Incident power in the wind

1
Wind Power
Incident power in the wind
Incident power P given by (mass/sec)(KE/mass)
P (dm/dt)½u2
(ruA)½u2
P ½ rAu3 Note strong
dependence on wind speed

• Example r 1 kg m-3 , A 1 m2 , u 12 m s-1
  (rated wind speed)
• P 864 W 1 kW
• Typically turbine efficiency 40 so power
  output 300-400 W

Why are wind turbines not more efficient?
2
Theoretical limit to Efficiency of a Wind
Turbine Betz Limit
Consider streamtube of air passing through
turbine The turbine extracts energy from the air
so the air speed decreases across the turbine
and the cross-sectional area of the streamtube
increases u0 A0 u1 A1 u2 A2
Force on turbine rate of loss of momentum
of air F
(dm/dt)(u0 - u2) Power extracted P Fu1
(dm/dt)(u0 - u2)u1 (rate of work done) Also
P ½(dm/dt)(u02 - u22 )
(rate of loss of KE)
3
P (dm/dt)(u0 - u2)u1 ½(dm/dt)(u02 - u22 )
(u0 - u2)u1 ½(u0 - u2) (u0 u2 )
u1 ½(u0 u2 ) or u2
2 u1 - u0 now dm/dt ruA ru1A1
so P ru1A1(u0 - u2)u1 or
P 2ru12 A1(u0 - u1)
Let u1 (1- a) u0 then
P ½ru03 A14a(1 - a)2 or
P P(wind)F(a) where P(wind) ½ru03
A1 and F(a) 4a(1 - a)2
4
P max/min when d4a(1 - a)2/da 0
(4 - 12a)(1-a)
0 so
a 1 (min) or 1/3 (max) At max F(a)
16/27 so Maximum efficiency is 59
(Betz criterion) Power coefficient CP P/
½ru03 A1 Lost power is due to fact that air
needs KE to go downstream
Thrust dF dLcosf Power dP dLsinfv dP
dFtanfv tanf u1/v so dP dFu1
5
Materials absorb radiation differently so
temperature gradients arise causing convection
and pressure changes which result in winds. A
simple example being the off-shore night-time
wind often found on coasts, caused by the sea
retaining the heat from the sun better than the
land.
6
Simplified representation of world wind
circulation
7
Global distribution of Wind Speeds
Wind Speeds in Western Europe
8
Persian Windmill
Windmills thought to have been in existence for
about 4000 years
9
Some examples of Wind Turbines
10
Flow around an Aerofoil
11
Forces on an Aerofoil
Lift Force ½CLrAu2 Drag Force ½CDrAu2

• CL, CD functions of non-dimensional parameters,
  ie
• Reynolds number Re r u l/h , where l is a
  characteristic length
• Shape of the aerofoil
• Angle of attack a

12
Horizontal Axis Wind Turbine (HAWT)
Angular velocity w
Wind speed u
Tip-speed vtip wR
Tip-speed ratio l vtip/u
Blade speed v wr so angle of wind f depends on
radius r Twist of blade changes with radius to
optimise a. Betz condition u1 2u/3 cot f
v /u1 3rl/2R Stall regulated- as u increases a
increases and blade stalls
13
Effect of Drag on CP
As a result of drag rotational force becomes L
sin f - D cos f ? L sin f (1-g cot f) where g ?
CD/CL .
cot f v /u1 3rl/2R , so reduction decreases
with r Typical r 2R/3 so cot f ? l and CPmax
? (1-gl)CP(Betz) g 1/40 and l 10 so CPmax
45
14
Modern wind turbine CP l curve
Turbine designed to have maximum efficiency at l
10 The width W and angle of attack a are for a
particular l. If the wind speed alters, then
angle f of wind to motion of blade and therefore
the lift L changes. This changes the thrust from
it optimal value and CP decreases.
15
Thrust on a Wind Turbine
A0
A2
A1
p0 u0
p1 u1
p2 u2
p1 u1
upstream
turbine
downstream
Consider streamtubes of air before and after
turbine, not across turbine because flow
unsteady and not streamlined.
p0 /r ½ u02 p1 /r ½ u12 p1 /r ½ u12
p2 /r ½ u22
Conservation of mass u1 u1 and p0 p2
atmospheric. So (p1 - p1) /r ½ (u02 - u
22) Fthrust ½ r (u02 - u22)A1

Thrust is maximum when u2 is minimum this
corresponds to maximum power extraction for which
u2 equals u0 / 3. Therefore Fthrust ½ r
u02A18/9
Similar to a circular disc of area A1 which has a
drag force FD ½ CD r u02A1 and CD 1
16
Probability distribution for wind speed at North
Ronaldsay, Orkney
Wind speed increases with height z uz
u10(z/10)0.14 where z is in metres

• Probability distribution F(u) can often be
  approximated by
• Rayleigh distribution
• F(u) (2u/c2)
  exp-(u/c)2
• where c 2uaverage /(p)1/2
• P ½rAltu3gt rAltugt3 as ltugt3 2ltu3gt

17
Wind Farm
(rated power)
Wind Turbine chosen to have output capacity 3
times average power output to take advantage of
high wind speed periods. Cut-out value to protect
turbine installation.
P 0.2D2ltugt3, D diameter of turbine
Typical spacing of turbines on a wind farm is
4D(crosswind) x 7D(downwind), where D is the
diameter of the turbine.
18
Wind Power
Lake Benton Minnesota
Wind farm off Denmark
19
Engineering Designs Pros/Cons
Vertical axis Advantages a) No Yaw
necessary b)
Direct coupling to electrical generator Disadvant
ages a) Many natural resonances leading to
vibration
and fatigue b)
Variable torque leading to uneven output
c) Less cost-effective than HAWTs
Horizontal axis Advantages Low solidity
machines (few blades)
a) low moment of inertia hence fast
b) high frequency good for
power generation High
solidity machines (many blades) (American
style farm windmills)
a) high moment of inertia hence slow
b) low frequency good for
battery charging
or water lifting
20
Horizontal axis Disadvantages a) Upwind
blades need Yaw (motor for alignment)
b) Downwind blades
self-orientate but tower blocks
some wind- suffers from fatigue
from turbulence
Fatigue Many revolutions gives rise to fatigue
which gives rise to cracks Wind turbines 108
cycles - very demanding on materials
Environmental impact Appearance matter of
opinion Noise gearbox, electrical generator,
aero noise (swish) eg Denmark has
requirement that wind turbines are located
gt150 m from houses and that noise level lt
45dB noise ? u5 ? low rotational speeds E.M.
Interference reflection of em waves/TV signals
from metal
blades Wildlife wind farms are a hazard for birds
21
Relative Noise Levels
I(dB) 10 log10(I/I0), where I0 is the threshold
of hearing (at 1000 Hz I0 10-12 Wm-2)
22
Economics Capital cost 600-1000/ kW California
(Reagan) tax breaks (most imported from
Denmark) Cost of production goes down as demand
goes up Best sites competitive with fossil fuels
Applications Battery chargers 105 in World
(China mostly) Wind pumps gt106 worldwide
(fast growth in developing world) Electricity
generators increasing worldwide. Low carbon
so very important as alternative to fossil fuel
Future Potential Could reach 10-20 of
electricity needs of World 2020-2050 Higher
needs more spinning reserve unless good
energy storage developed (eg Fuel Cells) due to
wind variability Electrical transmission lines
from windy sites to main population centres also
required
23
Annual incremental installed capacity (GW)
24
Land-based wind energy resources in TWh per year
Worlds land-based wind energy resources
estimated as 53,000 TWh per year World
electricity demand by 2020 estimated as 26,000
TWh per year (equivalent to 3 TW continuous cf
20 TW continuous for estimated total energy
demand)
25
Estimates of renewable-energy resources for 2025
in the UK
Current UK electrical energy demand is 350 TWh
per year (1.3 ? 1018 J per year or 40 GW
continuous power)
26
Theoretical or Gross potential Estimate of total
annual energy that could be produced
Technical potential Maximum annual energy that
could be extracted taking into account
practical, environmental, and social constraints
(estimates of 4 by WEC, and 10 of the land area
with suitable winds have been made)
Economic potential Amount of the technical
potential that is economically viable Depends on
the cost of alternative supplies, on the cost of
borrowing, and on policies such as a carbon tax
Practicable or accessible potential The amount of
the technical potential that can be utilized by
a particular time
(nb definitions of potential differ)
27
From The Sunday Times April 16, 2006 Home wind
turbines dealt killer blow Jonathan Leake,
Environment Correspondent ROOFTOP wind turbines
may have become the accessory of choice among
greener than thou politicians, but a new study
suggests that they are not only incapable of
saving the planet but may even damage your house.
David Cameron, the Tory leader, and Malcolm
Wicks, the energy minister, are two politicians
who plan to mount the devices on their London
homes in, respectively, Notting Hill and Croydon.
However, a study commissioned by Building for a
Future, a journal specialising in sustainable
construction techniques, has found that rooftop
turbines are plagued by technical problems and
seldom generate significant amounts of power,
especially in towns and cities. The report finds
that a typical rooftop turbine produces no more
than a quarter of the average homes power needs,
at best, and that in urban areas this is likely
to be more like 10-15, because wind blows
around towns in turbulent, unpredictable gusts.
In addition, older houses can face serious
structural damage from the powerful sideways
forces generated as the wind pushes against the
turbines. This can be a particular problem if the
turbines are mounted on chimneys.
Guardian Tuesday May 3rd 2005
28
Geothermal Energy
Origin Heat from a) cooling of
Core (loss of heat of formation) b)
decay of radioactive isotopes, 232Th, 238U and 40K
Total geothermal power 1021 J/yr 30 TW
cf total solar power 5.4 1024 J/yr
Mantle (depth gt 30 km), temperature 1000
oC Convective heat flow gt 100 km depth
Outer shell (depth 30 km) fissures
(volcanoes/geysers) Thermal conduction k 2 W
m-1K-1, no convection

• Heat flux q - k dT/dr - k (Tc- Ts)/d
• q 6.10-2 W/m2
• Total geothermal power q? 4pR2 1021 J/yr

29
Forms of exploitation
Warm water springs Spa towns (eg Bath) New
Zealand (Maoris)
Geysers Eg Italy, New Zealand, Iceland,
USA(California) All situated in geological fault
regions Total output 6 GW
Aquifers Porous layer sandwiched between
non-porous rock
Hot dry rock mining Like aquifers but water
pumped through natural fissures (cracks) in rocks
30
Map of the Earths plates
Movement generally 1-10cm per annum
31
Aquifer Properties
Porous medium (e.g. sand, gravel) Define porosity
(Volume of cavities)/(Total volume)
Pressure difference, DP rgH H Head of
water kw Hydraulic conductivity
Porosity (?)
kw(m/day) Clay 50
lt 10-2 Silt
40
10-2 1 Sand
30 1
500 Gravel 30
103 104
32
Heat Extraction from Aquifer
3km
Water volume V flows per second through narrow
porous layer area A and thickness h and is heated
by surrounding rock at temperature TT1
Heat lost by rock plus water Heat gained by
water -(1-f)rrcr frwcwAh d T VrwcwT dt
dT/dt -T/t T Toexp(-t/t) where t
C/Vrwcw and C (1-f)rrcr frwcwAh
33
Lifetime of an Aquifer Heat extracted is stored
energy- time to replace is longer than
lifetime if used as a power source- so not
renewable, but hot dry rock could be.
Example A1 km2, h 0.5 km, f5, rr 2700
kg/m3, cr 840 J/kg/K rw 1000 kg/m3, cw 4200
J/kg/K, V 100 l/s, T 100 C Lowest useful
temperature Tl 40 C C (1-f)rrcr frwcwAh
1.2 1015 J/K t (1.2 1015)/(4.2 105) 90 years

• Energy stored, Es CTo
• E(t) Esexp(-t/t)
• so dE/dt -(Es/t)exp(-t/t)
• Substituting Initial Power 44 MW

34
Hot Dry Rock Mining
USA, UK (Canborne, Cornwall), Germany, Japan Look
for high temperature gradients (so less
drilling) Granite good higher than average
radioactivity Depths 3 - 6 km Temperatures
200 - 300 C Resource in Cornwall approximately
equal to UK coal reserves, but currently too
expensive to exploit
Commercial Exploitation Southampton (Hampshire
Geological Basin) 1980/81 Depth 1.7 km
Temp 74 C Vol/s 12 l/s
Lifetime 20 years District
heating Civic Centre, Swimming Baths, Department
Stores Cost 3.5p/kWh
Output 1 MW Capital Cost
Government EC as a demonstration project
35
Hot Dry Rocks in the UK
a) Predicted temperature-depth curves in
parts of the UK
b) Projected temperature contours in
centigrade at 6 km depth in the SE
36
Extraction Techniques
a) Hot dry rock system
b) Hyperthermal power station (temperature
gradient gt 80 C/km- tectonic plate boundaries)
37
Geothermal
One thousand km³ of granite at 200C cooled by
20C delivers about 1 GWe for 250 years
By 2050 100 GWe in US Potential in US several
1000 GWe for centuries
38
Geothermal Power Plants
gt150 C
100-150 C
CO2 (lbs/MW-hr)
SO2 (lbs/MW-hr)
39
Geothermal Heat Pumps

• Take advantage of relatively constant temperature
  below ground- typically 100-400 ft

• Heat pump either extracts or transfers heat Q
  using a compressor (as in a refrigerator) that
  requires work W . The ratio Q/W is the
  coefficient of performance, COP. For an ideal
  pump heating a building COP T1/(T1 - T2)

Q1/T1 Q2/T2 Q1 W Q2

• eg for DT ? (T1 - T2) 31 C and a ground
  temperature T2 6 C 279 K, then COP 10
• Actual COPs are typically 3 - 4.5

• Over 40 of CO2 emissions in the US are from
  space
• heating and cooling- geothermal heat pumps
  powered
• by green electricity are an important
  source of very-
• low carbon energy

40
(No Transcript)
41
MegaWatts
42
Economics
Drilling costs increase exponentially with depth
increasing which results in deep-mined
geothermal energy not economic (but drilling
technology is getting cheaper)
Main potential in exploiting surface or
near-surface fissures Global potential 12 GW by
2005
Environmental factors
Drilling noisy Waste disposal of spoil, water
loss down cracks H2S (bad eggs) from
geysers Geothermal brine corrosive/toxic _
secondary heat exchangers
Safety Ok if properly managed
Future Development Restricted to geologically
unstable regions, particularly developing World
and Pacific Rim. BUT USA evaluating hot dry rock
drilling Global potential EGS several TW for
centuries.