# 4.14: Untitled Page 69

- Page ID
- 18202

## Chapter 4

where *x *

*A b * is the cup mixed mole fraction of species *A*. The definition of the mole fraction requires that

(4‐85)

and it will be left as an exercise for the student to show that *A * *N*

*x *

1

*A b*

(4‐86)

*A * 1

This type of constraint on the mole fractions (and mass fractions) applies at every entrance and exit and it often represents an important equation in the *set of* *equations* that are used to solve macroscopic mass balance problems.

**4.7 Species Mole/Mass Balance **

In this section we examine the problem of solving the *N* equations represented by either Eq. 4‐7 or Eq. 4‐17 under steady‐state conditions in the *absence of chemical reactions*. The distillation process illustrated in Figure 4‐8

*Figure 4‐8* Distillation column

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provides a simple example; however, most distillation processes are more complex than the one shown in Figure 4‐8 and most are integrated into a chemical plant as discussed in Chapter 1. It was also pointed out in Chapter 1

that complex chemical plants can be understood by first understanding the individual units that make up the plants (see Figures 1‐5 and 1‐6), thus understanding the simple process illustrated in Figure 4‐8 is an important step in our studies. For this particular ternary distillation process, we are given the information listed in Table 4‐1. Often the input conditions for a process are completely specified, and this means that the input flow rate and all the compositions would be specified. However, in Table 4‐1, we have not specified since this mole fraction will be determined by Eq. 4‐85. If we list this mole *C*

*x*

fraction as

0 5 , we would be *over‐specifying* the problem and this would lead *C*

*x * *. *

to difficulties with our degree of freedom analysis.

Table 4‐1. Specified conditions

Stream #1

Stream #2

Stream #3

*M* 1 = 1200 mol/h

x *A* = 0.3

x *A* = 0.6

x *A* = 0.1

x *B* = 0.2

x *B* = 0.3

In our application of macroscopic balances to single component systems in Chapter 3, we began each problem by identifying a control volume and we listed rules that should be followed for the construction of control volumes. For multicomponent systems, we change those rules only slightly to obtain **Rule I**. Construct a *primary cut* where information is *required*.

**Rule II**. Construct a *primary cut* where information is *given*.

**Rule III**. Join these cuts with a surface located where **v ** **n** *A*

is *known*.

**Rule IV**. When joining the primary cuts to form control volumes, minimize the number of new or

*secondary cuts* since these introduce information that is neither given nor required.

**Rule V**. Be sure that the surface specified by Rule III encloses regions in which volumetric information

is either *given* or *required*.

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