kWh = (1/2)(p)(V^3)(A)(E)(H), where
"p" (normally "rho", which looks like a small "p") is the density of air
"V^3" is the CUBE of the wind velocity
"A" is the area swept by the turbine rotors
"E" is the efficiency of capturing the kinetic energy that exists in a unit area of intercepted wind (such as "A", above) for the given wind power capturing device at the given wind speed. This,in theory, can never exceed 59.3 percent, the "Betz Limit". "Power Coefficient" is the technical term used for this parameter in the wind industry. "Power Coefficient" should not be confused with "Capacity Factor", which is the proportion of energy actually captured compared to what would be captured if running at rated capacity full time. In theory, this could be 100 percent.(Please see the "Capacity Factor" section).
"H" is the number of hours for which the power was captured
However, since the wind velocity is constantly changing, the total kWh is an integration of this formula with respect to time, i.e. delta time instead of "H" in the above formula.
While the most important element in the above wind power math formula is the cube of the wind velocity, air density, which is a linear factor, does decrease with altitude. At 15,000 feet it is typically 57 percent of its density at sea level and at 30,000 feet it is typically about 31 percent of its density at sea level.
The capacity factor calculations in the next section for the Roberts' rotorcraft designed to operate at whatever altitude up to 15,000 feet, 30,000 feet or 35,000 feet, will produce the most power on a given day take into account all the factors applicable to those situations.
An interesting calculation for the total amount of wind energy which theoretically could have been captured in the year 2001 from a column of air only one meter wide between ground and 29,000 feet, assuming a power coefficient of thirty percent for the capturing means, comes out:
For San Diego....30,015 megawatt hours
For Oakland......47,286 megawatt hours
For Topeka.......66,405 megawatt hours
That gives an idea about how little intercepted vertical area can potentially produce so much power, even though practically many factors reduce that potential.
But the fact that wind velocity is typically two to three times as great at 30,000 feet, in some cases even at 15,000 feet, as it is at 100 feet above ground means a factor of eight to twenty-seven as great for these high altitude winds, reduced only by the lesser air density factors at those altitudes.
When the higher "capacity factor" figures, discussed in the next section, are taken into account as well, captured energy factors per square meter of intercepted wind get further multiplied by a factor of two to three as compared to average ground level winds.
It is more expensive to capture the wind energy from a square meter of wind using a flying electric generator than using a ground based wind turbine, but since the energy captured from each square meter of wind is so much greater at high altitude, the cost per kilowatt hour calculates out to be far less.
The use of ground based wind turbines is highly to be recommended for relatively local use near sites where excellent ground level winds are available. But utilizing the energy from high altitude winds, available virtually everywhere between the latiudes of thirty and fifty degrees, will be able to replace oil and other fossil fuels more generally on the basis of economics.
