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What's All This About?

This module on lake pollution is about using differential equations to model lake pollution. To model lake pollution can be complicated, because there are many different factors which can be taken into account. We will use a simplified model, and consider only a few basic factors, which we list below:
Q The volumetric flow rate through the lake
V The volume of the lake
k (t ) The reaction rate coefficient of the contaminant
C (t ) The concentration of the contaminant both within and exiting the lake
Cin(t ) The concentration of the contaminant entering the lake

We will therefore be looking for solutions to the differential equation

tex2html_wrap_inline11

that gives a mathematical model for the concentration of contamination in a lake. (Explain why.) The volume V of water in the lake is assumed to be constant. The contaminant is carried into the lake by an incoming flow of water at a rate Q (volume per time unit) and with a concentration tex2html_wrap_inline13 (mass per unit of volume). There is a corresponding outflow at rate Q from the lake, and this outflow has the same concentration C of contaminant as the lake (as is the case for a well-mixed lake). C and tex2html_wrap_inline13 will depend on time, in general. We allow for a chemical reaction, with rate constant k, in which the contaminant in the lake decay into other substances that are of no concern.

Here are some scenarios for what happens in the lake, together with the choices you should make in the module in order to simulate each one. Choose "No Reaction [k(t)=0]" if the decay reaction is negligible, and choose "First Order Constant Reaction [k(t)=k, for k constant]" if you would like to have the option of choosing a rate constant in your simulations.

Scenario 1. (Impulse)
At t = 0, a load of pollutant is suddenly dumped into an otherwise clean lake. tex2html_wrap_inline13 is zero, so the flow of clean water into the lake gradually flushes the pollutant.

Scenario 2. (Step Input)
The lake is free of pollutant until time t = 0, when the plant begins operating. From that time on, tex2html_wrap_inline13 has a constant positive value.

Scenario 3. (Sinusoidal Input)
This is the same as Scenario 2, except that tex2html_wrap_inline13 varies periodically and we allow the initial concentration to be positive. We would especially like to know where the variations in the input concentration will have any significant effect or whether, for practical purposes, we could just as well simplify the calculations by assuming that tex2html_wrap_inline13 remains constant at its average value. In this case of course, we also have a period and an amplitude for the sinusoidal function which must be taken into account.
next up previous Index
Next: Contents Up: Beginning Previous: Explanation

John Hoggard <hoggard@math.vt.edu>
Last modified: Sun Aug 24 14:26:44 EDT 1997