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Concentrating Salt Tank

  1. Jun 8, 2015 #1
    I have a variation on the concentrating tank problem that I'm having a bit of trouble solving. I have a tank of 10 kg of pure water at time 0. I add a time dependent concentration of salt and remove the same volume of pure water so that the tank volume never changes. Once the tank has 1 kg of salt, I stop the problem. So unlike the typical problem where I have a known initial concentration, I instead have a target final concentration, but the time of that final concentration is unknown in that it depends on the flow rate.

    $$ \frac{dm_{salt}}{dt} = in - out $$
    There's no salt exiting, so it becomes,
    $$ \frac{dm_{salt}}{dt} = \dot{m}_{in} \frac{m_{salt}(t)}{m_{tank}} $$
    Separate the variables and integrate to get
    $$ln(m_{salt}) = \frac{\dot{m}_{in} t}{m_{tank}} + C$$
    $$m_{salt}(t) = Ae^{\frac{\dot{m}_{in} t}{m_{tank}}}$$

    Now I need an initial condition to get the particular solution, but I can't use
    $$m_{salt}(0) = 0$$
    and I'm not sure how to use my knowledge of the final concentration, since I don't know what time it will occur.

    Your help is appreciated.
     
    Last edited by a moderator: Jun 8, 2015
  2. jcsd
  3. Jun 8, 2015 #2

    vela

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    Are you sure you're removing only pure water? If so, the problem is pretty trivial. You would just calculate what volume of salt solution added that contains 1 kg of salt. It has nothing to do with the tank, etc. It only depends on the concentration of the incoming solution.
     
  4. Jun 8, 2015 #3
    Your response made me realize I was missing part of the picture. There's also the rest of the loop (including a pool) which contains all of the salt at time zero. I tried solving the problem from the perspective of the pool concentration depleting rather than the tank concentration accumulating. I then took the inverse as the rate of accumulation in the tank, which seems to give the right answer. I've attached some notes on my thought process.

    $$m_{salt}(t) = m_0 \left(1-e^{\frac{\dot{m}_{in}t}{m_{system}}}\right)$$

    Is this how it's supposed to be done?

    Thanks.
     

    Attached Files:

  5. Jun 9, 2015 #4

    Ray Vickson

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    I cannot make any sense of your solution, in part because you never define your terms. Exactly what are ##m##, ##m_{salt}(t)##, ##m_{tank}##, etc.? Also, if ##\dot{m}_{in}## is a constant, your DE solution is correct, but not if ##\dot{m}_{in}## depends on ##t##. Depending on exactly what your ##m##-variables mean, the DE itself may not even be correct.
     
  6. Jun 12, 2015 #5
    Let C be the concentration of salt in the tank and Cin represent the concentration of salt in the inlet stream. Let V be the volume of the tank and f be the volumetric flow rate into and out of the tank. Then,
    $$V\frac{dC}{dt}=fC_{in}$$
    Chet
     
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