Wind turbines, blade pitch and useful energy: sources & model

Turn a source-shaped blade, change the generator load and follow the energy. Discover why pitch, torque and speed must work together—and compare a model with a real wind-tunnel experiment.

What supports the explanation?

What this model assumes

1. One computational reference table at 11.4 m/s; no site forecast, yearly yield or homes-powered estimate.
2. Manual, fixed-pitch and fixed-load trials; no full ROSCO controller, actuator response or startup.
3. Rigid drivetrain and source-model constant efficiencies; no shaft flexibility, detailed bearing losses, wakes or blade pressure field.
4. Original aerodynamic loft from source stations; no blade interior, structural load rating or manufacturing CAD.
5. A resolved exercise boundary freezes the trial; it is not a universal machine protection threshold.
6. Real evidence is a different 1:100 IEA 15 MW model. No raw measurement archive was acquired or digitized.
7. An aerodynamic blade, reconstructed from source data: The NREL-5MW AeroDyn file declares 19 stations. Each gives span, curvature, sweep, twist, chord and a profile ID. The last declared span is 61.4999 m from the root; a later commented numeric line is not an extra station. Eight profiles each provide a separate reference point and 399 perimeter coordinates. We loft those declared sections and add display caps; this is not certified manufacturing CAD.
8. Pitch is a geometry change: In our source frame, X points downwind, Y sideways and Z upward. Positive pitch rotates the leading edge toward upwind. Each profile’s own reference is subtracted before scaling by chord. Source twist and offsets are retained; collective pitch then rotates the entire blade about its pitch axis. Pitch changes between trials are reconfigurations, not measured actuator motion.
9. One fixed source wind condition: The ROSCO performance file provides 26 tip-speed-ratio rows, 36 pitch columns and one wind-speed plane: 11.4 m/s. It contains calculated Cp, Ct and Cq values, not field measurements. The main trials use the same density, geometry and wind condition. Startup, no wind, reversal, yawed inflow and complete feathering are outside this model.
10. Two radius conventions that must not be mixed: The nominal radius is 63 m. With a 2.5° precone, the projected radius is 63 cos(2.5°) m and the reference swept area is about 12,445.257 m². Table TSR uses the nominal radius: ωRnom/U. The coefficient relation at source nodes uses projected TSR: Cp approximately equals TSRtable × cos(2.5°) × Cq. The small remaining difference follows from the published rounding.
11. Interpolation is a declared approximation: Four surrounding source Cp values are combined with bilinear weights. At TSR 6.25 and pitch 2.5°, those values average to Cp = 0.41332725. We calculate aerodynamic power from that Cp, then torque as power divided by angular speed. Separately interpolating Cp and Cq would not guarantee their identity between grid nodes.
12. The reference power is not a wake measurement: The coefficient reference is one-half density × projected area × wind speed cubed. Here it is about 11.293 MW. It describes an undisturbed-flow normalization through the selected area, not a direct measurement of flow through the slowed rotor. Unconverted reference power is not all frictional heat. This lookup model does not calculate a wake or local blade pressure.
13. A gearbox trades torque for speed: The reference ratio is 97:1. Generator speed is 97 times rotor speed; generator torque is the low-speed equivalent load divided by 97. In the source’s ideal 100%-efficient gearbox, their products are equal. The generator then has a separate modeled 94.4% efficiency. Neither assumption describes all real machinery.
14. The generator also contributes inertia: Rotor inertia is 38,677,040.613 kg·m². Generator inertia is 534.116 kg·m² on its faster shaft. Referred to the low-speed shaft, total inertia is Jr + G²Jg = 43,702,538.057 kg·m². The squared ratio follows from the generator’s rotational energy, not an arbitrary correction.
15. A ledger that includes the rotor: The rigid-drivetrain equation is Jdω/dt = Qa − QL. Electrical power is ηQLω; modeled generator loss is (1−η)QLω. Stored energy is Jω²/2. We integrate air work, electrical energy and modeled losses separately, then check that their difference equals the change in stored rotation. The numerical residual is exported alongside the values.
16. Why the 5° trial can give back energy: After one model second at 5° with the original load, incoming aerodynamic work is about 4.007 MJ, electrical energy is 4.594 MJ and modeled loss is 0.273 MJ. Stored rotation decreases by about 0.859 MJ. Rounding aside, the complete ledger balances. The finite reservoir cannot sustain unchanged output indefinitely.
17. Pitch does not have one effect everywhere: At table TSR 4, Cp increases from 0.212709 at 0° to 0.244105 at 5°. At TSR 6, the same pitch change decreases Cp from 0.434596 to 0.356023. Some source cells are negative: at TSR 6 and 15°, Cp is −0.039710. That sign represents the rotor giving mechanical energy back toward the air in this model; it is preserved.
18. Stop where the model stops: Our exercise uses 6.9–12.1 rpm, pitch 0–15°, a maximum low-speed load of 4.59 MN·m and electrical output at most 5 MW. These are a selected study envelope and authored load cap, not universal failure thresholds. A boundary event freezes the trial and its ledger. It does not add a hidden brake or continue generating at an artificially clamped speed.
19. Reference models and real experiments: The NREL-5MW reference was documented in 2009; our pinned ROSCO input deck contains a computational performance table dated January 2022. Fontanella and colleagues published the separate wind-tunnel study in 2023. Such comparisons help examine control and modeling. They do not make every later model or material automatically better.
20. What the wind tunnel actually measured: The evidence figures concern a 1:100 IEA 15 MW model, not the NREL 5 MW rotor. Instruments measured generator speed, incoming wind and tower-top loads; aerodynamic loads were obtained through processing. Actual blade-pitch feedback was unavailable, so a commanded pitch set point was used. Distinguish the recorded quantity, its processing and its interpretation.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Actual rendered NREL-5MW rotor reconstructed from pinned ROSCO/OpenFAST blade stations and airfoil profiles (Apache 2.0). Tower, nacelle, materials and display geometry are authored teaching representations, not manufacturer CAD. Separate IEA 15 MW wind-tunnel photographs are credited inside the lesson.