Page 69 - Wind Energy Handbook

THE ACTUATOR DISC CONCEPT 43

Stream-tube

Velocity +
U ∞ p d
Velocity U
p ∞ Pressure U d w
Pressure p ∞

Actuator disc
p –
d

Figure 3.2 An Energy Extracting Actuator Disc and Stream-tube

flow velocity. The mass flow rate must be the same everywhere along the stream-
tube and so

rA 1 U 1 ¼ rA d U d ¼ rA w U w (3:1)

The symbol 1 refers to conditions far upstream, d refers to conditions at the disc
and w refers to conditions in the far wake.
It is usual to consider that the actuator disc induces a velocity variation which
must be superimposed on the free-stream velocity. The stream-wise component of
this induced flow at the disc is given by aU 1 , where a is called the axial flow
induction factor, or the inflow factor. At the disc, therefore, the net stream-wise
velocity is
U d ¼ U 1 (1 a) (3:2)

3.2.1 Momentum theory

The air that passes through the disc undergoes an overall change in velocity,
U 1 U w and a rate of change of momentum equal to the overall change of velocity
times the mass flow rate:

Rate of change of momentum ¼ (U 1 U w )rA d U d (3:3)
The force causing this change of momentum comes entirely from the pressure
difference across the actuator disc because the stream-tube is otherwise completely
surrounded by air at atmospheric pressure, which gives zero net force. Therefore,

þ

( p p )A d ¼ (U 1 U w )rA d U 1 (1 a) (3:4)
d d
þ

To obtain the pressure difference ( p p ) Bernoulli’s equation is applied sepa-
d d
rately to the upstream and downstream sections of the stream-tube; separate equa-