# 6.11: Untitled Page 134

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## 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

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