Thursday, June 26, 2014

Thinking about Brownian Motion, Air Molecules, and Airplane Wings

In 1827, botanist Robert Brown peered into a microscope and observed the jittery motion of pollen particles in a fluid. This then-unexplained (therefore magical) jitteriness has since been known as Brownian Motion.

In 2012, pedestrian Brent Noorda peered out a window and observed the non-falling motion of an airplane up in the sky. This then-unexplaned (therefore magical) suspension has since been known as Airplane Aerodynamics.

(In 1738 Daniel Bernoulli published some book about fluid dynamics, including Bernoulli’s principle, but we’re going to ignore him.)

In 2014, this blog post attempted to connect these two phenomena. Can the reason that particles jiggle in Brownian Motion be the same reason that airplanes don’t fall from the sky?

Brownian Motion

Brownian Motion (Wikipedia article) is often observed by dirty daydreamers with nothing better to do than to watch dust motes float around in the sunlight (see dust-mote video), and by smoking chemists watching particles of soot through a microscope (see soot video).

It wasn’t clear what caused the random jostlings of Brownian Motion until Albert Einstein published his “Theory of Brownian Motion” in 1905 (known as Einstein’s “miracle year” because in that one year he released this and 3 other groundbreaking papers in physics, and still found time to record both Electric Ladyland and Sergeant Pepper). Einstein’s paper used the behavior of these microscopically visible particles to provide convincing evidence of the existence of atoms, too small to see in then-existing microscopes, and even to accurately count those invisible atoms.

What’s causing the Brownian Motion of visible soot particles in the air are their frequent and random-like collisions with very numerous, very small (even compared to soot particles), and very fast air molecules. If you could see the air molecules you’d be amazed at how many of those little things there were zipping about and bouncing against the relatively-giant bits of soot.

Interactive web page at MinuteLabs.io

You MUST now play with this interactive web page explaining Brownian Motion at MinuteLabs.io. I mean it. You are not allow to read further on this blog until you use that interactive web site. INTERACT NOW!

At the MinuteLabs.io page, if you are tracing paths of the big balls you’ll see their erratic Brownian Motion looking something like this:


If you adjust the “Drag to see what’s actually going on” slider to make the invisible air molecules visible, you’ll see that there’s a lot more going on there:



Upward-Only Brownian Motion (ignoring collisions from above)

I hope you played with that interactive web page at MinuteLabs.io long enough to really understand it, and to notice that those big colored dots are randomly bouncing all over the screen, and to understand why. Because now we’re going to change the rules a little bit.

I’ve altered that interactive page to create a new page showing Brownian Motion (with upward-only-option). This adds the option “Ignore Falling Air”, changing the rules so that collisions only have an effect if they are from below. In other words, tiny air particles coming from above are ignored (they pass right through), while air particles from below are the same old collisions (they continue to bounce off).

When you play Brownian Motion (with upward-only-option), which I sincerely hope you do, you’ll notice a decidedly different pattern of paths, e.g.:

There’s still a little bit of erratic behavior, but clearly, when collisions from above are ignored, these colored balls just want to rise.

The Upward-Only Brownian-Motion Wing

As we’ve seen, if an object could receive collisions only with air particles moving up, but was invisible to air particles falling down, then that object would rise, as if pushed from below (because it is being pushed from below). Such a magical device would be able to stay aloft in air.

We’ll now attempt to invent such a magical device, and we’ll call it a “Brownian Motion Wing” or just “wing”.

Imagine this “wing” object, having a right- triangular cross-sectional shape, moving from right to left at (oh... let’s just pull a number out of the air) 570 miles per hour. This wing is traveling through a sky full of very many super-tiny air molecules, a few of which are represented in this drawing. Those air molecules are moving very very quickly in random directions, with an average velocity of (oh…) twice the speed of the wing, so that relative to the wing the slowest ones at any instant are running away from the wing at about 500mph and the fastest ones are coming toward the plane at about 1500mph. Here is what I hope you’re imagining:


Notice first the level bottom of the wing compared to the sloped top. There are air molecules bouncing off the wing bottom, but no air molecules bouncing off of the top. This is because the front of the wing is pushing away most of the molecules whose trajectory would have hit the top of the wing. In a sense, the top of the wing is in the random-air-molecule-trajectory shadow of that front edge.

We have our magic wing!

Air molecules are now bouncing against the bottom but not against the top; in other words, there is force on the bottom of the wing pushing up, but no force on the top pushing down. That wing wants to fly!

Notice also that if the wing were not moving, but were stationary relative to the air, the front edge would not be preventing air molecules from hitting the top of the wing, and so there would be just as much downward force of air molecules bouncing against the top of the wing as there is against the bottom. So if this wing is not moving, it won’t fly.

This fast-moving “wing” is what we wanted. It is an object that collides only with air molecules moving up, but not with air molecules moving down. Success!!!!

Wing force-per-square-meter of Upward-Only Brownian Motion

Having invented this “wing” device, which is effected by the upward-pushing air molecules but not the downward-pushing ones, let’s calculate how much force is pushing up on the bottom of our wing.

Force per-square-meter of upward-only air molecules

First lets determine the force of a single air molecule bouncing off the bottom of our wing. To make this calculation easy, we’ll make the following reasonable assumptions:
  • the average air molecule weighs about 1e-25 lbs
  • the average air molecule is traveling at about 1100 miles/hour (in some random direction)
  • the wing weighs much (much much) more than an air molecule
Note about assumptions and estimates: These, and all of the rest of the numbers in this post, are reasonable but inexact estimates, but still reasonable enough to suggest whether further calculations are warranted.

I won’t show all my math, but here are things calculated along the way:
  • for air molecules moving in any non-downward direction (i.e. Vy>0) the upward component of velocity averages 635 mph (Vy=sqrt((V^2)/3), or about 284 meters/second
  • the change in momentum for each upward collision averages about 284e-25 (lb m/s)
  • there are about 2.69e+25 air molecules per cubic meter (at sea level, so we’ll assume plane is flying low)
  • in one second about 3.82e+27 collisions will happen between air molecules and the bottom of the wing (that’s half of all the molecules in a 284 meter tall column of air, or 1/2 * 284 x 2.69e+25, if assumptions about elasticity are made)
  • in each second, the change in momentum is 108,420 lb m/s (2.69e+25 * 284e-25) or 49179 kg m/s
Therefore, the upward force on a square meter of our magic wing is 49179 N/m^2 (where N is a Newton, which is 1 kg m/s^2).

How big must our magic wing be to support the weight of a 747?

A Boeing 747-400’s mass is 396,890 kg (875,000 lb), and gravity near earth is about 9.81 m/s^2. So the downward force on a 747 is up to 3893491 N (396,890 x 9.81).

To counteract this force of gravity, and keep the 747 in the air, our magic wing’s bottom surface area must be (3893491 / 49129 =) 79 m^2.

Boeing reports the 747 wing area as 525 m^2, which is 6.6 times larger than what is required by our estimate for what is required. That is much closer than I expected to be when I started this theory. Still, why are we off by 6.6X? Bad estimates? Whacky assumptions? Turbulance is a bitch? Boeing over-engineers?

The shape of our right-trangle wing?

In the previous right-triangle cross section drawing of our wing, I just guessed at the dimensions, especially the height relative to the width. So let’s get a better guess for our 747 magic wing.

The 747 cruises at about 913 km/h (567 mph) = 254 m/s. If we want the front edge to block our average downward air molecules (with average downward velocity of 284m/s) at that speed, the height/width ratio should be 284/254, making the cross-section of our triangle wing look like this:


That’s a HORRIBLE looking wing. Horrible, in a lot of ways, but the worst is all those air molecules that are pushing up against the gigantic front of the wing. How are the plane engines expecting to provide enough force to push all those molecules off the front?!

Even if each front-hitting particle would just magically “go away” after it hit the front, how much pressure is that?

Quick Calculation: Taking our wing bottom area to be 79 m^2, the wing front would be about 88 (79x284/254) m^2.  Each second, this would be colliding against 6e+29 (254*88x2.69e+25) air molecules each averaging 538 (284+254) m/s relative to the wing and so each changing momentum by 244e-25 kg m/s. The final force is therefore about 1.46e+7 N.

So to collide against all this air, our engines have to provide about 1.46e+7 N thrust. The combined engines on a 747 generate about 1.1e+6 N, which is about 13 times too little for our wing. The engines to push this thing through the air would have to be massive!

We could start modifying the front of the wing to be more aerodynamic, being angled to push more air down and out of the way. If we deflect the incoming air up we are getting pushed down, and if we deflect the air down there is some benefit, but that is offset by pushing away many of the upward-rising molecules we need to bounce against the underside of the plane.

In the end, this idea of bouncing away all downward molecules is requiring way more energy than current engines supply.

Conclusion: This wing idea doesn’t fly!

This upward-only browning wing is a failure. So the traditional explanation is correct, right?

The traditional explanation for what makes a wing fly is something like this: "The path around a wing is longer above the wing than it is below the wing; therefore the molecules above the wing must travel faster to keep up with the molecules below the wing; therefore, Bernoulli (who we almost got away with not mentioning) says the air pressure above the wing is lower than that below, so the wing is pushed up."

The problem with this traditional example is the idea that for some reason the molecules above the wing go faster than those below the wing. That’s just silly. It’s as if runners around a race track will run faster at the curved ends of the track than they do during the straight sections, just because something about curvature makes a person run faster.

If you want to know what really makes a plane fly, see this superbly excellent article: A Physical Description of Flight; Revisited, then watch some water wrap around a glass under the sink, especially if you can find a wing-shaped glass, and say to yourself “Oh, now I get it!” (But then again, maybe that article and the water-under-glass example are improperly crediting the Coandă effect, and so maybe you still don't get it at all, and neither do I.)

So, does air flight have anything to do with randomly-moving air molecules?

Yes, the flow of air around a wing is completely governed by those random air molecules bouncing around. From a macro level, watching smoke wrap around a wing, or streamline paths in our drawings, it can seem like air is a fluid macro thing. But air is really mostly empty space, if you look at it closely enough, with lots of tiny tiny balls of air molecules bouncing and colliding. It is their collective behavior that gives the characteristics of a fluid.

Take, for instance, the idea “low pressure” that often comes up when describing air flow, streamlines, and “lift”, and especially the extreme of low pressure  of a “vacuum”, which is an area of no pressure.  Lower pressure simply means there are fewer particles bouncing around (or slower particles); while a vacuum means there are no particles.

Nature doesn’t care about a vacuum

You’ll often hear that “nature abhors a vacuum”. But that isn’t at all true. Vacuums are fine, they just don’t tend to last for very long. This isn’t because the surrounding area abhors (or even notices) a vacuum, or is being sucked in by the vacuum, but simply because the vacuum is a place where randomly-moving particles that happen to be headed that way or not going to meet with any other particles to keep them away.

To demonstrate why vacuums don’t last very long, even thought they don’t express any force themselves, I’ve modified the interactive web page one more time, creating Push air molecules around, so it will only show the air molecules, and to allow you to push those molecules around.  If you’re quick about it you can create a vacuum region like this one:


But the vacuum region won’t stick around very long.

Go ahead, play with Push air molecules around. It’s fun, and it’s all I’ve got left.

1 comment:

  1. You had me convinced until the result that your wing is off by 13X. But, wait, would it also be fair to say that 1/13 of the lift of a wing is caused by this process of selectively hiding from the downward-flying air particles?

    ReplyDelete