Example Question #2 : Math Modeling
An oil spill occurs at a factory 12 kilometers up stream from a town. One hour after the spill happens it reaches the stream and a 1600 meters long patch of oil begins to flow towards the town at a rate of 2 kilometers per hour. The maximum concentration of the oil in the water is 15 times the acceptable level. What is the maximum concentration expected in town and when will it arrive?
Identify what is known and the assumptions.
This is a diffusion problem and thus a diffusion equation will be used with a term of relative concentration.
The relative concentration is denoted as
This function has been normalized resulting in the following.
The law of conservation of mass will also be useful is solving this problem.
The goal is to calculate the maximum pollution level in town.
The diffusion equation is,
Using the Fourier transforms to solve the diffusion equation is as follows.
For this particular function
so the interval will be
Now find where the maximum occurs in time.