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Lesson - Wind farms power calculation
Problem Statement: Wind power in Massena: Is it a good or bad thing?
Background
The use of formulas and measurement conversions are an important part of mathematics
on the state exams and in various careers. Formulas vary on the calculation of power for
wind turbines and using the formula given will show students why height, rotor length
along with wind speed are just a few of the things that cause the power calculation to be
complicated. Students will prove that given any formula with variables defined they will
be able to solve it.
Standards
8.PS.4 Observe patterns and formulate generalizations
8.PS.5 Make conjectures from generalizations
8.PS.6 Represent problem situations verbally, numerically, algebraically,
and graphically
8.R.7 Investigate relationships between different representations and
their impact on a given problem
8.R.1 Use physical objects, drawings, charts, tables, graphs, symbols,
equations, or objects created using technology as representations
8.CN.9 Recognize and apply mathematics to other disciplines, areas of
interest, and societal issues
8.CN.6 Recognize and provide examples of the presence of mathematics
in their daily lives
Objectives:
Use of formulas
Understanding of wind power
Collecting data
Convert measurements
Constant verses variable
Reading tables
Time required
At least 3 classes
Materials
Worksheets(2) / Handouts(2)
Computer lab
Wind with Miller program
Model wind turbine
Rulers
Lesson Plan (Day 1)
Introduce wind farms buy playing audio from North Country radio
(http://www.northcountrypublicradio.org/news/newstopics.php?tid=79) and
have pictures(my pictures ―wind farms‖) of wind farms during audio.
(Local wind farms and impact.) 10 minutes
-Have worksheet 1 to fill out during presentation.
-Short discussion
-Go to Wind with Miller
(http://www.windpower.org/en/kids/assign/turbine.htm)
and give worksheet 2 to fill in using the turbine simulator to see differences
in height, wind, and turbines.
Assign homework worksheet 1.14 Bloodhound.
Day 2
Introduce the power formula
P = 0.5 x rho x A x Cp x V3 x Ng x Nb
where:
P = power in watts (746 watts = 1 hp) (1,000 watts = 1 kilowatt)
rho = air density (about 1.225 kg/m3 at sea level, less higher up)
A = rotor swept area, exposed to the wind (m2)
Cp = Coefficient of performance (.59 {Betz limit} is the maximum
thoretically possible, .35 for a good design)
V = wind speed in meters/sec (20 mph = 9 m/s)
Ng = generator efficiency (50% for car alternator, 80% or possibly more for
a permanent magnet generator or grid-connected induction generator)
Nb = gearbox/bearings efficiency (depends, could be as high as 95% if
good)
If there is any single equation that the beginning wind enthusiast should
memorize, this is it.
Discuss the different parts of the turbine (Handout with picture and information.) and
look at what information is the constant and what are the variables in the formula.
Science – air density (use 1.2 Kg/m for Massena)
Area of rotor – (Pi) r2 r = radius of rotor - constant for each turbine
Cp – a guess below .59 All wind power cannot be harnessed
V = wind speed is a variable
Ng – Not all generator are going to be efficient all the time (.6)
Nb – Parts of turbine are not always effient (.8)
Two formulas are used: Area of a circle and the power formula.
Worksheet 3 will be used to calculate power on their own wind mill along with other
turbines given. They will calculate the area of the rotors and the air density based on
numbers for that given day
Assessment
Worksheets will be graded as homework. 10 points for completion.
5 points for partially done.
0 points for not done.
They would be given a test that would check their use of any formulas along with other
math skills that have been covered during the year.
Evaluation
We were unable to initiate the lesson due to teaming issues. The tech teacher just joined
our team and the Science teacher is at a different level. I plan on using the lesson soon
and feel it will reinforce the use of formulas and increase their knowledge of power
issues in the North Country.
Reflection
This lesson can be expanded to the use of graphs and the Excel program. The English
department can expand on this topic also by asking students to choose a side on the
debate of wind power. I feel students will enjoy the hands on aspects and if possible the
whole team could take a field trip to one of the turbines now in the North Country.
Hand out #1
Turbine Info
Modern wind turbine generators are robust, sophisticated high-
tech machines designed to convert the power of the wind into
electricity.
Main Components: The tower, the nacelle (machine house atop
the tower), and the rotor
Height of Flat Rock Wind Turbine Towers: 260 feet (100 m)
Rotor Blade Length: 130 feet (50 m)
Rotor Blade Speed: 14 RPM (revolutions per minute)
How Electricity Leaves the Turbine and Brings Us Power:
Electricity from each 1.65 MW wind turbine generator is fed
through numerous 34.5-kilovolt power underground cables that
come together at the wind farm substation near Rector Road.
These cables channel the electricity via a step-up transformer and
dedicated ten-mile power line into the New York electricity grid at
the 230-kilovolt Niagara Mohawk Adirondack line, feeding power to
towns and cities across New York's North Country and beyond.
Sophisticated computer control systems run constantly to ensure
that the machines are operating efficiently and safely.
Pollution Offset: The American Wind Energy Association
estimates that 1 MW of wind generation capacity is the equivalent
of 1 square mile of new forest, in terms of offsetting or displacing
carbon dioxide from conventional generating sources.
Maple Ridge Wind Farm: Tug Hill, New York
Wind Farm Handout #2
MGE built and operates a wind farm The total cost of the project was $14.5
consisting of 17 turbines (each 660 kW) million. The noise level is 47 decibels at
to provide renewable, Wisconsin-based 800' downwind.
electricity.
Annual Energy Production
23,000,000 kilowatt-hours annually—
enough electricity for roughly 3,300
homes (assumes 98% availability and
Rotors
Three 77' blades (154' rotor diameter)
23% capacity factor). The rated power
constructed of fiberglass reinforced
output is 660 kW at 33 mph (optimal
epoxy and other composite materials.
operating wind speed).
They weigh 7 tons and spin at 28.5
Wind RPMs.
Average wind speed is 13 to 14 mph (at
110 feet aboveground). Wind blows Turbines and rotors are manufactured by
predominantly from a west (from the Vestas Wind Systems A/S, Lem,
northwest through the southwest) Denmark.
direction. The cut-in wind speed is 9
mph and the cut-out wind speed is
56 mph.
Yaw Drive System
Electric motor driven with wind
Towers direction sensor and cable dewinding
Each tower weighs 73.5 tons with a control. Braking accomplished through
height of 213' and diameter of 12' at the friction pads.
base and 6.5' at the top. Tubular steel
tower assembled in three sections with
internal safety ladder and nacelle access.
Gear Box
High-performance planetary/helical.
Towers are manufactured by Beaird
Industries, Shreveport, Generator
LA.Turbines
The nominal voltage is 690 VAC for the
single-speed, variable slip,
Each turbine weighs 20 tons (complete
asynchronous, four-pole induction
nacelle without rotor) and has an
generator.
expected life of 30 years. They are
Vestas V47-660 kW pitch-regulated
wind turbines with OptiSlip and OptiTip. Foundation Design
These are horizontal axis, up-wind Concrete pier 15' to 30' deep with 14'
turbines. external diameter. Anchor bolts attach
OptiSlip generator system allows rotor tower to the foundation.
and generator to vary their revolutions
per minute (RPM) by up to 10% during Control System
wind gusts. This minimizes wear and Computer-controlled, automatic,
improves quality and supply of independent operation and remote
electricity produced. supervision from operations center in
Madison. Microprocessor-based
OptiTip pitch-regulated system controls monitoring of yaw, hydraulic, ambient
blades so they are always tilted to the conditions, rotation, generator, pitch
optimal angle for wind conditions. system, grid, power factor correction,
Optimizes production and minimizes thyristors (generator cut-in) and remote
noise level. monitoring.
Worksheet 1 Name_________________________
Answer questions from audio.
1. Describe how high the wind turbines are. _______________________
2. Describe how wide the blades are. _______________________
3. Describe how long the blades are. _______________________
4. How long does it take to bolt the blades on. _______________
5. Give two reasons stated for having the wind farm.
6. Give two reasons stated for not having the wind farm.
Worksheet 1 Name_________________________
Answer questions from audio.
1. Describe how high the wind turbines are. _______________________
2. Describe how wide the blades are. _______________________
3. Describe how long the blades are. _______________________
4. How long does it take to bolt the blades on. _______________
5. Give two reasons stated for having the wind farm.
6. Give two reasons stated for not having the wind farm.
Worksheet 2 Name_______________________
Wind with Miller
Go to http://www.windpower.org/en/kids/assign/turbine.htm . Scroll to the bottom
of the page and uncheck graph paper and calculator. Chose the turbine in the table
below then use the red arrows to find out how much power will be produced and
write it in the table. (The ? will explain each part if you click on it)
Turbines
Generator Roughness Hub Height Wind Speed Power
Size (kW)/ class (0 – 4) (in meters) (in meters/sec.) (in Kilowatts)
Rotor size
(Diameter)
Bonus 1000 kW 0 70 10
54 m rotor
1 70 10
2 70 10
3 70 10
4 70 10
1 60 5
1 60 10
1 60 15
1 50 15
Vestas 850 kW 0 70 10
52 m rotor
1 70 10
2 70 10
3 70 10
4 70 10
1 60 5
1 60 10
1 60 15
1 50 15
1 40 15
Answer the following questions.
1. Describe the 5 roughness classes used for table.
0___________________________________________________________________
1 ___________________________________________________________________
2 ___________________________________________________________________
3 ___________________________________________________________________
4 ___________________________________________________________________
2. Which class would the Massena area be? Why?
3. What happens when the wind doubles? Does the power double? Why?
4. Does the height make a difference in the power? How much?
5. Describe the differences in the two Turbines on the graph. (Give 3 differences)
Worksheet 3 Name_______________________
Write in the values for the turbines that you constructed in the first column.
Use the formulas for density, area of a circle, and power to calculate the power from the
other turbines given.(Use your calculators)
P = .5 x rho x A x Cp x V3 x Ng x Nb
Rho = Air Density of 1.2 Kg/m for Massena
A = Area of rotor that is exposed to the wind (area of a circle)
So A = Π r2 Where r = radius of the rotor.
Cp , Ng, Nb will be given amounts
The first three rows you will measure your windmill rotor radius in meters(convert cm to
meters) and calculate your possible power wattage.
(Air Rotor Area Amount Wind Generator Turbine Power
density) radius (r) (A) of wind Speed Efficiency Efficiency (in watts)
Rho (in meters) (Cp) (m/sec)
.35 2 .6 .8
.35 4.5 .6 .8
.35 9 .6 .8
30 .35 2 .6 .8
30 .35 4.5 .6 .8
30 .35 9 .6 .8
50 .35 2 .6 .8
50 .35 4.5 .6 .8
50 .35 9 .6 .8
40 .35 4.5 .6 .8
Name one other variable in building a Wind turbine that would effect the power output
(according to ―Wind with Miller) That is not used in the formula.
Name all the variables you can think of that affect how much power a wind turbine
produces. (At least 5)
