# OPTIMAL ROTOR TIP SPEED RATIO

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2 where: V v r r speed of rotor tip v r Tip speed ratio: = (1) wind speed V V is the wind speed [m/sec] is velocity of rotor tip [m/sec] is rotor radius [m] =2 f is the angular velocity [radian/sec] f -1 is the frequency of rotation [Hz], [sec ] This dimensionless factor arises from the detailed treatment of the aerodynamic theory of wind power extraction. EXAMPLE At a wind speed of 15 m/sec, blade radius of 10 m, rotating at 1 rotation per second: EXAMPLE rotation f 1[ ], sec radian 2 f 2 [ ] sec m v r [ ] sec r V The Suzlon S.66/1250, 1.25 MW rated power at 12 m/s rated wind speed wind turbine design has a rotor diameter of 66 meters and a rotational speed of rpm. Its angular speed range is: 2 f revolutions minute 2 [radian.. ] 60 minute second radian [ ] sec The range of its rotor s tip speed can be estimated as:

3 v r 66 ( ) 2 m [ ] sec The range of its tip speed ratio is thus: r V OPTIMAL ROTOR TIP SPEED RATIO The optimal tip speed ratio for maximum power extraction is inferred by relating the time taken for the disturbed wind to reestablish itself t w to the time taken for a rotor blade of rotational frequency to move into the position occupied by its predecessor ts. For an n bladed rotor, the time period for the blade to move to its predecessor s position is given by: t s 2 [sec] (2) n If the length of the strongly disturbed air stream upwind and downwind of the rotor is s, then the time period for the wind to return to normal is given by: t w s [sec] (3) V If t s > t w, then some wind is unaffected. If t w > t s, then some wind is not allowed to flow through the rotor. The maximum power extraction occurs when these two time periods are about equal: t s t w 2 s n 2 n V V s (4) From which the optimal rotational frequency is: 2V opt (5) ns

5 4 opt ( ) If poorly designed blades are used resulting in a tip speed ratio that is too low, the wind turbine would have a tendency to slow and to stall. If the tip speed ratio is too high, the turbine will rotate very fast through turbulent air, and the power will not be only optimally extracted from the wind stream, but the turbine will be highly stressed at the risk of catastrophic failure. POWER COEFFICIENT, Cp by: The power generated by the kinetic energy of a free flowing wind stream is given 1 [ ] (8) 2 3 P SV Watt The cross sectional area S of the turbine in terms of its blade radius R is given by: 2 2 [ ] (9) S R m From which the power P becomes: P R V (10) The power coefficient is defined as the power extracted by the turbine relative to that available in the wind stream: C p Pt Pt P 1 (11) 2 3 RV 2 The maximum achievable power factor is percent, and is designated as the Betz limit. In practice, values of obtainable power coefficients are in the range of 45 percent. This value below the theoretical limit is caused by the inefficiencies and losses attributed to different configurations, rotor blades and turbine designs.

6 Power Coefficient Betz limit = 0.59 Uncaptured power Generated power TSP=constant 0.1 Cut in speed Cut out speed Tip Speed Ratio. TSP Figure 1. Power coefficient as a function of tip speed ratio for a two bladed rotor. Maximum power extraction occurs at the optimal tip speed ratio. The uncaptured power is caused by the fact that the tip speed ratio is not constant as well as the inherent inefficiencies and losses in different turbine designs. INEFFICIENCIES AND LOSSES The inefficiencies and losses encountered in the operation of wind turbines include the blade number losses, whirlpool losses, end losses and the airfoil profile losses. AIRFOIL PROFILE LOSSES The slip or slide number s is the ratio of the uplift force coefficient of the airfoil profile used CL to the drag force coefficient CD is: s C C L (12) D Accounting for the drag force can be achieved by using the profile efficiency that is a function of the slip number s and the tip speed ratio λ as: profile s 1 (13) s s

7 ROTOR TIP END LOSSES At the tip of the rotor blade an air flow occurs from the lower side of the airfoil profile to the upper side. This air flow couples with the incoming air flow to the blade. The combined air flow results in a rotor tip end efficiency, ηtip end. WHIRLPOOL LOSSES In the idealized derivation of Betz law, the wind does not change its direction after it encounters the turbine rotor blades. In fact, it does change its direction after the encounter. This is accounted for by a modified form of the power coefficient known as the Schmitz power coefficient CpSchmitz if the same airfoil design is used throughout the rotor blade. Table 1. Whirlpool losses Schmitz power coefficient as a function of the tip speed ratio. Tip speed ratio TSR λ Whirpool Schmitz Power Coefficient CpSchmitz

8 Figure 2. Schmitz power coefficient as a function of the tip speed ratio, TSR. ROTOR BLADE NUMBER LOSSES A theory developed by Schmitz and Glauert applies to wind turbines with four or less rotor blades. In a turbine with more than four blades, the air movement becomes too complex for a strict theoretical treatment and an empirical approach is adopted. This can be accounted for by a rotor blades number efficiency ηblades. In view of the associated losses and inefficiencies, the power coefficient can be expressed as: Cp CpSchmitz profile tip endblades (14) There are still even more efficiencies involved: 1. Frictional losses in the bearings and gears: ηfriction 2. Magnetic drag and electrical resistance losses in the generator or alternator: ηelectrical. C' p C pschmitz profile tip end blades friction electrical (15) In the end, the Betz limit is an idealization and a design goal that designers try to reach in a real world turbine. A Cp value of 35 percent is a realistic design goal for a workable wind turbine. This is still reduced by an intermittency factor accounting for the periods of wind flow as the intermittency factor. TIP SPEED RATIOS OF DIFFERENT DESIGNS

9 The theoretical maximum efficiency of a wind turbine is given by the Betz limit around 59 percent. Practically, wind turbines operate below the Betz limit. In Fig. 1 for a two bladed turbine, if it is operated at the optimal tip speed ratio of 6, its power coefficient would be around At the cut-in speed, the power coefficient is just 0.10, and at the cut-out speed it is This suggests that for maximum power extraction a wind turbine should be operated around its optimal wind tip ratio. Modern horizontal axis wind turbine rotors consist of two or three thin blades and are designated as low solidity rotors. This implies a low fraction of the area swept by the rotors being solid. Its configuration results in an optimum match to the frequency requirements of modern electricity generators and also minimizes the size of the gearbox required as well as increases efficiency. Such an arrangement results in a relatively high tip speed ratio in comparison with rotors with a high number of blades such the highly successful American wind mill used for water pumping in the American West and all over the world. The latter required a high starting torque. The relationship between the rotor coefficient C p and the tip speed ratio is shown for different types of wind machines. It can be noticed that it reaches a maximum at different positions for different machine designs. Figure 3. The power coefficient Cp as a function of the tip speed ratio for different wind machines designs. Note that the efficiency curves of the Savonius and the American multi-blade designs were inadvertently switched in some previous publications, discouraging the study of the Savonius design.

10 The maximum efficiencies of the two bladed design, the Darrieus concept and the Savonius reach levels above 30 percent but below the Betz limit of 59 percent. The American multibladed design and the historical Dutch four bladed designs peak at 15 percent. These are not suited for electrical generation but are ideal for water pumping. DISCUSSION Wind turbines must be designed to operate at their optimal wind tip speed ratio in order to extract as much power as possible from the wind stream. When a rotor blade passes through the air stream it leaves a turbulent wake in its path. If the next blade in the rotating rotor arrives at the wake when the air is still turbulent, it will not be able to extract power from the wind efficiently, and will be subjected to high vibration stresses. If the rotor rotated slower, the air hitting each rotor blade would no longer be turbulent. This is another reason for the tip speed ratio to be selected so that the rotor blades do not pass through turbulent air. EXERCISE 1. For a wind speed of 15 m/s and a 3 bladed rotor radius of 10 meters rotating at 1 rotation / sec, calculate: 1. The angular rotational frequency, 2. The rotor tip speed, 3. The tip speed ratio. Compare this value to the optimal tip speed ratio. Repeat the comparison for a 2 bladed turbine. REFERENCES 1. N. S. Çetin, M. A Yurdusev, R. Ata and A. Özdemir, Assessment of Optimum Tip Speed Ratio of Wind Turbines, Mathematical and Computational Applications, Vol. 10, No.1, pp , E. Hau, Windkraftanlangen, Springer Verlag, Berlin, Germany, pp , B. Jones, Elements of Aerodynamics, John Wiley and Sons, New York, USA, pp , 1950.

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