Reactions and Separations
Visualizing the
McCabe-Thiele Diagram
Use this spreadsheet-based visualization
and interactive analysis of the
McCabe-Thiele diagram to understand the
foundations of distillation engineering.
Paul M. Mathias
Fluor Corp.
M
ore than 80 years ago, McCabe and Thiele
developed a creative graphical solution technique based on Lewis’s assumption of constant
molal overflow (CMO) for the rational design of distillation columns (1). The McCabe-Thiele diagram enabled
decades of effective design and operational analysis of
distillation columns, and has been used to teach several
generations of chemical engineers to design and troubleshoot distillation and other cascaded processes.
Simplified methods such as McCabe-Thiele are rarely
used today for detailed design. Modern tools are based
on rigorous solution of the equations governing cascaded
separations, and properly deal with multicomponent systems, heat effects, chemical reactions and mass-transfer
limitations (2). Commercially available software enables
simulation of entire chemical plants for the purposes of
design, optimization and control — but it is important
to ensure the discriminating and competent use of these
sophisticated tools. The McCabe-Thiele visual approach
provides a powerful way to attain this judgment and
competency.
Software applications based on rigorous calculations
do not explain how distillation really works and how the
numerous process variables interact to yield a distillation column that is energy efficient, stable, and produces
products with the desired purities. If the design engineer specifies an impossible set of inputs, even the best
commercial software either crashes or returns vague and
unhelpful messages (e.g., “Tray j dried up”). Engineers
who have only performed simulations resort to arbitrary
and frantic changes in input variables to obtain a simulation with a feasible specification. In addition, rigorous,
computerized calculations do not reveal design problems
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December 2009 CEP
such as composition pinches and unforgiving composition
profiles (i.e., a steep peak in the composition profile of one
or more components in multicomponent distillation) —
until they are converged. For these reasons, textbooks on
staged separations stress the visual approach and present
the McCabe-Thiele diagram as an essential tool for understanding and analyzing cascaded operations. Experienced
process engineers often use McCabe-Thiele diagrams to
understand or help debug simulation results (3). Yet some
consider graphical techniques tools of the past, and as a
result, the distillation column has become a black box and
engineers’ understanding of distillation has suffered (4).
Even though the construction of McCabe-Thiele diagrams is straightforward, it is a tedious and error-prone
process. Hence, engineers rarely study the large number
of cases needed to understand the interactions of the many
process variables. This article introduces a spreadsheet-based
approach that readily produces McCabe-Thiele diagrams for
binary systems so that the interacting effect of process variables can be visualized easily and interactively; the Excel file
is provided as a supplement to the online version of this article (www.aiche.org/cep). Because it is based on the widely
available Excel software and uses standard techniques, this
method is a useful advance over existing software tools that
enable visualization of the McCabe-Thiele diagram.
Excel spreadsheets are increasingly used in chemical
engineering education and industrial practice (5). This
article demonstrates how Excel may be used for chemical
engineering computation and visualization. Through the
example presented in the sidebar, the article also demonstrates how the McCabe-Thiele method, and, in particular,
the spreadsheet application introduced here, answers typical design questions.
Q
Example: The Problem Statement
An existing distillation column consists of a total
condenser and 10 equilibrium stages, with the feed
inlet on the fifth stage from the top. The 10 equilibrium
stages include a partial reboiler. The column needs to be
reused to separate an acetone-ethanol binary mixture
at a pressure of 1 atm. The feed is 20% vaporized and
contains 50 kmol/h acetone and 50 kmol/h ethanol. The
desired distillate and bottoms compositions of acetone
are 90 mole% and 3 mole%, respectively. Vapor-liquid
equilibrium (VLE) data are provided for this binary
system, and the enthalpies of vaporization of acetone
and ethanol may be assumed to be constant at 29.6 and
38.9 kJ/mol, respectively. Assume that the constantmolal-overflow (CMO) assumption applies.
a. Is the CMO approximation valid?
b. Calculate the number of stages needed at total
reflux. If the minimum number of stages exceeds 10, the
separation is impossible.
c. Your supervisor, a distillation expert, has
answered question b, and assures you that the minimum number of stages needed at total reflux is less
than 10. Calculate the reflux ratio needed to achieve the
desired separation. How does the reflux ratio change as
the thermal state of the feed is varied?
d. If it is possible to move the feed inlet stage, would
you recommend that this be done? What is the optimum
feed location? How is the separation affected as the
feed inlet is changed?
e. How should the design be changed if the tray
efficiency decreases below unity?
The approach
As far as possible, this application uses Excel’s
standard capabilities. The exception is a function to
perform data interpolation, called Interp, which was
developed using Visual Basic for Applications (VBA)
and is provided as part of the online version of the article.
This function is used, for example, to interpolate vaporpressure and x-y equilibrium data. This feature enables
the spreadsheet application to accept x-y data from any
source, including experimental data, calculated values
derived from a thermodynamic model, or values from
commercial software.
Figure 1 is a schematic diagram of a distillation column with a total condenser and a partial reboiler. The partial reboiler is an equilibrium stage, but the total condenser
is not. The stages are counted from the top down, with
stage 1 at the top of the column where the reflux enters.
(Note that if a partial condenser is used, it will have to be
counted as the first equilibrium stage.) According to the
CMO approximation, each stage is at phase equilibrium,
and the vapor and liquid flows are constant in both the rectifying section (above the feed tray) and the stripping section (feed tray and below). Following the usual practice,
QC
V
1
F0
(feed with q = 1)
D
R
F
QF
(heat rate required
to change its
thermal state
to the specified q)
N
Q
QR
Q
B
S Figure 1. The example distillation column has a total condenser, a partial
reboiler, and N stages; the first stage is at the top of the column where the
reflux enters, and the Nth stage is the partial reboiler.
the compositions refer to the more-volatile component.
The vapor and liquid flows in the rectifying section (above
the feed tray) are denoted as V and L, respectively.
The equations resulting from the McCabe-Thiele
technique are available in textbooks on separations and
are only summarized here. The constant vapor and liquid
flows resulting from the CMO approximation lead to the
following component-balance equation (or operating line)
in the rectifying section:
L
L
y j + 1 = V x j + (1 - V ) x D
^1 h
The relationship between the reflux ratio (R = L/D)
and the liquid-to-vapor flow ratio (L/V) is:
L
R
V = 1+R
^2 h
Analogous to Eq. 1, the operating line for the stripping section is:
yj+1 =
L
L
x j + ( - 1) x B
V
V
^3 h
The feed line, or “q-line,” starts from a point on the
x = y diagonal line where x = xF and has a slope based
on q or the liquid fraction of the feed. In the distillation
literature, q is usually referred to as the thermal state of
the feed. Note that q can be less than zero (superheated
vapor) or greater than unity (subcooled liquid). The slope
of the q-line is q/(q – 1).
The construction of the two operating lines and the
q-line is performed in Excel as follows:
1. The rectifying-section operating line is drawn using
Eqs. 1 and 2 and the specified values of xD and R.
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37
Reactions and Separations
2. The q-line is constructed by drawing a straight line
from (xF, xF) with a slope equal to q/(q – 1).
3. The stripping-section operating line is the straight
line from (xB, xB) to the intersection of the rectifyingsection operating line and the q-line.
The function Interp is useful in implementing the
operating lines and the q-lines even though all three are
straight lines. For the purpose of calculating equilibrium
compositions, x-y data have been entered as a table with
101 points (x_Data, y_Data). This spacing of data points
is expected to be adequate for accurate interpolation of
the acetone-ethanol x-y diagram. For another system with
an x-y diagram that has a more complex shape, more data
points in the table may be needed.
The McCabe-Thiele diagram is constructed by “stepping
off stages.” The starting point is (xD, xD), which is the vapor
composition ascending from tray 1. The liquid composition
descending from tray 1 is the mole fraction in equilibrium
with a vapor with composition xD, which is obtained using
the function Interp: x1 = Interp (xD, x_Data, y_Data, 0).
Next, the vapor composition rising from tray 2 is calculated
from the rectifying-section operating line (or componentbalance equation), again using Interp. The stepping off
of stages continues over the entire column, switching to
the stripping-section operating line at the feed tray. In this
way, the entire McCabe-Thiele diagram can be computed
and graphically represented using standard Excel charting
capability.
Standard Excel what-if analysis capability can be used
to evaluate design results. For example, Goal Seek or
Solver may be used to calculate the reflux ratio to obtain
the liquid composition (xB) descending from tray 10.
(Solver seems to converge more reliably and accurately
than Goal Seek.)
As was first suggested by Murphree, the McCabeThiele procedure may be made more realistic by relaxing the approximation that vapor-liquid equilibrium
is achieved on each stage. The Murphrey efficiency is
defined as:
E ML =
x j - x j-1
x *j - x j - 1
^4 h
In Eq. 4, xj* is the liquid mole fraction that is in equilibrium with the vapor ascending from tray j and xj is the
actual mole fraction of the liquid descending from tray j.
As the efficiency EML decreases to values less than unity,
the decrease in liquid mole fraction from the distillate to the
bottom will be reduced. The concept of Murphree efficiency
is equally applicable to the vapor or liquid phase, and here
it is convenient to apply it to the liquid phase. Note that if
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December 2009 CEP
Eq. 4 is be applied, the use of Murphree efficiencies requires
a modification in the way that the liquid mole fraction is
calculated from the vapor mole fraction.
It is useful to estimate heat effects consistent with the
CMO approximation. Here, it is assumed that sensible
heat effects and heats of mixing are negligible and that
heat duties are only associated with evaporating and condensing binary mixtures based on their (fixed) enthalpies of vaporization. The condenser duty, QC, is the heat
removal rate required to condense a vapor with flowrate V
and composition xD:
vap
vap
- Q C = V 6 x D DH 1 + (1 - x D) DH 2 @
^5 h
The CMO approximation implies that the sum of the
condenser and reboiler duties is zero for a saturated-liquid
feed (q = 1). The heat duty required to change the thermal
state of the feed from the saturated-liquid state is:
vap
vap
Q F = F (1 - q) 6 x F DH 1 + (1 - x F) DH 2 @
^6 h
The reboiler duty (QR) is calculated from QC and QF:
QR = QC - QF
^7 h
Evaluating binary separations
A spreadsheet is used to analyze the separation of the
acetone-ethanol mixture and other binary mixtures using
the McCabe-Thiele approach.
The McCabe-Thiele diagram is constructed by interpolation of x-y data. These data may be obtained from
a variety of sources, such as standard thermodynamic
models or commercial process-simulation software. The
advantage of this approach is that the source may be
sophisticated models and databases that are difficult and
time-consuming to program in Excel. However, x-y data
may not always be available for the system of interest, so
the use of Excel for regression of phase-equilibrium data
to generate the x-y data is demonstrated here.
VLE data for the acetone-ethanol system at 50°C,
71°C and 80°C (6, 7) have been used to regress the
parameters of the NRTL activity-coefficient model.
Since the pressure is low (about 2 bar or less), pressure
effects on the liquid fugacity may be neglected and the
vapor phase can be treated as an ideal gas. Thus, the total
pressure of the binary mixture is given by Eq. 8, and the
vapor composition for the specified liquid composition
may be calculated from Eq. 9:
The Role of the McCabe-Thiele Method in Distillation Engineering
T
he availability of digital computers and the intense
development of solution algorithms, beginning in about
1951, eliminated the need for approximate solutions in
equilibrium-stage distillation calculations and enabled more
rigorous simulations. A comprehensive suite of powerful
rigorous methods is available today in commercial software,
which has completely replaced simplified methods for the
design of distillation towers. Unfortunately, these rigorous,
computerized calculations are often used as a black box,
and the intuitive, visualization benefits inherent in the simplified procedures are in danger of being lost. Kister noted that
despite the huge progress in distillation, the number of tower
malfunctions is not declining (12).
Experts recognize that an effective distillation design
and analysis toolkit must have more than the capability for
rigorous, computerized calculations. Accurate calculation of
thermodynamic properties (especially vapor-liquid equilibrium) is crucial to producing correct designs (13). Residue
curve maps (RCMs) (14) provide visualization of feasible and
infeasible separation sequences, and have been extended to
multicomponent systems and pressure variations. Graphical
techniques, like McCabe-Thiele and Hengstebeck diagrams,
multicomponent distillation profiles and RCMs, are excellent
troubleshooting tools because they uncover design problems, such as composition pinches and unforgiving composition profiles (15). Good plant data are difficult to obtain, but
P=
y1 =
P 1sat x 1 c1 + P 2sat x 2 c2
P
P 1sat x 1 c1
^8 h
^9 h
Any suitable activity-coefficient model may be used
to represent the liquid nonideality, and here the NRTL
model has been chosen:
GE
G 21 x21
G 12 x12
x 1 x 2 RT = x 1 + x 2 G 21 + x 2 + x 1 G 12
^10h
G 12 / exp (- ax12) and G 21 / exp (- ax21)
^11h
x12 = A 12 + B 12 /T and x21 = A 21 + B 21 /T
^12h
The equations for the resulting activity coefficients
(γ1 and γ2) are available in standard references. At each
temperature, the vapor pressures of acetone and ethanol
have been estimated using the function Interp and
assuming that the logarithm of the vapor pressure is
are well worth the time and effort required to collect them,
since they are the prime tool of the troubleshooter. Visualization tools capture the fundamentals of distillation. Such a
diverse and comprehensive toolkit gives engineers detailed
results, insight, and understanding to develop superior
designs, as well as the judgment to diagnose and resolve
operational problems.
Today, implementation of the McCabe-Thiele graphical
procedure does not require the CMO approximation, since
the diagram can easily be constructed from a rigorous distillation calculation. In addition, the McCabe-Thiele diagram
has been extended to multicomponent systems by Hengstebeck. When used in this manner, McCabe-Thiele/Hengstebeck diagrams are highly effective as design and troubleshooting tools for analyzing new energy-saving technologies
(16), designing steam-stripping systems (17), improving
energy efficiency (18), evaluating revamp improvements (19),
and explaining counter-intuitive observations in a multi-feed
distillation tower (20).
“Perry’s Chemical Engineers’ Handbook” (21) states:
“With the widespread availability of computers, the preferred
approach to design is equation based … Nevertheless,
diagrams are useful for quick approximations, for interpreting
results of equation-based methods, and for demonstrating
the effect of various design variables. The x-y diagram is the
most convenient for these purposes.”
linear in reciprocal temperature.
The optimum values of the NRTL parameters
(A12, A21, B12 and B21) have been determined using Solver
to minimize the sum-squared error between the measured
and calculated total pressures (i.e., Barker’s method).
The optimum values of the NRTL parameters are:
α = 0.3; A12 = 0.724358; B12 = –159.176; A21 = –2.33876;
B21 = 921.128.
Using the NRTL model with Eqs. 8–12, an x-y diagram is generated. At each value of x, the temperature
must be found such that the total pressure (Eq. 8) is equal
to the stage pressure (constant at 1 atm or 101.325 kPa).
Although Goal Seek may be used to solve this equation, it is tedious, since the calculation will have to be
repeated 101 times. A better approach is to calculate the
sum-squared error of the difference between the calculated pressures and 101.325 kPa, and then use Solver to
minimize the sum-squared error by changing the 101 cells
that contain the temperatures corresponding to the liquid
compositions. The latter methodology (provided with the
online article) is efficient in terms of human effort, since
a single Excel step generates the entire x-y table.
Total reflux
The total-reflux calculation, which corresponds to
infinite reflux ratio, is useful because it determines the
minimum number of stages needed for the target separaCEP
December 2009
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39
Reactions and Separations
Design Specifications
1
0.9
2
XD
0.8
3
0.7
0.6
0.5
Feed Stage
0.8
xD
EML
1.00
Calculated
xB Stage 10
0.0300
0.90
QC, GJ/h
–4.984
xB
0.03
QR, GJ/h
4.299
Reflux Ratio
2.025
5
xB-error
4
0.5
0.4
5
0.3
0.2
6
0.1
XB
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x = Mole Fraction of Acetone in Liquid
S Figure 2. Acetone-ethanol separation at 1 atm and total reflux requires
between six and seven stages to achieve xD = 0.09 and xB = 0.03.
tion. At infinite reflux, both operating lines reduce to
the diagonal (x = y) line. The total-reflux diagram for
acetone-ethanol is presented as Figure 2, which indicates
that between six and seven stages are needed to achieve
xD = 0.9 and xB = 0.03. Hence, it is possible to use a distillation column with 10 stages for the desired separation.
Effect of q and comparison with rigorous calculation
Figure 3 presents the reflux ratio needed to achieve
the desired separation. As shown in the figure, the desired
reflux ratio is readily calculated using Solver to determine
the reflux ratio that gives the target bottom concentration,
xB = 0.03.
Figure 3 demonstrates that the optimum feed location
is tray 6 rather than tray 5, but the penalty for the nonoptimum feed location is fairly small.
A study of the effect of varying the thermal state of
the feed is summarized in Table 1 (using the optimum
feed location, tray 6). The required reflux ratio decreases
as the thermal state of the feed (q) increases, and the
reboiler duty (QR) increases as q increases. The reboiler
duty in Table 1 is not the entire heat load, since it does
not include any heat treatment of the feed. Thus, the table
includes QR + QF, where QF is the heat load required to
modify the thermal state of the feed from q = 1 (saturated
liquid). (Based on the Lewis CMO approximation,
QR + QF is exactly equal to –QC.) On this basis, the total
heat load decreases as q increases.
The analysis demonstrates that the effect of an
increase in q results in a decrease in capital costs (due to
a lower reflux ratio and column flows and hence a smaller
column diameter), as well as a decrease in operating
costs (mainly a lower heating duty, which is usually the
40
xF
q
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December 2009 CEP
y = Mole Fraction of Acetone in Vapor
y = Mole Fraction of Acetone in Vapor
1
8.9 E–23
1
0.9
Bottom
Operating
Line
0.8
0.7
0.6
4
6
3
2
1
5
7
0.5
Top
Operating
Line
8
0.4 11
0.3 10
9
0.2
Feed Line
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x = Mole Fraction of Acetone in Liquid
S Figure 3. The McCabe-Thiele diagram for acetone-ethanol at 1 atm
shows the input variables (design specifications) and key calculated
variables. xB-error is the square of the difference between the desired and
calculated values of xB. Solver is used to vary reflux ratio to drive xB-error to
a minimum value (effectively zero).
Table 1. The thermal state of the feed affects the
reflux ratio and the reboiler and condenser duties.
xF = 0.5, xD = 0.9, xB = 0.03, EML = 1.0
(QR + QF) is the reboiler duty plus the heat required to raise
the feed from a saturated liquid to its feed thermal state.
q
Reflux
Ratio
Optimum
Tray
Q R,
GJ/h
Q C,
GJ/h
Q R + Q F,
GJ/h
–0.1
2.54
6
2.06
–5.83
5.83
0
2.44
6
2.25
–5.68
5.68
0.2
2.27
6
2.65
–5.39
5.39
0.4
2.12
6
3.09
–5.15
5.15
0.6
2.00
6
3.56
–4.93
4.93
0.8
1.89
6
4.07
–4.75
4.75
1
1.79
6
4.60
–4.60
4.60
1.1
1.75
6
4.88
–4.53
4.53
major operating cost). The value of this analysis is that
the design engineer can easily understand the benefits of
varying q and make the best design choice.
This McCabe-Thiele method was tested by comparing
its calculated q variations with rigorous results from the
Aspen Plus process simulator. The Aspen Plus simulation
uses the same thermodynamic model (ideal vapor phase,
no pressure effects on the liquid fugacity, and the NRTL
activity-coefficient model), but rigorously deals with heat
effects. It does not make the Lewis approximations (constant molal overflow or constant heat of vaporization even
as the liquid composition changes, and negligible sensible
Error bars show ±10% variance
from rigorous calculation
Rigorous
x = Mole Fraction of
Acetone in Liquid
Reflux Ratio
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
McCabe-Thiele
0
0.2
0.4
0.6
q = Thermal Quality of Feed
0.8
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
McCabe-Thiele
Rigorous
0
S Figure 4. The required reflux ratio is a function of the thermal quality
of the feed.
2
4
6
Tray Number
8
10
S Figure 6. The liquid composition profiles obtained using the McCabeThiele techinique and rigorous calculations are similar.
5
200
Rigorous
4
Vapor
175
3
2
McCabe-Thiele
Error bars show ±15%
variance from rigorous calculation
1
0
0
0.2
0.4
0.6
0.8
1
Flow, kmol/h
Reboiler Duty, GJ/h
6
150
75
0
S Figure 5. The reboiler duty is a function of the thermal quality
of the feed.
2
4
6
Tray Number
8
10
Reboiler Duty, GJ/h
S Figure 7. There is poor agreement between the Lewis approximations
and rigorous calculations for the column flow profiles.
9
8
7
6
5
4
3
3
4
5
6
7
8
9
Feed Tray
S Figure 8. A nonoptimum feed tray location (such as tray 3 or 9) would
require a larger reboiler duty to achieve the target separation.
y = Mole Fraction of Acetone in Vapor
Effect of feed location
Figure 8 shows the effect of nonoptimum feed
tray location on the required reboiler duty. Note that a
severely nonoptimum feed location (e.g., tray 9 vs. tray
6) will require the reboiler duty to be doubled in order
to achieve the target separation. Figure 9 illustrates the
pinch point that occurs when the feed point is too low.
Figure 10 provides further illustration of the pinch point,
showing that the liquid mole fraction reaches an asymp-
Rigorous: Solid lines
McCabe-Thiele: Dashed lines
100
q = Thermal Quality of Feed
heat and heat of mixing), and hence provides a way to
estimate the errors caused by the approximations.
Figures 4 and 5 indicate that the Lewis approximations cause errors of about 10% in the reflux ratio and up
to 15% in the reboiler duty. The Lewis approximations
are in excellent agreement with the rigorous calculation
for the liquid composition profile (Figure 6), but poor
agreement with the rigorous calculation for the flow
profiles (Figure 7).
Today, rigorous calculations rather than shortcut methods are used for detailed design of distillation columns.
However, Figures 4 and 5 clearly demonstrate that the
McCabe-Thiele method captures the trends reasonably
well, and hence remains important for understanding the
foundations of distillation engineering. As discussed in
the sidebar on p. 39 (“The Role of the McCabe-Thiele
Method in Distillation Engineering”), the McCabe-Thiele
diagram, without the CMO approximation, has excellent
present-day value as a design and troubleshooting tool.
Liquid
125
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Bottom
Operating
Line
2
1
3
4
5
11
10 7
8
9
0
0.1
Top
Operating
Line
6
Feed Line
0.2
0.3 0.4 0.5 0.6 0.7 0.8
x = Mole Fraction of Acetone in Liquid
0.9
1
S Figure 9. A pinch point occurs when a feed point is too low, such as
tray 9 rather than the optimum tray 6.
CEP
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x = Mole Fraction of
Acetone in Liquid
Reactions and Separations
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Table 2. The Murphree efficiency (EML) affects the reflux
ratio and reboiler duty.
xF = 0.5, xD = 0.9, xB = 0.03, q = 1.0
EML
Reflux Ratio
Optimum Tray
QR, GJ/h
1.00
1.89
6
4.07
0.95
2.10
6
4.42
0.90
2.41
6
4.93
0.85
2.88
6
5.71
0.80
3.70
6
7.05
0.75
5.38
6
9.83
0.74
5.95
6
10.8
totic value of about 0.09 at tray 9, and the addition of
more stages above the feed will have no effect on improving product purities.
The Excel file can be used to further study the analogous
detrimental effect that occurs if the feed point is too high.
0.73
6.68
6
12.0
0.72
7.57
5
13.4
0.71
8.81
5
15.5
0.70
10.6
5
18.4
0.69
13.3
5
22.9
Effect of reduced tray efficiency
It is usually recommended that the sectional column
efficiency be defined as the ratio of the theoretical number of stages to the actual number of stages to achieve
a particular separation (8). Since efficiencies vary from
one section to another, it is best to apply the efficiency
separately for each section (i.e., rectifying and stripping).
The concept of Murphree efficiency (EML, Eq. 4) has been
used to investigate the effect of tray efficiency. Table 2 and
Figure 11 show that the required reflux ratio and the resulting
reboiler duty increase substantially as EML decreases below
unity. Note also that the optimum feed location changes as
efficiency changes, although the results are insensitive to the
feed location when the reflux ratio is high.
Figure 12 presents the McCabe-Thiele diagram for the
case where EML = 0.7. The reduced Murphree efficiency
effectively reduces the relative volatility, which makes the
separation more difficult, requiring a higher reflux ratio. In
fact, the desired separation is barely possible for EML = 0.67,
and further reduction in the efficiency will make the desired
separation impossible, even at total reflux.
It should be emphasized that the Murphree efficiency
is only a crude description of the performance of real
trays, and hence the effect of varying EML should be
interpreted with caution. The effects shown here are only
qualitatively applicable, but useful to illustrate the effects
of reduced tray efficiency.
0.68
18.2
5
30.9
0.67
28.8
5
48.1
McCabe-Thiele
Rigorous
0
2
4
6
Tray Number
8
10
S Figure 10. When the feed point is too low (tray 9 rather than the optimum tray 6), the liquid composition reaches an asymptotic value.
Solution to the acetone-ethanol example problem
a. Constant molal overflow is a reasonably good
approximation even for this case, where the enthalpy of
vaporization of ethanol is 32% higher than that of acetone.
The CMO approximation is especially useful because it
42
www.aiche.org/cep
December 2009 CEP
Reflux Ratio
20
15
10
5
0
0.65
0.70
0.75
0.80
0.85
Efficiency
0.90
0.95
1.00
S Figure 11. The Murphree efficiency (EML) substantially decreases with a
higher reflux ratio.
captures trends, and thus serves as an aid to understand the
fundamentals of distillation engineering. But CMO does
not yield reliably accurate results, and is not recommended
for detailed design calculations, especially since software
employing rigorous methods is widely available.
b. The number of stages needed at total reflux is
between six and seven. Thus, the available 10-stage column is likely to be adequate for the desired separation.
c. The reflux ratio needed for the specified separation
is 2.0. The reflux ratio decreases with increasing q, and the
required heat duty decreases as q increases. Therefore, it is
preferable to operate the column at high values of q.
d. The existing feed location (tray 5) is only slightly
suboptimal, with a required reflux ratio of 2.0, compared
with 1.9 if the feed location is lowered to tray 6. Figure 8
shows the negative effects that will occur for a poorly
located feed stage.
e. The separation becomes far more difficult as EML
y = Mole Fraction of Acetone in Vapor
1
1
2
0.9
Bottom
3
Operating
4
0.8
Line
0.7
5
0.6
6
Top
0.5
Operating
0.4 11
7
Line
0.3
10
Feed Line
8
0.2
9
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6 0.7
0.8
0.9
x = Mole Fraction of Acetone in Liquid
Nomenclature
1
S Figure 12. The effective equilibrium curve (green) at the reduced
efficiency of EML = 0.7 on the McCabe-Thiele diagram reveals that the
required reflux ratio increases from 1.89 (EML = 1.0, optimum feed
location) to 10.6.
decreases below unity (Figure 11). Note also that low
values of EML may cause pinch points that do not exist at
higher efficiencies.
Other binary separations
Other binary systems may easily be studied by replacing the x-y table used here for the acetone-ethanol binary
mixture. The Excel file supplied online provides two
additional examples of x-y diagrams for binary systems.
The first example is representative of the benzenetoluene system and assumes constant relative volatility, α:
a/
y=
y/x
(1 - y) / (1 - x)
xa
1 + (a - 1) x
^13h
^14h
The value of the simple constant-α system is that the
effects of close-boiling and wide-boiling systems may
easily be studied. Detailed study of this system is left as
an exercise for the reader.
In the second example, x-y data from an external
source are used — x-y data for the ethanol-water binary
system at 1 atm from the Aspen Plus process simulator.
The McCabe-Thiele diagram indicates that (1) a reflux
ratio of 5.4 is required for the target separation, and (2)
the optimum feed tray is low in the column because the
separation is very difficult at high ethanol concentrations
due to the formation of an azeotrope (at x ≈ 0.9). Hence,
more stages are needed above the feed stage. The reader
is urged to perform another calculation by increasing xD
slightly, from 0.83 to 0.84 (leaving xB unchanged at 0.01).
The required reflux ratio almost doubles (to 10.2), which
A12, A21, B12, B21
=
B
D
EML
=
=
=
GE
G12, G21
=
=
ΔH1vap, ΔH2vap
=
L
=
L
=
P
sat,
P1
sat
P2
=
=
q
=
Q
R
R
T
V
=
=
=
=
=
V
=
x
=
x*
=
y
=
Greek Letters
α
=
α
γ1, γ2
=
=
τ
=
Subscripts
1, 2
B
C
D
F
j
R
=
=
=
=
=
=
=
CEP
temperature-dependence
parameters in NRTL model
(Eq. 12)
bottom flowrate, kmol/h
distillate flowrate, kmol/h
Murphree efficiency, applied to
the liquid compositions (Eq. 4)
excess Gibbs energy (Eq. 10)
terms in NRTL model
(Eqs. 10 and 11)
enthalpy of vaporization of
components 1 and 2, kJ/mol
liquid flowrate in rectifying
section, kmol/h
liquid flowrate in stripping
section, kmol/h
pressure, kPa
vapor pressures of components
1 and 2, kPa
liquid fraction or thermal state
of the feed; q = 1 corresponds to
saturated liquid
heat rate, GJ/h
gas constant
reflux ratio
temperature, K
vapor flowrate in rectifying
section, kmol/h
vapor flowrate in
stripping section, kmol/h
liquid mole fraction (of the morevolatile component)
liquid composition in equilibrium
with y (Eq. 4)
vapor mole fraction (of the morevolatile component)
nonrandomness parameter in
NRTL model (Eq. 11)
relative volatility (Eq. 13)
activity coefficients of
components 1 and 2
interaction-energy parameter in
NRTL model (Eq. 10)
components 1 and 2
bottom
condenser
distillate
feed
stage number
reboiler
December 2009
www.aiche.org/cep
43
Reactions and Separations
highlights the difficulty of achieving higher product purity
in the vicinity of the azeotrope. The visual approach
clearly provides insight and understanding of the special
considerations required for the purification of a mixture
that forms an azeotrope.
Closing thoughts
The ability to easily vary input specifications in the
Excel spreadsheet and to visualize the effects on column
performance has significant value in teaching distillation fundamentals. Engineers are better able to grasp the
effects of various inputs, and they become better design
engineers and more competent, discriminating users of
commercial software. The understanding gained from
the spreadsheet is a valuable first step in using a rigorous
simulation as a design and troubleshooting tool.
But how robust is the McCabe-Thiele approach, which
is the foundation of this Excel application? In particular,
can it give results that will confuse or even mislead? In the
author’s experience, the McCabe-Thiele approach captures
trends well and thus is a valid and useful teaching tool.
So, should this tool be extended to improve accuracy
by eliminating the poor approximations and thus become
applicable to more realistic distillation situations (e.g., heat
effects, complex column configurations, multi-component
systems, mass transfer limitations, chemical reactions and
kinetics, etc.)? Extensions should be limited and should
be done with caution. Rigorous modeling methods are
very powerful for describing real distillation systems. The
computer code based on these methods supports flexible
design requirements very well. Thus, the best use of the
spreadsheet method presented here is to produce engineers
who understand the fundamentals, have good engineering
judgment, and become discriminating users of sophisticated
detailed methodologies.
As Kister noted, “the two can coexist” (4). In fact,
it is good practice to benefit from both the accuracy and
flexibility of rigorous calculations and the insight and
understanding gained by visualization of the venerable
CEP
McCabe-Thiele diagram.
Literature Cited
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
PAUL M. MATHIAS is a technical director at Fluor Corp. (47 Discovery, Irvine,
CA 92618; Phone: (949) 349-3595; Fax: (949) 349-5058); E-mail: Paul.
Mathias@Fluor.com), and previously worked on the ASPEN Project
(MIT), at Air Products and Chemicals, and at Aspen Technology. He is
a chemical technologist with more than 30 years of broad experience,
specializing in properties and process modeling. He has 50 publications and 75 presentations at technical conferences, and has been a
member of the editorial advisory boards of two journals: Chemical &
Engineering Data and Industrial & Engineering Chemistry Research.
He occasionally teaches chemical engineering courses at the Univ. of
California, Irvine. He is a member of AIChE. He earned a BTech from
the Indian Institute of Technology, Madras and a PhD from the Univ. of
Florida, both in chemical engineering.
44
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December 2009 CEP
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20.
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