RENEWABLE AND CLEAN ENERGY
Wind Turbines
Problem Set Solutions
Problem 1: Using the data on the Vesta V90 – 3.0 MW (turbine diametre = 90 m) in Figure 1, find the
turbine’s efficiency for A) just above the cut-in speed (5m/s), B) the nominal speed (15 m/s) C) the cutout speed (25 m/s) and compare. D) Why does the power output level off at 3,000 kW?
Figure 1. The optimal wind speeds needed to extract varying amounts of power
using a Vesta V90 – 3.0 MV wind turbine.
Solution 1A):
Approach: Find the mass of the air going through the turbine each second, use that mass to find the
kinetic energy and power of the air, and from that, calculate the efficiency.
What we know:
Wind speed = 5 m/s
Diametre of the turbine = 90 m
Actual power output = 250 kW
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Finding the mass:
The volume of the air passing through the turbine per second
The mass of air per second
Finding the power:
The Kinetic energy of this air per second
Finding the efficiency:
Efficiency
Solution 1B):
This is solved in exactly the same way as part A.
Approach: Find the mass of the air going through the turbine each second, use that mass to find the
kinetic energy and power of the air, and from that calculate the efficiency.
What we know:
Wind speed = 15 m/s
Diametre of the turbine = 90 m
Actual power output = 3,000 kW
Finding the mass:
The volume of the air passing through the turbine per second
The mass of air per second
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Finding the power:
The Kinetic energy of this air per second
Finding the efficiency:
Efficiency
Solution 1C):
Approach: Find the mass of the air going through the turbine each second, use that mass to find the
kinetic energy and power of the air, and from that calculate the efficiency.
This is solved in exactly the same way as part A.
What we know:
Wind speed = 25 m/s
Diametre of the turbine = 90 m
Actual power output = 3,000 kW
Finding the mass:
The volume of the air passing through the turbine per second
The mass of air per second
Finding the power:
The Kinetic energy of this air per second
The efficiency:
Efficiency
Solution 1D):
Why does the power output level off at 3,000 kW?
We have to limit the turbine from rotating faster so that there is not too much force on it,
leading to damage.
References:
Vestas. 3.0 MW - An Efficient Way to More Power (Online). http://www.vestas.com/en/wind-powersolutions/wind-turbines/3.0-mw.aspx [22 May 2009].
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Problem 2: Compare the yearly greenhouse gas emissions from 1 GWe (gigawatt-electric) power
stations powered by coal, natural gas and wind turbines. The energy content of coal is 30 MJ/kg, of
natural gas is 55 MJ/kg, and the electrical converting efficiency is 40% for both. Wind turbines do not
emit CO2 as they harness the wind but emissions do occur during the manufacturing of the turbine. For
Canada, the CO2 emission per GDP is approximately 600 Tonnes / $ 1 million of GDP. For wind turbines,
it costs $2 M for a 1 MWe plant, which has a capacity factor of 0.3, and the yearly maintenance is
approximately 2% of the initial cost. Assume the average lifetime of a wind turbine is 20 years.
Solution 2:
Part 1: A coal-fired plant
Approach: Based on the efficiency, we will find the input energy needed. Using the total energy of
coal, we will find the mass of coal needed. Using conversion units, we will find the mass of CO 2
produced.
What we know:
Total energy of coal ≈ 30 MJ/kg
Electrical conversion efficiency ≈ 40%
Finding the input needed in joules:
We want an output of 1 GWe.
GWe input needed
Joules input needed
Finding the mass of coal needed, in kilograms and tonnes:
Mass in kilograms
Mass in tonnes
Finding the mass of CO2 produced.
We know that the conversion from coal to CO2 is C (12 g/mol) + O2  CO2 (44 g/mol)
CO2 produced in tonnes
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Part 2: A natural gas plant
Approach: Based on the efficiency, we will find the input energy needed. Using the total energy of
natural gas, we will find the mass of natural gas needed. Using conversion units, we will find the mass of
CO2 produced.
What we know:
Total energy of natural gas ≈ 55 MJ/kg
Electrical conversion efficiency ≈ 40%
Finding the input needed in joules:
We want an output of 1 GWe.
GWe input needed
Joules input needed
Finding the mass of natural gas needed, in kilograms and tonnes:
Mass in kilograms
Mass in tonnes
Finding the mass of CO2 produced.
We know that the conversion from natural gas to CO2 is CH4 (16 g/mol) + O2  CO2 (44 g/mol) + 2H2O
CO2 produced in tonnes
Part 3: A wind turbine farm
What we know:
It costs $ 2 million for a 1 MWe farm
Capacity factor is 0.3
Maintenance costs 2% of the initial cost per year
For Canada, the CO2 emission per GDP is 600 Tonnes / $ 1 million of GDP.
Finding the yearly cost:
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We want an output of 1 GWe.
GWe input needed
Finding the total cost:
Initial costs
Yearly maintenance costs
Total costs over lifetime of farm
Finding the mass of CO2 produced:
CO2 produced
Conclusion: Each year 0.28 M tonnes of CO2 is emitted. Comparing this to the other power sources, we
find that this is only 3% of the emissions of coal and 7.5% of the emissions of natural gas!
If you compare actual CO2 emissions per kilowatt-hour, you find that wind power produces 5% of the
CO2 that natural gas does and 2% of the CO2 that coal does. Our estimates produced reasonable
answers!
References:
Andrews J, and Jelley N. Energy Science – Principles, Technologies and Impacts. New York, NY: Oxford
University Press, 2007, chapt. 1, p 11.
Problem 3: Design a set of turbines of a reasonable size that could produce 1GW of electrical power
under common wind conditions at a site you can choose in BC (for now you can neglect other factors
such as land formations and wind consistency). You need to specify the number of turbines and land
area needed.
Solution 3:
Approach: A city with one of the highest wind speeds in BC is Summit Lake, located in Northern BC,
which has a mean wind speed of 11.94 m/s. Let’s use the Vesta V90 – 3.0 MW wind turbine (Fig. 1)
operating at approximately 30% efficiency. We will find the power output for one turbine, and then
Physics and Astronomy Outreach Program at the University of British Columbia
determine how many turbines are needed. We will then find the land area so that no wind turbine is
within 5 rotor diametres from another.
What we know:
Mean wind speed: 11.94 m/s
Rotor diametre: 90 m
Finding the power output by one turbine:
Finding the number of turbines required:
Number of turbines needed
Finding the land area needed:
The area, A, needed is found as A = (5 n d) 2.
For 4 turbines (n=1)
For 9 turbines (n=2)
:
For N turbines (
)
So, for 513 turbines...
Area
References:
Environment Canada. Canadian Wind Energy Atlas (Online). http://www.windatlas.ca/en/maps.php [22
May 2009].
Physics and Astronomy Outreach Program at the University of British Columbia
Environment Canada. Canadian Atlas Level 0 (Online).
http://collaboration.cmc.ec.gc.ca/science/rpn/modcom/eole/CanadianAtlas0.html [20 May 2009].
Problem 4: Cape Sutil, located on Vancouver Island, is a potential site for a turbine farm. Assuming that
wind speeds are constant during each season at 8.57 m/s in winter, 5.75 m/s in spring, 3.71 m/s in
3
summer, and 6.64 m/s in fall. Find the mean speed cubed ( v ) and the mean cubed speed (
).
Solution 4:
To find the mean speed cubed, first calculate the
mean speed, then cube it.
To find the mean cubed speed, take the mean of the cubed velocities.
The mean of the cubed speeds is larger than the mean speed cubed. In power calculations, we would
rather use the mean cubed speed as it more accurately reflects the average power which is able to be
harnessed from the wind.
References:
Environment Canada. Canadian Wind Energy Atlas (Online). http://www.windatlas.ca/en/maps.php [22
May 2009]. http://collaboration.cmc.ec.gc.ca/science/rpn/modcom/eole/CanadianAtlas0.html [20 May
2009].
Holt JA, Eaton DJ. Assessment of the Energy Potential and Estimated Costs of Wind Energy in British
Columbia (Online). Garrad Hassan Canada Inc.
http://www.bchydro.com/etc/medialib/internet/documents/info/pdf/rou_wind_garrad_hassan_report.
Par.0001.File.rou_wind_garrad_hassan_report.pdf [25 May 2009].
Brittany Tymos 2009/05/29
Physics and Astronomy Outreach Program at the University of British Columbia