Let us get back to the integral for *C _{p,max}*:

If we use the transformation *x = 1- 3a* and its derivative *dx/da = -3* we obtain:

If we use this to look at the distribution, we find the induction factors *a* and *a'* as functions of the local tip speed ratio, *λ _{r}*, for maximum power for an example reference case of a tip speed ratio that equals 7.5.

Let us recall these numbers...

The axial induction factor $a=1-\frac{u}{{V}_{0}}$ where

uis the axial velocity of the actuator or rotor disk, andVis the wind speed._{0}${a}^{\prime}=\frac{\omega}{2\Omega}$ where

Ωis the rotor speed.And for the optimum case ${a}^{\prime}=\frac{(1-3a)}{4a-1}$

The graph below (Fig. 2a-10) illustrates this distribution. Keep in mind that this is for a specific tip speed ratio that is fairly high at 7.5. The graph shows *a* and *a'* versus the radial position along the blade (multipy this number by the tip speed ratio 7.5. to get *λ _{r}.*)

The solid line represents the axial induction factor ** a**. At the root where

*r*or

*λ*equals zero, the induction factor a is 0.25, and it approaches the ideal value of 1/3 towards the tip.

_{r}The dotted line represents the angular induction factor *a'*. In the relationship ${a}^{\prime}=\frac{(1-3a)}{4a-1}$ , *a* approaches 1/3 when ** a'** goes to zero. Out at the tip the effect or the loss due to rotation becomes zero, and

*a*approaches indeed the Betz limit. However, when

*a*becomes close to 1/4, as it does at the root, we're dividing by zero and

*a'*actually has a singularity.

**Video question:** In Fig. 2a-10, why does *a'* have a singularity?

*Click here for a video transcript.*

#### Transcript: a' singularity

Now if you fix the tip speed ratio, the highest rotation and where you work most is out at the tip, but as you go inboard, the rotational speed becomes lesser and lesser. However, in rotor disk theory, you still have to add the swirl to the flow inboard. However, that swirl the local little omega, has a rather big portion of the actual velocity at a more inboard station. So you have capitol omega here and you add the swirl, little omega, but down here you have capitol omega times r/r, so the little omega that's acting there is larger to the actual rotation. Or another analogy, you just say well along the blade I'm sweeping over a range of tip speed ratios and you get the closest match to the Betz limit out at the tip and inboard it's less than that. Just because the local tip speed ratio is smaller there.

The plot below (Fig. 2a-11) illustrates the maximum power we can get including the wake rotation. We have the tip speed ratio as a free parameter *λ,* which equals the velocity at the tip *Ω **• **R*, divided by the wind speed *V _{0}* : $\lambda =\frac{(\Omega \cdot R)}{{V}_{0}}$

We're plotting power coefficient versus tip speed ratio. The solid line represents actuator disk theory, or the the Betz limit, which is 16/27 or approximately equal to 0.593. This is the maximum you can acheive. If your tip speed ratio is 0, the power will be zero. As you increase the tip speed ratio, the power coefficient gets closer to the Betz limit.

**Video question:** Is there an ideal tip speed ratio?

*Click here for a video transcript.*

#### Transcript: Max Power

Tip speed ratio is the ratio of a tip speed to the incoming wind speed. How fast or how quick does a wind turbine have spin in order to get close to the Betz limit? Looks like from this plot, the tip speed ratio doesn't have to be 100 or 1000. This is good news. Because as you can see even for a tip speed ratio of about 5, you're already at 0.57 or so, getting pretty close to the Betz limit for a fairly low tip speed ratio. If you look at a wind turbine blade that's about 40 meters long, it weighs between 8000 and 9000 pounds. So if you double the rotor speed the centrifugal force goes with the square of the rotor speed, so that's immense. If you would have to go from 5 to 10, you're centrifugal force of this 8000 pounds that spins around gets multiplied by a factor of 4. That's significant. You don't want to do that, you want to get by with as small of a tip speed ratio as you can. Wind turbine noise is proportional between the 5th and 6th power of the tip speed. That's enormous. Any reduction in tip speed because it goes at least to the fifth power, reduces the noise significantly. And that is why for most onshore wind turbines a tip speed ratio of 4 is pretty common. Offshore in European developments in the north sea, they spin the rotors quicker. Nobody cares about the noise. But they would never go up to 10 because of structural considerations.

Validity of the Rotor Disk Model

Parameters:

$$a=1-\frac{u}{{V}_{0}}$$

$${a}^{\prime}=\frac{\omega}{2\Omega}$$

$${a}^{\prime}=\frac{(1-3a)}{4a-1}$$

The two relations between *a* and *a'*:

$$\frac{a(1-a)}{{a}^{\prime}(1+{a}^{\prime})}={\lambda}_{r}^{2}$$

$${\lambda}_{r}^{2}=\frac{(1-a){(4a-1)}^{2}}{(1-3a)}$$

Eliminate the *λ _{r}^{2}* by subtracting the two equations from one another and you'll find:

$${a}^{\prime}=\frac{1-3a}{4a-1}$$

As we approach locally the ideal axial induction factor of *a = 1/3*, *a'* naturally goes to zero. It will also do this for high tip speed ratios. An interesting twist is that the effects of wake rotation are smallest for a quickly spinning rotor. This seems counter-intuitive at first as the loss due to wake rotation is related to the 'swirl' that's being added to the downstream flow.

In general, the higher the rotor speed, the closer one approaches the limit of the uniform-flow actuator disk. And the faster the tip speed, the smaller the relation between the *ω* (the swirl you're adding to the wake) and the actual rotor speed *Ω*.

### Rotor Disk Model Summary

Let us summarize a few lines about the rotor disk model:

- 1-Dimensional, Inviscid, Irrotational, Steady
- Includes Effect of Tip Speed Ratio
*λ = (Ω·R) / V*_{0} - Rotor Power still remains a function of the 3-2-1 law:
*P = P(V*_{0}^{3}, D^{2}, C_{P}(λ) ) - Approaches “Betz Limit” of
*C*≈ 0.59 for high_{P,max}*λ*

This is the essense you should you know as well as how to sketch the relative distributions of *a* and *a'* prime versus radius and for different tip speed ratios.