Calculating Electric Power from Wind Turbines

[Bizkit]

Homework Statement

Consider a wind turbine farm that uses turbines with a blade span diameter of 80 m at a site subjected to wind speeds over the range of 4 to 32 m/s. Taking the overall efficiency of each turbine to be 25% and the air density to be 1.112 kg/m³, determine the electric power generated by a single turbine.

Homework Equations

I know that, for an average wind speed, the equation I need is: W = m(V²/2)*η (the bold variables are per unit time), but I'm not sure exactly how to modify this equation for a range of velocities. I at first thought that I could take the integral of the equation with respect to velocity, but then I end up with the wrong units.

W = m(V²/2)*η = ρVA(V²/2)*η = ρπr²(V³/2)*η = ρπ(d/2)²(V³/2)*η = ρπ(d²/4)(V³/2)*η = ρπ(d²/8)V³*η

The Attempt at a Solution

Since I'm not sure what equation to use, I can't attempt to find the solution.If someone could help me, I would really appreciate it. Thanks.

 

[willem2]

I think they just want the power as a function of v. You already gave this in your last equation.
You might think that you could find the average power by integrating, but this only works if
- the wind is always between 4 and 32 m/s
- all wind speeds between 4 and 32 m/s are equally likely.
Only if these two are valid you could find the avergage power by integrating from v_min to v_max and then dividing by the range of v_max - v_min.

 

[Bizkit]

Thanks for the help. I still have a question, though. How does leaving the power as the function of V I gave above (W = ρπ(d²/8)V³*η) show that V has a range of values, rather than just being an average? I mean, shouldn't there be like a V_low and V_high or something like that to show that V has a range values, or is that not how it works? Hopefully this question doesn't confuse you. Sorry if it does.

 

[jdwood983]

Thanks for the help. I still have a question, though. How does leaving the power as the function of V I gave above (W = ρπ(d²/8)V³*η) show that V has a range of values, rather than just being an average? I mean, shouldn't there be like a V_low and V_high or something like that to show that V has a range values, or is that not how it works? Hopefully this question doesn't confuse you. Sorry if it does.

You kinda answered your own question there. By defining the power as a function of V it follows that V can take a range of values and not just one set value.

 

[Bizkit]

Ok, I understand what you are saying, but I just thought of something else. If the people who wrote the problem wanted me to just define the power as a function of V, then why didn't they just ask me to find the power over a range of wind speeds, without giving any specific values. Since they gave specific values, that makes me think that I need to integrate and divide. What do you think?

 

[MarkP]

Mark: Semiconductor manufacturers are now producing chips that will generate a constant voltage from a source that has varying voltage or current capacity. Some even match their input impedance to maximize total power, rather than just current.

Although you may still wish to explore the mathematical theory regarding such things, these new chips will help eliminate some of those calculation problems.