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9.15: Untitled Page 231

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    18364
  • Chapter 9

    For the atomic matrix represented by Eq. 9‐117 the row reduced echelon form is given by

    1 0 0

    1 2

    1 2

    1

    A

    0 1 0

    5 4

    7 4

    1

    (9‐121)

    0 0 1  3 2  3 2

    1

     

    The primary application of Axiom II takes the form of the pivot theorem that involves the pivot matrix.

    Pivot matrix

    For the partial oxidation of ethane represented by Eq. 9‐116, we can use Eqs. 9‐118 and 9‐121 to represent Eq. 9‐120 in the form

    R

    C2H6

    R

    O2

    1 0 0

    1 2

    1 2

    1 

    0

      R H O 

     

    2

    0 1 0

    5 4

    7 4

    1

     

     

    0

     

    (9‐122)

    R

      

    CO

    0 0 1

    3 2

    3 2

    1

    0

     

    R CO

    2

    R

    C

    2H4O 

    Referring to the developments presented in Sec. 6.2.5, we note that a column / row partition of this result can be expressed as (9‐123)

    Carrying out the matrix multiplication illustrated by this column / row partition leads to a special case of the pivot theorem given by

    R

    C

    

    1

    R

    2H6

    1 2

    1 2

    CO

    R

     

    1

      R

    (9‐124)

    O

    5 4

    7 4

    2

    CO2

     3 2

    3 2

    1

    R

      R

     CO 

    C

    2H4O 

    Reaction Kinetics

    431

    The general representation of the pivot theorem takes the form Pivot Theorem:

    R

     P R

    (9‐125)

    NP

    P

    in which P is the pivot matrix. The pivot theorem is ubiquitous in the application of the concept that atomic species are neither created nor destroyed by chemical reactions. In the analysis of chemical reactors presented in Chapter 7

    the global form of Eq. 9‐125 was applied repeatedly, and we can express the global form as

    Global Pivot Theorem:

    R

     P R

    (9‐126)

    NP

    P

    Here

    represents the column matrix of non‐pivot species global net rates of NP

    R

    production while

    represents the column matrix of pivot species global net P

    R

    rates of production.

    Mechanistic matrix

    In the design of chemical reactors, one needs to know how the local net rates of production are related to the concentration of the chemical species involved in the reaction. In the development of this relation, we encountered the mechanistic matrix that maps reference chemical reaction rates (see Eq. 9‐18) onto all net rates of production. The general form is given by

    R

     M r

    (9‐127)

    M

    in which R is the column matrix of all net rates of production, M is the M

    mechanistic matrix, and r is the column matrix of elementary chemical reaction rates. For the hydrogen bromide reaction, Eq. 9‐127 provides the detailed representation given by

    k c

    R

    I Br

    Br

    2

    2

    1

    0 1

    0

    1 2  

      k c c

    R

    II Br H

    H

    0

    1

    0

    1

    0

    2

    2

     

     0

    1

    1

    1

    0   k

    c c

    R

    III H Br

    (9‐128)

    HBr

    2

     

    0

    1 1 1

    0

    R

      k c c

    H

    IV H HBr

     2 1 1 1

    1

     

    2

    R

    

    

    Br 

    k c

    

    V Br

    mechanistic matrix

    

    all species

    chemical

    reaction rates

    In many texts on chemical reactor design the mechanistic matrix is referred to as the stoichiometric matrix. However, when Bodenstein products (Aris, 1965) are

    432