Three years ago Inc magazine praised a recently-funded startup called WindTronics. Their energy claims for their $5500 rooftop wind turbine seemed so absurd that I suspected Inc had botched the technical details. Since then I’ve followed the Michigan firm. Their rooftop wind turbine was awarded “Best of What’s New” by Popular Science magazine last November. It was called “one of the 10 most brilliant products of 2009” by Popular Mechanics. In 2009 they moved their production to Ontario. They recently closed operations in Ontario and moved back to Michigan. Reports say Canadians aren’t happy about the $2.7 million Canada gave the company as an incentive to set up operations there. The Windsor Star reports that WindTronics left without making good on its debts.

There may be two sides to the financial issues; I didn’t dig very deep. The technical claims, however, are another matter. Some basic analysis reveals big problems with the claims.

Windtronics make a 6-foot diameter rooftop wind turbine. They claimed the device could supply 18% of an average household’s electricity, based on a 12.8 mph wind speed. Without knowing a thing about their technology, it’s very easy to debunk this. They also claim it generates power down to a wind speed of two miles per hour. This is true, but highly deceptive.

The wind in Chicago, the windy city, averages about 10 mph. Kinetic energy is equal to ½ the mass of the moving matter times its velocity squared. So wind energy extracted from moving air – if you could catch it all – would be proportional to the square of the wind speed. Cut the speed in half and you end up with one fourth of the energy. – You’d cut the ideal maximum by 75 percent, assuming the turbine were equally efficient at both wind speeds – which is impossible. At two mph wind speed, the maximum theoretical power would be 4% of the power at 10 mph. But a few more details will show it to be even far less than that.

Large modern wind turbines have an efficiency of about 40%, but they reach this maximum at the specific wind speed for which they were designed. The efficiency is constrained by frictional losses at low speeds and back pressure (the “lift” that makes an aircraft fly) on the blades above the design speed. Above or below the optimum wind speed, efficiency drops off steeply. For example, at twice their design wind speed, the efficiency of commercial wind turbines drops to about 10%.

Betz’ Law, a principle of hydraulics, shows that the maximum energy that a turbine of any design can extract from such a wind turbine is exactly 16/27 (~59%) of the kinetic energy of wind. The Windtronics machine is six feet in diameter. Assuming its blades go to the very outer diameter of their housing, its wind area is 28 square feet. Using average air pressure, temperature and humidity and a Rayleigh distribution of wind speed, one can then calculate the energy in a 6-foot diameter tube of air moving at 12.8 miles per hour. 59% of that will be the maximum possible energy that the Windtronics machine could produce if it were a perfect machine. That equates to 2000 kWh per year. But that value is for a machine that is frictionless.

At an optimistic efficiency of 50% and a wind velocity of 6.5 miles per hour, the calculated yearly output of the WindTronics turbine is 404 kWh, which is about 4.0% of the average household’s electrical usage, based on Department of Energy usage numbers.

Also per the DOE, the average cost of residential electricity in the United States was (and still is) 12 cents per kWh when WindTronics released their turbine. The average household uses 11,000 kWh per year, and therefore, pays about $1300 for all their electricity. If the rooftop turbine supplies 4% of that and costs $5500, you could amortize your purchase in a mere 100 years, assuming your installation costs are zero and the unit lasts a century without maintenance.

Consumer Reports evaluated the turbine in October 2011 and reported an installation cost of about $11,000. They said they got only a fraction of the power WindTronics told them to expect and noted that it would not pay for itself in its expected 20-year life. My quick analysis suggests they put it mildly.

Windtronics explains the magic of their gizmo:

Our wind turbine utilizes a system of magnets and stators surrounding its outer ring, capturing power at the blade tips where speed is greatest, practically eliminating mechanical resistance and drag. Rather than forcing the available wind to turn a generator, the perimeter power system becomes the generator by swiftly passing the blade tip magnets through the copper coil banks mounted onto the enclosed perimeter frame.

While there’s nothing actually false in those words, they seem to aim at baffling more than illuminating. Elegant words whose meaning is lost somewhere in a vast windswept expanse.