Power of a Fan and WindMill using Energy vs dynamic method)

[unscientific]

Homework Statement

Determine the power of a windmill using constants where appropriate.

Homework Equations

P = Fv
K = (1/2)mv²

The Attempt at a Solution

Dynamics Method
P = Fv

In time Δt, Δm amount of wind passes through the windmill, at speed of v.

Δp = vΔm
= v(AρΔx)

Dividing both sides by Δt,

F = Aρv²

P = Fv = Apv³Energy Method

In time Δt, Δm amount of wind passes through the windmill, with KE of (1/2)(Δm)v² .

Energy Transferred = (1/2)(Δm)v²
= (1/2)(ρAΔx)v²

Dividing both sides by Δt,

Power = (1/2)ρAv³ I've thought a long time about this; both methods make sense, but why are they different? Appreciate any help guys!

 

[LawrenceC]

If you combine the Bernoulli equation with the momentum equation (force) across the blades by relating to pressure differential, you will see that you get the same result as you get with energy considerations.

 

[unscientific]

If you combine the Bernoulli equation with the momentum equation (force) across the blades by relating to pressure differential, you will see that you get the same result as you get with energy considerations.

so do is the dynamics method here wrong?