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What is the Betz Limit?

The Betz Limit ( or the Betz Law ) describes the
theoretical limit of the percentage of power which can be extracted
from fluid moving through a turbine. Every particle in motion has an associated kinetic energy proportional to its
mass and the square of its speed (*0.5). Turbines convert this kinetic energy to a mechanical motion and during
the conversion process the speed of the flow is reduced. Is it possible to convert 100% of the kinetic energy to
mechanical power? The short answer is NO, because that would require reducing the flowing particles speed to zero,
which in turn means the flowing particles would not be flowing. Therefore, the theoretical efficiency of the turbine
should be less than 100%, however what percentage of the kinetic energy should be converted to mechanical power?
The answer is 59.3%, which is the "theoretical maximum" and one measures the success of their design by how close
they get to this figure.

So how did Betz come up with the magical percentage of 59.3%? If you are good at math and physics here is the
Wikipedia link for the Betz Limit.
For the rest of us, here is a simple explanation:

I find the best way to explain this concept is with a familiar analogy. There will be some deviation from the
real world and I will impose some constraints on my model. I will highlight, in red, the wind mill properties
versus my model properties.

Picture a stream of cars (air particles) approaching a toll booth
(wind turbine) and they are on a
single lane highway (air stream). All of the cars have the same SMALL distance between them
(air molecular distance) and are traveling 80 mph (wind speed).
Each car driver has $4
( air kinetic energy ) to pay the toll, allowing the state to generate a revenue
( air sturbine to generate power). However, there is a catch...the drivers drive their cars
proportional to the money they have in their pockets, therefore for every quarter they drive 5 mph ($4 => 4*4*5=80).
If they pay 25 cents for the toll their speed will drop to 75 mph as they leave the booth. In reality we speed up
after we leave the toll booth, but to make the analogy work we will be reducing our speed proportional to the money
we pay at the toll booth, because air particles reduce their speed after going through the turbine. So, we know that
if we pay 25 cents our speed with drop to 75 mph and if we pay 50 cents our speed will drop to 70 mph and so on...
Now we have a problem, the cars coming from behind to toll booth are coming 80 mph, so we need a two lane to compensate
for the reduced speed of the vehicles moving through the toll booth.

We know each driver has $4 ( maximum kinetic energy of the air ) and we could ask each
driver for that much at the toll booth, but this would create a problem because if we asked for $4 the speed of
the cars (air particles ) would be reduced to zero. So, we could get all $4 from one driver and that's it,
the next vehicle would not be able to approach the toll booth. The question then becomes: how much money
should we ask from each driver so that the system works as efficiently as possible without interrupting flow of the cars?

To find the answer I made the following flash animation, please change the amount of money you can charge and watch
the cars leaving the booth (there should be no fender benders or major collisions). If you are above the
theoretical limit you can't get money ( mechanical power) without interrupting the system.
Whatever limit you find to be the one which does not interrupt the cars movement is the theoretical Betz Limit
for our system. This limit is calculated by dividing the amount of money you paid (
the amount of kitnetic energy lost by the air) in the booth to 4, which is amount of money the driver
had when approaching the toll booth (amount of kinetic energy air had before the turbine).
Please note this is not the real Betz Limit but a limit for our hypothetical model.

I like to learn a subject by doing a thought experiment, I hope my little experiment makes sense and helps
you understand the Betz Limit. If you have any questions please feel free to send us an email: admin@flapturbine.com.

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