Wind Energy Handbook. Michael Barton Graham

alt="a overbar"/>(r) is the average value of the flow factor at radius r.

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 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).

Figure 3.35 Spanwise variation of combined tip/root loss factor for a three blade turbine optimised for a tip speed ratio of 6 and with a blade root at 20% span.

If μ_R is the normalised root radius, then the root loss factor can be determined by modifying the tip‐loss factor of Eq. (3.82):

      (3.84)

      If Eq. (3.82) is now termed f_T(r) the complete tip/root loss factor is

      (3.85)

      3.9.5 Effect of tip‐loss on optimum blade design and power

      With no tip‐loss the optimum axial flow induction factor is uniformly 1/3 over the whole swept rotor. The presence of tip‐loss changes the optimum value of the average value of a, which reduces to zero at the edge of the wake but local to the blade tends to increase in the tip region.

      For the analysis involving induction factors from here on in this chapter, only the azimuthal averages and the local values at the blade are required so it is convenient to use a(r) and a′(r) to mean azimuthal averages at radius r with a_b = and a^′_b = for the local values at the blade, thus avoiding the need for the overbar and subscript b notation in the algebraic expressions. The inflow angle ϕ at the blade is from Eq. (3.62):

      (3.86)

      but Eq. (3.61)