
 HOW MUCH WATTAGE CAN A WIND TURBINE
MAKE?

 Any moving object has kinetic energy.
In classical mechanics, its amount E in joules is given by the
equation E=mv2/2,
 where m is the mass in kg, and v is
the speed in m/sec. Air molecules have mass, and when they are
in motion, they
 contain kinetic energy that can be
converted into other forms for practical use.
 When molecules hit a surface of any
object that is allowed to move, their motion is partially transfered
to the moving
 object. Particularly, in wind turbines
the energy is extracted from the air as it moves through the
"swept area" of the
 turbine's blades. During this process
the air turns the aerodynamically designed blades, which transfer
this harvested
 energy into a spinning shaft. The shaft
is connected to a generator's rotor whose motion makes electricity.
HOW MUCH POWER IS IN THE WIND

 The wind energy diagram below illustrates
the process of energy transfer. If D is the diameter of the turbine's
blades, they
 intercept the air in the crosssectional
area A=(D/2)2.
In a time t, the mass of the air that will pass through this
area is m=?xAxvxt, where ? is the density of the air
 (approximately 1.2kg/m3 at sea level).
 By combining the above formulas, we
can calculate the energy of the air that passes through an area
A in a time t:
 E=?xAxv3xt/2
Then power in watts being E per unit time is given by:
 P= ?xAxv3/2
Note that to get the result in watts, all the values in these
formulas have to be expressed in SI units (for nonSI units we
 would need to add some conversion coefficients).
We see that power available in the wind is proportional to the
cube of
 its speed and the size of the turbine's
blades. If for example, the speed doubles, the available watts
increases by a factor of
 eight.

 HOW MUCH ELECTRICITY A TURBINE CAN
GENERATE

 The above formula for P represents
the amount of power in the imaginary tube of the air that flows
through the turbine's
 swept area A. However, only a fraction
of this wind power can be actually extracted there is no way
to harvest all of it. If
 all of air's energy was transfered
to the turbine, the air molecules that hit the blades would have
to come to a complete
 stop. This is impossible since for
continuous operation since the molecules that already hit the
blades need to get out of
 the way to let the air that is behind
them hit the blades. If all the air motion was transferred to
the blades, the air would pile
 up in front of the turbine, and then
the wind would have to blow around the turbine. The fact is,
the air that hits the blades
 must keep some speed to move out of
the way to allow continuous air flow into the turbine. According
to physics, the
 theoretical limit of wind energy that
can be transfered to the shaft is 59.26%. This fact is known
as the Betz Limit. In
 practice, the collection efficiency
of commercially manufactured rotors for home use is typically
25% to 45%. Small
 models for homes tend to have the efficiency
at the lower end of this range.
Example. Suppose you have a micro turbine with blade diameter
1 m (about 3 ft) and efficiency 20%. Let's calculate how
 much electricty it can generate for
your home at the air speed 6 m/sec (which is about 13.4 mph).
 Rotor swept area: A= (D/2)2 = 3.14x(1/2)2
= 0.785 m2

 Available power in the wind: Pwind=
?xAxv3/2 = 1.2x0.785x63/2 = 101.7 watt

 Then the power that can be extracted
at that speed is: Pturbine=0.20x101.7=20.3 watt.

 Note: Betz law says a turbine can only
convert 59% of it's energy into power.
