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Homework Help: Mixing Question - Linear ODE

  1. Sep 21, 2010 #1
    1. The problem statement, all variables and given/known data
    A stirred tank reactor that initially contains a volume V(0) = V_0 of water. Suppose that a stirred solution of salt at concentration S is pumped in at a rate of F_in = F litres/hr and the well-stirred mixture is pumped out at a slight faster rate of F_out = (F + f) litres/hr where f > 0. Let C(t) denote the concentration of salt inside the tank. Find C(t).

    2. Relevant equations

    3. The attempt at a solution
    V(t) = V_0 + F_in*t - F_out*t = V_0 + F*t - F*t + f*t = V_0 + f*t

    (C*V)' = C'*V + C*V' = C'(V_0 - f*t) - Cf = SF - C(F + f)
    C'(V_0 - f*t) = S*F - C*F
    C' + C*F/(V_0 - f*t) = S*F/(V_0 - f*t)

    Do I use integration factor to solve for C? I tried it and it was really complicated but the final answer is unusually simple. So I don't think it is right.
  2. jcsd
  3. Sep 22, 2010 #2
    Why don't you try to find C' and not (C*V)' ?

    Try, C'=(salt enters)- (salt leaves).

    And, Vo is pure water, right ?
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