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Homework Help: Modelling DE mixture problem

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data

    A solution of material A of concentration C0 kg/m3 flows into a tank at the flow rate F m3/min. The tank has fixed volume V m3 and is continuously stirred. The solution on the tank is of concentration C and flows out the top of the tank at the flow rate F. The material A in solution in the tank reacts with air and breaks down such that the amount present in the tank decreases at the rate kCV km/min, where k is constant. At time t=0, the inlet concentration undergoes a step change given by

    C0(t) = Cinit for t<0 and
    C0(t) = C0 for t>0

    a) Write the balance equation for material A in the tank.

    b) Find the solution C(t) of the model. What is the time constant and the limiting concentration in the tank?

    2. Relevant equations

    Rate of change of mass = (rate in) - (rate out)

    3. The attempt at a solution

    a)
    d/dt(CV) = C0F - CF - kCV

    dC/dt + (F/V+k)C = FC0/V

    b)
    Let CP(t)=A (constant)

    0=(F/V+k)A=FC0/V

    => A=FC0/(F+Vk)

    :. C(t) = FC0/(F+Vk) + me-(F/V+k)t for arbitrary m

    Initially C(0)=FC0/(F+Vk) + m =Cinit

    m=Cinit-FC0/(F+Vk)

    :. C(t)=FC0/(F+Vk) + (Cinit-FC0/(F+Vk))e-(F/V+k)t, t>0

    As t->infinity, C(t)->FC0/(F+Vk)

    My problem is the very last part, or at least, that's where I noticed I had a problem. Shouldn't the concentration approach C0? Or does the loss of material A into the atmosphere change that?
     
  2. jcsd
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