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5 November, 17:54

A brine solution of salt flows at a constant rate of 9 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.1 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.02 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.01 kg/L?

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  1. 5 November, 20:50
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    a) C = 0.02 - ((e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾) / 9)

    b) The concentration of salt in the tank attains the value of 0.01 kg/L at time, t = 0.0713 min = 4.28s

    Explanation:

    Taking the overall balance, since the total Volume of the setup is constant, then flowrate in = flowrate out

    Let the concentration of salt in the tank at anytime be C

    Let the Concentration of salt entering the tank be Cᵢ

    Let the concentration of salt leaving the tank be C₀ = C (Since it's a well stirred tank)

    Let the flowrate in be represented by Fᵢ

    Let the flowrate out = F₀ = F

    Fᵢ = F₀ = F

    Then the component balance for the salt

    Rate of accumulation = rate of flow into the tank - rate of flow out of the tank

    dC/dt = FᵢCᵢ - FC

    Fᵢ = 9 L/min, Cᵢ = 0.02 kg/L, F = 9 L/min

    dC/dt = 0.18 - 9C

    dC / (0.18 - 9C) = dt

    ∫ dC / (0.18 - 9C) = ∫ dt

    (-1/9) In (0.18 - 9C) = t + k

    In (0.18 - 9C) = - 9t - 9k

    -9k = K

    In (0.18 - 9C) = K - 9t

    At t = 0, C = 0.1/100 = 0.001 kg/L

    In (0.18 - 9 (0.001)) = K

    In 0.171 = K

    K = - 1.766

    So, the equation describing concentration of salt at anytime in the tank is

    In (0.18 - 9C) = - 1.766 - 9t

    In (0.18 - 9C) = - (9t + 1.766)

    0.18 - 9C = e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾

    9C = 0.18 - (e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾)

    C = 0.02 - ((e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾) / 9)

    b) when C = 0.01 kg/L

    0.01 = 0.02 - ((e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾) / 9)

    0.09 = (e ⁻ ⁽⁹ᵗ ⁺ ¹•⁷⁶⁶⁾)

    - (9t + 1.766) = In 0.09

    - (9t + 1.766) = - 2.408

    (9t + 1.766) = 2.408

    9t = 2.408 - 1.766 = 0.642

    t = 0.642/9 = 0.0713 min = 4.28s
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