Page 181 - Fluid Power Engineering

154 Chapter Eight

turbines is in the wake of the first row of turbines. The third row of
turbinesisinthewakeofthefirstandsecondrowofturbines.Wakeim-
pacts turbines in two primary ways: Lower wind speed and increase
in turbulence. Both the affects result in reduced energy production,
while an increase in turbulence causes greater structural loading of
the turbines. The reduction in energy in a wind farm because of wake
can be in the range of 2 to 20% depending on the distances and ambi-
ent turbulence. The impact of wind speed deficit and turbulence are
essentially eliminated at a distance of 20D in the wake of the rotor,
where Dis the diameter of the rotor. However, for efficient use of prop-
erty, guidelines that are more practical have been developed. In wind
farm design, a guideline of 9D distance along the primary direction
of wind and 3D distance perpendicular to the primary direction of
wind is used in the industry to locate turbines.
There are two prominent models for computing the deficit in wind
speed: Linear model by N.O. Jensen, and eddy viscosity model by
Ainslie.
N.O. Jensen Model for Wake
According to the actuator disk theory in Chapter 2, the thrust on a
turbine is given by Eq. (2-31). Restating the equation in terms of the
thrust coefficient C T yields:
1
2 2
F = 2ρ A r v a(1 − a) = ρ A r v C T (8-10)
0 0
2
where
C T = 4a(1 − a) (8-11)

From Eq. (2-30)

v 2 = (1 − 2a)v 0 = 1 − C T v 0 (8-12)
where v 0 is the free stream horizontal wind speed, v 2 is the down-
stream wake wind speed, and a is the axial induction factor.

v 2
Deficit in speed = 1 − = 1 − 1 − C T (8-13)
v 0
Assuming a linearly expanding wake with slope of k, the deficit as a
function of x becomes:

2
1 − v x /v 0 = (1 − 1 − C T ) ./(d + 2kx) (8-14)
where d is the rotor diameter, k is the slope or wake decay constant,
and x is the distance from the rotor. Onshore value of k = 0.075 and
for offshore value of k = 0.04 are commonly used. 3