### Rotor Thrust and Power

Next, we use the derived equations for *u* and *u _{1}* in those for the thrust

*T*and power

*P*and obtain:

We hence obtained thrust *T* and power *P* as a function of air density *ρ*, wind speed *V*_{0}, 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:

For the power *P*, we use the available ‘power in the wind’ P_{avail} passing through the rotor disk area A as the normalization factor:

We can thus define a thrust coefficient *C _{T}* and power coefficient

*C*as:

_{P}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:

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 *C _{T}* 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 *C _{P}* 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

*V*and rotor size

_{0}^{3}*D*.

^{2}Let us consider again the relations for the thrust coefficient *C _{T}* and power coefficient

*C*as sole functions of the axial induction factor

_{P}*a*.

We note that *C _{T}* is a quadratic function in

*a*, while

*C*is a cubic function in

_{P}*a*. Next, we plot the dimensionless quantities

*C*and

_{T}*C*versus their dependent

_{P}*a*in order to find some basic limitations on rotor thrust and power.

As for the thrust coefficient *C _{T}*, we find it to be of parabolic shape symmetric about

*a=0.5.*Its maximum value is

*C*, which makes the rotor thrust being equal to the dynamic pressure force on a solid actuator disk. (Figure 2a-5.) See definition of

_{T}=1*C*in equation (2.16).

_{T}*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 C*

_{T}.

As for the power coefficient *C _{P}*, 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 C*

_{P}.

With reference to the definition of the power coefficient *C _{P}* 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 *C _{P}*,

_{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 *u _{1}* and the wind speed

*V*. From Figure 2a-4, we can see that

_{0}*u*depends linearly on the axial induction factor

_{1}/V_{0}*a*. It is not surprising that

*u*, as in this case of zero induction both rotor thrust and power equal zero. However, the ratio

_{1}/V_{0}=1 for a=0*u*becomes negative for

_{1}/V_{0}*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.