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Easy Tank Model? Outflow of tank proportional to volume of tank.

  1. Sep 18, 2011 #1
    1. Suppose we have a tank partially fi lled with water. There is a pipe
    feeding water to the tank as a variable
    ow rate and there is also a drain
    pipe with a computer controlled variable valve hooked to a sensor in
    the tank. The valve opens exactly enough to let water drain from the
    tank at a rate proportional to the volume of the tank. The program
    allows for us to set one number: the constant of proportionality. Write
    a model for this physical problem. Be sure to de ne all the variables
    in your model. (b) Suppose the in
    ow rate is constant. How should
    the proportionality constant in the control mechanism be set to keep
    the tank near a constant desired volume? (c) Suppose the in flow rate
    is periodic. To be de nite let's say the flow rate is sinusoidal and
    known exactly, how should the constant of proportionality be set for
    the controller to best keep the tank at a constant desired volume.




    2. Relevant equations
    flow in = flow out (if desired in this case)



    3. The attempt at a solution

    I call f0 the flow out and fi the flow in.
    fi varies with, say, t.

    f0 is proportional to V, the volume of the tank. The volume of the tank is: V = the volume initially in the tank, Vi, + fi(t) - f0.

    f0 is proportional to V by c., but in my statement about the V, f0 is on that side so it can't really be in the model. If I could get help figuring out the model, I could answer parts (b) and (c) pretty easily it seems.

    My first proportion was f0=Vc thus,
    f0 = (Vi+fi(t))c

    But I know this can't be right because in answering part b, fi would need to be as close as possible to f0, but any amount for c would mean that the amount out was equal t the entire amount in the tank.

    help?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 18, 2011 #2

    uart

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    Science Advisor

    Use the fact that the "inflow minus outflow" is equal to the rate of change in volume to write a simple DE (differential equation) for the system. The DE is the system model.
     
  4. Sep 19, 2011 #3
    dV/dT = c(fi(t)-fo) ?

    On second thought,

    dV/dt = fi(t) - c*fo(t) ??

    seems better, can someone verify this or otherwise please? Thanks!
     
    Last edited: Sep 19, 2011
  5. Sep 19, 2011 #4

    rude man

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    Gold Member

    What does "constant of proportionality" mean? Is it h, the target height of water?
     
  6. Sep 19, 2011 #5

    uart

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    That's on the right track, but use the fact, in the problem statement, that the outflow is proportional to V so as to write your DE with just one input variable (f_i) and one state variable (V). The state variable V also happens to be the output variable in this case, which is nice.
     
  7. Sep 20, 2011 #6
    I'm sorry, I'm confused. They are proportional so f_out = kV, thus,

    dV/dt = fin(t) - k*V(t)

    which is not right? >=\
     
  8. Sep 20, 2011 #7
    Well I know it's not right. I'm just coming to the same answers over and over again because I've done it so much I can't think outside my current train of thought :S

    It must be

    q_in(t) = dV/dt + kV ... one final check on this please??
     
    Last edited: Sep 20, 2011
  9. Sep 20, 2011 #8

    uart

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    Science Advisor

    No that's the correct DE for the system.

    Now for part b) you can take F_in as a constant and look at what value of "k" you require to keep V constant, say V_desired. Note that V = const means dV/dt = 0.
     
  10. Sep 20, 2011 #9
    thanks a lot for your help, uart. I got part b, K would = Qin/V. Working on last part.. if it is a sinusoid nothing changes, you still want amount in to equal amount out, so just sinusoid(t)_in/V = k for part (c) I would think
     
    Last edited: Sep 20, 2011
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