter rises the Reynolds number of the blade.

#### **3.4 Power coefficient (***Cp***)**

The turbine performance is given by the power coefficient *Cp*. This coefficient represents the energy produced by the turbine as part of the total wind energy that passes through the swept area. Claessens [3], the *Cp* is represented as follow the Eq. (9):

$$C\_p = \frac{P\_T}{P\_{wind}} = \frac{P\_T}{\frac{1}{2}\rho V^3 A} \tag{9}$$

where *PT* are the total energy, ρ is the air's density, *V* is the velocity of the wind and *A* is the swept area for the turbine. Hansen et al. [10] express *PT* as it shown the Eq. (10):

$$P\_T = \mathbf{M}.\tag{10}$$

where *M* is the instantaneous momentum and *ω* is the angular velocity. Another parameter is the instantaneous moment coefficient (*Cm*) indicating the torque generated by the blades in the Eq. (11):

$$C\_m = \frac{M}{\frac{1}{2}\rho V^2 AR} \tag{11}$$

#### **3.5 Solidity and tip speed ratio**

The solidity and Tip Speed Ratio of the turbine are directly related with the *Cp* as will be seen in this section, therefore, they are crucial in the design of VAWT. With the correct relation of these, it can obtain the maximum *Cp*. These parameters are shown below.

#### *3.5.1 The solidity of the turbine (σ)*

The solidity of the turbine (σ) it can see in Eq. (12), is defined as the developed surface area of all blades divided by the swept area [11].

$$
\sigma = \frac{Nc}{R} \tag{12}
$$

σ has a strong influence on VAWT performance. High solidity machines reach optimum efficiency at a low Tip Speed Ratio (λ) and efficiency drops away quickly on either side of this optimum [12].

A low solidity results in less total blade area, therefore, the blade is lighter. This benefits wind turbine performance as higher rotation speeds can be reached [6].

Paraschivoiu [11] establishes that a maximum *Cp* value rise a pick in a range of σ between 0.3 and 0.4, and then, it decreases. This peak value is not higher than *Cp* in the proposed solidity of 0.2. This statement is closely linked to the result obtained by [13], which proposes through its computational tool an optimal value of σ between 0.25 and 0.5.

In **Figure 5**, it can observe when the solidity is increased from 0.05 to 0.2, the static torque coefficient will increase by a factor of approximately 4 for an H-Darrius wind turbine. Therefore, for a high solidity, the turbine has a selfstarting capability, because it has a higher static torque coefficient than the low solidity turbines [15].

Increasing *σ* can decrease the negative *Cp* region (i.e, when the turbine is not self-starting) and even make the *Cp* values completely positive for solidity values of 0.6 or more [3]. However, the result of this is very large blades that will increase

*Vertical Axis Wind Turbine Design and Installation at Chicamocha Canyon DOI: http://dx.doi.org/10.5772/intechopen.99374*

**Figure 5.** *Solidity effect on the static torque coefficient [14].*

their manufacturing cost, so a balance must be chosen when defining the robustness of the turbine.

## **3.6 Tip speed ratio (λ or TSR)**

The speed ratio (λ) is a ratio between the tip blade speed (ω.R) and the freestream wind velocity, and this ratio is defined as following in Eq. (13):

$$
\lambda = \frac{aR}{V} \tag{13}
$$

In **Figure 6** it can see a relation between the azimuth angle (ɵ), the angle of attack (α), and the speed ratio (λ), this relation is as follow in Eq. (14):

$$a = \tan^{-1} \left[ \frac{\sin \theta}{\lambda + \cos \theta} \right] \tag{14}$$

**Figure 6.** *Forces and velocities distribution on Darrius rotor airfoil [14].*

Zouzou et al. [16] conclude in his investigation that a variable pitch VAWT has a major advantage respectively to fixed pitch VAWT in the case of high solidity rotor where the blade wake is large. That is because the pitch variation of the blade reduces flow separation and as result, the drag forces are lower. **Figure 7** shows the relationship between the drag force and the λ and the comparison between the fixed and variable pitch.

The *Cp* of a VAWT increases with an increase in the λ and reaches a peak, after which it takes a dip as larger λ are attained. **Figure 8** shows the angle of attack (α) is evaluated at different values of λ. To higher λ the value of α is smaller for a λ = 0.5 and λ = 1.5.

For VAWT λ is lower the common range is (λ = 1; λ = 5), this ranges of λ values refer to the Wind Turbine peak (*Cpmax*). **Figure 9** shows the performance of main wind turbines and some possible areas for new designs.

**Figure 7.** *The drag force of the different wind turbine configurations depends on the specific speed TSR.*

**Figure 8.** *Attack angle variation vs. azimuthal angle for two tip speed ratios of 0.5 and 1.5 at θp=0° [17].*

*Vertical Axis Wind Turbine Design and Installation at Chicamocha Canyon DOI: http://dx.doi.org/10.5772/intechopen.99374*

**Figure 9.** *Performance of main conventional wind machines and possible areas for new hybrid designs [18].*

The *Cp* can be expressed in function of the λ and the *Cm*, replacing and solving Eq. (12) in (9), the *Cp* follow the Eq. (15) as follow:

$$Cp = \frac{Cm.R.\alpha}{V} = Cm \, \*\,\lambda\tag{15}$$

According to Posa [19] there is a relation between λ and the establishment of the flow downstream of a VAWT, this is related to the optimal distance between turbines in wind farm configurations. Establishing the downstream flow of a VAWT to its far-wake behavior takes a shorter distance at higher λ values.
