CASE STUDY: River Toxins

Source:

Dynamic Modeling, Bruce Hannon and Matthias Ruth, University of Illinois/National Center for Supercomputing Applications

Overview:

In this problem, we wish to look at the effect of a pollutant being released into a river at some point upstream. We want to know two things (i.e., TWO independent variables): the concentration of the pollutant as it changes over time AND the concentration of the pollutant at a number of different places downstream. Therefore, we need a model that keeps track of distance and time simultaneously.

To solve such a problem we establish a chain of stocks of the pollutant which represent connected sections of the river, 1, 2, 3, ... 6, and we connect each stock to the next one with a controlled transfer variable, f1, f2, f3, ....f6. f0 is the initial pollutant injection. Each of the f1, f2, ...f6 are controlled by a dwell time, T1, T2, T3, ...T6 which represents how long a molecule of pollutant and of river water stays in that particular section. The chain continues until the length of river is described with sufficient accuracy. The stocks give the amount of pollutant in each section at any time. To find the concentration in any section at any time (our goal), we must divide the amount of pollutant in a section by the volume of water in that section. The volume of the section is determined by the flow rate Q which changes after section 2. The volume flow relations are empirically determined. The dwell or residence time T in any section is just the volume of that section divided by the flow in that section.

Building the Model:

For purposes of this model, we'll divide the river into six "segments". In each of the sections we wish to measure the amount of pollutant (NOT the concentration of the pollutant) in that section of the river at any given time. The volume of water in each section is determined by a flow rate (labeled "Q"). The flow rate Q will determine the volume through a graphical converter. The volume-flow relationships are assumed to be empirically determined and, in principle, vary over time in response to changes in the flow rates.

Initially, each of the segments of the river contains no pollutant (toxin). At flow point F0 (so labeled because it is coming INTO river segment 1), we will pulse in 100 pollutant units at the first DT (in other words, at the start of the run). Injections of pollution can be made repeatedly and at various stations at various times.

The flow from segment 1 to segment 2 is labeled F1. It is defined as the volume of the pollutant in segment 1 divided by the dwell time. A dwell time is the time that a pollutant stays in a segment or compartment before it moves on. So, F1 will be defined as Segment One / T1, where T1 is the first dwell time. T1, the dwell time for segment one, is defined as the volume of water at segment 1 divided by the flow rate of water (Q1) at the first segment: T1 = V1/Q1. The volume of water V1 and V2 will be determined by the flow rate Q1, which is the same for segments 1 and 2. The volume of water V3, V4, V5, and V6 will be determined by the flow rate of Q2. Based on this reading, you should be able to build a complete model of the flow of pollutants and the concentrations at each segment.

Units:

Volume of toxin in each segment: cubic meters
Flow of toxin from one segment to the next: cubic meter of toxin per minute
Flow rate of water (Q): cubic meters of water per minute
Concentration of toxin: cubic meters of toxin per cubic meters of water
Dwell time (T): minutes
Volume of water (V): cubic meters of water

Parameters:

Q1 = 1 cubic meters of water per minute
Q2 = 1.6 cubic meters of water per minute

Graphs for Volumes of water (dependent variable) versus Flow rate of water (Q):

V1	Q1
0.05	0.00
0.3	0.167
0.35	0.333
0.55	0.5
1.00	0.667
1.80	0.833
2.80	1.00
4.60	1.17
6.80	1.33
8.25	1.50
9.05	1.67
9.70	1.83
9.95	2.00

V2	Q1
0.00	0.00
0.7	0.167
1.15	0.333
1.50	0.5
1.75	0.667
1.95	0.833
2.15	1.00
2.50	1.17
2.85	1.33
3.50	1.50
4.85	1.67
7.90	1.83
10.0	2.00

V3	Q2
0.075	0.00
1.35	0.167
2.10	0.333
2.48	0.5
2.92	0.667
3.30	0.833
3.90	1.00
4.88	1.17
6.83	1.33
9.07	1.50
11.6	1.67
13.3	1.83
14.8	2.00

V4	Q2
0.00	0.00
2.02	0.167
3.38	0.333
4.72	0.5
5.92	0.667
6.67	0.833
7.80	1.00
8.55	1.17
9.07	1.33
9.90	1.50
12.3	1.67
14.0	1.83
15.0	2.00

V5	Q2
0.09	0.00
2.07	0.167
4.05	0.333
5.85	0.5
6.39	0.667
6.75	0.833
7.47	1.00
8.28	1.17
9.63	1.33
11.7	1.50
14.7	1.67
17.0	1.83
18.0	2.00

V6	Q2
0.1	0.00
1.20	0.167
1.40	0.333
2.10	0.5
2.70	0.667
3.80	0.833
5.40	1.00
8.50	1.17
11.7	1.33
15.4	1.50
17.4	1.67
18.2	1.83
19.8	2.00

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