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8.5: Untitled Page 194

  • Page ID
    18327
  • Chapter 8

    R

    CH4

    0

    1 0 1 2

    R

     

    H

    0

    Axiom II:

    2

    4 2 0 6

     

     

    (8‐27)

    R

    0

    CO

    0 0 1 1

     

    0

    R

     C

    2H6O 

    Making use of the row reduced echelon form of the atomic matrix and applying the pivot theorem given in Sec. 6.4 leads to

    R

    CH

      1 

    4 

    R

     

     1  R

    (8‐28)

    H

    2

    C

    2 H6 O 

      1 

    R

     CO 

    in which C H O has been chosen as the pivot species. Hinshelwood and Asky 2

    6

    (1927) found that the reaction could be expressed as a first order decomposition providing a rate equation of the form

    Chemical reaction rate equation:

    R

      k c

    (8‐29)

    C2H6O

    C2H6O

    If we let dimethyl ether be species A, we can express the reaction rate equation as R

     

    A

    k cA

    (8‐30)

    Use of this result in Eq. 8‐25 leads to

    dc

    A

      k c

    A

    (8‐31)

    dt

    and we require only an initial condition to complete our description of this process. Given the following initial condition

    I.C.

    o

    c   c ,

    t  0

    A

    A

    (8‐32)

    we find the solution for  c  to be

    A

    o

    k t

    c   c e

    (8‐33)

    A

    A

    This simple exponential decay is a classic feature of first order, irreversible processes. One can use this result along with experimental data from a batch reactor to determine the first order rate coefficient, k. This is often done by

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    Transient Material Balances

    363

    plotting the logarithm of

    o

    c / c versus t, as illustrated in Figure 8‐6, and A

    A

    noting that the slope is equal to  k .

    Figure 8‐6. Batch reactor data, logarithmic scale

    If the rate coefficient in Eq. 8‐33 is known, one can think of that result as a design equation. The idea here is that one of the key features of the design of a batch reactor is the specification of the process time. Under these circumstances, one is inclined to plot

    o

    c / c as a function of time and this is done in A

    A

    Figure 8‐7. The situation here is very similar to the mixing process described in the previous section. In that case the characteristic time was the average residence time, V / Q , while in this case the characteristic time is the inverse of the rate

    coefficient,

    1

    k . When the rate coefficient is known one can quickly deduce that

    the process time is on the order of

    1

    3 k

    to

    1

    4 k . Very few reactions are as simple

    as the first order irreversible reaction; however, it is a useful model for certain decomposition reactions.

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    364