Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 32

Fig. 2.1 Control volume for
wind turbine of radius R in
steady uniform flow

Q R

Cylindrical face of CV at radius R CV

U 0

bounding streamline

U 0
entering
upstream
face
of
CV

R

∞

U

∞
leaving
downstream
face of
CV

R

R 0

Axis of rotation

rotor is comprised of helical vortices trailed from the blade tips in much the same
way that nearly straight tip vortices are shed at the tips of aircraft wings. The key
difference between helical and straight vortices is that the former can induce an
axial flow whereas the latter cannot. Trailing vortices are a consequence of
Kelvin's theorem that circulation must be continuous in an otherwise inviscid
fluid, so the ''bound'' vorticity of turbine blades and aircraft wings must be shed
into their wakes. For blades, this shedding occurs at the hubs as well as the tips but
the hub vorticity does not appear to have a leading-order effect on the flow.
The figure shows that there is expansion of the flow before the blades; in fact,
about one-half the expansion, as measured by the cross-sectional area of the
bounding streamtube, occurs in the upstream flow. This is one reason why the
turbine can never convert all the kinetic energy that would pass through the blade
area in the absence of the blades.
It is further assumed that U ? and the pressure in the far-wake are uniform, and
that the latter is equal to atmospheric pressure. Furthermore, the presence of any
swirl, or circumferential velocity generated by the blades, is ignored, even though
the torque on the blades must result in a change in the angular momentum of the
air. The accuracy of this assumption is examined in Chap. 4 . Briefly, for normal
wind turbine values of the tip speed ratio, k, the circumferential velocity is so low
that it can be neglected when considering conservation of momentum and energy,
and, in any case, it does not enter the conservation of mass equation.
The three conservation equations for an incompressible airflow (constant
density) are now applied by assuming that the flow is uniform and steady, which
means that there is no accumulation of mass, momentum, angular momentum, and
energy within the CV.

2.3 Conservation of Mass

When divided through by the constant density, q, the vector form of the conser-
vation of mass equation for a steady flow is