Lesson - Wind farms power calcul by fjzhangweiqun


									              Lesson - Wind farms power calculation
Problem Statement: Wind power in Massena: Is it a good or bad thing?

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.


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


Use of formulas
Understanding of wind power
Collecting data
Convert measurements
Constant verses variable
Reading tables

Time required
At least 3 classes

Worksheets(2) / Handouts(2)
Computer lab
Wind with Miller program
Model wind turbine
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
 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

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

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

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.

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.

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

                   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.
Annual Energy Production
23,000,000 kilowatt-hours annually—
enough electricity for roughly 3,300
homes (assumes 98% availability and
                                             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
                                             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
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)

Generator         Roughness        Hub Height       Wind Speed         Power
Size (kW)/        class (0 – 4)    (in meters)      (in meters/sec.)   (in Kilowatts)
Rotor size
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.
  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)

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