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6.14: Untitled Page 137

  • Page ID
    18270
  • Chapter 6

    net rates of production. This case is illustrated in Eq. 6‐8 and discussed in detail in Example 6.2. When we have multiple independent stoichiometric reactions, such as the partial oxidation of carbon (Example 6.3) or the partial oxidation of ethane (Example 6.4), we need to measure more than one net rate of production in order to determine all the net rates of production.

    6.2.6 Matrix partitioning

    Axiom II provides an example of the multiplication of a T N matrix with a 1 N column matrix. Multiplication of matrices can also be represented in terms of submatrices, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation

    b

    1

     

    a

    a

    a

    a

    a b

    c

    11

    12

    13

    14

    15  2 

    1

     

    a

    a

    a

    a

    a

    b  

    c

    (6‐52)

     21

    22

    23

    24

    25 

    3

     2

     

    a

    a

    a

    a

    a b

    c

    31

    32

    33

    34

    35   4 

     3

    b

     5 

    which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation 6‐52 represents the three individual equations given by

    a b a b a b a b a b

    c

    (6‐53a)

    11 1

    12 2

    13 3

    14 4

    15 5

    1

    a b a b a b a b a b

    c

    (6‐53b)

    21 1

    22 2

    23 3

    24 4

    25 5

    2

    a b a b a b a b a b

    c

    (6‐53c)

    31 1

    32 2

    33 3

    34 4

    35 5

    3

    which can also be expressed in compact form according to

    AB

     C

    (6‐54)

    Here the matrices A, B, and C are defined explicitly by

    b

    1

     

    a

    a

    a

    a

    a

    b

    c

    11

    12

    13

    14

    15

     2 

    1

     

    A

    a

    a

    a

    a

    a

    B

      b

    C

    c

    (6‐55)

     21

    22

    23

    24

    25 

    3

     2

     

    a

    a

    a

    a

    a

    b

    c

    31

    32

    33

    34

    35 

     4 

     3

    b

     5 

    In addition to the matrix multiplication that we have used up to this point, matrix multiplication can also be carried out in terms of partitioned matrices.

    index-262_1.png

    Stoichiometry

    253

    If we wish to obtain a column partition of the matrix A in Eq. 6‐52, we must also create a row partition of matrix B in order to conform to the rules of matrix multiplication that are discussed in Appendix C1. This column / row partition takes the form

    (6‐56)

    and the submatrices are identified explicitly according to

    a

    a

    a

    b

    a

    a

    11

    12

    13

    1

    14

    15

     

    b

    4

    A

    a

    a

    a

    B

    b

    A

    a

    a

    B

    (6‐57)

    11

     21

    22

    23 

    1

     2

    12

     24

    25 

    2

     

    b

     5 

    a

    a

    a

    b

    a

    a

     31

    32

    33 

     3

     34

    35 

    Use of these representations in Eq. 6‐56 leads to

    B 

    A

    A

     1

    C

    (6‐58)

    11

      

    12  B2

    and matrix multiplication in terms of the submatrices provides A B

     A B

     C

    (6‐59)

    11 1

    12

    2

    To verify that this result is identical to Eq. 6‐53, we use Eqs. 6‐57 and the third of Eqs. 6‐55 to obtain

    a

    a

    a   b

    a

    a

    c

    11

    12

    13

    1

    14

    15

      

      b

     1

    4

    a

    a

    a

    b

    a

    a

      

       

    c

    21

    22

    23

    2

    24

    25

     2 

    (6‐60)

    b

     5 

    a

    a

    a   b

    a

    a

    c

     31

    32

    33   3 

     34

    35 

     3

    Carrying out the matrix multiplication on the left hand side of this result leads to

    a b

    a b

    a b

    a b

    a b

    c

    11 1

    12 2

    13 3

    14 4

    15 5

     1

    a b

    a b

    a b

    a b

    a b

    c

     21 1

    22 2

    23 3 

     24 4

    25 5 

     2

    (6‐61)

    a b a b

    a b

    a b

    a b

    c

     31 1

    32 2

    33 3 

     34 4

    35 5 

     3

    At this point we add the two matrices on the left hand side of this result following the rules for matrix addition given in Sec.2.6 in order to obtain

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