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

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5.6 The Schmitz dimensioning taking into account the rotational wake

192

be adjusted further for the profile and tip losses, see Fig. 5-25. The immense loss

of 30% due to the torque reaction also affects the profile geometry of an optimally

designed wind turbine. The blade chord and the blade twist along the blade will be

different from the Betz dimensioning, as this is too simple for this case. As the

losses due to rotational wake are, in any case, relevant for the optimum blade

geometry, its calculation will be discussed in the next section.

5.6 The Schmitz dimensioning taking into account the rotational

wake

Betz assumed that the air flow is retarded from the far upstream velocity v

via the

velocity v

= (v

+ v

) /2 in the rotor plane, to the far downstream velocity

= v

/3, without any change to its exclusively linear axial direction.

However, Schmitz (and before him Glauert for his propeller calculation) takes into

account the circumferential component ¨u of the wake rotation which is an inevi-

table result of action = reaction.

Fig. 5-20 Plan view of the air flow in a ring section, downstream flow with circumferential

component ǻu

This circumferential component is equal to zero far upstream of the rotor and ¨u

far downstream of the rotor, Fig. 5-20. It is created when the flow passes the

blade.

Fig. 5-21 shows the relative velocity w at the airfoil, which is made up of

Betz’s axial velocity v

2.ax

= v

– 'v/2 according to Betz, and the increased

circumferential velocity

5 Blade geometry according to Betz and Schmitz

193

u = ȍ r +

. (5.49)

Since the additional circumferential component ¨u is created as the flow passes

over the blade, the mean value ¨u/2 of the value “before” and “after” is used (This

assumption is an analogy to the Froude-Rankine theorem, section 5.1. A more pro-

found theoretical consideration and proof is found in [2]).

The magnitude of ǻu will depend on the design tip speed ratio Ȝ

of the wind

turbine with optimum power extraction. The velocity triangle formed by v

and

ȍ r, giving w

and

, is that which would arise if the flow is not decelerated at

all by the rotor, Fig. 5-21.The change ¨w from the relative velocity w

far up-

stream the rotor to the relative velocity w

far downstream is caused by the airfoil

effect. According to the principle of momentum “mass flow rate times change in

velocity equals the force” this creates in the rotor ring section of the width dr the

lift force dL which is perpendicular to w and in parallel to ¨w,

mdwdL



. (5.50)

Machine axis

Rotor plane

Schmitz Betz

ȍr

2.ax

3.ax

Machine axis

Rotor plane

Schmitz Betz

ȍr

2.ax

3.ax

Fig. 5-21 Triangles of relative velocity upstream (subscript 1), in the rotor plane (no subscript),

and downstream (subscript 3) of the rotor; angle

of relative velocity direction in the rotor plane 

The mass flow rate through the ring section is

vdrrmd



. (5.51)

The power at the ring section, ignoring the drag, is obtained by

5.6 The Schmitz dimensioning taking into account the rotational wake

194

rdU





rdL

sin . (5.52)

Fig. 5-21 gives the following geometrical relations:

Relative velocity in the rotor plane: w = w

cos (



-

)

Axial component of w in the rotor plane:

2.ax

= w sin

= w

cos(



-

) · sin

I

Velocity change ¨w in the rotor plane: ¨w = 2 w

sin(



-

) (5.53)

Hence, the power depends on the angle

sinwmdrdP



)sin()(sin2)sin()cos(2

1111



wwdrrr  5.54)





IIISU

sin2sin2 : wdrrdP (5.55)

The derivation dP/d

= 0 gives the relative velocity angle

providing the maxi-

mum power



IIIIIIISU

cossin2sin2sin2cos22

: wdrr













@>@

IIIIIIISU

sin2coscos2sinsin22

: wdrr







IIISU

32sinsin22

: wdrr

(5.56)

For optimal power extraction this yields the angle



, (5.57)

5 Blade geometry according to Betz and Schmitz

195

which is related to the design tip speed ratio by

tan

. (5.58)

With this relative velocity angle

= 2

/3, the lift force dL in equation (5.50) can

be obtained by

wmddL '











IIIIISU



111

sin2sincos2 wwdrr

cos

sin42

wdrrʌȡ

(5.59)

/3 , and sin (2/3

) = 2 sin (

/3) cos (

/3) .

Now, the airfoil theory is required to determine the chord length necessary for the

optimum power extraction via the lift force

Ltot

cdrcwdL

. (5.60)

After rewriting this as

Ltot

cos

cdrcw

dL (5.61)

and equating the expressions (5.55) and (5.52), we can derive the Schmitz blade

chord length formula

tot

sin

)(

. (5.62)

Distributing the total chord length

tot

over z blades, we obtain the chord length

per blade

Schmitz

sin

161

)(

. (5.63)

5.6 The Schmitz dimensioning taking into account the rotational wake

196

For small angles

i.e. for high tip speed ratios, the values obtained from the

Schmitz chord formula are equal to those of the Betz equation (5.36). This is illus-

trated in Fig. 5-22 which shows the dimensionless total blade chord length c for

one-bladed turbines as per Betz and per Schmitz. This diagram was developed by

J. Maurer, and we followed his concise presentation of the Schmitz dimensioning

in this section [8].

BetzBetz



(5.64)

DDL

SchmitzSchmitz

sin

SOO

cc (5.65)

with

arctan

From Fig. 5-22, the development of the chord length c(r) along the radius can be

identified for wind turbines of any tip speed ratio Ȝ

. If, for instance, a turbine has

a design tip speed ratio of Ȝ

= 7, and an inner radius of r

= 0.1 R, we obtain the

(dimensionless) optimum blade chord length through the curve(s) in the range of

values between



r/R = 0.7 (at inner radius) and

r/R = 7.0 (at blade tip).

After determining a lift coefficient c

and the number of blades z, the correspond-

ing real blade chord length can be derived from the above definition

Taking into account the wake rotation, the smaller the local speed ratio Ȝ

r/R, the

more the obtained optimum blade chord length differs from that of Betz. For tur-

bines with a high tip speed ratio, this only affects the near hub area. This pleases

the designers, since at this location it is difficult to implement a large chord length.

Both functions of the total blade chord length now only depend on one single

r/R. parameter:

5 Blade geometry according to Betz and Schmitz

197

0.1 1 10

r/R

Betz

Schmitz

Betz

Schmitz

0.1 1 10

r/R

Betz

Schmitz

Betz

Schmitz

0.1 1 10

Fig. 5-22 Comparison of the dimensionless blade chord length c according to Betz and to

Schmitz versus the local speed ratio Ȝ

r/R

For turbines with a low tip speed ratio, such as the American farm windmill with

§ 1, the chord is completely different if the wake rotation is taken into account:

it tapers towards the hub. The manufacturers of Western mills seem to have had

the right intuition!

Fig. 5-23 shows the curve of the relative velocity angle

for the triangle of

velocities in the rotor plane, taking into account the wake rotation (Schmitz theory),

equation (5.57) and (5.58), and ignoring it (Betz theory). The graph illustrates that

the Schmitz design results in less twist: for wind turbines with a high tip speed

ratio in the inner area only, and for turbines with a low tip speed ratio along the

entire blade length.

The total twist angle

of the blade results from the difference between the

angle of the relative velocity

and the angle of attack Į

used for obtaining the

lift coefficient c

in equations (5.64) and (5.65)

E



(5.66)

This is the local angle between the rotor plane and the chord line of the blade.

5.6 The Schmitz dimensioning taking into account the rotational wake

198

Betz

Schmitz

Betz

Schmitz

0.1 1 10

́ in °

Betz

Schmitz

Betz

Schmitz

0.1 1 10

́ in °

Fig. 5-23 Angle

of relative velocity in rotor plane without (Betz) and with (Schmitz) wake

rotation versus local tip speed ratio Ȝ

r/R

5.6.1 Losses due to wake rotation

According to Betz, without taking into account the downstream wake rotation, the

maximum extracted power is

BetzP,BetzP,

Betz

ccRvP

Taking into account the downstream wake rotation, the maximum power dP at a

ring section results from equation (5.55), provided that the optimum condition for

the relative velocity angle

= 2/3

, equation (5.54), is inserted. Integrating all

ring sections and taking into account the relations in equations (5.55), (5.57) and

(5.58), the power P

Schmitz

including the losses due to wake rotation is obtained with

arctan



Schmitz

sin

. (5.67)

5 Blade geometry according to Betz and Schmitz

199

This integral can be solved analytically [2], although the solution is very complex

and not at all transparent. Therefore, the result is given in a diagram, Fig. 5.24. It

shows that the power coefficient c

P, Schmitz

is now strongly dependent on the tip

speed ratio Ȝ

– in contrast to the constant Betz coefficient c

P, Betz

P,Schmitz

0510Ȝ

P,Betz

0.6

0.5

0.4

0.3

0.2

0.1

0.0

P,Schmitz

0510Ȝ

P,Betz

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Fig. 5-24 Power coefficient according to Betz (without wake rotation) and Schmitz (with wake

rotation). The hatched area shows the additional losses

5.7 Wind turbine design in practice

A first estimate of the maximum mechanical power to be expected for a wind tur-

bine provides

realP,

Real

(5.68)

with













zcc ,,

DTipDProfilDSchmitzP,RealP,

where the power coefficient c

P, real

depends only on the design tip speed ratio Ȝ

the lift-drag ratio İ = c

/ c

of the selected profile and the number of blades z,

affecting the tip losses. Schmitz developed a diagram [2] showing c

P, real

versus the

tip speed ratio. The parameters of the curves are the lift/drag ratio İ and the num-

ber of blades z, see Fig. 5.25. Without any calculation, this diagram provides the

5.7 Wind turbine design in practice

200

efficiency to be expected if the wake rotation, profile losses and tip losses are

taken into account.

The estimation of the annual energy production in kWh is also now possible for

a given wind velocity frequency distribution, see section 4.3.2 and 15.2.5.

Fig. 5-25 Schmitz diagram: real power coefficient taking into account the profile, tip and wake

rotation losses; number of blades z, lift/drag ratio

[2, modified]

The Hütter diagram, Fig. 5-15, gives an idea of the required blade surface area. It

shows that the required blade area diminishes substantially with increasing tip

speed ratio. Whereas the Western mill with Ȝ

§ 1 requires the full disk surface

(100%), wind turbines with Ȝ

= 6 need only 4 to 6 percent, depending on the cho-

sen lift coefficient c

. The chord lengths and the relative velocity angles along the

radius are shown in the Maurer diagram, Fig. 5.22 and 5.23, albeit in a logarithmi-

cally distorted form.

Having set the design tip speed ratio Ȝ

, the profile chord length c(r) and the

relative angles

(r) will be drawn to visualise the twist along the radius. The total

blade twist angle

(r)- Į

can now be obtained with the angle of attack Į

For

the Betz design, equations (5.36) and (5.39) are used, while the equations (5.63)

and (5.66) together with (5.57) and (5.58) are applied for the Schmitz design.

They give the blade twist



)(arctan

)(



(5.69).

5 Blade geometry according to Betz and Schmitz

201

Machine axis

Wind

Plane of rotation

Machine axis

Wind

Plane of rotation

Fig. 5-26 Profile cross sections aligned at 0.25 of chord length c; wind turbine designed per Betz

with a high tip speed ratio (Ȝ

= 6, Göttingen profile 797)

At some stage, the number of blades z must be determined. The dimensioning

theories provide little help for this: z is a weak parameter, which only affects the

tip losses. Therefore, manufacturing aspects (three blades are more expensive than

two), dynamic considerations (dynamically speaking rotors with three or more

blades run more smoothly, while two- and single-bladed rotors are jumpy and

noisy) and strength issues are decisive.

For the hub with rigid blade attachments, it is theoretically better to use few

blades because this reduces the bending moments at the blade root, caused by the

thrust. However, rotors with individual flapping hinges (cf. Fig. 3-17 and 3-20)

have zero bending moments at the root, so this is not an issue. But even for rotors

with rigid blade attachments, the problem of a high bending moment at the blade

root is reduced by using different aerodynamic profiles along the blade radius, see

Fig. 3-8. At the blade tip, for high lift/drag ratios, thin profiles are used. For the

middle part of the blade, the profiles are somewhat thicker, and at the root, the

profile used is very thick. Here, the comparatively low lift/drag ratio (cf. equation

5.42) is hardly relevant. Thus, at the blade root, there is a sufficiently high blade

thickness to provide high section modulus against the bending moment caused by

the aerodynamic forces. Often, a family of profiles with different thicknesses is

chosen, e.g. Delft University profiles DU… (cf. Fig. 3-8) or NACA 44… profiles

(cf. Fig. 5-7).

Moreover, c

need not remain constant along the entire blade length. Hence, the

blade chord c(r) can be manipulated, and it is, e.g., possible to obtain a linear in-

crease of the chord length in the direction from the tip to the hub, when designing

according to Betz or Schmitz.