Michael Graham - Wind Energy Handbook

(3.14)

but note that C Pis not the same as this efficiency.

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)

(3.16)

A problem arises for values of because the wake velocity, given by (1 − 2 a ) 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.

Figure 3.3 Variation of C Pand C Twith axial induction factor a .

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.

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 a velocity component in a direction that is tangential to the rotation as well as an axial component; see Figure 3.4.

The acquisition of the tangential component of velocity by the air means an increase in its kinetic energy that is compensated for by a fall in the static pressure of the air in the wake in addition to that which is described in the previous section.

The flow entering the actuator disc has no rotational motion at all. The flow exiting the disc does have rotation, and that rotation remains constant as the fluid progresses down the wake. The transfer of rotational motion to the air takes place entirely across the thickness of the disc (see Figure 3.5). The change in tangential velocity is expressed in terms of a tangential flow induction factor a ′. Upstream of the disc the tangential velocity is zero. Immediately downstream of the disc the tangential velocity is 2 r Ω a ′. In the plane of the disc the tangential velocity is r Ω a ′(see also Figure 3.10and the associated discussion). Because it is produced in reaction to the torque, the tangential velocity is opposed to the motion of the rotor.

Figure 3.4 The trajectory of an air particle passing through the rotor disc.

Figure 3.5 Tangential velocity grows across the disc thickness.

An abrupt acquisition of tangential velocity cannot occur in practice and must be gradual. Figure 3.5shows, for example, a sector of a rotor with multiple blades. The flow accelerates in the tangential direction through the ‘actuator disc’ as it is turned between the blades by the lift forces generated by their angle of attack to the incident flow.

The tangential velocity will not be the same for all radial positions, and it may well also be that the axial induced velocity is not the same. To allow for variation of both induced velocity components, consider only an annular ring of the rotor disc that is of radius r and of radial width δr .

The increment of rotor torque acting on the annular ring will be responsible for imparting the tangential velocity component to the air, whereas the axial force acting on the ring will be responsible for the reduction in axial velocity. The whole disc comprises a multiplicity of annular rings, and each ring is assumed to act independently in imparting momentum only to the air that actually passes through the ring.

The torque on the ring will be equal to the rate of change of angular momentum of the air passing through the ring.

Thus, torque = rate of change of angular momentum

= mass flow rate through disc × change of tangential velocity × radius

(3.17)

where δA Dis taken as being the area of an annular ring.

The driving torque on the rotor shaft is also δQ , and so the increment of rotor shaft power output is

The total power extracted from the wind by slowing it down is therefore determined by the rate of change of axial momentum given by Eq. (3.10)in Section 3.2.2:

Hence

and

Ω r is the tangential velocity of the spinning annular ring, and so is called the local speed ratio . At the edge of the disc r = R and is known as the tip speed ratio .