Relationships
When a wind generator is designed various factors are considered
(i)
The wind exposure of the site
(ii)
The length of the blades
(iii)
The number of generators require on the site to guarantee a certain output.
The power generated is related to the area swept by the arm of the windmill and
the speed of the wind.
Relationship 1
How is the area swept out by the arm related to its length?
The area is a circle; the radius is the length of the arm.
The concentric circles on the graph paper below
have radii 0·5 cm, 1 cm, 1·5 cm
2 cm, 2·5 cm, 3 cm, 3·5 cm.
(a)
(b)
Find the area of each by counting square millimeters (hint: try to simplify the
counting process).
[You can find out how many square millimetres makes a
square centimetre by counting too.]
Put your results in the table below.
Radius
(cm)
0·0
0·5
1·0
1·5
2·0
2·5
3·0
3·5
Area
(cm2)
0
(c)
Draw a graph of the data.
The shape of this graph suggests the following table … why?
Radius2
0·0
0·25
1·0
2·25
4·0
6·25
9·0
12·25
Area
0
Complete the table and draw a graph.
The features of the graph that there is a relationship between the radius and
the area, what is it?
(iii)
So A = r2 times a constant.
Use your table and graph to find the value of the constant.
[Hint: If you are using a spreadsheet to help you draw the graphs from tables, add a column
using the formula: = r2 ÷ A.]
(d)
(i)
(ii)
(e)
How many turbines with arm length 4 metres would sweep out the same area as a
turbine with arm length 8 metres?
Supplementary question
If you look at a wind turbine blade rotating it looks slow and harmless.
How fast is the tip of a turbine’s blade moving?
What factors will you need to consider in order to calculate this?
Hint: Some questions you might consider:
(i)
How far does the blade tip travel in one turn?
(ii)
How is this related to the length of the arm?
(iii)
How long does it take to make one turn?
(iv)
At what speed is the tip of a turbine’s blade moving?
Relationship 2
The table below gives the wind energy at different speeds for a turbine
with a 10 m2 sweep.
Wind speed
(Metres per second)
2 (light breeze)
5 (Gentle breeze)
10 (Fresh breeze)
15 (Near gale)
20 (Fresh gale)
25 (Strong gale)
1
(i)
(iii)
Wind energy
(approx)
(Joules per metre2)
5
76
610
2060
4880
9530
Draw a graph of wind speed v wind energy and comment on it.
Investigate a relationship between wind speed and Energy.
Relationship 3
This table below shows how different wind energies produce different amounts of power
(Watts) [based on a wind turbine with a 10 square metre area of sweep]
Wind energy
(Joules per metre2)
5
76
610
2060
4880
9530
(i)
(ii)
(iii)
Power
(Watts)
30
460
3700
12400
29300
57200
Use a graph to help you find a relationship between Energy and Power.
Relate your earlier work relate wind speed to power.
Use your findings to help you work out the energy and power in the wind at the
following wind speeds:
(a) 4 m/s
(b) 8 m/s
(c) 12 m/s
(d) 18 m/s (e) 24m/s
(f) 30m/s
Using the power.
Many ordinary
light bulbs have
a power rating
of 60 watts
A toaster
might have a
power
rating of
600 watts
An electric
kettle may have a
power rating
of 1000 watts
An electric fire
can have a power
rating of
3000 watts
1
A wind turbine has an area of sweep of 10 square metres (i)
How many light bulbs could be powered by a gentle breeze? A fresh breeze?
(ii)
How many toasters could be powered by a gentle breeze?
(iii)
What sort of wind would be needed to power an electric kettle? An electric
fire?
2
If you go round your house and look at the electrical appliances you will find
somewhere on them the power rating (Watts) each one needs.
Make up a table of 10 most used appliances in your house.
Appliance
light bulb
kettle
toaster
fire
TV
Power rating (Watts
60
1000
600
3000
Explain why you would need a wind turbine with an area of sweep greater than 10 square
metres to meet all your power needs.
Discuss your findings with others.
Formulae … Being efficient.
If the turbine took none of the wind’s energy away, it would be just as windy behind the
turbine as in front of it.
0% efficient
If the wind turbine could extract all the energy from the wind then the air would be totally still
immediately behind the turbine … having no energy left.
100% efficient
We know this doesn’t happen. The wind turbine can only extract a percentage of the energy
available.
This percentage is called the efficiency of the turbine.
The theoretical maximum for this is 59%.
This is known as the Betz limit.
Question.
What should you not do anywhere near a wind turbine?
Formulae
Mathematicians, Physicists and Engineers have developed the following formulae that relate
to wind turbines.
1
Area swept = πr2 where r is the arm length of the blade.
2
Energy in wind = 1/2 u3d where u is the wind speed in m/s and d is the air density
measured in kg/m3.
3
Power = Energy × Area × Betz limit.
[The units used are called Joules/second]
In the following questions, use air density, d = 1·23 kg/m3]
1
Derive a formula for calculating the power in terms of u, d and r.
2
(a)
If the wind speed is 10m/s, a fresh breeze, and the density of the air is
1·23 kg/m3, what is the energy in the wind?
(b)
What is the energy in the wind for the following wind speeds?
(i)
Gentle breeze – 5 m/s
(ii)
Strong gale – 25 m/s
(iii)
Strong breeze – 12 m/s
3
What area is swept out by a rotor blade of length
(a)
2m
(b)
3m
(c)
4 m?
4
(a)
A wind turbine has a radius of 4m and the wind is blowing at 10m/s.
What is the maximum theoretical power available?
(b)
The wind speed drops 5m/s.
What is the drop in the maximum theoretical power available?
5
The wind speed is 10m/s, the density of the air is 1·23 kg/m3 and the length of the
rotor is 3m.
Calculate:
(a)
The energy in the wind
(b)
The area swept by the rotor blade
(c)
The power produced if the turbine is only 50% efficient.
(d)
Using the Betx limit, the theoretical maximum po
power
wer that could be produced
6
The world’s largest wind turbine generator is an offshore wind turbine in the North
Sea, situated at sea-level.
It has a rotor blade radius of 63m and the air density is 1·23 kg/m3.
(a)
(b)
(c)
Calculate the theoretical power available from the turbine for a wind
speed of 14m/s
A megawatt is a million watts. Express your answer above in
megawatts.
This turbine is rated at 5,000,000 watts (5 megawatts) for a 14m/s
wind?
Discuss and explain why there might be such a big difference between the theoretical power
and the actual power.