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Math Help - Three Tanks

  1. #1
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    Three Tanks

    Suppose we have 3 tanks, A, B, and C that hold 200, 150, and 100 galons of brine, respectively. Also, suppose that the system is closed, and further that the liquid (which is well-stirred) is pumped between the tanks only.

    Tank A is connected to B, and tank B is connected to C. Tank C is connected to A.

    Mixture of 4 gal/min goes from tank A to tank B.
    Mixture of 4 gal/min goes from tank B to tank C.
    Mixture of 4 gal/min goes from tank C to tank A.

    (So the flow goes A -> B -> C -> A)

    1.) Construact a model for the number of pounds of salt x_1(t), x_2(t), x_3(t) at time t in the tanks A, B, C (respectively).

    2.) From #1, write a system in the form X' = AX.
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  2. #2
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    What are the sizes of the tanks.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    What are the sizes of the tanks.
    The tanks are all the same size.
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  4. #4
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    Okay so I will let V be the volume of each of the three tanks.
    Let A(t),B(t),C(t) be amount of salt in each each tank at any time t.
    The differencial equation for A(t) is:
    \frac{dA}{dt} = \mbox{rate in } - \mbox{ rate out}.
    The rate in is what is coming in from tank C. Now C is coming in at 4 gallons per minute, so we need to multiply this by its concentration which is B/V so that works out to be 4B/V. The rate out is what is going out of A. That is 4 gallons per minute times the concentration of salt in A which also is 4A/V.
    Thus,
    \frac{dA}{dt} = \frac{4}{V}C - \frac{4}{V}A.
    Similarly,
    \frac{dB}{dt} = \frac{4}{V}A - \frac{4}{V}B.
    \frac{dC}{dt} = \frac{4}{V}B - \frac{4}{V}C.
    To write in matrix form let,
    \bold{X}(t) = \left( \begin{array}{c}A(t)\\B(t)\\C(t) \end{array}\right).
    And let,
    A = \frac{4}{V} \left( \begin{array}{ccc} -1&0&1\\1&-1&0\\0&1&-1 \end{array} \right).

    So, (without initial conditions)
    \bold{X}' = A\bold{X}
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  5. #5
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    Suppose we have 3 tanks, A, B, and C that hold 200, 150, and 100
    gallons of brine, respectively.

    If they're all the same size, then what does this mean?.
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  6. #6
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    Wink Three Tanks

    Warning! Completely frivolous post ahead!

    I must be getting tired because I can't resist doing this.

    You're welcome, you're welcome, you're welcome!

    -Dan
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