# Wind power **Wind power** is the conversion of the [[Kinetic_energy|kinetic energy]] of moving air into electricity or mechanical work. Its whole character follows from one equation and one ceiling. The power streaming through a disc of area A is `P = ½·ρ·A·v³`, so output rises as the *cube* of wind speed — double the wind and you get eight times the power — and no machine can take all of it, because air that gave up all its energy would stop and block the disc behind it. [[Betz's_law|The Lanchester–Betz–Joukowsky limit]] caps extraction at 16/27 = 0.593 of what passes through, a number Thomas Kerlin rounds to 59 % and applies equally to [[Wind_turbine|wind turbines]] and water turbines.[^kerlin-221][^kerlin-242][^derived-wp] In the microsim below the reader sets wind speed and rotor radius. A marker rides the cubic curve while the gap between the full stream power and the Betz-capped 0.593·P is shaded, so the ceiling is visible rather than asserted. Beside it sits the machine's own power curve — flat below the cut-in speed, cubic through the middle, clipped flat at rated power, and cut to zero above the cut-out speed — and an annual-energy readout, `E = P_rated × 8,760 h × availability`. One 2.5 MWe machine at 30 % availability yields 6,570,000 kWh a year; a quad a year needs about 45,000 of them.[^kerlin-229][^spec-e41] The cube is also a warning: a 10 % error in the wind speed is a 33 % error in the power.[^derived-wp] On the [[Energy]] flagship this article serves *Wind* in Part V — Transformation, and it is the root of the shared fluid-power sim that the [[Wind_turbine|turbine]] and shear pages reuse. ## Wind energy resources A site is graded by how hard the wind blows, and the grading is coarse because the cube makes fine distinctions unnecessary. Kerlin's Table 8-1 divides sites into seven wind classes, quoted at two heights: Class 1 is 9.8 mph at 33 ft and 12.5 mph at 164 ft, Class 3 is 12.5 and 15.7 mph, and Class 7 is 21.1 and 26.6 mph.[^kerlin-220] Class 3 or better — at least 12.5 mph at 33 ft — is the usual threshold for a commercial machine.[^kerlin-220] The ratios between the two heights in that table run from 1.24 to 1.28, close to the one-seventh power law's (164/33)^(1/7) = 1.257, which is the subject of the [[Wind_turbine|hub-height]] page.[^derived-wp] The reason a one-class difference matters so much is the exponent. Between Class 1 and Class 3 the speed rises by a factor of 12.5/9.8 = 1.28, but the available power rises by 1.28³ = 2.08 — the better site yields twice the energy from the same machine.[^derived-wp] From Class 1 to Class 7 the speed doubles and the power goes up eightfold.[^kerlin-221] This is why wind development is so geographically concentrated, and why a mediocre site cannot be rescued by a better turbine: the resource enters cubed and the technology enters linearly. The same equation with a different [[Density|density]] describes water. Kerlin's Eq. 9-4 has the identical form for a current, and water is "over 800 times" denser than air, so in US units a hydrokinetic rotor obeys `P[W] = 4.13·r²·f³` before the cap and `P[W] = 2.44·r²·f³` after it — 4.13 × 0.59 = 2.44.[^kerlin-242] A 10 ft rotor in a 10 ft/s current therefore yields 244,000 W.[^kerlin-242] Air needs 833^(1/3) = 9.4 times that speed, about 94 ft/s or 64 mph, for the same power: a hurricane against a walking-pace tide.[^derived-wp] ## Wind farms Machines are built in groups because the expensive parts of a wind project — the survey, the road, the crane, the substation, the grid connection — are shared. A farm is also a [[Fluid_dynamics|fluid-dynamics]] problem: each rotor leaves a slower, more [[Turbulence|turbulent]] wake behind it, so downwind machines see a degraded resource and, through the cube, a much degraded output. Spacing is therefore a trade between land cost and array loss. ### Offshore wind power Offshore sites offer a smoother, stronger and steadier wind — less [[Boundary_layer|boundary-layer]] drag over water than over ground — than almost any land site, and they are close to coastal demand. They also impose the [[Corrosion|corrosion]], foundation and access costs of working at sea, and the maintenance calculus inverts: onshore a technician drives, offshore a vessel waits on weather. ### Collection and transmission network Inside a farm, each [[Electric_generator|generator]]'s output is stepped up and collected on a medium-voltage network; the farm as a whole is then stepped up again for [[Electric_power_transmission|transmission]]. Because the best resource is often far from the best load, long-distance transfer — increasingly by [[High-voltage_direct_current|HVDC]] — is part of the cost of wind rather than an afterthought. ## Wind power capacity and production Nameplate capacity is what a fleet could produce; annual energy is what it does, and the conversion is one line: `E = P_rated × 8,760 h × availability`.[^kerlin-229] Kerlin works the arithmetic at three scales and it is worth following, because the gap between an impressive capacity figure and a modest energy figure is where most public confusion about wind lives. One contemporary machine is 2.5 MWe, on a 275 ft tower with 145 ft blades and about 500 tonnes of [[Steel|steel]].[^kerlin-223] At 30 % availability it makes 2.5 MW × 8,760 h × 0.30 = 6,570,000 kWh a year.[^kerlin-229] A world fleet of 10,000 MWe on the same assumption makes 26.3 billion kWh, which is 0.09 quad.[^kerlin-228] A quad a year — against a US electricity demand Kerlin puts at about 14 quads — needs about 45,000 such machines and roughly 2.25 million tonnes of steel.[^kerlin-221][^kerlin-229][^derived-wp] ### Growth trends Kerlin's US figure of 1,500 MWe works out to about 0.013 quad, roughly 0.1 % of national electricity, and he asks how long 30 % annual growth would take to reach a quad.[^kerlin-229][^derived-wp] The book prints "≈15 years"; the expression ln(1/0.01345)/ln(1.3) gives 16.4.[^kerlin-229][^manual10] Either way the lesson is the exponential one: a source at a tenth of a percent is three doublings from a percent and seven from ten. ### Capacity factor That 30 % is the [[Capacity_factor|capacity factor]], and it is not a measure of inefficiency. It is the consequence of the power curve meeting a wind-speed distribution: the machine is rated for a wind it sees only part of the time, spends much of the year below rated, and is idle below cut-in and above cut-out. A turbine whose rated speed were lowered would show a higher capacity factor and produce less energy — which is why the ratio is a description of the site, not a score. ### Variability Wind is not dispatchable. Output follows the [[Atmosphere_of_Earth|atmosphere]] on timescales from seconds to seasons, and the cube amplifies every fluctuation — a gust 10 % above the mean carries 33 % more power.[^derived-wp] Aggregating machines over a wide area smooths the fastest variation, since a gust is local while a pressure system is not, and this is one of the arguments for long transmission. ### Energy storage Because supply and demand are uncorrelated, high wind penetration eventually needs either flexible generation or [[Energy_storage|storage]]. [[Hydroelectricity|Pumped hydro]] pairs particularly well with wind, for the reason Kerlin gives: it suits plants that cannot cheaply throttle back at night, which is precisely a wind fleet's problem. ### Energy payback A 2.5 MWe machine embodies about 500 t of steel, and returns 6.57 GWh a year.[^kerlin-223][^kerlin-229] Against the energy cost of that steel and of the concrete, the payback is measured in months rather than years — the arithmetic is favourable enough that the interesting question is not whether a turbine repays its energy but how quickly, and on which site. ## Economics Wind's cost structure is the mirror of a [[Fossil_fuel|fossil]] plant's: nearly all capital, almost no fuel. That makes its delivered cost dominated by the discount rate, the capacity factor and the lifetime, and it makes comparisons based on fuel price meaningless. It also means the marginal cost of the next kilowatt-hour is close to zero, which depresses wholesale prices in high-wind hours and changes the economics of everything else on the [[Electrical_grid|grid]]. ### The value of wind power The value of a wind kilowatt-hour is not the same as its cost, because it arrives when the wind blows rather than when it is wanted. As penetration rises, wind's own output increasingly coincides with other wind's output, so its marginal value falls — an effect that has nothing to do with the technology and everything to do with correlation. ## Small-scale wind power Small machines face the cube from the wrong side. They sit low, where the wind is slow, and the one-seventh power law means a 33 ft tower sees a fraction of what a 275 ft tower sees.[^kerlin-220] Power also scales with swept area, so halving the radius quarters the output. A small turbine on a building — in turbulent, sheared, obstructed air — is the worst case of both effects at once, which is why small wind is a poor imitation of large wind rather than a smaller version of it. Where it works is where the alternative is worse: remote pumping, off-grid sites, and places with a good exposed resource and no wire. ## Impact on environment and landscape Wind power's impacts are mostly local and visible rather than global and invisible, which is an unusual profile for a generating technology and explains much of the argument around it. Turbines are tall, they move, they make an aerodynamic noise, they cast a moving shadow, and they occupy ridgelines and coastlines that people value for looking at. Bird and bat mortality is real, site-dependent, and best addressed by siting and by curtailment during migration. Against this stands the land arithmetic: a farm's turbines and roads occupy a small fraction of the site, and farming or grazing continues between them, so the land is shared rather than consumed — unlike the acreage a thermal or [[Solar_cell|photovoltaic]] plant of the same output covers. ## Politics Because wind's costs are local and its benefits diffuse, its politics is dominated by siting rather than by generation. Central governments set the framework — connection rules, planning law, support mechanisms — while the decisive arguments happen at the scale of a parish. Community ownership changes the answer more reliably than any technical mitigation, because it changes who receives the benefit that offsets the local cost. Geopolitically, wind inverts the fuel logic: a country with wind does not import it, so the strategic question moves from fuel supply to the supply chain for [[Neodymium|magnet]] materials, [[Steel|steel]] and [[Composite_material|composite]] blades. ## Turbine design Design is the art of getting as close to the [[Betz's_law|Betz]] value of 0.593 as materials allow while surviving weather that is not trying to be helpful. Rotor size is set by the cube and by the square of radius together: at a fixed wind, blade length scales as the square root of power, so Kerlin's exercise in doubling a 2.5 MWe machine to 5 MWe lengthens its 145 ft blades to 145 × √2 = 205 ft.[^kerlin-231][^derived-wp] Working backwards from the 2.5 MWe machine's 145 ft blades at a rotor efficiency around 0.45 gives a rated wind of about 11.5 m/s, or 26 mph — well up in Class 7 territory, which is why the machine spends most of the year below rated.[^derived-wp] Horizontal-axis machines dominate because they put the rotor high and can be pitched and yawed into the wind. Vertical-axis machines keep the generator at ground level, which simplifies maintenance, but they sit, as Kerlin notes, "where wind velocity is lowest" — the same shear that rewards a tall tower penalizes a ground-level rotor.[^kerlin-225] Cold-climate blades need heaters, which does not change the aerodynamics but does change the energy budget.[^kerlin-225] ## History Wind was the second prime mover humans harnessed after [[Hydroelectricity|falling water]], and for most of that history it did mechanical work directly — grinding, sawing, pumping — with no conversion to electricity and no cube-law arithmetic behind the design. The modern era begins when the machine becomes a generator and the output becomes a commodity that must be measured. That change brought the cube law with it: a mill that turned slowly on a poor day was merely slow, but a generator on a poor day produces a number, and the number falls faster than intuition expects. Kerlin's own fleet arithmetic — 45,000 machines and 2.25 million tonnes of steel per quad — is the modern form of the question every millwright once answered by eye.[^kerlin-229][^murphy-wind] ## See also - [[Betz's_law]] — the 16/27 ceiling the sim shades in - [[Wind_turbine]] — the machine, and the hub-height sibling sim - [[Wind_farm]] - [[Airborne_wind_energy]] - [[Capacity_factor]] — the 30 % that turns capacity into energy - [[Wind_profile_power_law]] — the one-seventh law behind Table 8-1's two heights - [[Tidal_power]] — the same cubic law with ρ 833 times larger - [[Electric_power_transmission]] ## Notes Book 048 states Eq. 8-1 without giving the units of ρ, and the air density itself does not appear in the extracts used for this page; the standard 1.2 kg/m³ used in the derived figures is therefore an external value, and the coefficients 4.13 and 2.44 hold only with r in feet and speed in feet per second. Kerlin's fleet figures are explicitly order-of-magnitude reasoning, and the Betz limit is a ceiling on top of which turbine and generator losses still apply. Two of the book's printed results differ slightly from its own expressions — the 30 %-growth time and the 45,000-machine count — and both are given here in printed and computed form rather than silently reconciled. Footnote definitions for this page are collected under References. ## References [^kerlin-220]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5 (pp. 217–231), pp. 220–221: Table 8-1, the seven wind classes at 33 ft and 164 ft (Class 1: 9.8/12.5 mph; Class 3: 12.5/15.7 mph; Class 7: 21.1/26.6 mph), and the Class 3 threshold of at least 12.5 mph at 33 ft for a commercial site. https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^kerlin-221]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 221: Eq. 8-1, `P = (1/2)·rho·A·v^3` with A the swept area (the book does not state ρ's units); doubling v multiplies P by 8; the Lanchester–Betz–Joukowsky limit caps extraction at 59 %; and US electricity of about 14 quads (2010). [^kerlin-223]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 223: a contemporary machine of 2.5 MWe with a 275 ft tower, 145 ft blades and about 500 tonnes of steel. [^kerlin-225]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 225: vertical-axis machines keep the generator at ground level but sit "where wind velocity is lowest"; cold-climate blades need heaters. [^kerlin-228]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 228: a world fleet of 10,000 MWe at 30 % availability gives 26.3 billion kWh = 0.09 quad (computed 0.0897). [^kerlin-229]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 229: annual energy is rated power × 8,760 h × availability; one 2.5 MWe machine gives 6,570,000 kWh/yr; 1 quad/yr needs about 45,000 machines and about 2.25 million tonnes of steel; the US 1,500 MWe gives about 0.013 quad, roughly 0.1 %; and at 30 %/yr growth 1 quad takes "≈15 years". The book calls this arithmetic order-of-magnitude reasoning. [^kerlin-231]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 5, p. 231: at a fixed wind speed, blade length scales as the square root of power (Exercise 8-4, unworked: a 5 MWe machine needs 145 × √2 = 205 ft blades). [^kerlin-242]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, Chapter 6 (pp. 232–248), pp. 241–243: Eq. 9-4 gives the same cubic form for water, which is "over 800 times" denser than air; in US units Eq. 9-5 is `P[W] = 4.13·r^2·f^3` and the Betz-capped Eq. 9-6 is `P[W] = 2.44·r^2·f^3`, with r in ft and f in ft/s, and 4.13 × 0.59 = 2.44; Example 9.2 gives 244,000 W for r = 10 ft at f = 10 ft/s, before turbine and generator losses. [^murphy-wind]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*, Chapter 12, "Wind", within Chapter 6 "Alternative Energy" (pp. 183–322): the wind resource and its scale against total demand (page to pin). https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^derived-wp]: Computed for this article from the equations and cited book values on this page: the Betz limit 16/27 = 0.5926, which the book rounds to 0.59; the Class 1 to Class 3 power ratio (12.5/9.8)³ = 2.08; the Table 8-1 height ratio (164/33)^(1/7) = 1.257 against the table's printed 1.24–1.28; the density ratio 4.13/0.00496 = 833 and its cube root 9.4, giving 94 ft/s ≈ 64 mph in air for the same power as 10 ft/s in water (the 0.00496 coefficient uses the external standard air density 1.2 kg/m³, and 4.13 = ½ × 1.94 slug/ft³ × π × 1.3558); the cube's error amplification, 1.1³ = 1.331; 2.5 MW × 8,760 h × 0.30 = 6.57×10⁶ kWh/yr and 1 quad ÷ that = 44,610 machines against the book's 45,000; 1,500 MWe → 0.01345 quad and ln(1/0.01345)/ln(1.3) = 16.4 years against the book's "≈15"; and the implied rated wind of a 145 ft-blade rotor, A = π(44.2 m)² = 6,136 m², at a rotor efficiency of 0.45 and ρ = 1.2 kg/m³, v = (2.5×10⁶/(0.5 × 1.2 × 6,136 × 0.45))^(1/3) = 11.5 m/s ≈ 26 mph. The 0.45 rotor efficiency is a representative design value, not a book figure. [^manual10]: Wikitube MICROSIM_GUIDE sub-manual 10, *Earth, Energy and Environment*, §3.2 (power in a moving fluid under the Betz cap) and §A.2: air density is external to the 048 extracts; growing 0.013 quad to 1 quad at 30 %/yr takes 16.4 years (computed), not the printed "≈15"; and the 4.13/2.44 coefficients are unit-bound. [^spec-e41]: Matter & Energy Cluster contract, `_registry/plans/ENERGY_SECTIONS.md` row E41: sim concept (`wind.betz`, `wind.powerCurve`), `P = 1/2·rho·A·v^3` with the Betz ceiling `C_p ≤ 16/27 = 0.593`; wind speed and rotor radius as controls; a marker riding the cubic curve with the Betz gap shaded; the machine's power curve (cut-in, rated, cut-out) and its annual energy `P_rated × 8,760 h × availability`, with the 2.5 MWe → 6.57 GWh/yr and 45,000-machines-per-quad anchors. ## External links - [*Future Energy: Opportunities & Challenges*](https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges), Kerlin (2013) — Chapters 5 and 6 carry every wind and hydrokinetic figure on this page - [*Energy and Human Ambitions on a Finite Planet*](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Murphy (2021) - The Wikipedia pair's *External links* section lists national capacity statistics and industry associations, which this page's shelf does not carry. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Wind_power.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Wind power* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Wind_power.html" data-title="Wind power"></div> *Built from `MICROSIM_GUIDE/specs/sims/Wind_power.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Wind_power) : [Wikitube](https://en.wikitube.io/wiki/Wind_power) · pinned revision [1371874629](https://en.wikipedia.org/w/index.php?oldid=1371874629) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E41 · sim pending (matter/Wind_power).*