### Introduction

Our next step is to develop “The Rotor Disk Model” that allows for rotation of the actuator disk and the downstream wake. The following assumptions, though, remain:

- One-Dimensional
- Inviscid & Irrotational
- Steady

In the simplified model of a rotor disk, we still consider a streamtube, however we proceed one step towards an actual wind turbine rotor by adding a ‘swirl’ or ‘rotation’ to the wake. Without further analysis, we surmise that the addition of wake rotation to the flow inside the streamtube is associated with a loss in power produced by the rotor disk as additional kinetic energy is needed to sustain the rotation of the wake.

The actuator disk theory revealed that the maximum power coefficient *C _{P}* is governed by the Betz limit of

*C*. We will find that adding wake rotation will decrease the theoretical maximum. – The actuator disk theory generated the rotor power via the rotor thrust and the axial induction factor a. In rotor disk theory, on the contrary, we will discover that rotor power is generated through rotor torque and the angular momentum in the wake via a combination of axial- and angular induction factors.

_{P,max}=0.59*Click here for a video transcript.*

#### Transcript: Actuator Disk to Rotor Disk - Introduction of Wake Rotation

The Betz limit is the limit, 59% of the energy that we can capture from the wind. Now, if you allow the actuator disk to become a rotor disk that's spinning at some rotor speed omega, the side of effect of doing that is that you add a swirl to the downstream wind. Adding a swirl means you have to pay for it. You have to pay for it with momentum and energy. In other words, that side of effect of the method that makes you adding that swirl, is that you cannot capture as much energy as in the actuator disk model. So we will get something that is less than the Betz limit for sure. The question is how much is that.

Rotor Torque and Power

At first, let us recall from classical mechanics that rotor torque is associated with a change in wake angular momentum:

Rotor Torque = Change in Wake Angular Momentum

We denote the rotor speed or the angular velocity of the rotor disk by * Ω *(Omega). The wake speed, respectively, is the sum of the rotor speed

*and the swirl speed ω (omega). We thus have:*

*Ω*Rotor Speed: *Ω*

Wake Speed:* Ω + ω*

Next, we consider an annulus of width dr of the rotor disk and write the incremental torque *dQ* acting on the annulus as...

*$$dQ=(d{m}^{*}\cdot {r}^{2})\cdot \omega $$*

...where the term in brackets represents a mass moment of inertia flowing through the considered annulus of the rotor disk.

*Click here for a video transcript.*

#### Transcript: Mass moment of inertia

The mass moment of inertial, you have to be careful. I cannot just take the entire disk to do it. Because the mass moment of inertia is something that changes with distance over mass to the center of rotation. So I take the ring or annulus out of that rotor disk and consider what will be the incremental torque distribution of that.

We make use of the definition of the mass flow rate in eqn. (2a.2) *dm* = ρ · u · dA* and write the incremental torque as:

We also remember from previous analysis that the velocity at the rotor disk is *u = (1-a) ·V_{0}_{}*, and note that an incremental area

*dA*of a rotor disk annulus is

*dA = 2 πr*

*·*drWe can also define an incremental power and an incremental power coefficient in the following way: