Wind turbine energy HW Question

[JuggerNauto]

Hi everyone. I am having extreme trouble finding the answer to this problem. I thought I originally had it, but nope. The original Q is: "Assume a wind turbine with a hub 50 meters above the ground, a rotor diameter of 65 meters and a wind-conversion efficiency of 25 percent. The turbine operates in an area with an average wind-power density of 300 watts/sq meter at 50 meters altitude. How much electricity (kwH) can the turbine generate per year? <--- First Q.

Wind densities greater than or equal to 500 watts/sq. meter at an altitude of 50 meters are present on 0.700 percent of the land area of the US. If, on average, wind farms contain 8 turbines/sq km, how much electricity (kwH/yr) could be generated from these wind turbines? (The area of the US is 7,827, 989 sq km)." <-- Second Q.

Here's my reasoning & thought:

The energy generated = efficiency * (power per unit area) * area * time
(In one year) = 0.25 * 300W/m2* (65/2)2m2*86400*365 Joules = 0.25 * 300 * 3318.31 * 86400 * 365 J
= 7.8485*10^12 J (1 J = 2.778*10^-7KWH)
= 2180313.3 KWH/Yr <--- 1st ANSWER

The total no. of turbines can then be installed by:

N = (Total area available) * no of turbines per unit area
= (7,827, 989 sq km * 0.007) * 8 per sq Km
= 438367(.384)

The energy generated by one turbine = efficiency*(power per unit area)*area* time
= 0.25 * 500W/m2* (65/2)2*86400*365 J = 0.25 * 500 * 3318.31 * 86400 * 365 J
= 1.308*10^13 J
= 3633855 KWH/year

therefore total power generated = 3633855*438367 = 1.592*10^12 KWH/year <--- 2nd ANSWER.

But the teacher tells me "this is not correct," and won't tell me why or what to do. I'm helplessly lost & confused. I thought I had it down. Please tell me what you guys think.

 

[JuggerNauto]

Any help please? I think I have gotten the numbers correct, but unsure of the formula or derivation. Thank you.

And...I realized I made a silly typo. Forgot to add in pi... for some reason my paste does NOT work. The pi is already factored into my original post as the #'s should come out with pi:

"(In one year) = 0.25 * 300W/m2*pi (65/2)2m2*86400*365 Joules = 0.25 * 300 * 3318.31 * 86400 * 365 J
= 7.8485*10^12 J (1 J = 2.778*10^-7KWH)
= 2180313.3 KWH/Yr <--- 1st ANSWER

The total no. of turbines can then be installed by:

N = (Total area available) * no of turbines per unit area
= (7,827, 989 sq km * 0.007) * 8 per sq Km
= 438367(.384)

The energy generated by one turbine = efficiency*(power per unit area)*area* time = 0.25 * 500W/m2*pi (65/2)2*86400*365 J = 0.25 * 500 * 3318.31 * 86400 * 365 J"

Sorry for the confusion.

 

[ogb p]

Hello there,

I can understand your frustration with this problem. It can be difficult to find the correct answer when there are multiple variables involved. However, let's break down the problem and see if we can find the correct solution together.

First, let's look at the initial question about the wind turbine. Your calculation for the energy generated per year is correct. However, the conversion from Joules to kWh is incorrect. The correct conversion factor is 2.778*10^-7 kWh/Joules, not the other way around. So the correct answer would be 0.218 kWh/year.

Moving on to the second question about the total number of turbines that can be installed, your calculation for the total number of turbines is correct. However, your calculation for the energy generated per turbine is incorrect. The wind density given in the problem is 300 watts/sq meter, not 500 watts/sq meter. So the correct calculation would be 0.25 * 300 * 3318.31 * 86400 * 365 J = 1.178*10^13 J = 3272206 KWH/year. And the total power generated would be 3272206*438367 = 1.436*10^12 KWH/year.

I hope this helps you understand where you went wrong in your calculations. It's important to pay close attention to units and conversion factors when solving scientific problems. Keep practicing and don't hesitate to ask for help when needed. Good luck!