CASE STUDY: Stream Assimilation Capacity for Waste Material (Dissolved Oxygen II)

Source:

Science from Chemodynamics, Louis J. Thibodeaux Š1979 by Wiley-Interscience Publication, New York, NY. Model design by Sanjit Mohapatra, Montgomery Blair High School, Š1994. Case study narrative by Robert Gotwals.

Goal:

To build an advanced model of the ability of a stream to assimilate waste through a measurement of the concentration of dissolved oxygen.

Background:

A measure of the "health" of a body of water, particularly a lake, river, or stream, is the measure of dissolved oxygen. In water, oxygen, O₂, gets mixed into water and dissolves into the water. This oxygen is used by fish and other animals as a source of oxygen to "breathe" through gills and other respiratory mechanisms. Dissolved oxygen also plays a role in plant growth. Low levels of dissolved oxygen can the growth of harmful algae and other plants, which can further cause problems for fish and other water-based organisms.

Dissolved oxygen concentration is affected by a number of parameters. In this case study, we wish to study the effect of the dumping of sewage and industrial wastes into a stream. Sewage and industrial waste are organic compounds that are characterized by the formula C_xH_yO_z, meaning they contain some amount of carbon, some amount of hydrogen, and some amount of oxygen. These organic compounds are broken down to carbon dioxide (CO₂) and water (H₂O) by microbes and other organisms in the water that "eat" these organic compounds and break them down to non-toxic chemicals. In order to do this, the microbes use the oxygen in the water to help "digest" the organic material. We can say that for some amount of harmful organic waste, some amount of oxygen in the water is needed. This is known as the "biochemical oxygen demand", or BOD. The oxygen used to break down organic pollutants must be replaced by reoxygenation. Most reoxygenation occurs from the dissolving of oxygen from the atmosphere at the surface of the water.

If the dissolved oxygen falls below a certain level, the fish in the water will die, harmful plants will grow rapidly, and the water will smell. A general number for this level of dissolved oxygen concentration is 5 parts per million (ppm). Parts per million means that for every million molecules of water, there is one molecule of oxygen.

Water can only dissolve a certain amount of oxygen. Typically, the value of 9.8 ppm is accepted as the maximum amount of oxygen from the atmosphere that will dissolve in the water. Once the concentration of dissolved oxygen is 9.8 ppm, no more oxygen will dissolve. To use an analogy, everyone has a limit on the number of hot dogs they can eat. Once you have reached this number of hot dots, there is simply no more room in your stomach to put more hot dogs!

In this model, the scenario is as follows: a factory is located near a stream. The stream flows into a larger body of water. The factory dumps waste into the stream in varying amounts, and the waste affects oxygen concentration in the water through the biodegradable processes. When the oxygen concentration gets below a certain concentration, the fish in the stream begin to suffocate. The goal of the model is to find the maximum amount of waste that can be released without killing the stream.

Building the Model:

The algorithm for this model comes from the work of Streeter Phelps in 1925. There are a variety of forms of this algorithm, but we'll use the primary Streeter Phelps equation in this case study. Only two mechanisms can affect the amount of oxygen dissolved in the water:

oxygen consumed in the biodegradation process (i.e., oxygen used in getting rid of the organic wastes)

reoxygenation from the atmosphere

We wish to measure the concentration of dissolved oxygen. We also only have two processes affecting the amount of waste present (also measured in parts per million):

waste being discharged into the water by the polluters

waste being removed through the biodegradation process

Some of the assumptions being made in this model are:

the temperature and pressure are constant (non-changing)
the waste and the water mix completely
the stream moves at a constant velocity
no other waste is dumped into the stream except the waste being modeled.

In general, we would call this stream "ideal". In real life, it is unlikely that the assumptions above would be true. Once this relatively simple model, using the above assumptions, is created, it is typical for a modeler to try to improve on the model by removing the assumptions one at a time!

We'll look at each of these four processes separately:

Waste coming in: we're going to assume, for this model, that the waste is being dumped into the stream one time, not continuously over time. As such, you might wish to consider the use of the PULSE function!
Waste being removed: there are two factors that influence how fast waste is removed. The first is time. The second is the waste removal coefficient. This is a number with units of ppm of waste per second. A typical value of this waste coefficient is 7.38 x 10^-5 ppm/sec. This value means that 7.38 x 10^-5 ppm of waste is being removed every second. The algorithm therefore is:

Waste removed = coefficient*concentration of waste*(time+dt)
Reoxygenation: reoxygenation depends on the volume of the stream and the height of the stream, as well as a reoxygenation rate constant, or reoxygenation coefficient. The amount of reoxygenation that occurs also depends on the surface area of the water. The more area, the easier/quicker it is to reoxygenate the water (up to a limit of 9.8 ppm, of course!). The algorithm for reoxygenation is the LOWEST (minimum) of these two algorithms:
- reoxygenation = area*(9.8-oxygen concentration)*coefficient*time
- reoxygenation = 9.8 - oxygen concentration
Note that this algorithm is ONLY TRUE if the concentration of oxygen is NOT 9.8 ppm. If the concentration is 9.8 ppm, then the amount of reoxygenation is 0 ppm.
Oxygen consumed: this depends, of course, on the amount of waste in the water. It also depends on the height of the stream, time, and the various coefficients. The algorithm for oxygen consumed is: oxygen consumed = (waste coefficient*waste concentration*time) - (reoxygenation coefficient/stream height * O₂concentration*time)

Other calculations you will need:

Grams of waste in: in this model, you will pulse in some amount of waste in grams at some point in time. This amount needs to be converted to parts per million (and it needs to be converted to a amount that is time-based):

amount of waste in = grams of waste*time/(volume of water*1000)

Initial Conditions:

Once you have built your model, enter the following data as initial conditions:

Initial waste: 0 ppm
Initial oxygen: 9.8 ppm
Waste coefficient: 7.38 x 10^-5 ppm/sec
Reoxygenation coefficient: 1.65 x 10^-3 ppm-ft/sec
Stream height: 20 feet
Area: 200 ft³

At 1 second, pulse in 2.5 x 10⁸ grams of waste.

Run the model for 100 seconds. Create a plot of the concentration of the dissolved oxygen and the concentration of waste versus time.

Using the model:

Use your model to answer these questions:

With the data given above, what is the lowest value of oxygen? At what point in time does the "oxygen sag curve" reach this lowest point?
The Durham Municipal Sewage plant reported that an accidental spill of 2.8 x 10⁸ grams of untreated sewage was spilled into the Eno River the previous day. Initially, the oxygen concentration of the Eno River was 9.37. The following day, a number of dead catfish were seen floating dead several miles downstream. Catfish need at least 3.8 ppm of dissolved oxygen to survive. The sewage plant contends that the accidental spill did not cause any fish deaths. They state their models and lab analysis of the dissolved oxygen content of the water at several points downstream consistently showed an oxygen content of at least 4.0 ppm, enough for the catfish to survive. Use your model to support or defend Durham's argument.

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