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29 December, 07:15

G a tank contains 300 liters of fluid in which 40 grams of salt is dissolved. pure water is then pumped into the tank at a rate of 5 l/min; the well-mixed solution is pumped out at the same rate. find the number a (t) of grams of salt in the tank at time t.

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  1. 29 December, 09:59
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    To solve this problem, we make a component mass balance. Let us say that the salt is called A. Therefore the mass of A in the tank over time is given by the differential equation:

    d (mA) / dt = ΦmA (in) - ΦmA (out)

    where mA is mass of A, ΦmA mass flow rate of A, and t is time

    Since the fluid being pumped into the tank is pure water, therefore:

    ΦmA (in) = 0

    so,

    d (mA) / dt = - ΦmA (out)

    We also know that:

    mass = concentration * volume

    m = A * V

    Φm = A * ΦV

    Therefore:

    d (A * V) / dt = - A ΦV

    Since V is constant, we take it out of the derivative:

    V dA / dt = - A ΦV

    Rearranging:

    dA / A = - ΦV dt / V

    Integrating with limits of:

    A = 40 to A

    t = 0 to t

    ln (A / 40) = ( - ΦV/V) t

    since it is given that ΦV = 5 L/min and V = 300 L therefore:

    ln (A / 40) = - (5/300) t

    Taking the e on both sides:

    A / 40 = e^ - (5t/300)

    A = 40 e^ - (5t/300)
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