# 6.17: Untitled Page 140

- Page ID
- 18273

## Chapter 6

*R*

*R*

0 2 *R*

(13b)

C2H4

H2O

C4H6

*R*

0

*R*

*R*

(13c)

CH3CHO

H2

C H

4

6

In this example we have provided a template for the solution of Eq. 6‐22 in which the mathematical steps are always the same, i.e., (I) begin with the atomic matrix given by Eq. 2 and develop the row reduced echelon form given by Eq. 3, (II) partition Axiom II as indicated by Eqs. 8 and 9, and (III) carry out the matrix multiplication to obtain the net rates of production for the non‐pivot species indicated by Eqs. 13.

In the design of a butadiene production unit, the representation for the net rate of production of the non‐pivot species is a crucial part of the analysis. In Chapter 7 we will apply this type of analysis to several rocesses.

p

*6.3.1* *Rank of the atomic matrix*

In the previous paragraphs we have seen several examples of the atomic matrix when the rank of that matrix was less than the number of molecular species, i.e., r *N *. Here we wish to illustrate two special cases in which r *N *

and r *N *. First we consider a reactor containing only methyl chloride ( CH Cl ), ethyl chloride ( C H Cl ), and chlorine ( Cl ). We illustrate the atomic 3

2

5

2

matrix by

Molecular Species CH Cl C H Cl

Cl

3

2

5

2

*carbon*

1

2

0

(6‐63)

*hydrogen*

3

5

0

*chlorine*

1

1

2

and use elementary row operations to obtain the *row reduced echelon form* given by

1 0 0

A

0 1 0

(6‐64)

0 0 1

Use of this result in Axiom II provides

259

1 0 0 *R* CH

0

3Cl

0 1 0 *R*

0

(6‐65)

C

2H5Cl

0 0 1

0

*R*

Cl

2

and we immediately obtain the *trivial solution* expressed as *R*

0

(6‐66a)

CH3Cl

*R*

0

(6‐66b)

C2H5Cl

*R*

0

(6‐66c)

Cl2

This indicates that *no net rate of production can occur* in a reactor containing only methyl chloride, ethyl chloride and chlorine. If we allow molecular hydrogen to be present in the reactor, our system is described by

Molecular Species CH Cl C H Cl

Cl

H

3

2

5

2

2

*carbon*

1

2

0

0

(6‐67)

*hydrogen*

3

5

0

2

*chlorine*

1

1

2

0

and the atomic matrix takes the form

1 2 0 0

A

3 5 0 2

(6‐68)

1 1 2 0

The *row reduced echelon form* of this atomic matrix can be expressed as

1 0 0

4

A

0 1

0 2

(6‐69)

0 0 1 1

and from Axiom II we obtain

*R*

CH3Cl

1 0 0

4

0

*R*

C H Cl

2

5

0 1

0 2

0

(6‐70)

*R * Cl

2

0 0

1

1

0

*R*

H

2

260