AERSP 583
Wind Turbine Aerodynamics

2a.3 The Actuator Disk Model

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Rotor Thrust and Power

Next, we use the derived equations for u and u1 in those for the thrust T and power P and obtain:

 T=2ρV02a(1a)AThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.12)
 
P=2ρV03a(1a)2AThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.13)
 

We hence obtained thrust T and power P as a function of air density ρ, wind speed V0, actuator disk area A, and the ‘axial induction factor’ a. It is convenient to write both physical quantities in an appropriate dimensionless form.

As for the thrust T, we use the dynamic pressure acting on the actuator disk as the normalization factor:

 12ρAV02This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.14)
 

For the power P, we use the available ‘power in the wind’ Pavail passing through the rotor disk area A as the normalization factor:

 12ρAV03This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.15)
 

We can thus define a thrust coefficient CT and power coefficient CP as:

 CT=T(12ρAV02)=4a(1a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.16)
 
 CP=P(12ρAV03)=4a(1a)2This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.17)
 

Both are dimensionless quantities and are sole functions of the axial induction factor a.

This means in particular that thrust T and power P can be written as:

 Thrust:T=T(V02,D2,CT)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.18)
 
 Power:P=P(V03,D2,CP)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.19)
 

The previous equation illustrates the key dependencies for thrust and power:

  • The thrust T is proportional to the wind speed squared, the rotor diameter squared, and the thrust coefficient.

In general, one aims at minimizing the rotor thrust for a given wind speed and rotor diameter, i.e. as small values of CT as possible are desired.

  • The power P is proportional to the wind speed cubed, the rotor diameter squared, and the power coefficient.

It is obvious that one would aim at maximizing the power coefficient CP given a wind speed and rotor diameter. However, we note that there is just a linear proportionality of rotor power to the power coefficient. The biggest effect on the total power P is due to the wind resource V03 and rotor size D2.

Let us consider again the relations for the thrust coefficient CT and power coefficient CP as sole functions of the axial induction factor a.

 CT=4a(1a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.20)
 
 CP=4a(1a)2This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.21)
 

We note that CT is a quadratic function in a, while CP is a cubic function in a. Next, we plot the dimensionless quantities CT and CP versus their dependent a in order to find some basic limitations on rotor thrust and power.

Graph showing that the maximum CT=1 at a=1/2. And maximum CP=0.59 at a=1/3. Refer to text for details.
Figure 2a-5. Plotting CT and CP against a to find the maximum for rotor thrust and power.
Source: Wind Energy Explained, Theory, Design, and Applications. Manwell. © 2015 Penn State. Reproduced with permission of John Wiley & Sons.

As for the thrust coefficient CT, we find it to be of parabolic shape symmetric about a=0.5. Its maximum value is CT=1, which makes the rotor thrust being equal to the dynamic pressure force on a solid actuator disk. (Figure 2a-5.) See definition of CT in equation (2.16).

Click for video transcript.

Transcript: Thrust Coefficient and Power Coefficient

Let's plot some of the results that we have learned. We want to plot the dimensionless magnitudes of the the thrust coefficient CT and the power coefficient CP verses the axial induction factor a. But first let us remember what these relations were. The thrust coefficient C sub T, we had 4a(1-a). And for the power coefficient C sub P, we found it to be 4a(1-a) squared. In other words, thrust coefficient exhibits quadratic dependence on the axial induction factor a, while the power coefficient shows a cubic dependence.

Let's start first with the thrust coefficient CT. The quadratic demands that it is a parabola. So if a is zero, naturally that becomes zero. Also the thrust coefficient is zero for a being equal to 1. The dotted line here represents the parabola. You can show easily that this parabola will have a maximum for an axial induction factor a=1/2, in which case the maximum thrust coefficient becomes one.

Things are a little different for the power coefficient CP because we have this cubic dependence. But again, if a is zero, the power coefficient is zero and also for a=1, the power coefficient will be become zero. There a are few subtleties to the power coefficient, that is the cubic dependence give us the maximum, the minimum and the inflection point somewhere in between. We will derive, later, that the maximum power coefficient of CP=0.59 is obtained for an axial induction factor a=1/3.

Now just remember that power is something that's good for us, we want to capture as much power as we can. Thrust, well, if we can reduce it to a minimum better, because the thrust force of the rotor is creating a large bending moment to the tower and foundation of a wind turbine.

One more thing I'd like to point out here is that we need to be aware of the exit velocity u sub 1 out of the streamtube. Why is that? We found earlier that u1/V knot is a fuction of A2 and that equals 1-2a. Now, if a, the axial induction factor, becomes larger than 1/2, something happens and that is u1/V knot, the wind speed, becomes less than zero. In other words, the exit velocity out of the stream tube is reversed. If the flow reverses, also called the turbulent wake state, that is something that is absolutely not compliant with the assumptions that we put into momentum theory. And this is why here is the diagram, it's indicated for a>1/2 the Betz theory or momentum theory becomes invalid. The maximum power coefficient, we shall see later, is also called the Betz limit.

So far so good, let's move on to the next subject.


Click here to view the derivation for the maximum of CT.


As for the power coefficient CP, we find that its maximum value is approximately 0.59 for an axial induction factor of a=1/3. (Figure 2a-5.)

Click here to view the derivation for the maximum of CP.


With reference to the definition of the power coefficient CP in equation (2a.17), it becomes clear that we can at the most harvest approximately 59% of all the kinetic energy in the wind that passes through the streamtube.

The upper limit for the power coefficient CP,max=0.59 is called the “Betz Limit”.

Figure 2a-5 also illustrates a diagonal dashed line representing the ratio of the streamtube exit velocity u1 and the wind speed V0. From Figure 2a-4, we can see that u1/V0 depends linearly on the axial induction factor a. It is not surprising that u1/V0=1 for a=0, as in this case of zero induction both rotor thrust and power equal zero. However, the ratio u1/V0 becomes negative for a>0.5. This means in particular that there is reverse flow at the exit of the streamtube. This flow state is called the “turbulent wake state” and violates the assumptions of inviscid and irrotational flow of the actuator disk model. We thus conclude that the actuator disk model is valid for axial induction factors a<0.5.