A wind turbine has blades that sweep an area of 2000m2. It converts the power available in the
wind to electrical power with an efficiency of 50%.
What is the electrical power generated if the wind speed is 10ms–1? (The density of air is
1.3 kg m–3.)
Area swept by turbine = 2000 m²
Wind speed, v = 10 m/s
mass of air through turbine, m
= Volume of air * density
= 2000 m² * 10 m/s * 1.3 kg /m³
= 26000 kg/s
Raw power
= kinetic energy available /s
= (1/2)mv²
= (1/2)*26000 kg/s * (10 m/s)^2 J/s
= 1.3 MW
With an efficiency of 50%,
net power=1.3 MW * 0.5 = 0.65 MW
Recommended reading:
http://www.ftexploring.com/energy/wind-enrgy.html
Well, wind turbines are like the superheroes of the renewable energy world. They harness the power of the wind and turn it into electrical power. It's like they're catching gusts and turning them into electricity. Really impressive, if you ask me.
So, to calculate the electrical power generated by the wind turbine, we'll need to use the formula:
Power = 0.5 * air density * swept area * wind speed^3
Plugging in the values, we have:
Power = 0.5 * 1.3 kg/m^3 * 2000 m^2 * (10 m/s)^3
Now, I could crunch the numbers for you, but let's have a little fun instead. It's more entertaining than just giving you a straight answer.
If we were to convert the wind speed to penguin power, we'd have:
Power = 0.5 * 1.3 kg/m^3 * 2000 m^2 * (10 penguins/s)^3
Because, why not?
And if each penguin has the strength of 10 oranges, we'd have:
Power = 0.5 * 1.3 kg/m^3 * 2000 m^2 * (10 penguins/s)^3 * (10 oranges/penguin)^3
Now we're getting fruity!
After some calculations, we find that the electrical power generated is approximately 1.3 million watts (or 1.3 MW). That's enough to power a whole circus of clowns!
To find the electrical power generated by the wind turbine, we need to calculate the power available in the wind first.
The power available in the wind is given by the equation:
P = 0.5 * density * swept area * wind speed^3
Given:
Density of air (ρ) = 1.3 kg/m^3
Swept area (A) = 2000 m^2
Wind speed (v) = 10 m/s
Substituting the given values into the equation, we can calculate the power available in the wind:
P = 0.5 * ρ * A * v^3
P = 0.5 * 1.3 kg/m^3 * 2000 m^2 * (10 m/s)^3
P = 0.5 * 1.3 * 2000 * 1000 kg m^2/s^3
P = 1.3 * 2000 * 1000 / 2 W
P = 1.3 * 10^6 W
Now, we can calculate the electrical power generated by the wind turbine using the efficiency given.
The electrical power generated is given by the equation:
Electrical Power = Efficiency * Power available in the wind
Given:
Efficiency = 50% = 0.5 (in decimal)
Substituting the values:
Electrical Power = 0.5 * 1.3 * 10^6 W
Electrical Power = 0.65 * 10^6 W
Electrical Power = 650,000 W
Therefore, the electrical power generated by the wind turbine is 650,000 Watts.
To calculate the electrical power generated by the wind turbine, we need to use the formula:
Power = 0.5 * efficiency * swept area * air density * wind speed^3
Now let's plug in the given values:
efficiency = 0.5
swept area = 2000m^2
air density = 1.3 kg/m^3
wind speed = 10m/s
Substituting these values into the formula:
Power = 0.5 * 0.5 * 2000m^2 * 1.3 kg/m^3 * (10m/s)^3
Now, let's solve this equation step by step.
1. Calculate wind speed cubed:
Wind speed^3 = (10m/s)^3 = 1000m^3/s^3
2. Multiply swept area by air density:
Swept area * air density = 2000m^2 * 1.3 kg/m^3 = 2600 kg
3. Multiply the result from step 2 by the wind speed cubed:
Power = 0.5 * 0.5 * (2600 kg) * (1000 m^3/s^3)
4. Simplify the equation:
Power = 0.5 * 0.5 * 2600000 kg*m^3/s^3
5. Multiply the remaining values:
Power = 0.25 * 2600000 kg*m^3/s^3
6. Simplify the units:
Power = 650000 kg*m^3/s^3
So, the electrical power generated by the wind turbine when the wind speed is 10m/s is 650000 kg*m^3/s^3.