# 4.16: Untitled Page 71

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## Chapter 4

along with the three mole fraction constraints that apply at the streams that are cut by the control volume illustrated in Figure 4‐9.

Mole fraction constraints:

(4‐90)

We list these specifications and constraints as

I. Balance equations for *three* molecular species

3

II. Mole fraction constraints for the *three* streams 3

and this leads to the *generic* specifications and constraints given by

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

**6 **

Moving on to the *third step* in our degree of freedom analysis, we list the *particular * specifications and constraints according to I. Conditions for Stream #1:

1

*M*

1200 mol/h *, x*

0 3

*. , x*

0 2

*A*

*B*

*. *

3

II. Conditions for Stream #2:

2

III. Conditions for Stream #3:

1

This leads us to the *particular* specifications and constraints indicated by

**Particular Specifications and Constraints**** (C)** **6 **

and we can see that there are *zero degrees of freedom* for this problem. We summarize our degree of freedom analysis in Table 4‐2 that provides a template for subsequent problems in which we have *N* molecular species and *M* streams.

When developing the particular specifications and constraints, it is extremely important to understand that the three mole fractions can be specified *only* in the following manner:

I. *None* of the mole fractions are specified in a particular stream.

II. *One* of the mole fractions is specified in a particular stream.

III. *Two* of the mole fractions are specified in a particular stream.

The point here is that one cannot specify all three mole fractions in a particular stream because of the constraint on the mole fractions given by Eq. 4‐90. If one specifies all three mole fractions in a particular stream, Eq. 4‐90 for that stream must be deleted and the generic specifications and constraints are *no longer* *generic*.

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*Table 4‐2*. Degrees‐of‐Freedom

Stream Variables

compositions

*N * x * M* = 9

flow rates

*M* = 3

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

( *N * x * M*) * + M* = **12**

Number of Independent Balance Equations

mass/mole balance equations

*N* = 3

Number of Constraints for Compositions

*M* = 3

**Generic Constraints**** (B)**

*N* + *M* = **6**

Specified Stream Variables

compositions

5

flow rates

1

Constraints for Compositions

0

Auxiliary Constraints

0

**Particular Specifications and Constraints**** (C) **

**6 **

**Degrees of Freedom**** (A ‐ B ‐ C) **

**0**

There are two important results associated with this degree of freedom analysis. First, we are certain that a solution exits, and this provides motivation for persevering when we encounter difficulties. Second, we are now familiar with the nature of this problem and this should help us to organize a procedure for the development of a solution.

*4.7.2* *Solution of macroscopic balance equations*

Before beginning the solution procedure, we should clearly identify what is known and what is unknown, and we do this with an extended version of Table 4‐1 given here as

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