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6.6: Untitled Page 129

  • Page ID
    18262
  • Chapter 6

    prove Eq. 6‐29 without imposing any constraints; however, one must accept the idea that different people state the

    s

    law of physics in different ways.

    6.2.2 Local and global forms of Axiom II

    Up to this point we have discussed the local form of Axiom II, i.e., the form that applies at a point in space. However, when Axiom II is used to analyze the

    eactors

    r

    shown in Figures 6‐1 and 6‐2, we will make use of a an integrated form of Eq. 6‐20 that applies to the control volume illustrated in Figure 6‐3.

    Figure 6‐3. Local and global rates of production

    Here we have illustrated the local rate of production for species A, designated by R , and the global rate of production for species A, designated by A

    A

    R . The latter

    is defined by

    net macroscopic molar rate 

    R

    R dV

    A

    A

    of production of species A

    (6‐30)

    V

    owing to chemical reactions

    and we often use an abbreviated description given by

    global rate of 

    R

     production of

    (6‐31)

    A

     species A

    When dealing with a problem that involves the global rate of production, we need to form the volume integral of Eq. 6‐20 to obtain

    A N

    N

    R dV

     0 ,

    J  1 , 2 ,...,T

     

    (6‐32)

    JA

    A

    A  1

    V

    Stoichiometry

    237

    The integral can be taken inside the summation operation, and we can make use f

    o the fact that the elements of N

    are

    independent of space

    to obtain

    JA

    A N

    N

    R dV

     0 ,

    J  1 , 2 ,...,T

     

    (6‐33)

    JA

    A

    A  1

    V

    Use of the definition of the global rate of production for species A given by Eq. 6‐30 leads to the following global form of Axiom II:

    A N

    Axiom II (global form):

    N

    R

     0 ,

    J  1 , 2 ,...,T

    (6

    ‐34)

    JA

    A

    A  1

    Here one must remember that

    R has

    A

    R has units of moles per unit time while A

    units of moles per unit time per unit volume, thus the physical interpretation of these two quantities is different as illustrated in Figure 6‐3. In our study of complex systems described in Chapter 7, we will routinely encounter global rates of production and Axiom II (global form) will play a key role in the analysis of those systems.

    6.2.3 Solutions of Axiom II

    In the previous paragraphs we have shown that Eq. 6‐20 and Eq. 6‐22

    represent the fundamental concept that atomic species are conserved during chemical reactions. In addition, we made use of the concept that atomic species are conserved by count

    ing atoms or balancing chemical equations (see Eqs. 6‐8, 6‐11, and 6‐12). The fact that the process of counting atoms is not unique for the partial oxidation of ethane is a matter of considerable interest that will be explored carefully in this chapter.

    In order to develop a better understanding of Axiom

    II, we carry out the

    matrix multiplication indicated by Eq. 6‐22 for a system containing three (3) atomic species

    and six (6) molecular species. This leads to the following set of ree

    th

    (3) equations containing six (6) net rates of production:

    Atomic Species 1:

    N R N R

    N R N R N R N R

     0

    (6‐35a)

    11

    1

    12

    2

    13

    3

    14

    4

    15

    5

    16

    6

    Atomic Species 2:

    N R N R

    N R N R N R N R

     0

    (6‐35b)

    21

    1

    22

    2

    23

    3

    24

    4

    25

    5

    26

    6

    238