Wind Energy Handbook. Michael Barton Graham

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By inspection, C_T1 must lie between 1.6 and 2: C_T1 = 1.816 would appear to be the best fit to the experimental data of Figure 3.16, whereas Wilson et al. (1974) favour the lower value of C_T1 = 1.6. Glauert fits a parabolic curve to the data [replacing a in the mass flow expression by 4a(1 − a)/(0.6 + 0.61a + 0.79a²) when a > 1/3] giving much higher values of C_T1 at high values of a but he was considering the case of an airscrew in the windmill brake state where the angles of attack are negative. De Vaal et al. (2014) suggest a be replaced by 0.25a(5 − 3a), similarly giving a somewhat lower windmill brake state result.

      The flow field through the turbine under heavily loaded conditions cannot be modelled easily, and the results of this empirical analysis must be regarded as being only approximate at best. They are, nevertheless, better than those predicted by the momentum theory. For most practical designs the value of the axial flow induction factor rarely exceeds 0.6 and for a well‐designed rotor will be in the vicinity of 0.33 for much of its operational range.

For values of a greater than a_T, it is common to replace the momentum theory thrust in Eq. (3.9) with Eq. (3.58), in which case Eq. (3.54a) is replaced by

(3.59)

      However, as the additional pressure drop is caused by breakdown of the streamline wake, this course of action is questionable, and it may be more appropriate to retain Eq. (3.54).

3.8 Blade geometry

      3.8.1 Introduction

      The purpose of most wind turbines is to extract as much energy from the wind as possible, and each component of the turbine has to be optimised for that goal. Optimal blade design is influenced by the mode of operation of the turbine, that is, fixed rotational speed or variable rotational speed and, ideally, the wind distribution at the intended site. In practice engineering compromises are made, but it is still necessary to know what would be the best design.

Optimising a blade design means maximising the power output, and so a suitable solution to BEM Eqs. (3.54 or (3.59) and (3.55)) is necessary.

      3.8.2 Optimal design for variable‐speed operation

      A turbine operating at variable speed can maintain the constant tip speed ratio required for the maximum power coefficient to be developed regardless of wind speed. To develop the maximum possible power coefficient requires a suitable blade geometry, the conditions for which will now be derived.

      For a chosen tip speed ratio λ the torque developed at each blade station is given by Eq. (3.49) and is maximised if

      giving

(3.60)

From Eqs. (3.51) and (3.52) a relationship between the flow induction factors can be obtained. Dividing Eq. (3.52) by the modified Eq. (3.51), modified to include the additional loss of axial momentum from the pressure drop term Δp_d2 in the far wake due to the centrifugal swirl generated radial pressure gradient, leads to:

(3.61)

      The flow angle ϕ is given by

(3.62)

Substituting Eq. (3.62) into Eq. (3.61) gives

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