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6.10: Untitled Page 133

  • Page ID
    18266
  • Chapter 6

    R

    1

    R

    2

    1

    0

    1

    0

    1

    1

      

    0

    1 1

    1 1

    3

    R

     3 

    1

    R R 2 :

     0

    (6‐46)

     0

    0

    0

    1 1

    1  R

     4 

     0

    0

    0

    0

    0

    0  

    R

    5

    R

     6 

    In this form the first two columns contain only a single entry along the diagonal, and we would like the third column to have this characteristic. Use of the following column / row interchange

    N R

    N R ,

    J  1 , 2 , 3

    (6‐47)

    J 3

    3

    J 4

    4

    provides a step in that direction given by

    R

    1

    R

     

    2

    1

    0

    0

    1

    1

    1  

    0

    1

    1

    1

    1

    3

    R

     4 

     0

    (6‐48)

     0

    0

    1

    0

    1

    1  R

     3 

    0

    0

    0

    0

    0

    0  R

    5

    R

     6 

    We now subtract row three from row two in order to obtain the following row reduced echelon form:

    R

    1

    R

     

    2

    1

    0

    0

    1

    1

    1  

    0

    1

    0 1

    0

    2

    R

     4 

    R 2  R 3

     0

    (6‐49)

     0

    0

    1

    0

    1

    1  R

     3 

    0

    0

    0

    0

    0

    0  R

    5

    R

     6 

    The last row of zeros produces the null equation that we express as 0  R  0  R

     0  R  0  R  0  R  0  R

     0

    (6‐50)

    1

    2

    4

    3

    5

    6

    thus we can discard that row to obtain

    index-254_1.png

    index-254_2.png

    index-254_3.png

    Stoichiometry

    245

    Axiom II:

    (6‐51)

    This form has the attractive feature that the submatrix located to the left of the dashed line is a unit matrix, and this is a useful result for solving sets of equations. Finally, it is crucial to understand that any atomic matrix can always be expressed in row reduced echelon form, and uniqueness proofs are available (Noble, 1969, Sec. 3.8; Kolman, 1997, Sec. 1.5).

    EXAMPLE 6.2. Experimental determination of the rate of production of ethylene

    Here we consider the experimental determination of a global rate of production for the steady‐state, catalytic dehydrogenation of ethane as illustrated in Figure 6.2. We assume (see Sec. 6.1.1) that the reaction produces ethylene and hydrogen, thus only C H , C H and H are 2

    6

    2

    4

    2

    present in the system. We are given that the feed is pure ethane and the feed flow rate is 100 kmol/min . The product Stream #2 is subject to a measurement indicating that the molar flow rate of hydrogen in that stream is 30 kmol/min , and we wish to use this information to determine Figure 6.2. Experimental reactor

    the global rate of production for ethylene. For steady‐state conditions, the axiom given by Eq. 6‐4 takes the form

    246