LITMIR.BIZ

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 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

       where g₁ = 0.1 + exp.{−c₁(Bλ‐c₂)} with c₁ = 0.125 and c₂ = 21.0.

      This formulation is similar to Glauert's (1935a) original simplification of the Prandtl tip‐loss correction but introduces a variable factor g₁ rather 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 (c₁ ∼ 0.122 and c₂ ∼ 21.5) and for the tangential force correction to be similarly defined but with (c₁ ∼ 0.1 and c₂ ∼ 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 3.4 can 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).