# 8.21: Untitled Page 210

- Page ID
- 18343

## Chapter 8

The analysis begins with the fixed control volume shown in Figure 8.16 and the general macroscopic balance given by

*d*

*c dV *

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

*R dV , *

*D * *A, B, C*

(1)

*D*

*D D*

*D*

*dt V*

*A*

*V*

The moles of species *A* (t‐butyl alcohol) in the phase can be neglected, thus the macroscopic balance for this species takes the form

*d*

t‐butyl alcohol:

*c*

*dV*

*R*

*dV *

(2)

*A*

*A*

*dt *

*V *( *t*)

*V *( *t*)

and in terms of average values for the concentration and the net rate of production of species *A* we have

*d*

t‐butyl alcohol:

*c *

*V *( *t*)

*R *

*V *( *t*)

(3)

*A*

*A*

*dt*

If we also assume that the moles of species *B* (isobutylene) and species *C* (water) are negligible in the phase , the macroscopic balances for these species take the form

*d*

isobutylene:

*c *

*V *( *t*) *M * *R *

*V *( *t*)

(4)

*B*

*B*

*B*

*dt*

*d*

water:

*c *

*V *( *t*)

*R *

*V *( *t*)

(5)

*C*

*C*

*dt*

This indicates that alcohol and water are retained in the system by the condenser while the isobutylene leaves the system at a molar rate given by *M*

. The initial

*B*

conditions for the three molecular species are given by

IC.

o

*c * *c , *

*t * 0

(6a)

*A*

*A*

IC.

*c * 0 *, *

*t * 0

(6b)

*B*

IC.

*c * 0 *, *

*t * 0

(6c)

*C*

Since the molar rates of reaction are related by

*R*

*R*

*, *

and

*R*

*R *

(7)

*C*

*A*

*B*

*A*

393

we need only be concerned with the rate of reaction of the *t*‐butyl alcohol. If we treat the reactor as *perfectly mixed*, the mole balance for t‐butyl alcohol can be expressed as

*dc*

*c*

*A*

*A*

*dV *( *t*)

*R*

(8)

*A*

*dt*

*V *( *t*)

*dt*

This indicates that we need to know both *cA* and *V* as functions of time in order to obtain *experimental values* of *RA* . The volume of fluid in the reactor can be expressed as

*V *( *t*) *n*

v

*n * v *n * v

(9)

*A* *A*

*B* *B*

*C * *C*

in which *n*

*n*

*n*

*A* ,

and

represent the moles of species *A*, *B* and *C* in the *B*

*C *

phase and v , v and v represent the partial molar volumes.

*A*

*B*

*C*

To develop a useful expression for *RA* , *assume* that the liquid mixture is ideal so that the partial molar volumes are constant. In addition, *assume* that the moles of species *B* in the liquid phase are negligible. On the basis of these assumptions, show that Eq. 8 can be expressed as

*d c*

*c*

*A*

*A* v

v

*A*

*C *

*R*

1

(10)

*A*

*dt*

1 *c*

*A* v

v

*A*

*C *

This form is especially useful for the interpretation of *initial rate data*, i.e., experimental data can be used to determine both *c*

*dc*

*/ dt*

*t *

*A* and

at

and

*A*

0

this provides an experimental determination of *RA* for the initial conditions associated with the experiment.

An alternate approach (Gates and Sherman, 1975) to the determination of *R*

is to measure the molar flow rate of species *B* that leaves the reactor in the *A*

phase and relate that quantity to the rate of reaction.

*Section* 8.4

8‐17. When Eqs. 8‐41 and 8‐42 are valid, Eq. 8‐43 represents a valid result for the chemostat shown in Figure 8‐10. One can divide this equation by a constant, *m*

, to obtain Eq. 8‐47; however, the average mass of a cell, *m*

, in the

*cell*

*cell*

chemostat may not be the average mass of a cell in the incoming stream. If *m* *cell*

394