Oxygen Sag in a River

by Ryan Eppler

Submitted : Fall 2011

Dissolved oxygen is very important to support fish populations in rivers, lakes and streams.  In particular, a fish population can become at risk if the dissolved oxygen content falls below 4 mg/L in the water. This can happen in highly polluted waters when microorganisms in the water use up the available oxygen. A commonly used model for the dissolved oxygen content in a river or stream is:

1. We would like to know the following:
2. Develop an expression for the time to reach minimum oxygen concentration.
3. Develop an expression for the concentration of oxygen at the time it reaches its minimum value.
4. Reasonable values of K_R and k_d are 0.1/hr and 0.02/hr, respectively.  The concentration of oxygen in saturated water is 8.0 mg/L at 27 ^oC, the temperature of the river. If L₀ is 50 mg/L, will the oxygen level drop to an unacceptable value?
5. What is the largest value for L₀ for which the fish population will not be at risk?
6. Verify your answer to (4) by making a graph of C versus t.

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