MATH 481, Project 3
Pollution in Lakes

Due Thursday March 28

Project description

The file proj3-outline.pdf describes the subject of this case study in an outline form.

This project was motivated by the article Natural Displacement of Pollution from the Great Lakes which appeared in Science in 1967, at the height of concern about pollution in the Great Lakes. Here is the full citation:

@article{science-1967,
  author = {Rainey, Robert H.},
  title  = {Natural Displacement of Pollution from the Great Lakes},
  journal= {Science},
  publisher = {American Association for the Advancement of Science},
  volume = {155},
  number = {3767},
  date   = {1967-03-10},
  pages  = {1242--1243},
}

You will find a great deal of information about relevant issues on searching the Web. Wikipedia, in particular, contains a lot of interesting reading (and data) about the great lakes. Also have a look at the Ohio Department of Natural Resources page Lake Erie Facts and Lake Erie Water Levels.

Lake Erie was considered a “dead lake” in the 1960s. Pollutant of all sorts, industrial and agricultural waste, and urban sewage were dumped into the lake with little concern. The lake harbored hardly any life. The Cuyahoga River, one of Lake Erie's major tributaries, has caught on fire several times in its history, the most famous episode being that of June 22, 1969. The article Cuyahoga River Fire recounts the event.

Lake Erie's ecological health improved tremendously after the establishment of the Clean Water Act of 1972, and by the turn of the millennium it had reached quite a respectable level of cleanliness. The more recent developments, however, point in the reverse directions. The New York Times reports of March 14, 2013 and August 4, 2014 paint a bleak picture of the lake's relapse into an ecological disaster. The August 2014 article of Wired Magazine, Don’t Drink the Water, provides yet another report on Lake Erie's current dire state.

How much water is there?

A beautiful picture produced by US Geological Survey (USGS) lumps all of Earth's water into a single sphere and contrasts its size to that of the Earth.

Typesetting Q_1,in and all that

To write Q_1,in in LaTeX, you would do Q_{1,\textrm{in}} (as a math expression, of course) but that becomes tiresome soon. Here is the easy way. Put the line

    \newcommand{\QinA}{Q_{1,\textrm{in}}}

in the document's preamble. This defines \QinA as a shorthand for Q_{1,\textrm{in}} .

While you are at it, also define \cinA, \QoutA, \QinB \cinB, \QoutB, as shorthands for c_1,in, Q_1,out, Q_2,in c_2,in, Q_2,out, respectively.

You may use these definitions anywhere within your document. For example you may write $\QoutA + \QinB = \QoutB$ .

Remark: Why did I write \QinA and \QinB instead of the more natural \Q1in and \Q2in? This is because in LaTeX (well, actually TeX) command names are limited to the letters of the alphabet; numbers are not allowed! So I used “A” and “B” instead of “1” and “2”.

The diagram

The diagram in the outline's Figure 1 is generated on the fly with the help of a LaTeX package called TikZ. Here is the LaTeX code for it:

\begin{figure}
  \centering
  \begin{tikzpicture}[>=latex]
    \tikzstyle{boxed}=[draw=black, minimum width=1.8cm, minimum height=1cm,
      rounded corners]
    \draw (0,0)   node[boxed, label=below:$c_1(t) \to \bar{c}_1$] (V1) {$V_1$};
    \draw (3.2,0) node[boxed, label=below:$c_2(t) \to \bar{c}_2$] (V2) {$V_2$};
    \draw[->] (V1.east) -- node[above] {$\QoutA$} (V2.west);
    \draw[->] (V2.east) -- node[above] {$\QoutB$} +(1.2,0);
    \draw[<-] (V1.north) -- ++(0,0.6)
      -- node[above] {$\QinA$} node[below] {$\cinA$} +(-1.4,0);
    \draw[<-] (V2.north) -- ++(0,0.6)
      -- node[above] {$\QinB$} node[below] {$\cinB$} +(-1.4,0);
  \end{tikzpicture}
  \caption{A schematic diagram of two interconnected reservoirs.
     If the volumes $V_1$ and $V_2$, the flow rates $\QinA$ and $\QinB$,
     and the input concentrations $\cinA$ and $\cinB$ are constants, then
     the concentrations $c_1(t)$ and $c_2(t)$ in the reservoirs
     tend to equilibrium values $\bar{c}_1$ and $\bar{c}_2$
     in the long term.}
  \label{fig:schematic_diagram}
\end{figure}

Copy and paste this into your LaTeX document, if you want. Please note that:

You need to put \usepackage{tikz} in the document's preamble to process this.
I have used the shorthands \QoutA, \cinB, etc., throughout that TikZ code, so you will have define them in your document as well. See the previous section for that.

TikZ is an extremely powerful and efficient way of producing graphical diagrams in LaTeX. Note that the rather complex diagram in Figure 1 is defined in a mere 12 lines of TikZ code! I will do brief tutorial on TikZ if there is enough time left toward the end of the semester.

Author: Rouben Rostamian

MATH 481, Project 3Pollution in Lakes