# Math 224--Differential Equations: Project 1, Spring 2000

## A Draining Project...

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, February 2000

©2000 Gavin LaRose (glarose@umich.edu)
permission granted to use and distribute free in an academic setting.

### Lonlinc CPE

the Lonlinc Building
South 9th Avenue
Lonlinc, SK 08685

14 February 1999

Rigorous Mathematical Contractors, Inc.
Suite 3, Strawmarket Business Plaza
Lonlinc, SK 04685

Dear Rimac:

The Lonlinc CPE (Council for the Protection of the Environment) is charged with monitoring the environmental impact of a wide range of human activities. We are contacting you to examine a particular case of one such activity, the chlorination of public water supplies. As you know, most communities kill bacteria in their water supply by adding trace amounts of Chlorine. In those cases in which the water supply is derived from a closed source from which water is removed and replenished as necessary, one way of doing this is to have a small amount of Chlorine added at a constant rate to the source. As the volume of water in the source is commonly maintained between an established minimum and maximum level, the Chlorine concentration in the water contained therein will remain between two known levels which may be chosen to fall between accepted bounds.

This scenario changes, however, if the source is being emptied---as, for example, for maintenance or if the source is being depleted in the event of a drought. In this case the volume of water in the source decreases continuously, and the Chlorine concentration in the water may increase to a level exceeding the accepted limit. It is to gain some insight on the nature of this phenomenon that we are contacting you.

For the sake of this example, we would like to consider a water tank of height approximately 40 feet and diameter 57 feet (about the size as might be used by a small community). We will assume that the water is draining out of the tank forced strictly by gravity, so that the rate at which it leaves is given by Torcelli's law. Torcelli's law states that the rate at which the water leaves the tank (in ft3/sec) is proportional to the square root of the water level, with constant of proportionality 25.21 (=sqrt(2 g) (cross-sectional area of outflow tube)). Given this information, we would like to find the amount (and hence concentration) of Chlorine in the tank as a function of time as it empties. You may assume that Chlorine is introduced to the tank at a constant rate of R = 0.085 lbs/hr.

It is, of course, possible to turn off the addition of Chlorine at some point in the draining process. We would also like to know when this should be done to keep the Chlorine concentration in the tank below the accepted maximum value which we take for our purposes here to be 3~ppm.

Because we need the data from this project to resolve a number of other outstanding issues, we need your report in fairly short order. In particular, we need your preliminary analysis of steps to be taken in the solution to this project no later than 21 February, with your final, typewritten report due by the 3rd of March. In order to expedite your accomplishment of this, we have arranged that a self-proclaimed mathematical expert in your area, the inscrutible Dr.~maybe-the-P.-stands-for-Pontificating Gavin LaRose to serve as a resource on the matter. You should contact him with any questions you may have as you work through the problem. He also has sample reports for your examination in the event that you should find it useful to see examples of acceptably formatted papers.

We look forward to receiving your work.

Sincerely,
Jack C. Ousteaux
Director, Lonlinc CPE

jco:glr

Gavin's DiffEq Project 1, Spring 2000
Last Modified: Wed Feb 16 20:42:46 CST 2000