Physics of Wind Parks - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 135

The derivation of the analytical wind park model shown here is an extension of
earlier versions of this model documented in Frandsen ( 1992 ), Emeis and Frandsen
( 1993 ) and Emeis ( 2010a ). The consideration of a simple, analytically solvable
momentum balance of large wind parks in this subchapter will show that
the design of a wind park and distance among each other has to take into account
the properties of the surface on which they are erected and the thermal stability of
the atmosphere typical for the chosen site. The momentum balance presented here
will indicate that the distance between turbines in an offshore wind park and the
distance between entire offshore parks must be considerably larger than for
onshore parks. Turbines will be characterized only by their hub height, rotor
diameter and thrust coefficient. Near wake properties are disregarded.
Starting point for the analytical wind park model is the overall mass-specific
momentum consumption m of the turbines which is proportional to the drag of the
turbines c t and the wind speed u h at hub height h:

m ¼ c t u h ð 6 : 7 Þ

In an indefinitely large wind park, this momentum loss can only be accom-
plished by a turbulent momentum flux s from above. Here, u 0 is the undisturbed
wind speed above the wind park, K m is the momentum exchange coefficient and
Dz is the height difference between hub height of the turbines and the undisturbed
flow above the wind park (see Fig. 6.1 ):
s

q ¼ K m u 0 u h

ð 6 : 8 Þ

Dz

The turbulent exchange coefficient K m describes the ability of the atmosphere to
transfer momentum vertically by turbulent motion. This coefficient describes an
atmospheric conductivity giving the mass-specific momentum flux (physical units:
m 2 /s 2 ) per vertical momentum gradient (unit: 1/s). Thus K m has the dimension of a
viscosity (unit: m 2 /s). Typical values of this viscosity are between 1 and 100 m 2 /s.
The main task in the formulation of the analytical park model is to describe the
exchange coefficient K m as function of the outer (surface roughness, thermal
stratification of the boundary layer) and inner (drag of the turbines, turbulence
generation of the turbines) conditions in the wind park. A major variable in this
context is turbulence intensity T i [see ( 3.10 ] for a definition) which is directly
proportional to K m . We obtain from the stability-dependent formulation of Monin-
Obukhov similarity in the surface layer (see Sect. 3.1.1 ) :

1
/ m

K m ¼ ju z ð 6 : 9 Þ

with the von Kármán constant j = 0.4, the friction velocity u * [see (A.13) in the
Appendix], the height z and the stability function / m :
1
x

z
L \0

for