STATEMENT OF PROBLEM

Industry Lake is bordered by a chemical plant which spews 3 tons (6000 pounds) of pollutants per minute into the lake. Numerous creeks run into the lake with a net inflow of 5000 gal/min, and a dam controls the outflow rate. The volume of the lake remains at approximately one million gallons.

(1) If the dam allows mixed lake water to flow out at 5000 gal/min, find a function that describes the amount (in pounds) of pollutants in the lake at any time. Also, determine the long term behavior of the pollutants in the lake. Clearly state the assumptions that you are making in your model, with one of your assumptions being that the pollutants mix throughout the lake instantaneously.

(2) Of course, the mixing is not instantaneous; in fact, studies have shown that the water in lakes is layered similar to the layering found in the atmosphere, and that small particulates actually stay close to the surface, while large bits of pollutants sink to the bottom.

For a great simplification (although not as simplified as before), assume that the lake has essentially 2 layers (of equal volume) and that mixing is instantaneous only within each layer.

Further assume that the transfer rate from one layer to the other is proportional to the rate is proportional to the pollution density level in the layer that the pollutants are LEAVING. (The constants of proportionality are called diffusivity constants.) You should still assume that mixing is instantaneous within each layer.

Note: The assumptions outlined above are more general than assuming that pollutants diffusion from one cell to the other is proportional to the DIFFERENCE in between pollutant levels. In the case where the diffusivity constants are the same, you are making this assumption.

(i) Write down the appropriate set of differential equations that models this situation. (You'll need two equations -- one for each layer.)

(ii) Show that the total pollution concentration in the lake was over estimated by assuming instantaneous mixing throughout the lake. Why would this be an important result if you worked for the company?

(iii) Assume that the diffusivity constants are the same.
- What happens as the diffusivity constant goes to zero?
- What happens as the diffusivity constant goes to infinity?
Give a physical explanation for the results.

(3) Describe a general model that can account for noninstantaneous mixing throughout the lake. You are not expected to solve the equations, but you should be able to formulate them and carefully explain what all of the variables and parameters are.

Hint: You might break the lake up into several compartments - a number in the upper level and a number in the lower level - all of which have diffusion with their neighboring compartment.