Aerodynamic Structural Noise - Hybrid Anisotropic Materials for Wind Power Turbine Blades - page 202

U 4 .=. U 1 (1.−.2 a ). (7.3)

U 4 .is.introduced.here.as.the.wind.velocity.in.the.far.wake..This.is.impor-
tant.because.the.power.extracted.from.the.turbine.is.deined.by.the.follow-
ing.equations..However,.the.Betz.limit.is.given.in.terms.of.the.coeficient.of.
power..The.coeficient.of.power.is.similar.to.eficiency,.but.not.the.same..The.
formula.for.the.coeficient.of.power.is.given.beneath.the.formula.for.power:

( )

P 0 5 . AU U U

2

-

2

= ρ . (7.4)

2 1 4

P
AU

. C ≡ 0 5 . (7.5)

P

3

. ρ

1

Betz.was.able.to.develop.an.expression.for. Cp .in.terms.of.the.induction.fac-
tors..This.is.done.by.the.velocity.relations.being.substituted.into.power.and.
power.is.substituted.into.the.coeficient.of.power.deinition..The.relationship.
Betz.developed.is:

C p .=.4 a (1.−. a ) 2. (7.6)

The. Betz. limit. is. deined. by. the. maximum. value. that. can. be. given. by. the.
above.formula.
This.is.found.by.taking.the.derivative.with.respect.to.the.axial.induction.
factor,. setting. it. to. zero,. and. solving. for. the. axial. induction. factor.. Betz.
was. able. to. show. that. the. optimum. axial. induction. factor. is. one-third..
The.optimum.axial.induction.factor.was.then.used.to.ind.the.maximum.
coeficient.of.power..This.maximum.coeficient.is.the.Betz.limit..Betz.was.
able.to.show.that.the.maximum.coeficient.of.power.of.a.wind.turbine.is.
16/27.. Airlow. operating. at. higher. thrust. will. cause. the. axial. induction.
factor.to.rise.above.the.optimum.value..Higher.thrust.causes.more.air.to.
be.delected.away.from.the.turbine..When.the.axial.induction.factor.falls.
below.the.optimum.value,.the.wind.turbine.is.not.extracting.all.the.energy.
it. can.. This. reduces. pressure. around. the. turbine. and. allows. more. air. to.
pass.through.the.turbine,.but.not.enough.to.account.for.the.lack.of.energy.
being.extracted.
The.derivation.of.the.Betz.limit.shows.a.simple.analysis.of.wind.turbine.
aerodynamics.. In. reality,. there. is. a. lot. more.. A. more. rigorous. analysis.
would.include.wake.rotation,.the.effect.of.variable.geometry..The.effect.of.
air.foils.on.the.low.is.a.major.component.of.wind.turbine.aerodynamics..
Within. airfoils. alone,. the. wind. turbine. aerodynamicist. has. to. consider.
the.effect.of.surface.roughness,.dynamic.stall.tip.losses,.solidity,.among.
other.problems.