Environmental Engineering Reference
In-Depth Information
Strip Theory for Horizontal-Axis Wind Turbines
In order to bridge the gap between actuator disk models of wind turbines and a rigorous
vortex theory , an intermediate theory known as strip theory has been developed that relates
rotor performance to rotor geometry. A particularly important prediction of strip theory is
the effect of finite blade number . We find that performance-optimized HAWTs do have
configurations similar to the Glauert-optimized rotor when finite blade number and drag are
included. In the development of strip theory, the relation between the local induction and the
local thrust coefficient in each streamtube is of paramount importance.
Background
A HAWT can be considered to be an airscrew that extracts kinetic energy from the
driving air and converts it into mechanical energy. The similarity of a HAWT to a propeller
(which puts energy into the air) enables the same theoretical development used for the pro-
peller to be followed for the HAWT. Propeller theory was developed along two independent
approaches: Actuator disk theory (which was discussed in the previous section) and blade-
element theory . The strip theory presented here has been called modified blade element
theory .
Blade-element theory was originated by Froude [1878] and later developed further by
Drzewiecki [1892]. The approach of blade-element theory is opposite that of momentum
theory since it is concerned with the forces produced by the blades as a result of the mo-
tion of the fluid. It was hampered in its original development by the lack of knowledge
of sectional aerodynamics and the mutual interaction of blades. Modern rotor theory has
developed from the concept of free vortices being shed from the rotating blades. These
vortices define a slipstream and generate induced velocities. This rigorous theory can
be attributed to the works of Lanchester [1907] and Flamm [1909], for the original con-
cept; to Joukowski [1912], for induced velocity analysis; to Betz [1919], for optimization;
to Prandtl [1919] and Goldstein [1929], for circulation distribution or tip-loss analysis;
and to Glauert [1922a, 1922b, 1935], Pistolesti [1922], and Kawada [1926], for general
improvements.
It has been found that strip-theory approaches are adequate for the analysis of wind
machine performance. One reason is that a wind turbine wake expands rather than con-
tracts. At high tip-speed ratios ( i.e. low advance ratios ), propellers and helicopter rotors
have been observed to shed strong tip vortices. Since these wakes are contracting, the shed
vortices are inboard of the tip and interact strongly with the flow through the rotor disk. The
resulting radial distribution of aerodynamic forces is found to be appreciably different from
that predicted by strip theory. Because most wind turbines operate at high tip-speed ratios,
they might be expected to experience the same strong interaction. However, because of the
expanding wake, the tip vortex moves outboard of the rotor, negating a strong interaction.
From an outboard position, the tip vortex generates induced velocities that decrease local
angles of attack and reduce aerodynamic loads.
Various forms of strip theory have been the standard methods of design and design
analysis of HAWTs. Strip theories are easy to program, inexpensive to run, and readily
adaptable to any size of computer. They are used with modest success to predict output
power. However, it is important to note that the largest sources of error in power prediction
have been in the airfoil lift and drag data, and these errors are frequently large enough to
mask the inaccuracies of the theory.
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