Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 39

community is that the simple analysis leading to ( 2.16 ), the Betz-Joukowsky limit,
and ( 2.18 ) is sufficiently accurate at least until the point of maximum power
production. After that point, which is admittedly the designer's first goal, little is
known about the details of the flow.
There have been a large number of suggested ''corrections'' to Eq. 2.19
advanced over the years, none of which are based on any proper physical con-
sideration of the origin for the additional thrust or on the nature of the wake. One
commonly used correction is due to Glauert; it consists of replacing ( 2.19 )by
C T ¼ 4aa ð Þþ 2 ð 2 : 20 Þ
whenever a [ 1/2. As shown in Fig. 2.2 , Eq. 2.20 is the mirror image of ( 2.19 )
about the line C T = 1. Another ''high thrust'' equation is

4a 1 a ð Þ a a c

C T ¼ ð 2 : 21 Þ

4a c þ 1 2a c ð Þ aa[ a c

where a c = 1/3, which is also shown in Fig. 2.2 . This equation was used by
Clifton-Smith [ 5 ] and others and will be used for the blade element calculations in
Chap. 5 . Fortunately for performance analysis and design, the high thrust cor-
rections are not critical. Another high thrust equation was proposed by Buhl [ 3 ]
and used by Lanzafame and Messina [ 6 ].
The high thrust region deserves detailed examination because of the importance
of the simple one-dimensional analysis for BET and the Betz-Joukowsky limit. It
is possible that the basic wake structure of helical tip vortices trailing from the
blades remains unaltered as the thrust increases, but that the internal structure of
these vortices changes to alter the axial induction through the rotor and to absorb
energy. Some evidence for this view was the finding of [ 7 ] that the near-wake of a
model rotor at runaway had a significant velocity deficit, showing that kinetic
energy had been extracted. However, the tip vortices also had sufficient angular
momentum to absorb nearly all that energy and leave none to produce power. In
other words, the simple assumption that the tip vortices are vortex lines whose
structure is not important, which is appropriate for the one-dimensional flow at low
thrust, quickly becomes incorrect. The importance of the wake structure in
determining turbine performance is the main reason why the discussion on opti-
mum performance in Chap. 5 starts with consideration of the far-wake.

2.7.1 Exercises

1. In deriving the equations of this chapter, it would have been much easier to use
a CV whose outer face coincides with the bounding streamtube. Can you think
of reasons why this CV was not used?
2. List the assumptions made in deriving the equations of this chapter. Which ones
do you consider important and possibly responsible for the breakdown of the
analysis at high wake expansion?