exercise of power coefficient Cp

Title: power coefficient （Cp）

Author:Weiwei Zeng

Content:The power coefficient limit(Betz’s limit)

Date:30/11/2010

As we all know, the turbine firstly transforms the power available in wind to the power output from wind machine. Then the mechanical energy is transformed to the electronic power by the generator.So there is a power coefficient during the wind power transformed to mechanical power..According to Betz's law, this power coefficient has a limit, which is

0.593.The ideal or maximum theoretical efficiency n max (also called power coefficient) of

a wind turbine is the ratio of maximum power obtained from the wind to the total power available in the wind. The factor 0.593 is known as Betz's coefficient (from the name of the man who first derived it). It is the maximum fraction of the power in a wind stream that can be extracted.

You can learn the basic of aerodynamic by the book named ‘wind energy handbook’ written by Tony Burton.

power coefficient = Cp =( power output from wind machine) / (power available in wind )

Betz's law:

First the Betz's law is an ideal model for energy transformation.

Figure 1: Schematic of fluid flow through a disk-shaped actuator.

This figure shows the fluid flow through a disk-shaped actuator.

Power available in wind=212/1v m ?? (1)

Power output from wind machine=)(2/12221v v m -?? (2)

The mass m= v ?ρ (3)

The wind flow volume V )/(3s m =

)()/(2m S s m v ? (4)

so the mass=S v ??ρ (5)

power available in wind=312/1v S ???ρ (6)

Power output from wind machine=)(2/12221v v v S -????ρ (7) In the ideal model,V=Vavg=)(2/121v v + (8)

So Power output from wind machine=)()(2/12/1222121v v v v S -?+???ρ

=)()(4/1222121v v v v S -?+???ρ =))(()(

1(4/11

221231231v v v v v v v S +--????ρ （9） Assume,V2/V1=a(a<1)

P=)1(4/12331a a a v S +--????ρ （10）

Figure 2：Cp-(v2/v1) ratio curve

When a=1/3,P reaches maximum.

So maximum Power output from wind machine=))3/1()3/1()3/1(1(4/12331+--????v S ρ

=312/127/16v S ????ρ （11）

Maximum power coefficient = Cp =( power output from wind machine) / (power available in wind )

=)2/127/16(31v S ????ρ/)2/1(31v S ???ρ

=16/27

≈0.593