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8.3: Untitled Page 192

  • Page ID
    18325
  • Chapter 8

    Figure 8‐2. Inlet concentration as a function of time In order to solve Eq. 8‐15a, we separate variables to obtain

    dc

    A

      dt

    (8‐16)

    1

    c  

    A

    cA

    The integrated form can be expressed as

       c

     

    A

    t

    d

       1

    d

    (8‐17)

    1

      c

    o

    A

     

     

    0

    cA

    in which  and  are the dummy variables of integration. Carrying out the integration leads to

     c   1 

    c

    ln  A

    A    t

    (8‐18)

     o

    c

     1

    c

    A

    A

    which can be represented as

    1

    c

     o 1 t

    cA

    A

    cA

    cA   

      

    e

    (8‐19)

    This result is illustrated in Figure 8‐3, and there we see that a new steady‐state condition is achieved for times on the order of three to four residence times. Even though perfect mixing can never be achieved in practice, and one can never

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    Transient Material Balances

    359

    change the inlet concentration instantaneously from one value to another, the results shown in Figure 8‐3 can be used to provide an estimate of the response time of a mixer and this qualitative information is extremely useful. Experiments can be performed in systems similar to that shown in Figure 8‐1 by suddenly Figure 8‐3. Response of a perfectly mixed stirred tank to a sudden change in the inlet concentration

    changing the inlet concentration and continuously measuring the outlet concentration. If the results are in good agreement with the curve shown in Figure 8‐3, one concludes that the system behaves as a perfectly mixed stirred tank with respect to a passive mixing process.

    The solution to the mixing process described in the previous paragraphs was especially easy since the inlet concentration was a constant for all times greater than or equal to zero. The more general case would replace Eqs. 8‐15 with dc

    A

      c   f ( t) ,

    t  0

    A

    (8‐20)

    dt

    I.C.

    o

    c   c ,

    t  0

    A

    A

    (8‐21)

    The solution of Eq. 8‐20 can be obtained by means of a transformation known as the integrating factor method and this is left as an exercise for the student (see Problems 8‐4 and 8‐5).

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    360