Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

330 Fluid Mechanics, Thermodynamics of Turbomachinery
electricity produced from wind energy are set to continue. They estimate that 22,000MW
of wind energy capacity, in the form of 40,000 wind turbines, will be installed in the next
ten years. However, in the trade journal Wind Power Monthly (March 2004), around
8200MW of new wind power capacity was installed in 2003, 21% more than the previous
year. This was enough to maintain the annual growth rate at almost the same level
of over 26%. The European growth rate in 2003 was similar to the global figure of around
24% while in the USA (up to the end of 2003) a surge of economic activity pushed
its growth rate up to 35%. In the 1990s the cost of manufacturing wind turbines declined
by 20% for each doubling of the number of wind turbines manufactured. Currently,
the production of large-scale, grid-connected wind turbines doubles about every three
years. A general survey of the economics of wind turbines and the forecasting of
cost reduction of electricity production is given by Ackermann and Söder (2002).
Outline of the theory
In the following pages the aerodynamic theory of the HAWT is gradually developed,
starting with the simple one-dimensional momentum analysis of the actuator disc and
followed by the more detailed analysis of the blade element theory. The flow state just
upstream of the rotor plane forms the so-called inflow condition for the rotor blades
and from which the aerodynamic forces acting on the blades can be determined. The
well-known blade element momentum (BEM) method is outlined and used extensively.
A number of worked examples are included at each stage of development to illustrate
the application of the theory. Detailed calculations using the BEM method were made
to show the influence of various factors such as the tip–speed ratio and blade number
on performance. Further development of the theory includes the application of Prandtl’s
tip loss correction factor that corrects for a finite number of blades. Glauert’s optimisation
analysis is developed and used to determine the ideal blade shape for a given lift
coefficient and to show how optimum rotor power coefficient is influenced by the choice
of tip–speed ratio.
Actuator disc approach
Introduction
The concept of the actuator disc was used in Chapter 6 as a method of determining
the three-dimensional flows in compressor and turbine blade rows. Betz (1926) in his
seminal work on the flow through windmill blades used a much simpler version of the
actuator disc. As a start to understanding the power production process of the turbine
consider the flow model shown in Figure 10.5 where the rotor of the HAWT is replaced
by an actuator disc. It is necessary to make a number of simplifying assumptions concerning
the flow but, fortunately, the analysis yields useful approximate results.
Theory of the actuator disc
The assumptions made are as follows:
(i) steady uniform flow upstream of the disc;
(ii) uniform and steady velocity at the disc;