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