Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 37

Þ U 0 ¼ 4a 1 a

P

2 qU 0 pR 2 ¼ 4U 1 U 0 U 1

Þ 2 C P ¼ ð ð ð 2 : 16 Þ

1

where a is the ''axial induction factor'' defined by

a ¼ 1 U 1 = U 0 ð 2 : 17 Þ

so that the larger the value of a the more deceleration occurs as the air goes
through the blades. Maximum performance will occur when dC P /da = 0. From
( 2.16 ), this occurs when a = 1/3, and it immediately follows that

C P ; max ¼ 16 = 27 0 : 593 ; when a ¼ 1 = 3 ; U 1 = U 0 ¼ 2 = 3 ; and U 1 = U 0 ¼ 1 = 3

ð 2 : 18 Þ

for optimum performance. This is the Betz-Joukowsky limit. Its derivation shows
that a turbine can never capture all the kinetic energy that would flow past the
blade disk in the absence of the blades. All it can possibly do, according to ( 2.17 ),
is to capture two-thirds of that wind (in terms of U 1 /U 0 ), and convert eight-ninths
of that into output power because 1 - (U ? /U 0 ) 2 = 8/9. To do so, there must be
significant expansion of the flow; the cross-sectional area of the far-wake when
a = 1/3 is twice the blade disk area and three times the area of the wind captured
by the blades. This expansion of an optimal wind turbine wake is large compared
to the contraction that occurs in the wake of an efficient propeller, which is
typically only 10%. Hovering rotors, which model helicopters in hover, have a
wake contraction comparable to an optimum wind turbine's expansion. These
comparisons are made to indicate that the operating range between the optimal k
and runaway, where wake expansion increases further with k, is difficult to ana-
lyse. This issue is raised again in Chap. 7 .
The derivation of the Betz-Joukowsky limit depends on major simplifications
and assumptions about the air flow, principally in terms of steadiness, uniformity,
and the neglect of viscosity, which cannot be strictly valid in practice. Never-
theless, no well-documented study of wind turbine power output has violated the
limit in ( 2.18 ). From Sect. 1.4 , the value of C P,max for modern wind turbines is
about 0.50. Furthermore, ( 2.18 ) is derived without any reference to the turbine
blades themselves, so it is reasonable to state that the first job of a blade designer is
to produce blades that result in an airflow as close as possible to the ideal. The next
chapter considers the relationship between the flow over the blades, the forces
acting on them, and a modification of the present analysis that is accurate at least
up until the optimum performance point.
To finish this discussion of maximum performance, it is important to emphasise
that C P is, strictly, not an efficiency, so that the Betz-Joukowsky limit is not a limit
on efficiency. Hopefully, the discussion in Sect. 1.1 of the amount of wind passing
through the blades emphasises this point; improved performance only requires an
increase in the amount of wind captured by the blades that outweighs any degra-
dation in efficiency of conversion. Several methods have been proposed to increase
flow capture, of which ''diffuser augmentation'' is probably the most promising.