Wind Speed Problem: Finding Total Electrical Output Proportional to v

[physixguru]

A windmill power generator converts a fixed fraction of wind energy into electrical energy..

Supposing the wind falls on the windmill at a speed v...[wind speed=v]

What is the total electrical output proportional to..?

1.v^2
2.v^3
3.v
4.v^4

 

[kamerling]

Find out how much air goes through the windmill each second and multipy by
the kinetic energy of that air.

 

[physixguru]

hey bro..pls be a bit more specific...

what do you mean by " how much air goes through the windmill'?

 

[physixguru]

can someone guide me through pls.

 

[kamerling]

Find the volume of air that will will go through the windmill in the next second. Assume for this that the windspeed is completely uniform, and the windmill won't influence it in any way. What shape is this volume? This volume will depend on the wind speed and the radius of the windmill
Now suppose we can convert all of the kinetic energy of this volume to electrical energy.

 

[physixguru]

game.

 

[MacG]

1D momentum theory:

T=0.5*rho*A*V^2*CT where CT=4*a*(1-a)
P=0.5*rho*A*V^3*CP where CP=4*a*(1-a)^2

rho: density of fluid kg/m^3
A: Area of turbine m^2
V: Free stream wind velocity m/s
a: axial induction factor, nondimensional. Axial flow velocity at rotor disc is V*(1-a).
T: Thrust force on rotor
P: Power form wind

a=1/3 maximizes power (for given A, rho and V), so this is usually close to what is aimed for on a wind turbine (below rated power).

So Trust force scales with V^2, and Power with V^3. (when running the turbine at the same design point, that is the same 'a')

-Mac G

 

[physixguru]

Thx all.