Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 34

Another important use of the conservation of mass equation is to fix the volume
flow rate within the bounding streamtube. By an appropriate change to the CV
shown in the figure, it is easy to deduce that

Q 0 ¼ U 0 pR 0 ¼ Q 1 ¼ U 1 pR 1 ¼ Q 1 ¼ U 1 pR 2 1 ð 2 : 3 Þ

so that the volume flow rate within the bounding streamtube at any axial location
in the flow, is constant.

2.4 Conservation of Momentum

Newton's law in CV form determines the force acting on the air, which is the
negative of the force (thrust) acting on the blades, T, in vector form. Thus the
equation for T is

T ¼ q Z

UU dA ð 2 : 4 Þ

Focusing only on T which is the force in the direction of the wind, it is easier to
revert to the scalar component of ( 2.4 ) in the direction of the wind. The pressure is
constant and equal at all CV faces, so it does not contribute to the momentum
equation, as has already been assumed in writing ( 2.4 ). Furthermore, the velocities
at the CV faces are uniform (even if the velocity at the downstream face is
discontinuous at R ? ) so the application of Eq. 2.4 proceeds in the same manner as
for Eq. 2.1 . The result is

qU 1 U 1 pR 2 1 qU 0 Q R

T ¼ qU 0 U 0 pR cv qU 0 U 0 p R cv R 2 1

ð 2 : 5a Þ

and is, therefore, positive in the direction of the wind. The most interesting term in
this equation is the last, representing the removal of momentum (equal to U 0 per
unit mass) by the volume flow rate (Q R ) out of the cylindrical face of the CV.
Using Eq. 2.2 this term can be removed and ( 2.5a ) rewritten as

T ¼ qU 1 pR 2 1 U 0 U 1

ð Þ ¼ qQ 1 U 0 U 1 ð Þ ð 2 : 5b Þ

Another equation can be derived for T by considering the flow through the
''disk'' representing the rotating blades. Imagine that the blades can be replaced by
a thin, uniform circular disk across which the velocity is continuous but the
pressure is discontinuous, then T can result only from the pressure difference
P 1 - P 2 . P 1 acts in the wind direction on the upwind side of the disk and P 2 acts
upwind on the downwind side. (Note that the symbol P is used both for power
when there is no subscript and pressure when it is subscripted.) Idealising the
blades as an infinitely thin porous disk—often called an ''actuator disk''—is a
common one in the analysis of fluid machines. It can be thought of as a model for a
rotor with an infinite number of infinitely thin blades. Thus