AERSP 583
Wind Turbine Aerodynamics

2a.6 The Rotor Disk Model

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Rotor Torque and Power

Our next task is to find an expression for the incremental power coefficient d CP. Before proceeding, though, let us define some additional dimensionless parameters that will he helpful in the analysis and useful to a thorough understanding of the effects of wake rotation.

Angular Induction Factor:  a' = ω / (2 Ω)

Tip Speed Ratio:  λ = (Ω · R) / V0

Local Tip Speed Ratio:  λr = λ · r/R

The ‘angular’ induction factor a’ is defined as being proportional to ω. It is apparent that in the special case of ω = 0 the angular induction factor becomes zero, and the rotor disk model reduces to the original actuator disk model. The next parameter, i.e. Tip Speed Ratio λ, involves the rotor speed Ω and describes the ratio of the blade tip speed Ω · R to the wind speed V0. We will see later in this course that typical values for λ range between 4 and 7. The third parameter shown is the Local Tip Speed Ratio λr, which is simply a fraction of λ based on the local blade position r/R.

Using these newly defined parameters and performing some algebra on the incremental power dP we obtain

 dP=Ω(ρ(1a)V02πrdr)2Ωar2=4πρa(1a)V0Ω2r3drThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.25)
 

...which gives us the incremental power dP as a function of wind speed V0, blade radius R, the rotational parameters λ and λr, and the dimensionless axial- and angular induction factors a and a’.

Click here to view the derivation for dP


Hence, the differential power coefficient dCP becomes

= dP / (½ ρ A V03) = dP / (½ ρ πR2 V03)
(2a.26)
 
Click for video transcript.

Transcript: differential power coefficient dCp

The incremental power coefficient of an annulus on the rotor disk is a function of the tip speed ratio, the angular induction factor, the axial induction factor, and the local tip speed ratio, which is simply a product of the actual tip speed ratio times you local relative location along the blade.


The power coefficient CP of the wind turbine is obtained by integrating the previous equation along the entire rotor disk, specifically for λr ranging between 0 and the tip speed ratio λ.

 CP=8λ20λa(1a)λr3dλrThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.27)
 

As the integral for the power coefficient CP is performed over the local tip speed ratio λr, our next task is to find two relations between λr and the axial- and angular induction factors a and a’, which will enable us to compute the integral in (2a.27). We will do so by considering the following:

First, let us remember equation (2a.9) from actuator disk theory that related the pressure jump Δpa across the actuator disk to the axial induction factor a.

Actuator Disk:   Δpa = ½ ρ (V02 - u12) = 2 ρ V02 a (1 - a)

What approach would we take for the Rotor Disk?

Think about this, then click here for the answer.

Our approach consists of having the same pressure jump cause the wake rotation ω. Hence, we are looking for Δpa’ = Δpa. In actuator disk theory, we used the Bernoulli equation applied to the axial velocity component through the streamtube.


We will use a similar approach here, however considering only the angular (or rotational) velocity. We write the Bernoulli equation as:

 p+12ρ(Ωr)2=(ppa)+12ρ[(Ω+ω)r]2This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.28)
 
Bounoulli equation as upstream and downstream of the actuator disk. Refer to text for detail
Figure 2a-9. Bernoulli equation as applied upstream and downstream of the actuator disk.
Source: Wind Energy Explained, Theory, Design, and Applications. Manwell. © 2015 Penn State. Reproduced with permission of John Wiley & Sons.

And substitute ω using the definition of the angular induction factor, i.e. a’ = ω / (2 Ω). Thus, we find for the pressure jump Δpa’:

 pa=ρωr2(Ω+12ω)=2ρΩ2r2a(1+a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.29)
 

Equating both formulations for the pressure drop across the rotor disk, i.e. Δpa = Δpa’ , we obtain the following after some algebra:

pa=pa2ρV02a(1a)=2ρΩ2r2a(1+a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

a(1a)=(ΩrV0)2a(1+a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

Firstrelationbetweena,a,λr:a(1a)a(1+a)=λr2This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.30)
 
Click here for a video transcript.

Transcript: First relationship between a and a'

And this omega times r/V0 square is again the square of the local tip speed ratio lambda r. So we isolate this on the right hand side and so on the left hand side you get the ratio of a(1-a) divided by a'(1+a').


The above equation constitutes a first relation between a, a’, and λr . In order to evaluate the integral for the power coefficient CP in (2a.27), we must find a second relation.

We mentioned earlier that the addition of wake rotation is likely to reduce the maximum power coefficient CP,max=0.59 due to Betz in actuator disk theory. One of our objectives in rotor disk theory is to understand to what extent do the newly introduced parameters a’, λ, and λr  reduce the Betz limit. By inspecting the integrand in (2a.27) we realize that the maximum power coefficient occurs for the integrand factor a’(1-a) attaining a maximum for each λr. We therefore want to maximize the function f(a, a’) = a’(1-a). As a first step, let us compute the first and second derivatives of f(a,a’) using the chain rule:

f(a,a)=a(1a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
dfda=dada(1a)aThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
d2fda2=d2ada2(1a)2dadaThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.31)
 

Next, we must perform the necessary algebra find the second relation between a and a'.

Click here to view the algebra.
dfda=0dada=a(1a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
Eqn.(1)a(1a)=λr2a(1+a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
dda(Eqn.1)12a=λr2(dada(1+a)+adada)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
12a=λr2dada(1+2a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
12a=a(1a)λr2(1+2a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
12a=a(1a)a(1a)(a(1+a))(1+2a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
12a=a(1+a)(1+2a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(12a)(1+a)=a(1+2a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
1+a2a2aa=a+2aaThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
a4aa=a(14a)=3a1This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
a=(3a1)(14a)(2)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
 Secondrelationbetweena,a:a=(3a1)(14a)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.32)