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8.1: Untitled Page 190

  • Page ID
    18323
  • Chapter 8

    to be constant, thus the control volume is fixed in space. However, the concentration of the inlet stream is subject to changes, and we would like to know how the concentration in the tank responds to these changes. Since no chemical reaction is taking place, we can express Eq. 8‐1 as

    d

    c dV

    c (

     )

     0

    (8‐6)

    dt

    dA

    A

    v w n

    A

    A

    V ( t)

    A( t)

    While the gas‐liquid interface may be moving normal to itself, it is reasonable to assume that there is no mass transfer of species A at that interface, thus (v w)  n  0 everywhere except at Streams #1 and #2. In addition, since the A

    volumetric flow rates entering and leaving the tank are equal, it is reasonable to treat the control volume as a constant so that Eq. 8‐6 simplifies to d

    c dV

    c

    dA  0

    v n

    (8‐7)

    A

    A

    A

    dt V

    A e

    Here A represents the area of both the entrance and the exit. Use of the e

    traditional assumption for entrances and exits, v n v n , along with the flat A

    velocity profile restriction, allows us to write Eq. 8‐7 as

    d

    c dV   c Q   c Q  0

    (8‐8)

    A

    A 2

    A 1

    dt V

    Here  c  and  c  represent the area‐averaged concentrations (see Sec. 4.5) at A 1

    A 2

    the entrance and exit respectively. The volume‐averaged concentration can be defined by

    1

    c  

    c dV

    (8‐9)

    A

    A

    V V

    and use of this definition in Eq. 8‐8 leads to

    dc

    A

    V

    c Q

    c Q

    (8‐10)

    A 1

    A 2

    dt

    

    

    

    rate at which species A

    rate at which species A

    rate of accumulation

    leaves the control volume

    enters the control volume

    of species A

    Here we have two unknowns,  c  and  c  , and only a single equation; thus we A

    A 2

    need more information if we are to solve this problem. Under certain

    Transient Material Balances

    357

    circumstances the two concentrations are essentially equal and we express this limiting case as

    c

    c

    (8‐11)

    A

    A 2

    