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8.2: Untitled Page 191

  • Page ID
    18324
  • volume average

    area average

    concentration in

    concentration

    the tank

    in the effluent

    This allows us to write Eq. 8‐10 in terms of the single unknown,  c  , in order to A

    obtain

    dc

    A

    V

      c Q   c Q

    (8‐12)

    A

    A 1

    dt

    One must be very careful to understand that Eq. 8‐11 is an approximation based on the assumption that the difference between  c  and  c  is small enough so A 2

    A

    that it can be neglected (Whitaker, 1988).

    It is convenient to divide Eq. 8‐12 by the volumetric flow rate and express the result as

    dc

    A

      c    c

    (8‐13)

    A

    A 1

    dt

    Here  represents the average residence time that is defined explicitly by

    average

    V

    residence 

     

    (8‐14)

    Q

    time

    In general, we are interested in processes for which the inlet concentration,

    c  , is a function of time, and a classic example is illustrated in Figure 8‐2.

    A 1

    There we have indicated that  c  undergoes a sudden change from o c to 1

    c ,

    A 1

    A

    A

    and we want to determine how the concentration in the tank,  c  , changes A

    because of this change in the inlet concentration. The initial value problem associated with the sudden change in the inlet concentration is given by dc

    A

    1

      c   c ,

    t  0

    A

    A

    (8‐15a).

    dt

    IC.

    o

    c   c ,

    t  0

    A

    A

    (8‐15b)

    Equations 8‐15 are consistent with a situation in which the inlet concentration is originally fixed at o

    c and then instantaneously switched from o

    c to 1

    c at t  0 .

    A

    A

    A

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    index-367_2.png

    index-367_3.png

    index-367_4.png

    358