Could the S.H.I.E.L.D. Helicarrier Fly?

This isn't just from The Avengers movie, it is in the comics also. Here an image of the S.H.I.E.L.D.'s helicarrier.

Could something like this really fly? Let me see if I can use my approximation from the human powered helicopter to estimate the amount of power needed to fly this thing. First, some assumptions.

• I will use the helicarrier shown above from the recent The Avengers movie. There are other variations of this thing in the comics.
• The expressions for force and power from my previous post are mostly valid. I know that some people freak out over that estimation - but it isn't terrible as far as estimations go.
• There are no special aerodynamic effects to help the helicarrier hover - like ground effects.
• The helicarrier in the movie is about the size and mass of a real aircraft carrier.
• The helicarrier stays in the air just from the rotors. It doesn't float like a lighter than air aircraft. I think this assumption does along with the movie since they show it sitting in water floating like a normal aircraft carrier.

Just as a reminder, for a hovering craft I estimated it the force from pushing the air down (and thus the lift) would be:

As a reminder, the A is the area of the air that is pushed down - which would be the size of the rotors and v is the speed that the rotors push the air.

Helicarrier Mass and Length

This helicarrier clearly isn't a Nimitz Class Carrier - but something else. However, it seems to be a good guess that they are the same size. Here is a comparison with a Nimitz class carrier.

The runways look about the same width, so I am going to say the length and the mass of the helicarrier is about the same. Wikipedia lists the length at 333 meters with a mass of about 10⁸ kg.

Using the length of the helicarrier, I can get an estimate for the size of the rotors. With each rotor having a radius of about 17.8 meters, this would put the total rotor area at 4000 m² (assuming all the rotors are the same size).

Thrust Speed and Power

When the helicarrier is hovering, the thrust force would have the same magnitude as the weight. From this, I can get an estimate of the speed the rotors would move the air down.

Just to make things easier, I will look at low level hovering. This means I can just use 1.2 kg/m³ for the density of air. Of course, at higher altitudes the density would be lower. Using the mass and rotor area from above, I get a thrust air speed of 642 m/s (1400 mph). Just to be clear, this is faster than the speed of sound. It is probably clear that I don't know much about real helicopters or jet engines, but I would suspect that a thrust this high would add other calculation complications. I will (as usual) proceed anyway.

With the air speed, I can now calculate the power needed to hover. Again, I am not going to go over the (possibly bogus) derivation of this power for hovering, it was in my huma-copter post.

With my values from above, I get a power of 3.17 x 10¹¹ Watts - quite a bit more than 1.21 giga watts. In horsepower, this would be 4.26 x 10⁸ horsepower. That's a lot of horses. Just for comparison, the Nimitz class carriers have a listed propulsion of 1.94 x 10⁸ Watts. I assume this is the maximum power, so it wouldn't be enough to lift the helicarrier. Obviously, the S.H.I.E.L.D. helicarrier has a better power source. I would guess it would have to be at least around 2 x 10⁹ Watts in order to operate. You don't want to use your maximum power just to sit still.

Really, I am surprised with my rough calculations that it is even partially close to the power output of a real carrier.

Real Helicopters

Why didn't I think to look at some real helicopters before? There are two things I can look up for different helicopters: the rotor size and the mass. Of course, I don't know the thrust air speed, but I can find that. Let me get the power needed to hover as a function of mass and rotor size. Starting with the force needed to hover, I know an expression for the thrust air speed. If I substitute this into the expression for the power, I get:

Now for some data. Here are some values I found on Wikipedia.

What if I look at the actual power for these aircraft compared to my "minimum power to hover"? Since my (possibly bogus) calculation just depends on the mass and the area of the rotors, there is nothing to stop me.

Honestly, I didn't expect this to turn out so nice and linear. The slope for this linear regression line is 0.41 and the intercept is 14.4 kW. So, what does this mean? For the slope, this means that my calculated power (based on the rotor area) is 41% of the actual maximum power available for these aircraft. Now, this doesn't exactly mean that a hovering helicopter would be running the engines at 41%. It could mean that there is also some other factor that should be in my calculation.

What about the 14.4 kW intercept? First, this is essentially zero in comparison to these engine powers. The smallest engine is 310 kilo watts. Second, I was going to say something about engine power just need to run the other stuff (overhead power) but the way I plotted that it would have to have a negative intercept. Let me just stick with "this is almost zero".

How about some other plots? Here is something interesting. This is a plot of thrust air speed vs. mass of the helicopter.

The cool part is that there doesn't seem to be a real pattern. The bigger helicopters push the air down (in my model) such that the air leaves with a speed around 28 m/s. This is much slower than than the calculated air speed for the helicarrier at 642 m/s. You know what comes next, right? Now I will calculate the size the rotors on the helicarrier would need to be to let it hover with a thrust air speed of 28 m/s. Let me go ahead and increase this to 50 m/s thrust speed - because it's S.H.I.E.L.D..

I don't need to power to find the area, I will just use the expression I used to find the velocity of the air and solve for the area of the rotors instead.

Now I just need to plug in my values for the mass of the helicarrier, the thrust air speed and the density of air (I am using the value at sea level). This gives a rotor area of 6.5 x 10⁵ m². This is quite a bit larger than my measured values from the image. I guess I will have to fix the image.

Yes, that looks crazy. But remember, I even used a higher than expected thrust speed. If I used 30 m/s, it would be even crazier big. Crazy.

Homework

Remember the rule with all assigned homework problems: if you wait too long to figure this out, I might do it instead.

1. This question is about the size of the helicarrier. Suppose the size is NOT the same as a Nimitz class carrier. Suppose it is smaller such that the rotor area is the correct size for a thrust air speed of 50 m/s. How big is the helicarrier in this case? (hint: assume a carrier density of about 500 kg/m³ since about half of it floats above the water line).

2. (SPOILER ALERT) When Iron Man tries to restart one of the rotors, he pushes it to get it going. Suppose the rotor pushes the air to a speed of 642 m/s - and this is the linear speed of the middle of the rotor. How fast was Iron Man flying around in a circle to get the thing started? You might want to assume the rotors at this point were only at half speed. What would be the g-force Iron Man would experience moving this fast in a circle? Would that kill him?

3. What about at operating speed for the rotors - would would the acceleration of the tip of the rotor blade be? Estimate the tension in the rotor blades (where would the tension be a maximum)? Is this too high of a tension for known materials?

Images courtesy Walt Disney Pictures