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Homework Help: Conservation of mass/flow through tanks

  1. Sep 15, 2011 #1
    1. Suppose that three tanks in series, connected by pipes of the same size are filled with water that is moved through the system with pumps at the rate of 5 gal/min. The first tank is 1000 gal. The second tank is 750 gal. And the third tank is 500 gal. The contents of each tank is vigorously stirred. The in-feed to Tank 1 is a constant flow of water in the same sized pipe from an external source at the same flow rate and the outflow from Tank 3 flows into a lake at the same flow rate. [Dye is added to the inflow at a constant rate for exactly one minute per hour over a long period of time (weeks, months,whatever)]

    What is the concentration of the dye RELATIVE to its input concentration as a function of time over a 10-hour period after the "long-time" has passed.

    Do you have to make further assumptions to solve the problem?
    How do you deal with the "relative" concentration?

    2. Okay so obviously this is a conservation of mass problem, so flow in - flow out = 0

    3. I have drawn an accurate picture, and I have set up what I believe to be the correct system of equations:

    x1' = (QsA - Q12x1)*1/V1
    x2' = (Q12x1 - Q23x2)*1/V2
    x3' = (Q23x2 - Q3Lx3)*1/V3

    x1 = Concentration in Tank 1, so on for x2 and x3
    A = Entrance rate of substance with units kg*t/v
    Q1 = rate at which the substance is transported by water flow (same throughout)

    V1 = Volume of tank 1, so on for V2, V3.

    Whatever you guys know what this stuff means these are straight forward normally but this one is throwing me off bad. I don't know if my system is right or what to solve I've been working on it and the rest of a huge problem set for too long.

    It is due tomorrow morning. Please help.

  2. jcsd
  3. Sep 17, 2011 #2

    rude man

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    Homework Helper
    Gold Member

    Well, input concentration is 100% for 1 minute and 0% for the next 59 minutes. So presumably you have to compute the concentration in each of the three tanks over the 1 hr period, second by second, after the steady state is reached.

    The concentration in each tank is probably represesnted by a first-order diff. eq. with constant coefficients. The forcing function is a square wave with 1/60 duty cycle. Sounds like a Fourier series application. If I think of any more brilliant suggestions I'll let you know, but probably too late ... :-)
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