Rankine Froude actuatordisk model
The simplest and oldest mathematical model which describes the wind turbine dynamics is the RankineFroude actuatordisk model. The concept was introduced by R.E. Froude in 1889 [68], after W.J.M. Rankine [231] had introduced the momentum theory. In this model the rotor is replaced by an "actuatordisk", which is a circular surface of zero thickness than can support a pressure difference, and thus decelerate the air through the disk. Physically, the disk could be approximated by a rotor with an infinite number of very thin, draggles blades rotating with a tip speed much higher than the wind velocity. The actuatordisk model is thus an approximation of a real wind turbine rotor (which has only a small number of blades). As a result the flow of the actuatordisk will be very different from that of a real rotor, which is unsteady, with a wake of discrete vorticity corresponding to the discrete loading.
The principal use of the actuatordisk model is to obtain a first estimate of the wakeinduced flow, and hence the total induced power loss. Note that the actual induced power loss will be larger than the actuatordisk result because of the nonuniform and unsteady induced velocity. The assumptions on which the RankineFroude actuatordisk theory are based are as follows:
1. Steady, homogeneous wind;
2. No obstructions to wind flow either upstream or downstream;
3. Uniform flow velocity at disk;
4. Wind flow passing through disk separable from remaining flow by welldefined streamtube (see Fig. 3.5);
5. Wind flow incompressible (i.e. air density, p, is constant);
6. No rotation of flow produced by disk.
Assumption 3 requires that the disk slows the wind equally at each radius, which is equivalent to assuming uniform thrust loading at the disk. Uniform thrust loading is, in turn, equivalent to considering an infinite number of rotor blades.
Figure 3.5: The energy extracting streamtube of a wind turbine. By removing some of the kinetic energy in the wind, the wind flow that passes through the disk plane will slow down. Assuming incompressible wind flow, the crosssectional area of the streamtube must expand in order to accomodate the slower moving air.
Figure 3.5: The energy extracting streamtube of a wind turbine. By removing some of the kinetic energy in the wind, the wind flow that passes through the disk plane will slow down. Assuming incompressible wind flow, the crosssectional area of the streamtube must expand in order to accomodate the slower moving air.
Now consider the flow diagram of Fig. 3.6 for a cylindrical control volume of crosssectional area S and note sections 0, 3, 2, and 1. Let A be the area of the rotor disk and p be the air density. Wind approaches the rotor at velocity Vw far upstream at section 0 at static pressure p0. Kinetic energy is extracted by the rotor, and the reduced velocity causes the the streamline to expand.
CONTROL VOLUME
CONTROL VOLUME
Figure 3.6: Onedimensional flow past the disk plane of an actuatordisk.
Figure 3.6: Onedimensional flow past the disk plane of an actuatordisk.
Fig. 3.7 shows this principle for a nonloaded and loaded machine. For instance, for a turbine with zero loading, the wind velocity in the rotor plane (Vax) is equal to the undisturbed wind velocity (Vw), while an operating and hence loaded turbine slows down the wind velocity to a lower value. If the velocity decrease induced by the rotor is v, then the velocity at the disk is Vw — v = Vax, while far downstream at section 1 the wind has been slowed further to velocity V^ and the pressure has returned to p0. The difference between the axial component of the wind velocity and the axial flow velocity in the rotor plane is usually called the "induced" velocity, the velocity induced by the presence of the turbine.
Figure 3.7: Wind turbine with (r) and without (I) loading. Vw is the undisturbed wind velocity, Vax is the wind velocity at the rotor disk position, and Vœ is the velocity far downstream, in the turbine wake. The unloaded wind turbine is transparent to the wind, the loaded turbine decelerates the wind. A change in loading implies a change in "induced" velocity.
The momentum loss of the fluid is the result of the thrust Dax that the rotor exerts against the flow, combined with the net resultant of the external pressure on the control volume, as shown in Fig. 3.6. Since the static atmospheric pressure, p0 acts on the entire control volume, its net resultant is zero.
Within the streamtube, continuity requires that Vw A0 = VaxA = V00Ai. Writing the continuity equation for flow outside the streamtube between sections 0 and 1, it follows that there must be a net flow, , out the sides of the control volume equal to the following:
Newton's second law or equivalently law of motion can be generalized from particles to fluids: "At any instant in steady flow the resultant force acting on the moving fluid within a fixed volume of space equals the net rate of outflow of momentum from the closed surface bounding that volume". This is known as the momentum theorem. Writing the momentum theorem for the cylindrical control volume, it follows pVW2S  Dax = pVW2(S  Ai) + pVlAi + f&f Vw (3.2)
Substituting from Eq. (3.1) and VwA0 = V00Ai gives the thrust as
To slow the wind, a force must be manifested as a pressure drop across the disk. After all, a sudden step change in velocity is not possible because of the enormous accelerations and forces this would require. The static pressure drop just ahead the disk is p3 and just behind the disk p2. Since it is assumed that these pressures do not vary with time, it is also assumed that there is no periodicity in the flow velocity at the rotor plane, a condition that is strictly true only for an infinite number of blades. Applying the Bernoulli theorem from section 0 to section 3 and again from section 2 to section 1, we have
The thrust on the rotor is then
Solving for the pressure difference using Eq. (3.4) and (3.5) gives
Equating Eq. (3.3) and (3.7) and using AVax = AiV00, we find that
Thus, the velocity at the disk is the average of the upstream and downstream velocities. Defining an axial induction factor, a, as the fractional decrease in wind velocity between the free stream and the rotor plane represented by a = — (3.9)
Vw it follows that
Also
For a = 0, the wind is not decelerated and no power is extracted, whereas for a = 0.5, the far wake velocity vanishes, and, without presence of flow behind the turbine, no power is generated. The power extracted from the wind by the rotor is:
Substituting Vax from Eq. (3.10) and Vo from Eq. (3.11), we find that
A power coefficient Cp is then defined as
where the denominator represents the kinetic energy of the freestream wind contained in a streamtube with an area equal to the disk area. Substituting Eq. (3.14) in Eq. (3.15) results in
The maximum value of the power coefficient Cp occurs when dcp = 4(1  a)(1  3a) = 0
which gives a value of a = 1. Hence
Thus the maximum amount of energy extraction from the wind equals the 57th part of the kinetic energy in the wind. This limit is often referred to as the "Betz limit", or more accurately the "LanchesterBetz limit". The power coefficient Cp versus the induction factor a is shown in Fig. 3.8. This plot illustrates that the sensitivity of Cp to changes in a in the region 0.2 < a < 0.5, is much less than for a < 0.2. The power coefficient Cp has proved to be the most useful measure of the effectiveness of a wind turbine [261]. The actuator disk "efficiency" nad (i.e. power output divided by power input), on the other hand, is nad = 1 vfv , = 4a(1  a) (3.20)
2 PVW VaxA
since the mass flow rate through the actuator disk is not pAVw, but pAVax or equivalently pAVw(1 — a) using Eq. (3.10). The maximum efficiency of 1 occurs at a = 2 implying zero velocity in the wake (Vœ = 0) and a power coefficient of 1. The actuator disk efficiency is § = 0.889 at the maximum power coefficient of 0.59259.
It must be noted that according to Van Kuik [139] the radial force assumption does not hold true owing to an edge singularity of the actuator disk flow, and that the "real" maximum of the power coefficient is to be expected to be slightly higher than the Betz limit. Apart from this, it is possible to reach much higher power coefficients (i.e. to bypass the optimum of Betz) with additional devices like tip vanes [262]. These devices are all based on the concentrator and/or ejector principle. But to date almost no such devices have progressed beyond the experimental stage, mainly due to their higher complexity and expense in comparison to the freerunning turbine.
The pressure and velocity relationships of an energy extracting actuatordisk are shown in Fig. 3.9.
Upstream Rotor disk downstream
Wind velocity
Po+kpVl total pressure
Pa+ÏPVl  
dynamic pressure iPK2 Static pressure Figure 3.9: Pressure relationships of an energy extracting actuatordisk. A wind turbine extracts kinetic energy from the wind by slowing down the wind. This results in a rise in the static pressure. Across the rotor swept area there is a drop in static pressure such that, on leaving, the air is below atmospheric pressure. As the air proceeds downstream the pressure climbs back to the atmospheric value causing a futher slowing down of the wind. Thus, between upstream and downstream conditions, no change in static pressure exists but there is a reduction in kinetic energy. Figure 3.9: Pressure relationships of an energy extracting actuatordisk. A wind turbine extracts kinetic energy from the wind by slowing down the wind. This results in a rise in the static pressure. Across the rotor swept area there is a drop in static pressure such that, on leaving, the air is below atmospheric pressure. As the air proceeds downstream the pressure climbs back to the atmospheric value causing a futher slowing down of the wind. Thus, between upstream and downstream conditions, no change in static pressure exists but there is a reduction in kinetic energy. The RankineFroude actuatordisk model has the following implications: • The wind velocity at the rotor plane is always less than the freestream velocity when power is being absorbed (i.e. Vax < Vw); • This model assumes no wake rotation, i.e. no energy wasted in kinetic energy of a twirling wake; • Even with the best rotor design, it is evidently not possible to extract more than about 60 percent of the kinetic energy in the wind. Note that the range of the axial induction factor, a, is from zero for no energy extraction to onehalf, at which point the wind theoretically slows to zero velocity behind the rotor. Outside this range, the assumptions made in deriving this model are violated. Additional data that can be derived from this model include the thrust loading on the rotor. The thrust on the rotor is: Dax = 2pVw2A [4a(1 — a)] = qA [4a(1 — a)] (3.22) where q is the dynamic pressure. If we were thinking of the rotor as a propeller, we would define a thrust coefficient, as follows: On the other hand, if we were to think of Dax as a drag force on an equivalent flat plate of area equal to that of a rotor disk, we can define a drag coefficient, as follows: In either case, it is apparent from Eq. (3.16) that for these definitions Since a flat plate has a drag coefficient of about 1.28, we can note that, for a = 1, we obtain an equivalent drag coefficient of  for a rotor operating at the maximum Cp condition. Thus the rotor thrust is about 30 percent less than that of a flat plate equal in diameter to the rotor. Therefore, it is easy to see that the thrust loads generated by continuing to operate in high winds can be very large, requiring a very strong rotor and tower. The Glauert limitThe RankineFroude actuatordisk model neglects both aerofoil drag and wake rotation (or "swirl"). As a result, the maximum possible level of extracted power will therefore be lower than that predicted by the LanchesterBetz limit. Glauert developed a simple model for a rotating actuatordisk (i.e. with an infinite number of draggles blades) that includes the effect of wake rotation. The interested reader is referred to Spera [279] for derivation of this model. The corresponding limit is referred to as the "Glauert limit" and is illustrated by the solid line in Figure 3.10. At low tipspeed ratios, the maximum possible power coefficient is reduced because of large rotational kinetic energy captured in the wake. At high tipspeed ratios, the power coefficient approaches Cp,max = 0.59259 (LanchesterBetz limit) and the wake rotation reduces to zero. The tipspeed ratio is defined here as the ratio between the rotor circumferential speed and the undisturbed wind velocity. Furthermore, typical effects of changing the number of blades Nb and changing the design drag to lift ratio D/L on the power coefficient Cp are added using the empirical relation of Wilson et al. [307]: XNb067 1.92A 2Nb Note that for an infinite number of draggles blades (i.e. Nb ^ < and D/L = 0) the maximum power coefficient predicted by the empirical relation (3.26) equals the LanchesterBetz limit for all tipspeed ratios A. 
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