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6.11: Untitled Page 134

  • Page ID
    18267
  • Chapter 6

    Axiom I:

    c v n dA

    R dV ,

    A  C H , C H , H

    (2)

    A A

    A

    2

    6

    2

    4

    2

    A

    V

    Application of this result to the control volume illustrated in Figure 6.2

    provides the following three equations:

    Ethane:

     ( y

    ) M

     ( y

    ) M

     R

    (3)

    C2H6 1 1

    C2H6 2

    2

    C2 H6

    Ethylene:

     ( y

    ) M

     ( y

    ) M

     R

    (4)

    C2H4 1 1

    C2H4 2

    2

    C2 H4

    Hydrogen:

     ( y ) M

    ( y

    ) M

     R

    (5)

    H

    1

    1

    H

    2

    2

    H

    2

    2

    2

    Here we have used R to represent the global net rate of production for A

    species A that is defined by (see Eq. 6‐30)

    R

    R dV ,

    A  C H , C H , H

    (6)

    A

    A

    2

    6

    2

    4

    2

    V

    The units of the global rate of production, R , are moles time while the A

    units of the rate of production, R , are moles ( time volume) , and one A

    must be careful to note this difference.

    At the entrance and exit of the control volume, we have two constraints on the mole fractions given by

    Stream #1:

    ( y

    )  ( y

    )  ( y

    )

     1

    (7)

    C2 H6 1

    C2 H4 1

    H2 1

    Stream #2:

    ( y

    )  ( y

    )  ( y

    )

     1

    (8)

    C H

    2

    C H

    2

    H

    2

    6

    2

    4

    2

    2

    For this particular process, the global form of Axiom II can be expressed as

    A N

    Axiom II

    N R

     0 ,

    J  C , H

    (9)

    JA A

    A  1

    The visual representation of the atomic matrix is given by Molecular Species  C H

    C H

    H

    2

    6

    2

    4

    2

    carbon

     2

    2

    0 

    (10)

    hydrogen

     6

    4

    2 

    and we express the explicit form of this matrix as

    Stoichiometry

    247

    2 2 0

    2 2 0

    A

    ,

    or

    N  

    (11)

    6 4 2

    JA

    6 4 2

    Use of this result for the atomic matrix with Eq. 9 leads to

    RC

    2 H6 

    2 2 0

    0

    R

     

    (12)

    H2

    6 4 2 

    0

    R

    C

    2 H4 

    At this point we can follow the development in Sec. 6.2.5 to obtain

    RC

    2 H6 

    1

    0

    1

    0

    R

    H

     

    (13)

    2

    0

    1

    1 

    0

    R

    C

    2 H4 

    in which C H has been chosen to be the pivot species (see Sec. 6.4).

    2

    4

    Carrying out the matrix multiplication leads to

    R

      R

    (14a)

    C2 H6

    C2 H4

    R

     R

    (14b)

    H

    C H

    2

    2

    4

    in which R

    is to be determined experimentally. A degree of freedom

    C2H4

    analysis will show that a unique solution is available and we can summarize the various equations as

    Ethane mole balance:  100 kmol/min 

    ( y

    ) M

     R

    (14)

    C2H6 2

    2

    C2 H6

    Ethylene mole balance:

    ( y

    ) M

     R

    (15)

    C2H4 2

    2

    C2 H4

    Hydrogen mole balance:

    30 kmol/min  R

    (16)

    H2

    Stream #1:

    ( y

    )  1 , ( y

    )  0 , ( y

    )

     0

    (17)

    C2 H6 1

    C2 H4 1

    H2 1

    Stream #2:

    ( y

    )  ( y

    )  ( y

    )

     1

    (18)

    C2 H6 2

    C2 H4 2

    H2 2

    Axiom II constraint:

    R

     R

    (19)

    C2 H6

    C2 H4

    Axiom II constraint:

    R

     R

    (20)

    H

    C H

    2

    2

    4

    The solution to Eqs. 14 through 20 is given by

    248