Heys, does anyone know how to perform this question to check if I did mt working out correctly? The question is ...

Consider the flow of air past an ideal turbine with a swept area of A.

The fraction of power that can be extracted from the power contained in the incoming wind is called the ‘power coefficient,’ Cp = 4a(1‐a)^2. This is effectively the aerodynamic efficiency of the blades turbine in extracting power out of the incoming wind.

(b) Find the maximum value of Cp and find out at what value of ‘a’ this occurs. Hint differentiate with respect to ‘a’.

Working:

Finding the extremum (max or min) of a function means taking the derivative and setting it to zero

dCp/da = 4(1-a)^2 + 8a(1-a)(-1) =0

4(1-a)^2 = 8a (1-a)

4(1-a) =8a

1-a = 2a or a = 1/3

the max value occurs at a = 1/3?

Therefore, Cp = 16/27?

Thanks