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8.21: Untitled Page 210

  • Page ID
    18343
  • Chapter 8

    The analysis begins with the fixed control volume shown in Figure 8.16 and the general macroscopic balance given by

    d

    c dV

    c v n dA

    R dV ,

    D A, B, C

    (1)

    D

    D D

    D

    dt V

    A

    V

    The moles of species A (t‐butyl alcohol) in the   phase can be neglected, thus the macroscopic balance for this species takes the form

    d

    t‐butyl alcohol:

    c

    dV

    R

    dV

    (2)

    A

    A

    dt

    V ( t)

    V ( t)

    and in terms of average values for the concentration and the net rate of production of species A we have

    d

    t‐butyl alcohol:

     c

    V ( t)

      R  

    V ( t)

    (3)

    A

    A

    dt

    If we also assume that the moles of species B (isobutylene) and species C (water) are negligible in the   phase , the macroscopic balances for these species take the form

    d

    isobutylene:

     c

    V ( t)  M    R  

    V ( t)

    (4)

    B

    B

    B

    dt

    d

    water:

     c

    V ( t)

      R  

    V ( t)

    (5)

    C

    C

    dt

    This indicates that alcohol and water are retained in the system by the condenser while the isobutylene leaves the system at a molar rate given by M

    . The initial

    B

    conditions for the three molecular species are given by

    IC.

    o

    c   c ,

    t  0

    (6a)

    A

    A

    IC.

    c   0 ,

    t  0

    (6b)

    B

    IC.

    c   0 ,

    t  0

    (6c)

    C

    Since the molar rates of reaction are related by

    R

      R

    ,

    and

    R

      R

    (7)

    C

    A

    B

    A

    Transient Material Balances

    393

    we need only be concerned with the rate of reaction of the t‐butyl alcohol. If we treat the reactor as perfectly mixed, the mole balance for t‐butyl alcohol can be expressed as

    dc

    c

    A

    A

    dV ( t)

    R

    (8)

    A

    dt

    V ( t)

    dt

    This indicates that we need to know both cA and V as functions of time in order to obtain experimental values of RA . The volume of fluid in the reactor can be expressed as

    V ( t)  n

    v

    n v  n v

    (9)

    AA

    BB

    C C

    in which n

    n

    n

    A ,

    and

    represent the moles of species A, B and C in the B

    C

      phase and v , v and v represent the partial molar volumes.

    A

    B

    C

    To develop a useful expression for RA , assume that the liquid mixture is ideal so that the partial molar volumes are constant. In addition, assume that the moles of species B in the liquid phase are negligible. On the basis of these assumptions, show that Eq. 8 can be expressed as

    d c

    c

    A 

    A  v

    v

    A

    C

    R

    1

     

    (10)

    A

    dt

    1  c

     

    A  v

    v

    A

    C  

    This form is especially useful for the interpretation of initial rate data, i.e., experimental data can be used to determine both c

    dc

    / dt

    t

    A and

    at

    and

    A

    0

    this provides an experimental determination of RA for the initial conditions associated with the experiment.

    An alternate approach (Gates and Sherman, 1975) to the determination of R

    is to measure the molar flow rate of species B that leaves the reactor in the A

      phase and relate that quantity to the rate of reaction.

    Section 8.4

    8‐17. When Eqs. 8‐41 and 8‐42 are valid, Eq. 8‐43 represents a valid result for the chemostat shown in Figure 8‐10. One can divide this equation by a constant, m

    , to obtain Eq. 8‐47; however, the average mass of a cell, m

    , in the

    cell

    cell

    chemostat may not be the average mass of a cell in the incoming stream. If m cell

    394