Design and Optimization of a Small Wind Turbine

by

John McCosker

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

December 2012

ii

© Copyright 2012

by

John J. McCosker

All Rights Reserved

iii

TABLE OF CONTENTS

LIST OF FIGURES ........................................................................................................... v

LIST OF SYMBOLS ........................................................................................................ vi

LIST OF ACRONYMS ................................................................................................... vii

ACKNOWLEDGMENT ................................................................................................ viii

ABSTRACT ..................................................................................................................... ix

1. Introduction .................................................................................................................. 1

1.1 Wind as a Resource ............................................................................................ 1

1.2 Overview of Aerodynamic Principals ................................................................ 2

2. Design Methodology and Theory ................................................................................ 6

2.1 Efficiency of Wind Turbine ............................................................................... 6

2.2 Turbine Style ...................................................................................................... 8

2.3 Blade Design .................................................................................................... 10

2.3.1 Defining the Chord Length and Blade Twist ....................................... 11

2.3.2 Airfoil Selection ................................................................................... 17

2.4 Blade Element Momentum (BEM) Theory...................................................... 20

2.5 Design Constraints ........................................................................................... 25

3. Results and Discussion .............................................................................................. 28

3.1 Blade Element Momentum (BEM) Theory Results ......................................... 28

3.2 Parametric Variable Sensitivity Study ............................................................. 31

3.2.1 Varying the Tip Speed Ratio ................................................................ 31

3.2.2 Varying the Airfoil ............................................................................... 33

4. Conclusions................................................................................................................ 36

5. References .................................................................................................................. 38

Appendix A- Airfoil Lift and Drag Data Extrapolation .................................................. 39

Appendix B- Blade Element Momentum (BEM) Spreadsheet........................................ 43

iv

LIST OF TABLES

Table 1: Optimized Dimensionless Wind Turbine Blade Geometry ........................ Page 29

v

LIST OF FIGURES

Figure 1: Chart of Global Installed Wind Power Capacity......................................... Page 1

Figure 2: The Angle of Attack and Chord Line of an Airfoil .................................... Page 3

Figure 3: Transformation of Lift and Drag into Torque and Thrust .......................... Page 4

Figure 4: Diagram of Wind Speed and Pressure Before, During, and After Crossing a

Wind Turbine .............................................................................................................. Page 6

Figure 5: Coefficient of Power for Lift- and Drag-Type Devices ............................. Page 9

Figure 6: Vertical-Axis Darrieus Wind Turbine (center) and Horizontal-Axis Wind

Turbine (right) .......................................................................................................... Page 10

Figure 7: Visual Representation of the Wind Velocity, Tangential Velocity, and Relative

Velocity ..................................................................................................................... Page 13

Figure 8: Comparison of Pitch Angles Calculated with Betz and Schmitz

Methods .................................................................................................................... Page 16

Figure 9: Comparison of Chord Length Distribution Calculated with Betz and Schmitz

Methods .................................................................................................................... Page 17

Figure 10: Performance Comparison between Wind Turbines with NACA and NREL

Airfoils ...................................................................................................................... Page 19

Figure 11: Tip Loss Flow Diagram ......................................................................... Page 22

Figure 12: Windmill Brake State Performance ........................................................ Page 23

Figure 13: Flow Diagram of the Iteration Process Used to Solve for Axial Induction

Factor a and the Tangential Induction Factor a’ ...................................................... Page 24

Figure 14: Wind Speed in Connecticut at a Height of 80 Meters ............................. Page 26

Figure 15: The Coefficient of Power for a Turbine Blade Optimized for X=7

.................................................................................................................................. Page 30

Figure 16: Comparison of Performance of Wind Turbine Blades Optimized for a Range

of Tip Speed Ratios .................................................................................................. Page 32

Figure 17: Comparison between Performance of NACA 23012 and NACA 4412

Airfoils ...................................................................................................................... Page 34

vi

LIST OF SYMBOLS

a - Axial Interference Factor

a’ - Tangential Interference Factor

A m^2 Area, area swept by turbine blades

B - Number of blades

CD - Coefficient of Drag

CL - Coefficient of Lift

CP - Coefficient of Power

Cy - Coefficient of axial forces

Cx - Coefficient of tangential forces

c m Chord length

P W Power

r m Radius to annular blade section

Th N Axial Force on Rotor, Thrust

T N*m Torque

U N Tangential Force on Rotor

u m/s Tangential Wind Speed in Rotor Plane

v m/s Axial Wind Speed in Rotor Plane

v1 m/s Wind Speed Upstream of Rotor

v3 m/s Wind Speed Downstream of Rotor

vtip m/s Speed of Blade Tip

w m/s Relative Wind Speed

X - Tip Speed Ratio

α deg Angle of Attack

β deg Pitch Angle of Blade to Rotor Plane

φ deg Angle of Relative Wind to Rotor Plane

ω s-1 Angular Velocity of Rotor

ρ kg/m^3 Density of Air

vii

LIST OF ACRONYMS

BEM Blade Element Momentum

HAWT Horizontal Axis Wind Turbine

NACA National Advisory Committee for Aeronautics

NREL National Renewable Energy Laboratory

VAWT Vertical Axis Wind Turbine

viii

ACKNOWLEDGMENT

To my father John McCosker Sr.: Thank you for your support not only in my pursuit of

higher academic achievements, but also in life. Laura, thank you for the sacrifices you

made in postponing your career while I finished my degree. Thank you to all my

professors at Rensselaer and Lehigh University for their time and effort spent to help me

grow academically. A special thanks to Professor Ernesto Gutierrez-Miravete for his

patience, understanding, and guidance throughout the masters project.

ix

ABSTRACT

The objective of this project is to design a small wind turbine that is optimized for the

constraints that come with residential use. The design process includes the selection of

the wind turbine type and the determination of the blade airfoil, pitch angle distribution

along the radius, and chord length distribution along the radius. The pitch angle and

chord length distributions are optimized based on conservation of angular momentum

and theory of aerodynamic forces on an airfoil. Blade Element Momentum (BEM)

theory is first derived then used to conduct a parametric study that will determine if the

optimized values of blade pitch and chord length create the most efficient blade

geometry. Finally, two different airfoils are analyzed to determine which one creates the

most efficient wind turbine blade. The project includes a discussion of the most

important parameters in wind turbine blade design to maximize efficiency.

1

1. Introduction

1.1 Wind as a Resource

By the end of 2011, it was reported by the World Wind Energy Association, that

there are over 238,351 MW of wind power capacity in the world, as illustrated in Figure

1. The same wind power advocacy group stated that wind power now has the capacity to

generate 500 TWh annually, which equates to about 3% of worldwide electricity usage

[1]. According to BTM Consult, a company that specializes in independent wind-

industry research, the level of annual installed capacity has grown at an average rate of

27.8% per year for the past five years [2]. These statistics demonstrate that wind energy

is already a vital source of energy production around the globe and that the demand for

wind energy solutions is increasing.

Figure 1: Chart of Global Installed Wind Power Capacity [1]

With such increasing demand, it is evident that the benefits of wind energy are

real. While wind turbine power capacity is increasing, not many are found in backyards

and on top of houses. However, depending on exactly where you live, there is usually an

appreciable amount of wind above the tree and houseline. The majority of power

2

generation from wind turbines is currently produced in wind farms, or large fields that

have several large commercial wind turbines. From an environmental standpoint, a wind

farm is much preferred to a coal burning plant because of carbon emissions and other

factors, but both methods of power generation require the consumer buy this power from

a utility company. What is stopping the average land owner from erecting his own wind

turbine? This project is aimed at determining how efficient the small wind turbine can

be given the space constraints of a residential area.

1.2 Overview of Aerodynamic Principals

Wind turbines are machines that remove energy from the wind by leveraging the

aerodynamic principals of lift and drag. Lift and drag forces move the turbine blades

which convert kinetic wind energy to rotational energy. The rotational energy can then

be transformed into electrical energy. The rate of energy extracted from the wind is

governed by Equation (1), where P is the power, T is the torque, and ω is the angular

velocity of the turbine blades.

(1)

Lift and drag forces are measured experimentally in a wind tunnel for airfoils as a

function of the angle of attack, α. The angle of attack is defined as the angle between the

chord line c of the airfoil and the direction of the wind, as shown in Figure 2. For

aircraft wing design, it is generally ideal to choose the airfoil that has the greatest lift-to-

drag ratio, since there will be the least amount of thrust required to maintain altitude.

The objective of turbine blade design is also to maximize the lift force on the blade and

reduce drag so that the force on the blade that acts in the tangential direction is

maximized. Lift acts in the direction normal to the fluid flow, which is not necessarily

acting in the tangential direction once the turbine blades begin to spin. In most wind

turbine designs, only the lift force on a blade creates a tangential force in the correct

direction, while the drag force creates a small tangential force in the opposite direction.

Other than the tangential force, another force, called thrust, is also comprised of lift and

drag and acts normal to the plane of rotation. In air turbine design, it is crucial to reduce

the thrust on the turbine blades because it wastes energy and it requires a stronger blade

to withstand its loading.

3

Figure 2: The Angle of Attack and Chord Line of an Airfoil [3]

The lift and drag forces on an airfoil are equal to the following functions,

respectively, where CL is the coefficient of lift, CD is the coefficient of drag, ρ is the

density of air, w is the relative wind speed, b is the length of the blade, c is the chord

length, and B is the number of blades.

(2)

(3)

Figure 3 shows how the lift and drag forces are transformed into torque T and thrust Th

forces, which are required to determine the power created by the turbine.

4

Figure 3: Transformation of Lift and Drag into Torque and Thrust [3]

The following equations define the torque T and thrust Th for a section of a turbine blade

with a width of dr, where φ is the angle between the relative wind speed and the plane of

rotation.

(4)

where (5)

(6)

where (7)

An additional difference between aircraft wing airfoils and those used in wind

turbines are the distributions of velocity from the base of the foil to the end. The wind

velocity relative to the wind turbine blade is comprised of two velocity components: the

wind velocity in the direction normal to the plane of rotation v and the tangential

velocity u of the blade due to its rotation about the hub. The tangential velocity is a

function of the distance r from the hub of the wind turbine and the angular velocity of

the turbine ω as shown in Equation (8).

(8)

Most aircraft wing designs assume the span of the wing has a uniform velocity

distribution, so as long as the angle of attack is correctly set, the performance of the

blade is optimized. In order to have the desired performance from the turbine blade, it

must be angled as a function of the blade radius so that the front of the blade is properly

5

angled into the wind. The farther from the hub the blade extends, the greater the

component of velocity becomes that is parallel to the plane of rotation. The efficiency of

most wind turbines can be defined as a function of the tip speed ratio X, which is the

speed of the tip of the blade divided by the wind speed.

(9)

6

2. Design Methodology and Theory

2.1 Efficiency of Wind Turbine

Wind turbine efficiency is quantified by a non-dimensional value called the

coefficient of power CP, which is the ratio of power extracted from the wind, P, to the

total power in wind crossing the turbine area. Equation (10) [3] shows that the

coefficient of power is a function of the air density ρ, the area inscribed by the turbine

blade A, and the wind speed v1.

(10)

The power extracted from the wind is derived using the Bernoulli equation on both sides

of a wind turbine as depicted in Figure 4.

Figure 4: Diagram of Wind Speed and Pressure Before, During, and After Crossing a

Wind Turbine [3]

By applying the Bernoulli equation to the flow upstream and downstream of the turbine

results in Equation (11) and (12), respectively.

(11)

7

(12)

By subtracting Equations (11) and (12), one arrives at the following expression:

(13)

Based on the change in linear momentum from v1 to v3, the change in pressure ∆p can

also be expressed as:

(14)

By solving equations (13) and (14) for v,

(15)

The power produced by the wind turbine is equal to the kinetic energy in the air.

(16)

The axial interference factor is a factor that represents the loss in wind speed as it

approaches the turbine blade. The axial interference factor is defined as:

(17)

or (18)

In terms of the axial interference factor, the power equation from Equation (16) can be

re-written as:

(19)

Using Equation (19) to further define the power extracted by the wind turbine in

Equation (10), the coefficient of power can be defined in terms of the axial interference

factor only.

(20)

The maximum theoretical value of the coefficient of performance is determined by

setting the derivative of Equation (20) equal to zero and solving for . Doing so results

in a root at =1/3, which corresponds to a maximum coefficient of power of 16/27.

This number, referred to as the Betz limit, represents the maximum theoretical

coefficient of power. Due to losses throughout the system in bearing friction, wing tip

vortices, hub losses, etc., the actual coefficient of power is expected to be less.

8

2.2 Turbine Style

The two dominant types of wind turbines are drag and lift devices. Power from a

drag device is calculated directly from the force of the wind on the device and the speed

of the device. As shown in Equation (21), the force on a drag device is a function of the

drag coefficient CD. For a drag device, the wind speed of the turbine is bound by the

speed of the wind. The following equations calculate the upper bound of the coefficient

of power for a drag-type wind turbine.

(21)

where w is the relative wind speed of the drag-type wind turbine, as governed by

Equation (22).

(22)

If,

, (23)

then, by combining Equations (21) and (23)

(24)

Using λ to represent the ratio of wind speed v1 to drag device speed u and by substituting

Equation (24) into Equation (10), one can arrive at the following definition of coefficient

or power for a drag device.

(25)

To find the optimal coefficient of power, the derivative of Equation (25) with respect to

λ is set to zero and solved for λ. The maximum value of the coefficient of power for a

drag device is 4/27*CD at a relative wind speed ratio λ of 1/3. Assuming that the

coefficient of drag is 1.0, the resulting maximum coefficient of power is 4/27.

Compared to the maximum coefficient of power derived for a lift type wind turbine, the

lift-type wind turbine is able to extract 4-times more power out of the air than a drag-

type turbine. Figure 5 shows the equations for the coefficient of power of lift and drag

devices plotted with respect to the corresponding non-dimensional wind velocity

coefficient.

9

Figure 5: Coefficient of Power for Lift- and Drag-Type Devices (assuming CD=1)

Another wind turbine characteristic that will affect the design of the turbine is the

orientation of the axis about which the blades rotate. Vertical Axis Wind Turbines

(VAWTs) such as the Darrieus wind turbine, shown in Figure 6, can operate in wind of

any direction, without having to adjust its own directionality. The downside to a VAWT

is that they do not reliably start without an additional motor. While powering an

additional motor may be cost effective for a large-scale wind turbine, it is certainly not

cost effective for a small wind turbine. Horizontal Axis Wind Turbines (HAWTs) are

the most popular lift-type devices. While HAWT’s do not require a starter motor to get

up to operating speed, they do require that the area projected by the blades is facing

perpendicular to the direction of the wind. This is accomplished most frequently on

small-scale wind turbines by including a tail that catches the wind.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Pow

er C

oef

fici

ent

Non-Dimensional Velocity Coefficient

Coefficient of Power- Drag vs. Lift Device

Lift-Type Device

Drag-Type Device

10

Figure 6: Vertical-Axis Darrieus Wind Turbine (center) and Horizontal-Axis Wind

Turbine (right), Gaspé peninsula, Quebec, Canada

The wind turbine chosen for this study is a lift-type HAWT because lift-type wind

turbines have the potential to produce more power than drag-type devices. The wind

turbine analyzed for this study will also have a horizontal axis so that a starting motor is

not required. Another benefit of choosing a horizontal axis, lift-type wind turbine is that

they are the most popular type of wind turbine which results in the most data supporting

its design.

2.3 Blade Design

In order to successfully design an efficient wind turbine, the blade contour must take

advantage of aerodynamic considerations while the materials it is made from provides

the necessary strength and stiffness. By investigating the aerodynamic characteristics of

a wind turbine blade, the parameters that make up the blade contour are optimized, and

11

the loads that test its structural adequacy are calculated. Only aerodynamic principles

are being analyzed in this study.

In order to define the power extracted from the wind by the wind turbine in Section

2.1, conservation of linear momentum and Bernoulli’s principle were used to arrive at

the Betz limit. Schmitz developed a more comprehensive model of the flow in the rotor

plane based on conservation of angular momentum [3]. This method of calculating

power will be reviewed in the following section and it will be utilized to determine the

most efficient chord length and pitch angle distribution along the radius of the blade.

Once the chord length and the pitch angle distributions are both defined, Blade Element

Momentum (BEM) theory can be utilized to determine the performance of the wind

turbine under a range of conditions.

2.3.1 Defining the Chord Length and Blade Twist

As shown in Equation (1), the power extracted from the air is the result of a

torque and angular velocity in the wind turbine. According to the conservation of

angular momentum, the torque in the wind turbine shaft can only be created if there is a

rotation in the downstream wake opposite the direction of the rotor’s rotation. By taking

account of the torque producing the wake in the opposing direction, the following

equation expresses the relative tangential speed of the rotating blade. The

term

accounts for the additional tangential wind speed that the blade experiences due to the

average counter-rotating wake velocity.

(26)

The additional tangential speed that the blade experiences due to the wake is defined as a

function of the tangential interference factor .

(27)

Figure 7 illustrates the components of the relative wind velocity w upstream of the wind

turbine plane, in the plane of the wind turbine, and downstream of the wind turbine

plane. Upstream of the rotor plane, the rotational velocity of the wake is zero. Down

stream of the wind turbine plane, the wake has a rotational velocity of Δw acting in the

12

opposite direction of the turbine motion. The average rotational velocity over the blade

due to wake rotation is therefore Δw/2.

Diagram b1) in Figure 7 shows the effect that the tangential and axial

interference factors have on the angle between relative wind velocity and the rotor plane.

The variables with a subscript of 1 denote the values before the plane of rotation, where

the variables without a subscript denote the values in the plane of rotation. By

comparing diagrams a) and b1), it is evident that the increase in tangential velocity

caused by the

term and the decrease in axial velocity caused by the axial

interference factor, cause the angle of relative wind to decrease. Using the geometric

relationship shown in diagram b4), the following equation defines the change in relative

wind speed Δw in terms of initial relative wind speed and the change in relative wind

speed angle φ.

(28)

13

Figure 7: Visual Representation of the Wind Velocity, Tangential Velocity, and Relative

Velocity a) Upstream, b) In the Plane of the Rotor, and c) Down Stream [3]

Using conservation of momentum, the following equation relates the lift force for a

section of the blade to the change in relative wind velocity Δw and mass flow rate dq of

air through a ring element of width dr at radius r from the hub.

(29)

In order to calculate the power created from the lift force for a segment of the foil, the

torque is first calculated by taking the tangential component of the lift force and

14

multiplying it by the differential blade segment’s radius. The assumption is made that

the drag of the airfoil is negligible which, if included, would create a torque in the

opposite direction and reduce the power generated.

(30)

By substituting the following expression for mass flow rate of air through the ring

element dq

(31)

and by substituting the expression for change in rotational velocity Δw from Equation

(28), the following expression for the power of a blade segment is produced.

(32)

In order to determine what the angle of relative wind to the rotor plane φ is that creates

the maximum power, the derivative of Equations (32) is taken with respect to φ and

solved equal to zero. When d(dP)/dφ=0,

(33)

Using the geometric relations found in Figure (2), and substituting for the tip speed ratio

X, the most power can be produced at the following angle between the relative wind and

the plane of rotation.

(34)

To transform the most efficient relative wind angle to the pitch angle of the blade β, φ

must be subtracted from the angle of attack α. The resulting equation for the optimal

pitch angle according to Schmitz theory is as follows.

(35)

Next, the optimal distribution of chord length as a function of radius from the hub can be

determined by substituting Equations (28) (30) and (31) into Equation (29).

(36)

Using an expression derived from diagram b4) in Figure 7 for axial velocity in the rotor

plane v and equating Equation (36) to the differential form of Equation (2) from

aerodynamic foil theory, Gundtoft [3] arrives at the following expression for optimal

chord length c as a function of blade radius r.

15

(37)

Based on the expressions derived in Equations (35) and (37) the blade is shaped

to provide maximum output. The pitch of the blade is distributed along its radius to

ensure the relative wind direction is intercepting the blade at the desired angle of attack.

And the chord length is optimized to provide maximum lift along the blade’s radius.

However, the output is only as good as the assumed values used in this equations. For

the value of parameters such as the tip speed ratio, where there is not one clear optimal

value, several values can be tested to determine a trend. Such a test is completed in

Section 3.2.1 to determine trends in maximizing turbine efficiency.

Figures 8 and 9 below compare the optimized pitch angle and chord length

distributions calculated by both Betz and Schmitz, respectfully. The difference in pitch

angle is greatest at the hub of the turbine blade, with a difference of about 20 degrees at

5% of the blade length. The difference decreases until after about 50% of the blade

length when the two lines are within a degree of one another. Since the hub of the

turbine will likely consume the first 10% of the blade, it appears that there is a small

variation in the results, regardless of the method.

16

Figure 8: Comparison of Pitch Angles Calculated with Betz and Schmitz Methods

Figure 9 below shows that the variation in chord length between the Betz and

Schmitz methods are greater than they were for the pitch angle distributions. Similar to

the pitch angle distributions, the difference is greatest near the base of the blade and

decreases going outward. According to the Betz method, the blade should become

increasingly thick as it approaches the hub, where the Schmitz method starts thin closest

to the hub, reaches a maximum at about 15% of the blade length and begins to decrease

again. Unlike the difference in pitch angles, the difference between chord length

distributions seem great enough outside of the hub area (<10% of the blade length) to

have an appreciable effect on turbine efficiency.

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Opti

miz

ed P

itch

Angle

,β (

deg

)

r/R

Comparison of Optimized Pitch Angle

Betz

Schmitz

17

Figure 9: Comparison of Chord Length Distribution Calculated with Betz and Schmitz

Methods

2.3.2 Airfoil Selection

In order to use the relationships derived in the previous section to arrive at the

most efficient blade design, the cross sectional properties of the wing must also be

defined. The decision of which airfoil to use over the turbine blade defines the

coefficients of lift and drag, which directly affect the forces produced on the blade.

Most airfoils used in airplane wing design have documented data from a wind tunnel of

the coefficients of lift and drag for a range of angles of attack. For aircraft wing design,

data is only required for angles of attack up to the first occurrence of a phenomena

known as stall, or the angle of attack where the lift coefficient is drastically reduced due

to flow separation. Generally, stall occurs in most airfoils between 15 and 20 degrees,

depending on the Reynolds number of the fluid. This data is easily found in many

handbooks, but since wind turbine blades operate at angles of attack up to 90 degrees,

lift and drag coefficient data is required for the angles of attack past 20 degrees.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Op

tim

ized

Ch

ord

Len

gth

, c/R

r/R

Comparison of Optimized Chord

Length

Betz

Schmitz

18

The National Renewable Energy Laboratory (NREL) has developed several

families of special-purpose airfoils for HAWTs. The NREL S-Series airfoils come in

both thin and thick families and within each family is a set of two of three different

airfoils that are designated “root”, “primary”, and “tip.” Each set of three airfoils is

defining a single blade with a variable cross section, such that the “root” airfoil is the

cross section shape at the location of largest chord length, the “primary” airfoil is the

shape at 75% of the radius, and the “tip” airfoil which occurs at 95% of the radius. The

cross section of the blade is interpolated between the three main airfoils.

The S-Series airfoils are classified according to their blade length. One family of

airfoils is made specifically for wind turbine blades ranging from 1 to 3 meters long.

This airfoil family, from root to tip, includes S835, S833, and S834. While this airfoil

family fits the intent of the small wind turbine design, sufficient experimental lift and

drag data does not yet exist, so it will not be used in this blade design study. The data

shown in Figure 10 demonstrates how wind turbine performance is drastically improved

by using an airfoil that is specifically tailored for use in a HAWT. Even though the

NACA airfoil has a greater maximum coefficient of power, the NREL airfoil in Figure

10 is designed to operate at a higher coefficient of power over a larger range of tip speed

ratios. While the NREL airfoils are superior to NACA airfoils for use in wind turbines,

wind tunnel lift and drag data is very scarce for NREL airfoils, especially those used in

small wind turbines. Since there is sufficient wind tunnel data for NACA airfoils, only

these will be considered in this analysis.

19

Figure 10: Performance Comparison between Wind Turbines with NACA and NREL

Airfoils [4]

In order to extend the given data for angles of attack well beyond the first

occurrence of stall, Viterna [5] provides a convenient approach to relating the post-stall

coefficient of lift and drag to overall blade geometry. Viterna’s equations for the

coefficient of lift and drag are as follows:

(38)

(39)

where

(40)

(41)

(42)

(43)

(44)

These equations will be used to calculate the coefficient of lift and drag between angles

of attack of 20 and 90 degrees. For angles less than 20 degrees, a polynomial will be fit

20

to the experimental data curves [6] so that the iterative solver in the BEM calculation

discussed below can continuously determine values without interpolating.

The airfoils chosen for use in this turbine blade are NACA 23012 and NACA

4412. The NACA 23012 is a 5-digit series NACA cambered airfoil which is known for

having a relatively high maximum coefficient of lift. The NACA 4412 is an airfoil used

in older wind turbines such as the Windcruiser turbine made by Craftskills Enterprises.

The lift and drag curves for these wind turbines are included in Appendix A.

2.4 Blade Element Momentum (BEM) Theory

BEM theory is a compilation of both momentum theory and blade element theory.

Momentum theory, which is useful in predicted ideal efficiency and flow velocity, is the

determination of forces acting on the rotor to produce the motion of the fluid. This

theory has no connection to the geometry of the blade, thus is not able to provide optimal

blade parameters. Blade element theory determines the forces on the blade as a result of

the motion of the fluid in terms of the blade geometry. By combining the two theories,

BEM theory, also known as strip theory, relates rotor performance to rotor geometry.

The assumptions made in BEM theory is the aggregate of the assumptions made for

momentum theory and blade element theory. The following assumptions are made for

momentum theory:

1. Blades operate without frictional drag.

2. A slipstream that is well defined separates the flow passing through the rotor

disc from that outside disc.

3. The static pressure in and out of the slipstream far ahead of and behind the

rotor are equal to the undisturbed free-stream static pressure (p1=p3).

4. Thrust loading is uniform over the rotor disc.

5. No rotation is imparted to the flow by the disc.

The following assumptions are made in the blade element theory:

1. There is no interference between successive blade elements along the blade.

21

2. Forces acting on the blade element are solely due to the lift and drag

characteristics of the sectional profile of a blade element.

By setting the expression for the differential thrust from blade element theory (Equation

(6)) equal to the following equation for differential thrust using momentum theory,

(45)

one is able to obtain the first of two relationships required for BEM theory.

(46)

Equating the expression for the differential torque from blade element theory (Equation

(4)) to the following equation for differential torque using angular momentum theory,

(47)

yields the second relation for BEM theory.

(48)

The solidarity ratio σ is defined as the following expression.

(49)

The axial and tangential interference factors are terms that are not known at the

beginning of the BEM calculation because they are both functions of the angle of

relative wind to the plane of rotation, which is also a function of the interference factors.

Physically, the axial interference factor is the fractional decrease in axial wind velocity

between the free stream and the rotor plane. The tangential interference factor is the

fractional increase in tangential wind speed due to the counter rotating wake experienced

by the blade. Guessing values for both interference factors is required to begin the BEM

calculation process, but with each iteration the interference factors converge onto certain

values.

Up to this point, BEM theory does not account for the interaction of shed vortices

with the blade flow near the blade tip. While air is flowing over the blade, the pressure

under the blade decreases relative to the pressure on the top of the blade. At the tip of

22

the blade, the air will flow radially inward over the tip, reducing the circulation of the

air, which reduces the torque and turbine efficiency, as shown in Figure 11.

Figure 11: Tip Loss Flow Diagram [7]

Even though the blade chord length is the least at the tip, because of its distance from the

hub, the tip loss contributes greatly to the overall performance of the wind turbine. In

order to account for the loss of torque at the tip, Prandtl developed a method to

approximate the radial flow effect near the blade tip which is sufficiently accurate for

high tip speed ratios for turbines with two or more blades. The factor Prandtl derived is

defined by

(50)

where

(51)

Solving Equations (46) and (48) for and , respectively, and including the Prandtl tip

loss correction factor, yields the final two equations for and used in the BEM

procedure [7].

(52)

(53)

23

Equation (52) is only accurate in computing axial interference factors for values less

than 0.2, above which simple momentum theory starts to break down. Figure 12

illustrates which theories are valid for a range of axial interference factors.

Figure 12: Windmill Brake State Performance [7]

Once a is calculated to be greater than 0.2, the following correction factor will be used

that was formulated by Glauert [8] and redefined in terms of the average axial

interference factor [9].

(54)

where

(55)

The following procedure takes the theory discussed thus far and uses it to

calculate the axial force and power of one ring element in the rotor. Figure 13 shows a

24

flow diagram outlining the process of calculating the axial induction factor and the

tangential induction factor for a single ring element. In order for the relative wind

speed angle φ to be calculated in the second step the following equation must be used,

which is derived from Figure 2.

(56)

Figure 13: Flow Diagram of the Iteration Process Used to Solve for Axial Induction

Factor and the Tangential Induction Factor

Once the values for and converge, the torque T and thrust Th for each blade segment

is calculated by using the following equations:

(57)

(58)

Choose guess

values for and (guess = 0)

Calculate φ from

Eqn. (56)

Calculate α (α = φ - β)

and find CL and CD from

the airfoil data

corresponding to α

Calculate

Cx and Cy from

Eqns. (5) and (7)

Calculate and

from Eqns (56),

(57), and (58)

Does new and differ by more than

the target % from

previous and ?

Yes

No Substitute

previous

and for

new values

Finished

Inputs: β, v1, ω, c,

ρair, B, R, dr

25

The total axial force and power are then calculated using the following summations:

(59)

(60)

2.5 Design Constraints

The size of the wind turbine is the first constraint in designing a residential-sized

wind turbine. Many towns have different zoning requirements for the maximum

allowable height of an erected structure and the minimum required lot size that contains

a wind turbine. Data shows that the higher a wind turbine sits off the ground, the greater

the wind speeds are, and the available power for a turbine increases with the cube of the

wind velocity (Equation (10)). The data in Figure 14 gives the annual average wind

speed at a height of 80 meters. According to the local municipal laws, structures in the

residential zone cannot be more than 40 feet (14.19m) from the ground [10]. Given this

residential zoning constraint, the wind turbine would not be able to operate at wind

speeds of 7 m/s as shown for New London in Figure 14.

26

Figure 14: Wind Speed in Connecticut at a Height of 80 meters

[http://www.windpoweringamerica.gov/wind_maps.asp]

Another parameter of the wind turbine design that is constrained by the allowable

height of the structure is the size of the blades. Since the maximum theoretical power

output of a wind turbine is proportional to the square of the blade length (Equation (19)),

it is also important to maximize the blade length as much as the zoning regulations

allow. There is a slight trade-off between the height of the turbine and the blade length

since the higher the blades are from the ground, the higher the wind speed is that they

will encounter. Assuming the zoning requirements from Waterford, CT, and allowing

sufficient space between the bottom of the inscribed area and the ground for safety, the

optimized turbine will have a 2.5 meter radius, allowing the center of the hub to be about

11.5 meters from the ground.

By using the power law equation for the vertical wind profile, the average wind

speed at the height of interest can be calculated. The following equation is the general

form of the power law which is a function of the wind speed at the known height v1*, the

27

corresponding height x1, the wind speed of interest v2*, the height of the wind speed of

interest, x2*, and an exponent α, which is determined experimentally [11].

(61)

Given an α value of 1/7, which is valid for general conditions [11], the average wind

speed at a height of 11 meters is about 5.0 m/s.

The final constraint regarding residential wind turbine use is the requirement that

it cannot be overly loud when operating. According to Tangler [12], airfoil shape pure-

tone noise can result from the presence of significant laminar separation bubbles

interacting with the trailing edges, which is more prevalent in small turbines because of

the lower Reynolds number. While the maximum sound level allowed for a wind

turbine is defined to be 60 dB for most turbines, the maximum tip speed that will create

that noise level is highly dependent on the blade. Based on the paper by Vick [13], three

different small turbine blades were tested and produced a noise level of 60dB at tip

speeds of 85, 95, and 100 m/s. Assuming the wind turbine being designed here operates

at the fastest of those speeds, the constraint on this turbine will be tip speeds of less than

100 m/s.

28

3. Results and Discussion

3.1 Blade Element Momentum (BEM) Theory Results

In order to start reducing the number of blade design variables, the constraints of a

small wind turbine must first be translated to input values of the BEM analysis. The

main constraint of a small wind turbine is the allowable height of the wind turbine which

constrains both the wind speed and the blade length. Based on the assumptions made in

the previous section, the average wind speed at the maximum allowable height of 11.5

meters is about 5 m/s with a corresponding blade radius of 2.5 meters.

The blade’s pitch, angle of attack, and chord length must be defined before

proceeding with BEM theory. The blade pitch and chord length for each segment of

blade are defined from Equations (35) and (37), respectively. The tip speed ratio X must

be chosen to calculate the pitch values and the coefficient of lift must be defined to set

the chord length. The tip speed ratio is initially defined as 7 to get a baseline value of

performance and will be varied in the parametric study to determine the ideal ratio. The

coefficient of lift CL is initially defined as 0.88 based on the value of the coefficient of

lift at the maximum glide ratio (CL/CD). The aerodynamic properties are based on the

lift and drag plots for NACA airfoil 23012. The angle of attack, 7 degrees, is chosen as

the angle of attack corresponding to the maximum glide ratio. The tip speed ratio and

the type of airfoil will both be revisited in the following section to determine how

sensitive the efficiency is based on these assumptions.

Table 1 below contains the pitch angle and relative chord length for each of the 9

blade segments (10 segments minus the inner-most segment for the hub). The values in

the table are dimensionless so that the distributions of pitch and chord length can be

applied to a blade of any size. Each segment is assumed to have constant aerodynamic

properties, pitch, and chord length, so having more blade segments creates a more

accurate analysis.

29

Blade Segment - 1 2 3 4 5 6 7 8 9

Relative radius r/R 0.150 0.250 0.350 0.450 0.550 0.650 0.750 0.850 0.950

Speed ratio X 1.050 1.750 2.450 3.150 3.850 4.550 5.250 5.950 6.650

Angle, optimal phi 29.069 19.830 14.802 11.742 9.707 8.264 7.190 6.360 5.701

Pitch beta 22.069 12.830 7.802 4.742 2.707 1.264 0.190 -0.640 -1.299

Rel. chord length c/R 0.180 0.141 0.111 0.090 0.075 0.064 0.056 0.050 0.045

Table 1: Optimized Dimensionless Wind Turbine Blade Geometry

Finally, using the spreadsheet shown in Appendix B, the power generated from the

wind turbine is calculated and the coefficient of power is then determined by comparing

the calculated power extracted by the wind turbine with the total power contained in the

wind. Using a constant wind velocity of 5 m/s, which was determined to be the average

wind speed for the southeast Connecticut shoreline at a height of 11.5 meters, the

rotational velocity of the turbine was changed until it created a tip speed ratio of about 7.

Since the blade was optimized for a tip speed ratio of 7, it should be the ratio that most

efficiently extracts power from the wind. Also, the tip speed of the turbine blade is

about 35 m/s which is sufficiently under 100 m/s to satisfy the allowable noise

requirement.

After the equations converged, the power extracted from the wind was computed to

be about 0.81 kW. Compared to the Betz limit of extractable power in the air of 0.89

kW, this turbine is calculated to have an efficiency of about 92%. Using Equation (10),

the coefficient of power is calculated to be 0.55, which is very close to the maximum

theoretical limit of 0.59. By increasing the wind velocity from 5 to 8 m/s and increasing

the rotational speed to maintain a tip speed ratio of about 7, the power production

roughly quadrupled to 3.2 kW, however, the efficiency and coefficient of power

remained unchanged.

In order to determine the turbine blade’s performance over a number of tip speed

ratios, the wind speed was kept constant and the rotational speed of the turbine was

increased until the desired tip speed ratio was computed. When the combination of

rotational speed and wind speed provided the correct tip speed ratio, the spreadsheet

would be iterated until the power output was computed. Using Equation (10), the

30

coefficient of power was calculated and recorded. This process was repeated for tip

speed ratios of 1 through 12.

As shown in Figure 15, when using a tip speed ratio of 7 or greater, the turbine

operated at a coefficient of power greater than 0.5. It is interesting to note that the

optimal performance of the turbine did not occur at the tip speed ratio of 7, but instead

the efficiency increased up to ratios of 9 and 10 before it began to decrease. From the

data in Figure 15, it is evident that in terms of designing turbine blades, the blades

should be optimized for tip speed ratios slightly less than what is anticipated. In addition

to operating at peak efficiency, if the wind turbine is operating at tip speed ratios greater

than what it was designed for, the decrease in performance for ratios greater than 12 is

much more gradual than the decrease for ratios less than 7.

Figure 15: The Coefficient of Power for a Turbine Blade Optimized for X=7

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11

Coef

fici

ent of

Pow

er, C

p

Tip Speed Ratio, X

Performance of the Initially Optimized

Wind Turbine

R=2.5,X=7

31

3.2 Parametric Variable Sensitivity Study

3.2.1 Varying the Tip Speed Ratio

Several assumptions are made for inputs to the BEM calculation of turbine

efficiency. The assumed values are chosen to maximize the power output of the turbine

or are constraints due to residential use. One of the variables that is assumed to

maximize efficiency is the tip speed ratio. In the previous section, it was demonstrated

that the turbine blade did reasonably well for the ratio it was design for, but performed

its best for a ratio slightly higher than that for which it was designed. During this

parametric study the tip speed ratio that the blade is designed for will be varied between

4 and 8. The performance of each blade will then be computed for a range of tip speed

ratios to determine if the same trend is observed.

Figure 16 contains the results of varying the blade pitch and chord length

distributions based on optimizing for a range of tip speed ratios. The blades that were

created with tip speed ratios less than 4 would not converge using the BEM solver. The

blade design for a tip speed ratio of 4 was only able to converge for two data points,

which did not include the supposed optimal conditions. However, the blades created for

tip speed ratios of 5 through 8 were able to converge for a range of speed ratios,

allowing a maximum coefficient of power for each blade to be calculated.

32

Figure 16: Comparison of Performance of Wind Turbine Blades Optimized for a Range

of Tip Speed Ratios

The trend observed for a blade optimized for a tip speed ratio of 7, where the peak

performance happens at a higher ratio, is common for all of the blades. One additional

pattern that is observed in Figure 16 is that as the blades increase from values of X=4 to

X=8, the peak performance occurs at increasingly greater ratios than the optimized ratio.

For example, the blade made for X=5 has a peak coefficient of power at X=6, where the

blade optimized for X=8 has a peak coefficient of power at X=10.5. The variable

causing the separation between designed-for and actual peak tip speed ratios has a

greater effect as the ratio increases. The increasing difference in theoretical and actual

coefficient of power may be due to the change in definition of the axial interference

factor a. The blade pitch is based on Equation (35) which accounts for the axial

interference factor as demonstrated in Figure 13, but only as a function coupled with the

actual tangential speed u. Since the axial interference factor a is based on the Glauert

equation after a=0.2, then the equation that couples the axial velocity and tangential

velocity may no longer be valid.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10 11 12

Coef

fici

ent of

Pow

er, C

p

Tip Speed Ratio, X

Comparison of Blades Optimized for

Varying Tip Speed Ratios

X=4

X=5

X=6

X=7

X=8

33

Another trend observed from Figure 16 is that the blades with higher tip speed ratios

have a more gradual slope of increasing coefficient of power, compared to the blades

made for lower tip speed ratios. Based on this trend, designing for higher tip speed ratios

is preferred because there is less of a penalty for having the tip speed ratio decrease

below the desired value.

The results of the variable tip speed ratio trade study have led to several conclusions

when considering how exactly to shape the blade for optimal performance. First, the tip

speed ratio of the turbine should be designed for a tip speed ratio less than what it will be

experiencing. The second conclusion is that blades designed for larger tip speed ratios

have a larger range of efficient speed ratios. While the average wind speed is known,

the tip speed ratio that corresponds to this speed cannot be known without further in-

depth analysis or testing. However, in order to obtain an approximate value of the

average tip speed ratio experienced by this turbine, one can be approximated from the

data shown in the paper by Vick [13]. According to Vick’s paper, at 5 m/s, one can

expect a tip speed of about 50 m/s, which means the average tip speed ratio is 10. Based

on a tip speed ratio of 10 and the conclusions mentioned above, designing the blade for a

tip speed ratio of 8 would create the optimal blade.

3.2.2 Varying the Airfoil

The airfoil is another parameter that can be varied to optimize a blade design.

Associated with the variation in airfoil is the change in optimal coefficient of lift and

optimal angle of attack. While the airfoil changes the blade cross section, it also alters

the optimal coefficient of lift and optimal angle of attack, which affects the pitch and

chord length distributions. The original airfoil used was NACA 23012, which is a

standard cambered airfoil. The second airfoil that will be used for comparison is the

NACA 4412.

The NACA 4412 airfoil is different than the NACA 23012 in that the maximum

glide ratio occurs at an angle of attack of 6 degrees, not 7 degrees like the NACA 23012.

Another difference between the two that will reshape the blade is the coefficient of lift at

34

the maximum glide ratio. The corresponding coefficient of lift for the NACA 4412 is

about 1.05 instead of 0.88.

The coefficient of power as output from the BEM spreadsheet will be compared

between the results of the NACA 23012 and NACA 4412 airfoils. Three different

blades were compared for each airfoil: the blades optimized for tip speed ratios of 6, 7,

and 8. Figure 17 shows the results of the comparison. For the range of tip speed ratios

between 1 and 7, the NACA 23012 airfoil is notably more efficient for all three blades.

For the blades optimized at a tip speed ratio of 8, while experiencing a tip speed ratio of

4, the NACA 23012 has a higher coefficient of power by almost 0.1. At a tip speed ratio

of 7, the airfoils have almost identical performance, but for all speed ratios greater than

7, NACA 4412 out-performs the NACA 23012. The improvement in performance for

the NACA 4412 increases with those blades optimized for greater speed ratios.

Figure 17: Comparison between Performance of NACA 23012 and NACA 4412

Airfoils

Both airfoils seem to have beneficial characteristics that are highly dependent on the

tip speed ratio which they encounter. However, since the ratios greater than 7 have the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Coef

fici

ent

of

Pow

er, C

p

Tip Speed Ratio, X

Comparison between NACA 23012 and

NACA 4412

NACA 23012, X=6

NACA 23012, X=7

NACA 20312, X=8

NACA 4412, X=6

NACA 4412, X=7

NACA 4412, X=8

35

highest coefficient of power, it is apparent that the NACA 4412 airfoil is the more

desirable of the two. The only detriment of the NACA 4412 airfoil is that if the ratio

reduces to 6 or less, there is more of an abrupt decrease in efficiency than with the

NACA 23012.

36

4. Conclusions

Optimizing the parameters that define a wind turbine blade is a process that requires

knowledge of both momentum theory and blade section aerodynamic theory. By

equating the thrust force on the rotor with the axial momentum force, one is able to solve

for the axial interference factor . By equating the torque force with the angular

momentum force on the rotor, one is also able to solve for the tangential interference

factor . And finally, one is able to calculate the power produced by the wind turbine,

by using an iterative process to solve for and . Using this process of determining the

efficiency of a wind turbine, one is able to test a range of values for any given parameter

in a design and determine which values optimize the output.

For a small wind turbine, the allowable size of the turbine creates constraints that

reduce the number of parameters required to maximize the efficiency of the turbine. The

main parameters constrained due to the size requirement are the length of the blade and

the height of the center of the hub. While it was shown that the coefficient of power is

not affected by either wind velocity or blade length alone, power output will increase

with an increase in both parameters. For a residential wind turbine, it is imperative to

maximize both blade radius and height, because from a cost efficiency perspective, one

is not optimizing the fixed cost of building a wind turbine if it is not reaching the fastest

winds or inscribing the largest area. Another constraint for a residential wind turbine

that is not necessarily size-driven, is the requirement that it not exceed a noise limit

above which is deemed irritating or harmful.

Wind turbine parameters that were not defined by the size limitations, such as the

pitch angle, chord length, and airfoil required analytical methods to determine optimal

values. Optimizing the pitch angle was accomplished by formulating an equation based

on conservation of angular momentum, which gave the pitch angle as a function of blade

radius. The function also requires assumptions of the tip speed ratio and the most

efficient angle of attack. The most efficient angle of attack was based on the angle of

attack corresponding to the greatest ratio of coefficient of lift to coefficient of drag,

which is a known value for any given airfoil.

37

The assumption which was made without much prior knowledge was the value of

tip speed ratio. Since the effect that the tip speed ratio would have on the turbine

performance was not known, a parametric study was conducted which demonstrated that

based on the methods of defining the pitch angle and chord length, the tip speed ratio

that is chosen to shape the blade should be less than the expected value that the turbine

encounters. Doing so will ensure the turbine operates at peak efficiency. Based on an

approximate value of tip speed that corresponds to the wind speed, the average tip speed

ratio was determined. From the average tip speed ratio calculated of about 10, and

following several of the observations that were concluded from the parametric study, it

was determined that the optimal blade should be designed for a tip speed ratio of 8.

The final parametric study was conducted to determine if the airfoil had an

appreciable effect on the efficiency of the turbine. Based on the data collected using

BEM theory it was confirmed that changing the airfoil could have an appreciable effect

on the turbine efficiency. Of the two airfoils that were analyzed in this study, the NACA

4412 airfoil was shown to have a higher efficiency at tip speed ratios greater than 7. The

NACA 4412 was also shown to have a higher maximum coefficient of power than the

NACA 23012. In choosing between the two airfoils, it is clear that the NACA 4412

creates a more efficient turbine blade than the NACA 23012. If additional work was

performed in continuation of this analysis, the effect of varying the cross section with the

blade radius would be analyzed. Additionally, continuation of this analysis would

include analyzing different airfoils such as the S-Series airfoils created specifically for

wind turbine blades.

38

5. References

[1] Gsanger, S., Pitteloud, J. The World Wind Energy Association. “Report 2011”.

<http://www.wwindea.org/webimages/WorldWindEnergyReport2011.pdf>

[2] MBT Consult. Viewed November 9, 2012. Web.

<http://www.btm.dk/special+issues/others+issues/the+wind+power+sector/?s=42

>

[3] Gundtoft, Soren, University of Aarhus. “Wind Turbines”. Copyright 2009

[4] Padmanabhan, K., Saravanan, R. “Study of the Performance and Robustness of

NREL and NACA Blade for Wind Turbine Applications”. European Journal of

Scientific Research, ISSN 1450-216X Vol.72 No.3 (2012), pp. 440-446.

<www.europeanjournalofscientificresearch.com/ISSUES/EJSR_72_3_11.pdf>

[5] Viterna, L., Janetzke, D. “Theoretial and Experimental Power from Large

Horizontal-Axis Wind Turbines”. U.S. Department of Energy- Wind Energy

Technology Division, 1982.

[6] Abbot, I.,von Doenhoff, A. “Theory of Wing Sections Including a Summary of

Airfoil Data”. Dover Publications, Inc. Copyright 1959.

[7] Wilson, R., Lissaman, P., Walker, S. “Aerodynamic Performance of Wind

Turbines”. 1976.

[8] Glauert, H., “The Analysis of Experimental Results in Windmill Brake and

Vortex Ring States of an Airscrew,” Reports and Memoranda, No. 1026 AE. 222,

1926.

[9] Berges, B. “Development of Small Wind Turbines”. Technical University of

Denmark. Copyright 2007.

[10] “Zoning Regulations” Town of Waterford, Connecticut. Web. December 22,

2011. <http://www.waterfordct.org/depts/pnz/zoning_regs.pdf>

[11] Spera, D., Richards, T. “Modified Power Law Equations for Vertical Wind

Profiles”. U.S. Department of Energy. 1979.

<http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800005367_1980005367.

pdf>

[12] Tangler, J. “The Evolution of Rotor and Blade Design”. National Renewable

Energy Laboratory. 2000. < www.nrel.gov/docs/fy00osti/28410.pdf>

[13] Vick. B. “Using Rotor and Tip Speed in the Acoustical Analysis of Small Wind

Turbines”. USDA- Agricultural Research Service. 2000.

39

Appendix A- Airfoil Lift and Drag Data Extrapolation

NACA 23012

All airfoil data for NACA 23012 is taken from “Theory of Wing Sections, Including a

Summary of Airfoil Data” by Ira H Abbott and Albert E. Von Doenhoff, 1959.

Alpha CL CD GR

0 0.12 0.006 20

2 0.3 0.0062 48.3871

4 0.55 0.0065 84.61538

6 0.77 0.0067 114.9254

8 1 0.0077 129.8701

10 1.15 0.0095 121.0526

12 1.38 0.0142 97.1831

14 1.56 0.0185 84.32432

16 1.7 0.02 85

18 1.75 0.021 83.33333

20 1.38 0.022 62.72727

Table A-1: Lift and Drag Data for NACA 23012

Coefficient CL CD

k0 0.1031 0.0061

k1 0.1051 -0.0004

k2 0.0011 5.43E-05

k3 7.35E-06 6.53E-06

k4 -6.58E-06 -2.80E-07

Table A-2: Polynomial Coefficients for Lift and Drag Properties

40

Figure A-1: Graph of Lift and Drag Coefficients and Glide Ratio for the NACA 23012

Airfoil

y = -5E-05x4 + 0.0015x3 - 0.0145x2 + 0.1525x + 0.0941

y = 9E-07x4 + 1E-06x3 - 0.0002x2 + 0.0012x + 0.0053

0

20

40

60

80

100

120

140

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12 14 16 18 20

Gli

de

Rat

io

Co

effi

cien

t o

f L

ift an

d D

rag

Angle of Attack (deg)

Airfoil Properties for NACA 23012

Coefficient of Lift

Coeffiient of Drag

Glide Ratio

41

NACA 4412

All airfoil data for NACA 4412 is taken from “Theory of Wing Sections, Including a

Summary of Airfoil Data” by Ira H Abbott and Albert E. Von Doenhoff, 1959.

Alpha CL CD GR

0 0.4 0.0061 65.57377

2 0.6 0.0061 98.36066

4 0.85 0.0065 130.7692

6 1.05 0.0075 140

8 1.25 0.011 113.6364

10 1.43 0.0135 105.9259

12 1.55 0.0175 88.57143

14 1.65 0.02 82.5

16 1.6 0.0185 86.48649

18 1.47 0.0162 90.74074

20 1.31 0.0115 113.913

Table A-3: Lift and Drag Data for NACA 4412

Coefficient CL CD

k0 0.4013 0.0064

k1 0.0935 -0.0009

k2 0.0052 0.0002

k3 -0.0005 -5.00E-06

k4 4.00E-06 -2.00E-07

Table A-4: Polynomial Coefficients for Lift and Drag Properties

42

Figure A-2: Graph of Lift and Drag Coefficients and Glide Ratio for the NACA 4412 Airfoil

y = 4E-06x4 - 0.0005x3 + 0.0052x2 + 0.0935x + 0.4013

y = -2E-07x4 - 5E-06x3 + 0.0002x2 - 0.0009x + 0.0064

0

20

40

60

80

100

120

140

160

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18 20

Glid

e R

atio

Coef

fici

ent o

f L

ift an

d D

rag

Angle of Attack (deg)

Airfoil Properties for NACA 4412

Coefficient of Lift

Coefficient of Drag

Glide Ratio

43

Appendix B- Blade Element Momentum (BEM) Spreadsheet

The following Appendix is included to document the spreadsheet used to compute the

power captured from the wind by the turbine using BEM Theory. In order to understand

the solver without having the spreadsheet file, the cell equations will be output for the

first column. The row and column headers will also be output. The values shown in the

following spreadsheet correspond to the initial computation of turbine efficiency in

Section 3.1.

Figure B-1: Initial Spreadsheet Used to Define Blade Shape [VALUES]

Figure B-2: Initial Blade Parameter Spreadsheet- Formulas for the First Two Blade

Segments [EQUATIONS]

Tip speed ratio X 7 -

Number of blades B 3 -

Angle of attack alpha 7 deg Optimum Glade Angle (CL/CD)max

Coef. of lift C_L 0.88 - CL @ (CL/CD)max

Blade Segment - 1 2 3 4 5 6 7 8 9

Relative radius r/R 0.150 0.250 0.350 0.450 0.550 0.650 0.750 0.850 0.950

Speed ratio X 1.050 1.750 2.450 3.150 3.850 4.550 5.250 5.950 6.650

Angle, optimal phi 29.069 19.830 14.802 11.742 9.707 8.264 7.190 6.360 5.701

Pitch beta 22.069 12.830 7.802 4.742 2.707 1.264 0.190 -0.640 -1.299

Rel. chord length c/R 0.180 0.141 0.111 0.090 0.075 0.064 0.056 0.050 0.045

Inputs

Partially Optimized Wind Turbine Blade Geometry

44

Figure B-3: Main BEM Calculation Spreadsheet [VALUES]

CL CD

Radius R m 2.5 k0 0.10318 6.04E-03

Wind speed v1 m/s 5 from wind data k1 0.10516 -3.63E-04

Rotational speed n min^-1 133.6 k2 0.001048 5.43E-05 for 0 < alpha <20 deg

Density of air rho kg/m^3 1.225 k3 7.35E-06 6.53E-06

Number of blades B - 3 k4 -6.58E-06 -2.80E-07

Angular speed omega s^-1 13.99

Thickness (1 element) dr m 0.25 alpha_s 16.00 deg

Inner radius RI m 0.25 C_LS 1.652801 -

Swept surface A m^2 19.44 C_DS 2.25E-02 - for 20< alpha < 90 deg

Max power Pmax kW 881.94 Betz B1 1.00 -

Tip Speed Vtip m/s 34.98 B2 -0.0556 -

Tip Speed Ratio X - 7.00 A1 0.500 -

A2 0.414 -

Ring. No. N_r - 1 2 3 4 5 6 7 8 9

Rel radius r/R - 0.150 0.250 0.350 0.450 0.550 0.650 0.750 0.850 0.950

Radius r m 0.38 0.63 0.88 1.13 1.38 1.63 1.88 2.13 2.38

Pitch beta deg 22.069 12.830 7.802 4.742 2.707 1.264 0.190 -0.640 -1.299

Chord c m 0.450 0.353 0.276 0.224 0.187 0.161 0.140 0.125 0.112

Pitch, no pitch control beta_0 deg 22.069 12.830 7.802 4.742 2.707 1.264 0.190 -0.640 -1.299

Chord c m 0.450 0.353 0.276 0.224 0.187 0.161 0.140 0.125 0.112

Pitch angle beta deg 22.1 12.8 7.8 4.7 2.7 1.3 0.2 -0.6 -1.3

Solid ratio sigma - 0.57 0.27 0.15 0.10 0.07 0.05 0.04 0.03 0.02

Speed of blade r*omega m/s 5.2 8.7 12.2 15.7 19.2 22.7 26.2 29.7 33.2

axial int. factor a - 0.30959739 0.31505567 0.31562 0.315 0.314202 0.313834 0.31566 0.327887 0.401473

Tang int. factor a' - 0.171443084 0.06820056 0.035656 0.021644 0.014433 0.010272 0.007699 0.006127 0.005694

angle of rel. wind PHI deg 29.3 20.1 15.1 12.0 10.0 8.5 7.4 6.4 5.1

Angle of Attack alpha deg 7.3 7.3 7.3 7.3 7.3 7.2 7.2 7.0 6.4

Coef of Lift C_L - 0.906 0.912 0.911 0.909 0.906 0.903 0.898 0.883 0.812

Coef of Drag C_d - 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007

y-comp C_y - 0.794 0.859 0.882 0.891 0.894 0.895 0.892 0.878 0.809

x-comp C_x - 0.437 0.306 0.230 0.182 0.149 0.126 0.107 0.091 0.065

Factor F - 1.000 1.000 1.000 1.000 0.999 0.997 0.987 0.940 0.729

Factor K - 2.111 2.048 2.042 2.049 2.058 2.062 2.042 1.908 1.275

Axial int. factor (1) a_1 - 0.321 0.328 0.329 0.328 0.327 0.327 0.329 0.344 0.439

Axial int. factor (2) a_2 - 0.310 0.315 0.316 0.315 0.314 0.314 0.316 0.328 0.401

Axial int. factor a - 0.310 0.315 0.316 0.315 0.314 0.314 0.316 0.328 0.401

Tang Int. factor a' - 0.171 0.068 0.036 0.022 0.014 0.010 0.008 0.006 0.006

error_a % 0 0 0 0 0 0 0 0 0

error_a' % -9.71363E-14 0 0 0 0 0 0 0 0

Rel speed w m/s 7.0 9.9 13.1 16.4 19.8 23.2 26.7 30.1 33.6

Tang. Force F_x N/m 6.0 6.5 6.7 6.7 6.7 6.7 6.6 6.3 5.0

Axial force F_y N/m 10.9 18.4 25.7 33.0 40.3 47.5 54.5 60.7 62.4

Power P_r W 23.50944125 42.9545156 61.57021 79.50039 96.86223 113.6076 129.1111 139.9798 125.3827

A_i m^2 0.589048623 0.9817477 1.374447 1.767146 2.159845 2.552544 2.945243 3.337942 3.730641

A_i % 3.03030303 5.05050505 7.070707 9.090909 11.11111 13.13131 15.15152 17.17172 19.19192

Axial Force F_yi N 8.145189899 13.7722288 19.30938 24.78403 30.21514 35.59727 40.84614 45.51783 46.80093

Air Foil Performance

NACA 23012

Blade Geometry and Wind Characteristics

Blade Pitch and Chord Distribution

Iterative Calculation of Tangential and Axial Force

Swept Area

45

Figure B-4: Main BEM Calculation Spreadsheet (1 of 3) [EQUATIONS]

Figure B-5: Main BEM Calculation Spreadsheet (2 of 3) [EQUATIONS]

46

Fig

ure B

-6: M

ain B

EM

Calcu

lation S

pread

sheet (3

of 3

) [EQ

UA

TIO

NS]

47

Figure B-7: BEM Results Table [VALUES]

Figure B-8: BEM Results Table [EQUATION]

Pitch Control wind

speed dbeta deg 0

wind speed v_1 m/s 5

rotational speed n min^-1 133.6

power P W 812.477961

efficieny eta_r % 92.12429558

torque M Nm 58.07330104

axial force T N 264.9881416

tip speed ratio X_act - 6.995279642

mean angle of attack alpha_m deg 7.1

coefficient of power Cp 0.55

Reults of Power Calculation