DESIGN OPTIMIZATION OF A WIND TURBINE BLADE by …

DESIGN OPTIMIZATION OF A WIND TURBINE BLADE

by

BHARATH KORATAGERE SRINIVASA RAJU

Presented to the Faculty of the Graduate School of

The University of Texas at Arlington in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN AEROSPACE ENGINEERING

THE UNIVERSITY OF TEXAS AT ARLINGTON

May 2011

Copyright © by Bharath Koratagere Srinivasa Raju 2011

All Rights Reserved

iii

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor Dr. B. P. Wang for his constant guidance and

support for my thesis work. I am indebted to him for his constructive criticism and patience in

guiding my thesis work. His knowledge and teaching skills are unique and I am totally motivated

and am thankful for him choosing me as his student.

I would like to thank my parents back in India and their constant support in every

aspect of my life, without whom I could have never achieved my masters. I am full of love and

honor for their sacrifice they have done for my benefit, it is something I can never return.

I would love to acknowledge all of my friends here in USA, UK, India and other relatives

who were there in times when i needed them for advice and guidance. It’s a warm feeling to

know all of them are there to support me

It is a privilege and honor that I am able to thank my GURUJI, Sri Sri Ravi Shankar,

because of whom I am inspired to do this work and all of my education has value only because

of my service to him. It is to Him I owe my entire life. Guruji thanks for choosing me to be a part

of your life.

April 11, 2011

iv

ABSTRACT

DESIGN OPTIMIZATION OF A WIND TURBINE BLADE

Bharath Koratagere Srinivasa Raju, M. S.

The University of Texas at Arlington, 2011

Supervising Professor: Dr. Bo Ping Wang

This work focuses on designing a blade of 45 meters in length that produces a power of

1.6 MW. The design of the blade was done using the Blade Element Momentum theory and the

Prandtl’s tip loss factor was used. The aerodynamic loads and differential power at are

tabulated and plotted.

The finite element method for analysis of the blade is used. As the chord lengths vary

decreasingly along the blade radii in order to use the simple beam theory the breath and height

of the blade is considered as a function of the chord length, hence the analysis is done

assuming the blade to be a tapered hollow beam. The first few natural frequencies in the axial

and transverse direction and mode shapes are calculated and plotted.

In order to reduce the weight of the blade designed and increase the power two sets of

optimization was done. The design variables are the chord lengths, with objective function as

power mass constraints was used. The other optimization was using the mass as objective

function and power as the constraint. The chord distribution results are plotted and discussed.

v

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ................................................................................................................iii ABSTRACT ..................................................................................................................................... iv LIST OF ILLUSTRATIONS..............................................................................................................vii LIST OF TABLES ............................................................................................................................ ix Chapter Page

1. INTRODUCTION……………………………………..………..….. ..................................... 1

2. LITERATURE REVIEW ................................................................................................. 10 3. AIRFOIL THEORY ........................................................................................................ 13

3.1 Introduction..................................................................................................... 13 3.2 Reynolds Number .......................................................................................... 14

4. AERODYNAMICS ......................................................................................................... 18

4.1 Introduction..................................................................................................... 18 4.2 Betz Limit ........................................................................................................ 21 4.3 Induction Factor ............................................................................................. 22 4.4 Tip Speed Ratio ............................................................................................. 24 4.5 Pitch, Twist and Chord Lengths ..................................................................... 24

4.5.1 Pitch ............................................................................................... 24 4.5.2 Twist ............................................................................................... 25 4.5.3 Chord Lengths ................................................................................ 25

5. BLADE ELEMENT MOMENTUM THEORY .................................................................. 26 6. RESULTS ...................................................................................................................... 32

vi

7. STRUCTURAL ANLYSIS .............................................................................................. 40

7.1 Simple Beam Theory ...................................................................................... 40 7.2 Stiffness Matrix ............................................................................................... 46 7.3 Mass Matrix .................................................................................................... 47 7.4 Mode Shapes ................................................................................................. 48 7.5 Breadth and Height ........................................................................................ 50

8. DESIGN OPTIMIZATION .............................................................................................. 54 9. CONCLUSION .............................................................................................................. 69 10. FUTURE WORK .......................................................................................................... 70

APPENDIX

A. LIST OF SYMBOLS ...................................................................................................... 71 REFERENCES ............................................................................................................................... 73 BIOGRAPHICAL INFORMATION .................................................................................................. 75

vii

LIST OF ILLUSTRATIONS Figure Page 1.1 Coriolis Force ............................................................................................................................. 2 1.2 Flow of Wind and Currents across the Globe ............................................................................ 3 1.3 Smock Mill Wind Turbine............................................................................................................ 4 1.4 Rotor Diameter vs Years ............................................................................................................ 6 1.5 Enercon and Gedser Wind Turbine ............................................................................................ 7 3.1 Airfoil......................................................................................................................................... 15 3.2 FX 66 S 196 V1 Coefficient of Lift Vs Angle of Attack ............................................................. 16 3.3 FX 66 S 196 V1 Coefficient of Drag Vs Coefficient of Lift ........................................................ 17 4.1 Ideal Rotor Velocity and Pressure Profiles............................................................................... 19 4.2 Ideal Rotor Control Volume ...................................................................................................... 20 4.3 Betz Limit .................................................................................................................................. 22 4.4 Velocity Triangle Over Airfoil .................................................................................................... 23 4.5 Pitch and Twist with Velocity Triangles .................................................................................... 25 5.1 Velocity Triangle of A Section of Wind Turbine Blade ............................................................. 27 5.2 Normal Forces Over Airfoil ....................................................................................................... 28 6.1 Chord Distribution ..................................................................................................................... 33 6.2 Twist Distribution ...................................................................................................................... 34 6.3 Differential Power ..................................................................................................................... 35 6.4 Differential Thrust ..................................................................................................................... 36 6.5 Differential Torque .................................................................................................................... 37 7.1 Reaction and Moments of A Beam .......................................................................................... 42

viii

7.2 Tapered Cantilever Beam ........................................................................................................ 43 7.3 First Mode Shape ..................................................................................................................... 52 7.4 Second Mode Shape ................................................................................................................ 52 7.5 Third Mode Shape .................................................................................................................... 53 7.6 Fourth Mode Shape.................................................................................................................. 53 8.1 Flow Chart of Optimization Process ......................................................................................... 55 8.2 Optimized Chord Distribution Problem 1 .................................................................................. 56 8.3 Optimized Twist Distribution Problem 1 ................................................................................... 57 8.4 Optimized Differential Thrust Problem 1 .................................................................................. 58 8.5 Optimized Differential Power Problem 1 .................................................................................. 59 8.6 Optimized Chord Distribution Problem 2 .................................................................................. 61 8.7 Optimized Chord Distribution Problem 2 .................................................................................. 62 8.8 Optimized Twist Distribution Problem 2 ................................................................................... 63 8.9 Optimized Differential Power Problem 2 .................................................................................. 64 8.10 Optimized Chord Distribution Problem 3 ................................................................................ 66

ix

LIST OF TABLES

Table Page

6.1 Distribution of Differential Thrust, Torque, Power and Flow Angles ........................................ 38

7.1 Natural Frequency in the Transverse Direction ....................................................................... 51

7.2 Natural Frequency in the Axial Direction .................................................................................. 51

8.1 Optimization problem definition ................................................................................................ 48

8.2 Original and Optimized Power and Mass Problem 1 ............................................................... 60

8.3 First three natural Frequencies Transverse Direction Problem 1 ............................................ 60

8.4 First three natural Frequencies Axial Direction Problem 1....................................................... 60


8.6 First Three Natural Frequencies Transverse Direction Problem 2 .......................................... 65

8.7 First Three Natural Frequencies Axial Direction Problem 2 ..................................................... 65


8.9 First three natural Frequencies Transverse Direction Problem 3 ............................................ 67

8.10 First three natural Frequencies Axial Direction Problem 3..................................................... 67

1

CHAPTER 1

INTRODUCTION

The need for electricity in our generation is of prime importance due to the sort of

evolved life mankind leads. The production of power using traditional methods has taken its toll

on the environment and the earth has been polluted to degrees beyond imagination. Alternative

energy and green energy from natural recourses is the need of the hour. Technology must be

used so as to provide human need and luxuries but still not affect our planet. With increasing

awareness about our needs and priorities one alternative source where we can draw power

would be the wind.

Wind is such a resource available that it just blows everywhere, from large areas to

local winds it just blows. There are various phenomenons that occur that makes the flow of wind

across the globe. Wind blows along the planet due to the difference in temperatures across the

surface of earth, the hot air rises up and cool air rushes to fill up the void. The equatorial region

of the earth gets heated up and in turn heats the air above it causing the wind to blow higher

due to which pressure drops and the air that’s cooler near to the poles rush towards the

equator, called the Geostrophic Wind. This occurs at higher altitudes of the atmosphere. There

is a Coriolis force due to the rotation of the earth, the northern hemisphere the winds move

counter clockwise and the southern hemisphere it rotates clockwise figure 1.1 shows this effect.

Surface winds are affected by the obstacles on the earth up to a height of 100 meters. There

are winds called sea breeze and land breeze which can also be a source of wind. The Danish

wind industry association [18] has documented these results in more detail

The local winds are also influenced by the global and local effects, the landscape of the

region. The seasonal winds change too at places in south Asia. The winds around mountain

2

regions due to the pressure differences and the height of the hills make up for these kinds of

winds that are strong. Hence based on research it is conclusive that winds across the globe are

consistent, depending on the region as well figure 1.2 shows the currents across the globe.

Something that is so freely available in nature is a source where enormous power can

be harnessed and used. The clean, inexhaustible, constant everyday occurrence and green

energy part of this source is the essence why we need to choose as a part of our large

consumption of energy needs.

Figure 1.1 Coriolis Force [19]

3

Figure 1.2 Flow of Wind and Currents across the Globe [20]

A brief history goes on to show that harnessing wind energy was done for a variety of

purposes in as early as 7th century. The use of wind energy in getting water out of wells and

grinding was a part where this source was of great significance for free power. Older wind

capturing machines developed in 200 BC is considered to be the first instance where wind was

as a power source for machines. The European countries had built smock mill type of turbines

which was mainly used for drawing water from wells and for agricultural purpose, figure 1.3.

4

Figure 1.3 Smock Mill Wind Turbine [21]

The power of wind if harnessed completely can actually power a whole nation, and if

used with other natural alternative energy we can create a pollution free green environment.

This energy is so important to third world countries where basic electricity is not available.

Power of wind turbines has increased 100 times compared to the wind mills those existed a

couple of decades ago.

In order to harness the wind effectively and for the low costs, the advancement of

technology over the last few decades has given rise to not individual turbines but wind farms in

5

general. Advances in materials and composites used for construction of turbines, the analysis

for efficiency of aerodynamics and structures, accurate prediction of winds and their direction

have provided for cost effective production of power. As technology in every area is advancing

the turbines go higher and grow powerful. As Greenpeace international puts it, “Behind the tall,

slender towers and steadily turning blades lays a complex interplay of lightweight materials,

aerodynamic design and computer controlled electronics”.

There are two types of wind turbines that are developed, one is the vertical axis wind

turbines and the other is the horizontal axis wind turbines. The vertical axis is the kind where the

main rotor shaft is set vertically and perpendicular to the ground. The horizontal axis wind

turbines are those where the main shaft and rotation of blades is perpendicular to the direction

of wind. The former type is highly useful due to its ease of construction and in small scale for

small wind farms and single buildings but their efficiency is low for large scale applications. The

horizontal axis is used for large scale production of power and can be used in offshore as well

as on shore and can be efficient in small scale production in farms as well. Although the

aerodynamics of both are the same, the most preferred in industry for large scale production of

power is the horizontal axis wind turbines (HAWT) and is used as a standard in this thesis.

The first wind turbine of modern type was produced by Johannes Juul named the

Gedser Wind Turbine figure 1.5 was the first one built for a power of 200 KW in 1904. HAWT

are of various types depending on the number of blades ranging from one blade to any ode

number of blades. The three bladed rotors are the most industry accepted design and version.

The largest wind turbine today is the ‘Enercon E-126’ figure 1.5, which produces an excess of 7

MW of power producing about 20 million KWh per year. As wind turbines go higher and wider,

these can be used only at certain places. The usage of wind turbine in wind farms are of each

producing 1.5 MW and around 40-50 meters in length. Figure 1.4 gives an approximate rotor

diameter and years in production.

6

Figure 1.4 Rotor Diameter vs Years [23]

The aerodynamic efficiency is lower on a two bladed rotor compared to a three bladed

rotor, the rotation speed needs to be higher so as to achieve the same power as that of the

three bladed rotor. The two and single bladed rotors need a special kind of arrangement that is

hinged or teetering hub. Each time the rotor passes the tower and in order to avoid heavy

shocks the rotor is to tilt away. Also the arrangement can have balance issues and in time the

blades are bound to hit the tower during operation. The three bladed rotors are effective to use

the yawing mechanism in them.

Analysis of blades using wind tunnel would be possible for small scale rotors, but the

increase in diameters has called for the use of Computational Fluid Dynamics for fluid flow over

blades and prediction of loads.

7

Figure 1.5 Enercon and Gedser Wind Turbine [22, 18]

8

The current energy needs of man are dependent on carbon based fuels that are cheap

and easily accessible, but the limitation and the environmental effects it has staggeringly

improvised the need for alternate cleaner energy. The advancement of technology and the

greener energy needed for the luxuries of mankind are the prime reason for this report and also

the advancements as seen from figure 1.5.

For the use of alternative sources of energy, we need to bring in more laws and the

promotion of this is definitely an added advantage considering what natural calamities can do to

power plants that are dangerous as in the case of nuclear energy or the dangers of burning coal

and exhausting the reserves of carbon based fuel. It takes commitment and action on part of all

of human kind for promoting these energies.

In order to produce larger wind turbines the efficiency of the blades designed must be

optimum. Since turbines growing larger in diameter, the rotation speed is slow and hence power

production is dependent on high performing aerodynamic design, a rigid structure, advanced

composite materials and optimization techniques to maximize power minimize cost of

production are of importance and to be scientifically studied and implemented.

There is no perfect rotor design, the choice of parameters are just optimized to obtain

one of a kind of rotor, the different airfoils and the choice of material with the speed of rotation

and the wind speeds for which the turbine is designed just leads an understanding that

optimization is critical in the design phase of the wind turbine rotors.

In order to produce power efficiently medium scale turbines that are of 1 MW to 5 MW

capacities are designed. In order to be efficient in drawing power from the wind optimization

techniques are needed at various stages in design of the rotors to the arrangement of the

turbines in the wind farms is of importance.

A survey of literatures for this thesis has yielded that structural optimizations on blades

of lengths of well between 10 to 30 meters has been optimized for maximizing power and

9

decreasing the weight of the blades. The blades of length greater than 50 meters has a different

design concept and the usage of CFD is needed for optimizing aerodynamically and the

structures of the blades are very different. In most cases the small turbines are scaled

dimensionally for designing medium scale turbines, hence an initial estimate of how designing

and optimizing of a medium scale turbine from beginning forms the basis for this research.

10

CHAPTER 2

LITERATURE REVIEW

A significant amount of exhaustive research has been done in the area of small and

medium scale wind turbine blades and most of them have used the classical blade element

momentum theory for designing the blades and calculating the forces acting on it. Lot of

research on finding the optimum chord lengths has been made using a variety of evolutionary

optimizing techniques. Some work that forms the background for this research is as follows.

Mahri and Rouabah [1] had calculated the dynamic stresses on a blade which was

designed using the blade element theory. The rotor diameter was 10 meters and the dynamic

analysis was made using the beam theory and the modal analysis is made using the finite

element modeling and also using the blade motion equation. Mickael Edon [2] had designed a

blade for 38 meters for a 1.5MW power using the BEM theory, and had suggested in his future

work the chord distribution formula which I have implemented. Since his blade was close to my

design I choose the same airfoil profile.

Philippe Giguere and Selig [3] had described blade geometry optimization for the

design of wind turbine rotors, pre-programmed software was used to optimize structures and

cost model. M. Jureczko, M. Pawlak, A. Mezyk [4] used the BEM theory to design and used

ANSYS for calculation of natural frequencies. They had found out the mode shape of the blades

by using the Timoshenko twisted tapered beam element theory. The genetic algorithm was

used to minimize blade vibration, maximize output, minimize blade cost and increase stability.

Tingting Guo, Dianwen Wu, Jihui Xu, Shaohua Li [5] developed a 1.5 MW turbine rotor

of 35 meters blade length, using Matlab programming for designing and concluded the

feasibility of Matlab for designing large wind turbines, further they had also compared with CFD

11

results and the found out Matlab was economical in artificial design and optimizing for

efficiency. Carlo Enrico Carcangiu [6] used CFD tool FLUENT to a better understanding of fluid

flow over blades.

Jackson, et.al [7] made a preliminary design of a 50 meters long blade, two versions

one of fiber glass and one with carbon composite was used to test the cost and thickness of

cross sections was changed in order to improve structural efficiency. The aerodynamic

performance was made using computational techniques and the computations were predicted

using clean and soiled surface.

Wang Xudong, et al [8] used three different wind turbine sizes in order to optimize the

cost based on maximizing the annual energy production for particular turbines at a general site.

In their research using a refined BEM theory, an optimization model for wind turbines based on

structural dynamics of blades and minimizes the cost of energy. Effective reduction of the

optimization was documented.

Karam and Hani [9] optimized using the variables as cross section area, radius of

gyration and the chord length, the optimal design is for maximum natural frequency. The

optimization is done using multi dimensional search techniques. The results had shown the

technique was efficient.

Ming-Hung Hsu [10] has given a model for analysis of twisted tapered beams using the

spline collocation method. The expressions for cross sectional area and moment of inertias are

given which are used in this present work.

Rao and Gupta [11] used the finite element method for the analysis of twisted tapered

rotating Timoshenko beams. The stiffness and mass matrices are derived using the shape

functions and the natural frequency is found out by converting the problem to an Eigen value

problem.

12

B. Hillemer,et al [12] designed wind turbines which were to output beyond 5 MW and

they had scaled up existing rotors and further calculated stress, moments and natural

frequencies. For the analysis they had used the simple beam theory. The scaled up blades

were optimized for minimizing weight by changing the airfoil shell thickness and web and flange

at every cross section. The constraints used for their work being structural strength and

minimum weight.

J.H.M. Gooden [13] investigated two dimensional characteristics of FX 66 S 196 V1

airfoil which is the airfoil that has been used in this report. The coefficients of lift and drag for

various Reynolds number.

This present work is done in designing a wind turbine blade using the Blade Element

Theory for a length of 45 meters. The chord lengths are calculated using the formula in

reference [2] and the chord distributions, flow angles, the differential power, thrust and torque

are all at discrete intervals of the blade are plotted. The blade is then assumed to be a tapered

hollow beam and the stiffness and mass matrix are derived as explained in the reference [11].

The natural frequency is found out by solving the Eigenvalue problem. The first six natural

frequencies for axial and transverse direction are calculated. The mode shapes are plotted as

well. The optimization involves chord length as the design variables and the power and mass

were used as objective functions. The constraints are the natural frequency along with the

power or the mass depending on the optimization problem.

13

CHAPTER 3

AIRFOIL THEORY

3.1 Introduction

The most important part in designing a wind turbine blade is the choice of airfoil, as the

entire blade is made up of airfoils sections and the lift generated from this airfoil at every section

causes the rotation of the blade, also the performance of the blade is highly dependent on this

choice making the selection and study of the airfoil of prime importance. From Figure 3.1 we

can define the chord to be a straight line connecting the leading edge to the trailing edge of the

airfoil. The angle of attack is defined as the angle between the chord line and the free stream

velocity of air. All the forces generated from the airfoil act on the aerodynamic center which is

located about a fourth of the chord length from the leading edge of the airfoil.

The forces generated by the airfoil is resolved into lift the force perpendicular to the

direction of free flow of wind and the drag force in direction of the free flow of wind. The lift and

drag force are given by the expression,

21

2lL C cVρ=

(3.1)

21

2dD C cVρ=

(3.2)

lC and

dC

are the coefficients of lift and drag.

14

3.2 Reynolds Number

The forces over the airfoil change with respect to the fluid properties the length of the airfoil and

surrounding temperature. Hence a non dimensional parameter called the Reynolds number is

defined. The ratio between the inertia and viscous forces, given by the expression,

ce

VLR

ν=

(3.3)

V is the free stream velocity,

cL is the characteristic length of the chord,

ν is dynamic viscosity of air

The choice of airfoils is such that the maximum lift is obtained for a given angle of

attack. The Reynolds number for aircrafts are really high compared to the wind turbine blades,

hence airfoils used in aircraft wings cannot be used to design wind turbine blades. The ‘Wind

Turbine Catalogue’ from The Riso national laboratory of Denmark had provided data for a

variety of airfoil families for designing and choosing airfoils in the design of wind turbine blades.

Based on the catalogue for large wind turbine blade design the FX66-S196-V1 type of airfoil is

given to be the best. Although in some cases of blade design mixtures of airfoils are used from

the root to the tip end of the blade, this is true in case if the airfoil type has a group. The FX66-

S196-V1 does not have a family group hence I have designed the blade with the same airfoil

throughout.

The coefficient of lift versus angle of attack and coefficient of lift versus coefficient of

drag for various Reynolds number is show in the graphs below. Form the graphs it is evident the

optimum angle of attack for this airfoil is 9°.

15

Figure 3.1 Airfoil [25]

L

c/4

16

Figure 3.2 FX 66 S 196 V1 Coefficient of Lift Vs Angle of Attack [13]

17

Figure 3.3 FX 66 S 196 V1 Coefficient of Drag Vs Coefficient of Lift [13]

18

CHAPTER 4

AERODYNAMICS

4.1 Introduction

The principle of wind turbine is that the kinetic energy from the wind is converted to

mechanical energy. Before understanding the blade element momentum theory a brief ideal

rotor case understanding is essential. An ideal rotor is assumed such that no friction and there

is no rotational velocity component in the wake. From the control volume figure below, the

velocity of air between upstream and downstream is reduced at the rotor. Expressions for thrust

and power with velocities are all derived. The pressure difference between upstream and

downstream is converted to thrust of the rotor given by expressionT A p= ∆ .

From Newton’s second law dp

Fdt

= where p is the momentum and t is time. Integrating over

control volume and applying the Newton’s second law is,

CV CS

dpF Vdv V VdA

dt tρ ρ

∂= = +

∂ ∫ ∫ ∫ ∫ ∫ (4.1)

21

2p V constρ+ =

(4.2)

From Bernoulli equation

2 2

1 1

1 1

2 2p V p Vρ ρ+ = + and

2 2

1 2

1 1

2 2p p V p Vρ ρ− ∆ + = +

2 2

1 2

1( )

2p V Vρ∆ = −

(4.3)

19

Figure 4.1 Ideal Rotor Velocity and Pressure Profiles [14]

Applying the conservation of mass from Figure 4.2

2 2 2

2 2 1 2 1 1 1( ( ) )CV s CVV A V A A V m V V A Tρ ρ ρ− + − + − =� (4.4)

Where 2 1 2( )sm A V Vρ= −�

The conservation of mass also gives the relation

2 2m AV A Vρ ρ= =� (4.5)

The above equations 4.4 and 4.5 give

1V

p

2V

p V

V

1V

2V

1p

20

1 2 1 2( ) ( )T m V V VA V Vρ= − = −� (4.6)

The thrust produced at the rotor

The velocity 1 2( )

2

V VV

−=

Figure 4.2 Ideal Rotor Control Volume [14]

T A 2

V CV

A

1V

2A

sm�

21

4.2 Betz Limit

A German physicist Albert Betz proved that the maximum that a wind turbine can

extract and convert the kinetic energy of wind to power is only 59.3%. This was termed as Betz’

limit and the proof is as follows. Assuming a rotor, the mass of air moving through the rotor is

1 2

2

V Vm Aρ

+ =

(4.7)

The power extracted at the rotor from wind is

2 2

1 2

1( )

2P m V V= −

(4.8)

From equation 4.7 and 4.8

2 2

1 2

1( )

2P AV V Vρ= −

(4.9)

The total power available in the wind is given

3

1

1

2oP AVρ=

(4.10)

The coefficient of power is the ratio between the actual power of rotor to available

power given by the expression

2

2 2

1 1

11 1

2P

o

V VPC

P V V

= = − +

Plotting a graph for PC versus 2

1

V

V

we get plot as

22

Figure 4.3 Betz Limit

4.3 Induction Factor

The drop in velocity of the free stream of air at the rotor in the axial direction we

introduce an axial induction factor a , expressed as

1(1 )V a V= − (4.11)

The thrust and power expressions in terms of axial induction factor are

3 2

1

2 2

1

2 (1 )

2 (1 )

P AV a a

T AV a a

ρ

ρ

= −

= − (4.12)

The aerodynamic shape of the blade causes a torque on the rotor when it comes in

contact with the wind. The thrust is produced as a consequence of the torque. The torque is a

force exerted by the wind on the blades. According to Newton’ third law an equal and opposite

reaction the blades exerts a force on the wind causing the air behind the rotor to rotate in the

opposite direction to that of the rotor. This induced tangential velocity in the wake of the rotor

gives the tangential induction factor given by

(1 ')rotV a rω= + , where r is the radial distance from the center of the rotor.

23

Figure 4.4 Velocity Triangle Over Airfoil

The relative velocity component is the actual velocity induced on the element of the

blade. Since the rotation of the blade and the velocity of free stream of air is perpendicular to

each other, the relative velocity of air hitting the blade is dependent on the radius of the blade,

in turn the induction factors.

W

1V

Rotor direction

'a rω

φ aV

rω

φ

L

relV

V

24

4.4 Tip Speed Ratio

The tip speed ratio is the ratio between the rotor rotational speed and the free stream

velocity of the wind given by

1

R

V

ωλ = . The local tip speed ratio is given by

1

rx

V

ω= . The tip

speed ratio affects the angular velocity of the rotor in turn the rotations per minute of the blade.

The tip speed ratio is of prime importance while designing a wind turbine as it affects the twist of

the blade and the power produced.

4.5 Pitch, Twist and Chord Lengths

Since the induction factors are dependent on radius of the element along the blade, it is

evident that the inductions factors changes. The velocity triangle for the induction factors are as

shown below explains the relative velocity component.

4.5.1 Pitch

As wind turbines go higher the range of operating speeds also needs to increase hence

the turbines are designed for a wide range of wind speeds. In figure 4.5, the top images show

the free stream velocity at both low and high speed and because of which the relative velocity

changes and in order for the optimum angle of attack of the blade to face the wind, the entire

blade has to be pitched.

25

Figure 4.5 Pitch and Twist with Velocity Triangles [2]

4.5.2 Twist

In the left half of figure 4.5 the top and bottom images show the relative velocity across

the length of the blade changes as the rotation speeds vary along the blade length, due to

which the entire section of the blade made up of a number of airfoil strips needs to face the wind

at optimum angle of attack of the incoming wind, hence the blades are twisted throughout.

4.5.3 Chord Length

In all of modern wind turbines chord length closer to the tip is longer and at the tip end it

is the shortest. The loads on the blade increase from the root end to the tip end and having the

same chord dimension all along the blade only increases the mass of the blade and not the

efficiency. Also in order to control the lift of the blade at all sections the chord is varied all along

the blade length. The chord length distribution is given by the expression in terms of radius.

2 3 4

5.957 3.1 .5433 .02917( ) 1.868

l

Rc r

BC x x x xλ

= + − + −

26

CHAPTER 5

BLADE ELEMENT MOMENTUM THEORY

The classical blade element momentum theory was developed in order to predict the

behavior of propellers. This method was also used to determine the loads on the blade. This

theory has been used constantly to design propeller blades of helicopters, aircrafts and also the

blades for harnessing wind power. The simplicity of this theory gives all the performance

parameters like thrust and power of any blade that is designed. The theory begins with the

conservation of momentum theory. The blade to be designed is divided into discrete elemental

sections, also the independency of each element, i.e. the neighboring elements are

independent of the forces acting on each element. Another assumption is the force from the

blades onto the flow is constant for each discrete element, this indicates an assumption of an

infinite number of blades. The correction for this assumption is incorporated in this thesis work

and explained later. This iterative method is used to calculate the flow angles, the differential

thrust, torque and power. Once the differential toque and power are known and integrated along

the blade element to obtain total power of the blade. This theory uses aerodynamic data from

the airfoils chosen and also the specific chord lengths. A basic flow chart is given later in this

chapter.

27

Figure 5.1 Velocity Triangle of A Section of Wind Turbine Blade

The differential thrust and torque is given by

2

1

3

1

4 (1 )

4 '(1 )

dT r V a a dr

dM r V a a dr

π ρ

π ρ ω

= −

= − (5.1)

From the given velocity triangle diagram we see that the angle of attack and twist of the blade

section is given by

1(1 )tan

(1 ')

a V

a r

α φ θ

φω

= −

−=

+ (5.2)

The lift and drag are given by equations

2

2

1

2

1

2

rel l

rel d

L V cC

D V cC

ρ

ρ

=

= (5.3)

1(1 )a V−

28

Figure 5.2 Normal Forces over Airfoil [14]

As the forces on in the normal and tangential direction to the rotor plane are important, the lift

and drag forces when resolved as shown in figure give expression as

cos sin

sin cos

N

T

P L D

P L D

φ φ

φ φ

= +

= − (5.4)

When the normal and tangential forces are normalized with respect to 21

2relV cρ the

expressions in terms of &l dC C are

cos sin

sin cos

n l d

t l d

C C C

C C C

φ φ

φ φ

= +

= − (5.5)

Where 2 2

&1 1

2 2

N Tn t

rel rel

P PC C

V c V cρ ρ= =

29

A parameter called the solidity is defined as the ratio of the area of the blades to the swept area

of the rotor, given by the expression ( )

( )c r B

rr

σπ

=

The differential thrust and torque in terms of normal and tangential forces on the control volume

of thickness dr are

N

T

dT BP dr

dM rBP dr

=

= (5.6)

The expression for differential thrust and toque on substitution for &N TP P

2 2

1

2

1

(1 )1

2 sin

(1 ) (1 ')1

2 sin cos

n

t

V adT B cC dr

V a r adM B cC rdr

ρφ

ωρ

φ φ

−=

− +=

(5.7)

On equating the thrust and toque equations we get expressions for the induction factors as

2

1

4sin1

1'

4sin cos1

n

t

a

C

a

C

φ

σ

φ φ

σ

=

+

=

− (5.8)

Since all the expressions are for an ideal rotor case and of infinite blades we introduce

a parameter called the Prandtl’s Tip Loss Factor. This correction changes the vortex system of

the wake for finite number of blades. The correction F incorporated into the thrust and torque

equations are,

2

1

3

1

4 (1 )

4 '(1 )

dT r V a a Fdr

dM r V a a Fdr

π ρ

π ρ ω

= −

= − (5.9)

30

The Prandtl’s correction factor F

12cos ( )

( )

2 sin

fF e

B R rf

r

π

φ

− −=

−=

(5.10)

The induction factors expressions change to

2

1

4 sin1

1'

4 sin cos1

n

t

aF

C

aF

C

φ

σ

φ φ

σ

=

+

=

− (5.11)

The flow chart for the Blade element momentum theory is given as follows

31

Once the values of & 'a a converge the differential thrust and torque can be calculated. The

power of the blade is computed by dP dMω=

Guess initial

value of a and a’

1(1 )tan

(1 ')

a V

a r

α φ θ

φω

= −

−=

+

cos sin

sin cos

n l d

t l d

C C C

C C C

φ φ

φ φ

= +

= −

2

1

4sin1

n

a

C

φ

σ

=

+

1

'4sin cos

1t

a

C

φ φ

σ

=

−

Check for convergence

of a and a’

a and a’

Y

N

32

CHAPTER 6

RESULTS

From the above concepts the blade of the wind turbine was designed and the

calculated values are as follows.

The length of the blade was chosen to be 45L = meters.

The radius of rotor is 46R = meters.

The tip speed ratio is 6TSR = .

The coefficient of performance was chosen to be as PC =0.4.

The free stream velocity of air is chosen as V =12 m/s.

The rotation speed of the blade based on the TSR was calculated to be

1.56V

TSRR

ω

= =

The rotation per minute 30

14.94RPM ωπ

= =

As the stresses on the root end of the blade will be high having an airfoil shape at the

root end is not feasible hence the root end is circular in shape and then transcends to an airfoil

shape is needed. This transition length of 4 meters was used.

I have taken 40 elements of 1 meter span across the entire blade and the chord lengths

were calculated but the distribution of chord lengths was from 9.5 m to 4.9 m. the length of the

chord near the root end was very large and after studying various literatures the size of the

chord at the root end for large wind turbines I came up with a distribution ranging from 3.72 m to

33

1.94 m. Using the blade element momentum theory, the twist of the blades, differential thrust,

torque and power are calculated.

The plots of chord distribution, twist or flow angle distribution, differential power, thrust

and torque are plotted with respect to radius.

Figure 6.1 Chord Distribution

The chord lengths in meters along the length of the blade element, from 3.72 meters at

4 metes of the blade length to 1.94 meters at 44 meters of the blade element.

34

Figure 6.2 Twist Distribution

The degrees of twist in degrees along the length of the blade element, from 42.210 at 4 metes of

the blade length to -3.430 at 44 meters of the blade element.

35

Figure 6.3 Differential Power

36

Figure 6.4 Differential Thrust

37

Figure 6.5 Differential Torque

38

Table 6.1 Distribution of Differential Thrust, Torque, Power and Flow Angles

radius (m) Chord lengths

(m)

Thrust x10^5

(N)

Torque

(MN.m) Power (MW)

twist

(deg)

4 3.7235 0.4969 0.1899 0.2971 42.2153

5 3.8331 0.6278 0.2589 0.405 38.022

6 3.7967 0.7554 0.3277 0.5126 34.2509

7 3.7009 0.8862 0.3971 0.6213 30.8194

8 3.5843 1.0228 0.4682 0.7324 27.7142

9 3.464 1.1667 0.5412 0.8468 24.9219

10 3.3475 1.3184 0.6166 0.9647 22.4206

11 3.2381 1.4781 0.6945 1.0866 20.183

12 3.1367 1.6459 0.7749 1.2124 18.1802

13 3.0434 1.8217 0.8578 1.342 16.3845

14 2.9577 2.0055 0.943 1.4753 14.7705

15 2.8791 2.1972 1.0305 1.6122 13.3153

16 2.8069 2.4043 1.1195 1.7514 11.9735

17 2.7406 2.6125 1.211 1.8946 10.778

18 2.6794 2.8282 1.3043 2.0405 9.6891

19 2.623 3.0513 1.3993 2.1891 8.6941

20 2.5708 3.2911 1.4944 2.3379 7.7608

21 2.5224 3.53 1.5919 2.4905 6.9207

22 2.4774 3.7759 1.6905 2.6448 6.1455

39

Table 6.1 continued

radius (m) Chord lengths

(m)

Thrust x10^5

(N)

Torque

(MN.m) Power (MW)

twist

(deg)

23 2.4355 4.0386 1.7881 2.7975 5.4103

24 2.3964 4.2994 1.8879 2.9537 4.7435

25 2.3598 4.5667 1.988 3.1103 4.1228

26 2.3255 4.8507 2.0859 3.2634 3.5276

27 2.2934 5.1311 2.1854 3.4191 2.9846

28 2.2631 5.4167 2.2842 3.5737 2.4753

29 2.2346 5.7187 2.379 3.7219 1.981

30 2.2078 6.0139 2.4744 3.8713 1.5285

31 2.1824 6.3236 2.5643 4.012 1.0867

32 2.1583 6.6237 2.6537 4.1517 0.6805

33 2.1356 6.935 2.7353 4.2794 0.2812

34 2.114 7.2315 2.8142 4.4029 -0.0872

35 2.0934 7.5332 2.882 4.5089 -0.4523

36 2.0739 7.8227 2.9397 4.5992 -0.8026

37 2.0553 8.0927 2.9845 4.6693 -1.1402

38 2.0376 8.3329 3.0124 4.7129 -1.4669

39 2.0206 8.5285 3.0179 4.7216 -1.7848

40 2.0045 8.6573 2.9937 4.6837 -2.096

41 1.989 8.6974 2.9253 4.5766 -2.4135

42 1.9742 8.5855 2.8005 4.3814 -2.7304

43 1.96 8.2444 2.5908 4.0533 -3.0626

44 1.9464 7.5165 2.2487 3.5181 -3.4317

40

CHAPTER 7

STRUCTURAL ANALYSIS

7.1 Simple Beam Theory

The structural analysis of the blade was done by performing the modal analysis of the

blade (calculating the natural frequency). This finite element method is used in order to obtain

the stiffness and mass matrices. The blades are assumed to be a hollow rectangular beam

element while performing the analysis with the breadth and depth as a function of the chord

length. Since the assumption involves the chord as a parameter for determining the dimensions

of the blade element and since chord length is largest at the root end and smallest at the tip,

hence the beam is to be analyzed as a tapered beam.

Considering a beam element and the displacements and moments at the nodes are as

shown. A linear elastic beam equation is derived as follows. The beam is subjected to a load in

the y direction, as ( )w x . Choosing an element of the beam and from force and moment

equilibrium,

ˆ( ) 0w dx dR− + =

ˆdRw

dx= −

(7.1)

ˆ ( ) 0R dx dM+ =

ˆ dMR

dx=

(7.2)

Curvature of the beam is given by

41

1 M

EIκ = =

Θ (7.3)

Where Θ is the radius of deflection. E is Young’s modulus, I is the moment of inertia, the

expressions for these is given later for tapered beams.

2

2

d v

dxκ =

(7.4)

v is the deflections in the axial and tangential direction.

2

2

d v M

dx EI∴ =

(7.5)

Substituting into equations 7.1 and 7.2 after solving for moment force M from 7.5,

2 2

2 2( )

d d vEI w x

dx dx

= −

(7.6)

42

Figure 7.1 Reaction and Moments of A Beam [15]

43

Figure 7.2 Tapered Cantilever Beam [11]

From figure 7.2 cross sectional area of the beam changes with respect to radius ‘r’. The area is

given as a function of radius as,

( ) ( ) ( )A r b r h r=

1 1( ) 1 1r r

A r b hR R

= + Μ + Ν

(7.7)

2 1 2 1

1 1

&b b h h

b h

− −Μ = Ν =

r

R

ez

x

r

y

1V 2V 3V 4V

44

&b h are the breadth and height.

The moment of inertia is given by

33

1 1( ) 1 112

xx

b h r rI r

R R

= + Μ + Ν

33

1 1( ) 1 112

yy

b h r rI r

R R

= + Μ + Ν

(7.8)

Choosing a displacement function as

3 2

1 2 3 4( )v x a r a r a r a= + + + (7.9)

As shown in the figure since the degrees of freedom is 4 the cubic displacement function is

chosen. Further expressing v as a function of nodal degrees of freedom 1 2 3 4, , ,v v v v as shown

in the figure 7.2,

1 4

2 3

3 2

3 1 2 3 4

2

1 2 3

(0)

(0)

( )

( ) 3 2

v v a

dv v a

dx

v r v a r a r a r a

dv r a r a r a

dx

= =

= =

= = + + +

= + +

(7.10)

Solving for 1 2 3 4, , ,a a a a and substituting in equation

3 2

1 3 2 4 1 3 2 4 2 13 2 2

2 1 3 1( ) ( ) ( ) ( )v v v v v x v v v v x v x v

L L L L

= − + − + − − − − + + (7.11)

In matrix form the expression is given by ˆ[ ]v N d=

45

1 2 3 4[ ] [ ]N N N N N= and

1

2

3

4

ˆ

v

vd

v

v

=

Where

3 2

3 21

2

3 23

4

3 2

2 3 1

2

2 3

x x

L L

N x xL L x

N L L

N x x

N L L

x xL L

L L

− +

− + =

− +

−

are the shape functions.

From beam element theory the assumption that the cross section does not deform in

shape even with the bending of the beam. Strain of the element is given as,

2

2( , )x

d vx y y

dxε = −

(7.12)

From the beam element theory, the traverse displacement function and the bending

moment and shear force are related.

2

2( )

d vM r EI

dx=

And

3

3ˆ d v

R EIdx

= (7.13)

46

7.2 Stiffness Matrix

Using the nodal and beam theory and the equation 7.11 and 7.13

3

1 1 2 3 43 3

22 2

2 1 2 3 42 3

3

3 1 2 3 43 3

22 2

4 1 2 3 42 3

(0)ˆ (12 6 12 6 )

(0)(6 4 6 2 )

( )ˆ ( 12 6 12 6 )

( )(6 2 6 4 )

d v EIf R EI v Lv v v

dx L

d v EIf m EI Lv L v Lv L v

dx L

d v L EIf R EI v Lv v v

dx L

d v L EIf m EI Lv L v Lv L v

dx L

= = = + − +

− −= = = + − +

= − = − = − − + −

= = = + − + (7.14)

ˆF Kd=

1 1

2 22 2

3

3 3

2 2

4 4

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

f vL L

f vL L L LEI

f vL L L

L L L Lf v

− − = − − −

−

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

L L

L L L LEIK

L L L

L L L L

− − = − − −

−

47

7.3 Mass Matrix

For modal analysis the usage of lumped mass matrix although easier, the mass matrix derived

from shape functions yields better results. The mass matrix is given by,

0

[ ] [ ] [ ]L

T

A

M N N dAdxρ= ∫ ∫∫

2 2

2 2

156 22 54 13

22 4 13 3[ ]

420 54 13 156 22

13 3 22 4

L L

L L L LALM

L L

L L L L

ρ

− − = − − − −

From the stiffness and mass matrix the natural frequency of the beam element can be

computed by solving it as an eigenvalue problem ( ) 0K M vλ− = .

In order to find the natural frequencies in the transverse and axial directions the following

expressions for stiffness matrix and mass matrix is used.

Transverse direction

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

xx

L L

L L L LEIK

L L L

L L L L

− − = − − −

−

and

2 2

2 2

156 22 54 13

22 4 13 3[ ]

420 54 13 156 22

13 3 22 4

L L

L L L LALM

L L

L L L L

ρ

− − = − − − −

Axial direction

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

yy

L L

EI L L L LK

L L L

L L L L

− − = − − −

−

and

2 2

2 2

156 22 54 13

22 4 13 3[ ]

420 54 13 156 22

13 3 22 4

L L

L L L LALM

L L

L L L L

ρ

− − = − − − −

7.4 Mode Shapes

In order to find the mode shapes of the beam, transverse vibration of Bernoulli-Euler beams,

48

( ")" 0EIV Avρ+ =�� (7.15)

An assumption of harmonic motions is made,

( , ) ( )cos( )v x t V x tω= − Θ (7.16)

Substituting equation 7.16 in 7.15

2( ")" 0EIv A Vρ ω− = (7.17)

This equation reduces to

44

40

d VV

dxλ− =

(7.18)

where

24 A

EI

ω ρλ =

The general solution to the above differential equation is given by

1 2 3 4( ) sinh( ) cosh( ) sin( ) cos( )V x C x C x C x C xλ λ λ λ= + + + (7.19)

The boundary conditions used for the beam is the cantilever beam conditions which are,

2 3

2 3

@ 0

(0)(0) 0; 0

@

( ) ( )0; 0

x

dVV

dx

x L

d V L d V L

dx dx

=

= =

=

= = (7.20)

On using these conditions in equation

49

2 4

1 2 3 4

22

1 2 3 42

33

1 2 3 43

( sinh( ) cosh( ) sin( ) cos( ))



V C C

dVC x C x C x C x

dx

d VC x C x C x C x

dx

d VC x C x C x C x

dx

λ λ λ λ λ

λ λ λ λ λ

λ λ λ λ λ

= +

= + + +

= + − −

= + − − (7.21)

In matrix form,

1

2

2 2 2 2

3

3 3 3 3

4

0 1 0 1 0

0 0 0

sinh cosh sin cos 0

sinh cosh sin cos 0

C

C

CL L L L

L L L L C

λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ

= − −

− − (7.22)

As the first two equations give a trivial solution, and in order to find the coefficients a

characteristic equation of the form

cos( )cosh( ) 1L Lλ λ = − (7.23)

The solution for this characteristic equation is solved in […] and the first four values are

1

2

1.8751

4.6941

L

L

λ

λ

=

=and

3

4

7.8548

10.996

L

L

λ

λ

=

=

From the four equation (7.22) we get,

1 3C C= − and 2 4C C= −

The third equation in equation (7.22) and the above equation give

1 2 3 4sinh( ) cosh( ) sin( ) cos( ) 0C x C x C x C xλ λ λ λ+ − − =

1 2(sinh( ) sin( )) (cosh( ) cos( ))C x x C x xλ λ λ λ+ + +

1 2 2

cosh( ) cos( )

sinh( ) sin( )r

x xC C C

x x

λ λχ

λ λ

+= − = −

+

50

The mode shape is given by the expression

( ) [(cosh( ) cos( )) (sinh( ) sin( ))]r rx C L L L Lφ λ λ χ λ λ= + − + (7.24)

7.5 Breadth and Height

In the analysis of the blades I have used the hollow beam theory and the breath and height of

the beam is considered as a function of the chord. The thickness of the element is 20% of the

original breath and height. Hence the area and moment of inertias are given as follows

1 1 2 2

1 1 2 2

0.8 & 0.8

0.8 & 0.8

B b B b

H h H h

= =

= =

2 1 2 11 1

1 1

2 1 2 12 2

1 1

&

&

b b h h

b h

B B H H

B H

− −Μ = Ν =

− −Μ = Ν =

33

1 11 1 1

33

1 12 2 2

( ) 1 112

( ) 1 112

xx

xx

b h r rI r

R R

B H r rI r

R R

= + Μ + Ν

= + Μ + Ν

1 2( ) ( )xx xxIxx I r I r= −

51

The first six natural frequencies are given as follows.

Table 7.1 Natural Frequency in the Transverse Direction

Frequency number Natural frequency

(rad/s)

1 301.1736

2 429.8942

3 498.3677

4 582.6563

5 677.6906

6 789.1982

Table 7.2 Natural Frequency in the Axial Direction

Frequency number Natural frequency

(rad/s)

1 1505.9

2 2149.5

3 2491.8

4 2913.3

5 3388.5

6 3946

The first four mode shapes of the beam are as follows given in the graphs below.

52

Figure 7.3 First Mode Shape

Figure 7.4 Second Mode Shape

53

Figure 7.5 Third Mode Shape

Figure 7.6 Fourth Mode Shape

54

CHAPTER 8

DESIGN OPTIMIZATION

The optimization process in the designing of wind turbines is always essential owing

due to the myriad of parameters that can be varied. As depending on what the design variables

and objective function determines the outcome of the optimization. There is no unique answer in

optimization problems; it is just the physics of the problem that yields the appropriate results

depending on the choice of input variables.

For this work the ‘fmicon’ function in Matlab was used as the optimization tool to

optimize the blade. The design variable was chosen to be the chord length. Since only Matlab

was used and a constant airfoil profile was used, the design variable had to only the lengths of

the chord across the blade element.

Various problems for constrained problems were performed, with only the chord lengths

as the design variable, the power could be maximized or the weight of the blade could be

minimized, where one is constraint and the other is the objective function could be used. As the

natural frequency is used constraints or the deflection to be minimized can also be done. This

report has a study of how the results are generated based on the following problems,

55

Table 8.1 Optimization problem definition

OBJECTIVE FUNCTION CONSTRAINTS

Maximize Power Mass

Minimize Mass Power

Minimize Mass Power and Natural Frequency

The flow chart as to how the optimization process was implemented is as follows.

Chord and radius

Thrust, Torque

Power and flow

angle

Natural Frequency

Constraint

Optimal Chord length

Blade

element

theory

Stiffness &

Mass matrix

N

Y

Figure 8.1 Flow Chart of Optimization Process

56

Problem 1: with the objective function is power and constraint of mass less than 7000

Kilograms, the optimization yielded, the following chord length distribution, twist, differential

power and thrust across the blade element.

Figure 8.2 Optimized Chord Distribution Problem 1

57

Figure 8.3 Optimized Twist Distribution Problem 1

58

Figure 8.4 Optimized Differential Thrust Problem 1

59

Figure 8.5 Optimized Differential Power Problem 1

60

The power and mass of the blade before and after optimization is tabulated,

Table 8.2 Original and Optimized Power and Mass Problem 1

Power MW Mass in Kilograms

Original 1.5787 13060

Optimized 1.7232 6806.8

Table 8.3 First Three Natural Frequencies Transverse Direction Problem 1

Optimized rad/sec Original rad/sec

269.3 301.1736

359.5 429.8942

389.5 498.3677

Table 8.4 First Three Natural Frequencies Axial Direction Problem 1


1346 1506

1798 2149

1947 2492

61

Problem 2: Objective function Mass and constraint of Power greater 1.7 MW, the optimization

yielded, the following chord length distribution, twist, differential power and thrust across the

blade element


From this problem it is evident that the physics of the problem yields results with constant chord

length and still satisfies the constraints of power. As the disadvantages of having a constant

chord throughout the length of the blade are higher, this result although right cannot be taken

into account. With constant chord the aerodynamics at the root end is not efficient and hence

practically not feasible. The lower bound on the problem is changed and the results are checked

again. The results after changing the lower bounds yield

62


63

Figure 8.8 Optimized Twist Distribution Problem 2

64

Figure 8.9 Optimized Differential Power Problem 2

65



Original 1.5787 13060

Optimized 1.6825 9106



288.9 301.1736

403 429.8942

448.6 498.3677



1444 1506

2015 2149

2243 2492

66

As seen from the results the constraints are violated in the frequency. Due to the fact

that we cannot predict the lower bounds for all the problems, we do not consider this

optimization to be precise, as only an initial guess is possible, but the results are not wrong they

are unique as it is for defining the optimization problem.

Problem 3: Objective function is the mass with the constraints natural frequency greater than

300 and power greater than 1.7 MW, the optimization results yielded, are


67



Original 1.5787 13060

Optimized 1.6823 8676.5


Optimized Original

293.8 301.1736

325.3 429.8942

367.1 498.3677


Optimized Original

1469 1506

1626 2149

1836 2492

This result also yields improper results, as it violates the constraints.

From the optimizations problems solved it is evident that there are multiple optimal

solutions but the solution to be chosen depends on factors like aerodynamic efficiency,

structural stiffness, manufacturing complexity. The choice of optimizing for power gave better

results compared to the other optimizations performed based on the factors mentioned. As

68

there is no perfect answer in optimizing and certainly not unique, the results can vary depending

on the problem formulation.

69

CHAPTER 9

CONCLUSION

In this thesis a wind turbine blade of 45 meters in length and a wind speed of 12 m/s

was designed using the blade element moment theory where the flow angles and differential

thrust, torque and power was calculated. The blade designed was further analyzed as a hollow

tapered beam and the stiffness and mass matrices were calculated with the cross sectional

area and moment of inertia was calculated as mentioned in reference [11], for a tapered beam.

The chord lengths at every section determined the taper and breadths and the height of the

beam was also a function of the chord length and assumed to be the thickness of the airfoil. The

natural frequencies in the axial and transverse direction are tabulated.

After analysis the blade was optimized using Matlab as a tool and fmincon as the

function. The design variables were the chord lengths and three different optimization problems

were solved. With using power as the objective function and mass as a constraint the first set of

optimized chord lengths were plotted as well as the differential thrust, torque and power. The

second and third optimization was done with the objective function as mass and constraint as

power and natural frequency.

70

CHAPTER 10

FUTURE WORK

Further studies on the optimum composite layering for the blades so that the optimal

design for stress and weight can be found out is one of the prime continuation of the work, with

the introduction of the composite layering gives a better understanding of the behavior of the

blades dynamic qualities.

This report gives only an understanding how we can start designing process of a wind

turbine blade and optimizing it for performance structurally. The future of this work could be to

include interactions between multiple airfoil data and aerodynamically optimizing the blade for

aerodynamic performance. More often the larger blades are scaled from smaller ones as they

are already proven to be efficient, so in order to improvise on aerodynamics inclusion of CFD

analysis simultaneously for determining the aerodynamic parameters and structural analysis of

the blade using Ansys for a better understanding of how the analysis and design of the blade

could be improved can be determined. If an airfoil database is created from where data can be

input, optimizations based on wider design variables can be formulated.

The optimization techniques used in this work was fmincon in Matlab, but there are

other evolutionary algorithms that can be used for optimizing the blades. If the design variables

are increased to incorporate other parameters of the blade the possibility of designing blades of

high efficiency at less time is possible, basically finding the global optimum value in a whole set

of optimum values.

71

APPENDIX A

LIST OF SYMBOLS

72

a Axial induction factor

'a Tangential induction factor

B Number of blades c Chord length

PC Coefficient of power

&n tC C Coefficient of normal and tangential force

dM Differential torque

dT Differential thrust

F Prandtl’s tip loss factor

&xx yyI I Moment of inertia

K Stiffness matrix

L Length of blade

M Mass matrix m Mass of air flowing over rotor

P Power

&n tP P Normal and tangential force over airfoil

R Radius of rotor

V Velocity of free stream air

1V Velocity of air farthest from the rotor

relV Relative velocity of air over blade/airfoil

x Local tip speed ratio ω Angular velocity of rotor

α Angle of attack

φ Flow angle

θ Local pitch angle

λ Tip speed ratio

( )xλ Eigenvalue

73

REFRENCES

[1] Z.L. Mahri, M.S. Rouabah, “Calculation of dynamic stresses using finite element

method and prediction of fatigue failure for wind turbine rotor“ Wseas Transactions

On Applied And Theoretical Mechanics, Issue 1, Volume 3, January 2008

[2] Mickael Edon, “38 meter wind turbine blade design, internship report“

[3] Philippe Giguere and Selig, “Blade Geometry Optimization For The Design Of Wind

Turbine Rotors” AIAA-2000-0045

[4] M. Jureczko, M. Pawlak, A. Mezyk, “Optimization of wind turbine blades“, Journal of

Materials Processing Technology 167 (2005) 463–471

[5] Tingting Guo, Dianwen Wu, Jihui Xu, Shaohua Li, “The Method of Large-scale Wind

Turbine Blades Design Based on MATLAB Programming”, IEEE

[6] Carlo Enrico Carcangiu, “CFD-RANS Study of Horizontal Axis Wind Turbines” ,Doctor

of philosophy Thesis report

[7] K.J.Jackson, et al., “Innovative design approaches for large wind turbine blades”, 43rd

AIAA Aerospace Sciences Meeting and Exhibit 10 - 13 January 2005, Reno, Nevada

[8] Wang Xudong, et al.,”Blade optimizations for wind turbines”, Wind Energy. 2009;

12:781–803, Published online 29 April 2009 in Wiley Interscience

[9] Karam Y, Hani M, ”Optimal frequency design of wind turbine blades”, Journal of Wind

Engineering and Industrial Aerodynamics 90 (2002) 961–986

[10] Ming-Hung Hsu, “Vibration Analysis Of Pre-Twisted Beams Using The Spline

Collocation Method”, Journal of Marine Science and Technology, Vol. 17, No. 2, pp.

106-115 (2009)

74

[11] S. S .Rao, R.K Gupta ,” Finite Element Vibration Analysis Of Rotating Timoshenko

Beams”, Journal of Sound and vibration (2001) 242(1), 103}124

[12] B. Hillmer et al.,” Aerodynamic and Structural Design of Multi MW Wind Turbine Blades

beyond 5MW”, Journal of Physics: Conference Series 75 (2007) 012002

[13] J.H.M. Gooden, ”Experimental Lowe speed Aerodynamic Characteristics of the

Wortmann FX 66-S-196 V1 Airfoil”, http://www.standardcirrus.org/FX66-S-196V1-

Gooden.PDF

[14] Martin O.L. Hansen, “Aerodynamics of Wind Turbines”, second edition.

[15] Daryl L. Logan, “A Finite Course in the Finite Element Method”, second edition.

[16] Roy R. Craig and Andrew J. Kurdila, “Fundamentals of Structural Dynamics” Wiley

publications

[17] Done and Balmford, “Bramwell’s Helicopter Dynamics” second edition, AIAA

[18] http://guidedtour.windpower.org/en/core.htm

[19] http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/circulation/coriolis_and%20frict

ion.html

[20] http://www.ic.ucsc.edu/~wxcheng/envs23/lecture6/lect6_prnt.htm

[21] http://en.wikipedia.org/wiki/Smock_mill

[22] http://www.treehugger.com/files/2008/02/enercon_e126_largest_wind_turbine.php

[23] http://eetweb.com/wind/wind-turbines-go-supersized-20091001/

[24] Guidelines for Design of Wind Turbines − DNV/Risø

[25] http://www.av8n.com/irro/conformi_e.html

75

BIOGRAPHICAL INFORMATION

Mr. Bharath Koratagere Srinivasa Raju completed his bachelor’s in Mechanical

engineering from Visveswaraiah Technological University, India. He has worked in National

Aerospace Laboratories, Bangalore, as a project graduate trainee and was a part of the team in

the first US Asian MAV demonstration held at Agra. He joined The University of Texas at

Arlington in 2009 for the Masters in Aerospace engineering program. He worked under Dr. B. P.

Wang for his thesis research. He is interested to pursue a career in wind energy and become

an entrepreneur in a few years time.

DESIGN OPTIMIZATION OF A WIND TURBINE BLADE by …

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