The Peclet Number, and Why Some People Shouldn't Get Degrees

Here's a scenario posed on a recent Safety quiz, with a few minor modifications. You're south of a town with an enormous chemical plant on its outskirts. You're not entirely sure what they make there, except that the end product is pretty nasty, and the intermediates they use to get to the end product are even nastier. You hear a loud boom, look to the north, and see a plume emerging from the plant. The wind is blowing strongly to the south (ie, toward you), and there's no distinguishing terrain near you in any direction. Which way do you run?

The answer relies on a dispersive model of fluid flow that compares diffusive to convective mass transfer. That's not nearly as complicated as it sounds. In fact, you've experienced both of them in your own kitchen. Imagine you're baking cookies, and within a few minutes, the delicious scent of cooking sugar and butter fills your kitchen. If you open the door to the oven, the smell gets stronger, as a puff of vapor is released. Obviously the smell comes from the oven, but in the absence of anything else acting on them, the particles that you smell would rather spread out (diffuse) than remain concentrated.

Now imagine you're sitting in a room upstairs far away from the kitchen. Eventually, if the cookies stay in the oven long enough (but hopefully not so long that they burn!) you'll smell them even upstairs, because a few scent-carrying particles reach you. There's a much faster way you might smell them, though. What if the oven is near an air vent, and so are you? It would be like the Seinfeld episode "The Calzone", where Steinbrenner knows George has his lunch just because of the smell coming out of the vent. In this case, the particles aren't merely diffusing, they're convecting, or being carried along by the air currents in your heating or cooling system.

Do diffusion and convection ever happen in the same system? Sure. At the same time you're smelling the cookies through the vent, someone downstairs can smell them too, and probably more strongly than you can.

Or, say you have some water in a bathtub. If you put some green food coloring in that water, you'd see it diffuse through the tub until all the water was about the same intensity green. But before you let it diffuse all the way, say you opened the drain. You'd observe water flow toward the drain--convection--as well as diffusion. Depending on how fast the water was draining, you might see a well-defined line of food coloring with a bit of fuzziness toward the edge, suggesting strong convection. Or, you might see a blur of green throughout with only slight movement toward the drain, suggesting strong diffusion.

Us engineering types use a number called the Peclet number to describe systems like this. It's merely a ratio of convective motion to diffusive motion. That's the key to answering the chemical plume problem. If you assume strong wind toward the south (ie, fast wind speed), the convective motion of the toxic cloud becomes much more important than the diffusive motion. That means the toxic particles in the cloud are going to be carried in one direction rather than spreading out in all directions. Unfortunately, that direction happens to be toward you, to the south. So which direction do you go?

You could try and go south, essentially trying to outrun the plume, but that might be a bad idea unless you can go really, really fast. "Strong" wind probably means something in the range of 20-30 miles an hour; if you don't have a car, there's no way you can hope to go this fast. The best choice is east or west--if you assume that all motion is in the convective direction, the effects of dispersion in any other directions is relatively small, so you minimize your chances of coming into contact with any of these particles.

Would you even think of standing still or running north? Both of these practically guarantee you're going to meet the plume head-on. And yet, some of my classmates, in their fourth year of education to be chemical engineers, thought that might be a prudent course of action.


Currently listening: Symphony No. 9, "From the New World", Antonin Dvorak