# 7.14: Untitled Page 159

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- 18292

## Chapter 7

( *M*

) 65 mol/h

(1c)

*C * 1

( *M*

) 4 mol/h

(1d)

*B * 3

( *M*

) 2 mol/h

(1e)

*B * 4

*Table 7.4*. Degrees‐of‐freedom: Four streams and three components Stream Variables

compositions (3 species)

*N* = 3

flow rates (4 streams)

*M* = 4

**Generic Degrees of Freedom**** (A)**

**12**

Number of Independent Balance Equations

balance equations (3 species)

*N* = 3

Number of Constraints for Compositions (4 streams)

*M* = 4

Number of Constraints for Reactions (zero)

0

**Generic Specifications and Constraints**** (B)**

**7**

**Degrees of Freedom**** (A ‐ B) **

**5**

The development given in Sec. 4.6 can be used to construct the general relations given by

*M*

(

)

( *x *) *M , *

*A * 1 *, * 2 *, * 3 *, i * 1 *, * 2 *, * 3

*A i*

*A i*

*i*

*, * 4

(2a)

*M*

*M*

*M*

(

)

(

)

( *M *) *, *

*i * 1 *, * 2 *, * 3

*i*

*A i*

*B i*

*C i*

*, * 4

(2b)

and from the data given in Eqs. 1a, 1b and 1c we have

( *x *)

0 *. * 1 *, *

( *x *)

0 *. * 25

( *x *)

0 *. * 65

(3)

*A * 1

*B * 1

*C * 1

This indicates that the conditions for Stream #1 are completely specified, and on the basis of Eq. 7‐33 all the mole fractions in the other streams are determined.

Equation 2a can be expressed in the form

*MA i*

(

)

*M*

*, *

*A * 1 *, * 2 *, * 3 *, i * 1 *, * 2 *, * 3 *, * 4

*i*

(4)

( *x *)

*A i*

*Material Balances for Complex Systems *

295

and this can be used for Stream #3 to obtain

( *M*

)

( *M*

)

4 mol/h

*B * 3

*B * 3

*M*

16 mol/h

(5a)

3

( *x *)

( *x *)

0.25

*B * 3

*B * 1

Given the total molar flow rate in Stream #3, we can use Eq. 2a to obtain ( *M*

)

3

( *x *)3 *M* 3

( *x * 1

) *M* 3

(0 1

*. *) (16 mol/h)

1 6

*. * mol/h

*A*

*A*

*A*

(5a)

( *M*

) ( *x *) *M* ( *x *) *M* (0 6

*. * 5) (16 mol/h) 10 4

*. * mol/h (5b)

*C * 3

*C * 3

3

*C * 1

3

indicating that all the molar flow rates in Stream #3 are determined.

Directing our attention to Stream #4, we repeat the analysis represented by Eqs. 5 to obtain

( *M*

)

( *M*

)

2 mol/h

*B * 4

*B * 4

*M*

8 mol/h

(6a)

4

( *x *)

( *x *)

0.25

*B * 4

*B * 1

( *M*

) ( *x *) *M* ( *x *) *M* (0 1

*. *)(8 mol/h) 0 8

*. * mol/h

(6b)

*A * 4

*A * 4

4

*A * 1

4

( *M*

) ( *x *) *M* ( *x *) *M* (0 *. * 65)(8 mol/h) 5 *. * 2 mol/h (6c) *C * 4

*C * 4

4

*C * 1

4

and we are ready to move on to determine all the molar flow rates in Stream #2. This requires the use of Eq. 7‐32 in terms of the species molar flow rates given by

( *M*

) ( *M* ) ( *M* ) ( *M* )

(7a)

*A * 1

*A * 2

*A * 3

*A * 4

( *M *)

( *M *) ( *M *) ( *M *)

(7b)

*B * 1

*B * 2

*B * 3

*B * 4

( *M *)

( *M *) ( *M *) ( *M *)

(7c)

*C * 1

*C * 2

*C * 3

*C * 4

and these results can be used to determine the species molar flow rates given by

( *M *)

7.6 mol/h *, *

( *M *)

19 mol/h *, *

( *M *) 49.4 mol/h (8)

*A * 2

*B * 2

*C * 2

Finally we see that the total molar flow rate in Stream #2 is given by *M*

76 mol/h

(9)

3

Note that if all the species molar flow rates are specified for a single stream, as they are by Eqs. 1a, 1b and 1c, then the additional specifications must be in the other streams. The above example illustrates why this is the case. Once all the species molar flow rates for a given stream are

296