Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 35

Þ pR 2 T ¼ P 1 P 2 ð ð 2 : 6 Þ

2.5 Conservation of Angular Momentum

The torque on the blades is equal and opposite to that acting on the air. The
equation for the vector torque Q is
Q ¼ q Z

r UU dA ð 2 : 7 Þ

If there is no swirl (or angular momentum) in the upstream flow, the only
contribution to ( 2.7 ) comes from the CV face in the far-wake. Equation 2.7 can
thus be turned into a scalar equation for the contribution to Q, the torque acting
around the axis of rotation, which is normally the only torque of interest. To do
this recognise that the magnitude of r 9 U is rW ? , where W ? is the swirl velocity
(about the turbine axis) in the far-wake. Furthermore, rW ? is related to a very
important quantity called the circulation around each blade, C,by

NC ¼ 2pr 1 W 1 ð 2 : 8 Þ

where N is the number of blades. It is shown in Chap. 4 that C is nearly constant in
the far-wake. Substitution into ( 2.7 ) leads to the following equation for C Q , the
torque coefficient

2 qU 0 pR 3 ¼ NCU 1

Q

C Q ¼ ð 2 : 9 Þ

1

p

The torque is related to the power by

P ¼ QX

ð 2 : 10 Þ

and is imparted to the blades by the aerodynamic forces (principally lift and drag)
generated by the flow through the blades.

2.6 Conservation of Energy

Finally, consider the energy equation for the CV used in the application of the
mass and momentum conservation equations. To start, recall the assumption that
the pressure in the far-wake is atmospheric (zero gauge pressure) so the pressure
on all faces of the CV is atmospheric. This means no net work is done by the
pressure forces in moving fluid into or out of the CV, so the only form of energy
that to be considered (in the ideal case) is kinetic energy. The conservation
equation gives the power output as