Question 6 of 6: Wind Turbines

Question 6: Wind Turbines (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Basic principles, and the pressure and velocity through the rotor

Wind is solar energy in mechanical form: roughly 2 % of the solar radiation reaching the earth is converted into atmospheric motion by differential heating of land, sea and the poles, modified by the earth's rotation. A wind turbine intercepts part of that kinetic energy. Air of density \(\rho\) crossing a swept area \(A\) at velocity \(V\) delivers a mass flow \(\rho A V\), each kilogram of which carries \(V^2/2\) of kinetic energy, so the power available in the wind is

Two features of that expression dominate wind engineering. The cube law means that a site with 25 % more wind yields nearly twice the energy, so siting matters more than any refinement of the machine; it also means the wind-speed distribution, not the mean, determines annual yield. The dependence on swept area, which goes as the square of the diameter, is why rotor diameters have grown from 15 m to over 150 m: doubling the diameter quadruples the output.

The turbine cannot simply "absorb" this energy, because to extract power it must slow the air, and slowing the air makes the stream tube expand. The standard idealisation is the actuator disc of Figure 6.1: the rotor is replaced by a permeable disc that extracts momentum uniformly, the flow is steady, incompressible, frictionless and non-rotating, and far upstream and far downstream the pressure is atmospheric.

The changes across that disc are as follows. Velocity falls continuously and smoothly from the free-stream value \(V_1\) far upstream, through \(V\) at the disc, to \(V_2\) in the far wake; it cannot jump at the disc, because mass continuity through a fixed area forbids it. Pressure, by contrast, behaves quite differently: it rises above atmospheric as the approaching air decelerates ahead of the rotor (Bernoulli applies in that region, where no work is done), then falls abruptly and discontinuously across the disc — this pressure difference \(\Delta p\), acting on the disc area, is the axial thrust and is the mechanism by which energy is taken out — and then recovers gradually back to atmospheric in the wake as the slowed air mixes with its surroundings. The stream tube widens throughout, since the same mass flow passes ever more slowly.

Applying momentum and energy to the control volume gives two results used in part (b): the disc velocity is the arithmetic mean of the far-field velocities,

and the extracted power is

At blade level, the mechanism is aerodynamic lift, not drag. Each blade element sees a relative wind that is the vector sum of the axial wind and the local blade speed \(\omega r\); the aerofoil section develops lift perpendicular to that relative wind, and because the relative wind is inclined, the lift has a component in the plane of rotation which drives the rotor. This is why modern machines have two or three slender blades turning fast rather than many broad ones turning slowly, and why the blades are twisted and tapered from root to tip — the inflow angle changes along the span as \(\omega r\) grows.

(b) The Betz limit, and why real machines reach only about 45 %

Define the axial induction factor \(a\) by \(V = V_1(1-a)\); from the mean-velocity result above, \(V_2 = V_1(1-2a)\). Substituting into the power expression,

so the power coefficient is \(C_p = 4a(1-a)^2\). Differentiating and setting \(dC_p/da = 0\) gives \(a = 1/3\), whence

This is the Betz limit. The physical argument behind the algebra is more illuminating than the calculus: the turbine can only take energy by slowing the air, but the air must keep moving to carry the next parcel through the disc. Extract nothing and the air passes at full speed with all its energy intact; extract everything, so that \(V_2 = 0\), and the flow stops — the stream tube would have to expand to infinite area and no mass would pass at all. Between those two useless extremes there is an optimum, and it lies at slowing the wind to one third of its upstream speed by the far wake, two thirds of it at the disc. At that condition the reference paper's own formula, \(P_{max} = 8\rho A V_1^3/27\), is exactly \(\tfrac{16}{27}\times\tfrac{1}{2}\rho A V_1^3\). Note that the limit is not a Carnot-like thermodynamic constraint and involves no losses whatever: it is a pure consequence of mass and momentum conservation on an ideal, frictionless, loss-free disc. No rotor of any design can exceed it.

A real machine reaches only about three quarters of 0.593, that is \(C_p \approx 0.45\), because the actuator disc idealises away five real effects:

• Wake rotation. The rotor applies a torque to the air, so by reaction the wake spins. That rotational kinetic energy is left in the flow and is lost. Its cost falls as the tip-speed ratio \(\lambda = \omega R/V_1\) rises — at \(\lambda = 1\) it removes nearly half the available power, at \(\lambda = 7\) only a few percent — which is the main reason modern rotors are designed to run fast.
• Blade drag. Aerofoils have a finite lift-to-drag ratio, typically 100–150 for a clean blade. Profile drag costs roughly \(C_p \times \lambda/(L/D)\), a few percent at design conditions and much more once the blades are roughened by rain erosion, insects or ice.
• Tip losses. The rotor has a finite number of blades, not the infinite number the disc assumes, so air leaks around each tip from the pressure to the suction side, shedding a vortex and unloading the outer part of the span. Prandtl's tip-loss factor accounts for this; three blades lose about 3–5 %, one blade far more.
• Off-design operation. \(C_p\) peaks at one tip-speed ratio only (Figure 6.2). Even a variable-speed machine is held at fixed speed below cut-in and above rated wind, and above rated the blades are deliberately pitched to spill power and protect the drive train, so \(C_p\) is intentionally reduced.
• Transmission and generator losses. Gearbox, generator and power converter together take a further 8–12 %, so the electrical coefficient at the terminals is lower again — typically 0.40–0.45 at best point for a large modern machine.

The arithmetic therefore runs roughly \(0.593 \times 0.97\ (\text{wake rotation}) \times 0.95\ (\text{drag}) \times 0.96\ (\text{tip}) \times 0.90\ (\text{drive train}) \approx 0.47\), which is the observed figure. Note carefully that this 45 % is a power coefficient at the best wind speed, not an annual capacity factor: because of the wind-speed distribution, a good onshore site yields 30–40 % of nameplate over a year and an offshore site 45–55 %.

(c) Operational limitations and environmental effects

Operational limitations. The governing difficulty is that the resource is variable and non-dispatchable. Output follows the cube of a wind speed the operator does not control; the machine produces nothing below the cut-in speed of about 3–4 m/s, holds rated output between roughly 12 m/s and the cut-out speed of 25 m/s, and shuts down entirely above that to protect itself — so the very stormiest hours contribute nothing. Forecast error means the system operator must carry reserve, and at high penetration must provide flexible generation, storage, interconnection or demand response to firm the supply. In British Columbia and Quebec the natural partner is large hydro with storage, which can be throttled to follow wind and effectively acts as a battery; in Alberta and Ontario the balancing duty falls on gas.

Beyond variability, the practical constraints are: low power density, so that a 100 MW wind farm occupies tens of square kilometres (although the turbines themselves take only 2–5 % of it and farming continues between them); siting, since the best winds are often remote from load and require new transmission, the single largest obstacle to Canadian wind development; grid stability, because inverter-connected machines contribute little inherent inertia or short-circuit strength; icing, a genuine and specifically Canadian problem that unbalances rotors, degrades the aerofoil and throws ice, requiring heated blades or winter shutdown rules; turbulence and wake interaction, which forces spacing of five to ten diameters and still costs 5–15 % array losses; and fatigue, since the blades see 10⁸ or more load cycles in a 25-year life, making gearbox and blade reliability the dominant maintenance cost, particularly offshore where access is weather-limited.

Positive environmental effects. Wind generation emits no carbon dioxide, sulphur dioxide, nitrogen oxides, mercury or particulate matter in operation, so it displaces exactly the emissions of whatever fossil plant it replaces; the life-cycle figure of about 10–20 g CO₂-equivalent per kWh, all of it embodied in manufacture and construction, is one to two orders of magnitude below coal's 900–1000 g. Energy payback is short — a modern turbine repays the energy used to build it in six to twelve months against a design life of 25 years. It consumes essentially no water, which by comparison with Question 4's 0.129 m³/kWh of once-through cooling or 3.4 L/kWh of tower evaporation is a substantial benefit in a drought-prone region. It creates no mine tailings, no ash ponds, no radioactive waste and no thermal discharge to lakes and rivers; the fuel is free and immune to price shocks; and the plant is modular, quick to build and almost entirely removable, with foundations that can be reclaimed at decommissioning.

Negative environmental effects. The most contested are visual and landscape impact, since 150 m machines on ridgelines are visible for tens of kilometres and provoke genuine opposition; noise, both the aerodynamic swish of the blades and low-frequency mechanical tones, generally managed by setback distances of 550 m or more, as required in Ontario; and shadow flicker from rotating blades crossing low sun, which is mitigated by siting and by scheduled shutdowns. Wildlife effects are real but need proportion: bird mortality per turbine is measurable and matters greatly if a farm is placed on a migratory corridor or near a raptor concentration, yet it is small compared with collisions with buildings, vehicles and domestic cats. Bat mortality, caused largely by barotrauma in the low-pressure region behind the blades, is proportionally more serious and is mitigated by raising the cut-in speed during low-wind late-summer nights. Further concerns are habitat fragmentation from access roads, offshore construction noise affecting marine mammals, land-use conflict, effects on radar and telecommunications, blade-disposal difficulty since thermoset composites are hard to recycle, and the mining impacts of the rare-earth magnets used in direct-drive generators.

The balanced judgement is that wind's operational impacts are real, local, largely manageable by siting and operating practice, and small in comparison with the global impacts of the fossil generation it displaces. In Canadian practice these are weighed through provincial environmental-assessment processes with mandatory consultation, and — where projects touch traditional territory — through Indigenous consultation and increasingly through Indigenous partnership ownership, which has become one of the more successful features of Canadian wind development.