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Chapter 2
Distillation Theory
by
Ivar J. Halvorsen and Sigurd Skogestad
Norwegian University of Science and Technology
Department of Chemical Engineering
7491 Trondheim, Norway
This is a revised version of an article accepted
for publication in the Encyclopedia of Separation
Science by Academic Press Ltd. (submitted in
1999). This article gives some of the basics of
distillation theory and its purpose is to provide
basic understanding and some tools for simple hand
calculations of distillation columns. The methods
presented here can be used for simple estimates and
to check more rigorous computations.
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2.1
Introduction
Distillation is a very old separation technology for separating liquid mixtures that
can be traced back to the chemists in Alexandria in the first century A.D. Today
distillation is the most important industrial separation technology. It is particu-
larly well suited for high purity separations since any degree of separation can be
obtained with a fixed energy consumption by increasing the number of equilib-
rium stages.
To describe the degree of separation between two components in a column or in
a column section, we introduce the separation factor:
(2.1)
where here x denotes mole fraction of a component, subscript L denotes light
component, H heavy component, T denotes the top of the section, and B the
bottom.
It is relatively straightforward to derive models of distillation columns based on
almost any degree of detail, and also to use such models to simulate the behaviour
on a computer. However, such simulations may be time consuming and often pro-
vide limited insight. The objective of this article is to provide analytical
expressions that are useful for understanding the fundamentals of distillation and
which may be used to guide and check more detailed simulations:
• Minimum energy requirement and corresponding internal flow
requirements.
• Minimum number of stages.
• Simple expressions for the separation factor.
The derivation of analytical expressions requires the assumptions of:
• Equilibrium stages.
• Constant relative volatility.
• Constant molar flows.
These assumptions may seem restrictive, but they are actually satisfied for many
real systems, and in any case the resulting expressions yield invalueable insights,
also for systems where the approximations do not hold.
S
x
L
x
H
⁄
(
)
T
x
L
x
H
⁄
(
)
B
------------------------
=
2.2 Fundamentals
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2.2
Fundamentals
2.2.1
The Equilibrium Stage Concept
The equilibrium (theoretical) stage concept (see Figure 2.1) is central in distilla-
tion. Here we assume vapour-liquid equilibrium (VLE) on each stage and that the
liquid is sent to the stage below and the vapour to the stage above. For some trayed
columns this may be a reasonable description of the actual physics, but it is cer-
tainly not for a packed column. Nevertheless, it is established that calculations
based on the equilibrium stage concept (with the number of stages adjusted appro-
priately) fits data from most real columns very well, even packed columns.
Figure 2.1: Equilibrium stage concept.
One may refine the equilibrium stage concept, e.g. by introducing back mixing or
a Murphee efficiency factor for the equilibrium, but these “fixes” have often rela-
tively little theoretical justification, and are not used in this article.
For practical calculations, the critical step is usually not the modelling of the
stages, but to obtain a good description of the VLE. In this area there has been
significant advances in the last 25 years, especially after the introduction of equa-
tions of state for VLE prediction. However, here we will use simpler VLE models
(constant relative volatility) which apply to relatively ideal mixtures.
2.2.2
Vapour-Liquid Equilibrium (VLE)
In a two-phase system (PH=2) with N
c
non-reacting components, the state is com-
pletely determined by N
c
degrees of freedom (f), according to Gibb’s phase rule;
y
x
P
T
Vapour phase
Liquid phase
Saturated vapour leaving the stage
Saturated liquid leaving the stage
with equilibrium mole fraction x
with equilibrium mole fraction y
and enthalpy h
L
(T,x)
and molar enthalpy h
V
(T,x)
Liquid entering the stage (x
L,in
,h
L,in
)
Vapour entering the stage (y
V,in
,h
V,in
)
Perfect mixing
in each phase
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(2.2)
If the pressure (P) and N
c
-1 liquid compositions or mole fractions (x) are used as
degrees of freedom, then the mole fractions (y) in the vapour phase and the tem-
perature (T) are determined, provided that two phases are present. The general
VLE relation can then be written:
(2.3)
Here we have introduced the mole fractions x and y in the liquid an vapour phases
respectively, and we trivially have
and
In ideal mixtures, the vapour liquid equilibrium can be derived from Raoult’s law
which states that the partial pressure p
i
of a component (i) in the vapour phase is
proportional to the vapour pressure (
) of the pure component (which is a func-
tion of temperature only:
) and the liquid mole fraction (x
i
):
(2.4)
For an ideas gas, according to Dalton’s law, the partial pressure of a component
is proportional to the mole fraction:
, and since the total pressure
we derive:
(2.5)
The following empirical formula is frequently used for computing the pure com-
ponent vapour pressure:
(2.6)
f
N
c
2 P
– H
+
=
y
1
y
2
…
y
N
c
1
–
T
, , ,
,
[
]
f P x
1
x
2
…
x
N
c
1
–
, , , ,
(
)
=
y T
,
[
]
f P x
,
(
)
=
x
i
i
1
=
n
∑
1
=
y
i
i
1
=
n
∑
1
=
p
i
o
p
i
o
p
i
o
T
( )
=
p
i
x
i
p
i
o
T
( )
=
p
i
y
i
P
=
P
p
1
p
2
…
p
N
c
+
+
+
p
i
i
∑
x
i
p
i
o
T
( )
i
∑
=
=
=
y
i
x
i
p
i
o
P
------
x
i
p
i
o
T
( )
x
i
p
i
o
T
( )
i
∑
---------------------------
=
=
p
o
T
( )
ln
a
b
c
T
+
------------
d
T
( )
ln
eT
f
+
+
+
≈
2.2 Fundamentals
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The coefficients are listed in component property data bases. The case with d=e=0
is called the Antoine equation.
2.2.3
K-values and Relative Volatility
The K-value for a component i is defined as:
. The K-value is some-
times called the equilibrium “constant”, but this is misleading as it depends
strongly on temperature and pressure (or composition).
The relative volatility between components i and j is defined as:
(2.7)
For ideal mixtures that satisfy Raoult’s law we have:
(2.8)
Here
depends on temperature so the K-values will actually be constant
only close to the column ends where the temperature is relatively constant. On the
other hand the ratio
is much less dependent on temperature which
makes the relative volatility very attractive for computations. For ideal mixtures,
a geometric average of the relative volatilities for the highest and lowest temper-
ature in the column usually gives sufficient accuracy in the computations:
.
We usually select a common reference component r (usually the least volatile (or
“heavy”) component), and define:
(2.9)
The VLE relationship (2.5) then becomes:
(2.10)
K
i
y
i
x
i
⁄
=
α
ij
y
i
x
i
⁄
(
)
y
j
x
j
⁄
(
)
------------------
K
i
K
j
------
=
=
α
ij
y
i
x
i
⁄
(
)
y
j
x
j
⁄
(
)
------------------
K
i
K
j
------
p
i
o
T
( )
p
j
o
T
( )
---------------
=
=
=
p
i
o
T
( )
p
i
o
T
( )
p
j
o
T
( )
⁄
α
ij
α
ij top
,
α
ij bottom
,
⋅
=
α
i
α
ir
p
i
o
T
( )
p
r
o
T
( )
⁄
=
=
y
i
α
i
x
i
α
i
x
i
i
∑
-----------------
=
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For a binary mixture we usually omit the component index for the light compo-
nent, i.e. we write x=x
1
(light component) and x
2
=1-x (heavy component). Then
the VLE relationship becomes:
(2.11)
This equilibrium curve is illustrated in Figure 2.2:
Figure 2.2: VLE for ideal binary mixture:
The difference y-x determine the amount of separation that can be achieved on a
stage. Large relative volatilities implies large differences in boiling points and
easy separation. Close boiling points implies relative volatility closer to unity, as
shown below quantitatively.
2.2.4
Estimating the Relative Volatility From Boiling Point Data
The Clapeyron equation relates the vapour pressure temperature dependency to
the specific heat of vaporization (
) and volume change between liquid and
vapour phase (
):
y
α
x
1
α
1
–
(
)
x
+
------------------------------
=
Increasing
α
Mole fraction
0
1
1
x
y
of light component
in liquid phase
Mole fraction
of light component
in vapour phase
α=1
y
α
x
1
α
1
–
(
)
x
+
------------------------------
=
H
vap
∆
V
vap
∆
2.2 Fundamentals
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(2.12)
If we assume an ideal gas phase, and that the gas volume is much larger than the
liquid volume, then
, and integration of Clapeyrons equation
from temperature T
bi
(boiling point at pressure P
ref
) to temperature T (at pressure
) gives, when
is assumed constant:
(2.13)
This gives us the Antoine coefficients:
.
In most cases
. For an ideal mixture that satisfies Raoult’s law we
have
and we derive:
(2.14)
We see that the temperature dependency of the relative volatility arises from dif-
ferent specific heat of vaporization. For similar values (
), the
expression simplifies to:
(2.15)
Here we may use the geometric average also for the heat of vaporization:
(2.16)
d p
o
T
( )
dT
------------------
H
vap
∆
T
( )
T V
vap
T
( )
∆
----------------------------
=
V
vap
∆
RT P
⁄
≈
p
i
o
H
i
vap
∆
p
i
o
ln
∆
H
i
vap
R
----------------
1
T
bi
--------
P
ref
ln
+
∆
H
i
vap
R
----------------
–
T
-------------------------
+
≈
a
i
∆
H
i
vap
R
----------------
1
T
bi
--------
P
ref
ln
+
=
b
i
,
∆
H
i
vap
R
----------------
–
=
c
i
,
0
=
P
ref
1 atm
=
α
ij
p
i
o
T
( )
p
j
o
T
( )
⁄
=
α
ij
ln
∆
H
i
vap
R
----------------
1
T
bi
--------
∆
H
j
vap
R
----------------
1
T
bj
--------
–
∆
H
j
vap
∆
H
i
vap
–
RT
---------------------------------------
+
=
∆
H
i
vap
∆
H
j
vap
≈
α
ij
ln
≈ ∆
H
vap
RT
b
------------------
β
T
bj
T
bi
–
T
b
----------------------
where T
b
T
bi
T
bj
=
∆
H
vap
∆
H
i
vap
T
bi
(
) ∆
H
j
vap
T
bj
(
)
⋅
=
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This results in a rough estimate of the relative volatility
, based on the boiling
points only:
where
(2.17)
If we do not know
, a typical value
can be used for many cases.
Example: For methanol (L) and n-propanol (H), we have
and
and the heats of vaporization at their boiling points
are 35.3 kJ/mol and 41.8 kJ/mol respectively. Thus
and
.
This gives
and
which is a bit lower than the experimental
value.
2.2.5
Material Balance on a Distillation Stage
Based on the equilibrium stage concept, a distillation column section is modelled
as shown in Figure 2.3: Note that we choose to number the stages starting from
the bottom of the column. We denote L
n
and V
n
as the total liquid- and vapour
molar flow rates leaving stage n (and entering stages n-1 and n+1, respectively).
α
ij
α
ij
e
β
T
bj
T
bi
–
(
)
T
b
⁄
≈
β
∆
H
vap
RT
B
----------------
=
∆
H
vap
β
13
≈
T
BL
337.8K
=
T
BH
370.4K
=
T
B
337.8 370.4
⋅
354K
=
=
H
vap
∆
35.3 41.8
⋅
38.4
=
=
β
∆
H
vap
RT
B
⁄
38.4
8.83 354
⋅
(
)
⁄
13.1
=
=
=
α
e
13.1 32.6
⋅
354
⁄
3.34
≈
≈
2.2 Fundamentals
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We assume perfect mixing in both phases inside a stage. The mole fraction of spe-
cies i in the vapour leaving the stage with V
n
is y
i,n
, and the mole fraction in L
n
is
x
i,n
.
Figure 2.3: Distillation column section modelled as a set of connected
equilibrium stages
The material balance for component i at stage n then becomes (in [mol i/sec]):
(2.18)
where N
i,n
in the number of moles of component i on stage n. In the following we
will consider steady state operation, i.e:
.
It is convenient to define the net material flow (w
i
) of component i upwards from
stage n to n+1 [mol i/sec]:
(2.19)
At steady state, this net flow has to be the same through all stages in a column sec-
tion, i.e.
.
The material flow equation is usually rewritten to relate the vapour composition
(y
n
) on one stage to the liquid composition on the stage above (x
n+1
):
L
n+1
L
n
V
n-1
V
n
y
n
x
n
y
n+1
x
n+1
y
n-1
x
n-1
Stage n-1
Stage n
Stage n+1
t
d
dN
i n
,
L
n
1
+
x
i n
1
+
,
V
n
y
i n
,
–
(
)
L
n
x
i n
,
V
n
1
–
y
i n
1
–
,
–
(
)
–
=
t
d
dN
i n
,
0
=
w
i n
,
V
n
y
i n
,
L
n
1
+
x
i n
1
+
,
–
=
w
i n
,
w
i n
1
+
,
w
i
=
=
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(2.20)
The resulting curve is known as the operating line. Combined with the VLE rela-
tionship (equilibrium line) this enables us to compute all the stage compositions
when we know the flows in the system. This is illustrated in Figure 2.4, and forms
the basis of the McCabe-Thiele approach.
Figure 2.4: Combining the VLE and the operating line to compute mole fractions
in a section of equilibrium stages.
2.2.6
Assumption about Constant Molar Flows
In a column section, we may very often use the assumption about constant molar
flows. That is, we assume
[mol/s] and
[mol/
s]. This assumption is reasonable for ideal mixtures when the components have
similar molar heat of vaporization. An important implication is that the operating
y
i n
,
L
n
1
+
V
n
------------- x
i n
1
+
,
1
V
n
------w
i
+
=
x
n-1
x
n
x
n
y
n-1
y
n
(2) Material balance
(1) VLE: y=f(x)
operating line
y=(L/V)x+w/V
Use (1)
Use (2)
(1)
(2)
x
y
L
n
L
n
1
+
L
=
=
V
n
1
–
V
n
V
=
=
2.3 The Continuous Distillation Column
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line is then a straight line for a given section, i.e
.
This makes computations much simpler since the internal flows (L and V) do not
depend on compositions.
2.3
The Continuous Distillation Column
We here study the simple two-product continuous distillation column in Figure
2.5: We will first limit ourselves to a binary feed mixture, and the component
index is omitted, so the mole fractions (x,y,z) we refer to the light component. The
column has N equilibrium stages, with the reboiler as stage number 1. The feed
with total molar flow rate F [mol/sec] and mole fraction z enters at stage N
F
.
Figure 2.5: An ordinary continuous two-product distillation column
y
i n
,
L V
⁄
(
)
x
i n
1
+
,
w
i
V
⁄
+
=
F
z
q
D
B
x
D
x
B
Q
r
Q
c
Rectifying
section
Stripping
section
x
F,
y
F
Condenser
Reboiler
L
T
Stage 2
V
T
L
T
Stage N
Feed stage N
F
V
B
L
B
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The section above the feed stage is denoted the rectifying section, or just the top
section, and the most volatile component is enriched upwards towards the distil-
late product outlet (D). The stripping section, or the bottom section, is below the
feed, in which the least volatile component is enriched towards the bottoms prod-
uct outlet (B). (The least volatile component is “stripped” out.) Heat is supplied
in the reboiler and removed in the condenser, and we do not consider any heat loss
along the column.
The feed liquid fraction q describes the change in liquid and vapour flow rates at
the feed stage:
(2.21)
The liquid fraction is related to the feed enthalpy (h
F
) as follows:
(2.22)
When we assume constant molar flows in each section, we get the following rela-
tionships for the flows:
(2.23)
2.3.1
Degrees of Freedom in Operation of a Distillation Column
With a given feed (F,z and q), and column pressure (P), we have only 2 degrees
of freedom in operation of the two-product column in Figure 2.5, independent of
the number of components in the feed. This may be a bit confusing if we think
about degrees of freedom as in Gibb’s phase rule, but in this context Gibb’s rule
does not apply since it relates the thermodynamic degrees of freedom inside a sin-
gle equilibrium stage.
L
F
∆
qF
=
V
F
∆
1
q
–
(
)
F
=
q
h
V sat
,
h
F
–
H
vap
∆
---------------------------
1
>
Subcooled liquid
1
=
Saturated liquid
0
q
1
< <
Liquid and vapour
0
=
Saturated vapour
0
<
Superheated vapour
=
=
V
T
V
B
1
q
–
(
)
F
+
=
L
B
L
T
qF
+
=
D
V
T
L
T
–
=
B
L
B
V
B
–
=
2.3 The Continuous Distillation Column
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This implies that if we know, for example, the reflux (L
T
) and vapour (V
B
) flow
rate into the column, all states on all stages and in both products are completely
determined.
2.3.2
External and Internal Flows
The overall mass balance and component mass balance is given by:
(2.24)
Here z is the mole fraction of light component in the feed, and x
D
and x
B
are the
product compositions. For sharp splits with x
D
≈
1 and x
B
≈
0 we then have that
D=zF. In words, we must adjust the product split D/F such that the distillate flow
equals the amount of light component in the feed. Any deviation from this value
will result in large changes in product composition. This is a very important
insight for practical operation.
Example: Consider a column with z=0.5, x
D
=0.99, x
B
=0.01 (all these refer
to the mole fraction of light component) and D/F = B/F = 0.5. To simplify
the discussion set F=1 [mol/sec]. Now consider a 20% increase in the dis-
tillate D from 0.50 to 0.6 [mol/sec]. This will have a drastic effect on
composition. Since the total amount of light component available in the
feed is z = 0.5 [mol/sec], at least 0.1 [mol/sec] of the distillate must now be
heavy component, so the amount mole fraction of light component in the
distillate is now at its best 0.5/0.6 = 0.833. In other words, the amount of
heavy component in the distillate will increase at least by a factor of 16.7
(from 1% to 16.7%).
Thus, we generally have that a change in external flows (D/F and B/F) has a large
effect on composition, at least for sharp splits, because any significant deviation
in D/F from z implies large changes in composition. On the other hand, the effect
of changes in the internal flows (L and V) are much smaller.
2.3.3
McCabe-Thiele Diagram (Constant Molar Flows, but any
VLE)
The McCabe-Thiele diagram where y is plotted as a function x along the column
provides an insightful graphical solution to the combined mass balance (“opera-
tion line”) and VLE (“equilibrium line”) equations. It is mainly used for binary
mixtures. It is often used to find the number of theoretical stages for mixtures with
constant molar flows. The equilibrium relationship
(y as a function
of x at the stages) may be nonideal. With constant molar flow, L and V are con-
F
D
B
+
=
Fz
Dx
D
Bx
B
+
=
y
n
f x
n
( )
=
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stant within each section and the operating lines (y as a function of x between the
stages) are straight. In the top section the net transport of light component
. Inserted into the material balance equation (2.20) we obtain the oper-
ating line for the top section, and we have a similar expression for the bottom
section:
(2.25)
A typical McCabe-Thiele diagram is shown in Figure 2.6:
Figure 2.6: McCabe-Thiele Diagram with an optimally located feed.
The optimal feed stage is at the intersection of the two operating lines and the feed
stage composition (x
F
,y
F
) is then equal to the composition of the flashed feed mix-
ture. We have that
. For q=1 (liquid feed) we find
and for q=0 (vapour feed) we find
(in the other cases we must solve the
w
x
D
D
=
Top:
y
n
L
V
----
T
x
n
1
+
x
D
–
(
)
x
D
+
=
Bottom: y
n
L
V
----
B
x
n
1
+
x
B
–
(
)
x
B
+
=
0
1
1
x
F
y
y=x
x
D
x
B
y
F
x
Top section operating line
VLE y=f(x)
Bottom section
Optimal feed
Reboiler
Condenser
z
The intersection of the
Slope (L
T
/V
T
)
operating line
Slope (L
B
/V
B
)
Slope q/(q-1)
along the “q-line”.
stage location
operating lines is found
z
qx
F
1
q
–
(
)
y
F
+
=
x
F
z
=
y
F
z
=
2.3 The Continuous Distillation Column
33
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equation together with the VLE). The pinch, which occurs at one side of the feed
stage if the feed is not optimally located, is easily understood from the McCabe-
Thiele diagram as shown in Figure 2.8:
2.3.4
Typical Column Profiles
—
Pinch
An example of a column composition profile is shown in Figure 2.7 for a column
with z=0.5,
=1.5, N=40, N
F
=21 (counted from the bottom, including the
reboiler), y
D
=0.90, x
B
=0.002. This is a case were the feed stage is not optimally
located, as seen from the presence of a pinch zone (a zone of constant composi-
tion) above the stage. The corresponding McCabe-Thiele diagram is shown in
Figure 2.8: We see that the feed stage is not located at the intersection of the two
operating lines, and that there is a pinch zone above the feed, but not below.
Figure 2.7: Composition profile (x
L
,x
H
) for case with non-optimal feed location.
α
0
5
10
15
20
25
30
35
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bottom Stages Top
Molfraction
α
=1.50
z=0.50
q=1.00
N=40
N
F
=21
x
DH
=0.1000
x
BL
=0.0020
Light key
Heavy key
34
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Figure 2.8: McCabe-Thiele diagram for the same example as in Figure 2.7:
Observe that the feed stage location is not optimal.
2.4
Simple Design Equations
2.4.1
Minimum Number of Stages
—
Infinite Energy
The minimum number of stages for a given separation (or equivalently, the max-
imum separation for a given number of stages) is obtained with infinite internal
flows (infinite energy) per unit feed. (This always holds for single-feed columns
and ideal mixtures, but may not hold, for example, for extractive distillation with
two feed streams.)
With infinite internal flows (“total reflux”) L
n
/F=
∞
and V
n
/F=
∞
, a material bal-
ance across any part of the column gives V
n
= L
n+1
, and similarly a material
balance for any component gives V
n
y
n
= L
n+1
x
n+1
. Thus, y
n
= x
n+1
, and with con-
stant relative volatility we have:
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Vapour Molfraction (y)
Liquid Molfraction (x)
α
=1.50
z=0.50
q=1.00
N=40
N
F
=21
x
DH
=0.1000
x
BL
=0.0020
Optimal feed
stage
Actual
feed stage
2.4 Simple Design Equations
35
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(2.26)
For a column or column section with N stages, repeated use of this relation gives
directly Fenske’s formula for the overall separation factor:
(2.27)
For a column with a given separation, this yields Fenske’s formula for the mini-
mum number of stages:
(2.28)
These Fenske expressions do not assume constant molar flows and apply to the
separation between any two components with constant relative volatility. Note
that although a high-purity separation (large S) requires a larger number of stages,
the increase is only proportional to the logarithm of separation factor. For exam-
ple, increasing the purity level in a product by a factor of 10 (e.g. by reducing x
H,D
from 0.01 to 0.001) increases N
min
by about a factor of
.
A common rule of thumb is to select the actual number of stages
(or
even larger).
2.4.2
Minimum Energy Usage
—
Infinite Number of Stages
For a given separation, an increase in the number of stages will yield a reduction
in the reflux (or equivalently in the boilup). However, as the number of stages
approach infinity, a pinch zone develops somewhere in the column, and the reflux
cannot be reduced further. For a binary separation the pinch usually occurs at the
feed stage (where the material balance line and the equilibrium line will meet),
and we can easily derive an expression for the minimum reflux with
. For
a saturated liquid feed (q=1) (King’s formula):
(2.29)
α
y
L n
,
y
H n
,
-----------
x
L n
,
x
H n
,
-----------
⁄
x
L n
1
+
,
x
H n
1
+
,
-------------------
x
L n
,
x
H n
,
-----------
⁄
=
=
S
x
L
x
H
------
T
x
L
x
H
------
B
⁄
α
N
=
=
N
min
S
ln
α
ln
---------
=
10
ln
2.3
=
N
2N
min
=
N
∞
=
L
Tmin
r
L D
,
α
r
H D
,
–
α
1
–
----------------------------------F
=
36
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where
is the recovery fraction of light component, and
of heavy component, both in the distillate. The value depends relatively weakly
on the product purity, and for sharp separations (where
and
), we have L
min
= F/(
α
- 1). Actually, equation (2.29) applies without
stipulating constant molar flows or constant
α
, but then L
min
is the liquid flow
entering the feed stage from above, and
α
is the relative volatility at feed condi-
tions. A similar expression, but in terms of
entering the feed stage from
below, applies for a saturated vapour feed (q=0) (King’s formula):
(2.30)
For sharp separations we get
= F/(
α
- 1). In summary, for a binary mixture
with constant molar flows and constant relative volatility, the minimum boilup for
sharp separations is:
(2.31)
Note that minimum boilup is independent of the product purity for sharp separa-
tions. From this we establish one of the key properties of distillation: We can
achieve any product purity (even “infinite separation factor”) with a constant
finite energy (as long as it is higher than the minimum) by increasing the number
of stages.
Obviously, this statement does not apply to azeotropic mixtures, for which
α
= 1
for some composition, (but we can get arbitrary close to the azeotropic composi-
tion, and useful results may be obtained in some cases by treating the azeotrope
as a pseudo-component and using
α
for this pseudo-separation).
2.4.3
Finite Number of Stages and Finite Reflux
Fenske’s formula S =
α
Ν
applies to infinite reflux (infinite energy). To extend this
expression to real columns with finite reflux we will assume constant molar flows
and consider below three approaches:
1. Assume constant K-values and derive the Kremser formulas (exact close to
the column end for a high-purity separation).
r
L D
,
x
D
D z
⁄
F
=
r
H D
,
r
L D
,
1
=
r
H D
,
0
=
V
Bmin
V
Bmin
r
H B
,
α
r
L B
,
–
α
1
–
---------------------------------F
=
V
Bmin
Feed liquid, q=1: V
Bmin
1
α
1
–
------------F
D
+
=
Feed vapour, q=0: V
Bmin
1
α
1
–
------------F
=
2.4 Simple Design Equations
37
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2. Assume constant relative volatility and derive the following extended Fen-
ske formula (approximate formula for case with optimal feed stage
location):
(2.32)
Here N
T
is the number of stages in the top section and N
B
in the bottom
section.
3. Assume constant relative volatility and derive exact expressions. The most
used are the Underwood formulas which are particularly useful for com-
puting the minimum reflux (with infinite stages).
2.4.4
Constant K-values
—
Kremser Formulas
For high-purity separations most of the stages are located in the “corner” parts of
the McCabe-Thiele diagram where we according to Henry’s law may approxi-
mate the VLE-relationship, even for nonideal mixtures, by straight lines;
Bottom of column: y
L
= H
L
x
L
(light component; x
L
→
0)
Top of column: y
H
= H
H
x
H
(heavy component; x
H
→
0)
where H is Henry’s constant. (For the case of constant relative volatility, Henry’s
constant in the bottom is
and in the top is
). Thus, with con-
stant molar flows, both the equilibrium and mass-balance relationships are linear,
and the resulting difference equations are easily solved analytically. For example,
at the bottom of the column we derive for the light component:
(2.33)
where
is the stripping factor. Repeated use of this equation
gives the Kremser formula for stage N
B
from the bottom (the reboiler would here
be stage zero):
(2.34)
(assumes we are in the region where s is constant, i.e.
).
S
α
N
L
T
V
T
⁄
(
)
N
T
L
B
V
B
⁄
(
)
N
B
-----------------------------
≈
H
L
α
=
H
H
1
α
⁄
=
x
L n
1
+
,
V
B
L
B
⁄
(
)
H
L
x
L n
,
B L
B
⁄
(
)
x
L B
,
+
=
sx
L n
,
1
V
B
–
L
B
⁄
(
)
x
L B
,
+
=
s
V
B
L
B
⁄
(
)
H
L
1
>
=
x
L N
B
,
s
N
B
x
L B
,
1
1
V
B
–
L
B
⁄
(
)
1
s
N
–
B
–
(
)
s
1
–
(
)
⁄
+
[
]
=
x
L
0
≈
38
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At the top of the column we have for the heavy component:
(2.35)
where
is the absorbtion factor. The corresponding
Kremser formula for the heavy component in the vapour phase at stage N
T
counted from the top of the column (the accumulator is stage zero) is then:
(2.36)
(assumes we are in the region where a is constant, i.e.
).
For hand calculations one may use the McCabe-Thiele diagram for the interme-
diate composition region, and the Kremser formulas at the column ends where the
use of the McCabe-Thiele diagram is inaccurate.
Example. We consider a column with N=40, N
F
=21,
=1.5, z
L
=0.5, F=1,
D=0.5, V
B
=3.2063. The feed is saturated liquid and exact calculations give
the product compositions x
H,D
= x
L,B
=0.01.
We now want to have a bottom product with only 1 ppm heavy product, i.e.
x
L,B
= 1.e-6. We can use the Kremser formulas to easily estimate the addi-
tional stages needed when we have the same energy usage, V
B
=3.2063.
(Note that with the increased purity in the bottom we actually get B=0.4949
and L
B
=3.7012). At the bottom of the column
and the
stripping factor is
.
With x
L,B
=1.e-6 (new purity) and
(old purity) we find by
solving the Kremser equation (2.34) with respect to N
B
that N
B
=33.94, and
we conclude that we need about 34 additional stages in the bottom (this is
not quite enough since the operating line is slightly moved and thus affects
the rest of the column; using 36 rather 34 additional stages compensates
for this).
The above Kremser formulas are valid at the column ends, but the linear approx-
imation resulting from the Henry’s law approximation lies above the real VLE
curve (is optimistic), and thus gives too few stages in the middle of the column.
However, if the there is no pinch at the feed stage (i.e. the feed is optimally
located), then most of the stages in the column will be located at the columns ends
where the above Kremser formulas apply.
y
H n
1
–
,
L
T
V
T
⁄
(
)
1 H
H
⁄
(
)
y
H n
,
D V
T
⁄
(
)
x
H D
,
+
=
ay
H n
,
1
L
T
V
T
⁄
–
(
)
x
H D
,
+
=
a
L
T
V
T
⁄
(
)
H
H
⁄
1
>
=
y
H N
T
,
a
N
T
x
H D
,
1
1
L
T
–
V
T
⁄
(
)
1
a
N
–
T
–
(
)
a
1
–
(
)
⁄
+
[
]
=
x
H
0
≈
α
H
L
α
1.5
=
=
s
V
B
L
B
⁄
(
)
H
L
3.2063 3.712
⁄
(
)
1.5
1.2994
=
=
=
x
L N
B
,
0.01
=
2.4 Simple Design Equations
39
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2.4.5
Approximate Formula with Constant Relative Volatility
We will now use the Kremser formulas to derive an approximation for the sepa-
ration factor S. First note that for cases with high-purity products we have
That is, the separation factor is the inverse of the product of
the key component product impurities.
We now assume that the feed stage is optimally located such that the composition
at the feed stage is the same as that in the feed, i.e.
and
Assuming constant relative volatility and using
,
,
and
(including
total reboiler) then gives:
(2.37)
where
(2.38)
We know that S predicted by this expression is somewhat too large because of the
linearized VLE. However, we may correct it such that it satisfies the exact rela-
tionship
at infinite reflux (where
and c=1) by
dropping the factor
(which as expected is always larger than 1). At
finite reflux, there are even more stages in the feed region and the formula will
further oversestimate the value of S. However, since c > 1 at finite reflux, we may
partly counteract this by setting c=1. Thus, we delete the term c and arrive at the
final extended Fenske formula, where the main assumptions are that we have con-
stant relative volatility, constant molar flows, and that there is no pinch zone
around the feed (i.e. the feed is optimally located):
(2.39)
where
.
S
1
x
L B
,
x
H D
,
(
)
⁄
≈
y
H N
T
,
y
H F
,
=
x
L N
B
,
x
L F
,
=
H
L
α
=
H
B
1
α
⁄
=
α
y
LF
x
LF
⁄
(
)
y
HF
x
HF
⁄
(
)
⁄
=
N
N
T
N
B
1
+
+
=
S
α
N
L
T
V
T
⁄
(
)
N
T
L
B
V
B
⁄
(
)
N
B
-----------------------------
c
x
HF
y
LF
(
)
------------------------
≈
c
1
1
V
B
L
B
-------
–
1
s
N
B
–
–
(
)
s
1
–
(
)
-------------------------
+
1
1
L
T
V
T
-------
–
1
a
N
T
–
–
(
)
a
1
–
(
)
-------------------------
+
=
S
α
N
=
L
B
V
B
⁄
V
T
L
T
⁄
1
=
=
1
x
HF
y
LF
(
)
⁄
S
α
N
L
T
V
T
⁄
(
)
N
T
L
B
V
B
⁄
(
)
N
B
-----------------------------
≈
N
N
T
N
B
1
+
+
=
40
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Together with the material balance,
, this approximate for-
mula can be used for estimating the number of stages for column design (instead
of e.g. the Gilliand plots), and also for estimating the effect of changes of internal
flows during column operation. However, its main value is the insight it provides:
1. We see that the best way to increase the separation S is to increase the
number of stages.
2. During operation where N is fixed, the formula provides us with the impor-
tant insight that the separation factor S is increased by increasing the
internal flows L and V, thereby making L/V closer to 1. However, the effect
of increasing the internal flows (energy) is limited since the maximum sep-
aration with infinite flows is
.
3. We see that the separation factor S depends mainly on the internal flows
(energy usage) and only weakly on the split D/F. This means that if we
change D/F then S will remain approximately constant (Shinskey’s rule),
that is, we will get a shift in impurity from one product to the other such
that the product of the impurities remains constant. This insight is very
useful.
Example. Consider a column with
(1% heavy in top) and
(1% light in bottom). The separation factor is then approxi-
mately
Assume we slightly
increase D from 0.50 to 0.51. If we assume constant separation factor
(Shinskey’s rule), then we find that
changes from 0.01 to 0.0236
(heavy impurity in the top product increases by a factor 2.4), whereas and
changes from 0.01 to 0.0042 (light impurity in the bottom product
decreases by a factor 2.4). Exact calculations with column data: N=40,
N
F
=21,
=1.5, z
L
=0.5, F=1, D=0.5, L
T
/F=3.206, give that
changes from 0.01 to 0.0241 and
changes from 0.01 to 0.0046 (sep-
aration factor changes from S=9801 to 8706). Thus, Shinskey’s rule gives
very accurate predictions.
However, the simple extended Fenske formula also has shortcomings. First, it is
somewhat misleading since it suggests that the separation may always be
improved by transferring stages from the bottom to the top section if
. This is not generally true (and is not really “allowed” as it
violates the assumption of optimal feed location). Second, although the formula
F z
F
Dx
D
Bx
B
+
=
S
α
N
=
x
D H
,
0.01
=
x
B L
,
0.01
=
S
0.99
0.99
0.01
0.01
×
(
)
⁄
×
9801
=
=
x
D H
,
x
B L
,
α
x
D H
,
x
B L
,
L
T
V
T
⁄
(
)
V
B
L
B
⁄
(
)
>
2.4 Simple Design Equations
41
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gives the correct limiting value
for infinite reflux, it overestimates the
value of S at lower reflux rates. This is not surprising since at low reflux rates a
pinch zone develops around the feed.
Example: Consider again the column with N=40. N
F
=21,
=1.5, z
L
=0.5,
F=1, D=0.5; L
T
=2.706 Exact calculations based on these data give x
HD
=
x
LB
=0.01 and S = 9801. On the other hand, the extended Fenske formula
with N
T
=20 and N
B
=20 yields:
corresponding to x
HD
= x
LB
= 0.0057. The error may seem large, but it is
actually quite good for such a simple formula.
2.4.6
Optimal Feed Location
The optimal feed stage location is at the intersection of the two operating lines in
the McCabe-Thiele diagram. The corresponding optimal feed stage composition
(x
F
,
y
F
)
can
be
obtained
by
solving
the
following
two
equations:
and
. For q=1 (liquid feed)
we find
and for q=0 (vapour feed) we find
(in the other cases we
must solve a second order equation).
There exists several simple shortcut formulas for estimating the feed point loca-
tion. One may derived from the Kremser equations given above. Divide the
Kremser equation for the top by the one for the bottom and assume that the feed
is optimally located to derive:
(2.40)
The last “big” term is close to 1 in most cases and can be neglected. Rewriting the
expression in terms of the light component then gives Skogestad’s shortcut for-
mula for the feed stage location:
S
α
N
=
α
S
1.5
41
2.7606 3.206
⁄
(
)
20
3.706 3.206
⁄
(
)
20
--------------------------------------------
×
16586000
0.34
18.48
-------------
×
30774
=
=
=
z
qx
F
1
q
–
(
)
y
F
+
=
y
F
α
x
F
1
α
1
–
(
)
x
F
+
(
)
⁄
=
x
F
z
=
y
F
z
=
y
H F
,
x
L F
,
------------
x
H D
,
x
L B
,
------------
α
N
T
N
B
–
(
)
L
T
V
T
-------
N
T
V
B
L
B
-------
N
B
-------------------
1
1
L
T
V
T
-------
–
1
a
N
T
–
–
(
)
a
1
–
(
)
-------------------------
+
1
1
V
B
L
B
-------
–
1
s
N
B
–
–
(
)
s
1
–
(
)
-------------------------
+
---------------------------------------------------------------
=
42
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
Version: 11 August 2000
(2.41)
where y
F
and x
F
at the feed stage are obtained as explained above. The optimal
feed stage location counted from the bottom is then:
(2.42)
where N is the total number of stages in the column.
2.4.7
Summary for Continuous Binary Columns
With the help of a few of the above formulas it is possible to perform a column
design in a matter of minutes by hand calculations. We will illustrate this with a
simple example.
We want to design a column for separating a saturated vapour mixture of 80%
nitrogen (L) and 20% oxygen (H) into a distillate product with 99% nitrogen and
a bottoms product with 99.998% oxygen (mole fractions).
Component data: Normal boiling points (at 1 atm): T
bL
= 77.4K, T
bH
= 90.2K,
heat of vaporization at normal boiling points: 5.57 kJ/mol (L) and 6.82 kJ/mol
(H).
The calculation procedure when applying the simple methods presented in this
article can be done as shown in the following steps:
1. Relative volatility:
The mixture is relatively ideal and we will assume constant relative volatil-
ity. The estimated relative volatility at 1 atm based on the boiling points is
where
,
and
. This gives
and we find
(however, it is generally recommended to obtain
from experimental VLE data).
N
T
N
B
–
1
y
F
–
(
)
x
F
--------------------
x
B
1
x
D
–
(
)
--------------------
ln
α
ln
---------------------------------------------------------------
=
N
F
N
B
1
+
N
1
N
T
N
B
–
(
)
–
+
[
]
2
---------------------------------------------------
=
=
α
ln
∆
H
vap
RT
b
----------------
T
bH
T
bL
–
(
)
T
b
------------------------------
≈
∆
H
vap
5.57 6.82
⋅
6.16 kJ/mol
=
=
T
b
T
bH
T
bL
83.6K
=
=
T
H
T
L
–
90.2
77.7
–
18.8
=
=
∆
H
vap
(
)
RT
b
(
)
⁄
8.87
=
α
3.89
≈
α
2.4 Simple Design Equations
43
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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2. Product split:
From the overall material balance we get
.
3. Number of stages:
The separation factor is
, i.e. ln S = 15.4.
The minimum number of stages required for the separation is
and we select the actual number of stages as
(
).
4. Feed stage location
With an optimal feed location we have at the feed stage (q=0) that y
F
= z
F
= 0.8 and
.
Skogestad’s approximate formula for the feed stage location gives
corresponding to the feed stage
.
5. Energy usage:
The minimum energy usage for a vapour feed (assuming sharp separation)
is
. With the choice
, the actual energy usage (V) is then typically about 10%
above the minimum (V
min
), i.e. V/F is about 0.38.
This concludes the simple hand calculations. Note again that the number of stages
depends directly on the product purity (although only logarithmically), whereas
for well-designed columns (with a sufficient number of stages) the energy usage
is only weakly dependent on the product purity.
D
F
----
z
x
B
–
x
D
x
B
–
------------------
0.8
0.00002
–
0.99
0.00002
–
------------------------------------
0.808
=
=
=
S
0.99
0.99998
×
0.01
0.00002
×
------------------------------------
4950000
=
=
N
min
S
ln
α
ln
⁄
11.35
=
=
N
23
=
2N
min
≈
x
F
y
F
α
α
1
–
(
)
y
F
–
(
)
⁄
0.507
=
=
N
T
N
B
–
1
y
F
–
(
)
x
F
--------------------
x
B
1
x
D
–
(
)
--------------------
ln
α
ln
(
)
⁄
=
0.2
0.507
-------------
0.00002
0.01
-------------------
×
1.358
⁄
ln
5.27
–
=
=
N
F
N
1
N
T
N
B
–
(
)
–
+
[
]
2
⁄
23
1
5.27
+
+
(
)
2
⁄
14.6
15
≈
=
=
=
V
min
F
⁄
1
α
1
–
(
)
⁄
1 2.89
⁄
0.346
=
=
=
N
2N
min
=
44
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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Remark 1:
The actual minimum energy usage is slightly lower since we do not have
sharp separations. The recovery of the two components in the bottom prod-
uct is
and
, so from the formulas given earlier the exact
value for nonsharp separations is
Remark 2:
For a liquid feed we would have to use more energy, and for a sharp
separation
Remark 3:
We can check the results with exact stage-by-stage calculations. With
N=23, N
F
=15 and
=3.89 (constant), we find V/F = 0.374 which is about
13% higher than V
min
=0.332.
Remark 4:
A simulation with more rigorous VLE computations, using the SRK equa-
tion of state, has been carried out using the HYSYS simulation package.
The result is a slightly lower vapour flow due to a higher relative volatility
(
in the range from 3.99-4.26 with an average of 4.14). More precisely, a
simulation with N=23, N
F
=15 gave V/F=0.291, which is about 11% higher
than the minimum value
found with a very large number
of stages (increasing N>60 did not give any significant energy reduction
below
). The optimal feed stage (with N=23) was found to be N
F
=15.
Thus, the results from HYSYS confirms that a column design based on the very
simple shortcut methods is very close to results from much more rigorous
computations.
2.5
Multicomponent Distillation — Underwood’s
Methods
We here present the Underwood equations for multicomponent distillation with
constant relative volatility and constant molar flows. The analysis is based on con-
sidering a two-product column with a single feed, but the usage can be extended
to all kind of column section interconnections.
r
L
x
L B
,
B
(
)
z
FL
F
(
)
⁄
0.9596
=
=
r
H
x
H B
,
B
(
)
z
FH
F
(
)
⁄
0
≈
=
V
min
F
⁄
0.9596
0.0
3.89
×
–
(
)
3.89
1
–
(
)
⁄
0.332
=
=
V
min
F
⁄
1
α
1
–
(
)
⁄
D F
⁄
+
0.346
0.808
+
1.154
=
=
=
α
α
V'
min
0.263
=
V'
min
2.5 Multicomponent Distillation — Underwood’s Methods
45
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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It is important to note that adding more components does not give any additional
degrees of freedom in operation. This implies that for an ordinary two-product
distillation column we still have only two degrees of freedom, and thus, we will
only be able to specify two variables, e.g. one property for each product. Typi-
cally, we specify the purity (or recovery) of the light key in the top, and specify
the heavy key purity in the bottom (the key components are defined as the com-
ponents between which we are performing the split). The recoveries for all other
components and the internal flows (L and V) will then be completely determined.
For a binary mixture with given products, as we increase the number of stages,
there develops a pinch zone on both sides of the feed stage. For a multicomponent
mixture, a feed region pinch zone only develops when all components distribute
to both products, and the minimum energy operation is found for a particular set
of product recoveries, sometimes denoted as the “preferred split”. If all compo-
nents do not distribute, the pinch zones will develop away from the feed stage.
Underwood’s methods can be used in all these cases, and are especially useful for
the case of infinite number of stages.
2.5.1
The Basic Underwood Equations
The net material transport (w
i
) of component i upwards through a stage n is:
(2.43)
Note that w
i
is constant in each column section. We assume constant molar flows
(L=L
n
=L
n-1
and V=V
n
=V
n+1
), and assume constant relative volatility. The VLE
relationship is then:
where
(2.44)
We divide equation (2.43) by V, multiply it by the factor
, and take
the sum over all components:
(2.45)
w
i
V
n
y
i n
,
L
n
1
+
x
i n
1
+
,
–
=
y
i
α
i
x
i
α
i
x
i
i
∑
-----------------
=
α
i
y
i
x
i
⁄
(
)
y
r
x
r
⁄
(
)
-------------------
=
α
i
α
i
φ
–
(
)
⁄
1
V
----
α
i
w
i
α
i
φ
–
(
)
-------------------
i
∑
α
i
2
x
i n
,
α
i
φ
–
(
)
-------------------
i
∑
α
i
x
i n
,
i
∑
---------------------------
L
V
----
α
i
x
i n
1
+
,
α
i
φ
–
(
)
----------------------
i
∑
–
=
46
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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The parameter
is free to choose, and the Underwood roots are defined as the
values of
which make the left hand side of (2.45) unity, i.e which satisfy
(2.46)
The number of values
satisfying this equation is equal to the number of com-
ponents, N
c
.
Comment:
Most authors use a product composition (x) or component recovery
(r) in this definition, e.g for the top (subscript T) section or the distillate product
(subscript D):
(2.47)
but we prefer to use the net component molar flow (w) since it is more general.
Note that use of the recovery is equivalent to using net component flow, but use
of the product composition is only applicable when a single product stream is
leaving the column. If we apply the product recovery, or the product composition,
the defining equation for the top section becomes:
(2.48)
2.5.2
Stage to Stage Calculations
By the definition of
from (2.46), the left hand side of (2.45) equals one, and the
last term of (2.45) then equals:
The terms with
disappear in the nominator and
can be taken outside the
summation, thus (2.45) is simplified to:
φ
φ
V
α
i
w
i
α
i
φ
–
(
)
-------------------
i
1
=
Nc
∑
=
φ
w
i
w
i T
,
w
i D
,
Dx
i D
,
r
i D
,
z
i
F
=
=
=
=
V
T
α
i
r
i D
,
z
i
α
i
φ
–
(
)
--------------------F
i
∑
α
i
x
i D
,
α
i
φ
–
(
)
-------------------D
i
∑
=
=
φ
α
i
2
x
i n
,
α
i
φ
–
(
)
-------------------
i
∑
α
i
x
i n
,
i
∑
---------------------------
1
–
α
i
2
x
i n
,
α
i
φ
–
(
)
-------------------
α
i
x
i n
,
–
i
∑
α
i
x
i n
,
i
∑
-----------------------------------------------------
α
i
2
x
i n
,
α
i
φ
–
(
)α
i
x
i n
,
–
(
)
α
i
φ
–
(
)
-------------------------------------------------------------
i
∑
α
i
x
i n
,
i
∑
---------------------------------------------------------------------
=
=
α
i
2
φ
2.5 Multicomponent Distillation — Underwood’s Methods
47
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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(2.49)
This equation is valid for any of the Underwood roots, and if we assume constant
molar flows and divide an equation for
with the one for
, the following
expression appears:
(2.50)
and we note the similarities with the Fenske and Kremser equations derived ear-
lier. This relates the composition on a stage (n) to an composition on another stage
(n+m). The number of independent equations of this kind equals the number of
Underwood roots minus 1 (since the number of equations of the type as in equa-
tion (2.49) equals the number of Underwood roots), but in addition we also have
. Together, this is a linear equation system for computing
when
is known and the Underwood roots is computed from (2.46).
Note that so far we have not discussed minimum reflux (or vapour flow rate), thus
these equation holds for any vapour and reflux flow rates, provided that the roots
are computed from the definition in (2.46).
2.5.3
Some Properties of the Underwood Roots
Underwood showed a series of important properties of these roots for a two-prod-
uct column with a reboiler and condenser. In this case all components flow
upwards in the top section (
), and downwards in the bottom section
(
). The mass balance yields:
where
.
Underwood showed that in the top section (with N
c
components) the roots ( )
obey:
(2.51)
L
V
----
α
i
x
i n
1
+
,
α
i
φ
–
(
)
----------------------
i
∑
φ
α
i
x
i n
,
α
i
φ
–
(
)
-------------------
i
∑
α
i
x
i n
,
i
∑
------------------------------
=
φ
k
φ
j
α
i
x
i n
m
+
,
α
i
φ
k
–
(
)
-----------------------
i
∑
α
i
x
i n
m
+
,
α
i
φ
j
–
(
)
-----------------------
i
∑
-------------------------------
φ
k
φ
j
-----
m
α
i
x
i n
,
α
i
φ
k
–
(
)
---------------------
i
∑
α
i
x
i n
,
α
i
φ
j
–
(
)
---------------------
i
∑
-----------------------------
=
x
i
∑
1
=
x
i n
m
+
,
x
i n
,
w
i T
,
0
≥
w
i B
,
0
≤
w
i B
,
w
i T
,
w
i F
,
–
=
w
i F
,
Fz
i
=
φ
α
1
φ
1
α
2
φ
2
α
3
… α
Nc
φ
Nc
>
>
>
>
>
>
>
48
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
Version: 11 August 2000
And in the bottom section (where
) we have a different set of
roots denoted (
) computed from
(2.52)
which obey:
(2.53)
Note that the smallest root in the top section is smaller than the smallest relative
volatility, and the largest root in the bottom section is larger then the largest vol-
atility. It is easy to see from the defining equations that as
and similarly as
.
When the vapour flow is reduced, the roots in the top section will decrease, while
the roots in the bottom section will increase, but interestingly Underwood showed
that
. A very important result by Underwood is that for infinite number
of stages;
.
Thus, at minimum reflux, the Underwood roots for the top ( ) and bottom (
)
sections coincide. Thus, if we denote these common roots
, and recall that
, and that
we obtain the fol-
lowing equation for the “minimum reflux” common roots ( ) by subtracting the
defining equations for the top and bottom sections:
(2.54)
We denote this expression the feed equation since only the feed properties (q and
z) appear. Note that this is not the equation which defines the Underwood roots
and the solutions ( ) apply as roots of the defining equations only for minimum
reflux conditions (
). The feed equation has N
c
roots, (but one of these is
not a common root) and the N
c
-1 common roots obey:
. Solution of the feed equation gives us
the possible common roots, but all pairs of roots (
) for the top and
bottom section do not necessarily coincide for an arbitrary operating condition.
We illustrate this with the following example:
w
i n
,
w
i B
,
0
≤
=
ψ
V
B
α
i
w
i B
,
α
i
ψ
–
(
)
--------------------
i
∑
α
i
r
i B
,
–
(
)
z
i
F
α
i
ψ
–
(
)
-------------------------------
i
∑
α
i
1
r
i D
,
–
(
)
–
(
)
z
i
F
α
i
ψ
–
(
)
----------------------------------------------
i
∑
=
=
=
ψ
1
α
>
1
ψ
2
α
2
ψ
3
α
3
… ψ
Nc
α
Nc
>
>
>
>
>
>
>
V
T
∞
→
⇒ φ
i
α
i
→
V
B
∞
→
⇒ ψ
i
α
i
→
φ
i
ψ
i
1
+
≥
V
V
min
→
⇒ φ
i
ψ
i
1
+
→
φ
ψ
θ
V
T
V
B
–
1
q
–
(
)
F
=
w
i T
,
w
i B
,
–
w
i F
,
z
i
F
=
=
θ
1
q
–
(
)
α
i
z
i
α
i
θ
–
(
)
-------------------
i
∑
=
θ
N
∞
=
α
1
θ
1
α
2
θ
2
… θ
Nc
1
–
α
Nc
>
>
>
>
>
>
φ
i
and
ψ
i
1
+
2.5 Multicomponent Distillation — Underwood’s Methods
49
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Assume we start with a given product split (D/F) and a large vapour flow
(V/F). Then only one component i (with relative volatility
) can be dis-
tributed to both products. No roots are common. Then we gradually reduce
V/F until a second component j (this has to be a component j=i+1 or j=i-
1) becomes distributed. E.g for j=i+1 one set of roots will coincide:
, while the others do not. As we reduce V/F further, more
components become distributed and the corresponding roots will coincide,
until all components are distributed to both products, and then all the N
c
-1
roots from the feed equation also are roots for the top and bottom sections.
An important property of the Underwood roots is that the value of a pair of roots
which coincide (e.g. when
) will not change, even if only one,
two or all pairs coincide. Thus all the possible common roots are found by solving
the feed equation once.
2.5.4
Minimum Energy
—
Infinite Number of Stages
When we go to the limiting case of infinite number of stages, Underwoods’s equa-
tions become very useful. The equations can be used to compute the minimum
energy requirement for any feasible multicomponent separation.
Let us consider two cases: First we want to compute the minimum energy for a
sharp split between two adjacent key components j and j+1 (
and
). The procedure is then simply:
1. Compute the common root (
) for which
from the feed equation:
2. Compute the minimum energy by applying the definition equation for
.
.
Note that the recoveries
For example, we can derive Kings expressions for minimum reflux for a binary
feed (
,
,
, and liquid feed (q=1)). Con-
sider the case with liquid feed (q=1). We find the single common root from the
α
i
φ
i
ψ
i
1
+
θ
i
=
=
φ
i
ψ
i
1
+
θ
i
=
=
r
j D
,
1
=
r
j
1 D
,
+
0
=
θ
j
α
j
θ
j
α
j
1
+
>
>
1
q
–
(
)
a
i
z
i
a
i
θ
–
(
)
------------------
i
∑
=
θ
j
V
Tmin
F
---------------
a
i
z
i
a
i
θ
j
–
(
)
--------------------
i
1
=
j
∑
=
r
i D
,
1 for i
j
≤
0 for i
j
>
=
z
L
z
=
z
H
1
z
–
(
)
=
α
L
α α
H
,
1
=
=
50
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feed equation:
, (observe
as expected). The
minimum reflux expression appears as we use the defining equation with the com-
mon root:
(2.55)
and when we substitute for
and simplify, we obtain King’s expression:
(2.56)
Another interesting case is minimum energy operation when we consider sharp
split only between the most heavy and most light components, while all the inter-
mediates are distributed to both products. This case is also denoted the “preferred
split”, and in this case there will be a pinch region on both sides of the feed stage.
The procedure is:
1. Compute all the N
c
-1 common roots ( )from the feed equation.
2. Set
and solve the following linear equation set
(
equations) with respect to
(
variables):
(2.57)
Note that in this case, when we regard the most heavy and light components as
the keys, and all the intermediates are distributed to both products and Kings very
simple expression will also give the correct minimum reflux for a multicompo-
nent mixture (for q=1 or q=0). The reason is that the pinch then occurs at the feed
stage. In general, the values computed by Kings expression give a (conservative)
upper bound when applied directly to multicomponent mixtures. An interesting
θ
α
1
α
1
–
(
)
z
+
(
)
⁄
=
α θ
1
≥ ≥
L
Tmin
F
--------------
V
Tmin
F
---------------
D
F
----
–
θ
r
i D
,
z
i
α
i
θ
–
(
)
-------------------
i
∑
θ
r
L D
,
z
α θ
–
-----------------
θ
r
H D
,
1
z
–
(
)
1
θ
–
--------------------------------
+
=
=
=
θ
L
Tmin
F
--------------
r
L D
,
α
r
H D
,
–
α
1
–
----------------------------------
=
θ
r
1 D
,
1 and r
N
c
D
,
0
=
=
N
c
1
–
V
T
r
2 D
,
r
3 D
,
…
r
N
c
1
–
,
,
[
]
N
c
1
–
V
T
a
i
r
i D
,
z
i
a
i
θ
1
–
(
)
---------------------
i
1
=
N
c
∑
=
•
•
V
T
a
i
r
i D
,
z
i
a
i
θ
N
c
1
–
–
(
)
-------------------------------
i
1
=
N
c
∑
=
2.5 Multicomponent Distillation — Underwood’s Methods
51
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result which can be seen from Kings’s formula is that the minimum reflux at pre-
ferred split (for q=1) is independent of the feed composition and also independent
of the relative volatilities of the intermediates.
However, with the more general Underwood method, we also obtain the distribu-
tion of the intermediates, and it is easy to handle any liquid fraction (q) in the feed.
The procedure for an arbitrary feasible product recovery specification is similar
to the preferred split case, but then we must only apply the Underwood roots (and
corresponding equations) with values between the relative volatilities of the dis-
tributing components and the components at the limit of being distributed. In
cases where not all components distribute, King’s minimum reflux expression
cannot be trusted directly, but it gives a (conservative) upper bound.
Figure 2.9 shows an example of how the components are distributed to the prod-
ucts for a ternary (ABC) mixture. We choose the overhead vapour flow (V=V
T
)
and the distillate product flow (D=V-L) as the two degrees of freedom. The
straight lines, which are at the boundaries when a component is at the limit of
appearing/disappearing (distribute/not distribute) in one of the products, can be
computed directly by Underwood’s method. Note that the two peaks (P
AB
and
P
BC
) gives us the minimum vapour flow for sharp split between A/B and B/C. The
point P
AC
, however, is at the minimum vapour flow for sharp
A/C split and this occurs for a specific distribution of the intermediate B, known
as the “preferred split”.
Kings’s minimum reflux expression is only valid in the triangle below the pre-
ferred split, while the Underwood equations can reveal all component recoveries
for all possible operating points. (The shaded area is not feasible since all liquid
and vapour streams above and below the feed have to be positive).
52
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Figure 2.9: Regions of distributing feed components as function of V and D for a
feed mixture with three components: ABC. P
ij
represent minimum energy for
sharp split between component i and j. For large vapour flow (above the top
“saw-tooth”), only one component distribute. In the triangle below P
AC
, all
components distribute.
2.6
Further Discussion of Specific Issues
2.6.1
The Energy Balance and the Assumption of Constant Molar
flows
All the calculations in this article are based on the assumption of constant molar
flows in a section, i.e
and
. This is a very
common simplification in distillation computations, and we shall use the energy
balance to see when we can justify it. The energy balance is similar to the mass
balance, but now we use the molar enthalpy (h) of the streams instead of compo-
sition. The enthalpy is computed for the actual mixture and will be a function of
composition in addition to temperature (or pressure). At steady state the energy
balance around stage n becomes:
0
1
V/F
D/F
1-q
P
AC
P
AB
P
BC
ABC
D
V
L
V=D (L=0)
ABC
AB
ABC
A
BC
A
BC
AB
C
ABC
C
AB
BC
ABC
ABC
ABC
“The preferred split”
Sharp A/BC split
Sharp AB/C split
(sharp A/C split)
Infeasible region
V/F=(1-q)
F
V
n
V
n
1
–
V
=
=
L
n
L
n
1
+
L
=
=
2.6 Further Discussion of Specific Issues
53
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(2.58)
Combining this energy balance with the overall material balance on a stage
(
where W is the net total molar flow through a
section, i.e. W=D in the top section and -W=B in the bottom section) yields:
(2.59)
From this expression we observe how the vapour flow will vary through a section
due to variations in heat of vaporization and molar enthalpy from stage to stage.
We will now show one way of deriving the constant molar flow assumption:
1. Chose the reference state (where h=0) for each pure component as saturated
liquid at a reference pressure. (This means that each component has a dif-
ferent reference temperature, namely its boiling point (
) at the
reference pressure.)
2. Assume that the column pressure is constant and equal to the reference
pressure.
3. Neglect any heat of mixing such that
.
4. Assume that all components have the same molar heat capacity c
PL
.
5. Assume that the stage temperature can be approximated by
. These assumptions gives
on all stages and
the equation (2.59) for change in boilup is reduced to:
(2.60)
6. The molar enthalpy in the vapour phase is given as:
where
is the
heat of vaporization for the pure component at its reference boiling temper-
ature (
).
L
n
h
L n
,
V
n
1
–
h
V n
1
–
,
–
L
n
1
+
h
L n
1
+
,
V
n
h
V n
,
–
=
V
n
1
–
L
n
–
V
n
L
n
1
+
–
W
=
=
V
n
V
n
1
–
h
V n
1
–
,
h
L n
,
–
h
V n
,
h
L n
1
+
,
–
------------------------------------
W
h
L n
,
h
L n
1
+
,
–
h
V n
,
h
L n
1
+
,
–
------------------------------------
+
=
T
bpi
h
L n
,
x
i n
,
c
PLi
T
n
T
bpi
–
(
)
i
∑
=
T
n
x
i n
,
T
bpi
i
∑
=
h
L n
,
0
=
V
n
V
n
1
–
h
V n
1
–
,
h
V n
,
------------------
=
h
V n
,
x
i n
,
H
bpi
vap
∆
i
∑
x
i n
,
c
PVi
T
n
T
bpi
–
(
)
i
∑
+
=
H
bpi
vap
∆
T
bpi
54
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Version: 11 August 2000
7. We assume that c
PV
is equal for all components, and then the second sum-
mation term above will become zero, and we have:
.
8. Then if
is equal for all components we get
, and thereby constant molar flows:
and also
.
At first glance, these assumptions may seem restrictive, but the assumption of
constant molar flows actually holds well for many industrial mixtures.
In a binary column where the last assumption about equal
is not fulfilled,
a good estimate of the change in molar flows from the bottom (stage 1) to the top
(stage N) for a case with saturated liquid feed (q=1) and close to pure products, is
given by:
. The molar heats of vaporization is taken at
the boiling point temperatures for the heavy (H) and light (L) components
respectively.
Recall that the temperature dependency of the relative volatility were related to
different heat of vaporization also, thus the assumptions of constant molar flows
and constant relative volatility are closely related.
2.6.2
Calculation of Temperature when Using Relative Volatilities
It may look like that we have lost the pressure and temperature in the equilibrium
equation when we introduced the relative volatility. However, this is not the case
since the vapour pressure for every pure component is a direct function of temper-
ature,
thus
so
is
also
the
relative
volatility.
From
the
relationship
we derive:
(2.61)
Remember that only one of P or T can be specified when the mole fractions are
specified. If composition and pressure is known, a rigorous solution of the tem-
perature is found by solving the non-linear equation:
(2.62)
h
V n
,
x
i n
,
H
bpi
vap
∆
i
∑
=
H
bpi
vap
∆
H
vap
∆
=
h
V n
,
h
V n
1
–
,
H
vap
∆
=
=
V
n
V
n
1
–
=
L
n
L
n
1
+
=
H
bpi
vap
∆
V
N
V
1
⁄
H
H
vap
∆
H
L
vap
∆
⁄
≈
P
p
i
∑
x
i
p
i
o
T
( )
∑
=
=
P
p
r
o
T
( )
x
i
α
i
i
∑
=
P
x
i
p
i
o
T
( )
∑
=
2.6 Further Discussion of Specific Issues
55
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However, if we use the pure components boiling points (T
bi
), a crude and simple
estimate can be computed as:
(2.63)
For ideal mixtures, this usually give an estimate which is a bit higher than the real
temperature, however, similar approximation may be done by using the vapour
compositions (y), which will usually give a lower temperature estimate. This
leads to a good estimate when we use the average of x and y, i.e:
(2.64)
Alternatively, if we are using relative volatilities we may find the temperature via
the vapour pressure of the reference component. If we use the Antoine equation,
then we have an explicit equation:
where
(2.65)
This last expression is a very good approximation to a solution of the nonlinear
equation (2.62). An illustration of how the different approximations behave is
shown in Figure 2.10: For that particular case (a fairly ideal mixture), equation
(2.64) and (2.65) almost coincide..
T
x
i
T
bi
∑
≈
T
x
i
y
i
+
2
---------------
T
bi
∑
≈
T
B
r
p
r
o
log
A
r
–
--------------------------
C
r
+
≈
p
r
o
P
x
i
α
i
i
∑
⁄
=
56
Draft - /home/ivarh/thesis/book/DistillationTheory_ch.fm
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Figure 2.10: Temperature profile for the example in Figure 2.7 (solid line)
compared with various linear boiling point approximations.
In a rigorous simulation of a distillation column, the mass and energy balances
and the vapour liquid equilibrium (VLE) have to be solved simultaneously for all
stages. The temperature is then often used as an iteration parameter in order to
compute the vapour-pressures in VLE-computations and in the enthalpy compu-
tations of the energy balance.
2.6.3
Discussion and Caution
Most of the methods presented in this article are based on ideal mixtures and sim-
plifying assumptions about constant molar flows and constant relative volatility.
Thus there are may separation cases for non-ideal systems where these methods
cannot be applied directly.
However, if we are aware about the most important shortcomings, we may still
use these simple methods for shortcut calculations, for example, to gain insight or
check more detailed calculations.
0
5
10
15
20
25
30
35
40
300
301
302
303
304
305
306
307
308
309
310
Bottom Stages Top
Temperature [K]
α
=1.50
z=0.50
q=1.00
N=40
N
F
=21
x
DH
=0.1000
x
BL
=0.0020
T=
Σ
x
i
T
bi
T=
Σ
y
i
T
bi
T=
Σ
(y
i
+x
i
)T
bi
/2
T=
f(x,P)
2.7 Bibliography
57
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Version: 11 August 2000
2.7
Bibliography
[1]
Franklin, N.L. Forsyth, J.S. (1953), The interpretation of Minimum Reflux
Conditions in Multi-Component Distillation. Trans IChemE, Vol 31,
1953. (Reprinted in Jubilee Supplement - Trans IChemE, Vol 75, 1997).
[2]
King, C.J. (1980), second Edition, Separation Processes. McGraw-Hill,
Chemical Engineering Series, , New York.
[3]
Kister, H.Z. (1992), Distillation Design. McGraw-Hill, New York.
[4]
McCabe, W.L. Smith, J.C. Harriot, P. (1993), Fifth Edition, Unit Opera-
tions of Chemical Engineering. McGraw-Hill, Chemical Engineering
Series, New York.
[5]
Shinskey, F.G. (1984), Distillation Control - For Productivity and Energy
Conservation. McGraw-Hill, New York.
[6]
Skogestad, S. (1997), Dynamics and Control of Distillation Columns - A
Tutorial Introduction. Trans. IChemE, Vol 75, Part A, p539-562.
[7]
Stichlmair, J. James R. F. (1998), Distillation: Principles and Practice.
Wiley,
[8]
Underwood, A.J.V. (1948), Fractional Distillation of Multi-Component
Mixtures. Chemical Engineering Progress, Vol 44, no. 8, 1948
58
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