Wind Energy Formula

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

C_t = 40% (Turbine efficiency rating)

C_a = 65% (Alternator/Generator efficiency rating)

Solution:

P = 1/2 x ρ x A x v³ x C_t x C_a

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

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