# 6.12: Untitled Page 135

## Chapter 6

M

 130 kmol/min

(21)

2

7

3

3

( y

)

,

( y

)

,

( y

)

(22)

C H

2

C H

H

2

6

13

2

4 2

13

2 2

13

R

 30 kmol/min

(23)

C2 H4

Here we see how the experimental system illustrated in Figure 6.2 can be used to determine the global rate of production for ethylene, R

.

C2 H4

In this example we have made use of the global form of Axiom II given by Eq. 6‐34 as opposed to the local form given by Eq. 6‐20. In addition, we can integrate the local form given by Eq. 6‐6 to obtain

A N

MW R

 0

(24)

A A

A  1

This form of Axiom II reminds us that, in general, moles are not conserved nd

a

they are certainly not conserved in this specific example.

In the previous example, we illustrated how a net rate of production could be determined experimentally for the case of a single independent stoichiometric reaction. When this condition exists for an N‐component system, we can express N  1 rates of production in terms of a single rate of production, R . For the N

complete combustion of ethane described in Example 6.1, there are four molecular species, and the rates of production for C H , O , H O can be 2

6

2

2

related to the rate of production for CO . For the rate of production of ethylene 2

described in Example 6.2, we have another example of a single independent reaction. In more complex systems, the stoichiometry is represented by multiple independent stoichiometric reactions, and we consider such a case in the following example.

EXAMPLE 6.3. Partial oxidation of carbon

Carbon and oxygen can react to form carbon monoxide and carbon dioxide, thus the reaction involves four molecular species and two atomic species as indicated by

Molecular Species:

C , O , CO , CO

(1)

2

2

Atomic Species:

C and O

(2)

Stoichiometry

249

A visual representation of the atomic matrix for this system is given by Molecular Species  C

O

CO CO

2

2

carbon

 1

0

1

1 

(3)

oxygen

 0

2

1

2 

and this can be used with Eq. 6‐22 to obtain

R

C

1 0 1 1  R O 

0

Axiom II:

2

 

   

(4)

0 2 1 2 R

0

 CO 

R

 CO

2 

A simple calculation shows that the rank of the atomic matrix is two r  rank N   2

(5)

JA

thus we have two equations and four unknowns. Here we note that the atomic matrix can be expressed in row reduced echelon form (see Eq. 6‐51) leading to

R

C

1 0

1

1  R

O

0

Axiom II:

2

 

   

(6)

0 1 1 2 1 R

0

CO

R

 CO

2 

and the homogeneous system of equations corresponding to this form is given by

R

 0  R

R

 0

(7a)

C

CO

CO2

1

0  R

R

R

 0

(7b)

O2

CO

CO

2

2

Given two equations and four rates of production, it is clear that we must determine two rates of production in order to determine all the rates of production. We will associate these two rates with two pivot species, and if we choose the pivot species to be carbon monoxide and carbon dioxide the rates of production for carbon and oxygen are given by

R

R

R

(8a)

C

CO

CO2

1

R

 

O

R

R

(8b)

2

CO

2

CO2

250