Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook

is important to note that where the incident velocity varies spatially, as here, W takes the value that would exist at the effective position of the bound vortex representing the local blade circulation excluding its own induced velocity.

Consequently,

(3.68) rho upper W normal upper Gamma sine phi equals rho normal upper Gamma upper U Subscript infinity Baseline left-parenthesis 1 minus a right-parenthesis equals 4 pi rho StartFraction upper U Subscript infinity Baseline Superscript 3 Baseline Over normal upper Omega EndFraction a left-parenthesis 1 minus a right-parenthesis squared

(3.69) normal upper Gamma equals 4 pi StartFraction upper U Subscript infinity Baseline Superscript 2 Baseline Over normal upper Omega EndFraction a left-parenthesis 1 minus a right-parenthesis

If, therefore, a is to take everywhere the optimum value (1/3), the circulation must be uniform along the blade span, and this is a condition for optimised operation.

To determine the blade geometry, that is, how should the chord size vary along the blade and what pitch angle β distribution is necessary, neglecting the effect of drag, we must return to Eq. (3.52) with C_D set to zero:

StartFraction upper W squared Over upper U Subscript infinity Baseline Superscript 2 Baseline EndFraction upper B StartFraction c Over upper R EndFraction upper C Subscript l Baseline sine phi equals 8 pi lamda mu squared a prime left-parenthesis 1 minus a right-parenthesis

substituting for sinϕ gives

(3.70) StartFraction upper W Over upper U Subscript infinity Baseline EndFraction upper B StartFraction c Over upper R EndFraction upper C Subscript l Baseline left-parenthesis 1 minus a right-parenthesis equals 8 pi lamda mu squared a prime left-parenthesis 1 minus a right-parenthesis

The value of the lift coefficient C_l in the above equation is an input, and it is commonly included as above on the left side of Eq. (3.70) with a ‘chord solidity’ parameter representing blade geometry. The lift coefficient can be chosen as that value that corresponds to the maximum lift/drag ratio StartFraction upper C Subscript l Baseline Over upper C Subscript d Baseline EndFraction _, as this will minimise drag losses: even though drag has been ignored in the determination of the optimum flow induction factors and blade geometry, it cannot be ignored in the calculation of torque and power. Blade geometry also depends upon the tip speed ratio λ_, which is also an input. From Eq. (3.70) the blade geometry parameter can be expressed as

StartFraction upper B Over 2 pi EndFraction StartFraction c Over upper R EndFraction equals StartStartFraction 4 lamda mu squared a prime OverOver StartFraction upper W Over upper U Subscript infinity Baseline EndFraction upper C Subscript l Baseline EndEndFraction

Hence

(3.71)

Introducing the optimum conditions of Eq. (3.67),

(3.72)

The parameter λμ is the local speed ratio λr and is equal to the tip speed ratio where μ = 1.

If, for a given design, C_l is held constant, then Figure 3.17 shows the blade plan‐form for increasing tip speed ratio. A high design tip speed ratio would require a long, slender blade (high aspect ratio) whilst a low design tip speed ratio would need a short, fat blade. The design tip speed ratio is that at which optimum performance is achieved. Operating a rotor at other than the design tip speed ratio gives a less than optimum performance even in ideal drag‐free conditions.

In off‐optimum operation, the axial inflow factor is not uniformly equal to 1/3; in fact, it is not uniform at all.

The local inflow angle ϕ at each blade station also varies along the blade span, as shown in Eq. (3.73) and Figure 3.18:

(3.73) phi equals tangent Superscript negative 1 Baseline left-brace StartFraction 1 minus a Over lamda mu left-parenthesis 1 plus a prime right-parenthesis EndFraction right-brace

Graph depicts the variation of blade geometry parameter with local speed ratio.

Figure 3.17 Variation of blade geometry parameter with local speed ratio.

Graph depicts the variation of inflow angle with local speed ratio.

Figure 3.18 Variation of inflow angle with local speed ratio.

which, for optimum operation, is

(3.74) phi equals tangent Superscript negative 1 Baseline left-brace StartStartFraction 1 minus one third OverOver lamda mu left-parenthesis 1 plus StartFraction 2 Over 9 lamda squared mu squared EndFraction right-parenthesis EndEndFraction right-brace

Close to the blade root the inflow angle is large, which could cause the blade to stall in that region. If the lift coefficient is

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