Calculating expected RPM for a turbine blade.

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1. Nov 21, 2016

MattHorbacz

For my senior design project, my group is tasked with creating a wind turbine. At the moment, I am trying to figure out what approximate RPM the blades will rotate at so we can select an appropriate gear ratio and generator. Our advisor instructed us to use conservation of momentum on the turbine blade to get the force imparted by the airflow. Since we dont need an exact number, he told us to assume that the blade deflects the air by 90 degrees, and that velocity_in = velocity_out.
I figure that once you get the force acting in the y direction, you can plug into P=Fv=F(ωr) to get the radial velocity of the blade.
And im not sure how to write "m dot" for mass flow rate, so i will use Ω instead.
Equations:
P=½ρAsweptV3Cp
P=Fv=F(ωr)
Known values:
Vair=10 m/s
ρair=1.225 kg/m3
max chord length=.089 m
Cp=.45 ( the betz coefficient)

Attempt at Solution:
Power is simple to calculate,
Power=.5*1.225kg/m*π*(.457m)2*(10 m/s)3*.45=181 W

Next, I solve the conservation of momentum equation in the y direction (the x direction doesn't tell me anything useful)

Now I plug that into P=Fv and find that v=36.35m/s. assuming the force acts in center of blade, ω=159 rad/s....I still convert to RPM even though this answer is obviously nowhere near correct....
(159 rad/s)(180 degrees/π rad)(1 rev/360 degrees)(60 s/min)=1518 RPM. Our advisor told us that the blade will have and RPM of around 60-120, which I feel is a bit low, but 1518 RPM is wayyyyy to high.

I would very much appreciate any advice, whether it be where I messed up, or alternative ways of calculating RPM. Let me know if you would like any clarifications​

2. Nov 21, 2016

Simon Bridge

The turbine and air system has cylindrical symmetry...
A bit of air velocity v has momentum p in the +z direction - strikes the blades at radius r and imparts angular momentum $L = I\omega = Iv/r$ to the turbine.

The rest is up to how many approximations you want to use. Make your assumptions explicit - and state the approximations.
Also take care to define your coordinates as you go.

Aside: you may want to learn LaTeX for equations like this.