Michael Graham - Wind Energy Handbook

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Discover this fully updated and authoritative reference to wind energy technology written by leading academic and industry professionals  The newly revised Third Edition of the 
 delivers a fully updated treatment of key developments in wind technology since the publication of the book’s Second Edition in 2011. The criticality of wakes within wind farms is addressed by the addition of an entirely new chapter on wake effects, including ‘engineering’ wake models and wake control. Offshore, attention is focused for the first time on the design of floating support structures, and the new ‘PISA’ method for monopile geotechnical design is introduced. 
The coverage of blade design has been completely rewritten, with an expanded description of laminate fatigue properties and new sections on manufacturing methods, blade testing, leading-edge erosion and bend-twist coupling. These are complemented by new sections on blade add-ons and noise in the aerodynamics chapters, which now also include a description of the Leishman-Beddoes dynamic stall model and an extended introduction to Computational Fluid Dynamics analysis. 
The importance of the environmental impact of wind farms both on- and offshore is recognised by extended coverage, which encompasses the requirements of the Grid Codes to ensure wind energy plays its full role in the power system. The conceptual design chapter has been extended to include a number of novel concepts, including low induction rotors, multiple rotor structures, superconducting generators and magnetic gearboxes.
References and further reading resources are included throughout the book and have been updated to cover the latest literature. Importantly, the core subjects constituting the essential background to wind turbine and wind farm design are covered, as in previous editions. These include: 
The nature of the wind resource, including geographical variation, synoptic and diurnal variations and turbulence characteristics The aerodynamics of horizontal axis wind turbines, including the actuator disc concept, rotor disc theory, the vortex cylinder model of the actuator disc and the Blade-Element/Momentum theory Design loads for horizontal axis wind turbines, including the prescriptions of international standards Alternative machine architectures The design of key components Wind turbine controller design for fixed and variable speed machines The integration of wind farms into the electrical power system Wind farm design, siting constraints and the assessment of environmental impact Perfect for engineers and scientists learning about wind turbine technology, the 
 will also earn a place in the libraries of graduate students taking courses on wind turbines and wind energy, as well as industry professionals whose work requires a deep understanding of wind energy technology.

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A particle path, as shown in Figure 3.32, may be interpreted as an average particle passing through the rotor disc at a given radius in the actual situation: the azimuthal variations of particle axial velocities at various radii are shown in Figure 3.28, and a ‘Prandtl particle’ would have a velocity equal to the azimuthal average of each. Figure 3.32depicts the wake model.

The mathematical detail of Prandtl's analysis is given in Glauert (1935a), and because it is based on a somewhat strangely simplified model of the wake will not be repeated here. It has, however, remained the most commonly used tip‐loss correction because it is reasonably accurate and, unlike Goldstein's theory, the result can be expressed in closed solution form. The Prandtl tip‐loss factor is given by

(3.80) R w r is a distance measured from the wake edge Distance d between the discs - фото 309

R w− r is a distance measured from the wake edge. Distance d between the discs should be that of the distance travelled by the flow between successive vortex sheets. Glauert (1935a), takes d as being the normal distance between successive helicoidal vortex sheets.

The helix angle of the vortex sheets ϕ sis the flow angle assumed to be the same as ϕ t, the helix angle at the blade tip, and so with B sheets intertwining from B blades and assuming that the discs move with the mean axial velocity in the wake, U ∞(1 – 381 Figure 332 Prandtls wakedi - фото 310):

(3.81) Figure 332 Prandtls wakedisc model to account for tiplosses Prandtls - фото 311

Figure 332 Prandtls wakedisc model to account for tiplosses Prandtls - фото 312

Figure 3.32 Prandtl's wake‐disc model to account for tip‐losses.

Prandtl's model has no wake rotation, but whether the discs are considered to spin is irrelevant to the flow field, as it is inviscid, thus a ′is zero and W sis the resultant velocity (not including the radial velocity) at the edge of a disc. Glauert (1935a) argues that Wind Energy Handbook - изображение 313, which is more convenient to use,

so and 382 Although the physical basis o - фото 314

so

and 382 Although the physical basis of this model is not correct it does - фото 315

and

(3.82) Although the physical basis of this model is not correct it does quite - фото 316

Although the physical basis of this model is not correct, it does quite effectively represent a convenient approximation to the attenuation towards the tips of the real velocities induced by the helicoidal vortex sheets.

The Prandtl tip‐loss factor for a three blade rotor operating at a tip speed ratio of 6 is compared with the tip‐loss factor of the helical vortex wake in Figure 3.33.

It should also be pointed out that the vortex theory of Figure 3.28also predicts that the tip‐loss factor should be applied to the tangential flow induction factor.

It is now useful to know what the variation of circulation along the blade is. For the previous analysis, which disregarded tip‐losses, the blade circulation was uniform [ Eq. (3.69))].

Following the same procedure from which Eq. (3.68)was developed:

Recall that a b r is the flow factor local to the blade at radius r and r - фото 317

Recall that a b( r ) is the flow factor local to the blade at radius r and r is the average value of the flow factor at radius r Figure 333 - фото 318(r) is the average value of the flow factor at radius r .

Figure 333 Comparison of Prandtl tiploss factor with that predicted by a - фото 319

Figure 3.33 Comparison of Prandtl tip‐loss factor with that predicted by a vortex theory for a three blade turbine optimised for a tip speed ratio of 6.

Figure 334 Spanwise variation of blade circulation for a three blade turbine - фото 320

Figure 3.34 Spanwise variation of blade circulation for a three blade turbine optimised for a tip speed ratio of 6.

Therefore,

(3.83) Γ r is the total circulation for all blades and is shown in Figure 334 - фото 321

Γ( r ) is the total circulation for all blades and is shown in Figure 3.34, and, as can be seen, it is almost uniform except near to the tip. The dashed vertical line shows the effective blade length (radius) R ef= 0.975 if the circulation is assumed to be uniform at the level that pertains at the blade sections away from the tip.

The Prandtl tip‐loss factor that is widely used in industry codes appears to offer an acceptable, simple solution to a complex problem; not only does it account for the effects of discrete blades, it also allows the induction factors to fall to zero at the edge of the rotor disc.

A more recently derived tip‐loss factor that has been calibrated against experimental data and appears to give improved performance was given by Shen et al. (2005). The spanwise distribution of axial and tangential forces is multiplied by the factor

Wind Energy Handbook - изображение 322 where g 1 01 expc 1B λ c 2 with c 1 0125 and c 2 210 This - фото 323where g 1= 0.1 + exp.{−c 1(B λ ‐c 2)} with c 1= 0.125 and c 2= 21.0.

This formulation is similar to Glauert's (1935a) original simplification of the Prandtl tip‐loss correction but introduces a variable factor g 1rather than unity. Wimshurst and Willden (2018) suggest that a better fit is given in the above by using different constants for the axial force correction (c 1∼ 0.122 and c 2∼ 21.5) and for the tangential force correction to be similarly defined but with (c 1∼ 0.1 and c 2∼ 13.0).

3.9.4 Blade root losses

At the root of a blade the circulation must fall to zero as it does at the blade tip, and so it can be presumed that a similar process occurs. The blade root will be at some distance from the rotor axis, and the air flow through the disc inside the blade root radius will be at the free‐stream velocity. Actually, the vortex theory of Section 3.4can be extended to show that the flow through the root disc is somewhat higher than the free‐stream velocity. It is usual, therefore, to apply the Prandtl tip‐loss function at the blade root as well as at the tip (see Figure 3.35).

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