# 7.1: Untitled Page 147

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## Chapter 7

choice of the control volumes that provides the most convenient analysis will be examined in this chapter.

In Sec. 4.7.1 we developed a degree‐of‐freedom analysis for systems with *N*

components, *M* streams, and no chemical reactions. Here we extend that analysis to include chemical reactions in systems for which the governing equations are given by

*d*

Axiom I:

*c dV *

*c ***v ****n ** *dA *

*R dV , A * 1 *, * 2

(7‐1)

*A*

*A A*

*A*

*, ..., N*

*dt V*

*A*

*V*

*A * *N*

Axiom II:

*N*

*R*

0 *, J * 1 *, * 2 *, ...,T *

(7‐2)

*JA A*

*A * 1

When Axiom II is applied to control volumes, we will make use of the species global net rates of production defined by

R

*R dV , A * 1 *, * 2

*A*

*A*

*, ..., N *

(7‐3)

*V*

and we will follow the development in Sec. 6.2.2 so that Eq. 7‐2 takes the form *A * *N*

Axiom II:

*N*

R

0 *, J * 1 *, * 2 *, ...,T *

(7‐4)

*JA A*

*A * 1

In terms of the global net rate of production, Axiom I takes the form *d*

Axiom I:

*c dV *

*c*

*dA * R *, A * 1 *, * 2

*A*

**v n**

*A*

*A*

*A*

*, ..., N *

(7‐5)

*dt V*

*A*

These two results are applicable to any fixed control volume and we will use them throughout this chapter to determine molar flow rates, mass flow rates, mole fractions, etc. In addition to solving problems in terms of Eqs. 7‐4 and 7‐5, one can use those equations to derive *atomic species balances*. This is done in Appendix D where we illustrate how to solve problems in terms of the *T * atomic species rather than in terms of the *N * molecular species.

*Material Balances for Complex Systems *

271

**7.1 Multiple Reactions: Conversion, Selectivity and Yield **

Most chemical reaction systems of industrial interest produce one or more primary or desirable products and one or more secondary or undesirable products. For example, benzene ( C H ) and propylene ( C H ) undergo 6

6

3

6

reaction in the presence of a catalyst to form both the *desired product*, isopropyl benzene or cumene ( C H ) and the *undesired product*, p‐diisopropyl benzene 9

12

( C H ) . This situation is illustrated in Figure 7‐2 in which the reactants, 12

18

benzene and propylene, also appear in the exit stream because the reaction does not go to completion. In addition, the undesirable product, p‐diisopropyl *Figure 7‐2*. Production of cumene

benzene appears in the exit stream. It is important to remember that this process should be considered in terms of the principle of *stoichiometric skepticism* described in Sec. 6.1.1.

The analysis of this reactor is based on Axioms I and II. For a single entrance (Stream #1) and a single exit (Stream #2), Axiom I takes the form Axiom I:

( *M* )

R

1

( *M *) 2

*, *

*A*

1 *, * 2 *, * 3

*A*

*A*

*A*

*, * 4

(7‐6)

For a system containing only two atomic species, the global form of Axiom II is given by

*A * 4

Axiom II:

*N*

R

0 *, *

*J * 1 *, * 2

(7‐7)

*JA A*

*A * 1

In both the experimental study of the reactor shown in Figure 7‐2 and in the operation of that reactor, it is useful to have a number of defined quantities that characterize the performance. The first of these defined quantities is called the *conversion* which is given by