Can Riemann Sums Accurately Account for Yearly Wind Turbine Hours?

[Physics197]

Homework Statement

The following table shows the power produced by a 600kW wind turbine at the given wind speed and the number of hours the wind blows at that speed.
a) Plot the power characteristic as a function of wind speed.
b) Plot the wind duration curve as a function of wind speed.
c) Use a Riemann sum to verify that all 8760 hours in a year are accounted for.
d) Use a Riemann sum to calculate the total yearly energy capture for this wind location (remember energy = power x time).
e) If the power curve is p(t) and the wind duration curve is w(t), write the energy as an integral.

Wind Speed Wind duration Turbine output
m/s hours/year kW
0 0 0
2.5 1880 0
5 2580 10
7.5 2000 80
10 1300 220
12.5 650 400
15 290 550
17.5 60 600

Homework Equations

The Attempt at a Solution

I have already graphed a&b, How would you use riemann sums to show that all the hours are accounted for? Wouldn't you just add up the number of hours?

d & e are going over my head, but they don't seem to be hard

 

[mighty2000]

, just a little confusing.

Hello! Thank you for your post. I can help you with your questions.

a) To plot the power characteristic as a function of wind speed, you can create a scatter plot with wind speed on the x-axis and turbine output on the y-axis. The points on the plot will represent the different wind speeds and the corresponding power output of the turbine.

b) To plot the wind duration curve as a function of wind speed, you can create a line graph with wind speed on the x-axis and wind duration on the y-axis. The line on the graph will show the relationship between wind speed and the number of hours the wind blows at that speed.

c) To use a Riemann sum to verify that all 8760 hours in a year are accounted for, you can divide the year into smaller time intervals (such as hours, days, or months) and calculate the total number of hours in each interval. Then, you can add up all the hours in each interval to make sure it equals 8760 hours. This will show that all the hours in a year are accounted for.

d) To use a Riemann sum to calculate the total yearly energy capture for this wind location, you can use the formula for Riemann sums, which is ∑ f(xi)Δx. In this case, f(xi) represents the power output at each wind speed and Δx represents the corresponding wind duration. You can calculate the Riemann sum by multiplying the power output at each wind speed by the wind duration and adding all the values together. This will give you the total energy captured by the wind turbine for the entire year.

e) The energy can be written as an integral by using the formula ∫ p(t)w(t)dt, where p(t) is the power curve and w(t) is the wind duration curve. This integral represents the area under the curve on a graph of power vs. time. By calculating this integral, you can find the total energy captured by the wind turbine for the entire year.

I hope this helps! Let me know if you have any further questions.