Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 78

A suitable straight line through the experimental points would appear to be possible, although Glauert proposed a parabolic curve, and provides an empirical solution to the problem of the thrust on a heavily loaded turbine (a rotor operating at a high value of the axial flow induction factor).

Most authors assume that the entire thrust on the rotor disc is associated with axial momentum change. Therefore, for the empirical line to be useful it must be assumed that it applies not only to the whole rotor but also to each separate streamtube. Let C_T1 be the empirical value of C_T when a = 1. Then, as the straight line must be a tangent to the momentum theory parabola at the transition point, the equation for the line is

(3.58)

and the value of a at the transition point is

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).