Laboratory A
Question Reservoir
With Answers
Mathematical Voyage
The Dimensions Program
Wind Power Project
Wind Turbines
Figure 1 The Maple Ridge Wind Project – Lowville, New York
Courtesy of DOE/NREL, Credit – Iberdrola Renewables, Inc.
Project Grant Team
Prof. Ellena Reda (Ret.)
Project Designer
Dennis C. Ebersole
Principal Investigator
John S. Pazdar
Program Director
Prof. Annette Benbow
Project Tester
Patricia L. Hirschy
Principal Investigator
Geometry
I.
Dimensional
A. Suppose the area swept by the blades of a wind turbine is 1,250 sq m.
1. If the wind speed it 7 m/s, what is the volume of air that flows through the swept
area in 1 second?
8,750 cu m
2. If the wind speed it 7 m/s, what is the volume of air that flows through the swept
area in 1 minute?
525,000 cu m
3. If the wind speed it 5.4 m/s, what is the volume of air that flows through the
swept area in 1 second?
6,750 cu m
4. If the wind speed it 5.4 m/s, what is the volume of air that flows through the
swept area in 1 minute?
405,000 cu m
B. 1 mile equals 1,609.344 m. Suppose the area swept by the blades of a wind turbine is
1,125 sq m.
1. If the wind speed is 12 mph, what is the volume (in cubic meters) of air that flows
through the swept area in 1 second?
6,035 cu m
2. If the wind speed is 12 mph, what is the volume (in cubic meters) of air that flows
through the swept area in 1 minute?
362,100 cu m
3. If the wind speed is 13.5 mph, what is the volume (in cubic meters) of air that
flows through the swept area in 1 second?
6,790 cu m
4. If the wind speed is 13.5 mph, what is the volume (in cubic meters) of air that
flows through the swept area in 1 minute?
407,400 cu m
Wind Power QR – Lab A - 2
Arithmetic
I.
Operations
A. How many kilowatt-hours of electricity are used when you light 1 50-watt bulb for 20
hours?
1 kWh
B. How many kilowatt-hours of electricity are used when you light 3 150-watt bulbs for
5 hr a day for one week?
15.75 kWh
C. Find the annual number of kilowatt-hours used by the following common household
appliances given the wattages used by that appliance.
1. A clock radio uses 10 watts and is on 24 hr a day.
87.6 kWh
2. A clothes dryer uses 5,000 watts and is used for 3 hr twice a week.
1,560 kWh
3. An electric blanket uses 100 watts and is on for 6 hr each night during the winter
months of December, January and February.
54 kWh
4. A ceiling fan uses 100 watts and is on for 12 hr a day during the summer months
of July and August.
74.4 kWh
5. A personal computer (120 watts) and monitor (150 watts) that is on 12 hr a day
each day of the year.
1,182.6 kWh
Wind Power QR – Lab A - 3
D. The wind turbines have power ratings ranging from 250 watts to 10-kW to 10 MW
annually. The average American household consumes about 10,000 kWh of
electricity each year.
1. Determine the number of American households that can receive power from a 5
MW wind turbine that produces more than 15,000,000 kWh of electricity in one
year.
1,500 Households
2. Determine the number of households that can receive power from a 250 – kW
wind turbine that produces 350,000 kWh of electricity per year.
35 Households
3. If a 3 MW wind turbine can provide enough power to supply approximately 1,300
homes, how many homes will receive power from a 1.5 MW wind turbine?
Round your answer to the nearest home.
650 Households
II.
Conversions
A. In order for a wind turbine to work efficiently, wind speeds usually must be above 12
to 14 mph.
1. Convert a 12 mph wind to meters per second.
≈ 5.364 m/s
2. Convert a 13 mph wind to meters per second.
≈ 5.812 m/s
3. Convert a 14 mph wind to meters per second.
≈ 6.259 m/s
4. Convert 5 m/s to mph. Will a wind turbine work efficiently at 5 m/s
≈ 11.185 mph
will probably not work efficiently
5. Convert 13 m/s to mph. Will a wind turbine work efficiently at 13 m/s?
≈ 29.080 mph
will work efficiently
6. Convert 20 m/s to mph. Will a wind turbine work efficiently at 20 m/s?
≈ 44.739 mph
will work efficiently
Wind Power QR – Lab A - 4
B. A kilowatt-hour (kWh) means one kilowatt (1,000 watts) of electricity produced or
consumed for one hour.
1. Convert 1,500 watts to kW.
1.5 kW
2. Convert 38,000 watts to kW.
38 kW
3. Converts 108 kW to watts.
108,000 W
4. Convert 23 kW to watts.
23,000 W
5. Convert 234,000 watts to MW.
0.234 MW
6. Convert 56 kW to MW.
0.056 MW
7. Convert 4 MW to watts.
4,000,000 W
8. Convert 16MW to kW.
16,000 kW
9. The average American household used approximately 10,000 kW annually.
Convert 10,000 kW to MW.
10 MW
Wind Power QR – Lab A - 5
C. In order for a wind turbine to work efficiently, wind speeds usually must be above 12
to 14 miles / hour.
1. Convert a 12 mph wind to meters per second.
≈ 5.364 m/s
2. Convert a 13 mph wind to meters per second.
≈ 5.812 m/s
3. Convert a 14 mph wind to meters per second.
≈ 6.259 m/s
4. Convert 5 m/s to mph. Will a wind turbine work efficiently at 5 miles / second?
≈ 11.185 mph
will probably not work efficiently
5. Convert 13 m/s to mph. Will a wind turbine work efficiently at 13 miles /
second?
≈ 29.080 mph
will work efficiently
6. Convert 20 m/s to mph. Will a wind turbine work efficiently at 20 miles /
second?
≈ 44.739 mph
will work efficiently
Wind Power QR – Lab A - 6
D. A kilowatt-hour (kWh) means one kilowatt (1,000 watts) of electricity produced or
consumed for one hour.
1. Convert 1500 watts to kW.
1.5 kW
2. Convert 38000 watts to kW.
38 kW
3. Converts 108 kW to watts.
108,000 W
4. Convert 23 kW to watts.
23,000 W
5. Convert 234000 watts to MW.
0.234 MW
6. Convert 56 kW to MW.
0.056 MW
7. Convert 4 MW to watts.
4,000,000 W
8. Convert 16MW to kW.
16,000 kW
9. The average American household used approximately 10,000 kW annually.
Convert 10,000 kW to MW.
10 MW
10. How many kilowatt-hours of electricity are used when you light 1 50-watt bulb
for 20 hours?
1 kWh
11. How many kilowatt-hours of electricity are used when you light 3 150-watt bulbs
for 5 hr a day for one week?
15.75 kWh
Wind Power QR – Lab A - 7
12. Find the annual number of kilowatt-hours used by the following common
household appliances given the wattages used by that appliance.
a. A clock radio uses 10 watts and is on 24 hr a day.
87.6 kWh
b. A clothes dryer uses 5,000 watts and is used for 3 hr twice a week.
1,560 kWh
c. An electric blanket uses 100 watts and is on for 6 hr each night during the
winter months of December, January and February.
54 kWh
d. A ceiling fan uses 100 watts and is on for 12 hr a day during the summer
months of July and August.
74.4 kWh
e. A personal computer (120 watts) and monitor (150 watts) that is on 12 hr a
day each day of the year.
1,182.6 kWh
Wind Power QR – Lab A - 8
Laboratory B
Question Reservoir
With Answers
Mathematical Voyage
The Dimensions Program
Wind Power Project
Nacelle
Figure 1 Rotor Hub and Nacelle Ready for Installation
http:://www.nacelleparts.com
Project Grant Team
Prof. Ellena Reda (Ret.)
Project Designer
Dennis C. Ebersole
Principal Investigator
John S. Pazdar
Program Director
Prof. Annette Benbow
Project Tester
Patricia L. Hirschy
Principal Investigator
Geometry
I.
Dimensional
A. Small wind turbines intended for residential or small business have a rotor diameter
of 25 ft or less. Determine the circumference of the circle formed by the blades as
they spin, in feet and meters.
78.540 ft
23.939 m
B. The largest wind machines in the world have blades longer than a football field (100
yd). Determine the circumference of the circle formed by the blades as they spin, in
feet and meters.
629.261 ft
191.799 m
C. The Mitsubishi MWT-600 model wind turbine has a blade length of 20.4 m.
Determine the circumference of the circle formed by the blades as they spin, in
meters and feet.
128.805 m
422.590 ft
D. The Mitsubishi MWT-100 model wind turbine has a blade length of 26.8 m.
Determine the circumference of the circle formed by the blades as they spin, in
meters and feet.
169.646 m
556.581 ft
E. Mariah Power, located in Reno, Nevada, has introduced a new vertical wind turbine
called the Windspire. This machine stands only 30 ft tall and has a diameter of 4 ft.
Determine the circumference of this new style wind turbine in feet and meters.
12.566 ft
3.830 m
Wind Power QR – Lab B - 2
F. Use this simplified diagram of a wind turbine blade and hub to answer questions
1 – 4.
1. Given the following blade dimensions: tip 3 in., end of blade 6 in., trailing edge
6.01 ft, and leading edge 6 ft.
a. Determine the perimeter of the blade, in feet.
12.76 ft
b. Determine the area of the blade, in square feet.
2.25 sq ft
2.
Suppose the hub has a diameter of 4 in.
a. Determine the circumference and the area of the hub.
circumference 12.566 in
area 12.566 sq in
b. Find the distance, in feet, the end of the 6 ft blade traverses in one revolution
(the circumference of the circle it forms)?
area 38.746 sq ft
3. Given the following blade dimensions: tip 7.5 cm, end of blade 15 cm, trailing
edge 1.802 m, and leading edge 1.8 m.
a. Determine the perimeter of the blade.
3.827 m
b. Determine the area of the blade.
0.203 sq m
4. Suppose the hub has a diameter of 26 cm.
a. Determine the circumference and the area of the hub.
circumference 81.681 cm
area 530.929 sq cm
b. Find the distance the end of the 1.8 m blade traverses in one revolution (the
circumference of the circle it forms)?
area 530.929 sq cm
Wind Power QR – Lab B - 3
5. Given the following blade dimensions: tip 3 in., end of blade 7.5 in., trailing edge
10.01 ft, and leading edge 10 ft.
a. Determine the perimeter of the blade.
20.885 ft
b. Determine the area of the blade.
4.375 sq ft
6. Suppose the hub has a diameter of 5.5 in.
a. Determine the circumference and the area of the hub.
circumference 17.278 in
area 23.758 sq in
b. Find the distance, in feet, the end of the 10 ft blade traverses in one revolution
(the circumference of the circle it forms).
area 23.758 sq in
7. Given the following blade dimensions: tip 25 cm, end of blade 55 cm, trailing
edge 64.001 m, and leading edge 64 m.
a. Determine the perimeter of the blade.
128.801 m
b. Determine the area of the blade.
25.6 sq m
8. Suppose the hub has a diameter of 3 m. Determine the circumference and the area
of the hub.
circumference 9.425 m
area 7.068 sq m
Wind Power QR – Lab B - 4
Arithmetic
I.
Operations
A. The usual hub height is around 1 to 1.2 times the rotor diameter.
1. The MWT-600 model turbine has a blade length of 20.4 m, hub diameter of 0.04
m, and a hub height of 45 m. Is the hub height in 1 to 1.2 times the rotor
diameter?
Yes, the hub height is approximately 1.1 times the rotor diameter.
2. The MWT-600 model turbine has a blade length of 20.4 m and a hub height
of 60 m. Is the hub height 1 to 1.2 times the rotor diameter?
No, the hub height is not approximately 1 to 1.2 times the rotor diameter.
3. The MWT-100 model turbine has a blade length of 26.8 m, hub diameter of 0.04
m, and a hub height of 45 m. Is the hub height in 1 to 1.2 times the rotor
diameter?
No, the hub height is not approximately 1 to 1.2 times the rotor diameter.
4. The MWT-100 model turbine has a blade length of 26.8 m, hub diameter of 0.04
m, and a hub height of 69 m. Is the hub height in 1 to 1.2 times the rotor
diameter?
No, the hub height is not approximately 1 to 1.2 times the rotor diameter.
Wind Power QR – Lab B - 5
II.
Conversions
A. Wind turbines come in various sizes.
1. A utility-scale turbine can range in size from about 160 ft to about 300 ft.
a. Change 160 ft to meters. Round your result to the nearest thousandths.
48.768 m
b. Change 300 ft to meters. Round your result to the nearest thousandths.
91.440 m
2. A wind turbine has a total height from the tower base to the tip of the rotor of
approximately 442 ft. What is the total height of the wind turbine in meters?
134.722 m
3. Small wind turbines intended for residential or small business have a rotor
diameter of 25 ft or less.
a. Change 25 ft to meters.
7.62 m
b. If the rotor diameter is 25 ft what is the length of one rotor blade? Give your
answer in both feet and meters.
3.81 m
12.5 ft
B. The largest wind machines in the world have blades longer than a football field (100
yards).
1. Change 100 yd to meters.
91.44 m
2. Change 100 yd to feet.
300 ft
C. The Mitsubishi MWT-600 model wind turbine has a blade length of 20.4 meters.
1. Change 20.4 meters to feet.
66.929 ft
2. Determine the rotor diameter for the MWT-600 in both meters and feet.
41 m
134.514 ft
Wind Power QR – Lab B - 6
D. The Mitsubishi MWT-100 model wind turbine has a blade length of 26.8 m.
1. Change 26.8 m to feet.
87.927 ft
2. Determine the rotor diameter for the MWT-100 in both meters and feet, if the hub
is 0.4 m.
54 m
177.165 ft
E. The Darrieus vertical wind turbine typically stands 100 ft tall and 50 ft wide.
Change these dimensions to meters.
Tall 30.48 m
Wide 15.24 m
F. Mariah Power located in Reno, Nevada has introduced a new vertical wind turbine
called the Windspire. This machine stands only 30 ft tall and has a diameter of 4 ft.
1. Change 30 ft to meters
9.144 m
2. Change 4 ft to meters.
1.219 m
Wind Power QR – Lab B - 7
G. Use this simplified diagram of a wind turbine blade and hub to answer questions
1 – 4.
1. Given the following blade dimensions: tip 3 in., end of blade 6 in., trailing edge
6.01 ft, and leading edge 6 ft. Convert all of the dimensions to feet.
3 1
tip
= ft or 0.25 ft
12 4
6 1
blade end
= ft or 0.50 ft
12 2
2. Given the following blade dimensions: tip 7.5 cm, end of blade 15 cm, trailing
edge 1.802 m, and leading edge 1.8 m. Convert all of the dimensions to meters.
tip 0.075 m
blade end 0.15 m
3. Given the following blade dimensions: tip 3 in., end of blade 7.5 in., trailing edge
10.01 ft, and leading edge 10 ft. Convert all of the dimensions to feet.
3 1
tip
= ft or 0.25 ft
12 4
7.5 625 5
blade end
=
= ft or 0.625 ft
12 1000 8
4. Given the following blade dimensions: tip 25 cm, end of blade 55 cm, trailing
edge 64.001 m, and leading edge 64 m. Convert all of the dimensions to meters.
tip 0.25 m
blade end 0.55 m
Wind Power QR – Lab B - 8
III.
Ratios & Proportions
A. The usual hub height is around 1 to 1.2 times the rotor diameter.
1. The MWT-600 model turbine has a blade length of 20.4 m, hub diameter of 0.04
m, and a hub height of 45 m. What is the ratio of hub height to rotor diameter?
45 1.1
≈
40.8 1
2. The MWT-600 model turbine has a blade length of 20.4 m and a hub height
of 60 m. What is the ratio of hub height to rotor diameter?
60
15 1.5
=
≈
40.8 10.2 1
3. The MWT-100 model turbine has a blade length of 26.8 m, hub diameter of 0.04
m, and a hub height of 45 m. What is the ratio of hub height to rotor diameter?
45
0.8
≈
53.6
1
4. The MWT-100 model turbine has a blade length of 26.8 m, hub diameter of 0.04
m, and a hub height of 69 m. What is the ratio of hub height to rotor diameter?
69 1.3
≈
53.6 1
Wind Power QR – Lab B - 9
Laboratory C
Question Reservoir
With Answers
Mathematical Voyage
The Dimensions Program
Wind Power Project
Power
Figure 1 Wind Turbine Power Diagram
http://wind-turbine.tripod.com/1d1.3.html
Project Grant Team
Prof. Ellena Reda (Ret.)
Project Designer
Dennis C. Ebersole
Principal Investigator
John S. Pazdar
Program Director
Prof. Annette Benbow
Project Tester
Patricia L. Hirschy
Principal Investigator
Geometry
I.
Dimensional
A. Being able to measure the swept area of your wind turbine is essential if you want to analyze
the efficiency of your machine.
1. Calculate the swept area of these five different rotor blade lengths, if the hub radius is 1
foot:
Blade length
Swept area
Blade
length
Swept
area
3 ft
5 ft
10 ft
12 ft
13 ft
3 ft
5 ft
10 ft
12 ft
13 ft
50.265
sq ft
113.097
sq ft
380.133
sq ft
530.929
sq ft
615.752
sq ft
2. Convert the blade lengths in problem 1 to meters, and then calculate the swept area, if the
hub radius is 0.3 meters.
Blade length
3 ft = ? m
5 ft = ? m
10 ft = ?m
12 ft = ? m
13 ft = ?m
Blade length
≈ 0.914 m
1.524 m
3.048 m
≈ 3.658 m
≈3.962 m
Swept area
4.630
sq m
10.452
sq m
35.214
sq m
49.215
sq m
57.066
sq m
Swept area
3. Calculate the swept area of a wind turbine with the following rotor diameters.
Rotor diameter
4ft
7ft
12ft
14ft
20ft
Swept area
Rotor
diameter
4ft
7ft
12ft
Wind Power QR – Lab C - 2
14ft
20ft
4. Convert the rotor diameter in problem 3 to meters, and then calculate the swept area.
Rotor
diameter
Swept
area
Rotor
diameter
Swept
area
4ft = ?m
≈1.219 m
11.167 sq m
7ft = ?m
12ft = ?m
14ft = ?m
20ft = ?m
≈2.134m
≈3.658 m
≈4.267 m
6.096 m
3.577 sq m
10.509 sq m
14.300 sq m
29.18 sq m
5. What impact does doubling the rotor diameter have on the wind swept area?
Doubling the rotor diameter increases the
wind swept area by a factor of four.
B. The amount of electricity (in watts) that a turbine is able to produce depends on the diameter
of the rotor and the speed of the wind that propels the rotor. To determine total potential
power use the formula P = 0.5 ⋅ ρ ⋅ A ⋅ V 3 where ρ = 1.225 , A is the swept area in square
meters, and V is the wind velocity in meters per second.
1. The following steps will determine the power of a small wind turbine with a rotor
diameter of 5 ft in a 10 mph wind. Determine the wind swept area, in square meters.
1.824 sq m
2. The following steps will determine the power of a small wind turbine with a rotor
diameter of 5 ft in a 20 mph wind. Determine the wind swept area, in square meters.
1.824 sq m
3. Determine the power of a small wind turbine with a rotor diameter of 10 ft in a 12 mph
wind. Determine the wind swept area.
7.297 sq m
4. Abundant Renewable Energy (www.abundantre.com) markets the ARE442 wind
generator with a rotor diameter of 7.2 m. Determine the wind swept area of the ARE442.
40.715 sq m
5. DC Power Systems (www.dcpower-systems.com) markets the Kestrel Wind Turbines.
Model E220 has a rotor diameter of 2.2 m.
7857.380 w
Wind Power QR – Lab C - 3
Arithmetic
I.
Operations
A. The amount of electricity (in watts) that a turbine is able to produce depends on the diameter
of the rotor and the speed of the wind that propels the rotor. To determine total potential
power use the formula P = 0.5 ⋅ ρ ⋅ A ⋅ V 3 where ρ = 1.225 , A is the swept area in square
meters, and V is the wind velocity in meters per second.
1. What impact does doubling the wind speed have in the amount of available power?
Doubling the wind speed increases the
amount of available power by a factor of 8.
2. Determine the monthly power generated by a small wind turbine with a blade length
diameter of 5 ft if the average wind speed is 10 mph.
5,172.714 w
3. Find the average power generated per day in a given month for the following wind
speeds if your rotor diameter is 20 ft long: five days of the month the wind speed is 15
mph, ten days of the month the wind speed is 9 mph and fifteen days of the month you
have a wind speed of 12 mph.
2,429.007 w
Wind Power QR – Lab C - 4
II.
Conversions
A. The amount of electricity (in watts) that a turbine is able to produce depends on the diameter
of the rotor and the speed of the wind that propels the rotor. To determine total potential
power use the formula P = 0.5 ⋅ ρ ⋅ A ⋅ V 3 where ρ = 1.225 , A is the swept area in square
meters, and V is the wind velocity in meters per second.
1. The following steps will determine the power of a small wind turbine with a rotor
diameter of 5 ft in a 10 mph wind.
a.
Convert 5 ft to meters.
1.524 m
b.
Convert 10 mph to miles / second.
4.47 m/s
c.
Find P, in watts.
99.782 w
2. The following steps will determine the power of a small wind turbine with a rotor
diameter of 5 ft in a 20 mph wind.
a.
Convert 5 ft to meters.
1.524 m
b.
Convert 10 mph to miles / second.
8.941 m/s
c.
Find P, in watts.
798.258 w
3. Determine the power of a small wind turbine with a rotor diameter of 10 ft in a 12 mph
wind.
a.
Convert 10 ft to meters.
3.048 m
b.
Convert 12 mph to miles / second.
5.364 m/s
c.
Find P, in watts
689.790 w
Wind Power QR – Lab C - 5
Laboratory D
Question Reservoir
With Answers
Mathematical Voyage
The Dimensions Program
Wind Power Project
Freestanding
Figure 1 Wind Farm – West of Mason City, Iowa
Courtesy of DOE/NREL, Credit – Todd Spink
Project Grant Team
Prof. Ellena Reda (Ret.)
Project Designer
Dennis C. Ebersole
Principal Investigator
John S. Pazdar
Program Director
Prof. Annette Benbow
Project Tester
Patricia L. Hirschy
Principal Investigator
Geometry
I.
Dimensional
A. Bergey Windpower manufactures the BWC XL.1 Tilt Tower. The tower sections are
4.5 inches in diameter that have been molded at one end so that they fit into each
other to form a cylinder. Each tower section is 10 ft in length.
1. Determine the volume of a wind turbine tower that is 60 ft high.
26.507 cu ft
2. Determine the volume of a wind turbine tower that uses 10 tower sections.
44.179 cu ft
B. Determine the volume of a tower for a wind turbine that is formed using two
cylindrical sections. The pipe used for the tower comes in 21 ft sections, so the tower
itself is made from two lengths of pipe. The bottom section has a diameter of 12
inches while the upper section has a diameter of 6 inches.
82.467 cu ft
C. Beaird Industries of Shreveport, LA manufactures wind turbine towers that are
truncated cones, have a height of 213 ft with a diameter of 12 ft at the base and 6.5 ft
at the top. The associated cone would have a height of 252 ft.
1. Determine the volume of the cone with base diameter 12 ft and height 252 feet.
8,482.3 cu ft
2. Determine the volume of the cone with base diameter 6.5 ft and height
252 - 213 = 39 ft.
431.380 cu ft
3. Determine the volume of the tower using your answers from Parts 1 and 2.
8,050.920 cu ft
Winder Power QR – Lab D - 2
D. Given Table 1.
Table 1
(3-piece tower)
Section / No
Top
/3
Intermediate / 2
Bottom
/1
Top End
(Diameters in mm)
2,316
2,821
3,488
Bottom End
(Diameters in mm)
2,821
3,488
4,190
1. Find the circumference for the top of each section in Table 1.
Table 1
All circumference answers are rounded to thousandths.
Top End
Top End
Section / No
(Diameters in mm)
(Circumference in mm)
Top
/3
2,316
7,275.929
Intermediate / 2
2,821
8,862.433
Bottom
/1
3,488
10,957.875
2. Find the area for the bottom of each section in Table 1.
Table 1
Section / No
Top
/3
Intermediate / 2
Bottom
/1
Bottom End
(Diameters in mm)
2,821
3,488
4,190
Winder Power QR – Lab D - 3
Bottom End
(Area in mm)
6,250,230.786
38,221,068.61
13,788,528.7
E. A wind generator foundation must be very strong. Pouring a concrete foundation that
extends down below the frost line is highly recommended.
1. A wind turbine assembly with a 30 ft freestanding pole that does not use guy
wires can have a support base (or foundation) as small as 7 feet by 10 feet and 3
feet deep. Find the volume of the concrete foundation.
210.0 cu m
2. Concrete piers are sometimes used as a foundation for wind turbines. Depending
on soil conditions these concrete piers are 15 ft to 30 ft deep with a 14 ft diameter.
Find the range in the volume of concrete needed to construct such a pier
foundation
The range in volume is 2,309.071 cu ft — 4,618.141 cu ft
3. Wind turbine foundations are generally octagonal in shape. Suppose a foundation
is built that is a regular octagon with each side measuring 4 ft and having a depth
of 2.5 ft. Find the amount of concrete needed in order to provide a stable base for
the turbine.
193.6 cu ft
Winder Power QR – Lab D - 4
F. Madison Gas and Electric, located in Madison, Wisconsin, operate a wind farm
with 17 wind turbines. The towers for these wind turbines have a height of 213 ft.
The diameter at the base of the tower is 12 ft and at the top of the tower it is 6.5 ft.
1. Find the circumference at the base and the top of the tower.
Botton ≈ 20.404 ft
Top ≈ 37.699 ft
2. Find the areas of the two bases at bottom and top of tower.
Botton ≈ 113.097 sq ft
Top ≈ 33/183 sq ft
3. The tower is a truncated cone. If the cone were completed, its height would be
252 ft. Find the volume of this cone with base diameter 12 feet.
≈ 9,500.176 cu ft
4. Find the volume of the small cone that would be at the top of the tower if the
cone had been completed. (The diameter is 6.5 ft and the height is the
difference between 252 ft and the height of the truncated cone.)
≈ 2,787.379 cu ft
5. Find the volume of the truncated cone by finding the difference of the volumes of
the two cones found in Questions 3 and 4 above.
≈ 6,712.798 cu ft
Winder Power QR – Lab D - 5
Laboratory E
Question Reservoir
With Answers
Mathematical Voyage
The Dimensions Program
Wind Power Project
Guyed
Figure 1 Guy Wire Wind Turbine – Flagstaff, Arizona
Courtesy of DOE/NREL, Credit – Robert Slack
Project Grant Team
Prof. Ellena Reda (Ret.)
Project Designer
Dennis C. Ebersole
Principal Investigator
John S. Pazdar
Program Director
Prof. Annette Benbow
Project Tester
Patricia L. Hirschy
Principal Investigator
Geometry
I.
Dimensions
A. Wind turbines may be supported using three or four guy wires anchored to a
foundation. If four guy wires are used, a
square-based pyramid is formed as shown in
the figure. If three guy wires are used, a
triangular pyramid is formed.
Find the area of the square base of the pyramid
formed for each of the towers listed if four guy
wires are used. (Hint: The length of a side of
the square is approximately 1.4 times the guy
wire radius.)
Guy
wire
radius
1. The Windspire wind turbine that stands 30 feet high.
992.25 sq ft
2. The ARE442 wind turbine that stands 64 feet high.
4,515.84 sq ft
3. The Mitsubishi MWT-600 that stands 45 m high. Express the answer in feet.
24,030.952 sq ft
4. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answer in feet.
2,670.058 sq ft
B. Find the volume of space inside the pyramid formed by the guy wires used to support
each of the wind turbine towers.
1. The Windspire wind turbine that stands 30 feet high.
7,938 cu ft
2. The ARE442 wind turbine that stands 64 feet high.
77,070.336 cu ft
3. The Mitsubishi MWT-600 that stands 45 m high. Express the answer in feet.
945,098.580 cu ft
4. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answer in feet.
35,040.061 cu ft
Wind Power QR – Lab E - 2
C. If three guy wires are used, the base of the pyramid is an equilateral triangle.
Find the area of the triangular base of the pyramid formed for each of the towers
when three guy wires are used.
1. The Windspire vertical wind turbine that stands 30 feet high.
658.125 sq ft
2. The ARE442 wind turbine that stands 64 feet high.
2,995.2 sq ft
3. The Mitsubishi MWT-600 that stands 45 m high. Express the answers in square
feet.
15,938.897 sq ft
4. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answers in square feet.
1,770.957 sq ft
D. Find the volume of space inside the triangular pyramid formed by the guy wires used
to support each of the wind turbine towers.
1. The Windspire vertical wind turbine that stands 30 feet high.
5,265 cu ft
2. The ARE442 wind turbine that stands 64 feet high.
51,118.08 cu ft
3. The Mitsubishi MWT-600 that stands 45 m high. Express the answers in cubic
feet.
627,514.375 cu ft
4. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answers in cubic feet.
23,240.859 cu ft
Wind Power QR – Lab E - 3
E. A lattice tower of a wind turbine structure creates a pyramid shape. A pyramid has a
base and triangular sides that rise to meet at the same point. The base of the wind
turbine may create a triangle (as in the case of a triangular pyramid lattice structure).
Determine the volume of an equilateral triangle pyramid lattice tower with
1. A height of 30 m and guy wire radius length of 2.4 m.
224.64 cu m
2. A height of 65 m and guy wire radius length of 5.5 m.
2,556.125 cu m
F. A lattice tower of a wind turbine structure creates a pyramid shape. A pyramid has a
base and triangular sides that rise to meet at the same point. The base of the wind
turbine may create a square (as in the case of a square pyramid lattice structure).
Determine the volume of a square pyramid lattice tower with
1. A height of 36 m and a bottom lattice length (side of the square-base) of 3 m.
324 cu m
2. A height of 100 m and a bottom lattice length of 7.5 m.
5,625 cu m
Wind Power QR – Lab E - 4
Arithmetic
I.
Operations
A. To supply the desired amount of support and keep the stresses on the tower to a
minimum, the guy lines should extend as far as possible around the tower. A good
rule of thumb is to have the guy wire radius (the distance from the center of the base
to where the guy line is anchored) be three-quarters of the tower height. Determine
the guy wire radius for each of the following tower heights.
1. The Windspire vertical wind turbine that stands 30 ft high.
22.5 ft
2. The ARE442 wind turbine that stands 64 ft high.
48.0 ft
B. Find the height at which the guy wires are attached to each of the towers listed if they
are attached at a fixed point that is 80% of the tower height. Express the answer in
feet. Determine the guy wire radius for each of the following tower heights.
1. The Windspire vertical wind turbine that stands 30 ft high.
24.0 ft
2. The ARE442 wind turbine that stands 64 ft high.
51.2 ft
Wind Power QR – Lab E - 5
II.
Conversions
A. To supply the desired amount of support and keep the stresses on the tower to a
minimum, the guy lines should extend as far as possible around the tower. A good
rule of thumb is to have the guy wire radius (the distance from the center of the base
to where the guy line is anchored) be three-quarters of the tower height. Determine
the guy wire radius for each of the following tower heights.
1. The Mitsubishi MWT-600 that stands 45 m high. Express the answers in feet.
110 728 ft
2. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answers in feet.
36.909 ft
B. Find the height at which the guy wires are attached to each of the towers listed if they
are attached at a fixed point that is 80 % of the tower height. Determine the guy wire
radius for each of the following tower heights.
1. The Mitsubishi MWT-600 that stands 45 m high. Express the answers in feet.
118.11 ft
2. The Kestrel Wind Turbine Model E220 that has a tower height of 15 m. Express
the answers in feet.
39.37 ft
Wind Power QR – Lab E - 6