Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook

2 a right-parenthesis upper U Subscript infinity"/>

That is, half the axial speed loss in the streamtube takes place upstream of the actuator disc and half downstream.

3.2.2 Power coefficient

The force on the air becomes, from Eq. (3.4),

(3.9) upper T equals left-parenthesis p Subscript upper D Superscript plus Baseline minus p Subscript upper D Superscript minus Baseline right-parenthesis upper A Subscript upper D Baseline equals 2 rho upper A Subscript upper D Baseline upper U Subscript infinity Superscript 2 Baseline a left-parenthesis 1 minus a right-parenthesis

As this force is concentrated at the actuator disc, the rate of work done by the force is TU_D and hence the power extraction from the air is given by

(3.10) italic Power equals italic upper T upper U Subscript upper D Baseline equals 2 rho upper A Subscript upper D Baseline upper U Subscript infinity Superscript 3 Baseline a left-parenthesis 1 minus a right-parenthesis squared

A power coefficient is then defined as

(3.11) upper C Subscript upper P Baseline equals StartFraction italic Power Over one half rho upper U Subscript infinity Superscript 3 Baseline upper A Subscript upper D Baseline EndFraction

where the denominator represents the power available in the air, in the absence of the actuator disc.Therefore,

(3.12) upper C Subscript upper P Baseline equals 4 a left-parenthesis 1 minus a right-parenthesis squared

3.2.3 The Betz limit

(This limit is also referred to as the Lanchester–Betz limit or the Betz–Joukowski limit).¹

The maximum value of C_P occurs when

StartFraction d upper C Subscript upper P Baseline Over italic d a EndFraction equals 4 left-parenthesis 1 minus a right-parenthesis left-parenthesis 1 minus 3 a right-parenthesis equals 0

that gives a value of a equals one third

Hence,

(3.13) upper C Subscript upper P max Baseline equals StartFraction 16 Over 27 EndFraction equals 0.593

The maximum achievable value of the power coefficient is known as the Betz limit after Albert Betz (1919), the German aerodynamicist. Frederic Lanchester (1915), a British aeronautical pioneer, worked earlier on a similar analysis and is sometimes given prior credit, and Joukowski (1920) also contributed an analysis. To date, no unducted wind turbine has been designed that is capable of exceeding the Betz limit. The limit is caused not by any deficiency in design because, as yet in our discussion, we have no design. However, because the streamtube has to expand upstream of the actuator disc, the cross‐section of the tube where the air is at the full, free‐stream velocity is smaller than the area of the disc.

The efficiency of the rotor might more properly be defined as

(3.14) StartFraction italic Power extracted Over italic Power available EndFraction equals StartStartFraction italic Power extracted OverOver StartFraction 16 Over 27 EndFraction period left-brace one half rho upper U Subscript infinity Superscript 3 Baseline upper A Subscript upper D Baseline right-brace EndEndFraction

but note that C_P is not the same as this efficiency.

3.2.4 The thrust coefficient

The force on the actuator disc caused by the pressure drop, given by Eq. (3.9), can also be non‐dimensionalised to give a coefficient of thrust C_T

(3.15) upper C Subscript upper T Baseline equals StartFraction italic Thrust Over one half rho upper U Subscript infinity Superscript 2 Baseline upper A Subscript upper D Baseline EndFraction

(3.16) upper C Subscript upper T Baseline equals 4 a left-parenthesis 1 minus a right-parenthesis

A problem arises for values of a greater-than-or-equal-to one half because the wake velocity, given by (1 − 2a)U_∞, becomes zero, or even negative: in these conditions the momentum theory, as described, no longer applies, and an empirical modification has to be made (Section 3.7).

The variation of power coefficient and thrust coefficient with a is shown in Figure 3.3. The solid lines indicate where the theory is representative and the dashed lines where it is not.

Graph depicts the variation of CP and CT with axial induction factor a.

Figure 3.3 Variation of C_P and C_T with axial induction factor a.

3.3 Rotor disc theory

The manner in which the extracted energy is converted into usable energy depends upon the particular turbine design. The most common type of wind energy converter, the horizontal axis wind turbine or HAWT, employs a rotor with a number of blades rotating with an angular velocity Ω about an axis normal to the rotor plane and parallel to the wind direction. The blades sweep out a disc and by virtue of their aerodynamic design develop a pressure difference across the disc, which, as discussed in the previous section, is responsible for the loss of axial momentum in the wake. Associated with the loss of axial momentum is a loss of energy that can be collected by, say, an electrical generator attached to the rotor shaft. As well as a thrust, the rotor experiences a torque in the direction of rotation that will oppose the torque that the generator exerts. The work done by the aerodynamic torque on the generator is converted into electrical energy. The required aerodynamic design of the rotor blades to provide a torque as well as a thrust is discussed in Section 3.5.

3.3.1 Wake rotation

The exertion of a torque on the rotor disc by the air passing through it requires an equal and opposite torque to be imposed upon the air. The consequence of the reaction torque is to cause the air to rotate in a direction opposite to that of the rotor; the air gains angular momentum, and so in the wake of the rotor disc the air particles have

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