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
© Copyright 2012
by
John J. McCosker
All Rights Reserved
ii
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
iii
LIST OF TABLES
Table 1: Optimized Dimensionless Wind Turbine Blade Geometry ........................ Page 29
iv
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
v
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
vi
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
vii
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.
viii
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.
ix
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 windindustry 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
1
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.
𝑃 = 𝑇𝜔
(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-todrag 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.
2
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.
1
(2)
𝐹𝐿 = 𝐶𝐿 𝜌𝑤 2 (𝑏𝑐)𝐵
2
1
(3)
𝐹𝐷 = 𝐶𝐷 𝜌𝑤 2 (𝑏𝑐)𝐵
2
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.
3
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.
where
1
𝑑𝑇 = 𝜌𝑤 2 𝑐 𝑟 𝑑𝑟(𝐶𝑥 )
2
(4)
𝐶𝑥 = 𝐶𝐿 𝑠𝑖𝑛(𝜑) − 𝐶𝐷 𝑐𝑜𝑠(𝜑)
(5)
𝑑𝑇ℎ =
where
1
𝜌𝑤 2 𝑐 𝑑𝑟(𝐶𝑦 )
2
𝐶𝑦 = 𝐶𝐿 𝑐𝑜𝑠(𝜑) + 𝐶𝐷 𝑠𝑖𝑛(𝜑)
(6)
(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).
𝑢 = 𝑟ω
(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
4
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.
𝑋=
5
𝑅ω
v
(9)
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.
𝑃
𝐶𝑝 =
(10)
1
3
2 ∗ 𝜌 ∗ 𝐴 ∗ 𝑣1
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.
𝑝1 +
1
1
=
𝑝
+
𝜌 𝑣2
+
2
2
2𝜌𝑣1
6
(11)
1
1
(12)
𝑝+ − ∆𝑝 + 𝜌 𝑣 2 = 𝑝1 + 𝜌 𝑣32
2
2
By subtracting Equations (11) and (12), one arrives at the following expression:
1
(13)
∆𝑝 = 𝜌(𝑣12 − 𝑣32 )
2
Based on the change in linear momentum from v1 to v3, the change in pressure ∆p can
also be expressed as:
∆𝑝 = 𝜌 𝑣(𝑣1 − 𝑣3 )
(14)
By solving equations (13) and (14) for v,
𝑣 = 1/2(𝑣1 + 𝑣3 )
(15)
The power produced by the wind turbine is equal to the kinetic energy in the air.
𝑃 = 1/2𝜌𝑣(𝑣12 − 𝑣32 )
(16)
The axial interference factor 𝑎 is a factor that represents the loss in wind speed as it
approaches the turbine blade. The axial interference factor is defined as:
or
𝑣 = (1 − 𝑎)𝑣1
(17)
𝑣3 = (1 − 2𝑎)𝑣1
(18)
In terms of the axial interference factor, the power equation from Equation (16) can be
re-written as:
𝑃 = 2𝜌𝑎(1 − 𝑎)2 𝑣13 𝐴
(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.
𝐶𝑃 = 4𝑎(1 − 𝑎)2
(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 𝑎. Doing so results
in a root at 𝑎 =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.
7
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.
𝐹𝐷
= 0.5𝜌𝑤 2 𝐶𝐷
𝐴
(21)
where w is the relative wind speed of the drag-type wind turbine, as governed by
Equation (22).
𝑤 = 𝑣1 − 𝑢
(22)
𝑃 = 𝐹𝐷 𝑢,
(23)
If,
then, by combining Equations (21) and (23)
𝑃 1
1
= 𝜌𝑤 2 𝐶𝐷 𝑢 = 𝜌(𝑣1 − 𝑢)2 𝐶𝐷 𝑢 .
𝐴 2
2
(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.
𝐶𝑃 = 𝐶𝐷 (𝜆 − 2𝜆2 + 𝜆3 )
(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 dragtype 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.
8
Coefficient of Power- Drag vs. Lift Device
0.7
Power Coefficient
0.6
Lift-Type Device
0.5
Drag-Type Device
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Non-Dimensional Velocity Coefficient
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.
9
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
10
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
1
equation expresses the relative tangential speed of the rotating blade. The 2 ∆𝑢 term
accounts for the additional tangential wind speed that the blade experiences due to the
average counter-rotating wake velocity.
1
(26)
𝑢 = 𝑟𝜔 + ∆𝑢
2
The additional tangential speed that the blade experiences due to the wake is defined as a
function of the tangential interference factor 𝑎′ .
𝑢 = 𝑟𝜔(1 + 𝑎′ )
(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
11
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
1
2
∆𝑢 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 𝑤1 and the change in relative wind
speed angle φ.
∆𝑤 = 2𝑤1 sin(𝜑1 − 𝜑)
12
(28)
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.
𝑑𝐹𝐿 = ∆𝑤 𝑑𝑞
(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
13
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.
𝑑𝑇 = 𝑑𝐹𝐿 sin(𝜑) 𝑟 = ∆𝑤 𝑑𝑞 sin(𝜑) 𝑟
(30)
By substituting the following expression for mass flow rate of air through the ring
element dq
𝑑𝑞 = 2𝜌𝜋𝑟𝑑𝑟𝑣
(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.
𝑑𝑃 = 𝑟 2 𝜔𝜌2𝜋𝑑𝑟𝑤12 sin[2(𝜑1 − 𝜑)] sin2 (𝜑1 )
(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,
𝜑=
2
𝜑
3 1
(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.
𝜑=
2
𝑣1
2
𝑅
arctan ( ) = arctan ( )
3
𝜔𝑟
3
𝑋𝑟
(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.
2
𝑅
𝛽(𝑟) = arctan ( ) − 𝛼𝐷
3
𝑋𝑟
(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).
𝑑𝐹𝐿 = ∆𝑤 𝑑𝑞 = [2𝑤1 sin(𝜑1 − 𝜑)](2𝜌𝜋𝑟𝑑𝑟)𝑣
(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.
14
𝑐(𝑟) =
1 16𝜋𝑟 2 1
𝑅
sin ( arctan ( ))
𝐵 𝐶𝐿
3
𝑋𝑟
(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.
15
Comparison of Optimized Pitch Angle
Optimized Pitch Angle,β (deg)
70
60
50
40
Betz
30
Schmitz
20
10
0
0
0.2
0.4
0.6
0.8
1
r/R
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.
16
Comparison of Optimized Chord
Length
Optimized Chord Length, c/R
0.7
0.6
0.5
0.4
Betz
0.3
Schmitz
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
r/R
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.
17
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.
18
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:
𝐴2 cos 2 (𝛼)
𝐶𝐿 = 𝐴1 sin(2α) +
sin2 (𝛼)
(38)
𝐶𝐷 = 𝐵1 sin2 ( 𝑎) + 𝐵2 cos(𝛼)
(39)
where
𝐴1 =
𝐵1
2
𝐴2 = (𝐶𝐿 − 𝐶𝐷𝑀𝐴𝑋 sin(𝛼𝑠 ) cos(𝛼𝑠 ))
(40)
sin(𝛼𝑠 )
cos2 (𝛼𝑠 )
(41)
𝐵1 = 𝐶𝐷𝑀𝐴𝑋
(42)
𝐶𝐷𝑀𝐴𝑋 sin2 (𝛼𝑠 )
𝐵2 = 𝐶𝐷𝑠 −
cos(𝛼𝑠 )
(43)
𝐶𝐷𝑀𝐴𝑋 = 1.11 + 0.18 𝐴𝑅
(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
19
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.
20
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,
𝑑𝑇ℎ = 2𝜋𝑟𝜌𝑣2 (𝑣1 − 𝑣3 )𝑑𝑟
(45)
one is able to obtain the first of two relationships required for BEM theory.
𝜎𝐶𝑦
𝑎
=
𝑎 − 1 4 sin2 (𝜑)
(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,
𝑑𝑇 = 2𝜋𝑟 2 𝜌𝑣2 𝑢2 𝑑𝑟
(47)
yields the second relation for BEM theory.
𝑎′
𝜎𝐶𝑥
=
𝑎′ + 1 4 sin(𝜑) cos(𝜑)
(48)
The solidarity ratio σ is defined as the following expression.
𝜎=
𝑐𝐵
2𝜋𝑟
(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 𝑎 is the fractional decrease in axial wind velocity
between the free stream and the rotor plane. The tangential interference factor 𝑎′ 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
21
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
2
cos −1(𝑒 −𝑓 )
𝜋
𝐵 𝑅−𝑟
𝑓=
2 𝑟𝑠𝑖𝑛(𝜑)
𝐹𝑃 =
where
(50)
(51)
Solving Equations (46) and (48) for 𝑎 and 𝑎′, respectively, and including the Prandtl tip
loss correction factor, yields the final two equations for 𝑎 and 𝑎′ used in the BEM
procedure [7].
1
𝑎=
(
𝑎′ =
4𝐹𝑃
𝑠𝑖𝑛2 (𝜑)
𝜎𝐶𝑦
+ 1)
1
4𝐹 𝑠𝑖𝑛(𝜑) cos(𝜑)
( 𝑃 𝜎𝐶
− 1)
𝑥
22
(52)
(53)
Equation (52) is only accurate in computing axial interference factors 𝑎 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].
1
𝑎 = ( 2 + 𝐾(1 − 2𝑎𝑐 )
2
(54)
− √(𝐾(1 − 2𝑎𝑐 ) +
where
𝐾=
2)2
4𝐹𝑠𝑖𝑛2 (𝜑)
𝜎𝐶𝑦
+
4(𝐾𝑎𝑐2
− 1))
(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
23
flow diagram outlining the process of calculating the axial induction factor 𝑎 and the
tangential induction factor 𝑎′ 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.
(1 − 𝑎)𝑣1
𝜑 = tan−1 (
)
(1 + 𝑎′ )𝑟𝜔
Inputs:
β, v1, ω, c,
ρair, B, R, dr
Choose guess
values for 𝑎 and 𝑎′
(guess = 0)
Calculate α (α = φ - β)
and find CL and CD from
the airfoil data
corresponding to α
Calculate φ from
Eqn. (56)
Substitute
previous 𝑎
and 𝑎′ for
new values
(56)
No
Does new 𝑎 and 𝑎′
differ by more than
the target % from
previous 𝑎 and 𝑎′?
Calculate
Cx and Cy from
Eqns. (5) and (7)
Calculate 𝑎 and
𝑎′ from Eqns (56),
(57), and (58)
Yes
Finished
Figure 13: Flow Diagram of the Iteration Process Used to Solve for Axial Induction
Factor 𝑎 and the Tangential Induction Factor 𝑎′
Once the values for 𝑎 and 𝑎′ converge, the torque T and thrust Th for each blade segment
is calculated by using the following equations:
1
𝜌𝑤 2 𝑐𝐶𝑥 𝑟
2
1
𝑇ℎ ∗ (𝑟) = 𝜌𝑤 2 𝑐𝐶𝑦
2
𝑇 ∗ (𝑟) =
24
(57)
(58)
The total axial force and power are then calculated using the following summations:
𝑅
𝑇ℎ = 𝐵 ∫ 𝑇ℎ ∗ (𝑟)𝑑𝑟
(59)
0
𝑅
𝑃 = 𝜔𝐵 ∫ 𝑇 ∗ (𝑟)𝑑𝑟
(60)
0
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.
25
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
26
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].
𝑥2 𝛼
𝑣2∗ = 𝑣1∗ ( )
𝑥1
(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 puretone 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.
27
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.
28
Blade Segment
Relative radius
Speed ratio
Angle, optimal
Pitch
Rel. chord length
r/R
X
phi
beta
c/R
1
2
3
4
5
6
0.150 0.250 0.350 0.450 0.550 0.650
1.050 1.750 2.450 3.150 3.850 4.550
29.069 19.830 14.802 11.742 9.707 8.264
22.069 12.830 7.802 4.742 2.707 1.264
0.180 0.141 0.111 0.090 0.075 0.064
7
8
9
0.750 0.850 0.950
5.250 5.950 6.650
7.190 6.360 5.701
0.190 -0.640 -1.299
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
29
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.
Performance of the Initially Optimized
Wind Turbine
Coefficient of Power, Cp
0.6
0.5
0.4
0.3
R=2.5,X=7
0.2
0.1
0
1
3
5
7
9
11
Tip Speed Ratio, X
Figure 15: The Coefficient of Power for a Turbine Blade Optimized for X=7
30
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.
31
Comparison of Blades Optimized for
Varying Tip Speed Ratios
Coefficient of Power, Cp
0.6
0.5
0.4
X=4
0.3
X=5
X=6
0.2
X=7
0.1
X=8
0
1
2
3
4
5
6
7
8
9
10
11
12
Tip Speed Ratio, X
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.
32
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 indepth 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
33
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.
Comparison between NACA 23012 and
NACA 4412
0.7
Coefficient of Power, Cp
0.6
0.5
NACA 23012, X=6
0.4
NACA 23012, X=7
NACA 20312, X=8
0.3
NACA 4412, X=6
NACA 4412, X=7
0.2
NACA 4412, X=8
0.1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Tip Speed Ratio, X
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
34
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.
35
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 𝑎. By equating the torque force with the angular
momentum force on the rotor, one is also able to solve for the tangential interference
factor 𝑎′. And finally, one is able to calculate the power produced by the wind turbine,
by using an iterative process to solve for 𝑎 and 𝑎′. 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.
36
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.
37
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.
38
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
0
2
4
6
8
10
12
14
16
18
20
0.12
0.3
0.55
0.77
1
1.15
1.38
1.56
1.7
1.75
1.38
CD
0.006
0.0062
0.0065
0.0067
0.0077
0.0095
0.0142
0.0185
0.02
0.021
0.022
GR
20
48.3871
84.61538
114.9254
129.8701
121.0526
97.1831
84.32432
85
83.33333
62.72727
Table A-1: Lift and Drag Data for NACA 23012
Coefficient
k0
k1
k2
k3
k4
CL
CD
0.1031
0.0061
0.1051
-0.0004
0.0011
5.43E-05
7.35E-06
6.53E-06
-6.58E-06
-2.80E-07
Table A-2: Polynomial Coefficients for Lift and Drag Properties
39
Airfoil Properties for NACA 23012
2
140
y = -5E-05x 4 + 0.0015x3 - 0.0145x2 + 0.1525x + 0.0941
1.8
100
1.4
1.2
80
1
60
0.8
Glide Ratio
Coefficient of Lift and Drag
120
1.6
Coefficient of Lift
Coeffiient of Drag
Glide Ratio
0.6
40
0.4
20
0.2
y = 9E-07x 4 + 1E-06x 3 - 0.0002x 2 + 0.0012x + 0.0053
0
0
0
2
4
6
8
10
12
14
16
18
20
Angle of Attack (deg)
Figure A-1: Graph of Lift and Drag Coefficients and Glide Ratio for the NACA 23012
Airfoil
40
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
0
2
4
6
8
10
12
14
16
18
20
CL
0.4
0.6
0.85
1.05
1.25
1.43
1.55
1.65
1.6
1.47
1.31
CD
0.0061
0.0061
0.0065
0.0075
0.011
0.0135
0.0175
0.02
0.0185
0.0162
0.0115
GR
65.57377
98.36066
130.7692
140
113.6364
105.9259
88.57143
82.5
86.48649
90.74074
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
41
1.8
160
1.6
140
1.4
120
1.2
100
1
80
0.8
y=
4E-06x 4 -
0.0005x 3 + 0.0052x 2 + 0.0935x + 0.4013
60
0.6
Glide Ratio
Coefficient of Lift and Drag
Airfoil Properties for NACA 4412
Coefficient of Lift
Coefficient of Drag
Glide Ratio
40
0.4
20
0.2
y = -2E-07x 4 - 5E-06x 3 + 0.0002x 2 - 0.0009x + 0.0064
0
0
0
2
4
6
8
10
12
14
16
18
20
Angle of Attack (deg)
Figure A-2: Graph of Lift and Drag Coefficients and Glide Ratio for the NACA 4412 Airfoil
42
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.
Tip speed ratio
Number of blades
Angle of attack
Coef. of lift
Blade Segment
Relative radius
Speed ratio
Angle, optimal
Pitch
Rel. chord length
Inputs
X
B
alpha
C_L
r/R
X
phi
beta
c/R
7
3
7
0.88
deg
-
Optimum Glade Angle (CL/CD)max
CL @ (CL/CD)max
Partially Optimized Wind Turbine Blade Geometry
1
2
3
4
5
6
0.150
0.250 0.350
0.450
0.550
0.650
1.050
1.750 2.450
3.150
3.850
4.550
29.069
19.830 14.802
11.742
9.707
8.264
22.069
12.830 7.802
4.742
2.707
1.264
0.180
0.141 0.111
0.090
0.075
0.064
7
0.750
5.250
7.190
0.190
0.056
8
0.850
5.950
6.360
-0.640
0.050
9
0.950
6.650
5.701
-1.299
0.045
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]
43
Air Foil Performance
NACA 23012
CL
CD
0.10318 6.04E-03
0.10516 -3.63E-04
0.001048 5.43E-05 for 0 < alpha <20 deg
7.35E-06 6.53E-06
-6.58E-06 -2.80E-07
Blade Geometry and Wind Characteristics
Radius
Wind speed
Rotational speed
Density of air
Number of blades
Angular speed
Thickness (1 element)
Inner radius
Swept surface
Max power
Tip Speed
Tip Speed Ratio
R
v1
n
rho
B
omega
dr
RI
A
Pmax
Vtip
X
m
m/s
min^-1
kg/m^3
s^-1
m
m
m^2
kW
m/s
-
Ring. No.
Rel radius
Radius
Pitch
Chord
N_r
r/R
r
beta
c
m
deg
m
Pitch, no pitch control
Chord
Pitch angle
Solid ratio
Speed of blade
beta_0
c
beta
sigma
r*omega
deg
m
deg
m/s
axial int. factor
Tang int. factor
angle of rel. wind
Angle of Attack
Coef of Lift
Coef of Drag
y-comp
x-comp
Factor
Factor
Axial int. factor (1)
Axial int. factor (2)
Axial int. factor
Tang Int. factor
error_a
error_a'
a
a'
PHI
alpha
C_L
C_d
C_y
C_x
F
K
a_1
a_2
a
a'
deg
deg
%
%
Rel speed
Tang. Force
Axial force
w
F_x
F_y
m/s
N/m
N/m
Power
P_r
A_i
A_i
F_yi
W
m^2
%
N
2.5
5 from wind data
133.6
1.225
3
13.99
0.25
0.25
19.44
881.94 Betz
34.98
7.00
k0
k1
k2
k3
k4
alpha_s
C_LS
C_DS
B1
B2
A1
A2
16.00
1.652801
2.25E-02
1.00
-0.0556
0.500
0.414
deg
-
for 20< alpha < 90 deg
Blade Pitch and Chord Distribution
1
0.150
0.38
22.069
0.450
2
0.250
0.63
12.830
0.353
3
0.350
0.88
7.802
0.276
4
0.450
1.13
4.742
0.224
5
0.550
1.38
2.707
0.187
6
0.650
1.63
1.264
0.161
7
0.750
1.88
0.190
0.140
8
0.850
2.13
-0.640
0.125
9
0.950
2.38
-1.299
0.112
1.264
0.161
1.3
0.05
22.7
0.190
0.140
0.2
0.04
26.2
-0.640
0.125
-0.6
0.03
29.7
-1.299
0.112
-1.3
0.02
33.2
Iterative Calculation of Tangential and Axial Force
Swept Area
Axial Force
22.069
0.450
22.1
0.57
5.2
12.830
0.353
12.8
0.27
8.7
7.802
0.276
7.8
0.15
12.2
4.742
0.224
4.7
0.10
15.7
2.707
0.187
2.7
0.07
19.2
0.30959739 0.31505567 0.31562
0.315 0.314202 0.313834 0.31566 0.327887 0.401473
0.171443084 0.06820056 0.035656 0.021644 0.014433 0.010272 0.007699 0.006127 0.005694
29.3
20.1
15.1
12.0
10.0
8.5
7.4
6.4
5.1
7.3
7.3
7.3
7.3
7.3
7.2
7.2
7.0
6.4
0.906
0.912
0.911
0.909
0.906
0.903
0.898
0.883
0.812
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.007
0.794
0.859
0.882
0.891
0.894
0.895
0.892
0.878
0.809
0.437
0.306
0.230
0.182
0.149
0.126
0.107
0.091
0.065
1.000
1.000
1.000
1.000
0.999
0.997
0.987
0.940
0.729
2.111
2.048
2.042
2.049
2.058
2.062
2.042
1.908
1.275
0.321
0.328
0.329
0.328
0.327
0.327
0.329
0.344
0.439
0.310
0.315
0.316
0.315
0.314
0.314
0.316
0.328
0.401
0.310
0.315
0.316
0.315
0.314
0.314
0.316
0.328
0.401
0.171
0.068
0.036
0.022
0.014
0.010
0.008
0.006
0.006
0
0
0
0
0
0
0
0
0
-9.71363E-14
0
0
0
0
0
0
0
0
7.0
6.0
10.9
9.9
6.5
18.4
13.1
6.7
25.7
16.4
6.7
33.0
19.8
6.7
40.3
23.2
6.7
47.5
26.7
6.6
54.5
30.1
6.3
60.7
33.6
5.0
62.4
23.50944125 42.9545156 61.57021 79.50039 96.86223
0.589048623 0.9817477 1.374447 1.767146 2.159845
3.03030303 5.05050505 7.070707 9.090909 11.11111
8.145189899 13.7722288 19.30938 24.78403 30.21514
113.6076
2.552544
13.13131
35.59727
129.1111
2.945243
15.15152
40.84614
139.9798
3.337942
17.17172
45.51783
125.3827
3.730641
19.19192
46.80093
Figure B-3: Main BEM Calculation Spreadsheet [VALUES]
44
Figure B-4: Main BEM Calculation Spreadsheet (1 of 3) [EQUATIONS]
Figure B-5: Main BEM Calculation Spreadsheet (2 of 3) [EQUATIONS]
45
Figure B-6: Main BEM Calculation Spreadsheet (3 of 3) [EQUATIONS]
46
Reults of Power Calculation
Pitch Control wind
speed
wind speed
rotational speed
power
efficieny
torque
axial force
tip speed ratio
mean angle of attack
coefficient of power
dbeta
v_1
n
P
eta_r
M
T
X_act
alpha_m
Cp
deg
m/s
min^-1
W
%
Nm
N
deg
0
5
133.6
812.477961
92.12429558
58.07330104
264.9881416
6.995279642
7.1
0.55
Figure B-7: BEM Results Table [VALUES]
Figure B-8: BEM Results Table [EQUATION]
47