2a.3 The Rotor Disk Model

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_P,max=0.59. 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.

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

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.

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)·V₀ , and note that an incremental area dA of a rotor disk annulus is dA = 2 πr · dr

We can also define an incremental power and an incremental power coefficient in the following way:

Rotor Torque and Power

Our next task is to find an expression for the incremental power coefficient d C_P. 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:  λ_r = 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 λ_r, 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

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

Hence, the differential power coefficient dC_P becomes

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

As the integral for the power coefficient C_P is performed over the local tip speed ratio λ_r, our next task is to find two relations between λ_r 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 Δp_a across the actuator disk to the axial induction factor a.

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

What approach would we take for the Rotor Disk?

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

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

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

$$△p_a=△p_{a^{\prime }}\Rightarrow 2\rho V_0{}^{2}a\left(1-a\right)=2\rho {\Omega }^2{r}^2{a}^{\prime }(1+a^{\prime })$$

$$\Rightarrow a\left(1-a\right)={\left(\Omega \cdot \frac{r}{V_0}\right)}^2{a}^{\prime }(1+a^{\prime })$$

The above equation constitutes a first relation between a, a’, and λ_r . In order to evaluate the integral for the power coefficient C_P 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 C_P,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 λ_r  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 λ_r. 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 product rule:

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