Discharge of pollutants and toxic materials into rivers and lakes have put the a

Question

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, metabolism. Growth dilution is assumed negligible. As a first approximation it can be assumed that the rates of uptake and discharge are directly proportional to pollutant concentration in water and in the fish body respectively. ( , 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 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

Explanation / Answer

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 proportional to pollutant concentration in water and in the fish body respectively. (, 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

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

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