I am obsessed with helicopters. You probably already knew that. In my previous post, I looked at the power and force for a hovering helicopter. The basic assumption was that by pushing air down, the air pushed back up. From this, I obtained the following expression for the force from the helicopter rotor:
I am calling this Fair because it is the force of the air pushing on the helicopter. Don't confuse this with the air resistance for an object moving through the air (which I often call the same thing). In this model, ρ is the density of air (about 1.2 kg/m2), A is the area of the rotors and v is the speed of the air after it passes through the fan.
For the hovering helicopter, this air force would be equal to the weight of the helicopter. Using this, I could get an expression for the velocity of the air. With the velocity of the air and the force on the air, I could calculate the power needed to hover. This is what I got (without putting a value in for the air speed).
Now for some more data. I found this old fan cart in one of the labs.
Can I measure the force this fan exerts on the cart? Yup. Can I measure the power going into the fan motor? Yup. So, even though it isn't really a helicopter, it is sort of like one. First, let's get to the force.
Measuring the Force
You can put this cart on a track and let it go. In order to get the acceleration, I will create a plot of position vs. time using the Vernier motion sensor. So, before I do this, let me get an estimate of the amount of friction in this system. If I just give the cart a push with the fan off, I get an average acceleration of about 0.027 m/s2. I suspect this will be small enough to ignore, so I will for now.
Here is a plot for the motion of the cart with the fan on after it is released from rest.
The fitting parameter in front of the t2 term is 0.1757 m/s2. Since this is the same as the (1/2)*a term in the kinematic equation:
Then the acceleration is twice that parameter giving it a value of 0.3514 m/s2. I am sure I have said this point about finding the acceleration before, but it is easy to forget. I repeated the process a few times and found an average acceleration of 0.354 m/s2. This seems large enough for me to neglect the effects of friction - if this were a real lab report, I would include friction.
If I consider this fan to be the only horizontal force on the cart, then I can find the value of this force from the acceleration and the mass. The cart plus its batteries has a mass of 0.576 kg. This puts the fan force at:
But wait! There's more. The fan has a "high" and a "low" setting. The above force is for the "high" setting. If I repeat the experiment on the "low" setting, I get a force from the fan of 0.122 Newtons.
Oh, I guess I should be clear about my assumptions. I already said I was ignoring friction. The other assumption is that the force from the fan doesn't change with the speed of the cart. Of course, this isn't really true. As the carts goes faster, it won't push the air as hard. Also, as it it goes faster there would be an air resistance force. In this case, the car is going pretty slowly so it shouldn't matter too much. Also, for the case of the hovering helicopter the speed would be zero.
Power
I want to know the power going into this motor. The simplest way is to measure the change in electric potential (voltage) across the battery at the same time as measuring the current through the battery. With this, the power would be:
In "high" mode, the fan has 4.22 volts across it with a current of 2.12 Amps. This gives a power of 8.95 Watts. In "low" mode, the fan potential is 3.44 volts with a current of 1.59 Amps. This gives a power of 5.47 Watts.
Perhaps I should make one change to my expression for power:
The power the motor gets is indeed IΔV. But not all of this power goes into pushing the air. There is some loss. So, the e is the efficiency of this power transfer.
Oh, there is one more thing to measure - the fan size. In my helicopter analysis, I used this for the rotor area. This fan has a radius of 7.5 cm which gives it a rotor area of 0.0176 m2.
Comparing Models to Data
I really measured two things. I measured the force from the fan and the power of the air. I don't know the speed of the air coming off the fan. Let me solve the force expression for the velocity and plug that into the power equation. This gives:
I already see that I am going to have a tough time here with just two data points. Ok. What if I solve for the efficiency for both high and low modes?
With this I get an efficiency of (remember the density of air is about 1.2 kg/m3) 0.0378 for low power and 0.0500 for high power. Odd. I thought it would be much higher than that. At least the efficiency is in the same ball park for high and low setting. Still, I am troubled. Maybe these tiny fan blades just don't work as well as larger helicopter blades. Maybe I am an idiot and messed up somewhere.
Even More Data
I couldn't leave it alone. I had to get more data. So, I stuck some more batteries on the cart.
With this, I could run the fan in "high" and "low" mode with 1 extra battery and 2 extra batteries. This gives me a total of 6 different settings. Of course, the mass of the cart changes with more batteries. That just means that I will have to multiply the acceleration by a different value to get the force of the fan.
Let me just throw that efficiency thing out. Here is a plot of the measured fan power vs. the calculated power.
At least it looks linear. However, the slope of this fitting function is only 0.0618. If I interpret this as the efficiency, it would only be about 6% efficient. I don't know. Maybe these small fans just aren't the same as the big helicopter rotors. Clearly, I have no idea what I am doing.
You know, it would be cool if I repeated this with a very large low friction cart with a large (person-sized) fan. Maybe.
