Wind Turbine

I

Introduction Since 3000 B.C, wind energy is used, sails captured the energy wind to pull a boat across the water. Then the wind power was used to grind grain in the windmill. Now we use this energy to produce electricity thanks to wind turbine.

II

The elements of a wind turbine

A wind turbine works the opposite of a fan. Instead of using electricity to make wind, a turbine uses wind to make electricity. The main composants of a wind turbine are the blades, the shaft, the generator and the tower. The wind turns the blades, which spin a shaft, which connects to a generator and makes electricity. The electricity is sent through transmission and distribution lines to a substation, then on to homes, business and schools. A generator consists of magnets and a conductor (the conductor is typically a coiled wire). Inside the generator, the shaft connects to an assembly of magnets that surrounds the coil of wire. When the shaft spins the assembly of magnets, electricity is produced in the coil of wire.

III

A function

The formula for calculating the power from a wind turbine is : P =

1 ρACp v 3 2

where, – P is the power produced in Watt. – ρ is the air density in kg/m3 . – A is the swept area in m2 . If l is the lenght of a blade, then A = πl2 . – Cp is the power coefficient. According to the Bezt’ law, no more than 59% of the energy carried by the wind can be extracted by a wind turbine. The real world limit is well below the Betz Limit with values of 0.35-0.45 common even in the best designed wind turbines. – v is the wind speed in m/s. We are given the following data : Blade length, l = 52m Air density, ρ = 1.23kg/m3 Power Coefficient, Cp = 0.4 Questions 1. Calculate the power in M egaW att(M W ) produced by the wind turbine when the wind speed is 12m/s. 2. Let P be the function which associates the wind speed v to the power (in M W ) produced by the wind trubine. Give P (v) in terms of v and give the value of P (12). 3. Make a table of values of P with v going from 0 to 27, then plot the graph of P . 4. Graphically, solve the equation P (x) = 12. What does the result represent ? 5. Graphically, solve the inequation P (x) > 24. What does the result represent ? 6. Give the variations of the function P . Extra question : 7. In USA, the air density unit is pounds per cubic foot (lb/f t3) . Knowing that 1 foot = 30.5 cm and 1 pound = 0.454 kg, convert the air density 1.23kg/m3 into lb/f t3 .