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Wind Energy Handbook. Michael Barton Graham
Читать онлайн.Название Wind Energy Handbook
Год выпуска 0
isbn 9781119451167
Автор произведения Michael Barton Graham
Жанр Физика
Издательство John Wiley & Sons Limited
Again, the effects of tip‐loss are confined to the blade tip.
Figure 3.37 Variation of blade geometry parameter with local speed ratio, with and without tip‐loss for a three blade rotor with a design tip speed ratio of 6.
Figure 3.38 Variation of inflow angle with local speed ratio, with and without tip‐loss for a three blade rotor with a design tip speed ratio of 6.
The power coefficient for an optimised rotor, operating at the design tip speed ratio, without drag and tip‐losses is equal to the Lanchester–Betz limit 0.593, but with tip‐loss there is obviously a reduced optimum power coefficient. Equation (3.20) determines the power coefficient; see Figure 3.39:
for which a′ and a are obtained from Eqs. (3.89) and (3.90). This differs from the result given by Eq. (3.20) by the term a′/f in the denominator, which is very small except close to the root at low tip speed ratio.
Figure 3.39 Spanwise variation of power extraction in the presence of tip‐loss for three blades with uniform circulation and of optimised design for a tip speed ratio of 6.
Figure 3.40 The variation of maximum CP with designλ for various lift/drag ratios and including tip‐losses for a three bladed rotor.
The maximum power coefficient that can be achieved in the presence of both drag and tip‐loss is significantly less than the Betz limit at all tip speed ratios. As is shown in Figure 3.40, drag reduces the power coefficient at high tip speed ratios, but the effect of tip‐loss is most significant at low tip speed ratios because the pitch of the helicoidal vortex sheets is larger.
An alternative formulation for incorporating tip‐loss effect is to assume that tip‐loss effect may be applied to correct the blade section forces directly ensuring that they fall to zero at the blade root and tip. Thus the tip‐loss only appears as a factor f multiplying the right hand sides of Eqs. (3.48) and (3.49), which predict δT and δQ in terms of the momentum losses in the wake; see, for example, Wilson et al. (1974) and Jamieson (2018). This formulation if used simplifies the foregoing analysis of power coefficient because f only appears as a factor multiplying the expression for CP.
But in the following analysis, we will continue to follow the method of applying the tip‐loss factor derived in Section 3.9.2.
3.9.6 Incorporation of tip‐loss for non‐optimal operation
The BEM Eqs. (3.54a) and (3.55) are used to determine the flow induction factors for non‐optimal operation. With tip‐loss included the BEM equations have to be modified. The necessary modification depends upon whether the azimuthally averaged values of the flow factors are to be the determined or the maximum (local to a blade element) values. If the former alternative is chosen, then, in the momentum terms, the averaged flow factors a and a′ remain unmodified, but in the blade element terms, the flow factors must appear as the average values divided by the tip‐loss factor. Choosing to determine the maximum values of the flow factors, i.e. ab and ab′, means that they are not modified in the blade element terms but are multiplied by the tip‐loss factor in the momentum terms. The former choice allows the simpler modification of Eqs. (3.54a) and (3.55):
(3.54c)
(3.55a)
where the flow factor values determined are the averaged values a and a′.
There remains the problem of the breakdown of the momentum theory when wake mixing occurs. The helicoidal vortex structure may not exist, and so Prandtl's approximation is less physically appropriate. Nevertheless, due to the finite length of the blades and radius of the vortex wake, the application of a tip‐loss factor is necessary. Prandtl's approximation is the only practical method available and so is commonly used. In view of the manner in which the experimental results of Figure 3.16 were gathered, it is the average value of a that should determine at which stage the momentum theory breaks down.
3.9.7 Radial