# 7.12: Untitled Page 157

## Chapter 7

are joined to form a single stream, and we have illustrated a splitter in which a single stream is split into S streams. Both accumulation and chemical reaction can be neglected in mixers and splitters since these devices consist only of tubes joined in some convenient manner. This means that they can be analyzed in Figure 7‐6. Mixer and splitter

terms of the steady form of Eq. 7‐1 that simplifies to

c

dA  0 ,

A  1 , 2

v n

A A

,...N

(7‐30)

A

For the mixer shown in Figure 7‐6, this result can be expressed as i S

Mixer ( species balances): 

( x ) M  ( x ) M

 0 , A  1 , 2 ,...,N (7‐31)

A i

i

A o

o

i  1

This result also applies to the splitter shown in Figure 7‐6, and for that case the macroscopic balance takes the form

i S

Splitter ( species balances):

( x ) M

( x ) M  0 , A  1 , 2 ,..., N (7‐32) A o

o

A i i

i  1

The physics of a splitter require that the compositions in all the outgoing streams be equal to those in the incoming stream, and we express this idea as Splitter ( physics):

( x )

 ( x ) ,

i  1 , 2 ,...,S , A  1 , 2 ,...,N

(7‐33)

A i

A o

Material Balances for Complex Systems

291

In addition to understanding the physics of a splitter, we must understand how this constraint on the mole fractions is influenced by the constraints that we apply in terms of our degree of freedom analysis. Table 7‐1 indicates that all streams cut by a control surface are required to satisfy the constraint on the mole fractions given by

( x )  ( x )  ( x )  .....  ( x )

 1 ,

i  0 , 1 , 2 ,....,S

(7‐34)

A i

B i

C i

N i

This means that only N  1 mole fractions can be specified in the outgoing streams of a splitter. To illustrate how these constraints influence our description of a splitter, we consider Stream #2 of the splitter illustrated in Figure 7‐6. For that stream, Eq. 7‐33 provides the following N  1 equations: ( x )

 ( x )

(7‐35a)

A 2

A o

( x )

 ( x )

(7‐35b)

B 2

B o

( x )

 ( x )

(7‐35c)

C 2

C o

… … …

(x

)

 ( x

)

(7‐35n)

N1 2

N1 o

When we impose Eq. 7‐34 for both Stream #2 and Stream #0 we obtain 1  ( x )

 1  ( x )

(7‐36)

N 2

N o

and this leads to the result that the mole fractions of the th N component must be

equal.

( x )

 ( x )

(7‐37)

N 2

N o

This indicates that we should impose Eq. 7‐33 on only N  1 of the components so that our degree of freedom representation of the splitter takes the form Splitter ( degree of freedom / mole fractions):

( x )

 ( x ) ,

i  1 , 2 ,..,S , A  1 , 2 ,..., N  1

(7‐38)

A i

A o

For many situations, a splitter may be enclosed in a control volume, as illustrated in Figure 7‐6. In those situations, we will impose the macroscopic species balance given by Eq. 7‐32, and we must again be careful to understand how this effects our degree of freedom analysis. In particular, we would like to prove that the splitter condition indicated by Eq. 7‐38 need only be applied to S  1 streams

292