Calculating expected RPM for a turbine blade

In summary, the blades on a wind turbine rotate at a certain speed based on the amount of air they deflect. The power of the turbine is based on the amount of air it can move and the amount of momentum it can generate.
  • #1
MattHorbacz
18
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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=½ρAsweptV3Cp
P=Fv=F(ωr)
Ω=ρAbladeV
∑FxinVin-Fx,blade=0
∑Fy=-ΩoutVout+Fy,blade=0​
Known values:
Vair=10 m/s
ρair=1.225 kg/m3
lengthblade (swept radius)=.457 m
max chord length=.089 m
Ablade=lengthblade*chord length (assuming blade to be a rectangle)
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)
Fy,blade=ΩVout=ρAbladeV2=(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​
 
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  • #2
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.
 
  • #3
MattHorbacz said:
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=½ρAsweptV3Cp
P=Fv=F(ωr)​
Ω=ρAbladeV​
∑FxinVin-Fx,blade=0​
∑Fy=-ΩoutVout+Fy,blade=0​
Known values:
Vair=10 m/s
ρair=1.225 kg/m3
lengthblade (swept radius)=.457 m
max chord length=.089 m
Ablade=lengthblade*chord length (assuming blade to be a rectangle)
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)​
Fy,blade=ΩVout=ρAbladeV2=(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.
 
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FAQ: Calculating expected RPM for a turbine blade

1. How do you calculate the expected RPM for a turbine blade?

The expected RPM for a turbine blade can be calculated using the formula: RPM = (B x V) / (60 x P), where B is the number of blades, V is the velocity of the fluid in meters per second, and P is the number of poles on the generator. This formula takes into account the number of blades and the speed of the fluid to determine the expected RPM.

2. What factors affect the expected RPM for a turbine blade?

The expected RPM for a turbine blade can be affected by various factors such as the number of blades, the speed of the fluid, the size and shape of the blades, the angle of the blades, and the number of poles on the generator. These factors can impact the efficiency and performance of the turbine blade, thus affecting the expected RPM.

3. How does the velocity of the fluid impact the expected RPM for a turbine blade?

The velocity of the fluid has a direct impact on the expected RPM for a turbine blade. The higher the velocity of the fluid, the higher the expected RPM will be. This is because the speed of the fluid determines the force that is applied on the blades, which in turn affects the rotational speed of the turbine.

4. What is the significance of the number of blades in calculating the expected RPM for a turbine blade?

The number of blades is an important factor in calculating the expected RPM for a turbine blade. The more blades a turbine has, the higher the expected RPM will be. This is because more blades can capture more energy from the fluid, resulting in a higher rotational speed of the turbine.

5. How can the expected RPM for a turbine blade be optimized?

The expected RPM for a turbine blade can be optimized by considering various factors such as the design and shape of the blades, the speed of the fluid, and the number of blades and poles. Additionally, regular maintenance and adjustments can help to ensure that the turbine is functioning at its maximum efficiency, thus optimizing the expected RPM.

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