Batch Distillation
SHR Chapter 13
1
BatchDistillation.key - April 3, 2014
SHR §13.1
Simple Binary Batch Distillation Analysis
D, y
W, x
QB
D
Distillate molar flow rate
W
Moles of “residue” in the still
x
Mole fraction of light key in
the still (assumed spatially
homogeneous)
y
Mole fraction of light key in
the distillate (assumed
spatially homogeneous)
Light key mole balance:
d (W x)
= Dy
dt
dx
dW
W
+x
= Dy
dt
dt
W
dx
dW
=
(y
dt
dt
Z
x
x0
dx
y
x
Relates time rate of change of
x to time rate of change of W.
x)
W dx = dW (y
=
x)
Z
Overall mole balance:
dW
= D
dt
W
W0
Relates change in x to change in W.
No connection to time evolution.
Think “projectile trajectory.”
dW
W
= ln
W
W0
Sometimes this is
referred to as “phase
space” or “state space”
Rayleigh
Equation
To integrate the LHS term, we need to know y(x).
2
BatchDistillation.key - April 3, 2014
Z
x
dx
y
x0
x
=
Z
W
W0
dW
W
= ln
W
W0
Rayleigh
Equation
Consider a single-stage with no reflux.
Option 1: assume that K = y/x is constant.
Z
x
x0
dx
= ln
x (K 1)
✓
W
W0
◆
1
K
1
ln
✓
x
x0
◆
= ln
✓
W
W0
◆
(constant K)
Option 2: assume that α is constant.
KA
↵⌘
=
KB
Z
x
x0
✓
1 + x(↵
↵x
yA/xA
yB/xB
1)
=
x
◆
y/x
y=
(1 y)/(1 x)
1
dx = ln
✓
W
W0
◆
↵x
1 + x (↵
ln
✓
W0
W
◆
1)
=
1
↵
 ⇣ ⌘
✓
◆
x0
1 x
ln
+ ↵ ln
1
x
1 x0
(constant α)
Option 3:
if T-x-y data are available, integrate
numerically using the trapezoid rule.
3
BatchDistillation.key - April 3, 2014
SHR Example 13.1
Example: Constant Boilup
A batch still is loaded with 100 kmol of a binary mixture
of 50 mol% benzene in toluene. As a function of time,
make plots of: (a) still temperature, (b) instantaneous vapor composition, (c) still-pot composition, and (d) average total-distillate composition. Assume a constant boilup rate, 10 kmol/h of product, and
a constant α of 2.41 at a pressure of 101.3 kPa (1 atm).
W0
W
◆
=
1
↵
 ⇣ ⌘
✓
◆
x0
1 x
ln
+ ↵ ln
1
x
1 x0
It is easier to solve for W(x) than x(W)...
How do we connect W and x to t?
dW
=
dt
D
constant boilup...
W0
Dt ) t =
W = W0
Total amount distilled: W0 - W
Amount of light key distilled: x0W0 - xW
x0 W0 xW average distillate composition
y avg =
(see SHR eq. (13-6))
W0 W
1
x
W
D
W/W0
yavg
y
Mole Fraction
0.8
110
T (K)
ln
✓
From the T-x-y data in
105 SHR Table 13.1, we get T
from x.
100
0.6
0.4
0.2
95
0
2
4
t(h)
6
8
0
0
10
4
2
4
t(h)
6
8
10
BatchDistillation.key - April 3, 2014
SHR Example 13.2
Example: Using Tabular Data
A batch still is loaded with 100 kmol of a binary mixture of 50
mol% benzene in toluene. As a function of time, make plots of: (a) still temperature, (b) instantaneous vapor composition, (c) still-pot composition, and (d) average total-distillate composition. Assume a constant boilup rate, 10 kmol/h of product, and use
the T-x-y data in SHR Table 13.1.
Z
Z
x
x0
b
a
dx
y
W
Different approach to get this term.
= ln
x
W0 Otherwise it is the same as the last example.
f (x)dx ⇡
b a
2
[f (a) + f (b)]
x
y
T
0.0
0.0
111
0.10
0.208
105.3
0.20
0.372
101.5
0.30
0.507
98
0.40
0.612
95.1
0.50
0.713
92.3
0.60
0.791
89.7
0.70
0.857
87.3
0.80
0.912
85
0.90
0.959
82.7
0.95
0.980
81.4
1.0
1.0
80.1
trapz in matlab...
x decreases in
time from 0.5.
5
BatchDistillation.key - April 3, 2014
SHR §13.2.1
Batch Distillation with Constant Reflux Ratio
Z
Assumptions:
• McCabe-Thiele analysis (see notes on binary
distillation)
• Total condenser
• Constant molar overflow
• Only heat transfer is in condenser and
reboiler (not in the rectifying section)
• Constant V and R (implies constant L and D)
V, yD
L, xD
xW
x W0
dxW
W
= ln
xD xW
W0
came from the
mole balance
(see slide 2)
xD is a nontrivial function of xW, depending on
# stages, reflux ratio, equilibrium, etc.
That means that this integral is not easy to get!
Z
dW
dt
=
L
dW
=
(L
W
=
W0 + (L
=
W0 + V (L/V
V
W0
t
R+1
V =
D
W
V )t
D, xD
=
W0 W
V (1 L/V )
R+1
=
(W0 W )
V
Time to reach
amount W in still.
t=
W0
V )t
1) t
Concept:
W, xW
QB
• Use McCabe-Thiele to get the relationship between xD and xW.
• Given the graphically-obtained integrand, carry out the
integration to find W(xW).
• Choose xW - find W - find t.
6
BatchDistillation.key - April 3, 2014
McCabe-Thiele to get (xD - xW)
Assume that the number of theoretical stages
(Nt) and the reflux ratio (R) are known.
L
R
=
V
R+1
Z
xW
x W0
dxW
W
= ln
xD xW
W0
1.Choose xW.
adjust xD
2. Construct an operating line with the prescribed
R such that in Nt stages, we hit the intersection
of the operating line and the 45º line. This
defines xD for the xW chosen in step 1.
3. Repeat steps 1-2 for the desired range of xW,
choosing “enough” points to do step 4.
4. Numerically integrate (trapezoid rule) to find
ln(W/W0).
5. From W, find t.
W0 W
V (1 L/V )
R+1
=
(W0 W )
V
xW(t)
t=
Calculate average
x W0 W 0
avg
distillate composition xD =
W0
over time t as before:
xW W
W
7
BatchDistillation.key - April 3, 2014
SHR §13.2.2
Batch Distillation with Constant xD
Idea: constant V and vary R in time to obtain a desired xD.
• More difficult than constant R, but typically more desirable.
• Implies that D is changing in time.
xavg
D =
x W0 W 0
W0
xW W
xD
) W = W0
W
xD
x W0
xW
dW/dt, assume constant xD and rearrange...
(V
dW
xD
= W0
dt
(xD
xW0 dxW
2
xW ) dt
xD
xW0 dxW
2
xW ) dt
L) = W0
dt = V W0 (xD
t = V W0 (xD
(xD
x W0 )
x W0 )
dxW
1
Z
L
V
(xD
xW )
xW
x W0
2
Recall that xD is known
(constant) and V is constant, but
L/V is changing depending on xW.
dxW
1
L
V
(xD
xW )
8
2
Use McCabe-Thiele to get values of
the integrand graphically.
BatchDistillation.key - April 3, 2014
t = V W0 (xD
x W0 )
Z
xW
x W0
dxW
1
L
V
(xD
xW )
2
W = W0

xD
xD
x W0
xW
1.0
1. Choose xW.
R(t)
y
2. Construct an operating line with
R such that in Nt stages, we hit
the intersection of the operating
line and the 45º line. This defines
R for the xW chosen in step 1.
3. Repeat steps 1-2 for the desired
range of xW, choosing “enough”
points to do step 4.
4. Numerically integrate (trapezoid
rule) to find t for a given xW.
x=
y
xW(t)
0
9
xW2
x
xW1
xD
1.0
BatchDistillation.key - April 3, 2014