How Does Cup Anemometer Design Utilize Fluid Mechanics Principles?

In summary: For the second step, we will need to determine the drag on each cup as a function of the rotation angle.I drew a diagram of the setup.I'm assuming that the fluid velocity relative to the wind is what's most important, as that's what's causing the "drag" on each of the anemometer's cups.I also set up my equation for the drag force in such a way that only the component of the wind speed normal to the "face" of the cups are considered.Anyways, my initial force equations look like this.Note that "1" subscripts refer to the cup which has its opening facing
  • #1
KTBMedia
6
9
Homework Statement
"It is desired to design a cup anemometer for wind speed, with a more sophisticated approach than the “average-torque” method commonly used in other problems. The anemometer's angular speed, ω, is a function of the anemometer's rotation angle, and is constantly changing (even for a constant wind speed). The design should achieve an approximately linear relation between wind velocity and rotation rate in the range 20 < U < 40 mi/h, and the anemometer should rotate at about 360 rpm at U = 30 mi/h. All specifications—cup diameter D, rod length L, rod diameter d, the bearing type, and all materials—are to be selected through your analysis. Make suitable assumptions about the instantaneous drag of the cups and rods at any given angle θ(t) of the system. Compute the instantaneous torque T(t), and find and integrate the instantaneous angular acceleration of the device. Develop a complete theory for rotation rate versus wind speed in the range 0 < U < 50 mi/h. Include actual commercial bearing friction properties."
Relevant Equations
For a hemispherical cup shape, the drag coefficient c_d = 1.4 for flow traveling toward the cup's opening, and c_d = 0.4 for flow traveling towards the closed, rounded side of the cup.

A fluid drag force, F_d, is given by F_d = 1/2*c_d*rho*A*v^2, where v is the speed of the body relative to the fluid, A is the effective cross-sectional area, and rho is the fluid density.
First, this class uses Frank White's Fluid Mechanics textbook. This particular problem is taken straight from chapter 7, which is on "Flow Past Immersed Bodies" and is basically focused on external flow, geometry effects, and boundary-layer conditions. So I imagine that the problem makes use of those concepts. Concepts covered in the book prior to this chapter include internal flow through ducts, the Moody chart, the Navier-Stokes equation, the Reynolds transport theorem, the Bernoulli equation, dimensional analysis / the Pi theorem, and basic shear stress / viscosity. All of those topics are fair game.

Second, because this problem has a lot of elements to it and things going on (asking us to decide on materials and everything), it's somewhat open-ended as a result. My professor has said that it's up to us to choose how many cups we'd like to have the anemometer to have. For simplicity, I think we'll go with a design that includes two cups.

The professor recommended that our first step be to find a formula that estimates the instantaneous drag on the cups as a function of the rotation angle, θ. The first thing I did was draw a diagram of the setup. I'm assuming that the fluid velocity relative to the wind is what's most important, as that's what's causing the "drag" on each of the anemometer's cups. I also set up my equation for the drag force in such a way that only the component of the wind speed normal to the "face" of the cups are considered. I'm assuming that this is valid, though if there's a problem with that approach then please point it out to me!

Anyways, my initial force equations look like this. Note that "1" subscripts refer to the cup which has its opening facing the incoming wind (and is therefore rotating "with" the wind), and "2" subscripts refer to the cup with is faced in the opposite direction, and is therefore "fighting" the wind at any given angle.
$$
\begin{align*}
F_d1 & = \frac{1}{2} \rho A c_{d1} (v_{rel1})^2 \\
F_d2 & = \frac{1}{2} \rho A c_{d2} (v_{rel2})^2
\end{align*}
$$

Note that this doesn't account for the drag acting on the rotating rod between the two cups. Expanding it out a bit, accounting for the "relative" wind speeds as I mentioned earlier:
$$
\begin{align*}
F_d1 & = \frac{1}{2} \rho A c_{d1} (U \cos(\theta) - R \omega)^2 \\
F_d2 & = \frac{1}{2} \rho A c_{d2} (U \cos(\theta) + R \omega)^2
\end{align*}
$$

The values for cd are provided by the textbook. I've listed them in the "Relevant Equations" section.
I don't feel terrible about this approach, but I do admit that my logic might be kind of shaky. Due to the fact that the anemometer is rotating, technically a "1" cup will become a "2" cup and vice-versa after each 180-degree rotation of the device. I'm not sure if this hurts the validity of my equations.

The bigger problem I have with what I've done, is the fact that I don't think I've fully solved the "first" step of the problem. Sure, it's a formula for drag force with respect to theta, but it also includes omega - a non-constant. Not sure what to do about that.

I also tried to make a formula for the net torque of the system:

$$
\begin{align*}
M &= L (F_{d1} - F_{d2}) \\
&= \frac{1}{2} \rho R A ( 1.4(U \cos(\theta) - R \omega)^2 - 0.4(U \cos(\theta) + R\omega)^2 )
\end{align*}
$$

And that's about where I'm at right now. I feel like I probably want to find a formula for the angular velocity as a function of theta, but I've been playing around with it for a bit and I'm really not sure how to do that, and I kind of feel like I've hit a dead end right now. A classmate who's working on the same problem has said that he's going to try finding a function of the drag coefficient with respect to theta, as opposed to what I've done. Might his approach be better?

Any tips would be much appreciated!
 
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  • #2
KTBMedia said:
For simplicity, I think we'll go with a design that includes two cups.
The original meteorology standard anemometer had four cups. That was later reduced to three. I believe that was because two or four cups would not always start in light winds.
My quick experiment showed that the speed appeared to be independent of the number of cups, over the range from one to six.

KTBMedia said:
Note that this doesn't account for the drag acting on the rotating rod between the two cups.
That rod becomes elliptical over much of the rotation.
 
  • #3
The provided values of Cd are only valid for the tests completed in a wind tunnel for the cups being positioned perpendicularly to the airstream.
Therefore, those are only valid for those instantaneous positions.

As each cup rotates from 1 to 2, both change, the cross-section facing the wind and that Cd.
The Cd values for each position between 1 and 2 can only be assumed, or determined in a wind tunnel.

What angle is represented by omega?
 
  • #4
Baluncore said:
The original meteorology standard anemometer had four cups. That was later reduced to three. I believe that was because two or four cups would not always start in light winds.
Yeah, I recognize that having a two-cup anemometer is probably not the most efficient anemometer in the world, but the professor explicitly said that it would be okay for the purposes of the problem.

That said, I'm pretty interested in you saying that your "quick experiments" showed that the "speed appeared to be independent of the number of cups". Knowing more about how you found this might actually be helpful; have you got a bunch of different anemometers on hand? How did you go about experimenting with different wind speeds?

Lnewqban said:
The provided values of Cd are only valid for the tests completed in a wind tunnel for the cups being positioned perpendicularly to the airstream.
Therefore, those are only valid for those instantaneous positions.
Which is why I clarified, and showed in the equations I derived in the OP, that I specifically derived the drag force equation I found to be in terms of the perpendicular components of the wind, using a cos(theta) term.

Like I said in the OP, I'm open to the idea that my line of thinking there is flawed, but since I've solved other vaguely similar problems using the same idea, I'd like to hear an alternate suggestion if you think there's a better procedure that I'm overlooking.

Lnewqban said:
As each cup rotates from 1 to 2, both change, the cross-section facing the wind and that Cd.
The Cd values for each position between 1 and 2 can only be assumed, or determined in a wind tunnel.
Fair point, though I ran this by my professor and he encouraged me to use those values given in the book. I think we're supposed to assume that we're essentially designing this thing for a wind tunnel.

Lnewqban said:
What angle is represented by omega?
I've attached a photo of a cleaner version of my hand-drawn sketch of the situation. Essentially, it's just an angle of rotation for the anemometer itself. Omega is the angular speed, which the problem states should be constantly changing as a function of theta.
 

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  • #5
KTBMedia said:
Knowing more about how you found this might actually be helpful; have you got a bunch of different anemometers on hand?
25 years ago I made several different cup-count anemometers by drilling holes in small cylindrical hubs, then glued cut-in-half table-tennis-balls, onto satay sticks, that I plugged into the hubs. I made a couple with arms of twice the length.
I threaded all onto a stainless steel whip antenna on my car, mounted on the bull-bar in free air-flow where I could see it. Each was spaced and resting on an electrical terminal-block, screw-clamped, with a flat washer as a bearing.

It was windy the next morning, so I spun-up the stalled one and two cup units, before I drove to work along the back roads. I noticed that even though the units were spaced and not connected, they seemed to be synchronised, and the whip was not visibly vibrating due to unbalance. It gave me confidence that the number of cups was not critical, it was the length of the arm from the axis to cup-hemisphere-central that was critical.

At work, I went up in the cherry picker, then dropped facial tissues that were wind blown past the rebuilt reference-anemometer on a mast. The time and distance the tissues travelled, before being caught in a wire fence, or a bush, gave me a good estimate of wind speed. I read the RPM of the three-cup reference and checked it against the tachometer voltage that went to the control room, that was used to trigger the emergency automatic parking of a big radio astronomy dish.

I used an ancient meteorology standard equation for a four-cup, to verify my calibration of the three-cup unit. That just happened to have the exact same radius and hemisphere diameter. I cannot now find that old equation, it corrected for the wind velocity to RPM curve.

I think the following part of my later interesting notes file, may have come originally from; http://au.omega.com/prodinfo/anemometers.html
and possibly; "The Working of the Cup Anemometer" by Leif Kristensen, Ole Frost Hansen and Svend Ole Hansen. June 19, 2014, (with bibliography).
You will need to verify the contents if you use them.
A simple type of anemometer is the cup anemometer. It consisted of three or four hemispherical cups each mounted on one end of horizontal arms, which in turn were mounted at equal angles to each other on a vertical shaft. The air flow past the cups in any horizontal direction turned the cups in a manner that was proportional to the wind speed. Therefore, counting the turns of the cups over a set time period produced the average wind speed for a wide range of speeds. On an anemometer with four cups it is easy to see that since the cups are arranged symmetrically on the end of the arms, the wind always has the hollow of one cup presented to it and is blowing on the back of the cup on the opposite end of the cross.

When Robinson first designed his anemometer, he wrongly claimed that no matter how big the cups or how long the arms, the cups always moved with one-third of the speed of the wind. This was apparently confirmed by some early independent experiments, but it was very far from the truth. It was later discovered that the actual relationship between the speed of the wind and that of the cups, called the anemometer factor, depended on the dimensions of the cups and arms, and may have a value between two and a little over three. Every single experiment involving an anemometer had to be done all over again.

The three cup anemometer developed by the Canadian John Patterson in 1926 and subsequent cup improvements by Brevoort & Joiner of the USA in 1935 led to a cupwheel design which was linear and had an error of less than 3% up to 60 mph. Patterson found that each cup produced maximum torque when it was at 45 degrees to the wind flow. The three cup anemometer also had a more constant torque and responded more quickly to gusts than the four cup anemometer.

The three cup anemometer was further modified by the Australian Derek Weston in 1991 to measure both wind direction and wind speed. Weston added a tag to one cup, which causes the cup wheel speed to increase and decrease as the tag moves alternately with and against the wind. Wind direction is calculated from these cyclical changes in cup wheel speed, while wind speed is as usual determined from the average cup wheel speed.

Three cup anemometers are currently used as the industry standard for wind resource assessment studies. The NRG Systems #40C is the most commonly used cup anemometer for this purpose.

For historical reasons, anemometer sizes are measured in crows.

Crow Instability. https://en.wikipedia.org/wiki/Crow_instability
Crow, S. C. (1970). "Stability theory for a pair of trailing vortices".
 
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