## Friday, September 16, 2005

### Salt and Toxins

This semester I'm teaching a differential equations course. It's kind of fun. (Don't tell my students, but I didn't take this as an undergraduate. I actually learned most of this when I taught the course for the first time about four years ago.)

We just recently finished a section on modeling, which includes so-called mixing problems. Generally, there is a tank containing some solution (often salt dissolved in water), and some inflow and outflow of solution with differing concentrations. The idea is to model how the concentration of the solution in the tank changes based on the concentration of the incoming solution and what is allowed to flow out of the tank. A common problem is a "flushing" problem, where we start with a mixture in the tank at some concentration, and have clean water flowing in while the mixture flows out. Over time, the concentration in the tank will approach zero, "flushing" the tank.

It's a nifty application, and the approach to modeling with differential equations is (I think) pretty clever. One thing I like about this application is it has a nice application to environmental issues. The same idea is often used to model pollution levels in a lake, given that there is some inflow and outflow to the lake in the form of rivers or streams. In fact, I was once involved with creating a web-based module focusing on the application. I like the fact that in some sense this is a mathematical problem that might lead people to think about broader social implications about clean water and environmental protection.

There's a problem like this in our text that talks about two inter-connected lakes, one feeding into the next. At the start of the problem, it says 1000 pounds of a toxin is spilled into the first lake, and the task is to model the levels of the toxin in the two lakes over time. However, I noticed something in the solution set developed in our department: The solution immediately starts talking about the concentration of salt in the lakes. I found this oddly disturbing, but no more so than discovering many of my students did the same thing when solving the problem.

Now while I know salt is not really considered a health food, I think switching from "toxins" to "salt" is a pretty significant change. Obviously the model looks the same either way, but it means very different things for actual lakes. Perhaps nobody really takes this problem particularly seriously. It sort of bothers me that there's a glib attitude towards polluting lakes, even if they're imaginary ones.