# 7.25: Untitled Page 170

## Chapter 7

At this point we make use of Eq. 7‐34 to obtain the standard constraints on mole fractions when only ethylene ( C H ) , carbon dioxide ( CO ) and 2

4

2

nitrogen ( N ) are present. This provides

2

( x

)

 ( x

)  ( x

)

 1

(37a)

C2H4 4

CO2 4

N2 4

( x

)  ( x

)  ( x

)

 1

(37b)

C2H4 6

CO2 6

N2 6

and subtracting Eq. 37a from Eq. 37b leads to

( x

)

 ( x

)

(38)

C2H4 6

C2H4 4

We now make use of Eq. 7‐45 to represent the constraints on the three mole fractions as

( M

)

M

M

( M

)

(39a)

CO

6

 6 4

2

CO2 4

( M

)

M

M

( M

)

(39b)

N

6

 6 4

2

N2 4

( M

)

M

M

( M

)

(39c)

C H

6

 6 4

2

4

C2H4 4

In Eq. 1 note that  is the fraction of Stream #4 that needs to be purged, i.e.,

  MM

(40)

6

4

thus the molar flow rates in Stream #6 are given by

( M

)

 ( M

)

(41a)

CO2 6

CO2 4

( M

)

 ( M )

(41b)

N2 6

N2 4

( M

)

 ( M

)

(41c)

C2H4 6

C2H4 4

Use of these results with Eqs. 35 leads to

C H :

( M

)

 (1  )( M

)

(42a)

2

4

C2H4 5

C2H4 4

CO :

( M

)

 (1  )( M

)

(42b)

2

CO2 5

CO2 4

N :

( M

)

 (1  )( M )

(42c)

2

N2 5

N2 4

Material Balances for Complex Systems

317

and we can use Eq. 34b to represent the molar flow rate of ethylene in Stream #5 as

( 1  )( 1   )

( M

C

)

(43)

C2H4 5

CY

At this point we need to return to Control Volume I in order to acquire additional information needed to determine the parameter,  . Beginning with Eq. 30a, we use Eq. 34a and Eq. 43 to obtain

C H :

( M

)

1  1 C 1 



(44a)

C H

1



2

4

2

4

CY

Moving on to Eq. 30b, we use Eq. 32d along with Eq. 34a and Eq. 42b to obtain

2 1  1  

CO :

( M

Y

)

(44b)

2

CO2 2

Y

Next we use Eq. 30c along with Eq. 34d to obtain

O :

5

( M

)

 ( M )

 (3  Y) Y

(44c)

2

O2 1

O2 2

2

and finally we make use of Eq. 30c along with Eq. 31 to express the molar flow rate of nitrogen entering the system as

N :

5

( M

)

 (3  Y) Y

(44d)

2

N2 1

2

At this point we are ready to return to Eq. 34g and make use of Eq. 32d and Eq. 33 to obtain

( M

)

 ( M

)  2  1  Y Y

(45)

CO

4

CO

2

2

2

This result can be used with Eq. 44b to provide the molar flow rate of carbon dioxide ( CO ) entering the splitter

2

( M

)

 2  1  Y Y

(46)

CO

4

2

This result can be used with Eq. 34g to determine the molar flow rate of nitrogen ( N ) entering the splitter as

2

2  1  

Y

1

( M

)

1   (1  C  CY) 

(47)

N2 4

2

CY

Y

318