Slide1: 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? Wind Power
Slide2: 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)
Slide3: 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 = a u0
then P = ½ru03 A1{4a2(1 - a)}
or P = P(wind)F(a)
where P(wind) = ½ru03 A1 and F(a) = 4a2(1 - a)
Slide4: P max/min when d{4a2(1 - a)}/da = 0
8a - 12a2 = 0
so a = 0 (min) or 2/3 (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
Slide5: 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.
Slide6: Simplified representation of world wind circulation
Slide7: Persian Windmill Windmills thought to
have been in existence
for about 4000 years
Slide8: Some examples of
Wind Turbines
Slide9: Flow around an Aerofoil
Slide10: 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
Slide11: Blade speed v = wr so angle of attack a 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 Tip-speed vtip = wR Wind speed u Tip-speed ratio l = vtip/u Horizontal Axis Wind Turbine (HAWT) Angular velocity w
Effect of Drag on CP: 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%
Modern wind turbine CP – l curve: Modern wind turbine CP – l curve Turbine designed to have maximum efficiency at l ~ 10
Slide14: Rotor efficiencies of some wind turbines l 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.
Slide15: Thrust on a Wind Turbine p0 u0 p1 u1 p2 u2 A0 A1 A2 upstream turbine downstream p0 /r + ½ u02 = p1 /r + ½ u12
p1* /r + ½ u1*2 = 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 p1* u1* Consider streamtubes of air before and after turbine, not across turbine because flow
unsteady and not streamlined. Thrust is maximum when u2 is minimum – this corresponds to maximum power extraction
for which u2 equals u0 / 3. Therefore
Fthrust = ½ r u02A1×8/9 Similar to a circular disc of area A1 which has a drag force FD = ½ CD r u02A1 and CD ~1
Slide16: Probability distribution for wind speed at North Ronaldsay, Orkney 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 = ½rA<u3> ≈ rA<u>3 as <u>3 ≈ 2<u3> Wind speed increases with
height z : uz ≈ u10(z/10)0.14
where z is in metres
Slide17: 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. Typical spacing of turbines on a wind farm is 4D(crosswind) x 7D(downwind), where D is the diameter of the turbine. Wind Farm P ≈ 0.2D2<u>3, D diameter of turbine
Slide18: Engineering Designs : Pros/Cons Vertical axis
Advantages: a) No Yaw necessary
b) Direct coupling to electrical generator
Disadvantages: 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)
a) high moment of inertia hence slow
b) low frequency good for battery charging
or water lifting
Slide19: Horizontal axis
Disadvantages: a) Upwind blades need Yaw (fan-tail for alignment)
b) Downwind blades self-orientate but tower blocks
some wind 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
>150 m from houses and that noise level < 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
Slide20: Relative Noise Levels I(dB) = 10 log10(I/I0), where I0 is the threshold of hearing
(at 1000 Hz I0 = 10-12 Wm-2)
Slide21: 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: >106 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 increasing more ‘spinning reserve’ unless good
energy storage developed (eg Fuel Cells) due to wind variability
Electrical transmission from windy sites to main population
centres also required
Slide22: Annual incremental installed capacity (GW)
Slide23: 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) Land-based wind energy resources in TWh per year
Slide24: Potential UK offshore wind generation resource Current UK electrical energy demand is ~350 TWh per year
(1.3 1018 J per year or ~40 GW continuous power)
Slide25: Estimates of renewable-energy resources for 2025 in the UK
Slide26: 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) Practicable or accessible potential
The amount of the technical potential that can be utilized
by a particular time 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 (nb definitions of potential differ)
Slide27: Guardian Tuesday May 3rd 2005
Slide28: 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
cf total solar power ~5.4 1024 J/yr Mantle (depth > 30 km), temperature ~ 1000 oC
Convective heat flow > ~ 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
Slide29: 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
Slide30: Map of the Earth’s plates Movement generally 1-10cm per annum
Slide31: 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 < 10-2
Silt 40 10-2 – 1
Sand 30 1 – 500
Gravel 30 103 – 104
Slide32: Heat Extraction from Aquifer Water volume V flows per second through narrow porous layer area A and thickness h and is heated by surrounding rock at temperature T+T1 Heat lost by rock plus water = Heat gained by water
-[(1-f)rrcr+ frwcw]Ah d T = VrwcwT dt dT/dt = -T/t
T = Toexp(-t/t)
where t = C/Vrwcw and C = [(1-f)rrcr+ frwcw]Ah ~3km
Slide33: 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 Example: A=1 km2, h= 0.5 km, f=5%, 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+ frwcw]Ah = 1.2 1015 J/K
t = (1.2 1015)/(4.2 105) = 90 years Energy stored, Es = C(T - Tl)
E(t) = Esexp(-t/t)
so dE/dt = -(Es/t)exp(-t/t)
Substituting: Initial Power = 25 MW
Slide34: Volcanic Geothermal System
Slide35: 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 1p/KW
Output 1 MW
Capital Cost: Government + EC as a demonstration project
Slide36: 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
Slide37: Extraction Techniques a) Hot dry rock system b) Hyperthermal
power station
(temperature gradient
> 80 C/km- tectonic plate
boundaries)
Geothermal Power Plants: Geothermal Power Plants >150 C ~ 100-150 C SO2 (lbs/MW-hr) CO2 (lbs/MW-hr)
Slide39: Potential UK geothermal energy resource
at different temperatures Current UK electrical energy demand is ~ 350 TWh per year
(1.3 1018 J per year or ~ 40 GW continuous power)
Slide40: 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
Slide41: MegaWatts
Geothermal Heat Pumps: 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)
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