8.14: Untitled Page 203
- Page ID
- 18336
Chapter 8
d
x M
y c dA
(8‐68)
dt
v
n
0
A
A
Aexit
and we can express the flux at the exit in the traditional form to obtain d
x M y c Q 0
(8‐69)
A
A exit
dt
This represents the governing equation for x and it is restricted to cases for A
which c c
.
In addition to x , there are other unknown terms in Eq. 8‐69, and the total A
mole balance will provide information about one of these. Returning to Eq. 8‐62
we apply that result to the control volume illustrated in Figure 8‐11 to obtain
d
c dV
c dV
c
(8‐70)
dt
v n dA 0
V ( t)
V( t)
Aexit
At the exit of the control volume, we again ignore diffusive effects and replace
v
n with v
n so that this result takes the form
d
M M
c Q
0
(8‐71)
dt
At this point, we again impose the restriction that c c
which allows us to
simplify this result to the form
d M
c Q
0
(8‐72)
dt
We can use this result to eliminate c Q
from Eq. 8‐69 so that the mole balance
for species A takes the form
d M
d
x M y
0
(8‐73)
A
A exit
dt
dt
At this point we have a single equation and three unknowns: x , M
A
and
y
, and our analysis has been only moderately restricted by the condition A exit
that c c
. We have yet to make use of the equilibrium relation indicated by
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Eq. 8‐59, and to be very precise in the next step in our analysis we repeat that equilibrium relation according to
Equilibrium relation:
y
x ,
at the vapor ‐ liquid interface
(8‐74)
A
AB A
In our macroscopic balance analysis, we are confronted with the mole fractions indicated by x and y
, and the values of these mole fractions at the
A
A exit
vapor‐liquid interface illustrated in Figure 8‐11 are not available to us. Knowledge of x and y at the interface can only be obtained by a detailed analysis of A
A
the diffusive transport (Bird et al., 2002) that is responsible for the separation that occurs in batch distillation. In order to proceed with an approximate solution to the batch distillation process, we replace Eq. 8‐74 with
Process equilibrium relation:
y
x
(8‐75)
A exit
eff
A
Here we note that Eqs. 8‐74 and 8‐75 are analogous to Eqs. 5‐49 and 5‐50 if the approximation
is valid. The process equilibrium relation suggested by eff
AB
Eq. 8‐75 may be acceptable if the batch distillation process is slow enough, but we do not know what is meant by slow enough without a more detailed theoretical analysis or an experimental study in which theory can be compared with experiment.
Keeping in mind the uncertainty associated with Eq. 8‐75, we use Eq. 8‐75 in Eq. 8‐73 to obtain
dM
d x
A
M
(1 ) x
0
(8‐76)
eff
A
dt
dt
The initial conditions for the mole fraction, x , and the number of moles in the A
still, M( t) , are given by
I.C.1
o
x x ,
t 0
(8‐77)
A
A
I.C.2
o
M
M ,
t 0
(8‐78)
At this point we have a single differential equation and two unknowns, x and A
M( t) . Obviously we cannot determine both of these quantities as a function of time unless some additional information is given. For example, if M ( t)
were
specified as a function of time we could use Eq. 8‐76 to determine x as a A
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