Environmental Engineering Reference
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parallel plates. He developed a simple and ingenious approximation for
F
expressed solely
in terms of local flow parameters. While the Prandtl model was useful, questions as to its
accuracy remained. Goldstein solved the problem of the flow about a lightly-loaded opti-
mum rotor (with a constant wake diameter), and this “exact” solution was used as a test of
the accuracy of the Prandtl model. Comparison showed that the Prandtl model gave results
very close to the Goldstein solution at high tip-speed ratios, and qualitative agreement at all
tip-speed ratios.
Comparisons between the Betz and Prandtl tip-loss models have been made for wind
turbines [Wilson and Lissaman 1974], and agreement was also good. However, since an
expanding wind turbine wake is different from the constant-diameter wake of an optimum
propeller, it is not apparent which model is more accurate. In the absence of test data on lo-
cal flow quantities near HAWT blade tips, the Prandtl tip-loss model is recommended. This
recommendation is based on the following:
·
The Prandtl model predicts a continuous change in circulation, in qualitative agree-
ment with the behavior of wind turbine rotors.
·
Calculations of rotor power and thrust made with the Prandtl model are in good
agreement with test data.
·
Strip theory calculations made with the Prandtl model show good agreement with
calculations made with free wake vortex theory.
·
The Prandtl model is easier to program and use.
It should be noted that, as shown in Figure 5-16, the agreement between test and calculations
using the Prandtl tip-loss model is poorest at the blade tip.
In Prandtl's tip-loss model, the
vortex sheets
generated by the blades are replaced (for
purposes of analysis) with a series of parallel planes at a uniform spacing equal to the normal
distance between successive vortex sheets at the slipstream boundary. Thus
D
z
=
2 p
R
B
sin f
R
where
D
z
= axial spacing between planes representing vortex sheets (m)
f
R
= angle between relative wind vector and plane of rotation at the tip (rad)
Using this spacing, the expression obtained by Prandtl for the tip-loss factor is
2
p
arc cos
B
(
R
-
r
)
2
R
sin f
R
F
=
exp
-
In practice, Prandtl's expression is modified to be
2
p
arc
cos
exp
-
B
(
R
-
r
)
2
r
sin f
(5-35)
F
=
since, as the tip-loss factor
F
goes to zero at the tip, the tip flow angle f
R
goes to the blade
pitch angle at the tip, which may be zero. Equation (5-35) avoids a potential tip singularity
and, more significantly, produces good results when compared with theoretical approaches.
The tip-loss factor
F
enters the determination of the induced flow calculation through the
local thrust coefficient,
C
t
. Recall from Figure 5-11 that the overall thrust coefficient departs
from the momentum theory value of 4
a
(l -
a
) when
a
exceeds about 0.4. It has been shown
[Wilson and Walker 1984] that the following formulation for the local thrust coefficient
compares well with vortex theory calculations and produces good results in the correlation of
calculated and measured performance and loads:
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