Calculating expected RPM for a turbine blade

[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 don't 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 I am not sure how to write "m dot" for mass flow rate, so i will use Ω instead.
Equations:

P=½ρA_sweptV³C_p
P=Fv=F(ωr)
Ω=ρA_bladeV
∑F_x=Ω_inV_in-F_x,blade=0
∑F_y=-Ω_outV_out+F_y,blade=0​

Known values:
V_air=10 m/s
ρ_air=1.225 kg/m³
length_blade (swept radius)=.457 m
max chord length=.089 m
A_blade=length_blade*chord length (assuming blade to be a rectangle)
C_p=.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)
F_y,blade=ΩV_out=ρA_bladeV²=(1.225 kg/m³)*(.457 m)*(.089 m)*(10 m/s)^2=4.98 N

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​

 

[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.

 

[molsen81]

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 don't 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 I am not sure how to write "m dot" for mass flow rate, so i will use Ω instead.
Equations:

P=½ρA_sweptV³C_p​

P=Fv=F(ωr)​

Ω=ρA_bladeV​

∑F_x=Ω_inV_in-F_x,blade=0​

∑F_y=-Ω_outV_out+F_y,blade=0​

Known values:
V_air=10 m/s
ρ_air=1.225 kg/m³
length_blade (swept radius)=.457 m
max chord length=.089 m
A_blade=length_blade*chord length (assuming blade to be a rectangle)
C_p=.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)​

F_y,blade=ΩV_out=ρA_bladeV²=(1.225 kg/m³)*(.457 m)*(.089 m)*(10 m/s)^2=4.98 N​

​

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​

You might find some utility/help using an application called "phyphox" or something similar. It's a free smartphone physics app that utilizes all of the sensors within your smartphone to do a large variety of testing/data analysis/projection. It's incredibly intuitive and easy to use. I teach HS physics and have had my students using it for a very similar project; though there's is a bit more oversimplified and focuses on efficiency gains using data analysis in order to inevitably design/3d print a mini-turbine using TinkerCAD. There are other free apps that offer a large number of uses by simply using the sensors already within your smartphone. Hope this helps a bit.