Yes, this is cool. A human powered helicopter. Check out the video.
For this particular flight, they managed to hover for around 50 seconds but just barely off the ground.
I am not going to really talk about the details of this project (Gamera 2 from the University of Maryland). You know that's not what I like to do. Instead, let me try to calculate the power needed to do this. Clearly, it isn't an easy task. If it was, I would be flying my helibike to work instead of road bike. The problem seems quite difficult. Let me see if I can make a simpler model.
Lifting Force
In the most basic case (no ground effect or wind or anything), the person is supported because the machine "throws" air down. This is sort of like a rocket, but the fuel is not carried on board. Let me start with a simple diagram. Suppose my helicopter "throws" a chunk of air down. In this case, my chunk of air will be a cylinder.
This huma-copter takes air with some air and exerts a force on it to increase it's speed to some value (v). According to the momentum principle, this means the huma-copter has to exert a force on the air (downward) so that the air exerts a force on the huma-copter upward. In the y-direction, I can write this as:
I know what you are thinking. You are thinking: "well, we can find the mass and the velocity - but not the change in time". It seems like a lost cause - but has that ever stopped me before? What if the huma-copter is continually throwing down air. For this chunk of air going at a speed v, it would take a time Δt to get through. I can write this as:
Now I can put this expression in for the time.
Notice that the top doesn't have the "change in" anymore since I am assuming the starting speed of the chunk of air is zero m/s. Since I know the height of this column of air, I can find the mass assuming a density of ρ.
I know, this seems crazy enough to work. But let's check somethings. First, increasing either the density, the area or the air speed will increase the "lift" force. That makes sense. What about units? Here is a check.
Looks like we are all set.
Power
I didn't want the force. I wanted the power. To get the power, I guess I need to look at how much energy it takes to move this air and then divide by the time it takes to move it. Let me take the same amount of air as I had above. If I assume this air starts at rest, then the work to get this stuff up to speed would be the final kinetic energy of the the chunk of air.
For the power, I just need the time. Again, I will use the same time as before and I can get the power.
And yes, I already know that I cheated. Once a cheater, always a cheater. So, the expression for time assumes the stuff is moving at a constant speed and yet I wanted to find the time to get this stuff up to speed v. I guess to be fair, I should use the average velocity (v/2) in the time calculation. This would make the force have the value:
And the power would also be different:
In order to remove the dependence on the height of the air (h), I can write the mass in terms of the density and volume.
You can consider it a homework assignment to show that this does indeed have the correct unit of Watts.
Real Data
Now, back to Gamera II huma-copter. It has a mass of 32 kg and consists of 4 rotors each with a radius of about 6.5 meters. I don't immediately see the mass of the huma-copter pilot, so I will just guess a value of 60 kg.
The speed of the air used as thrust can be calculated from a hovering position. I know the weight of the aircraft and I know an expression for the "thrust" force. At equilibrium, these must be equal and I can solve for the speed of the air.
The area of course will be 4 times the area of a circle with the 6.5 meter radius. For the density of air, I will use 1.2 kg/m3. Putting in these values, I get an air speed of 1.68 m/s (3.8 mph). That is a low air speed, but it seems reasonable after looking at the video.
Now, what about the power? All I have to do is to put my values into the power formula (along with the air speed of 1.68 m/s) and I get 755 Watts - just over 1 horse power. A quick reminder. I have assumed everything is perfectly efficient. I suspect the actual power requirement could be over 1000 watts. Is this a human accessible power? According to Wikipedia, the maximum power output for elite cyclists is around 2,000 watts (of course for very short periods). But overall, this seems like a plausible thing to get working.
Rotor Size
Perhaps you can already see that bigger rotors are better (wider is better). If you need to support a huma-copter, you will need a force from the air equal to the weight of the huma-copter. You can't really change the density of the air, so don't worry about that. The two things you can change are the area of the rotors and the speed of the air coming out of the rotors. If you double the area of the rotors, you can decrease the air speed by a factor of the square root of 2.
Lets say that you want a smaller huma-copter. Say you want to use a rotor area that is half the size of the one above. To compensate for the smaller rotor, you will need to push the air faster - faster by a factor of the square root of 2. Fine. But now, what about the power? Since the power depends on area and the air speed cubed, this will take 40% more power. When you are at the limit of human power output, 40% can make a big difference.
Do I think someone will eventually succeed with a huma-copter hovering 3 meters for over 60 seconds? Yes. Will this be some practical form of transportation in the future? No.
One last note: I have thought about that factor of 2 in the force and power equations. I am still not certain. At this point, I am going to leave the "2" in there.
