Wind Energy Harvesting
One field of great interest and promise for students is “energy harvesting”. Energy
harvesting is the science, technology, and execution of collecting usable energy from a
variety of sources typically outside of fossil fuels. Some examples would be wind energy,
thermal energy, kinetic energy, solar energy, and tidal energy. There is a substantial push
at the state and national levels regarding the exploration and research of such alternative
energy sources. The political and corporate pipeline is becoming filled with policies,
subsidies, and research grants to incentivize entities to help diversify America’s energy
base.
The goal of this project is to examine one particular source of energy: wind energy.
The science of wind energy is quite complex so our treatment here will be to give you a
glimpse into this growing field.
We will
 Examine a typical wind turbine.
 Examine how the area swept out by the rotor affects energy harvesting ability.
 Examine how wind speed affects energy harvesting ability.
 Examine how turbine height affects energy harvesting ability.
 Explore the energy harvesting ability of a typical wind turbine through elementary
mathematical models.
 Discuss how the harvesting ability of a wind turbine is affected by the proximity of
other turbines.
 Close with general comments about wind energy as an alternative energy source and
examine some of the innovative approaches to capturing wind energy.
Peppered throughout this project are web-links and references for you to enjoy and
enhance your understanding of the various topics presented.
The Local Scene
Arizona has two major utilities: Salt River Project and Arizona Public Service.
When it comes to wind power, Salt River Project has just completed the first stages of
Arizona’s first wind farm (2009). The Dry Lake Wind Power Project will be a full-fledged
wind turbine farm which is located north of Heber, Arizona and northwest of Snowflake,
Arizona.
Task 1: Logon to: http://www.srpnet.com/about/stations/drylakewind.aspx and do the
following.
 Watch the short video.
Task 2: Logon to the links below and look through the fact sheet, community brochureand
videos to answer the following.
http://iberdrolarenewables.us/cs_drylake.html
http://www.srpnet.com/environment/windfarm.aspx
1. How many turbines will the Dry Lake Wind Power Project have in place when it is
finished?
2. What is the anticipated yearly energy output in megawatts?
3. How many average-sized homes could the Dry Lake Wind Power Project potentially
supply power to?
4. What maximum height will the turbines reach?
5. How many tons is the combined weight of the tower and nacelle?
6. At what speed will a turbine spin in order to first be able to generate electricity?
7. Typically, wind turbines contain sensors to shut the turbine down (braking) or alter
the blade angle (decreases the blade rotation speed) to account for high winds which
may damage the system. Here are a couple of videos which show the incredibly
destructive forces that can occur if such precautions are not taken or if such
mechanical precautions fail.
http://www.youtube.com/watch?v=sbCs7ZQDKoM
http://www.youtube.com/watch?v=CqEccgR0q-o
Now that you have a general feel for how large and powerful wind turbines actually are,
let’s examine a typical wind turbine more closely.
The typical horizontal axis wind turbine (HAWT) has a tubular tower (smaller
towers may have a lattice frame) with three rotor blades connected to a housing (called a
nacelle) of the gear-box, drive-chain, generator, and a few other items.
The wind will strike the rotor blades causing them to turn, thus causing some of the wind’s
kinetic energy to be converted into mechanical energy which when transferred to the
generator (located in the nacelle) creates electricity.
The amount of energy “harvested” from the wind is dependent on several factors. Three
main ones are:
 The area swept out from the rotating blades.
 The speed of the wind striking the blades.
 The efficiency of the gear-box generator devices.
Let’s first look at the area swept out by the blades. According to the Danish Wind Energy
Association, the following graphic shows kilowatt output for various rotor diameter sizes.
Task 3: Answer/do the following.
8. How long is the rotor diameter for a typical 600 KW electrical generator?
9. The largest rotor diameter pictured above is 80 meters. What is the area of the
circle swept out with that rotor diameter?
10. Create a spreadsheet of ordered pairs with rotor diameter as your independent
variable and KW produced as your dependent variable. Graph your data and
comment on the nature of the data points. Are they linear?
11. Run a quartic regression on your data and comment about the fit of your 4th degree
polynomial.
12. Show mathematically that if you double your diameter, you’ll multiply your swept
out area by a factor of four.
Now that we know the rotor diameter will affect the amount of energy harvested from the
wind, let’s talk about the speed of the wind.
Wind speed usually increases with an increase in elevation. This phenomenon is called
“wind shear”. Why does this happen? Think of the ground as causing friction, so the wind
travels slower near the ground (rubbing against the ground slows it down). There are
specific “roughness” values for different types of ground surfaces. There exist several
mathematical models to determine the speed of the wind at varying heights above the
ground. Two very common formulas for predicting how a change in elevation will affect the
speed of the wind are the Wind Speed Logarithmic Equation and the Wind Speed Power
Equation.
The Wind Speed Logarithmic Equation:
Velecityunknown  Velocityknown
 heightdesired 
ln 




 heightknown 
ln 




The value,  , is called the roughness constant and has typical values shown in the table
below.
Surface Roughness Length,  , (m)
0.00001
0.0002
0.0005
0.003
0.008
0.01
0.03
0.05
0.1
0.25
0.5
1.5
3.0
Terrain Description
Very smooth, ice or mud
Calm open sea
Blown sea
Snow surface
Lawn grass
Rough pasture
Fallow ground
Crops
Few trees
Many trees, hedges, few buildings
Forest and woodlands
Suburbs
Centers of cities with tall buildings
The Wind Speed Power Equation:

Velecityunknown
 heightdesired 
 Velocityknown 

 heightknown 
The exponent,  , is called the wind shear exponent. A table of common wind shear exponent
values follows;
Terrain Description
Smooth, hard ground, lake or ocean
Short grass on untilled soil
Level country with foot-high grass
Tall row crops, hedges, a few trees
Many trees and occasional building
Wooded country – small towns and suburbs
Urban areas with tall buildings
Wind Shear Exponent, 
0.10
0.14
0.16
0.20
0.22 – 0.24
0.28 – 0.30
0.4
Task 4:
A generally recognized 'rule of thumb' is that wind speed increases as the 1/7th power of the
height above ground. This fits quite nicely for the Great Plains in the U.S. A typical large
utility-sized wind turbine has a hub height of 80 m and a rotor diameter of 77 m. Let’s
suppose we observe that the average wind speed is 10 m/s at a height of 10 m at a proposed
wind turbine site. Use the typical hub height, rotor diameter, and observed speed/height
values to do the following:
13. Using the Wind Speed Power Equation with  
1
calculate the following:
7
a. Speed at the rotor height.
b. Speed at the lowest blade tip height.
c. Speed at the highest blade tip height.
14. Instead of using the Wind Speed Power Equation repeat number 15 using the Wind
Speed Logarithmic Equation with   0.02 .
15. How do your results compare in parts 1 & 2?
16. In order to determine if a proposed site offers enough wind a “wind map” is
consulted. Examine Arizona’s wind map using the link below. Where are the windiest
places in Arizona?
http://www.windpoweringamerica.gov/wind_resource_maps.asp?stateab=az
17. Another useful item in analyzing the wind is a “wind rose”. A wind rose shows the
percentage of time winds flow from particular directions. This helps the wind farm
designers position the turbines so that they capture the most wind. As a little different
twist in this project, read and examine the wind rose for Chicago’s O’Hare Airport.
Go to http://www.wrcc.dri.edu/cgi-bin/rawMAIN.pl?azADRL and create the wind rose
for Dry Lake, Arizona from Jan. 1, 2009 through December 31, 2009. You’ll need to
click on “Wind Rose Graph and Tables” in the left-hand column. Copy the picture and
then answer the following:
a. In which direction does the greatest amount of yearly wind flow from and
what percentage is that?
b. In which direction quadrant (NE, NW, SE, SW) does the least amount of
wind flow from.
c. Would you characterize the winds at this location as coming from a
general direction or would you consider them variable?
How Much Energy and Power Does the Wind Have?
As wind strikes the rotor, the kinetic energy of the wind is converted into
mechanical energy as the rotor turns. We now ask this simple question: “How much kinetic
energy and power does the wind contain for our rotor diameter?”
From physics, the amount of kinetic energy a moving amount of air has is given by:
Ekinetic 
1 2
mv
2
Where m  mass and v  velocity .
The mass of flowing air per unit time (mass flow rate) that strikes our rotor is given by;
mass  air _ density  (area of rotor )  velocity   Av
If we substitute our mass flow rate into our kinetic energy equation we get the available
power of the wind:
Pavailable 
1
1
  Av  v 2   Av3
2
2
Task 5:
Answer the following:
18. What is the affect on the power available if we double our rotor’s radius? What is
the percentage increase?
19. What is the affect on the power available if we double our wind speed? What is the
percentage increase?
20. It is said “Use as tall a wind turbine as possible.” Explain why that would be the
case.
21. Using our Wind Speed Power Equation, show that our power equation from above
can be written as;
h
1
P   Av 0 3  
2
 h0 
3
v0  known speed for height h0
Where h  desired height
  wind shear exponent value
22. Suppose we currently have a wind turbine with hub height of 50 m on a flat grassy
plain. Furthermore we observe that at a height of 10 m the average wind speed is 10
m/s. If we swap out this turbine with one that is 100 m tall at the hub height, what
increase in power will the wind hold (assume same rotor diameter and air density and

1
)?
7
How Much Power Can a Wind Turbine Harvest From the Wind?
In 1919, Albert Betz concluded that no wind turbine can convert more than 16/27
(about 59.3%) of the kinetic energy of the wind into mechanical energy at the rotor. What
this means is that the theoretical maximum power efficiency of any design of wind turbine is
about 59%. This is called the Betz Limit or Betz’ Law. In reality, current wind turbines are
only capable of extracting somewhere between 35 – 45% of the wind’s power by the
turbine. Taking into account the gearbox, bearings, generator, and other elements, only
about 30% of the wind’s power is actually converted into usable electricity. Let’s derive
Betz’ Limit.
Consider the diagram shown below:
As wind moves from left-to-right, Betz proved that the mass of air passing through the
rotor S is given by;
v v 
m   A 1 2 
 2 
(1)
Where  is the air density, A is the area swept out by the rotor, and
v1  v2
2
is the wind
velocity at the rotor (note: Betz showed that this is just the average velocity of the
undisturbed wind velocities before and after the rotor).
The wind’s change in kinetic energy is given by:
KEin  KEout 
1
1
1
mv12  mv2 2  m  v12  v2 2 
2
2
2
(2)
Task 6.
23. Using equations (1) and (2) from above, show that the power extracted by the rotor
is given by.
Pextracted 
A
4
v
3
1
 v1v2 2  v2 v12  v23 
The equation above shows us that the power extracted from the wind is determined by
the density of the air (  ), the area swept out by the rotor (A), and the velocity of the
moving air before and after the rotor ( v1 & v2 ).
24. Suppose the velocity after the rotor is zero. What implications does this have for the
turbine’s rotor and the volume of air after the turbine? Is any power extracted?
25. Suppose the velocity of the air after the turbine is the same as the velocity of air
before the turbine. What implication does this hold for our model?
26. Show that the ratio of the power extracted from the wind to the power of
undisturbed wind is given by:
Pextracted 1  v2 2 v2 v23 
C  v1 , v2  
 1  2   3 
Pwind
2  v1
v1 v1 
We call C  v1 , v2  the “power coefficient” and Betz established that this value is
maximal at 16/27. Each turbine design has a power coefficient associated with it.
27. Using the partial derivatives of C  v1 , v2  show that C  v1 , v2  is maximized when
v2 
v1
16
and thus Cmax 
. Make sure you employ all algebraic steps in your work.
3
27
28. There is a simpler method to arrive at our result. Define the variable, t, as follows:
t  v1 , v2  
v2
v1
Show that our above function C  v1 , v2  can be written more compactly as:
C t  
Pextracted 1
 1  t 2  t  t 3 
Pwind
2
29. Graph the function
1
1 t2  t  t3 

2
0  t  1; 0  C  t   1 /* Do you know why ?
C t  
where
30. Using calculus methods again, show that the maximum of this function yields the
same results as #29. Does your graph of this new compact function illustrate this?
The power we actually get from a wind turbine is most often written as:
1
Pextracted  C p Av 3
2
where,
C p  turbine ' s power coefficient :
C p  Betz Limit
  the turbine ' s mechanical & electrical efficiencies :  75  85%
How Should Wind Turbines Be Arranged?
Wind turbines are large mechanical devices which dramatically slow down the natural flow
of the wind. Because of this, we cannot place one turbine slightly behind the other. The
wind’s speed is reduced behind the rotor blades and turbulence (wake) is caused. Thus,
there must be sufficient space so that the wind can “recover” before it strikes the next
turbine. Although accused of being doctored, this photo of the Horns Rev Off-Shore Wind
Farm in Denmark is quite interesting! You can observe the “wake” behind each turbine.
There exist very complex mathematical models for both the modeling of the wind’s wake
after passing through a turbine and the placement of the individual turbines to optimize the
energy harvested for a particular locale. There are, however, some elementary practical
guidelines. For example, a simple rule of thumb is to space the turbines with horizontal &
vertical spacings of so many rotor diameters. For the farm pictured below, the turbines
follow the spacing rule of 4 rotor diameters apart horizontally and 7 rotor diameters apart
vertically (in the wind’s direction).
Generally, the lay of the land and the direction of the prevailing wind will determine
how the turbines will be placed. On ridgelines or off-shore at a particular water depth, it is
not uncommon to see a long row of turbines.
For large flat areas on-shore or stable depths off-shore, arrayed patterns are often seen.
Papalote Creek Wind Farm, San Patricio, Texas
Horse Hollow Wind Energy Center - is the world's largest wind farm at 735.5
megawatt (MW) capacity. It consists of 291 GE Energy 1.5 MW wind turbines and
130 Siemens 2.3 MW wind turbines spread over nearly 47,000 acres (190 km²) of
land in Taylor and Nolan County, Texas.
General Comments about Wind Energy
There are many wind farms being built both on land and off-shore throughout the world.
For each site, a few of the important questions to be answered include:


What is the topography of the site?
How should the turbines be placed in order to maximize the harvesting potential of
the site?
Arizona’s Dry Lake Wind Power Project


What environmental impact will the site impart?
What costs are associated with this site and will the harvesting potential lend this
site feasible from a cost-benefit standpoint?
Wind energy will continue to share a portion of America’s energy base. It won’t become the
prevailing energy source due to the large swaths of land or sea needed to create substantial
farms but it does offer significant benefits over traditional fossil fuels. According to the
Department of Energy, about 9% percent of America’s energy consumption is from wind
energy while wind production composes less than 1% of our energy sources.
33. Despite the limited amount of wind energy used at present, the U.S. Department of
Energy has sought an ambitious goal for wind energy’s contribution by 2030. Use the
following website below to answer this question: Want percentage of America’s energy
demand does the DOE hope wind energy can meet?
http://www1.eere.energy.gov/windandhydro/wind_2030.html
To Close: Creative Minds Wanted!
The push for alternative sources of energy and energy harvesting has opened the
door for creative thinkers. As a student, you may wish to explore options in this growing
field. Keep in mind that traditional engineering fields are being off-shored by major
companies (opinions aside) and it is the new cutting edge technologies which offer the most
promising job prospects. Not only will they emerge at the forefront of policy decisions but
they will also offer the most flexibility and entrepreneurial opportunities. Who would have
thought (except perhaps Benjamin Franklin) that something as simple as a kite may become
a major player in the future of energy production? But “airborne wind turbine technology
“is not a myth. Enjoy the video and company sites linked below.
Wind captured by kites:
http://kitegen.com/ Cool Italian wind kite company (link for English at the right)
http://www.makanipower.com/ Wind Kite company
Video from U.K.
http://www.guardian.co.uk/environment/2008/aug/03/renewableenergy.energy