The method of using wind to generate electricity is known as wind energy. The kinetic energy in the wind is converted into mechanical power by wind turbines.
Wind energy is a renewable energy source that determines the wind’s entire power. Wind turbines convert kinetic energy to mechanical power, which is then transformed into electricity, which is then used as a source of energy.
Wind Energy Formula
P = 1/2ρAv3
Where,
ρ = Density (kg/m3)
A = Swept Area (m2)
v = Wind Speed (m/s)
P = Power (W)
Derivation of Wind Energy Formula
The kinetic energy of an item with mass m and velocity v under constant acceleration is equal to the work done W in displacing that object from its original position.
Under a force F, rest to a distance s, i.e.
E = W = Fs
According to Newton’s Law, we have:
F = ma
Hence,
E = mas … (1)
Using the third equation of motion:
v2 = u2 + 2as
we get:
a = (v2 – u2)/2s
Since the initial velocity of the object is zero, i.e.
u = 0 , we get:
a = v2/2s
Substituting it in equation (1), we get that the
kinetic energy of a mass in motions is:
E = 1/2mv2……….(2)
The power in the wind is given by the rate of
change of energy:
P = dE/dt = 1/2v2dm/dt ……..(3)
As mass flow rate is given by:
dm/dt = ρAdx/dt
and the rate of change of distance is given by:
dx/dt = v
we get:
dm/dt = ρAv
Hence, from equation (3), the power can be
defined as:
P = 1/2ρAv3
Sample Problems
Problem 1: If the wind speed is 20 m/s and the blade length is 50 m, calculate the power in the wind.
Solution:
Given:
Wind speed v = 20 m/s,
Blade length l = 50 m,
Air density ρ = 1.23 kg/m.
The area is given by, A = πr2
A = π × 2500= 7850 m2
The wind power formula is given as,
P = 1/2ρAV3
P = 1/2 x (1.23) x (7850) x 203
P = 38622 W
Problem 2: A wind turbine has a blade length of 20 metres and runs at a speed of 10 metres per second. Determine the amount of wind power available.
Solution:
Given:
Wind speed v =10 m/s,
Blade length l = 20 m,
air density ρ = 1.23 kg/m3,
area , A = πr2
= π × 400
= 1256 m2
The wind power formula is given as,
P = 1/2ρAV3
= 0.5 × 1.23 × 1256 × 1000
P = 772440 W.
Problem 3: Calculate the wind power. Given:
Blade length, l = 22 m
Number of blades = 3
Average Island Wind speed, v = 10 m/sec
Air Density, ρ = 1.23 kg/m3
Solution:
Area, A = πr2
= π x 484
= 1520.5 m2
The wind energy formula is given by,
P = 1/2ρAV3
= 1/2 x (1.23) x (1520.5) x 103
P = 935107.5 W
Problem 4: Determine a realistic power output (in megawatts) for your client that the wind turbine could deliver.
Blade length, l = 22 m
Number of blades = 3
Average Island Wind speed, v = 10 m/sec
Air Density, ρ = 1.23 kg/m3
Ct = 40% (Turbine efficiency rating)
Ca = 65% (Alternator/Generator efficiency rating)
Solution:
P = 1/2 x ρ x A x v3 x Ct x Ca
P = 1/2 x 1.23 kg/m3 x (π x 222) x (10m/sec)3 x 0.4 x 0.65
P = 0.24 MW
Problem 5: The actual available power from a wind mill with diameter 1 m, efficiency 0.2 (20%) – with wind velocity 10 m/s
Solution:
P = 1/2ρAV3
Pa = (0.2) x (1.2 kg/m3) x π x (1 m)2 x (10 m/s)3 / 8
= 94.2 W