Michael Graham - Wind Energy Handbook

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 Ris 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)

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)derives tan ϕ from the ratio of the non‐dimensional rate of change of angular momentum to the non‐dimensional rate of change of axial momentum, which is not changed because it deals with the average flow through the disc and so uses average values. If drag is ignored for the present, Eq. (3.62)becomes

(3.87)

Hence

which becomes

(3.88)

A great simplification can be made to Eq. (3.88)by ignoring the first term because, clearly, it disappears for much of the blade, where f = 1, and for the tip region the value of a ′2is very small. For tip speed ratios greater than 3, neglecting the first term makes negligible difference to the result:

(3.89)

As before, Eq. (3.60)still applies,

From Eq. (3.89),

Consequently,

which, combined with Eq. (3.89), gives

so

(3.90)

The radial variation of the average value of a , as given by Eq. (3.90), and the value local to the blade is shown in Figure 3.36. An exact solution would also have the local induced velocity falling to zero at the blade tip.

Figure 3.36 Axial flow factor variation with radius for a three blade turbine optimised for a tip speed ratio of 6.

Clearly, the required blade design for optimal operation would be a little different to that which corresponds to the Prandtl tip‐loss factor because a b= ; the local flow factor does not fall to zero at the blade tip. The use of the Prandtl tip‐loss factor leads to an approximation, but that was recognised from the outset.

The blade design, which gives optimum power output, can now be determined by adapting Eqs. (3.70)and (3.71), noting that the left hand side of Eq. (3.70)refers to a local inflow angle at the blade, hence the factor becomes ( 1 – a/f ):

Introducing Eq. (3.89)gives

(3.91)

The blade geometry parameter given by Eq. (3.91)is shown in Figure 3.37compared with the design that excludes tip‐loss. As shown, only in the tip region is there any difference between the two designs.

Similarly, the inflow angle distribution, shown in Figure 3.38, can be determined by suitably modifying Eq. (3.73):

(3.92)

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.