# 8.13: Untitled Page 202

- Page ID
- 18335

## Chapter 8

*Figure 8‐11*. Batch distillation unit

The control volume illustrated in Figure 8‐11 is fixed in space and can be separated into the volume of the liquid (the phase ) and the volume of the vapor (the phase ) according to

*V*

*V *( *t*) *V *( *t*)

(8‐60)

Under normal circumstances there will be no chemical reactions in a distillation process, and we can express the macroscopic mole balance for species *A* as *d*

*c dV *

*c*

*dA * 0

**v n**

(8‐61)

*A*

*A A*

*dt V*

*A*

In addition to the mole balance for species *A*, we will need either the mole balance for species *B* or the total mole balance. The latter is more convenient in this particular case, and we express it as (see Sec. 4.4)

*d*

*c dV *

*c*

*dA * 0

**v n**

(8‐62)

*dt V*

*A*

377

For the control volume shown in Figure 8‐11, the molar flux is zero everywhere except at the exit of the unit and Eq. 8‐61 takes the form

*d *

*c * *dV *

*c * *dV *

*c *

(8‐63)

*dt*

**v n** *dA*

0

*A*

*A*

*A*

*A*

*V *( *t*)

*V *( *t*)

*A exit*

Here we have explicitly identified the control volume as consisting of the volume of the liquid ( phase ) and the volume of the vapor ( phase ) At the exit of the control volume, we can ignore diffusive effects and replace **v** **n ** with

*A*

**v**

**n **, and the concentration in both the liquid and vapor phases can be

represented in terms of mole fractions so that Eq. 8‐63 takes the form

*d *

*x*

*c dV *

*y c* *dV *

*y c*

(8‐64)

*dt*

**v**

**n ** *dA*

0

*A*

*A*

*A*

*V *( *t*)

*V *( *t*)

*A exit*

If the total molar concentrations, *c* and *c* , can be treated as constants, this result can be expressed as

*d*

*d*

*x *

*M *

*y * *M*

*y c* *dA *

(8‐65)

*dt*

*dt*

**v**

**n**

0

*A*

*A*

*A*

*A exit*

in which *x * and *y * are defined by

*A*

*A*

1

1

*x *

*x dV , *

*y *

*y dV *

(8‐66)

*A*

*V *( *t*)

*A*

*A*

*V *( *t*)

*A*

*V *( *t*)

*V *( *t*)

In Eq. 8‐65 we have used *M* and *M* to represents the total number of moles in the liquid and vapor phases respectively. We can simplify Eq. 8‐65 by imposing the restriction

*d*

*d*

*y * *M *

*x * *M *

(8‐67)

*A*

*A *

*dt*

*dt*

since *c* is generally much, much less than *c* . Given this restriction, Eq. 8‐65

takes the form

378