Physics Factsheet
www.curriculum-press.co.uk
Number 112
Practicalities Of Domestic Wind Energy Generation
Generating your own electricity using a private wind turbine is now
a popular project. Is it good idea, and how practical is it?
•
If our average consumption is 4450 kWh per year, since there are
365 x 24 = 8760 hours in one year, we use 4450/8760 or about 0.5 kW
or 500 watts all the time. Obviously this fluctuates according to the
time of day. Do you know which appliances in your home use less
than 500 watts and which contribute most to your electricity bill?
Britain is a very windy country. We get 40% of Europe’s wind
energy, yet only 0.5% of our electricity comes from the wind.
The wind is renewable and free (although building and
maintaining your wind turbine can be expensive), so should
save money in the long term.
•
Payback time is time taken to save money you spent
installing the wind turbine. The savings come from electrical
energy you generated yourself rather than buying it from the
Electricity Board. 3 months is very good; 70 years far too
long.
•
An electric fence of 12 V has a current of 400 mA (0.4 A). Its
power is I × V watts = 4.8 watts; very low compared to some
appliances in the home.
How much energy is stored in the wind?
This depends, critically, on how hard the wind is blowing. The
Beaufort scale is no use when it comes to calculations. A fresh
breeze has a wind speed of about 10 m s-1, and a gale that may
damage your home about 20 m s-1. The density of air is 1.2 kg m-3.
At present, large power stations, mostly burning fossil fuels,
generate our electricity. A lot of heat energy is wasted, and the
efficiency cannot be improved unless the heat is used close to
the power station. More heat is lost as the electricity travels
many miles to our homes, and we consume usefully about 20 –
30% of the original energy stored in the fuel. We pay for all the
waste energy in our electricity bill, and pollute the atmosphere,
heat up the rivers, contribute to global warming etc. at the same
time. Producing our own electricity would help save the
environment as well as money.
A
v
1. The cylinder of length v contains all the air that will pass through
area A in one second, where v is the velocity of the wind.
2. Its volume is A × v,
3. Hence its mass is A × v × d where d is the density.
1. Velocity is the distance something moves in one second,
(in a particular direction since velocity is a vector).
2. Volume = base area x height.
3. Density = mass / volume, so mass = volume x density.
4. Power is the rate at which energy is transferred. Power =
Energy/time. 1 watt = 1 joule/second. 1 kWatt = 1000
joules/second.
Efficiency is the fraction of the original energy changed
into useful energy (in this case electrical). In the above example
it’s 0.2 – 0.3, also expressed as 20 – 30%.
In this fact sheet, we shall look at some of the practical details of
generating electricity yourself from the wind.
How much energy do we need?
NB More air will pass through area A in one second if v is larger,
(the cylinder is longer) and you will also get more energy if A is
larger. Suppose A is a circle of radius 1.6 m, (a value used for some
wind turbines).
A = π × r2 = 8.0 m2
Volume passing through cylinder at wind velocity 10 m s-1 = A × v = 80 m3 s-1
Mass of air passing through cylinder = A × v × d = 80 × 1.2 = 96 kg s-1
So kinetic energy of air passing through in one second
= ½ m v2 = 0.5 × 96 × 102 = 4800 J s-1 or 4800 watts (4.8 kW).
This is the power in 8.0 m2 of wind.
Do we want our wind turbine to power the lights for a caravan, run
an electric fence for keeping animals in, or make a significant
difference to our domestic electricity bill? We need to estimate how
much energy we want. Do you know how much your home or
school uses in one year? One estimate is that the average UK home
uses 4450 kWh per year. Check your electricity bills and see how
they match up.
Remember this from GCSE?
•
•
•
The Electricity Board uses kilowatt-hours as energy units because joules are too small to be practical. A joule is one watt-second.
(Energy used = power × time). There are 1000 × 3600 joules in a kWh, or 3.6 × 106 joules.
Gemma watches 2 hours of TV every night, 6 days a week, 4 on Sundays. How many kWh does she use in one year? Her TV is rated
at 72 Watts; 72/1000 or 0.072 kW, and she uses it for 16 hours a week, or 16 × 52 = 832 hours per year. 0.072 × 832 = 60 kWh per year.
So Gemma’s TV habit is 60/4450 x 100, 1.3% of her family’s electricity consumption.
Asif is training to be a chef in his parents’ restaurant. He often cooks a meal using an electric hob of 2000 Watts for about 20 minutes
each night. How many kWh does he use in one year? The hob is 2 kW, and the number of hours is 365/3. So Asif clocks up 243 kWh,
which is 243/4450 × 100 = 5.5% of his family’s consumption.
1
Physics Factsheet
112. Practicalities Of Domestic Wind Energy Generation
For a wind velocity of 20 m s-1 you will get 38 400 watts, which is 8 times greater, because the wind velocity which has doubled appears in
the calculation 3 times. There is twice as much air, with four times as much kinetic energy per kg. The full formula is ½ Adv3, so the power
available and damage the wind can cause are proportional to the cube of the wind velocity. Only a small difference in the wind speed makes
a big difference to the power available.
Fig 1 Wind power in kw for 8 m2 wind turbine
If the 10 m s-1 breeze blew constantly, the wind’s 4.8 kW is nearly 10
times as much power as our home’s average requirements. The
wind does not blow equally hard all the time, but does blow more
strongly in winter when we need more energy. Also the air does not
give all its energy to the turbine and there are other inefficiencies
affecting the amount of electrical energy you get. Some people sell
some electricity back to the Electricity Board, but they still need
mains electricity when using high power appliances or on still days.
40
wind power in kW
35
A graph of wind power against wind speed for the 8 m2 turbine
shows how steeply the power increases for only small changes in
wind speed. At wind speeds below about 3 m s-1 the turbine may not
start and won’t yield enough power to be worthwhile. At very high
wind speeds the turbine will need to be shut down or operate at
vastly reduced efficiency to avoid structural damage.
30
25
20
15
10
5
00
0
22
44
66
10
12
88
10
12
wind speed in m/s
14
14
16
16
18
18
20
20
Choosing your site
Ideal sites need the highest possible prevailing wind speed for maximum power. These are in Scotland and the North. The Met. Office’s
computer model of the country means that you can look up your home’s likely wind speed on the internet, by typing in your postcode*.
However, there must be nothing causing air turbulence such as large trees or tall buildings. You probably don’t have an ideal site, and
need to do accurate measurements before spending a lot of money. Also the wind speed increases with height. (See the next graph for 3
sites). Experts suggest you use a tower at least twice as tall as any major obstacle, and ten times as far away from it as the height, so wind
turbines are fashionable among rich people with an acre of land to spare!
*Go to the British Wind Energy Association Website (BWEA) and follow the links.
Flow over hills and obstacles
Good site
Obstacles
10 M or more
turbulence
Height of
obstacle(s)
(M)
turbulence
speed up effect over smooth hills
site clear of obstacles by at least 10 x the height of the obstruction
(or use a very tall tower)
Even if you live in a terraced city house, however, wind turbines to be attached to your roof or a wall are being developed. Some are no
bigger than TV aerials and won’t make much noise or worry the neighbours. They will not produce all the electricity you need, but they
could power low power devices such as fridge, TV, radio alarm, and significantly reduce your electricity bill. Some buildings even funnel
the wind to strengthen it in certain places, and can be designed for that purpose. If you are considering architecture as a career there may
be interesting possibilities ahead.
variation of wind speed with height for 3 sites
Fig 2
20.0
18.0
X
Wind speed (m/s)
16.0
14.0
12.0
Y
10.0
8.0
Z
6.0
4.0
2.0
0.0
0
5
10
15
20
2
25
30
35
40 Height (m)
Physics Factsheet
112. Practicalities Of Domestic Wind Energy Generation
What size and shape of wind turbine?
•
Old-fashioned windmills had 4 sails with flat slats made of wood.
The speed could be adjusted in light or strong winds by varying
the angle of the slats. The same is true for modern wind turbines.
Wooden slats are very heavy and the forces and efficiency not
very great. Modern turbines use the lift force like that on an
aeroplane wing, and the turbine blades are shaped accordingly.
Even so the air has to have some speed after power has been extracted
in order to move away from the turbine. The most energy that can
ever be extracted from the wind is about 0.59 (59%) theoretically,
whereas 0.35 (35%) can be achieved with good design.
Torque is the combined turning moment of the forces on all
the blades, acting so as to rotate the turbine. To get the
biggest torque, the forces should act at the tips, but this will
never happen.
Force F
r
lift force
wind direction
•
drag force
Wind flowing over the aerofoil provides the lift force that makes an
aeroplane fly, and also a drag force (friction). Aerofoils are designed
to make the lift as great and the drag as small as possible. In a wind
turbine, there are usually two or three blades. The forces are still
called lift forces even though they now act in directions other than
vertically upwards.
If there are 5 blades, each 1 metre long, with forces of 400N
acting at the mid-point of each, the total torque will be
5 × 400 × 1 × 0.5 = 1000 N m.
The formula is n × F × r, where n = number of blades, F = force
on each, and r = distance from the pivot to the point of action
of the force.
Will you or won’t you go for wind power?
The full formula for the amount of electrical power you can get from
the wind is ½ Adv3 × 0.59 × εb × εg. Here A is the area of the wind
turbine, d the density of air (1.2 kg m-3), v is the velocity of the wind
in ms-1, 0.59 is the theoretical maximum of the energy that you can
extract, εb is the efficiency of the gear-box if you have one (could
be 0.9 or 90%), and εg is the efficiency of the generator which could
be 0.8 or 80%.
lift force
To make the final decision, you need to know:1. how much power you will need,
2. the average wind-speed at the place where you plan to build the
turbine,
3. the size of turbine you will need at that wind speed,
4. how much it will cost,
5. the savings you will make on electricity you won’t need to buy
6. the payback time (how long it will take you to recover the initial
costs)
7. Decision!
lift force
Example
1. You want a 50 W (max) turbine to power an electric fence.
2. The average wind-speed at the site is 4.5 m s-1 (quite low).
3. Use the formula to calculate the area of the turbine that you
need. We don’t know A, but all the other factors multiplied
together = 23. So 50 /23 = 2.18 m2 for the area of your turbine.
Using πr2, the radius of the turbine needs to be 0.83 m.
4. Typical cost for a turbine of that size; £500.
5. The electric fence uses 10 W for all 365 × 24 hours in the year.
Number of kWh = 0.01 × 8760= 88 kWh. At 12p per kWh, this
would save you £10.56 per year.
6. Payback time = £500 / £10.56 = 47 years.
7. Is it worth it? That’s a very long time – you need to compare
the cost of cables or batteries.
Two or three blades are chosen for the biggest turbines, because
they are lighter and rotate faster than those with more blades, as
each blade causes less turbulence for the one behind. However,
they may need a motor to get them going. Smaller, lower power
turbines often have more blades, as they produce more torque (see
below) in light winds, start easily but then rotate more slowly. One
award-winning design for roof-top use has 5 blades with tips
attached to a ring. This improves air flow, and reduces stresses on
the blades and noise from the vibrating tips.
3
112. Practicalities Of Domestic Wind Energy Generation
Practice Questions
Where necessary, take the density of air d as 1.2 kg m-3.
1. You leave your 90 Watt computer on standby all the time.
(a) How many kWh do you use in one year?
(b) How much does it cost per year at 13p per unit?
(c) What percentage is that of your family’s electricity bill if they use 4450 kWh altogether?
2. You put up a £1000 wind turbine to supply your computer from question 1. What’s the payback time?
3. A power station produces 100 Mwatts of electricity, but is only 35% efficient.
(a) How much energy was transferred from the original coal every second?
(b) How much energy was wasted every second?
(c) A further 6 Mwatts are wasted in the overhead power lines, and the electrical appliances in people’s homes waste 9 Mwatts. What
is the overall efficiency of the complete process?
4. (a) A battery charger needs to produce 20V with a maximum current of 500 mA. What is its power?
(b) A washing machine uses 230V with a maximum current of 9.5 Amps. What is its power?
5. Suppose the wind speed near your home is 8 ms-1.
(a) Use the first graph to find the power of the wind hitting an 8.0 m2 turbine.
(b) If the radius of the wind turbine is halved, what difference does this make to (i) the area (ii) the power you can get from the wind?
6. (a) A small turbine of radius 1 m receives wind of speed 7 m s-1. Use the formula ½ Adv3 to find the power of the wind hitting it.
(b) If the wind speed increases to 9 m s-1, what is the power now?
7. (a) To get 1 MWatt (106 watts) of power in the wind hitting one turbine, what radius would you need, if the prevailing wind speed is
12 m s-1?
(b) If the wind turbine is only 25% efficient, what radius would you need to get 1 MWatt of power out?
8. A certain wind turbine of area 8.0 m2 will not self-start or operate reliably if the wind-speed is less than 7 m s-1, and has to be shut down
for safety reasons if the wind speed is higher than 17 m s-1. A 20 m tower costs 4 x as much as a 10 m tower, and a 40 m tower 16 x as much.
(a) You require an output of 8 kW. Use the first graph (Fig 1 Page 2) to decide on the ideal wind speed.
(b) Now use the second graph (Fig 2 page 2) to comment on the suitability of this wind turbine for each of sites X Y and Z.
9. (a) What is the torque on a 3-blade wind turbine of radius 8 metres, if forces of 1000 N act at the mid-point of each blade?
(b) What is the torque on a 12-blade turbine of radius 0.8 m, if forces of 50 N act at the mid-point of each blade?
10. You want a 500 W turbine to run low-power domestic appliances. The average wind speed near your home is 7 m s-1.
(a) Use the formula ½ Adv3 × 0.59 × εb × εg to calculate the area and radius of wind turbine that you will need. (Assume that εb = 0.9
and εg = 0.8).
(b) A turbine of suitable size plus electrical accessories costs £1500. Calculate the savings you will make if the turbine operates 365
days of the year, and 1 kWh of bought electricity costs 12p.
(c) What would be the payback time of this wind turbine?
1. (a) 788 kWh (b) £102
(c) 18%
2. 9.8 years
3. (a) 286 Mwatts or MJ s-1. (286 × 106 J s-1) (b) 186 × 106 J s-1 (c) Only 85 Mwatts are used usefully so overall efficiency = 30 %.
4 (a) 10 watts
(b) 2185 watts, or 2.2 kW.
5 (a) About 2.5 kW (b)(i) It will reduced to 2.0 m2. (c) About 0.6 kW, or ¼ of (a)
6 (a) 647 watts
(b) 1370 watts
7. (a) 17.5 m (b) 35 m
8 (a) 12 m s-1. (b) X – the wind speeds are much higher than required. You would get all you need even at a height of 5 metres but the
turbine might sometimes have to be shut down. Y – It depends whether 8 kW is the absolute maximum or the average that you
require. If it’s the average, you would need a 35 metre tower so a bigger area wind turbine might work out cheaper. Z – no use, as
you will never get enough power.
9 (a) 12000 Nm (b) 240 Nm.
10 (a) 5.7 m2, radius 1.3 m. (b) £526 (c) 2.9 years.
Answers
Acknowledgements:
This Physics Factsheet was researched and written by Hazel Lucas
The Curriculum Press,Bank House, 105 King Street,Wellington, Shropshire, TF1 1NU
Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber.
No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher.
ISSN 1351-5136
4