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  • 3.2 Compartmental Analysis

    Mixture Problems

    In these mixture problems a substance dissolved in a liquid is entering and exiting a holding tank. The

    differential equation modeling this situation is:

    π‘…π‘Žπ‘‘π‘’ π‘œπ‘“ πΆβ„Žπ‘Žπ‘›π‘”π‘’ π‘œπ‘“ π‘₯(𝑑) = π‘…π‘Žπ‘‘π‘’ π‘₯(𝑑) π‘’π‘›π‘‘π‘’π‘Ÿπ‘  βˆ’ π‘…π‘Žπ‘‘π‘’ π‘₯(𝑑) 𝑒π‘₯𝑖𝑑𝑠

    𝑑π‘₯

    𝑑𝑑 = π‘Ÿπ‘–π‘› βˆ’ π‘Ÿπ‘œπ‘’π‘‘

    where,

    π‘₯(𝑑) = π‘Žπ‘šπ‘œπ‘’π‘›π‘‘ π‘œπ‘“ π‘Ž π‘ π‘’π‘π‘ π‘‘π‘Žπ‘›π‘π‘’ π‘‘π‘–π‘ π‘ π‘œπ‘™π‘£π‘’π‘‘ 𝑖𝑛 π‘Ž π‘™π‘–π‘žπ‘’π‘–π‘‘ π‘Žπ‘‘ π‘‘π‘–π‘šπ‘’ 𝑑

    π‘Ÿπ‘–π‘› = (π‘“π‘™π‘œπ‘€ π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘™π‘–π‘žπ‘’π‘–π‘‘ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘”) Γ— (π‘π‘œπ‘›π‘π‘’π‘›π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ π‘’π‘π‘ π‘‘π‘Žπ‘›π‘π‘’ 𝑖𝑛 π‘™π‘–π‘žπ‘’π‘–π‘‘ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘”)

    π‘Ÿπ‘œπ‘’π‘‘ = (π‘“π‘™π‘œπ‘€ π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘™π‘–π‘žπ‘’π‘–π‘‘ 𝑒π‘₯𝑖𝑑𝑖𝑛𝑔) Γ— (π‘π‘œπ‘›π‘π‘’π‘›π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ π‘’π‘π‘ π‘‘π‘Žπ‘›π‘π‘’ 𝑖𝑛 π‘™π‘–π‘žπ‘’π‘–π‘‘ 𝑒π‘₯𝑖𝑑𝑖𝑛𝑔)

    Examples

    1. Water flows into a pond at a rate of 10 π‘š3 π‘šπ‘–π‘›β„ and out at the same rate. The volume of the pond is 1000 π‘š3.

    Initially, the pond contains 100 grams of pollutants. The concentration of pollutants in the water flowing into the

    pond is 2 𝑔 π‘š3⁄ . Assuming immediate and uniform dispersion of the pollutants, find a formula for the number

    of grams of pollutants in the pond, π‘₯ = π‘₯(𝑑).

    πΆπ‘œπ‘›π‘π‘’π‘›π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ π‘’π‘π‘ π‘‘π‘Žπ‘›π‘π‘’ 𝑖𝑛 π‘™π‘–π‘žπ‘’π‘–π‘‘ 𝑒π‘₯𝑖𝑑𝑖𝑛𝑔 = π‘Žπ‘šπ‘œπ‘’π‘›π‘‘ π‘œπ‘“ π‘ π‘’π‘π‘ π‘‘π‘Žπ‘›π‘π‘’ 𝑖𝑛 π‘‘β„Žπ‘’ π‘‘π‘Žπ‘›π‘˜ π‘Žπ‘‘ π‘Žπ‘›π‘¦ π‘‘π‘–π‘šπ‘’, 𝑑

    π‘£π‘œπ‘™π‘’π‘šπ‘’ π‘œπ‘“ π‘€π‘Žπ‘‘π‘’π‘Ÿ 𝑖𝑛 π‘‘β„Žπ‘’ π‘‘π‘Žπ‘›π‘˜ π‘Žπ‘‘ π‘Žπ‘›π‘¦ π‘‘π‘–π‘šπ‘’, 𝑑

  • 2. The air in a small room 12 ft by 8 ft by 8 ft is 3% carbon monoxide. Starting at 𝑑 = 0, fresh air containing no

    carbon monoxide is blown into the room at a rate of 100 𝑓𝑑3 π‘šπ‘–π‘›β„ . If air in the room flows out through a vent

    at the same rate, when will the air in the room be 0.01% carbon monoxide?

  • Population Models

    𝑑𝑝

    𝑑𝑑 = π‘˜π‘, 𝑝(0) = 𝑝0

    Where k is a constant of proportionality

    Note: π‘˜ > 0 for growth and π‘˜ < 0 for decay

    3. Solve the differential equation for population growth.

    𝑑𝑝

    𝑑𝑑 = π‘˜π‘, 𝑝(0) = 𝑝0

    This is the Malthusian or exponential, law of population growth and it assumes that members of a population only die

    by natural causes. This means it is not very accurate for a large population or over a long period of time.

  • The Logistic Equation

    The logistic equation has proven to be more accurate in predicting population growth because it takes into account

    competition within a population.

    𝑑𝑝

    𝑑𝑑 = βˆ’π΄π‘(𝑝 βˆ’ 𝑝1) , 𝑝(0) = 𝑝0

    where a and b are constants.

    Solving this equation gives us the logistic function

    𝑝(𝑑) = 𝑝0𝑝1

    𝑝0 + (𝑝1 βˆ’ 𝑝0)𝑒 βˆ’π΄π‘1𝑑

    4. In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006

    the population had grown to an estimated 6000.

    a. Use the Malthusian law for population growth to estimate the alligator population on the Kennedy

    Space Center grounds in the year 2020.

  • b. Suppose we have the additional information that the population of alligators on the grounds of the

    Kennedy Space Center grounds in 1993 was estimated to be 4100. Use the logistic model to estimate the

    population of alligators in the year 2020. What is the predicted limiting population?

    𝑝1 = [ π‘π‘Žπ‘π‘ βˆ’ 2𝑝0𝑝𝑏 + 𝑝0π‘π‘Ž

    π‘π‘Ž 2 βˆ’ 𝑝0𝑝𝑏

    ] π‘π‘Ž

    𝐴 = 1

    𝑝1π‘‘π‘Ž ln [

    𝑝𝑏(π‘π‘Ž βˆ’ 𝑝0)

    𝑝0(𝑝𝑏 βˆ’ π‘π‘Ž) ]

  • 5. If initially there are 300g of a radioactive substance and after 5 years there are 200g remaining, how much time

    must elapse before only 10g remain?

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