For a machine which is assumed to operate in an ideal manner ( h = 1, C P,Betz =

Question

For a machine which is assumed to operate in an ideal manner (h = 1, CP,Betz =16/27) and in a wind regime assumed to have a wind speed probability given by the Rayleigh distribution, the average power for output this situation can be solved for and reduced to a relatively simple equation (based on the work of Carlin, 1997 and derived on page 64 of the textbook Wind Energy Explained, by Manwell), as follows:

Pw=rhp*(2/3*D)^2*U^3

Where rho is the average air density for the site, D is the diameter of the rotor of the machine, and U is the annual average wind speed for the site.

Given this information, estimate the annual energy production of a D m diameter horizontal axis wind turbine operating at standard atmospheric conditions (density = rho kg/m^3) in a U m/s average wind speed regime. You are to assume that the site wind speed probability density is given by the Rayleigh distribution.  

Explanation / Answer

Since it is already given [taking into acount the Rayleigh distribution] that

Pw = rho [kg/m^3] * 4/9 * {D[m]}^2 * {U[m/s]}^3 so that Pw is in kg-m^2/s^2 i.e. N-m/s^2 i.e. [W]

Hence, enery production in a standard year PwAnn= 365 [day] * 24 [hr/day] * [3600 [s/hr] * {rho*4/9*D^2*U^3] [J/s]

That is PwAnn = 876000*{rho*4/9*D^2*U^3] [J]

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