Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation

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6.3 Dimensionless characteristic curves of a turbine with a high tip speed ratio

212

Thus we find



sin

cos











dyn





drc

Ld 

and

thrust coefficient:



cos

dc 













torque moment coefficient:



sin

dc 













power coefficient:





























Summing up the coefficients of all blade elements along the radius, one point of

the dimensionless characteristic curve is obtained.

6.3 Dimensionless characteristic curves of a turbine with a high

tip speed ratio

The coefficients for power, torque and thrust were calculated for a wind turbine

with a high design tip speed ratio



= 7, using the method described in the previ-

ous section. The three blades of the rotor are equipped with the profile FX 63-173

(e.g. in [1]), and they were designed according to Schmitz. In the calculation of

the characteristics, tip and profile losses are taken into account, although they will

only be discussed in section 6.8.

The maximum power coefficient of wind turbines with a high tip speed ratio is

found at the optimum tip speed ratio



opt

close to



and is clearly lower than the

ideal value c

= 16/27, Fig. 6-2 gives



opt

= 6.5 and c

P.opt

= 0.52. This is due to the

actually occurring profile drag and tip losses which was not considered during the

choice of the design tip speed ratio



= 7.0. For turbines with a high tip speed

ratio, the profile drag is nearly parallel to the circumferential speed but has the

opposite orientation; therefore it reduces the extracted power.

Typically, turbines with a high tip speed ratio have a low torque coefficient

during start-up at



= 0 (rotational speed n = 0 rpm), Fig. 6-3. This is the reason

for their poor self-starting behaviour. Due to c

= c



, the torque moment coef-

ficient can be read directly from the c



curve. For



= 0, the torque moment

coefficient is the slope of the c



curve.

6 Calculation of performance characteristics and partial load behaviour

213

The maximum torque moment coefficient is obtained by drawing the tangent

starting from the origin of coordinates against the c



curve. This torque moment

coefficient is important for the dimensioning of an emergency brake.

As the tip speed ratio increases, the power and torque moment coefficient

become smaller again. At approximately twice the design tip speed ratio both co-

efficients are zero. This (load-free) idling tip speed ratio of the turbine is reached

if the turbine is over speeding when it runs without load.

The thrust coefficient increases steadily with higher tip speed ratios, Fig. 6-4.

Turbines with high tip speed ratios have few and slender blades. During start-up,

almost all the wind can pass undisturbed through the plane of rotation, hence c

very small. However, as the tip speed ratio increases, the swept rotor area is pro-

gressively blocked. At the design tip speed ratio



opt

= 7, the thrust coefficient is

= 8/9 according to Betz, even after taking into account the profile and tip losses.

u = 7 v

Fig. 6-2 Wind turbine with a high design tip speed ratio (λ

= 7); power coefficient c

versus tip

speed ratio

6.3 Dimensionless characteristic curves of a turbine with a high tip speed ratio

214

Fig. 6-3 Wind turbine with a high design tip speed ratio (λ

= 7); torque moment coefficient c

versus tip speed ratio

Fig. 6-4 Wind turbine with a high design tip speed ratio (λ

= 7); thrust coefficient c

versus tip

speed ratio

6 Calculation of performance characteristics and partial load behaviour

215

When idling, the thrust coefficient is c

= 1.25 which is roughly similar to the drag

coefficient of a solid disk. It should be emphasized that for turbines with a high tip

speed ratio the large thrust force at idling is not caused by the profile drag but by

the lift which acts in this case almost parallel to the wind direction, i.e. perpen-

dicular to the swept rotor area.

6.4 Dimensionless characteristic curves of a turbine with a low

tip speed ratio

In the following, the power, torque moment and thrust coefficients for a turbine

with a low tip speed ratio of



= 1 will be discussed. The 15 blades of the rotor

are also designed with the high-quality Wortmann profile FX 63-137. Though this

profile would never be used for the design of a turbine with a low tip speed ratio,

it is applied in our case to allow performance comparison and to illustrate the

general differences to turbines with a high tip speed ratio.



Fig. 6-5 Wind turbine with a low design tip speed ratio (λ

= 1); power coefficient c

versus tip

speed ratio

6.4 Dimensionless characteristic curves of a turbine with a low tip speed ratio

216

Fig. 6-6 Wind turbine with a low design tip speed ratio (λ

= 1); torque moment coefficient c

versus tip speed ratio

Fig. 6-7 Wind turbine with a low design tip speed ratio (λ

= 1); thrust coefficient c

versus tip

speed ratio

6 Calculation of performance characteristics and partial load behaviour

217

The maximum power coefficient of 0.43 (Fig. 6-5) is well below c

= 16/27 and

much lower than for the turbine with a high tip speed ratio. This time, it occurs at

a tip speed ratio of



opt

>



and is due to the wake rotation losses considered in

the calculations which decrease significantly for tip speed ratios of



>1, cf. Fig.

5-24. The profile drag has only little influence on these characteristics.

The high torque moment coefficient during start-up at



= 0 (Fig. 6-6) is a typical

feature of turbines with a low tip speed ratio. This results from the large blade

twist angle



and from the high solidity. The large twist angle causes large lift

forces at the profile, even during start-up. Turbines with a low tip speed ratio,

therefore, are suitable for driving piston pumps (cf. chapter 10) or other machines

which require a high starting torque.

The high solidity also causes high thrust coefficients during start-up, Fig. 6-7. The

reduction of the thrust coefficients of turbines with a low tip speed ratio during

idling is due to the fact that some regions of the blades with a disadvantageous in-

flow show negative angles of attack and therefore also negative lift coefficients

when idling. These regions then accelerate the air flow (i.e. “acting as a ventila-

tor”) instead of decelerating them (i.e. “acting as a turbine”), cf. section 6.6.2. A

summarizing overview of the characteristic features of wind turbines with high

and low tip speed ratio is given in section 6.6.1.

6.5 Turbine performance characteristics

How can we now use the dimensionless curves presented to determine power,

torque moment and thrust for given rotational speeds and wind speeds?

- First, the tip radius R of the rotor is selected.

- The wind speed v

is also given, e.g. the wind speed with the highest con-

tribution to the energy yield, cf. chapter 4. Now the reference force F

dyn

calculated

dyn

vRF





 . (6.10)

- Then, a choice is made of the rotational speed n at which the turbine

should run, e.g. according to the chosen generator and gearbox. From the

rotational speed n, the angular speed



is obtained which is used to deter-

mine the tip speed ratio:

R





with



 in 1/s or rad/s and n in rpm.

6.5 Turbine performance characteristics

218

- The characteristic curves give us the dimensionless coefficients for the

determined tip speed ratio, and thrust force, torque moment and power can

be calculated:

If for the same wind speed the values of T, M or P are now calculated for different

rotational speed, the same reference force F

dyn

can be used. If the wind speed is

changed, the reference force F

dyn

must be recalculated.

Using this method, the power curve, power versus rotational speed for different

constant wind speeds in Fig. 6-8, of a wind turbine with a high tip speed ratio and

a rotor diameter of D = 4 m was determined. The used dimensionless c



curve of

the turbine with a high tip speed ratio is the one shown in Fig. 6-2, section 6.3. For

the same turbine Fig. 6-9 shows the power versus the wind speed for different

constant rotational speeds.

Fig. 6-8 Power versus rotational speed at different wind speeds for a wind turbine with D = 4 m,

-λ characteristics from Fig. 6-2

Thrust : T = F

dyn

(

)

Torque moment: M = R F

dyn

(

)

Power: P = v

dyn

(

)

6 Calculation of performance characteristics and partial load behaviour

219

02468101214

Fig. 6-9 Power versus wind speed at different rotational speeds for a wind turbine with D = 4 m,

-λ characteristics from Fig. 6-2

6.6 Flow conditions

6.6.1 Turbines with high and low tip speed ratio: a summary

This section summarizes once again the findings of chapters 5 and 6, and high-

lights the differences between turbines with a high tip speed ratio and those with a

low tip speed ratio. The power and torque moment coefficients of wind turbines

with different design tip speed ratios are given in Fig. 6-10.

6.6 Flow conditions

220

Turbines with a low

tip speed ratio

Turbines with a high

tip speed ratio

AB C D E

1345678910

0.1

0.2

0.3

0.4

0.5

0.6

0.3

0.4

0.2

0.1

Fig. 6-10 Power and torque moment coefficients of different wind turbines varying in design

and in tip speed ratio (Fateev [6], modified)

Power

- The power of a wind turbine increases proportionally to v

, equation

(6.9).

- The maximum power coefficient c

is always smaller than 16/27.

- For turbines with a high tip speed ratio, c

Pmax

is basically only affected by

the profile drag. The optimum tip speed ratio reached is always slightly

smaller than the design tip speed ratio,



opt



- For turbines with a low tip speed ratio, c

Pmax

is mainly reduced by the wake

rotation of the air; hence in this case



opt

is always slightly larger than



6 Calculation of performance characteristics and partial load behaviour

221

Torque moment

- The torque moment of a wind turbine increases proportionally to v

- The maximum moment coefficient is always found at tip speed ratios of



<



, based on c

= c





- Turbines with a high tip speed ratio have small torque moment coefficients

during start-up at



= 0.

- Turbines with a low tip speed ratio have large torque moment coefficients

during start-up.

Thrust force

- The thrust force of a wind turbine increases proportionally to v

- Turbines with a high and a low tip speed ratio both have approximately the

same thrust coefficient at the design point, c

 8/9 (according to Betz).

- Turbines with a high tip speed ratio have small thrust coefficients at start-

up, and the thrust coefficient increases for higher tip speed ratios (c

is at

load-free idling nearly as high as for a solid disk).

- Turbines with a low tip speed ratio have large thrust coefficients at start-

up; these decrease with larger tip speed ratios (at load-free idling partly

“acting as a ventilator”).

Idling

- In general, the idling tip speed ratio is approximately twice the design tip

speed ratio.

- For turbines with a high tip speed ratio, the idling tip speed ratio is limited

by the profile drag.

- For turbines with a low tip speed ratio, the profile drag has little effect on

the idling tip speed ratio.

Blades

- Turbines with a high tip speed ratio have few and slender blades, which are

smaller at the outer radius than at the inner radius. The blades require a

high-quality surface; airfoil profiles with a high lift/drag ratio have to be

used. There is heavy stress on the few blades due to aerodynamic and iner-

tia forces.

- Turbines with a low tip speed ratio have a larger number of blades that

have constant width, or are even wider at the outer radius than at the inner

radius. There are no special requirements for the blade surface and the