next up previous Index
Next: Lakes as Well-Mixed Flow Up: Solutions Previous: Solutions

(Want to get to the simulators? Go on and skip ahead to the next page. ->
We'll still be waiting for you here when you get back.)

Lake Pollution Solutions

In this part of the module, we will look at solutions for the differential equation

tex2html_wrap_inline11

that gives a mathematical model for the concentration of contamination in a lake. 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. For more details on the assumptions behind this equation and the derivation of the terms, please go back to the previous screen and choose one of the explanatory options.

Here are some scenarios for what happens in the lake, together with the choices you should make in the following screens 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.


next up previous Index
Next: Lakes as Well-Mixed Flow Up: Solutions Previous: Solutions
John Hoggard <hoggard@math.vt.edu>
Last modified: Sun Aug 24 14:33:00 EDT 1997