# 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

*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