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8.13: Untitled Page 202

  • Page ID
    18335
  • Chapter 8

    Figure 8‐11. Batch distillation unit

    The control volume illustrated in Figure 8‐11 is fixed in space and can be separated into the volume of the liquid (the   phase ) and the volume of the vapor (the   phase ) according to

    V

    V ( t)  V ( t)

    (8‐60)

    Under normal circumstances there will be no chemical reactions in a distillation process, and we can express the macroscopic mole balance for species A as d

    c dV

    c

    dA  0

    v n

    (8‐61)

    A

    A A

    dt V

    A

    In addition to the mole balance for species A, we will need either the mole balance for species B or the total mole balance. The latter is more convenient in this particular case, and we express it as (see Sec. 4.4)

    d

    c dV

    c

    dA  0

    v n

    (8‐62)

    dt V

    A

    Transient Material Balances

    377

    For the control volume shown in Figure 8‐11, the molar flux is zero everywhere except at the exit of the unit and Eq. 8‐61 takes the form

    d

    c dV

    c dV

    c   

    (8‐63)

    dt

    v n dA

    0

    A

    A

    A

    A

     

    V ( t)

    V ( t)

    A exit

    Here we have explicitly identified the control volume as consisting of the volume of the liquid (  phase ) and the volume of the vapor (   phase ) At the exit of the control volume, we can ignore diffusive effects and replace v n with

    A 

    v

    n , and the concentration in both the liquid and vapor phases can be

     

    represented in terms of mole fractions so that Eq. 8‐63 takes the form

    d

    x

    c dV

    y cdV

    y c  

    (8‐64)

    dt

    v

    n dA

    0

    A

    A

    A

     

    V ( t)

    V ( t)

    A exit

    If the total molar concentrations, c and c , can be treated as constants, this result can be expressed as

    d

    d

      x  

    M  

      y M 

    y c   dA

    (8‐65)

    dt

    dt

    v

    n

    0

    A

    A

    A

    A exit

    in which  x  and  y  are defined by

    A

    A

    1

    1

    x  

    x dV ,

    y  

    y dV

    (8‐66)

    A

    V ( t) 

    A

    A

    V ( t) 

    A

    V ( t)

    V ( t)

    In Eq. 8‐65 we have used M and M to represents the total number of moles in the liquid and vapor phases respectively. We can simplify Eq. 8‐65 by imposing the restriction

    d

    d

     y M 

    x M

    (8‐67)

    A

     

    A 

    dt

    dt

    since c is generally much, much less than c . Given this restriction, Eq. 8‐65

    takes the form

    378