Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 74

The calculation of torque and power developed by a rotor requires a knowledge of the flow induction factors, which are obtained by solving Eqs. (3.54a or b) and (3.55). The solution is usually carried out iteratively because the 2‐D aerofoil characteristics are non‐linear functions of the angle of attack.

To determine the complete performance characteristic of a rotor, that is, the manner in which the power coefficient varies over a wide range of tip speed ratio, requires the iterative solution.

The iterative procedure is to assume a and a^′ to be zero initially, determining ϕ, C_l, and C_d on that basis, and then to calculate new values of the flow factors using Eqs. (3.54a or b) and (3.55). The iteration is repeated until convergence is achieved.

From Eq. (3.49), the torque developed by the blade elements of spanwise length δr is

If drag, or part of the drag, has been excluded from the determination of the flow induction factors, then its effect must be introduced when the torque is calculated [see Eq. (3.49)]:

The complete rotor, therefore, develops a total torque Q:

(3.57)

The power developed by the rotor is P = QΩ

The power coefficient is, therefore,

Solving the blade element − momentum Eqs. (3.54a or b) and (3.55) for a given, suitable blade geometrical and aerodynamic design yields a series of values for the power and torque coefficients that are functions of the tip speed ratio. A typical performance curve for a modern, high‐speed wind turbine is shown in Figure 3.15.

The maximum power coefficient occurs at a tip speed ratio for which the axial flow induction factor a, which in general varies with radius, approximates most closely to the Betz limit value of . At lower tip speed ratios the axial flow induction factor can be much less than and aerofoil angles of attack are high, leading to stalled conditions. For most wind turbines stalling is more likely to occur at the blade root because, from practical constraints, the pitch angle β due to built‐in twist of a blade is not large enough in that region. At low tip speed ratios blade stalling is the cause of a significant loss of power, as demonstrated in Figure 3.15. At high tip speed ratios a is high, angles of attack are low, and drag begins to predominate. At both high and low tip speed ratios, therefore, drag is high and the general level of a is non‐optimum so the power coefficient is low. Clearly, it would be best if a turbine can be operated at all wind speeds at a tip speed ratio close to that which gives the maximum power coefficient.

Figure 3.15 Power coefficient – tip speed ratio performance curve.