Betz' law

The proof of the pudding

Musing on the wind turbine post below,  it occured to me that the derivation of Betz' law could be made into a good class room exercise. All maths is GCSE level if you stick to graphical methods, though you can use calculus if the students have studied it to get an exact result. Some the algebra could be a bit tricky for school children - you need to use the difference of two squares twice.

Consider a wind turbine in the form of a disk, with area A. Wind traveling at a speed v1 hits the turbine, and leaves the turbine at speed v2.

Undisturbed wind flowing through the area of the turbine is 1/2 m v2. In one second, the mass of air flowing through the turbine is given by ρAv where ρ is the density of the air. 

Accordingly, the power - kinetic energy per second - in the undisturbed wind is P0=1/2 ρAv3. 

The wind speed through the turbine is (v1+v2)/2, the average of the the entrance and exit speeds.  

The mass of air flowing in one second is ρA(v1+v2)/2

The power imparted to the turbine is the kinetic energy of the wind imparted to the wind turbine in one second, P1

The efficiency of the turbine, η, is given by the ratio of extracted power to the power in the undisturbed wind

We can use the difference of two squares [(A2 - B2) = (A+B)(A-B)] to re-write this as 

We can write the efficiency as a function of the ratio R=v1/v2 :

We have two ways of proceeding: the first is to simply plot out η as a function of R, and see that the maximum occurs at just under 60% and for a ratio of about 1/3.

More rigorously, we can differentiate the efficiency with respect to R, and set the derivative equal to zero to calculate the value of R such that η is maximized. 

While we could plug our a,b, and c coefficients into the quadratic formula, a more elegant approach is to factorize the quadratic by inspection to obtain:

Taking the positive root (as the negative root is unphysical), the maximum occurs when R=1/3, and the efficiency η=14/24 =0.58 (2 s.f.)

Questions:

• What slightly dubious assumption has been invoked? Can you justify it?
• Could you confirm this result with a dimensional analysis?
• Any other suggestion in the comments.