Sunday, January 27, 2013

Snow Crash

Update: Because I'm neurotic and check my math again and again, I discovered that I made a slight error in my calculations. Somewhere along the way I incorrectly converted some units and ended up with an answer that I'm pretty sure is a million times smaller than the actual value. That is, it takes way more energy to melt (or sublimate) a snowflake than I said, such that my car would have to be going something like a thousand times faster in order to effect a phase change.

Which means my prediction was egregriously wrong. I guess it's a good thing I didn't try to get my research published in a peer-reviewed journal. (I'll note, however, that being off by six orders of magnitude is not quite as bad as being off by 120 orders of magnitude.)

So where does that leave us? I have no idea, really. Perhaps the difference between sticking and not sticking is not related to melting or sublimation. Perhaps the air resistance a moving car experiences pushes snowflakes away from the windshield. I'll have to give this a little more thought...

And now for the original entry:

(Sorry for the month-long absence, my numerous and devoted fans. I'll try to post more frequently. However, school starts back up again on Monday. I might have more to write about, but I'll have less time to write about it. We'll see what happens.)

So we had the first real snowfall of the year this week, and I noticed a rather interesting phenomenon while driving home from work the other day. No, it's not that storm intensity appears to be inversely correlated with driver intelligence; that's depressingly typical.

Rather, I noticed that the flakes hitting my windshield reacted differently at different vehicle speeds. That's not too surprising. In the rain, driving faster means more rain hits the windshield per second, and the rain will hit harder, so the end result is you need to speed up the wipers. But this didn't hold true for snow.

When my car was stopped at a light, the snow would accumulate on the windshield and I'd have to hit the wipers before I started driving again. But at a high enough speed, the flakes wouldn't accumulate. They'd strike my windshield, and then be gone. As best as I could determine while driving down a busy road in the middle of a snowstorm and trying not to reduce the average driver intelligence, the critical speed was about 10 mph.

So what's the explanation? I'll admit upfront that my physics class this past semester was a little light on the thermo, so I'm going to be making a lot of assumptions and ignoring (or being ignorant of) a lot of complicated factors. But my theory is that when my car is stationary, the kinetic energy of a falling snowflake is enough to melt the flake from ice to water, but no more. When my car's forward velocity is added on, however, the snowflake has enough kinetic energy to sublimate directly into gas without first passing through the liquid stage, and thus does not stick to my car.

Materials are only able to sublimate when the temperature and pressure are below that material's triple point. The triple point is a specific temperature and pressure at which a material can exist as solid, liquid, or gas. Water's triple point (snow is really just frozen water with air mixed in) is about .01 &degC and a pressure of .006 atm. You'll notice that Earth's atmosphere has an atmospheric pressure of 1 atm (funny, that), which is significantly higher than .006. The answer to this is that the pressure close to a solid is a result of the solid's vapor rather than the atmosphere as a whole. The upshot is that water can easily sublimate at low enough temperatures.

Okay, then, how much kinetic energy does a falling snowflake have? Our old friend terminal velocity makes an appearance again. According to this paper, the terminal velocity of a snowflake in the conditions I was driving through is about 1 m/s. Wikipedia tells me that a typical snowflake consists of 1019 water molecules, which is 1.66x10-5 mol of water. The molar mass of water is 18 g/mol, so a typical snowflake comes in at 3x10-7 kg. Our formula for kinetic energy is &frac12mv2, so a falling snowflake packs a whopping 150 nanojoules.

Let's figure out what we can do with 150 nJ. The energy required to melt ice into water is known as the specific heat of fusion. Under ordinary pressures, this occurs at the familiar 0 &degC. At other temperatures, adding energy changes the temperature, but at 0, it causes a phase transition. The explanation for why this occurs at a particular temperature for a particular material is well beyond my understanding, but the basic idea is that the average kinetic energy of an ice molecule is enough to break the intermolecular bonds holding the ice together. Water's specific heat of fusion is 334 kJ/kg, which means that it requires 334 kJ of energy to transform 1 kg of ice into water.

Now, there are a lot of big assumptions here. Some of them are: all of the snowflake's kinetic energy goes into melting it, the air inside a snowflake doesn't change anything, the temperature of my windshield has no effect on the transition, none of the snowflakes interact with each other, etc. Anywho, if you run the numbers, it looks like it takes 100 nJ of energy to melt 300 &#956g of ice. Thus, falling alone is enough to melt the ice. But is it enough to sublimate the ice?

To sublimate, one need only add the specific heat of fusion onto the specific heat of vaporization (the heat required to turn a liquid into a gas). Water's specific heat of vaporization at 0 &degC is 2500 kJ/kg -- way higher than fusion. So to sublimate, you need a total of 2835 kJ/kg (there's some invisible rounding go on here), which is roughly 8.5 times as much energy. We only have 1.5 times the energy required to melt, so we cannot sublimate from falling alone.

I guesstimated that my car going 10 mph was the critical speed. Let's see what that gives us. 10 mph is 4.5 m/s. We can treat my car as stationary and the snowflake as moving at 4.5 m/s toward my car horizontally. This is easy to believe. When you're driving into a snowstorm, even if the snow is falling straight down, it appears to be moving almost directly at you. The total speed of the snowflake relative to the car is calculated from the Pythagorean theorem: it's the hypotenuse of the right triangle formed from the vertical and horizontal components of speed. This amounts to 4.6 m/s.

Our snowflake is now traveling 4.6 times faster than it was when it was just falling, which makes it 21.25 times as energetic. This means it has more than enough energy to sublimate itself. In fact, we can calculate what the critical speed really is (given our assumptions). The snowflake needs to be traveling √8.5 times faster, or 2.9 m/s. The horizontal component of that speed is 2.7 m/s, or about 6 mph.

I think that was a pretty darn good estimate, if you consider the vast number of factors I was hand-waving away. It's possible that there are a whole bunch of factors swinging one way and an equal number swinging the opposite way, making me only accidentally correct, but regardless, it's pretty nice to have your "theory" match up with your "data." Go Team Science!


  1. I would have thought this was an aerodynamics problem...

  2. It probably is. At the time, my knowledge of aerodynamics was limited. (It still is, really.) There is a followup post that looks at this from a drag/friction angle.

  3. I'm just pointing out the intuitive 'truth' that a low-density object like a snowflake -already traveling at a terminal velocity greatly reduced by the atmosphere- is surely going to find the push of the air the car displaces a more significant factor by something like several orders of magnitude...