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Wind Energy Simulator
Betz Limit Simulator — Theoretical Maximum Wind Turbine Efficiency
Vary wind speed, rotor diameter, axial induction factor a, and air density to compute the power coefficient C_P(a) = 4a(1-a)² and captured power in real time. The optimum a = 1/3 reproduces the Betz upper bound C_P,max = 16/27 ≈ 0.593.
Parameters
Wind speed V
m/s
Rotor diameter D
m
Axial induction factor a
Air density ρ
kg/m³
Sweep: a runs continuously from 0.00 to 0.50 to highlight the C_P peak at a = 1/3.
Results
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Power coefficient C_P
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Available wind power
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Captured power
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Betz ratio
Turbine and streamtube
Top: front view of a 3-bladed rotor. Bottom: top view of the streamtube. As a increases, the wind slows at the rotor and the streamtube widens downstream.
Power coefficient C_P(a) curve
Horizontal axis: axial induction factor a. Vertical axis: C_P. The dashed line marks the Betz optimum at a = 1/3 (C_P = 16/27 ≈ 0.593). The yellow marker tracks the current setting.
Theory & Key Formulas
From axial momentum theory with upstream wind $V_\infty$, rotor velocity $V=(1-a)V_\infty$, wake velocity $V_w=(1-2a)V_\infty$ and area $A=\pi D^2/4$:
$$C_P(a) = 4a(1-a)^2 \,, \qquad P = C_P \cdot \tfrac{1}{2}\rho A V_\infty^3$$
Differentiating, $dC_P/da = 4(1-a)(1-3a) = 0$, gives the maximum at $a = 1/3$:
$$C_{P,\max} = \frac{16}{27} \approx 0.5926$$
This is the Betz theoretical upper bound. The Betz ratio $C_P / C_{P,\max}$ measures how close a real machine comes; modern turbines reach 75–85%.
What is the Betz Limit Simulator
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I read that a wind turbine cannot capture 100% of the wind's energy. Why is there an upper limit?
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Good question. If the rotor tried to stop the wind completely, the air behind it would have nowhere to go and no new air would arrive — flow would stall. So you have to slow the wind 'just enough' while still letting mass flow through. The German physicist Albert Betz worked out in 1919 that the optimum is a = 1/3, giving C_P,max = 16/27 ≈ 59.3%. We call that the Betz limit.
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What is 'a'? The slider calls it the axial induction factor.
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It is the fraction by which the wind has slowed at the rotor relative to the upstream velocity. a = 0 means the air sails through unaffected (no power), a = 1 means it stops dead (no flow, no power). The trade-off has a single optimum. The captured power is P = 2ρAV_∞³ a(1-a)², and differentiating gives a maximum at a = 1/3.
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In the chart C_P peaks at a = 1/3 and falls off after that. Why does it drop at a = 0.5?
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Because the wake speed is V_w = (1 - 2a)V_∞, so at a = 0.5 the wake is at zero velocity — the model breaks down. Beyond a = 0.5 the formula gives a negative C_P, which means momentum theory no longer applies. Real turbines run at a ≈ 0.33 and operators don't try to push beyond it; it is much more profitable to enlarge the rotor or pick a windier site than to chase a higher a.
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How efficient is a real wind turbine compared to the Betz limit?
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Modern three-bladed horizontal-axis machines reach C_P ≈ 0.45–0.50, that is 75–85% of the Betz limit. That is already after blade drag, wake rotation, tip vortices, and generator losses. From a fluid-mechanics standpoint, they are nearly as good as the theory allows. The remaining few percent come from blade-shape optimization, active pitch control, and choosing the best offshore sites — which is exactly where modern wind R&D is focused.
FAQ
The Betz limit applies to a one-dimensional, axisymmetric, frictionless, non-rotating-wake actuator-disk model: incompressible air, the rotor as a sudden pressure jump, mass and momentum conservation. From these alone you derive C_P = 4a(1-a)², so no axisymmetric frictionless turbine can exceed its maximum 16/27 ≈ 59.3%. Once you include wake rotation (Glauert correction), the upper bound drops further; with viscous and mechanical losses, real machines reach 0.45–0.50. Vertical-axis Darrieus rotors, sails, and tethered kites have different upper bounds derived from different theories and are not bounded by Betz directly.
The mass flow per second is ṁ = ρAV and the kinetic energy per unit mass is (1/2)V², so the available wind power is P_wind = ṁ·(1/2)V² = (1/2)ρAV³. Mass flow scales linearly with V and kinetic energy scales quadratically, multiplying to V³. This cubic dependence dominates wind economics: a site with 8 m/s mean wind generates (8/6)³ = 2.37× more energy than 6 m/s on the same turbine. Site selection — onshore vs offshore, hub height, complex terrain — is therefore the most important economic decision. Modern turbine towers exceed 100 m, with 150 m hub heights becoming common, to reach the higher winds above the surface boundary layer.
The standard CFD approach is the Actuator Disk Model (ADM): instead of resolving the rotor geometry, a streamwise pressure jump ΔP is imposed as a boundary condition, and C_P is computed from the flow rate and momentum balance. OpenFOAM's actuationDiskSource, Ansys Fluent's fan model, and STAR-CCM+'s swirl/blade-element solvers all support this. Sweeping a from 0.0 to 0.49 reproduces the analytic curve C_P = 4a(1-a)² to three decimal places — the benchmark for any new ADM implementation. More refined methods are Blade Element Momentum (BEM) theory, the Actuator Line Model (ALM), and full 3D LES, but all use Betz theory as their reference.
Betz theory ignores wake rotation, but in practice the rotor produces torque, so by Newton's third law the air spins behind the disk. Glauert added the angular-momentum balance and showed that the upper bound depends on the tip-speed ratio λ = ωR/V_∞. As λ → ∞ the limit converges to 16/27; at λ = 7 it is 0.59; at λ = 1 only about 0.42. Modern large turbines run at λ = 6–10, so the Glauert loss is only a few percent. This tool plots the λ → ∞ limit (pure Betz). To compare with real machines, also subtract Glauert wake-rotation, viscous blade drag, and 3D tip effects.
Real-world applications
Large offshore turbines (GE Haliade-X, Siemens Gamesa SG 14): The current offshore market leaders have rotor diameters around 220 m and rated capacity 14–15 MW. With V = 11 m/s, D = 220 m, a = 0.33, ρ = 1.225 this tool gives roughly 25 MW of theoretical aerodynamic power; mechanical and electrical losses (generator, gearbox, grid) plus Glauert effects reduce that to around 14 MW rated — a Betz ratio of 83%. Energy scales with the square of rotor diameter, which explains the rapid upscaling of offshore turbines; 18–20 MW machines are on the roadmap for the early 2030s.
Onshore micro-wind turbines for homes: Small turbines (1–10 kW, 2–7 m diameter) typically operate in turbulent residential terrain with C_P = 0.20–0.35 — a Betz ratio of 34–59%. They underperform utility-scale machines, but at sites with mean wind below 6 m/s, the economic payback is poor regardless. The formulas in this tool give a fast 'site mean wind × swept area × C_P' first estimate to evaluate whether a residential wind turbine makes sense for a given location.
Hydro turbines and tidal stream generators: Run-of-river hydro and tidal stream turbines obey the same Betz theory but in water (ρ ≈ 1000 kg/m³, ~800× air). At identical flow speed and diameter you get 800× more power, so tidal turbines around 15–20 m diameter at 2–3 m/s currents can rate at 1–2 MW. The Betz limit transfers directly as a design ceiling, with extra constraints from cavitation and marine-life impact specific to the underwater environment.
Diffuser-Augmented Wind Turbines (DAWTs): Adding a divergent diffuser around the rotor creates designs that appear to exceed Betz when normalized by rotor area. However, when normalized by the diffuser exit area, these still respect 16/27. Companies like FloDesign and FlowTubine have built prototypes, but the structural weight and cost burden have so far prevented commercial uptake. DAWTs are mainly useful as a concrete illustration of what 'the Betz limit' actually constrains: momentum theory, not engineering ingenuity.
Common misconceptions and caveats
The most common error is to think 'the Betz limit is the limit of mechanical efficiency'. It is not. It is the upper bound on the aerodynamic power coefficient of an axisymmetric, frictionless rotor — fundamentally distinct from generator and gearbox efficiency. Total efficiency multiplies as C_P × η_gen × η_gear × η_grid; with C_P = 0.50, η_gen = 0.95, η_gear = 0.97, η_grid = 0.98, the overall efficiency is about 0.45. This tool reports only C_P. Annual energy production (AEP) requires combining C_P with the site's Weibull wind-speed distribution.
A second pitfall is the intuition that 'a higher induction factor a captures more power'. The chart shows the opposite: C_P peaks at a = 1/3, drops to 0.5 at a = 0.5, and goes negative beyond — meaning the simple momentum model breaks down. Physically, slowing the wind too much chokes the mass flow. Real control systems implement maximum-power-point tracking (MPPT) by jointly adjusting rotor speed and blade pitch; that is precisely what variable-speed drives and hydraulic pitch actuators are for in modern turbines.
Finally, 'the Betz limit applies to every kind of wind machine' is wrong. Betz holds for an axisymmetric actuator disk. Savonius (drag-driven) rotors have C_P,max ≈ 0.30; Darrieus (lift-driven vertical-axis) rotors reach about 0.40; sails, kite-energy systems, and ducted designs each have their own upper bounds derived from different physical models. 'Betz limit 16/27' is the classical figure for horizontal-axis propeller turbines, and applying it to a different topology can give the wrong answer.