Formulario y conceptos de Energa ElicaA wind turbine is a device for extracting kinetic energy from the wind. By removing some of its kinetic energy the wind must slow down but only that mass of air which passes through the rotor disc is affected.

3.2 The actuator disc conceptEnergy extraction processThe mass of air which passes through a given cross-section of the stream-tube in a unit length of time is AU, where is the air density, A is the cross-sectional area and U is the flow velocity. The mass flow rate must be the same everywhere along the stream-tube and so

The symbol refers to conditions far upstream, D refers to conditions at the disc and W refers to conditions in the far wake.The actuator disc induces a velocity variation which must be superimposed on the free stream velocity. The stream-wise component of this induced flow at the disc is given by aU, where a is called the axial flow induction factor, or the inflow factor. At the disc, therefore, the net stream-wise velocity is

3.2.1 Simple momentum theoryThe air that passes through the disc undergoes an overall change in velocity, UUW and a rate of change of momentum equal to the overall change of velocity times the mass flow rate:

The force causing this change of momentum comes entirely from the pressure difference across the actuator disc:

To obtain the pressure difference (pD+pD) Bernoullis equation is applied separately to the upstream and downstream sections of the stream-tube:

Upstream, therefore, we have

Assuming the flow to be incompressible (=D) and horizontal (h=hD) then,

Similarly, downstream,

Subtracting these equations we obtain

Equation 3.4 then gives

And so,

That is, half the axial speed loss in the stream-tube takes place upstream of the actuator disc and half downstream.3.2.2 Power coefficientThe force on the air becomes, from Equation 3.4

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

A power coefficient is then defined as

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

3.2.3 The Lanchester-Betz limitThe maximum value of CP occurs when

That gives a value of a= 1/3Hence,

CP could, perhaps, more fairly be defined as:

but this not the accepted definition of CP.3.2.4 The thrust coefficientThe force on the actuator disc caused by the pressure drop, given by Equation 3.9, can also be non-dimensionalised to give acoefficient of thrust CT

A problem arises for values of a because the wake velocity, given by (12a) 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 3.6).The variation of power coefficient and thrust coefficient with a is shown in Figure 3.3.3.3 Rotor disc theoryThe manner in which the extracted energy is converted into usable energy depends upon the particular turbine design. Most wind energy converters employ a rotor with a number of blades rotating with an angular velocity about an axis normal to the rotor plane and parallel to the wind direction.3.3.1 Wake rotationThe exertion of a torque on the rotor disc by the air passing through requires an equal and opposite torque to be imposed upon the air. The the air to rotates in a direction opposite to that of the rotor, gains angular momentum and so in the wake of the rotor disc the air particles have a velocity component in a direction which 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.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 2ra. At the middle of the disc thickness, a radial distance r from the axis of rotation, the induced tangential velocity is ra. Because it is produced in reaction to the torque the tangential velocity is opposed to the motion of the rotor.An abrupt acquisition of tangential velocity cannot occur in practice and must be gradual. Figure 3.5 shows the flow accelerating in the tangential direction as it is squeezed between the blades: the separation of the blades has been reduced for effect, but it is the increasing solid blockage that the blades present to the flow close to the blade roots that causes high values of tangential velocity.

3.3.2 Angular momentum theoryThe 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 change of tangential velocity radius

where AD is taken as being the area of an annular ring.The driving toque on the rotor shaft is also Q 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 Equation 3.10 in Section 3.2.2

Hence,

And

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

The area of the ring is AD=2rr, therefore, the incremental shaft power is, fromEquation 3.17,

The term in brackets represents the power flux through the annulus, the term outside the brackets, therefore, is the efficiency of the blade element in capturing the power.Blade element efficiency

In terms of power coefficient

Where =r/R.Knowing how a and a vary radially, Equation 3.20 can be integrated to determine the overall power coefficient for the disc for a given tip speed ratio .The lift forces on the blades forming the rotor disc are normal to the resultant velocity relative to the blades and so no work is done on or by the fluid. Therefore, Bernoullis theorem can be applied to the flow across the disc, relative to the disc spinning at angular velocity , to give

Consequently,

The pressure drop across the disc clearly has two sources. The first source

is shown to be, from Equation 3.18, the same as that given by Equation 3.9 in the simple momentum theory in which rotation plays no part. The second source is

pd2 can be shown to be caused by a radial, static pressure gradient in the rotating wake that balances the centrifugal force on the rotating fluid. The kinetic energy per unit volume of the rotating fluid in the wake is also equal to the drop in static pressure of Equation 3.22; and so the two are in balance and there is no loss of available kinetic energy. However, does the pressure drop of Equation 3.22 cause an additional thrust on the rotor disc? No, because whereas the pressure drop of Equation 3.21 applies only to points immediately downstream of the disc, with a positive axial pressure gradient between the disc and the fully developed wake (Figure 3.2), the pressure drop of Equation 3.22 applies to the entire wake and there is no pressure gradient causing a further slowing down of the axial velocity. If there is no change of axial momentum of the fluid there can be no corresponding force on the rotor because that would violate Newtons third law of motion! There appears to be an anomaly and it is probable that this is caused by the concept of an actuator disc itself because an actuator disc is not physically realisable.However, the concept does produce useful results and is not to be dismissed.

3.3.3 Maximum powerThe values of a and a which will provide the maximum possible efficiency can be determined by differentiating Equation 3.19 by either factor and putting the result equal to zero.Whence

From Equation 3.18

Giving

The combination of Equations 3.18 and 3.21 gives the required values of a and a which maximise the incremental power coefficient.

And

The axial flow induction for maximum power extraction is the same as for the non-rotating wake case, that is, a=1/3 and is uniform over the entire disc. On the other hand a varies with radial position.From Equation 3.20 the maximum power is

Substituting for the expressions in Equation 3.23

Which is precisely the same as for the non-rotating wake case.

3.4 Vortex cylinder model of the actuator disc3.4.1 Introduction