Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 31

Chapter 2
Control Volume Analysis for Wind
Turbines
2.1 Introduction
This chapter uses the control volume (CV) method introduced informally in
Sect. 1.1 , to develop the basic equations for conservation of mass ( Sect. 2.3 ),
momentum ( 2.4 ), angular momentum ( 2.5 ), and energy ( 2.6 ). The forms of the
equations to be used for this finite CV are derived in any standard undergraduate
textbook on fluid mechanics such as [ 1 ]. It must be remembered that the basic
analysis undertaken here (and up to the end of Chap. 4 ) assumes steady, uniform
flow—conditions that rarely occur in any real wind! This chapter makes the further
assumption that the flow through the blades is radially uniform, that is, the
velocities (and therefore pressures) do not vary with radius.
In Sect. 2.7 , these equations are used in conjunction with the turbine operating
parameters to determine the Betz-Joukowsky limit, the theoretical upper limit on
the power extraction capability of a wind turbine.
2.2 The Control Volume
Figure 2.1 shows the CV to analysis wind turbine behaviour. The CV is a cylinder
whose radius, R CV , is very much larger than the blade tip radius, R. The upstream
face of the CV (at the left-hand side) is located far enough upstream that the
velocity entering the CV is the wind speed U 0 and the pressure is ambient or zero
gauge pressure. In other words, the presence of the blades does not influence the
flow entering the upstream face of the CV. Because the turbine extracts energy
from the wind, the velocity in the far-wake, U ? , is always less than U 0 . The radius
of the far-wake is R ? . Thus the wake expands as shown by the ''bounding
streamline'' which is part of the ''bounding streamtube'', the boundary between the
flow passing through the blades and the ''external'' flow. This boundary is sharp
and can support a discontinuity in the velocity and pressure across the bounding
streamline. It will be shown in Chap. 5 that the bounding streamtube behind the