Blade Element Theory for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 42

This chapter is a brief introduction to the theory. A much more extensive
explanation is given in Chap. 3 of [ 1 ]. For this chapter there are no significant
features that are specific to small turbines.
3.2 Some Assumptions of Blade Element Theory
To extend the analysis of Chap. 2 , use is made of the following assumptions:
• The flow in each streamtube is independent of that in other streamtubes, and
• The forces acting on each blade element are the same as those on an aerofoil of
the same section, angle of attack, and effective velocity.
It is easy to demonstrate that both these assumptions can be in error. The first
assumption is violated if there is a radial variation in the velocity through the blades,
as this will cause a radial pressure gradient and each streamtube will therefore exert a
force on its two neighbours. Fortunately this force is redistributive, i.e. it sums to zero
over the whole flow, and so ignoring it should not cause too severe an error.
The second assumption is much more intriguing. As will be made clear in
Chap. 4 , an aerofoil is a two-dimensional body in an infinite flow that is uniform
away from the region influenced by the body. Such a situation can never occur in a
wind turbine because the blades are always separated by a finite distance in the
azimuthal direction. The measure of the importance of this effect is the solidity
which is defined in Sect. 3.5 . Another difference from aerofoil behaviour is that
the flow over blade elements can remain attached at angles of attack that would
cause an aerofoil to stall, see Sect. 3.12 of [ 1 ]. This so-called ''stall delay'' is an
empirical fact, but its cause is not clear. It is usually argued that the Coriolis and
centrifugal forces in the boundary layers on the rotating blades are responsible.
However, the most common correction of aerofoil lift and drag for stall delay, see
Eq. 3.190 of [ 1 ], involves only the solidity. In other words, stall delay supposedly
scales on a parameter that does not measure the centrifugal and Coriolis forces
because it does not contain X. Furthermore, stall delay predominantly occurs
towards the hub, so it is likely that solidity, which is usually larger near the hub
and also delays separation, is at least partially responsible. Fortunately, stall delay
does not significantly influence optimum turbine performance, so no attempt will
be made to include its effects in the blade element calculations.
3.3 The Conservation Equations for Annular Streamtubes
These equations can be obtained by applying the vector equations in the last
chapter to the annular streamtube at radius r, and radial extent dr, as shown in
Fig. 3.1 .
The blade tip radius is R. The streamtube thickness is dr 0 upstream, dr at the
blades, and dr ? in the far-wake. Note that the streamtube is annular and it is