Wind Energy Handbook. Michael Barton Graham
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 2rΩa′. In the plane of the disc the tangential velocity is rΩa′ (see also Figure 3.10 and 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 3.5 shows, 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.
3.3.2 Angular momentum theory
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
where δAD is 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
Ωr is the tangential velocity of the spinning annular ring, and so
Thus
The area of the ring is δAD = 2πrδr, therefore the incremental shaft power is, from Eq. (3.17),
The first term in brackets represents the power flux through the annulus in the absence of any rotor action; the term outside these brackets, therefore, is the efficiency of the blade element in capturing that power.
Blade element efficiency is
in terms of power coefficient