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# 7.44: Untitled Page 189

## Chapter 8

### Transient Material Balances

In Chapter 3 we studied transient systems that involved only a single molecular species. In this chapter we extend our original study to include multicomponent, multiphase, reacting mixtures. First we introduce the concept of a perfectly mixed stirred tank and study some simple mixing processes. This analysis of mixing is followed by a study of a batch reactor as an example of a transient process with chemical reaction. We conclude our studies of batch processes with an analysis of biomass production in a chemostat and then move on to the transient process of batch distillation.

Our analysis of transient systems begins with the molar form of Axiom I for species A given by

d

Axiom I:

c dV

c v

(

w)n dA

A

A A

R dV

dt

A

(8‐1)

V ( t)

A ( t)

V ( t)

a

a

a

Here one must remember that V ( t) represents an arbitrary, moving control a

volume having a surface A ( t) that moves with a speed of displacement given a

by w n . The second axiom requires that atomic species be conserved and is stated as

A N

Axiom II:

N

R

 0 ,

J  1 , 2 ,... , T

(8‐2)

JA A

A  1

One can use this form to develop (see Sec. 6.1) a useful constraint on the net molar rates of production given by

A N

MW R

 0

(8‐3)

A A

A  1

The mass form of Axiom I will be useful in our analysis of biomass production in Sec. 8.4 and this form is obtained from Eq. 8‐1 by multiplying by the molecular mass. The result is given by

354

Transient Material Balances

355

d

dV

 (v w) n dA

(8‐4)

dt

r dV

A

A A

A

V ( t)

A ( t)

V ( t)

a

a

a

and it is often used with a constraint on the species mass rates of production that takes the form

A N

r

 0

(8‐5)

A

A  1

We will use all of these forms in our study of transient systems.

8.1 Perfectly Mixed Stirred Tank

In the chemical process industries, one encounters a system used for mixing which is referred to as a “completely mixed stirred tank” or a “perfect mixer.”

When used as a continuous reactor, such a system is often identified as a continuous stirred tank reactor or CSTR as an abbreviation. The essential characteristic of the perfectly mixed stirred tank is that the concentration in the tank is assumed to be uniform and equal to the effluent concentration even when the inlet conditions to the tank are changing with time. While this is impossible to achieve in any real system, it does provide an attractive model that represents an important limiting case for real stirred tank reactors and mixers.

As an example of a mixing process, we consider the system illustrated in Figure 8‐1. The volumetric flow rate entering and leaving the system is assumed Figure 8‐1. Perfectly mixed stirred tank

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