Water Pollution Flow Mathematical Modeling Matt Gilbride, Butler High School Kelsey Brown, DH Conley High School 2008

Matt Gilbride and Kelsey Brown

Outline What is water pollution? What is the problem? Explaining the model Tests Results Conclusion

Water pollution is the contamination of water by foreign materials. It became a problem in the 19 th century, during the Industrial Revolution

About 40% of America’s rivers are unusable due to pollution. 860 billion gallons of sewage are dumped into rivers yearly.

Only 3 percent of Earth’s Water is freshwater and of that only 0.3 percent is rivers and lakes. Although rivers are a small percentage, rivers run into lakes and bring the pollution there.

What is the Problem? We have created a three dimensional river in VPython. We are attempting to measure the amount of water and pollution collected by an intake valve. We will change variables in the experiment such as pollution concentration, initial location, and the size of the intake valve. With the collected data we will analyze trends in the and make predictions.

The Question Can the random motion of river flow be consistent enough to derive trends and equations in order to predict how variables will affect data output?

Our Project Our Project was modeled in VPython. Lets go through each of its components:

“Start!”= Begins the model and the graph “Stop!” = Stops the graph from recording data “Restart!”= Resets to initial conditions Green Slider = Determines depth of pollution Yellow = Determines how far down the river the pollution will be placed White Slider = Determines pollutions distance from the river bank

Data Acquisition All data was extracted from the computer program. Multiple variables were tested and changed in this computer simulated experiment. Here is a list of the variables as we used them: x = The x axis position of the pollution y = The y axis position of the pollution z = The z axis position of the pollution p = The amount of pollution placed in the river w = The number of water molecules (corresponds to volume) t = Time (in seconds)

Assumptions To run this module there are several assumptions one must take into account: 1. The water and pollution movement is completely random, with the exception of flow direction. 2. No objects in or around the river demean its flow other than the intake valve. 3. Pollution is only removed by the intake valve, what is not collected flows further downstream. 4. The river modeled does not have complex topography or bends as most real rivers do. 5. Based on the data acquired percentages will be used to extrapolate data.

Testing Methods All tests were run in VPython Five tests were run for each variable and results were averaged together. Data was imported to Excel in order to create graphs We then compared and interpreted the data

Control Group Here are the values for the control group x = 0,y = 7,z = -50, p = 50, w = 700 We ran five simulations and found the average number of each of the particles collected. Avg. n collected = 88, Avg. n2 collected = 12 Avg. t = 96.8(seconds) Based on the initial conditions, about twelve percent of the water collected is contaminated. If the average city uses about 40,000 gallons of water a day: The city would withdraw 4,800 gallons of polluted water.

Control Group When all the variables keep their initial values, 24% of the pollutant is collected by the intake valve:

Changing the Intake Valve When the intake valve is decreased by a scale of 1:2, and the other variables remain constant: Only 8% of all the pollution is collected by the intake valve. 4 of the water molecules are collected. Only 4% of the water collected is polluted. If the average city uses 40,000 gallons of water a day: The city would save 3,200 gallons of water from being polluted.

Testing Multiple Variables at Once If the intake valve is half size and the pollution concentrations is doubled, how will the data produced compare with the control group? Avg. n collected = 92, Avg. n2 collected = 8 compared to 12 for control Avg. t= 225 (seconds) % Collected = 16% of pollutant compared to 24% for control.

Is There a Trend? What happens when the intake valve is 1:3 its original size, and the pollution concentration is tripled? Avg. n collected = 96, Avg. n2 collected = 4 Avg. t = 234 (seconds) % Collected = 8% of pollutant

What is the Trend? When the intake valve is scaled down by a fraction, and the pollution concentration is multiplied by the inverse; The percentage of pollutant collected appears to follow: The time on average, increases by about 9

Our Prediction We predict that if the intake valve is scaled down to 1:4, only 4% percent of the original pollution (2 pollutant molecules) will be collected. The t value will be around 243 seconds.

Testing the Prediction When intake valve = ¼ original size: Avg. n collected = 98, Avg. n2 collected = 2 Avg. t = 241 (seconds) % Collected = 4% of pollutant Our Time (t) estimate was off by 1%.

Testing Once More If the fraction is After averaging five tests: Avg. n collected = 99, Avg. n2 collected = 1 Avg. t= 245 (seconds) % Collected = 2% of pollutant

Predicting One could use probability to figure out the percentage of original pollutant that would be collected.

Using the Formula

According to probability, one pollution molecule would be collected.

Pollution Position How does the position of the pollution affect the number of molecules collected? -The depth is insignificant If the intake valve was smaller, the depth would be more important. When the pollution is in line with the pollution more of it is collected

The x Value x is the location of the pollution between the riverbanks. When x is nearer to the center, more pollution is collected. This is due to how the pollution disperses. When x is nearer to a river bank the pollution percentage collected decreases by about 50%.

The z Value The z value corresponds to the length of the river, and how far down it the pollution is placed. Let’s observe the graph

Summary Background information on water pollution. Discussion about the model Data acquisition and data collection Trends and prediction equation Variables that affect the data such as pollution position.

Acknowledgements We would like to thank our Mathematical Modeling Professor, Dr. Russ Herman and our Master Teacher, Mr. David Glasier, and Nathan Jones for helping with our code and other problems we experienced along the way. We would also like to thank our parents for all they do and the SVSM Staff for the opportunities they’ve given us. We would also like to thank the audience for taking the time to listen to our presentation.