### Bernoulli Equation

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 Δ*p*, then the pressure just downstream of the actuator disk becomes *p*-Δ*p*. We can then write the ‘downstream’ Bernoulli equation as:

Subtracting the upstream and downstream Bernoulli equations from one another relates the pressure drop Δ*p* to the unknown velocities *u* and *u _{1}*:

We can use the relation for the pressure drop Δ*p* 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)⋅V _{0}* which describes the velocity

*u*at the actuator disk as a fraction of the free stream wind speed

*V*.

_{0}Using *u=½⋅(V _{0}+u_{1})* we can write the velocity

*u*at the exit plane as

_{1}*u*It becomes evident that in the special case of

_{1}=(1-2a)⋅V_{0}.*a=0*, the unknown velocities

*u*and

*u*simply equal the free stream wind speed

_{1}*V*and both thrust

_{0}*T*and power

*P*are zero.

A couple things about thrust and power:

*Click for video transcript.*

#### 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.