Mathematical Modeling Assignment (group assignment)
(February, 2010)
Due date: week of March 1st
Problem 1. (30 points)
Discharge of pollutants and toxic materials into rivers and lakes have put the aquatic life at risk.
Longtime exposure of a fish to these pollutants results in accumulation of pollutants in the body
of fish. When the concentration of pollutants exceeds the standard limits, the fish cannot be used
as food.
Pollutants may enter and leave fish by various mechanisms. The uptake of pollutants is carried
out through gills and from food. The pollutants can be lost from fish via gills, egestion, and
metabolism. Growth dilution is assumed negligible.
As a first approximation it can be assumed that the rates of uptake and discharge are directly
dC
 kC ,
proportional to pollutant concentration in water and in the fish body respectively. (
dt
first order)
Develop a model to estimate concentration of pollutant against time in a fish in a lake where the
pollutant concentration is equal to Cp mg/L. The concentration of pollutant in fish body is CF.
State your assumptions.
Verify the model.
If the concentration of the pollutant in water Cp = 0.1 mg/L and
kf = 0.01 d-1, k1 = 0.006 d-1, k2 = 0.0001 d-1 , ke= 0.00004 d-1 , km= 0.00002 d-1
Calculate the time required to reach to CF = Cp and CF = 1.478 mg/L
Loss by metabolism,
km
Loss via gills, k2
Uptake from
Food, kf
Loss via egestion, ke
Uptake via
gills, k1
Problem 2. (30 points)
Develop a mathematical model for the prediction of temperature for a cup of coffee. Assume that
your coffee cup is cylindrical in shape and it is full of coffee. The cap is always on. Heat transfer
from your coffee to surrounding is by free convection and can be calculated by Q = hA(T – Ts)
where, A is heat transfer area (no heat loss from the bottom surface), h is the heat transfer
coefficient, T is the surface temperature and Ts is the surrounding temperature. Also assume that
the surface temperature of the cup is equal to the coffee temperature.
- List your assumptions
- Develop your mathematical model
- Validate your model
Prepare the graph of coffee temperature against time if the initial temperature of the coffee is 80
0
C, and the cup is 6 cm in diameter and 12 cm tall. Density of coffee = 1000 kg/m3. The
surrounding temperature is 20 C. The specific heat of coffee is 4 kJ/kg.C and the heat transfer
coefficient is 1 W/m2C.
Heat transfer by
free convection
Heat loss
Problem 3 (30 points)
A company makes computer desks (C.D.) and regular desks (R.D.) made of Oak and Maple. C.D.s
and R.D.s consumes different amounts of lumber. A desk requires 6 board-feet (B.F.) each of
oak and maple, while a desk requires 3 B.F. of Oak and 9 B.F. of maple. The cost associated
with the raw materials and manufacturing is $6/B.F. for Oak and $4/B.F. for maple.
The company can purchase Oak in one time delivery of 1200 B.F. to pay the minimum price of
6$ per B.F. minimum purchase price for maple is 4$/B.F. in one time delivery of one load 1800
B.F.
The company cannot sell or return the remaining due to the high cost of shipping and handling. If
not used the company must pay for the storage in a warehouse, which is also costly.
The market for C.D.s and R.D.s are such that they can be sold for 90 dollars and 84 dollars
respectively.
Develop a mathematical model to maximize the profit for this company. Determine the number of
C.D.s and R.D.s and calculate the profit.