The Physics of Wind Power

The mass of air passing through area A in small time dt is rAdx, where r is mass/volume and dx is distance traveled in time dt, so dx = vdt. Its kinetic energy, ¹/₂mv², is therefore ¹/₂rAv³ dt, and the rate of energy passage, or power, is ¹/₂rAv³. We cannot capture all of this energy, because in doing so we would need to bring the wind to a halt, so no more wind, and it makes no sense. We need a continuous flow. In this note we outline a method for finding the theoretical maximum power output of a wind machine.

Define velocities: far upstream of a wind machine, call it v₁; at the machine, v; and far downstream, v₂. Following streamlines, the relevant areas are A₁, A and A₂. The continuity equation is r₁A₁v₁ = rAv = r₂A₂v₂, or in other words, mass flow rate is a constant. A wind machine can get a force equal to the rate of momentum flow into A₁ minus the rate through A₂ , and power is Fv. Momentum flow rate is mass flow rate times velocity, or rAv². Thus,

P = (r₁A₁v₁² - r₂A₂v₂²)v

Power is also the rate of kinetic energy in minus KE rate out, or

P = ¹/₂r₁A₁v₁³ - ¹/₂r₂A₂v₂³
so we equate these expressions, use the continuity equation to get rid of the rA terms, then do some algebra, and we find that v is the average of v₁ and v₂,
v = (v₁ + v₂)/2. By combining this and the continuity equation with one of the power equations, one can show

P = (1/4)rA(v₁³ – v₂³ + v₁²v₂ - v₁v₂²)

Nature gives us v₁ and the machine takes energy from the wind, reducing its velocity. At maximum power, dP/dv₂ = 0. Do this and it reduces to a quadratic in v₂, from which show v₂ = v₁/3. Insert this above and get

P_max = (16/27) ¹/₂rAv₁³

The coefficient 16/27 or 0.593 is known as the Betz coefficient, after the guy who discovered the derivation. Real coefficients are necessarily much lower, but it is good to have an upper limit to shoot for. The r is not the r_o of still air, but close. It varies with pressure, so using precision like 0.593 is unjustified. I tried to find how r changes with changes of v₂, but could not find it. The derivation above treats r as a constant, a good approximation. So the best we can say is that the upper limit of power is in the neighborhood of (0.6) ¹/₂rAv₁³. With a good wind machine, the typical power coefficient is 0.4. Area A is the area swept out. In physics lab at Cuyahoga Community College, we have measured the power output of a tiny propeller-generator combination, and we found a coefficient of about 0.004. The main problem, we think, was due to an inefficient generator.

Some useful numbers: Typical air density is around 1.2 kg/m³. Using a power coefficient of 0.4, with A = 1 m² and v=1 m/s, we get P = 0.24 W. So if A = 4 m² and v = 5 m/s for example, just multiply the 0.24 times 4 times 5³ and get 120 W power output. Similarly if the power coefficient is 0.4 and A = 1 ft² and v = 1 mph, P = 0.0020 W. Multiply this by the area in ft² times the cube of the wind speed in mph. For example at 10 mph we get 2 W per square foot and at 20 mph we get 16 W/ft².

The Savonius Machine

Conventional wind machines use propellers, as you know. This is an old, well developed technology, and these machines are reliable and efficient. The Savonius machine, described below, is less efficient, but it has two advantages: 1) It can cover a larger area than a propeller type. 2) It is a vertical axis machine, so the natural location for the generator is at or near ground level, an advantage for maintenance. The machine consists of two vertical cylinder surface portions, top view shown here (a split Savonius):

Say the wind is going left to right. Then the upper half cylinder feels a force toward the top of this page and the lower toward the bottom, and the structure rotates clockwise. If the top view looks like this, ⁽ ₎ the rotation is counterclockwise and this ~ would rotate clockwise. The Bernoulli effect creates a low pressure on the outer curved surfaces, and the wind going between the vanes pushes on one and then the other. Both of these effects help turn it.

The propeller wind machine has a blade length limited by material strength. The Savonius could be as tall as a skyscraper and as wide as the space available, but then the bearings would be a major challenge. Design a big one having the weight of a locomotive, then use locomotive wheels and a circular track. A cheap, rugged Savonius for the back-yard mechanic: cut a 55 gallon drum in half and arrange the two halves as in the sketch above. I read somewhere about one that was built, and I think it had a power coefficient of about 0.2.

The Darrieus Machine

Another vertical axis wind turbine (VAWT) is the Darrieus, with eggbeater-like blades having a teardrop cross-section.

An Enhancement

The efficiencies of both the Savonius and Darrieus machines can be enhanced with a shield to divert wind away from the part that is moving against the wind. For example the dotted circle below is the top view of a clockwise rotor and wind is going from left to right. The pointed thing diverts wind away from the side moving against the wind, so it makes the rotor spin faster. If the wind direction changes, the diverter must rotate to always point into the wind, a difficulty, but probably worth the trouble.

You can find more information on vertical axis machines at http://www.windmillworld.com/links/verticalaxis.htm (with some defunct links) or try the American Wind Energy Association at http://www.awea.org/default.htm . For the Darrieus machine, look at http://windturbine-analysis.com/ . Also try http://www.eia.doe.gov/cneaf/solar.renewables/renewable.energy.annual/backgrnd/tablecon.htm for renewables in general.

Questions, comments: fredrick.gram @ tri-c.edu

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