# 4.15: Untitled Page 70

- Page ID
- 18203

## Chapter 4

In Rule III it is understood that **v ** represents the species velocity for all *A*

*N* species, and in Rule II it is assumed that the *given* information is necessary for the solution of the problem. For the system illustrated in Figure 4‐8, it should be obvious that we need to cut the *entrance* and *exit* streams and then join the cuts as illustrated in Figure 4‐9 where we have shown the details of the cut at stream #2, and we have illustrated that the cuts at the entrance and exit streams are joined by a surface that is coincident with the solid‐air interface where

.

*Figure 4‐9*. Control volume for distillation column

4.7.1 *Degrees‐of‐freedom analysis*

In order to solve the macroscopic balance equations for this distillation process, we require that the number of constraining equations be equal to the number of unknowns. To be certain that this is the case, we perform a *degrees‐of-freedom analysis* which consists of three parts. We begin this analysis with a *generic* *part* in which we identify the process variables that apply to a *single* *control volume* in which there are *N* molecular species and *M* streams. We assume that every molecular species is present in every stream, and this leads to the *generic degrees of freedom*. Having determined the generic degrees of freedom, we

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direct our attention to the *generic specifications and constraints* which also apply to the control volume in which there are *N* molecular species and *M* streams.

Finally, we consider the *particular specifications and constraints* that reduce the generic degrees of freedom to zero if we have a well‐posed problem in which all process variables can be determined. If the last part of our analysis does not reduce the degrees of freedom to zero, we need *more information* in order to solve the problem. The inclusion of chemical reactions in the degree of freedom analysis will be delayed until we study stoichiometry in Chapter 6.

The *first step* in our analysis is to prepare a list of the process variables, and this leads to

Mole fractions:

(4‐87)

Molar flow rates:

(4‐88)

For a system containing three molecular species and having three streams, we determine that there are twelve generic process variables as indicated below.

I. Three mole fractions in each of three streams

9

II. Three molar flow rates

3

For this process the generic degrees of freedom are given by

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

**12 **

In this first step, it is important to recognize that we have assumed that all species are present in all streams, and it is for this reason that we obtain the *generic* degrees of freedom.

The *second step* in this process is to determine the *generic specifications and* *constraints* associated with a system containing three molecular species and three streams. In order to solve this ternary distillation problem, we will make use of the three molecular species balances given by

Species balances:

(4‐89)

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