Module 5.2 Wind Turbine Design (Continued) Lakshmi N Sankar lsankar@ae.gatech.edu

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.

Selection of Planform Once the airfoils are chosen, and the best lift coefficient (yielding highest Cl/Cd)at which the airfoil will operate are known, we can determine how chord “c” should vary with r. The idea is to set the axial induction factor to be equal to 1/3 – equal to the Betz limit- from root to tip. This value of induction factor yields the highest possible power from actuator disk model studies in Module 2. Optimum planforms are possible for a given tip speed ratio, but not for all tip speed ratios.

Recall Thrust Produced by an Annulus of the Rotor Disk Area = 2prdr Mass flow rate =2prr(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) = 4prr(U∞ -v)vdr dT = 4prr U∞2(1-a)adr (1) dr r

Blade Elements Captured by the Annulus Thrust generated by these blade elements: dr 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 r

Optimum variation of Chord with r Equate 1 and 2 (neglecting drag effects, which are small): small

Optimal Variation of Chord vs r 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 Number of Blades In the previous derivation of optimum variation of chord with radius, Bc/pR is a non-dimensional combination, where B is the number of blades. This quantity is called local solidity. If solidity is high, Cl can be low and the rotor can operate away from stall. On the other hand, if solidity is too high, blades are subject to extreme wind loads. This equation says we can have a large number of blades (B) with small c, or vice versa. In practice fewer number of blades (2 or 3 at most) with a large chord is preferred, both from a cost and strength perspective Blades and appendages are costly!) Larger chord, implies thicker blades that are structurally stronger.

Optimum Variation of Twist with r Angle of attack For best L/D

Selection of Tip Speed Ratio Best tip speed ratio WR/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.

Variation of Power Coefficient with Tip Speed Ratio for a Representative Rotor NREL Phase VI Rotor Recall: 16/27 is the maximum Power Coefficient (Betz Limit)

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.