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
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
(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
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.