We begin by writing the Bernoulli equation ‘upstream’ of the actuator disk as:
In this equation, p is the pressure slightly upstream of the rotor disk. If we define the pressure drop across the actuator disk as Δp, then the pressure just downstream of the actuator disk becomes p-Δp. We can then write the ‘downstream’ Bernoulli equation as:
Subtracting the upstream and downstream Bernoulli equations from one another relates the pressure drop Δp to the unknown velocities u and u1:
We can use the relation for the pressure drop Δp in the equation for the thrust:
...to obtain u:
It is interesting to observe that the velocity u at the actuator disk is the arithmetic average of the velocities entering and exiting the streamtube.
It is practical in many engineering problems to use dimensionless coefficients (figure 2a-4). Let us define an ‘axial induction factor a through u=(1-a)⋅V0 which describes the velocity u at the actuator disk as a fraction of the free stream wind speed V0.
Using u=½⋅(V0+u1) we can write the velocity u1 at the exit plane as u1=(1-2a)⋅V0. It becomes evident that in the special case of a=0, the unknown velocities u and u1 simply equal the free stream wind speed V0 and both thrust T and power P are zero.
A couple things about thrust and power:
Transcript: Thrust and Power
We note a couple of very important things. Number one is thrust is proportional to the square of the wind speed. And that is also reflected in a quadratic dependence on the axial induction factor a. While the the power P is proportional the cube of the wind speed. We had mentioned it earlier. And that is also reflected in a third order dependence on the axial induction factor a.