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TRANSCRIPT
A Project on Wind Turbine Energy
Required Calculus I Project 2 – 50 points
Directions:
1. Complete each task and the respective questions 1 – 34. You may skip #29.
2. Use full sentences when answering questions and show all algebraic steps for those requesting math answers.
3. Make sure you include units if appropriate.
4. Make a note of any websites that are no longer visible.
Enjoy!
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/environment/windfarm.aspx
and
http://willowindenergy.com/Why-Wind-/FAQs/#q7
and do the following.
· Renewable Energy
· Sustainability
· Wind Turbines: Just how big are they?
· How much power can a turbine make?
Task 2: Answer the following.
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? The 2009 Annual Report site compares the maximum turbine height to what four things? How does the turbine’s height compare to them?
5. At what speed must a turbine spin in order to first be able to generate electricity?
6. What appears to be the maximum wind speed a turbine can handle before damage occurs?
7. What would this mean for the feasibility of a wind farm being constructed in an area with very heavy sustained winds?
8. 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.
View a nice animation here!
http://www1.eere.energy.gov/wind/wind_animation.html
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.
9. How long is the rotor diameter for a typical 600 KW electrical generator?
10. The largest rotor diameter pictured above is 80 meters. What is the area of the circle swept out with that rotor diameter?
11. 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?
12. Run a quartic regression on your data and comment about the fit of your 4th degree polynomial.
13. 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:
ln
ln
desired
unknownknown
known
height
VelecityVelocity
height
b
b
æö
ç÷
èø
=
æö
ç÷
èø
The value,
b
, is called the roughness constant and has typical values shown in the table below.
Terrain Description
Surface Roughness Length,
b
, (m)
Very smooth, ice or mud
0.00001
Calm open sea
0.0002
Blown sea
0.0005
Snow surface
0.003
Lawn grass
0.008
Rough pasture
0.01
Fallow ground
0.03
Crops
0.05
Few trees
0.1
Many trees, hedges, few buildings
0.25
Forest and woodlands
0.5
Suburbs
1.5
Centers of cities with tall buildings
3.0
The Wind Speed Power Equation:
desired
unknownknown
known
height
VelecityVelocity
height
a
æö
=
ç÷
èø
The exponent,
a
, is called the wind shear exponent. A table of common wind shear exponent values follows;
Terrain Description
Wind Shear Exponent,
a
Smooth, hard ground, lake or ocean
0.10
Short grass on untilled soil
0.14
Level country with foot-high grass
0.16
Tall row crops, hedges, a few trees
0.20
Many trees and occasional building
0.22 – 0.24
Wooded country – small towns and suburbs
0.28 – 0.30
Urban areas with tall buildings
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:
15. Using the Wind Speed Power Equation with
1
7
a
=
calculate the following:
a. Speed at the rotor height.
b. Speed at the lowest blade tip height.
c. Speed at the highest blade tip height.
16. Instead of using the Wind Speed Power Equation repeat number 15 using the Wind Speed Logarithmic Equation with
0.02
b
=
.
17. How do your results compare in parts 1 & 2?
18. 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
19. 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:
2
1
2
kinetic
Emv
=
Where
and
mmassvvelocity
==
.
The mass of flowing air per unit time (mass flow rate) that strikes our rotor is given by;
_( )
massairdensityareaofrotorvelocityAv
r
=××=
If we substitute our mass flow rate into our kinetic energy equation we get the available power of the wind:
(
)
23
11
22
available
PAvvAv
rr
==
Task 5:
Answer the following:
20. What is the affect on the power available if we double our rotor’s radius? What is the percentage increase?
21. What is the affect on the power available if we double our wind speed? What is the percentage increase?
22. It is said “Use as tall a wind turbine as possible.” Explain why that would be the case.
23. Using our Wind Speed Power Equation, show that our power equation from above can be written as;
0
3
3
0
1
2
h
PAv
h
a
r
æö
=
ç÷
èø
Where
00
known speed for height
desired height
wind shear exponent value
vh
h
a
=
=
=
24. 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
a
=
)?
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 the 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;
12
2
vv
mA
r
+
æö
=
ç÷
èø
(1)
Where
r
is the air density,
A
is the area swept out by the rotor, and
12
2
vv
+
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:
(
)
2222
1212
111
222
inout
KEKEmvmvmvv
-=-=-
(2)
Task 6.
25. Using equations (1) and (2) from above, show that the power extracted by the rotor is given by.
(
)
3223
112212
4
extracted
A
Pvvvvvv
r
=-+-
The equation above shows us that the power extracted from the wind is determined by the density of the air (
r
), the area swept out by the rotor (A), and the velocity of the moving air before and after the rotor (
12
&
vv
).
26. 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?
27. 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?
28. Show that the ratio of the power extracted from the wind to the power of undisturbed wind is given by:
(
)
23
222
12
23
111
1
,1
2
extracted
wind
P
vvv
Cvv
Pvvv
æö
==-+-
ç÷
èø
We call
(
)
12
,
Cvv
the “power coefficient” and Betz established that this value is maximal at 16/27. Each turbine design has a power coefficient associated with it.
29. In Calc III we can show that
(
)
12
,
Cvv
is maximized when
1
2
3
v
v
=
and thus
max
16
27
C
=
.
30. There is a simpler method to arrive at our result that Calc I students can employ. Define the variable, t, as follows:
(
)
2
12
1
,
v
tvv
v
=
Show that our above function
(
)
12
,
Cvv
can be written more compactly as:
(
)
(
)
23
1
1
2
extracted
wind
P
Ctttt
P
==-+-
31. Graph the function
(
)
(
)
23
1
1
2
Ctttt
=-+-
where
(
)
01; 01 /* ?
tCtDoyouknowwhy
££££
32. Taking the first derivative, 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:
3
1
2
,
' :
' & : 7585%
extractedp
pp
PCAv
where
CturbinespowercoefficientCBetzLimit
theturbinesmechanicalelectricalefficienc
ies
hr
h
=
=<
=»-
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/wind/resources.html
34. What did you like most about this project? Did you learn some new things?
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