Sub-Module 3A, Page 5

Variations with time

Concentrations of some contaminants will not vary with time, or vary so slowly we may consider that they are constant. Some long lived radioactive isotopes, some metals, for example, change very little. But the concentration of most contaminants will change with time. Let's look at a lake that has been contaminated with MEK from a barge spill. The lake has a stream running in and a stream running out. At the start of our analysis, we will assume that the lake has the same concentration of MEK and the lake is "well-stirred." By well-stirred, we mean that the concentration of MEK in the entire lake is the same as the concentration in the outflow stream. If MEK diffuses rapidly in the the lake and the streams are small compared to the volume of the lake, this is not a bad assumption. In any case, if we don't assume it the math gets impossible.

Here is a side view of the lake:

So if the volume of the lake is 5,000 m^3 and the initial spill of MEK is 500 kg, what is the initial concentration of MEK in the lake in mg / L . [HW problem #1]

Now let's suppose we had a method of instantly putting in 1000 m^3 of clean water and removing 1000 m^3 of the 100 mg/L contaminated water. What is the concentration of MEK now? You might see directly that you have removed 20% of the contaminated water, so the concentration is now 80 mg / L. Or you might proceed as follows.

There was 1000 m^3 removed at a concentration of 100 mg /L so the mass removed was 100 mg/L times 1000 m^3 , remembering there are 1000 L in a cubic meter and 1 million milligrams in a kilogram, we get that there was 100 kg removed. So if we start with 500 kg, we now have 400 kg left. Our new concentration must be 400 kg / 5000 m^3 and after converting units 80 mg/L is our new concentration.

So, suppose the flow from the streams is 100 m^3 per hour. In 10 hours we remove 1000 m^3 of contaminated water and replace it with 1000 m^3 of clean water. So if those were the flows, would the concentration of the lake now be 80 mg/L as it was if we instantly changed 1000 m^3? Here's why

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