2.2.1 Power Conversion and Power Coefficient
From the expression for kinetic energy in flowing air, the power contained in the wind passing an area A with the wind velocity v1 is:
𝑃𝑃𝑊𝑊 = 𝜌𝜌2𝐴𝐴𝐴𝐴13 (2-1) where ρ is air density, depending on air pressure and moisture. For practical calculations it may be assumed ρ ≈ 1.225kg/m3. The air streams in axial
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direction through the wind turbine, of which A is the swept area. The useful mechanical power obtainment is expressed by means of the power coefficient Cp:
𝑃𝑃 = 𝐶𝐶𝑝𝑝𝜌𝜌2𝐴𝐴𝐴𝐴13 (2-2) Supposing the wind velocity of airflow is homogeneous, the value ahead of the turbine plane is v1. After passing through the retardation of wind turbine, suffers retardation due to the power conversion to a speed v3 far behind the wind turbine. 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 has shown by a simple calculation that the maximum useful power can be obtained for v3/v1 = 1/3 in 1920; where the power coefficient Cp = 16/27 ≈ 0.593. In reality, wind turbine displays the maximum values Cp, max = 0.4 ~ 0.5 due to losses, such as profile loss, tip loss and loss due to wake rotation. In order to determine the mechanical power available for the load machine, such as electrical generator or pump, Eq.
(2-2) has to be multiplied with 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:
𝜆𝜆 = 𝑢𝑢 𝐴𝐴� =1 𝐷𝐷2 ∙𝑣𝑣𝜔𝜔
1 (2-3) where D is the outer turbine diameter and ω 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
𝐶𝐶𝑝𝑝 =𝑃𝑃𝑃𝑃
𝑊𝑊 = 1𝑇𝑇⋅𝜔𝜔
2𝜌𝜌𝐴𝐴𝑣𝑣13 (2-4)
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Fig. 2.2 shows typical characteristics Cp (λ) for different types of wind rotors. Besides 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 tip speed ratios.
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. The momentum theory by Betz indicates the physically based, ideal limit value for the extraction of mechanical power from free-stream airflow without considering the design of the energy converter. However, the power which can be achieved under real conditions cannot be independent of the characteristics of the energy converter. The fundamental difference for various rotor blade designs depends on what kind aerodynamic force is utilized to produce the mechanical power. The first fundamental difference which considerably influences the actual power depends on which aerodynamic forces are utilized for producing mechanical power. As the wind rotor blades are subjected to airflow, the generated aerodynamic drag is parallel 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
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surface area A, the wind velocity u and the aerodynamic drag coefficient CD as 𝐷𝐷 = 𝐶𝐶𝐷𝐷12𝜌𝜌𝐴𝐴𝐴𝐴𝑟𝑟2 = 𝐶𝐶𝐷𝐷12𝜌𝜌𝐴𝐴(𝐴𝐴 − 𝑢𝑢)2 (2-5) The relative velocity, vr = v – u, which effectively impinges on the drag area, is determined by wind velocity v and blade rotating speed u = ωRM, in which RM is the mean radius. Then the resultant power is
𝑃𝑃 = 𝐷𝐷 ⋅ 𝑢𝑢 = 12𝜌𝜌𝐴𝐴𝐴𝐴3�𝐶𝐶𝐷𝐷�1 −𝑢𝑢𝑣𝑣�2 𝑢𝑢𝑣𝑣� =12𝜌𝜌𝐴𝐴𝐴𝐴3𝐶𝐶𝑝𝑝 (2-6) Analog to the approach 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
𝐶𝐶𝑝𝑝,𝑚𝑚𝑚𝑚𝑚𝑚 =274 𝐶𝐶𝐷𝐷 (2-7) It is taken into account that the aerodynamic drag coefficient CD of a concave surface curved against the wind direction can hardly exceed a value of 1.3. Thus, the maximum power coefficient Cp, max of a general drag-type wind rotor becomes about 0.2, only one third of Betz’s ideal Cp value of 0.593.
2.2.2.2 Lift Devices
Utilization of aerodynamic lift on wind rotor blade can achieve much higher power coefficients. 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
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𝐿𝐿 = 𝐶𝐶𝐿𝐿12𝜌𝜌𝐴𝐴𝐴𝐴2 (2-8) 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 suited for this purpose is with a horizontal rotational axis. The aerodynamic force created is 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-to-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, it is no longer possible to calculate the power coefficients of lift-type wind rotors quantitatively with the aid of elementary physical relationships alone.