Optimizing Wind Turbine Design: Chord Length, Blade Number, and Tip Speed Ratio, Slides of Environmental Law and Policy

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Module 5.

Wind Turbine Design (Continued)

OVERVIEW

• In Module 5.1, we gave preliminary comments about rotor design.
• We reviewed the possible approaches to rotor design (parametric sweep, optimization, inverse design, genetic algorithm). - These may be combined. - For example, a response surface (or a carpet plot) of the power production as a function of design variables may be curve fitted, and searched for an optimum combination.
• While increasing the rotor radius is a good way of increasing power (since power varies as swept area) this greatly increases the weight and ultimately the cost of the system.
• Other parameters should also be optimized.
• In Module 5.1, we also looked at some available airfoils and their characteristics.

Recall Thrust Produced by an Annulus

of the Rotor Disk

r

dr

Area = 2πrdr

Mass flow rate =2πrρ(U∞ -v)dr

Change in induced velocity = 2v

Thrust produced over this annulus= dT dT = (Mass flow rate) * (2v, i.e. Twice the induced velocity at the annulus) = 4πρr(U∞ -v)vdr dT = 4πρr U∞^2 (1-a)adr (1)

Blade Elements Captured by the

Annulus

r

dr

Thrust generated by these blade elements:

dT B VTotal cCl dr 2

=^1 ρ^2

Some blade sections near the root and tip may not behave like 2-D sections. This is due to a loss of lift as pressure Tends to equalize between upper and lower sides of the rot and tip. We correct this with a loss factor F

[ cos ( ) sin( )] F (2) 2

dT B^1 V^2 c C C dr = ρ (^) Total l φ + d φ

Optimal Variation of Chord vs r

Tip Speed Ratio U

R

where

1 1

9

16 2

=

Ω

=

∞

λ

π R Cl λ

Bc

Local solidity

Variation of Chord with r for

Optimum Rotors

• The previous slide states that chord should

vary as 1/r , large near the root and small near the tip.

• In practice, linear tapered blades are easier to

manufacture.

• The design variables – root and tip chord- are

parametrically varied, with a linear taper, to find optimum combinations.

Optimum Variation of Twist with r

R

r

R

r a r

U a

1 1 3

2 tan

1 1 3

2 1

( 1 ) tan

λ

β α

λ

φ

• =

≈ Ω + ′

− = ∞

Twist

Angle of attack For best L/D

Selection of Tip Speed Ratio

• Best tip speed ratio ΩR/U∞ may be found by a parametric sweep, using a computer code such as WT_PERF or a spreadsheet based analysis.
• Initially, as tip speed increases, for a fixed wind speed, f increases increasing the propulsive force. - Power increases, but optimum induced velocity has not been realized yet. Efficiency is low.
• As tip speed further rises, efficiency rises and peaks.
• At higher tip speeds, the airfoil sections begin to operate at non-optimum angles of attack, and propulsive force decreases. - Power decreases.

In summary..

• Keep number of blades small (2 or 3).
• Keep solidity sufficiently high to avoid stall, but small

enough to avoid extreme airloads as well.

• Use linear taper ratio for simplicity in manufacturing.
• Consider nonlinear twist to keep induction factor

close to 1/3 over most of the rotor.

• Nonlinear twist is easily accommodated in modern wind turbines.
• Operate, if possible, at optimum speed ratios where

power production peaks.