Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 36

P ¼ q Z 1

2 U UU dA

ð 2 : 11 Þ

and it should be straightforward (after experience with the mass and momentum
balances) to show that

P ¼ 1

2 qQ 1 U 0 U 2 1

ð 2 : 12 Þ

An alternative form of ( 2.12 ) can be found by applying Bernoulli's equation
from the upstream face of the CV to a position just upwind of the blades to give
P 1 ¼ 1

2 q U 0 U 1

Similarly from just downwind of the blades to the far-wake, and noting that the
velocity exiting the disk is the same as that entering;

P 2 ¼ 1

2 q U 2 1 U 1

(It is important to understand that Bernoulli's equation cannot be applied across
the disk as the energy extracted from the air alters the Bernoulli constant on each
streamline.) Using these two equations with ( 2.12 ) gives

Þ pR 2

P ¼ Q 1 P 1 P 2 ð Þ ¼ U 1 P 1 P 2 ð ð 2 : 13 Þ

Equations 2.6 and 2.13 can be combined to give

P ¼ TU 1 ð 2 : 14 Þ

showing that the power is the product of the force on the disk and the air velocity
through it. The correspondence between ( 2.14 ) and the relation between power,
force, and velocity in engineering dynamics is obvious but it must be emphasised
that ( 2.14 ) applies only to an ideal flow.
Combining ( 2.14 )with( 2.5a , b ) and ( 2.12 ) gives the very interesting result that
U 1 ¼ U 0 þ U ð Þ= 2 ð 2 : 15 Þ
which is, because of the restriction on ( 2.14 ), applicable only to ideal flow.
Equation 2.15 shows that half the expansion of the flow in terms of the velocity
changes occurs before the blades and half in the wake, behind the blades.

2.7 Turbine Operating Parameters and Optimum
Performance

The analysis of the previous section leads to equations for the turbine operating
parameters that were introduced in Sect. 1.4 , of which the most important is the
power coefficient, C P . From ( 2.3 ), ( 2.5a , b ), ( 2.14 ) and ( 2.15 ):