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

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5.7 Wind turbine design in practice

202

There is also no recommendation in the theory on how to arrange the individual

blade elements (sections) along the radius with respect to the radial line. Only the

blade twist angle

is determined in relation to the plane of rotation.

For a steam turbine, where massive blades are used, the centres of gravity are

arranged along the radial line due to the large centrifugal forces.

For wind turbines, the elastic axis of the spar is often located in the range

between 0.25 c and 0.30 c of the chord length. Here the point of attack of the aero-

dynamic forces is located for the attached flow (cf. Fig. 3-9, 5-26 and 5-27). These

forces then bend the spar, but they do not elastically twist it. That way, the angle

of attack remains unchanged.

0.2

0.4

0.6

0.8

Fig. 5-27 Alignment of blade sections

The result of this design procedure is a rotor that has an optimum power coeffi-

cient when running at its design tip speed ratio. This is an advantage for variable-

speed turbines, as their electronics allow the optimum rotational speed to be

adjusted for the actual wind velocity, cf. chapters 11, 12, 13. For stall turbines,

further design parameters for the rotor geometry should be included. In that case,

the goal is a wide, flat c

maximum, and a controlled flow separation (stall) at low

tip speed ratios. Therefore, these rotor blades are to some extent different to the

Betz/Schmitz design.

For wind turbines with a blade pitch mechanism, it is best to keep the torque

around the pitch axis caused by the aerodynamic forces small, since it stresses the

pitch mechanism. With common pitch-controlled wind turbines which turn their

leading edge into the wind for power reduction, the air flow stays attached to the

blade. Therefore, during control the point of action of the aerodynamic forces

5 Blade geometry according to Betz and Schmitz

203

remains near to c/3 where the centre of elasticity of the spar was laid down during

the design. Hence, the axis of rotation of the pitch mechanism is also located here.

However, if the pitch direction is set to “active-stall”, i.e. to flow separation,

the axis of rotation of the pitch mechanism will be located near to c/2 because the

point of action of the aerodynamic forces is found here for separated flow (Fig.

5-28).

Further aspects regarding the choice of the design tip speed ratio, the number of

blades and the profiles will be discussed at the end of the next chapter.

0.0

0.2

0.4

0.6

0.8

1.0

Regeln auf Fahne

ctive-Stall

Pitch to feather Active Stall

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Regeln auf Fahne

ctive-Stall

Pitch to feather Active Stall

1.0

0.8

0.6

0.4

0.2

0.0

Fig. 5-28 Alignment of blade sections (blade pitch axis), left: pitch to feather, right: active-stall

(LM34.0P and LM35.0P)

5.8 Final remark

The optimisation of the extracted power including the effects of the wake rotation

in the downstream flow was presented here according to J. Maurer’s description of

Schmitz publication from 1956 [2]. It is easy to understand since only the relative

velocity angle

I

of the attacking flow is varied in order to find the maximum

power with equation (5.56).

The Anglo-American literature [15, 16, 17] is mostly based on the earlier

works of Glauert 1935 [7]. There, as already presented in Fig. 5-3, the axial decel-

eration a in the plane of rotation of the flow and the radial a’ are considered sepa-

rately as induction factors.

5.8 Final remark

204

and

' . (5.70)

This complicates the algebra needed to find the optimum, but is comprehensive

from the historical point of view: first came Betz who considered only the down-

stream losses due to the axial wind velocity deceleration 'v, and later this was fol-

lowed by the more precise consideration of the wake rotation losses from 'u.

However, both factors a and a’ are interdependent, see Fig. 5-21. Here the flow

angle

of the relative velocity is, on the one hand

tan

2.ax

:

(5.71)





and on the other

tan

. (5.72)

So the relative velocity angle

determines both a and a’. Equating the above

expressions gives the relation between the induction factors (cf. equation (5–22) in

Wilson, [15])



aaaa

)1(''1

 

, (5.73)

factors, however, which are not needed in the presentation of the theory by

Schmitz.

Since the physical assumptions of Glauert and Schmitz are the same, the results

of the optimisation are identical! The somewhat more complicated older presenta-

tion of Glauert blurs this fact a little. The easy to remember result optimisation of

Schmitz (and Glauert) is visualised in the following table:

5 Blade geometry according to Betz and Schmitz

205

Betz Glauert - Schmitz

Far upstream of the rotor



wind

In the rotor plane

III

Far downstream of

the rotor



The induction factors can be derived from Fig. 5-21 and equation (5.53) from the

Schmitz and Maurer descriptions:

'

= 2 w

sin(

–

) cos

(5.74)

'u = 2 w

sin(

–

) sinI (5.75)

They are the axial and radial components of 'w. Replacing w

with

= v

/ sin

= :r / cos

, a and a’ are obtained by rearranging the definition,

equation (5.70)





cos

sin



a and (5.76)





sin

cos

sin



. (5.77)

Design tip speed ratio

at the maximum radius R and the local tip speed ratio

in the ring section at the radius r and the relative velocity angle

are related

tan

. (5.78)

Inserting the result of the optimisation by Schmitz

= 2

/3 , the induction

factors are determined:



3/2cos

sin

3/sin

a and







3/2sin

cos

3/sin

a . (5.79)

5.8 Final remark

206

They are presented in Fig. 5-29 (cf. table 5-2 in [15]). The curve of the axial

induction factor a clearly shows, on the one hand, that for higher tip speed ratios the

consideration a = 1/3 of Betz is valid. On the other hand, it is obvious how strong

the wake rotation is when

r/R < 1. That is the reason for the drastic increase of

the wake rotation losses in this range, cf. Fig. 5-24, as well as the quite different

blade shape near to the hub, cf. Fig. 5-22.

0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 10

Induction factors a and a'

1/3

0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 10

Induction factors a and a'

1/3

Fig. 5-29 Induction factors a and a’ according to Glauert and Schmitz

Note:

In his original paper, [1], Betz counted the angle of the relative velocity w not in

relation to the rotor plane (angle

, e.g. in Fig. 5-13) but relative to the rotation

axis of the machine (angle

= 90°-

I

). For historical reasons, this nomenclature

was used in the first three German editions and the first English edition of this

book. But this caused a break in the description since, from chapter 5.6 on, the

angle

according to Schmitz is more suitable when including the effects from wake

rotation in the calculations. For didactical reasons, we now use the relative velo-

city angle

in the section on the simpler Betz theory. The main difference is that

now in the equations (5.24) to (5.35) all the terms of cos

J

are replaced by sin

I

5 Blade geometry according to Betz and Schmitz

207

References

[1] Betz, A.: Wind-Energie und ihre Ausnutzung durch Windmühlen (Wind energy and its

utilization through windmills), Vandenhoeck & Ruprecht, Göttingen 1926, reprint:

Öko-Buchverlag Kassel 1982

[2] Schmitz, G.: Theorie und Entwurf von Windrädern optimaler Leistung (Theory and

design of windwheels with an optimum performance), Wiss. Zeitschrift der Universität

Rostock, 5. Jahrgang 1955/56

[3] Riegels: Aerodynamische Profile (Aerodynamic profiles), R. Oldenbourg Verlag,

München 1958

[4] Althaus, D.: Profilpolaren für den Modellflug (Profile polar lines for aeromodelling),

Neckar-Verlag, VS-Villingen 1980

[5] Althaus, D.: Stuttgarter Profilkatalog (Stuttgart profile catalogue), Vieweg-

Verlagsgesellschaft, Braunschweig 1981

[6] Betz, A.: Einführung in die Theorie der Strömungsmaschinen (Introduction to the

theory of turbo machinery), Verlag G. Braun, Karlsruhe 1959

[7] Glauert, H.: Section "Airplane Propellers" in: Durand W.F.: Aerodynamic Theory,

Springer Verlag, Berlin 1935, Reprint Verlag Peter Smith. Mass. US. 1976

[8] Maurer, J.: Wind turbinen mit Schlaggelenkrotoren - Baugrenzen und dynamisches

Verhalten (Wind turbines with flapping hinge rotors - constructional limits and

dynamic behaviour), VDI Verlag, Reihe 11, Nr. 173, Düsseldorf 1992

[9] Paulsen, U.S.: Aerodynamics of a full-scale, non rotating wind turbine blade under

natural wind conditions, Risø National Laboratory, Roskilde Denmark 1989

[10] Hau, E.: Windkraftanlagen (Wind turbines), Springer-Verlag, Stuttgart 1988

[11] Sass, F. (Hrsg.): Dubbels Taschenbuch für den Maschinenbau 11 Aufl. (Dubbel’s

paperback for mechanical engineering, 11th edition), Springer-Verlag, Berlin 1955

[12] Miller, R.v (Hrsg.): Energietechnik und Kraftmaschinen 6 (Power engineering and

power engines 6, Technical encyclopedia), Techniklexikon

Rowohlt Taschenbuch-Verlag, Hamburg 1972

[13] Smith, H.: The illustrated guide to aerodynamics, TAB Books Inc, USA 1985

[14] Le Gourierès, D.: Wind Power Plants, Pergamon Press, Oxford a. New York, 1982

[15] Spera, D.A. (editor): Wind Turbine Technology, ASME Press, New York, 1994, 2nd

printing 1995

[16] Burton, T. et al.: Wind Energy Handbook, John Wiley & Sons, Chichester (UK), New

York, 2001

[17] Manwell, I.F. et al.: Wind Energy Explained, John Wiley & Sons, Chichester (UK),

New York 2002

[18] Hansen, Martin O.L.: Aerodynamics of Wind Turbines, James & James, London (UK),

2000

[19] Althaus, D.: Niedriggeschwindigkeitsprofile (Aerodynamic profiles for low Reynolds

numbers), Vieweg Verlag, Wiesbaden 1996

R. Gasch and J. Twele (eds.), Wind Power Plants: Fundamentals, Design, Construction 208

and Operation, DOI 10.1007/978-3-642-22938-1_6, © Springer-Verlag Berlin Heidelberg 2012

6 Calculation of performance characteristics

and partial load behaviour

6.1 Method of calculation (blade element momentum method)

Calculating the forces and the relative velocities at the rotor blades for tip speed

ratios other than the design tip speed ratio requires a substantial effort. In the

following a method is described which is comparatively simple to understand: the

blade element momentum method.

When dimensioning the blade geometry according to Schmitz-Glauert (cf. sec-

tion 5.6), the relative velocity angle



at the plane of rotation for a given design tip

speed ratio



was determined. At this



the maximum power can be extracted

from the wind using the optimum angle of attack



. Afterwards, the blade chord

c and the twist (the total blade twist angle



) were dimensioned so that the opti-

mum flow conditions are reached during operation at design tip speed ratio.

Now blade chord and twist are given. For tip speed ratios other than



, there

are different relative velocity angles



in the plane of rotation, Fig. 6-1. For the

calculation of these relative velocity angles



, we use the same equations as for the

dimensioning of the blade chord: the lift force formula obtained using airfoil

theory and the principle of linear momentum.

We use the airfoil theory to obtain the lift force at a blade section:



cdrcwdL







(6.1)

with









cosww

c = blade chord length

dr = width of the blade section

and











= lift coefficient

  



= air density

The lift force at a blade section can also be obtained by using the principle of

linear momentum:

wmddL







(6.2)

with







sin

wdr

md 



= mass flow rate, r = local radius of the blade section,

z = number of rotor blades and













sin2 ww

6 Calculation of performance characteristics and partial load behaviour

209

 r



w

 r



w

Fig. 6-1 Triangles of velocity upstream (subscript 1), in-plane (no subscript), and downstream

(subscript 3) of the rotor,



and w at the plane of rotation

By equating the two lift forces, we obtained an equation that was used to calculate

the blade chord length c for the given relative velocity angle



. Now the blade

chord c is given, and the relative velocity angle



is unknown.



 wmdcdrcw







(6.3)

Inserting all values of equations (6.2) and (6.3), we obtain

 



AL1

cos





cdrcw

 

0sin2sincos









wwdr

. (6.4)

This equation can be reduced and simplified. The result is an equation without the

air density



, the width dr of the ring section and the undisturbed relative velocity

. Using the formula of the angle of attack



with the known twist angle



, we

obtain an equation with only one unknown variable left, the relative velocity angle



in the rotor plane.

   

0tansin









(6.5)

Unfortunately, this equation cannot be solved directly for the unknown relative

velocity angle



, but must be determined by iteration. Moreover, there are still some

defects in this equation: reasonable calculation results are only obtained in the

range around the design point



. During start-up (



) and close to load-free

idling there are additional aerodynamic effects which have not been considered up

6.1 Method of calculation (blade element momentum method)

210

to now. Moreover, the profile drag is at this stage not included in the iteration.

Section 6.8 treats the necessary extensions to equation (6.5) introducing additional

terms to transform it into an equation suitable for a performance characteristics

calculation.

The results and diagrams discussed in the following were calculated with this

extended equation (6.23). However, for the discussion it is enough to have under-

stood equation (6.5). So, the description of the calculation procedure including

losses is found in later sections.

If the relative velocity angle



is determined by the iteration, the relative velo-

city w and the lift force dL at the blade section dr considered can be calculated.

From the lift force, we obtain the blade element’s contribution to thrust, circum-

ferential force, and torque moment of the rotor:









cosww







cdrcwdL 

thrust force: dT(r) = dL cos





circumferential force: dU(r) = dL sin



(6.6)



torque moment: dM(r) = dU r

After that, we repeat the procedure for the next blade element, calculate the rela-

tive velocity angle which gives then the forces. The iteration must be repeated for

each blade element! A computer is used for this complex calculation.

The forces and torque of the entire rotor result from summing up the results of

all blade elements:

circumferential force of a blade:





)(rdUU

thrust force of the rotor:





)(rdTzT (6.7)

torque moment of the rotor:





)(rdUrzM

power of the rotor:





(6.8)

If the losses resulting from the air flow around the blade tip and the profile drag,

cf. section 6.8, and others are to be included, only equations (6.4) and (6.5) have

to be extended. The calculation procedure for the iteration remains the same.

6 Calculation of performance characteristics and partial load behaviour

211

6.2 Dimensionless presentation of the characteristic curves

As illustrated above, the most demanding calculation is the iterative determination

of the relative velocity angle



. The equation (6.4) of the lift contains neither wind

speed nor rotational speed. As a first step, dimensionless values for forces,

moments and power can be calculated for a set undisturbed relative velocity angle



These dimensionless factors can be used to calculate the thrust force, torque

moment and power of any combination of rotational speed and wind speed by

simply multiplying them with the reference values.

Two procedures are commonly used for the representation by dimensionless

characteristics:

For the wind turbine characteristics, the selected reference speed is the wind

speed v

far upstream, and the dimensionless characteristic curve is plotted versus

the tip speed ratio





R/v

(where R is the tip radius of the rotor). The charac-

teristics of propellers and helicopter rotors are calculated analogously, but the

chosen reference speed is the circumferential speed



R, and the dimensionless

curve is plotted versus the inverse tip speed ratio 1/



= v



R (degree of propaga-

tion).

In the following, we consider only the representation commonly used for wind

turbines. The chosen reference force F

dyn

is the product of dynamic pressure



/2 and the swept rotor area



dyn

vRF







reference force

 







Tdyn

cvRcFT  thrust coefficient: c

(



)

 







Mdyn

cvRcRFM  moment coefficient: c

(



)

 







Pdyn1

cvRcFvP 

power coefficient: c

(



)

The power P =



M is calculated with equation (6.8) from torque and rotational

speed. Therefore, equation (6.9) gives for the power coefficient





. (6.10)

In order to calculate the dimensionless coefficients, equation (6.5) is used once

more to determine the relative velocity angle



for each blade element. Then, it

follows from the triangles of velocities in Fig. 6-1 that











cosww and

111

sin





(6.9)