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8.14: Untitled Page 203

  • Page ID
    18336
  • Chapter 8

    d

      x M 

    y c   dA

    (8‐68)

    dt

    v

    n

    0

    A

    A

    Aexit

    and we can express the flux at the exit in the traditional form to obtain d

     x M   y c Q  0

    (8‐69)

    A

    A exit

    dt

    This represents the governing equation for  x  and it is restricted to cases for A

    which c  c

     .

    In addition to  x  , there are other unknown terms in Eq. 8‐69, and the total A

    mole balance will provide information about one of these. Returning to Eq. 8‐62

    we apply that result to the control volume illustrated in Figure 8‐11 to obtain

    d

    c dV

    cdV

    c  

    (8‐70)

    dt

    v n dA 0

     

    V ( t)

    V( t)

    Aexit

    At the exit of the control volume, we again ignore diffusive effects and replace

    v

    n with v

    n so that this result takes the form

     

     

    d

    M M

       c Q

     0

    (8‐71)

    dt

    At this point, we again impose the restriction that c  c

     which allows us to

    simplify this result to the form

    d M

    c Q

     0

    (8‐72)

    dt

    We can use this result to eliminate c Q

     from Eq. 8‐69 so that the mole balance

    for species A takes the form

    d M

    d

     x M   y

     0

    (8‐73)

    A

    A exit

    dt

    dt

    At this point we have a single equation and three unknowns:  x  , M

    A

     and

    y

    , and our analysis has been only moderately restricted by the condition A exit

    that c  c

     . We have yet to make use of the equilibrium relation indicated by

    Transient Material Balances

    379

    Eq. 8‐59, and to be very precise in the next step in our analysis we repeat that equilibrium relation according to

    Equilibrium relation:

    y

     

    x ,

    at the vapor ‐ liquid interface

    (8‐74)

    A

    AB A

    In our macroscopic balance analysis, we are confronted with the mole fractions indicated by  x  and  y

    , and the values of these mole fractions at the

    A

    A exit

    vapor‐liquid interface illustrated in Figure 8‐11 are not available to us. Knowledge of x and y at the    interface can only be obtained by a detailed analysis of A

    A

    the diffusive transport (Bird et al., 2002) that is responsible for the separation that occurs in batch distillation. In order to proceed with an approximate solution to the batch distillation process, we replace Eq. 8‐74 with

    Process equilibrium relation:

    y

       x

    (8‐75)

    A exit

    eff

    A

    Here we note that Eqs. 8‐74 and 8‐75 are analogous to Eqs. 5‐49 and 5‐50 if the approximation 

      is valid. The process equilibrium relation suggested by eff

    AB

    Eq. 8‐75 may be acceptable if the batch distillation process is slow enough, but we do not know what is meant by slow enough without a more detailed theoretical analysis or an experimental study in which theory can be compared with experiment.

    Keeping in mind the uncertainty associated with Eq. 8‐75, we use Eq. 8‐75 in Eq. 8‐73 to obtain

    dM

    d x

    A

    M

     (1   ) x

     0

    (8‐76)

    eff

    A

    dt

    dt

    The initial conditions for the mole fraction,  x  , and the number of moles in the A

    still, M( t) , are given by

    I.C.1

    o

    x   x ,

    t  0

    (8‐77)

    A

    A

    I.C.2

    o

    M

    M ,

    t  0

    (8‐78)

    At this point we have a single differential equation and two unknowns,  x  and A

    M( t) . Obviously we cannot determine both of these quantities as a function of time unless some additional information is given. For example, if M ( t)

    were

    specified as a function of time we could use Eq. 8‐76 to determine  x  as a A

    380