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# A Single Lake

They say the longest journey begins with a single step. We begin our study of the five Great Lakes by considering a single lake. Pick your favorite, and keep it in mind.

Our immediate goal is to develop a difference equation which models the amount of pollutant in a single lake. The amount of water and the amount of pollutant will both play important roles in our study. To help in our formulation we define the following parameters:

1. Let V be the volume of water in our lake, in cubic miles.
2. Let be the volume of water flowing into the lake, in cubic miles per year.
3. Let be the volume of water flowing out of the lake, in cubic miles per year.
4. Let be the concentration of pollutant in the incoming water, in units per cubic mile.
5. Finally, let be the amount of pollutant in the lake at time t. The initial condition, which must be known, is the amount of pollutant in the lake at time zero, namely The quantities, are the variables in this problem.

With these parameters it is possible to formulate a difference equation for the amount of pollution in the lake at time t. The basic idea for this equation is the same as for all difference equations, namely that the change in the pollution level per unit of time, is the amount of pollution flowing into the lake each time period less the amount of pollution leaving the lake each time period. The challenge in setting up this difference equation is to determine the amounts of pollution entering and leaving the lake.

Since we have defined to be the concentration of pollutant in the water entering the lake, the annual inflow of pollution is given by the product of and .

The amount of pollution leaving the lake per unit time is also the product of the pollution concentration and the amount of water leaving the lake. We assume the amount of water leaving the lake is a known quantity, a quantity we have called What we do not know in advance is the concentration of pollutant in the outflow at every time t. We can, however, write an expression for this concentration. It is . Since is the total pollution in the lake at time t and V is the volume of water in the lake, the given quotient is the concentration of the pollutant in the lake. Since lake water flows out of the lake, the quotient is also the concentration of pollutant in the outflow.

Once we know the amount of outflow () and the concentration of pollutant in the outflow, the total pollution leaving the lake is given by the product of these two quantities, namely

The total change in the amount of pollution within the lake is then given by the difference equation

Once an initial condition is given for this equation is in the form to be analyzed with Stella.

Next: An Example: Lake Up: No Title Previous: The Great Lakes

Steve McKelvey
Fri Jul 5 08:01:19 CDT 1996