Aerodynamics of wind Turbine

Wind Energy Assignment II – [group 1] Page 1

Contents Lists of Figures ................................................................................................................................ 2

List of Table ..................................................................................................................................... 3

Introduction ...................................................................................................................................... 4

Objective .......................................................................................................................................... 5

Specific objectives ..................................................................................................................................... 5

The specification of the turbine is ............................................................................................................ 5

Momentum Theory and Blade Element Theory .............................................................................. 5

Momentum Theory ................................................................................................................................... 6

Rotating Annular Stream tube .............................................................................................................. 8

Blade element theory ............................................................................................................................... 9

Blade Shape for Ideal Rotor without Wake Rotation .................................................................... 10

The blade layout of an optimum rotor ................................................................................................... 10

Result and Discussion .................................................................................................................... 12

Twist and chord distributions as a function of r/R for DU93-W210 airfoil ........................................... 12

Linearization of the twist angle and chord distribution .................................................... 14

Turbine Data (angle of twist and chord Distribution) for optimal blade ................................................ 16

Twist angle and chord distribution comparison ............................................................................. 17

Drag distribution of the blade ........................................................................................................ 19

Reduction of power coefficient due to viscous drag ...................................................................... 22

Power velocity (P-V) curve ........................................................................................................... 23

Annual yield for your turbine (annual Energy production) ........................................................... 25

The Weibull distribution of the wind speed ........................................................................................... 27

Conclusions .................................................................................................................................... 30

References ...................................................................................................................................... 31


Lists of Figures

Figure 1A Geometry for rotor analysis. ........................................................................................... 7

Figure 1b Wind turbine blade .......................................................................................................... 9

Figure -2: Blade geometry for analysis of a horizontal axis wind turbine ..................................... 10

Figure 3 Schematic ddiagram of chord and Twist angle distribution for an Optimum blade

DU93-W210 airfoil ........................................................................................................................ 12

Figure 4 Chord distributions for an Optimum blade DU93-W210 airfoil ..................................... 13

Figure 5 Twist angle distributions for an Optimum blade DU93-W210 airfoil. ............................ 13

Figure 6 Linearization of the chord distribution ............................................................................ 12

Figure 7 linearized distribution of chord........................................................................................ 15

Figure 8 Linearization of the twist angle and chord distribution ................................................... 15

Figure 9 linearized Twist angle distribution .................................................................................. 16

Figure 10 Linear, optimum and design chord distribution for an Optimum DU93-W210 ............ 17

Figure 11 Linear, optimum and design twist angle distribution for an Optimum DU93-W210 .... 18

Figure 12 Variation angle of attack over the blade length ............................................................. 18

Figure 13 angle of attack versus drag coefficient and it polynomial fit ......................................... 19

Figure 14 Linear Cd distributions for an Optimum blade DU93-W210 airfoil .......................... 20

Figure 15 Linear and design Fd distribution for an Optimum blade REPOWER 5 MW turbine .. 21

Figure 16 Variation of Cp max with r/R. ............................................................................... 23

Figure 17 Power curve ................................................................................................................... 25

Figure 18 The Weibull distribution of the wind speed at 100 meter height................................... 27

Figure19 power comparisons ......................................................................................................... 28

Figure 20 Annual power distribution curve ................................................................................... 28

Figure 21 The distribution of the energy available in wind, extractable and can be harnessed by

Turbine REPOWER 5M. ............................................................................................................... 29


List of Table

Table 1: Twist angle and chord distribution for an Optimum blade DU93-W210 airfoil ............. 12

Table 2 linearization of twist angle and chord ............................................................................... 14

Table 3 non dimensional factor(r/R) versus Chord(c) ................................................................... 15

Table 4 Design chord and twist angle distribution of DU93-W210 airfoil ......................... 16

Table 5 Linear, optimum and design chord, twist and α-linearized distribution for an Optimum

DU93-W210 ................................................................................................................................... 17

Table 6 Relative velocities ............................................................................................................. 20

Table 7 Drag forces for different relative velocity along the blade length .................................... 21

Table 8 Cp max for Cl/Cd of 143.3, 100, and 25 versus local tip speed ratio .............................. 22

Table 9 Power versus wind speed .................................................................................................. 24

Table 10 calculated value for different parameter ......................................................................... 26

Table 11 Weibull distribution, power and energy .......................................................................... 26


Introduction

A wind turbine is a device that extracts kinetic energy from the wind and converts it into

mechanical energy. Wind flows over the rotor of a wind turbine, causing it to rotate on a shaft.

The resulting shaft power can be used for mechanical work, like pumping water, or to turn a

generator to produce electrical power. Therefore wind turbine power production depends on the

interaction between the rotor and the wind. So the major aspects of wind turbine performance

like power output and loads are determined by the aerodynamic forces generated by the wind.

Depending on their rotor orientation wind turbines are classified as Horizontal axis wind turbines

(HAWT) and vertical axis wind turbines (VAWT), but the first one mostly used in worldwide.

Modern HAWTs usually feature rotors that resemble aircraft propellers, which operate on similar

aerodynamic principles, i.e., the air flow over the airfoil shaped blades creates a lifting force that

turns the rotor. Wind turbine blades use airfoils to develop mechanical power. The cross-sections

of wind turbine blades have the shape of airfoils. The width and length of the blade are functions

of the desired aerodynamic performance, the maximum desired rotor power, the assumed airfoil

properties and strength considerations. In this assignment we followed simple procedure for an

approximate design of a wind rotor is analyzed, based on the fundamental aerodynamic theories

(Momentum Theory and Blade Element Theory).

The Airfoil characteristics DU‐ 93‐W210 such as Radius of the rotor (R), Number of blades (B),

Tip speed ratio of the rotor at the design point λ, Design lift-drag coefficient ratio of the airfoil

,

Angle of attack of the airfoil lift α were calculated. The theoretically analyze parameters were

compared with the REPOWER 5M turbine characteristics. The power curve, energy production

and capacity factor of the turbine at hub height of 100m are estimated.


Objective

The main objective of this assignment is to understand and verify the rotor design choices made

by manufacturers. Basic parameters real machine is used for calculating wind turbine of

REPOWER 5M which is a variable speed machine with pitch controlled blade.

Specific objectives

1. To estimate blade layout of an optimum rotor

Determine the twist and chord distributions as a function of r/R, using the DU‐ 93‐

W210 airfoil characteristics

To give the optimum chord distribution in terms c/R as a function of r/R using an

expression for the axial force on a blade strip.

Linearization of the twist and chord distribution

2. To Linearize the rotor design

Linearize both twist and chord distribution between 0.20R and 1.0R

Compare the results with the distribution with real data

3. To Calculate the PV curve for wind speeds between 0 and 25 m/s Comparing the results

with the REPOWER 5M turbine

To Calculating the P-V curve and annual energy production

To give the electrical power curve (P as a function of V) assuming that the losses in

the drive train and generator is: 3 % base loss and 3% power dependent losses.

To Calculate the cut-in wind speed and the rated wind speed

The specification of the turbine is

Turbine REPOWER 5MW, diam. 126 m, hub height 100m

Optimal tip speed ratio λ=7.4

Design angle of attack α = 6.17o

Design lift coefficient = 1.234

Maximum design lift-drag coefficient ratio

Number of blades B = 3

Momentum Theory and Blade Element Theory

The analysis of the aerodynamic behavior of wind turbines can be started without any specific

turbine design just by considering the energy extraction process. A simple model, known as


actuator disc model, can be used to calculate the power output of an ideal turbine rotor and the

wind thrust on the rotor. Additionally more advanced methods including momentum theory,

blade element theory and finally blade element momentum (BEM) theory are introduced. BEM

theory is used to determine the optimum blade shape and also to predict the performance

parameters of the rotor for ideal, steady operating conditions. Blade element momentum theory

combines two methods to analyze the aerodynamic performance of a wind turbine. These are

momentum theory and blade-element theory which are used to outline the governing equations

for the aerodynamic design and power prediction of a HAWT rotor. Momentum theory analyses

the momentum balance on a rotating annular stream tube passing through a turbine and blade-

element theory examines the forces generated by the aerofoil lift and drag coefficients at various

sections along the blade. Combining these theories gives a series of equations that can be solved

iteratively.

Momentum Theory

The forces on a wind turbine blade and flow conditions at the blades can be derived by

considering conservation of momentum since force is the rate of change of momentum. The axial

and angular induction factors are assumed to be functions of the radius, r. A simple model may

be used to determine the power from an ideal turbine rotor, the thrust of the wind on the ideal

rotor and the effect of the rotor operation on the local wind field. The analysis assumes a control

volume, in which the boundaries are the surface of a stream tube and two cross-sections of the

stream tube.


Figure 1A Geometry for rotor analysis

Applying the conservation of linear momentum to the control volume enclosing the whole

system, one can find the net force on the contents of the control volume. That force is equal and

opposite to the thrust, T, which is the force of the wind on the wind turbine. From the

conservation of linear momentum for a one-dimensional, incompressible, time-invariant flow,

the thrust is equal and opposite to the rate of change of momentum of the air stream:

…………………………………………………………………(1)

Where r is the air density, A is the cross-sectional area, U is the air velocity, and the subscripts

indicate values at numbered cross-sections. For steady state flow,

…………………………………………………………………………….(2)

Assuming that no work is done on either side of the turbine rotor, Bernoulli function can be used

in the two control volumes on either side of the actuator disc: Assume p1 = p4 and that V2 = V3.

We can also assume that between 1 and 2 and between 3 and 4 the flow is frictionless so we can

apply Bernoulli’s equation. In the stream tube upstream of the disk:

…………………………………………………………………(3)

In the stream tube downstream of the disk:

………………………………………………………………….(5)

…………………………………………………………………(6)

………………………………………………………………………………(7)

The thrust can also be expressed as the net sum of the forces on each side of the actuator disc:

Substituting (P2-P3) into equation 7

)……………………………………………………………………........(8)

Equating equation 2 and 8


…………………………………………………………………………………(9)

The axial induction factor, a, as the fractional decrease in wind velocity between the free stream

and the rotor plane:

……………………………………………………………………………………(10)

And also

…………………………………………………………………………..…(11a)

……………………………………………………………………. …….(11b)

Differential force dFx;

; Where

And substituting equation 11a, 11b and 6 into dFx

……………………………………………………………(12)

Applying linear momentum conservation to the control volume of radius r and thickness dr gives

the thrust contribution as:

…………………………………………………………………….(13)

Rotating Annular Stream tube

Define angular induction factor a’:

………………………………………………………………………………………(14)

Similarly, from conservation of angular momentum, the differential torque, Q, imparted to the

blades (and equally, but oppositely, to the air) is:

………………………………………………………………(15)

Together, these define thrust and torque on an annular section of the rotor as functions of axial

and angular induction factors that represent the flow conditions.


Blade element theory

The forces on the blades of a wind turbine can also be expressed as a function of Cl, Cd and α.

For this analysis, the blade is assumed to be divided into N sections (or elements).

Assumptions:

There is no aerodynamic interaction between elements.

The forces on the blades are determined solely by the lift and drag characteristics of the

airfoil shape of the blades.

Lift and drag forces are perpendicular and parallel, respectively, to an effective, or relative, wind.

The relative wind is the vector sum of the wind velocity at the rotor, U (1 - a), and the wind

velocity due to rotation of the blade. This rotational component is the vector sum of the blade

section velocity, 𝜴 r, and the induced angular velocity at the blades from conservation of angular

momentum, ɷ r / 2, or

(

) 𝜴 𝜴 ……………………………………………………………...(16)

Figure 1b Wind turbine blade


Blade Shape for Ideal Rotor without Wake Rotation

The problem can be solved by simplification of certain parameters such as

1. No wake rotation, thus a’ = 0

2. No drag, thus CD = 0

3. No tip losses

4. For Betz optimum rotor, a =

in each annular stream tube

The blade layout of an optimum rotor

Considering the following Blade section geometry for analysis of a horizontal axis wind turbine

parameters,

Figure -2: Blade geometry for analysis of a horizontal axis wind turbine

Where φp,o is the blade pitch angle at the tip. The twist angle is, of course, a function of the blade

geometry, whereas φp changes if the position of the blade, φp,o, is changed. Note, also, that the

angle of the relative wind is the sum of the section pitch angle and the angle of attack:

……………………………………………………………………………………(17)

Φ can also be determined from the figure 1

……………………………………………………………(18)


The blade twist angle which is a function of the blade geometry is defined relative to the blade

tip as:

………………………………………………………… (19)

By definition, the angle of the relative wind is the sum of the section pitch angle and the angle of

attack

α ………………………………… ……………………. (20)

From the relation on figure 1,

For a’ = 0 and a = 1/3, the angle of relative wind can be determined as:

λ ………………………………………………………… (21)

Where λ is local speed ratio which is defined by the relation, λ λ ⁄

λ⁄ ………………………………………………………… (22)

Substituting equation 22 in to equation 20 gives the section pitch:

λ ⁄⁄ α ………………………………………………………… (23)

Now by substituting equation 23 into 19, the twist angle can be determined as:

λ ⁄⁄ α ………………………………………………… (24)

But the twist angle is assumed to start at 0 at the tip of the blade and hence from equation 19, the

blade pitch angle can be obtained as:

λ⁄ α …………………………………………………………………… (25)

Finally combining equations 24 and 25, the twist distribution along the blade length is given as:

⁄⁄ ⁄

………………………………………………………… …………………………… (26)

Again from figure 19, the cord distribution is related with the blade radius according to the

following relation.

…………………………………………………… (27)


Result and Discussion

Twist and chord distributions as a function of r/R for DU93-W210 airfoil

Table 1: Twist angle and chord distribution for an Optimum blade DU93-W210 airfoil

r/R λr ϕ Өp ӨT C/R chord(c)

0.1 0.74 42.01572 35.84572 36.86783 0.204689 12.89538

0.2 1.48 24.2492 18.0792 19.10132 0.125598 7.912656

0.3 2.22 16.71503 10.54503 11.56714 0.087954 5.541121

0.4 2.96 12.69267 6.522666 7.544781 0.067193 4.233146

0.5 3.7 10.21397 4.043973 5.066088 0.054228 3.416335

0.6 4.44 8.539179 2.369179 3.391294 0.045408 2.860721

0.7 5.18 7.333661 1.163661 2.185776 0.039036 2.45925

0.8 5.92 6.425158 0.255158 1.277273 0.034222 2.155964

0.9 6.66 5.716272 -0.45373 0.568387 0.030459 1.918936

1 7.4 5.147885 -1.02212 0 0.027439 1.728673

Figure 3 Schematic ddiagram of chord and Twist angle distribution for an Optimum blade

DU93-W210 airfoil

From the above table as well as figure 4 and figure 5 it was observed that both chord and Twist

angle gets the maximum value at sholder and minimum at tip of blade. Moreover we can observe

from the graph of chord distribution that the graph look like reperesention a higher polynimal

function with order of three or above, this implies the shape is complicated for manufacturing.


Figure 4 Chord distributions for an Optimum blade DU93-W210 airfoil

Figure 5 Twist angle distributions for an Optimum blade DU93-W210 airfoil

0

1

2

3

4

5

6

7

8

9

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C (

m)

Non-dimensionalized blade radius, r/R

chord(C)

0

5

10

15

20

25

30

35

40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Bla

de

tw

ist

angl

e [

de

gre

e]


Twist angle distribution as a function of r/R(ӨT)

ӨT


C=-6.77(r/R) +7.6

0

1

2

3

4

5

6

7

8

9

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c(m

)


chord(C)

Linear (chord(C))

Linearization of the twist angle and chord distribution

Table 2 linearization of twist angle and chord

r/R C-linearized linearized ӨT linearized

α

0.2 6.2456 19.91 13.74 6.2

0.3 5.5684 16.64 12.1 6.2

0.4 4.8912 13.37 10.46 6.2

0.5 4.214 10.1 8.82 6.2

0.6 3.5368 6.83 7.18 6.2

0.7 2.8596 3.56 5.54 6.2

0.8 2.1824 0.29 3.9 6.2

0.9 1.5052 -2.98 2.26 6.1

1 0.828 -6.25 0.62 6.1

Figure 6 Linearization of the chord distribution

Function of linearized chord

Using linear polynomial function f(x) =p1(r/R) +P2 we can get better fit Where p1=-6.77 and p2=7.6 and

the function becomes C=-6.77(r/R) +7.6


Table 3 non dimensional factor(r/R) versus Chord(c)

r/R 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c 6.2456 5.5684 4.8912 4.214 3.5368 2.8596 2.1824 1.5052 0.828

Figure 7 linearized distribution of chord

Figure 8 Linearization of the twist angle and chord distribution

Function ӨT=-16.4(r/R) +17.2

ӨT = -16.4(r/R)+ 17.02

-10

-5

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Twis

t an

gle

in D

egr

ee


ӨT

ӨT

Linear (ӨT)


Figure 9 linearized Twist angle distribution

Turbine Data (angle of twist and chord Distribution) for optimal blade

Table 4 Design chord and twist angle distribution of DU93-W210 airfoil

ӨT = -16.4(r/R)+ 17.02

0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9

Twis

t an

gle

in D

egr

ee


ӨT linearised

ӨT linearised


Table 5 Linear, optimum and design chord, twist and α-linearized distribution for an Optimum

DU93-W210

r/R ӨT-Design ӨT-Optimum ӨT linearized C-design C-optimum C-linearized α

0.2 10 19.10132 13.74 4 7.912656 6.2456 6.19

0.3 10 11.56714 12.1 5 5.541121 5.5684 6.19

0.4 6.6 7.544781 10.46 4.6 4.233146 4.8912 6.19

0.5 4.3 5.066088 8.82 3.8 3.416335 4.214 6.19

0.6 2.7 3.391294 7.18 3.4 2.860721 3.5368 6.19

0.7 1.6 2.185776 5.54 3 2.45925 2.8596 6.18

0.8 0.8 1.277273 3.9 2.6 2.155964 2.1824 6.18

0.9 0.2 0.568387 2.26 2.2 1.918936 1.5052 6.18

1 0 0 0.62 0.1 1.728673 0.828 6.18

Twist angle and chord distribution comparison

Figure 10 Linear, optimum and design chord distribution for an Optimum DU93-W210

From the above figure we observe that the distribution of design and linearized chords

are much similar for r/R in between 0.4 and 0.9. At tip of the blade the chord is 0 for

design and below one for linearized and above one for optimum.

0

1

2

3

4

5

6

7

8

9

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C(m

)


c-optimum

C-linearized

C-design


Figure 11 Linear, optimum and design twist angle distribution for an Optimum DU93-W210

The twist angle distribution above shows that for r/R between 0.3-1 the optimum and design fit each

other, but for the table 5 we can observe that the linearized is more close to the design than the

optimum.

Figure 12 Variation angle of attack over the blade length

From the above figure we can observe that the linearized angle of attack distribution has a negligible

decrease as it goes to the tip and it has no significant difference with the optimal design angle of attack

which is 6.170.

0

5

10

15

20

25

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ӨT


ӨT-Design

ӨT-Optimum

ӨT linearised

6.18

6.18

6.18

6.18

6.19

6.19

6.19

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

α i

n d

egr

ee

s


α


Drag distribution of the blade

There are two basic aerodynamic forces exerted on the blade surfaces: the pressure distribution

and the Frictional shear stress distribution exerted by the airflow on the body surface. The

pressure exerted by the air at a point on the surface acts perpendicular to the surface at that point;

and the shear stress, which is due to the frictional action of the air rubbing against the surface,

acts tangentially to the surface at that point. The net aerodynamic force on the body is due to the

net imbalance between these distributed loads as they are summed (integrated) over the entire

surface.

Figure 13 angle of attack versus drag coefficient and it polynomial fit

The drag force can be obtained using the following relations

…………………………………………………….. (28)

Cd = 0.0001x2 - 0.0037x + 0.0278

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

‐8.2

3

‐6.7

‐5.1

6

‐3.6

1

‐2.0

6

‐0.5

2

1.0

3

2.5

7

4.1

2

5.6

6

7.2

8.7

4

10

.22

11

.71

13

.22

16

.23

Dra

gco

ffic

en

t(C

d)

Attack angle(α)

cd

Poly. (cd)


Figure 14 Linear Cd distributions for an Optimum blade DU93-W210 airfoil

The above figure shows that the drag coefficient varies along the length of blade with varying the

angle of attack, but very small when keeping the angle of attack constant.

From the equation 28, it was observed that the magnitude of the aerodynamic force R is

governed by the density ƍ and velocity of the free stream, the size of the body, and the angle of

attack (α). So we first calculate the relative velocities along the length of the blade then, we

estimated the drag forces.

Table 6 Relative velocities

λr Vrel5 Vrel10 Vrel15

1.48 8.93084542 17.86169 26.79254

2.22 12.1741529 24.34831 36.52246

2.96 15.6217797 31.24356 46.86534

3.7 19.1637679 38.32754 57.4913

4.44 22.7560981 45.5122 68.26829

5.18 26.3782107 52.75642 79.13463

5.92 30.0193271 60.03865 90.05798

6.66 33.6732832 67.34657 101.0198

7.4 37.3363094 74.67262 112.0089

0.086995

0.087

0.087005

0.08701

0.087015

0.08702

0.087025

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cd


Cd


Table 7 Drag forces for different relative velocity along the blade length

Fd Linear,5m/s Fd Linear,10m/s FdLinear,15m/s Fd Design5m/s Fd ,Design,10m/s Fd ,Design,15m/s

18.6240559 74.49622 167.6165 13.35783 53.43130775 120.2204424

34.2637667 137.0551 308.3739 22.83579 91.34315185 205.5220917

55.8526075 223.4104 502.6735 34.33136 137.3254279 308.9822127

83.2002356 332.8009 748.8021 46.74402 186.9760821 420.6961846

116.116308 464.4652 1045.047 58.97327 235.8930606 530.7593864

154.410483 617.6419 1389.694 69.91858 279.6743098 629.267197

197.892417 791.5697 1781.032 78.47944 313.9177757 706.3149954

251.230499 1004.922 2261.074 83.55535 334.2214047 751.9981606

305.567792 1222.271 2750.11 4.66921 18.67684127 42.02289287

Figure 15 Linear and design Fd distribution for an Optimum blade REPOWER 5 MW turbine

From table 7 and figure15 we observe that the magnitude of drag force Fd for both optimum and

design increases with increase of both relative velocity and drag coefficient, which are increasing

with increase of r/R. the optimum shows high increase than the design, this is because in design

we kept the angle of attack and Cd constant throughout the blade length.

0

500

1000

1500

2000

2500

3000

Fd

Drag cofficient

Fd ,Linear,5m/s

Fd ,Linear,10m/s

Fd ,Linear,15m/s

Fd ,Design5m/s

Fd ,Design,10m/s

Fd ,Design,15m/s


Reduction of power coefficient due to viscous drag

The power confident is expressed using the following formula

……………………………………. (29)

Where

λ is blade tip ratio = 7.4

B=number of blade =3

The following table and figure are obtained for Cl/Cd of 143.3, 100, and 25

Table 8 Cp max for Cl/Cd of 143.3, 100, and 25 versus local tip speed ratio

The power coefficient varies with the variation of local tip speed ratio; it gets its maximum value

at tip where r/R is one. The maximum power coefficient that can be achieved in the presence of

drag is significantly less than the Betz limit at all tip speed ratios.

Cp max, for Cl/Cd=143.3 Cp max, for Cl/Cd= 100 Cp max ,for Cl/Cd=25

0.289029088 0.276564572 0.152812

0.384951367 0.372486852 0.248734

0.431231892 0.418767377 0.295015

0.458484867 0.446020351 0.322268

0.47644375 0.463979234 0.340226

0.489170301 0.476705785 0.352953

0.498660429 0.486195914 0.362443

0.506009424 0.493544908 0.369792

0.511868442 0.499403927 0.375651

0.516648907 0.504184391 0.380432


Figure 16 Variation of Cp max with r/R

Power velocity (P-V) curve

The wind Power increases with the cube of the wind speed when the power coefficient is

constant, but when velocity increases the drag force also increases. So the power will be limited

at certain level where we cannot further increase it with increasing the wind velocity. This power

is called rated power and the wind speed, at which the rated generator power is achieved, is

called the rated wind speed. For the wind speeds form 0 till 25 m/s the electrical power

curve can be calculated with the assumed losses in the drive train and generator are

3% base loss and 3% power dependent losses. The following equation can be used to

calculate the real power output.

………………………………(30)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cp


Cp max,for Cl/Cd=143.3

cp max, forCl/Cd= 100

cp max ,for Cl/Cd=25


Table 9 Power versus wind speed

U P(U) Preal

0 0 0

1 0.00398048 0

2 0.03184387 0

3 0.10747305 0

3.386839 0.15463918 0

4 0.25475092 0.097108

5 0.4975604 0.332634

6 0.85978437 0.683991

7 1.36530574 1.174347

8 2.0380074 1.826867

9 2.90177225 2.664719

10 3.9804832 3.711069

11.00779 5.30928339 5.000005

12 6.87827497 5

13 8.74512159 5

14 10.9224459 5

15 13.4341308 5

16 16.3040592 5

17 19.556114 5

18 23.214178 5

19 27.3021343 5

20 31.8438656 5

21 36.8632549 5

22 42.3841851 5

23 48.4305391 5

24 55.0261998 5

25 62.19505 5

The cut in speed is calculated Preal=0,

Vcut in =3.39 m/s

The rated speed is calculated Preal= Prated,

Vrealed= 11 m/s

And the cut out speed = 25 m/s


Figure 17 Power curve

Annual yield for your turbine (annual Energy production)

The annual wind energy in particular site depends on may factor, among this the yearly average wind

velocity and its distribution, topography and others. In this assignment we only focus on the velocity

distribution and we take the wind shear factor α=0.24 and average wind speed at 10 meter U=5.47 m/s

for assignment I.

Modify the average wind speed for the hub height of your turbine, which is 100m for the 5MW Repower

turbine.

…………………………………………………………………(31)

Considering the shape factor at a reference height of 10m, recalculate the shape factor for the required

hub height, using: ‐

………………………………………………………………………………….(32)

The Weibull distribution can be determined with the following formula

……………………………………………………..(33)

Where


The annual power can be determined by:

………………………………………………………………………………….(34)

Annual energy production is given by

……………………………………………………………………(35)

Where T (T=8760) is the number of hours in a year, and Vci, Vco is the cut-in and cut-out wind

speed respectively.

Table 10 calculated value for different parameter

Uref 5.473 k10 1.782895

Zo 10 cp 0.516973

Z 100 ᴧk 0.756

α 0.24 K100 2.538895

U100 9.510984 Prated 5

c 10.71686 η 0.97

Table 11 Weibull distribution, power and energy

U(m

/s)

We

ibu

ll P

rob

abili

ty

P(U

)rea

l

P(U

)win

d

Pan

nu

al

E-re

al

E-W

ind

P-e

xtra

ctab

le

E-ex

trac

tab

le

1 0.006143 0 0.0077 0 0 0.414314 0.00398 0.214189

2 0.017642 0 0.061597 0 0 9.519308 0.031844 4.92122

3 0.032101 0 0.207889 0 0 58.45884 0.107473 30.22161

3.386839 0.038141 0 0.299124 0 0 99.94234 0.154639 51.66744

4 0.047901 0.097108 0.492774 0.004652 40.74799 206.7747 0.254751 106.8968

5 0.063441 0.332634 0.962449 0.021102 184.8578 534.8718 0.49756 276.514

6 0.077147 0.683991 1.663113 0.052768 462.2461 1123.944 0.859784 581.0481

7 0.087628 1.174347 2.640961 0.102905 901.4488 2027.248 1.365304 1048.031

8 0.093852 1.826867 3.942193 0.171455 1501.944 3241.043 2.038005 1675.53

9 0.095302 2.664719 5.613005 0.253954 2224.635 4686.006 2.901769 2422.536

10 0.092049 3.711069 7.699596 0.341599 2992.406 6208.539 3.980479 3209.644

11.00779 0.084649 5.000005 10.26994 0.423246 3707.639 7615.441 5.309278 3936.974

12 0.074375 5 13.3049 0.371873 3257.606 8668.424 6.878268 4481.337

13 0.062303 5 16.91601 0.311513 2728.857 9232.276 8.745113 4772.833


14 0.0498 5 21.12769 0.249 2181.243 9216.926 10.92244 4764.897

15 0.037973 5 25.98614 0.189864 1663.213 8644.095 13.43412 4468.76

16 0.027608 5 31.53754 0.138039 1209.225 7627.2 16.30404 3943.053

17 0.019127 5 37.82811 0.095634 837.7543 6338.133 19.5561 3276.641

18 0.012618 5 44.90404 0.06309 552.6703 4963.427 23.21416 2565.955

19 0.00792 5 52.81153 0.039601 346.9075 3664.143 27.30211 1894.261

20 0.004726 5 61.59677 0.023631 207.011 2550.242 31.84384 1318.405

21 0.002679 5 71.30596 0.013394 117.3337 1673.318 36.86322 865.0597

22 0.001441 5 81.9853 0.007204 63.11118 1034.838 42.38414 534.9827

23 0.000735 5 93.68098 0.003674 32.18399 603.0056 48.43049 311.7373

24 0.000355 5 106.4392 0.001775 15.54581 330.9369 55.02615 171.0853

25 0.000162 5 120.3062 0.000811 7.105801 170.9744 62.19499 88.38904

0.99978 25235.69 90530.14 46801.59

The Weibull distribution of the wind speed

Figure 18 The Weibull distribution of the wind speed at 100 meter height

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526

Pro

bab

ility

Den

sity

Wind Speed (m/S)

Weibull Probability


Figure19 power comparisons

The above figure shows the power available in wind, power extractable by setting Betiz limit and real,

which can be harness able by Turbine REPOWER 5M. The power available in wind is much

greater than extractable by Turbine REPOWER 5M.

Figure 20 Annual power distribution curve

0

20

40

60

80

100

120

1 3 5 7 9 11 13 15 17 19 21 23 25

Po

we

r(M

W)

Wind speed(m/s)

Power

P(U)real

P(U)wind

P-extractable

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

1 3 5 7 9 11 13 15 17 19 21 23 25

Pro

bab

ility

Wind speed(m/s)

Pannual

Pannual


Figure 21 The distribution of the energy available in wind, extractable and can be harnessed by Turbine

REPOWER 5M.

From the figure 21 we observe that for speed up to rated which is 11 m/s the Turbine REPOWER 5M

can harness all the energy extractable but only some losses. Above the rated speed the Turbine

REPOWER 5M cannot harness the available power.

So the annual energy production of Turbine REPOWER 5M is

AEP= 25235.7 MWhr

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1 3 5 7 9 11 13 15 17 19 21 23 25

Ene

rgy(

MJ)

Wind Speed(m/s)

E-real

E-Wind

E-extractable


Conclusions

In this assignment we learned how to calculate airfoil parameters and how they affect the

performance of the aerofoil. We observed that the Blade layout of an optimum rotor is much

different from design, so to make them somehow close each other we linearized the optimum

Blade layout. In all we learned that beside the site resources (wind speed, Wind shear and

others) turbine blade layout affects the amount of energy can be harnessed.


References

1. J. F. Manwell and J. G. McGowan, Textbook-Wind-Energy-Explained-Theory- Design-and-

Application, 2nd edition

2. Gijs van Kuik, Wim. Bierbooms, Introduction to wind turbine design

Aerodynamics of wind Turbine

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