2.2.1 Power Conversion and Power Coefficient
The wind energy of the flowing air passing through an area A with velocity v1 is:
𝑃� = ��𝐴𝑣�� (2-1) where ρ is air density, depending on air pressure and moisture. It may be assumed ρ ≈ 1.225 kg/m3 for practical calculations.
If the airflow pass through the wind turbine in axial direction that the swept area is A, The useful mechanical power obtainment is expressed by means of the power coefficient Cp:
𝑃 = 𝐶���𝐴𝑣�� (2-2) Supposing the wind velocity of airflow is homogeneous, the value before the wind turbine is v1. After passing through the retardation of wind turbine, the speed value which is well behind the wind turbine, reduce to v3. Due to the power conversion, wind velocity v1 reduce to a velocity v3, as shown in Fig. 2.1, a simplified theory could be claimed that the velocity can be represented in an average value v2, where v2= (v1+v3)/2, in the retardation where the moving blades located. On this basis, Betz in 1920 has shown by a simple calculation that the maximum useful power can be obtained for v3/v1 = 1/3; where the power coefficient Cp = 16/27 ≈ 0.593. On account of profile loss, tip loss and wake rotation loss, wind turbine displays the maximum values Cp, max within 0.4 ~ 0.5
14
in reality. In order to determine the mechanical power available for the load machine, such as electrical generator or pump, Eq. (2-2) has to take an efficiency of the drive train, taking losses in bearings, couplings and gear boxes into account.
An important parameter of wind rotor is the tip-speed ratio (TSR), λ. It is defined as a ratio of the circumferential velocity of blade tips to the wind speed:
𝜆 = 𝑢 𝑣� =� �� ∙��
� (2-3) where D is the outer turbine diameter and ω is the angular wind rotor speed.
Considering that in the rotating mechanical system, the power is the product of torque T and angular speed ω (P = T·ω), then Cp becomes
𝐶� = ��
� = ��⋅�
������ (2-4) Fig. 2.2 shows typical characteristics Cp (λ) for different types of wind rotor, includes the constant maximum value according to Betz, as well as the figure indicates a revised curve Cp by Schmitz, who takes the downstream deviation from axial air flow direction into account. The difference is notable in the region of lower TSRs.
2.2.2 Wind Rotor Blades Using Aerodynamic Drag or Lift
Extract the airflow power to mechanical power without considering design of wind rotor blades, Betz [17] showed the momentum theory with the corresponding physical based ideal limit value. However, the wind power generation unit cannot be without wind rotor blades in real conditions. The fundamental difference for various rotor blade designs depends on what kind aerodynamic force is utilized to produce the mechanical power. As the wind rotor blades are subjected to airflow, the generated aerodynamic drag is parallel
15
to the flow direction, whereas the lift is perpendicular to flow direction. The real power coefficients obtained are greatly dependent on whether aerodynamic drag or aerodynamic lift is used.
2.2.2.1 Drag Devices
The simplest type of wind energy conversion can be achieved by means of pure drag surfaces as shown in Fig. 2.3. The air impinges on the surface A with wind velocity v, and then the drag D can be calculated from the air density ρ, the surface area A, the wind velocity u and the aerodynamic drag coefficient CD as
𝐷 = 𝐶���𝜌𝐴𝑤� = 𝐶���𝜌𝐴(𝑣 − 𝑢)� (2-5) The relative velocity, w = v–u, which effectively impinges on the drag area, is determined by wind velocity v. The resultant power is
𝑃 = 𝐷 ⋅ 𝑢 = ��𝜌𝐴𝑣��𝐶��1 −���� ��� = ��𝜌𝐴𝑣�𝐶� (2-6) Resemblance to which is described in Chapter 2.2.1, it can be shown that Cp
reaches a maximum value with a velocity ratio of u/v = 1/3. The maximum value of Cp is then
𝐶�,��� =��� 𝐶� (2-8) If considerate that the aerodynamic drag coefficient of a concave surface curved against the wind direction can hardly exceed a value of 1.3. Thus, the maximum power coefficient of a general drag-type wind rotor becomes about 0.2, only one third of Betz’s ideal Cp value of 0.593.
16
2.2.2.2 Lift Devices
Utilization of aerodynamic lift on wind rotor blade can achieve power coefficients much higher. The lift blade design employs the same principle that enables airplanes to fly. As shown in Fig. 2.4, when air flows over the blade, a pressure gradient creates between the upper and the lower blade surfaces. The pressure at the lower surface is greater than upper surface. Thus, the difference of pressure produces a lift force to uplift the blade. The lift force occurred on a body by wind can be calculated from the air density ρ, acting area A, wind velocity v and aerodynamic lift coefficient CL as
𝐿 = 𝐶���𝜌𝐴𝑣� (2-9) When blades are attached to the central axis of a wind rotor, the lift force is translated into rotational motion. All of the modern wind rotor types are designed for utilizing this effect, and the best type of these kinds of wind rotor is suited with a horizontal rotational axis. The creation of aerodynamic force can be divided into a component in the direction of free-stream velocity, the drag force D, and a component perpendicular to the free-stream velocity, the lift force L. The lift force L can be further divided into a component Ltorque in the plane of rotation of the wind rotor, and a component Lthrust perpendicular to the plane of rotation.
Ltorque constitutes the driving torque of the wind rotor.
Modern airfoils, developed for aircraft wings and which are also applied in wind rotors, have an extremely favorable lift-drag ratio. It could show a qualitative utilization of how much an aerodynamic lift force uses as a driving force would have more efficiency. However, to calculate qualitatively of the power coefficients of lift-type wind rotor is no longer possible with just an aid of elementary physical relationships.
17