# 7.13: Untitled Page 158

## Chapter 7

when we make use of Eq. 7‐32. To see that this is true, we first note that Eq. 7‐32

can be summed over all N species to obtain the total molar balance given by

M  M  M  M  .... M

 0

o

 1

2

3

S

(7‐39)

Next, we rearrange Eq. 7‐32 in the form

i S1

( x ) M

( x ) M

( x ) M

 0 ,

A  1 , 2 , ... , N

(7‐40)

A o

o

A i i

A S

S

i  1

and then apply the constraints indicated by Eq. 7‐38 for only S  1 of the streams leaving the splitter in order to obtain

i S1

( x ) M

( x )

M

( x ) M

 0 ,

A  1 , 2 , ... , N

(7‐41)

A o

o

A o 

i

A S

S

i  1

This result can be used with the total molar balance given by Eq. 7‐39 to arrive at the condition

( x ) M

( x ) M

 0 ,

A

o

1 , 2

A

S

A S

S

, ... , N

(7‐42)

( x )

 ( x ) ,

A

o

1 , 2

A S

A

, ... , N

(7‐43)

Here we see that we have derived, independently, one of the conditions implied by Eqs. 7‐38 and this means that the splitter constraints indicated by Eqs. 7‐38

can be expressed as

Splitter ( degree of freedom / mole fractions / species balances): ( x )  ( x ) ,

i  1 , 2 , ... , S  1 ,

A  1 , 2 ,... , N  1

(7‐44)

A i

A o

Often it is more convenient to work with species molar flow rates that are related to mole fractions by

B N

x

MM  M

M

 

(7‐45)

A

A

A

B

B  1

Use of this result in Eq. 7‐44 provides the alternative representation for a splitter. Material Balances for Complex Systems

293

B N

B N

( M )

( M )

A i

( M )

B i

A

 ( M )

o

B o ,

B  1

B  1

(7‐46)

i  1 , 2 , ... , S  1 ,

A  1 , 2 , ... , N  1

To summarize, we note that the physical conditions associated with a splitter are given by Eqs. 7‐33. When we take into account the constraint on mole fractions given by Eqs. 7‐34, we must simplify Eqs. 7‐33 to the form given by Eqs. 7‐38 in order to be consistent with our degree of freedom analysis. When we further take into account the species mole balances that may be imposed on a control volume around the splitter, we must simplify Eqs. 7‐38 to the form given by Eqs. 7‐44 in order to be consistent with our degree of freedom analysis.

EXAMPLE 7.4. Splitter calculation

In this example we consider the splitter illustrated in Figure 7.4 in which Stream #1 is split into Streams #2, #3 and #4, each of which contain the Figure 7.4. Splitter producing three streams

three species entering the splitter. We will use ( M

 ) , ( M ) and ( M ) to

A j

B j

C j

represent the molar flow rates of species A, B and C in the th j stream and

we will use M

 to represent the total molar flow rate entering Stream #1.

1

The degree‐of‐freedom analysis given in Table 7.4 indicates that we have five degrees of freedom, and there are several ways in which the splitter problem can be solved. In this particular example we consider the case in which five molar flow rates are specified according to:

( M

 )  10 mol/h

(1a)

A 1

( M

 )  25 mol/h

(1b)

B 1

294