CE 170: Environmental Engineering

Dissolved Oxygen Sag Curve



  1. Derive a simple mathematical model the incorporates the factors that influence the oxygen resources of a stream.
  2. Practice solving problems using this model.
  3. Discover how using computer-based mathematical models of the environment can assist in solving engineering problems.



Cooper et al, Foundations of Environmental Engineering, Section 4.6 (page 130-137)


Background (Johnston)

Environmental models can be either physical of mathematical. An example of a physical model is the Bay-Delta model maintained by the Corps of Engineers in Sausalito. Physical models are based on the principles you studied in Engr 132 (Buckingham Pi theorem and others). They are good for modeling transport, but not so good at modeling chemical processes. They're also not very portable. The Bay-Delta model, for instance, is over an acre in size. With the increase in the availability of computers, mathematical models have become much more common. They are based on mass balance equations and equations describing the behavior of various physical and chemical processes.

Environmental models are very useful for: 1) predicting the effects of engineering activities on the environment, and 2) evaluating different alternative actions. In this second use, engineers can play "what if" games with the model -- what if the project was twice as large, or what if we treated the waste twice as much, or what if we moved the project somewhere else, etc. One of the first, and simplest environmental models was derived by Streeter and Phelps in 1925. It is a one-dimensional model of oxygen concentrations in a river. This is the model we will discuss today.

The Streeter-Phelps equation (also known as the "dissolved oxygen sag" equation) is based on a mass balance which is affected by two processes. One is that oxygen is removed from water by the degradation of organic materials. In other words, the biochemical oxygen demand of an organic waste is satisfied by oxygen taken from the water. The second process is "reaeration" by oxygen transfer into the water from the atmosphere.


The Streeter-Phelps equation is a simple differential equation. You can find the derivation in a separate file. Please download it, print it, and bring it to class.

Using the DO Sag Equation in a More Complex Environment

In the Streeter-Phelps equation can predict conditions downstream as long as physical parameters like temperature, kd (which depends on temperature), and kr (which depends on turbulence and temperature) remain constant, and there are no external flows into the stream. What happens if these conditions change? Suppose a tributary creek enters the main river, for instance?

It's possible to adapt the Streeter-Phelps equation to accommodate these situations. Start by dividing the river into small pieces, called "reaches", so that inside each reach conditions are constant. The reaches should be defined so that external flows occur only on reach boundaries. Calculate the starting conditions for the Streeter-Phelps equation (Lo and Do) using mixing equations on the output from the reach immediately upstream plus any external flows. (If there are no external flows, the output values (L and D) of the upstream reach become the starting values.) Then, use the Streeter-Phelps equation to predict the DO concentration at the downstream end of the reach. Use the first-order decay equation to calculate the BOD at the downstream end of the reach also. These values are then used in calculating the starting conditions of the next reach downstream, and so on.

A spreadsheet model based on this concept has been written. Bring a disk to class and to get a copy, or take it off the web page. You'll need it for your lab write-up.

One thing you should know about computer models is that they are only as good as the equations and coefficients used. Usually, before a model is used to predict something that hasn't yet happened, it is tested against past conditions. Then, if necessary, coefficients or equations are adjusted so that the model reproduces existing data. This is called "calibrating" the model. In you computer assignment, you are asked to calibrate the DOSAG model and then use it to solve a waste loading problem.

Limitations of the DO Sag Equation

As with all mathematical models, certain simplifications have been made in the derivation of the DO sag equation. Before reading on, look at the model derivation, think about the situation, and try to make a list of limitations yourself.

Here's the list that I came up with:

  1. Steady state -- Streams aren't steady state. Flows, velocities, geometries, and temperatures all vary with time. Dividing the stream into smaller reaches reduces this limitation, but steady state conditions are still assumed inside each reach. To the extent that the reach is not steady state, inaccuracies will be introduced.
  2. Plug flow -- Streams aren't really plug flow. The geometries of natural streams are not regular -- there are wide spots, pools, narrow chutes, sand bars, rocks -- so the flow doesn't move as a plug.
  3. Algae -- The model doesn't include algae which are a very important source of oxygen. Note that the effects of algae are very dependent on sunlight, which changes through the day. Modeling algae accurately would require a nonsteady-state model.
  4. Benthic organisms -- The model assumes that all the oxygen demand is from suspended organisms (i.e., bacteria living in the water column like they were in the BOD bottle). In fact, most natural bacteria live attached to surfaces in "biofilms" -- slimy coatings on rocks or soil particles. So a significant portion of the BOD is due to bottom-dwelling (benthic) organisms. The effect of benthic demand is especially strong if much of the organic material is in the form of particles that settle out. Benthic effects are not included in the model either.


Procedure for today's lab

  1. Review the derivation of the model. (Please print it out and bring it.)
  2. Play some "what-if" games using the spreadsheet model to reinforce concepts that go into the equation.
  3. Review a manually-solved example problem with the instructor.
  4. Discuss construction and use of the spreadsheet model.
  5. Review and discuss the computer assignment you will work on at home.
  6. Solve a problem manually in the lab and turn it in before you leave.