Lake Pollution

PREREQUISITE SKILLS:

1. MODELING WITH DIFFERENTIAL EQUATIONS

2. SOLVING FIRST-ORDER DIFFERENTIAL EQUATIONS USING NUMERICAL AND ANALYTIC METHODS

Modeling pollution in a river-bay system can be thought of as a mixing problem where the rate of a quantity Q of a pollutant is equal to the difference between the rate of input(s) and the rate of output(s):

Our particular problem concerns pollution in a small stream and bay connect-ed to the Great Lakes. The primary source of pollution is located upstream on the river. Additionally, there is an aluminum factory located directly on the bay that contributes to the problem. Environmentalists are very concerned about the level of pollutants in the bay and have lobbied for laws to protect the bay. The present law forces the aluminum factory to temporarily shut down anytime the average concentration of pollutants reaches 1.6 milligrams per liter of bay water.

To model the change over time of the concentration of pollutants in the bay, we will consider it to be of a constant volume V containing at time t an amount of pollutant Q(t). Assume the pollutant is evenly distributed throughout the bay with a concentration C(t) where . Also, assume that water from the river containing a constant concentration k of pol-lutant enters the bay at a rate r and that the polluted water in the bay flows out at the same rate. From this information, we determine that and 

REQUIREMENT 1.

Write a differential equation that models the concentration of pollutants in the bay. Include a boundary condition if the bay begins with N milligrams per liter of pollutants. Explain your model.

PROBLEM CONTINUATION:

We will assume that the approximate volume of the bay is four million liters. We further assume that the water flows into the bay at a rate of 40,000 liters per day, and that the water flows out of the bay at the same rate. The amount of pollutants Q(t) will be measured in milligrams and the concentration C(t) will therefore have units of milligrams per liter. Time will be measured in days. The bay begins with 0.8 milligrams of pollutants per liter of water. The stream normally has a pollution concentration of k = 0.5 milligrams per liter.

REQUIREMENT 2.

A. Using a numerical method, find the concentration of the pollutants in the bay for the first thirty days.

B. Solve the differential equation by the method of Separation of Variables to obtain the concentration of pollutants in the bay for the same data as in part (A.) of this requirement.

C. Compare the answers to the two methods. Which method is more accurate? Why?

D. Will the pollution concentration achieve a steady state value? If so, what is it?

REQUIREMENT 3.

Four million milligrams of pollution are instantaneously spilled into the stream. Assume that all of the pollution in the spill reaches the bay, that it arrives as one big input (or impulse), and that it is instantly mixed throughout the lake upon arrival. [Reminder: The bay has a pollution concentration of 0.8 mg/1 already.] For how long will the factory have to shut down? Assume that after the spill reaches the bay the amount of pollution in the stream returns to normal.

REQUIREMENT 4.

List and discuss at least four factors not taken into account by this model that would have an impact on the pollution level in the bay.