MA 252 Saint Louis Community College Dioxin Removal from Lakes Equations

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Department of Mathematics UAB
Differential Equations
MA252 Fall 2020
ASSIGNMENT 5
Due date: Tuesday 10/13/20
Dioxin Removal from Contaminated Lakes
Use MAPLE for this modeling assignment. Your report should conform to the requirements
given in the information sheet; you should in particular describe fully how you solved these
modeling problems. Use the text insertion capabilities of MAPLE to add text describing
each MAPLE command before you execute it. You can download the MAPLE worksheet
ass5-fall20.mw from Canvas to guide you here.
Two small lakes in South Alabama have been contaminated by the rather nasty carcinogen dioxin released from a local paper mill. The EPA has ordered the mill owners to
clean up the lakes within one year, and they in turn have hired you as a consultant to model
the clean up process and in particular to recommend an effective strategy for doing this in
the allotted time.
Lake A contains 900, 000 cubic meters of water contaminated with 110, 000 parts per trillion
of dioxin and lake B contains 600, 000 cubic meters of water contaminated with 1, 600, 000
parts per trillion of dioxin. Lake A is fed by freshwater stream 1 flowing at 30, 000 cubic
meters per day, and stream 2 flows from lake A to lake B at 30, 000 cubic meters per day;
stream 3 alllows water to exit lake B at 30, 000 cubic meters per day.
The First Proposal
The mill owners propose to pump water from lake B, at a rate a (cubic meters per day) to
be determined, filter out the dioxin, and pump the filtered water back into lake A, at the
same rate a, with the understanding that the flow rate in stream 2 from lake A to lake B
will naturally increase by the same amount so as to not effect the water levels in lake A
and lake B during the process. It is also understood that for technical reasons (to wit, the
company is too cheap to buy decent filters), at each instant of time only 80% of the dioxin
in the water being filtered is actually removed by the filters; the rest is recycled back to lake
A, to be subjected to further filtering later. The EPA has determined that the dioxin levels
in both lakes have to be reduced to 0.09 parts per trillion, or less (the level of a typical US
resident existing on a vegan diet by the way), and this must be achieved over a period of no
more than one year from the commencement of the clean up process. The faster one pumps
the contaminated water from lake B to the filtering plant, the faster the dioxin is removed
from the lake system. But, faster pumping uses more energy and raises the cost. Your task
is to compute the lowest flow rate a for pumping from lake B, in cubic meters per day, that
will effect the clean up in one year.
The Second Proposal
It turns out that there is a problem with this first solution. Preliminary flow tests by the
mill company indicate that stream 2, connnecting Lakes A and B, can handle at most an
additional flow of a = 13000 cubic meters per day. If this rate is exceeded, the water
level in Lake A would begin to rise, leading to flooding of lakefront properties owned by
one R. H. (“Bubba”) Bing, a powerful local real estate developer and county commissioner.
Fearful of the political consequences, the mill owners have told you to limit the pumping
of contaminated water from Lake B (and consequent recycling to Lake A) to a = 13000
cubic meters per day. The solution entails buying more expensive filters that use reverse
osmosis membrane technology, which removes a greater percentage of the dioxin. Adjust the
differential equations used above and find the smallest value of the dioxin removal percentage
b (currently, b = 80%) that will get the job done here.
The Report
Your report should include a one paragraph description of each problem in sections labelled
Background, together with sections labelled Differential Equations in which you carefully derive the differential equations used for each problem. The report should conclude
with a Summary section containing the recommended pumping rates and filter percentages
for each problem, as well as the final dioxin concentations (in ppt) for each lake in each case,
as laid out in the worksheet ass5-fall20.mw.

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Tags:
differential equations

pollution

Dioxin Removal

carcinogen dioxin

Contaminated Lakes

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