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Reactions and Separations

Paul M. Mathias

Fluor Corp.

Use this spreadsheet-based visualization

and interactive analysis of the

McCabe-Thiele diagram to understand the

foundations of distillation engineering.

Visualizing the

McCabe-Thiele Diagram

M

ore than 80 years ago, McCabe and Thiele developed a creative graphical solution tech-nique based on Lewis’s assumption of constant molal overfl ow (CMO) for the rational design of distilla-tion 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 trouble-shoot distillation and other cascaded processes.

Simplifi ed 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 sys-tems, 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 distilla-tion column that is energy effi cient, stable, and produces products with the desired purities. If the design engi-neer specifi es 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 simula-tion with a feasible specifi casimula-tion. In addisimula-tion, rigorous, computerized calculations do not reveal design problems

such as composition pinches and unforgiving composition profi les (i.e., a steep peak in the composition profi le 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 under-standing 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 dia-grams 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 vari-ables can be visualized easily and interactively; the Excel fi le is provided as a supplement to the online version of this arti-cle (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 demon-strates how the McCabe-Thiele method, and, in particular, the spreadsheet application introduced here, answers typi-cal design questions.

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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 vapor-pressure 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 col-umn with a total condenser and a partial reboiler. The par-tial 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 refl ux enters. (Note that if a partial condenser is used, it will have to be counted as the fi rst equilibrium stage.) According to the CMO approximation, each stage is at phase equilibrium, and the vapor and liquid fl ows are constant in both the rec-tifying section (above the feed tray) and the stripping sec-tion (feed tray and below). Following the usual practice,

the compositions refer to the more-volatile component. The vapor and liquid fl ows 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 fl ows resulting from the CMO approximation lead to the following component-balance equation (or operating line) in the rectifying section:

( )

yj+1=VLxj+ 1-VL xD ^ h1 The relationship between the refl ux ratio (R = L/D) and the liquid-to-vapor fl ow ratio (L/V) is:

R R

VL = 1+ ^ h2

Analogous to Eq. 1, the operating line for the strip-ping section is:

( )

y

VL x V

L 1 x 3

j+1= j+ - B ^ 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 specifi ed values of xD and R.

An existing distillation column consists of a total condenser and 10 equilibrium stages, with the feed inlet on the fi fth 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 constant-molal-overfl ow (CMO) assumption applies.

a. Is the CMO approximation valid?

b. Calculate the number of stages needed at total refl ux. 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 mini-mum number of stages needed at total refl ux is less than 10. Calculate the refl ux ratio needed to achieve the desired separation. How does the refl ux 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 effi ciency decreases below unity?

F

0

(feed with q = 1)

QF (heat rate required

to change its thermal state to the specified q) Q Q QR F V R D B N 1

S Figure 1. The example distillation column has a total condenser, a partial reboiler, and N stages; the fi rst stage is at the top of the column where the refl ux enters, and the Nth stage is the partial reboiler.

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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 rectifying-section 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 component-balance 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 refl ux ratio to obtain the liquid composition (xB) descending from tray 10. (Solver seems to converge more reliably and accurately than Goal Seek.)

As was fi rst suggested by Murphree, the McCabe-Thiele procedure may be made more realistic by relax-ing the approximation that vapor-liquid equilibrium is achieved on each stage. The Murphrey effi ciency is defi ned as:

E x x x x 4 * ML j j j j 1 1 = -^ h In Eq. 4, xj* is the liquid mole fraction that is in equi-librium with the vapor ascending from tray j and xj is the actual mole fraction of the liquid descending from tray j. As the effi ciency 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 effi ciency is equally applicable to the vapor or liquid phase, and here it is convenient to apply it to the liquid phase. Note that if

Eq. 4 is be applied, the use of Murphree effi ciencies requires a modifi cation 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 con-densing binary mixtures based on their (fi xed) enthalp-ies of vaporization. The condenser duty, QC, is the heat removal rate required to condense a vapor with fl owrate V and composition xD:

(1 )

QC V xDDH1vap xD DH2vap 5

- = 6 + - @ ^ 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:

(1 ) ( )

QF=F -q x6 FDHvap1 + 1-xF DHvap2 @ ^ h6

The reboiler duty (QR) is calculated from QC and QF:

QR=QC-QF ^ h7

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 inter-polation 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 diffi cult 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-coeffi cient 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 specifi ed liquid composition may be calculated from Eq. 9:

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P x P x P= sat1 1 1c + sat2 2 2c ^ h8 y P x P 9 sat 1 1 1 1c = ^ h

Any suitable activity-coeffi cient model may be used to represent the liquid nonideality, and here the NRTL model has been chosen:

x x RTG x G x G x G x G 10 E 1 2 1 2 21 21 21 2 1 12 12 12 x x = + + + ^ h ( ) ( ) exp exp G12/ -ax12 and G21/ -ax21 ^ h11 / / A B T and A B T 12 12 12 12 21 21 21 x = + x = + ^ h

The equations for the resulting activity coeffi cients (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

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 dia-gram 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 equa-tion, 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 calcu-lated 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 effi cient in terms of human effort, since a single Excel step generates the entire x-y table.

Total refl ux

The total-refl ux calculation, which corresponds to infi nite refl ux ratio, is useful because it determines the minimum number of stages needed for the target

separa-T

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 simplifi ed methods for the design of distillation towers. Unfortunately, these rigorous, computerized calculations are often used as a black box, and the intuitive, visualization benefi ts inherent in the simpli-fi ed 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 equilib-rium) 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 profi les and RCMs, are excellent troubleshooting tools because they uncover design prob-lems, such as composition pinches and unforgiving compo-sition profi les (15). Good plant data are diffi cult to obtain, but

since they are the prime tool of the troubleshooter. Visualiza-tion tools capture the fundamentals of distillaVisualiza-tion. 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 distil-lation calcudistil-lation. In addition, the McCabe-Thiele diagram has been extended to multicomponent systems by Hengste-beck. When used in this manner, McCabe-Thiele/Hengste-beck diagrams are highly effective as design and trouble-shooting tools for analyzing new energy-saving technologies

(16), designing steam-stripping systems (17), improving energy effi ciency (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.”

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Reactions and Separations

tion. At infi nite refl ux, both operating lines reduce to the diagonal (x = y) line. The total-refl ux 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 distil-lation column with 10 stages for the desired separation.

Effect of q and comparison with rigorous calculation

Figure 3 presents the refl ux ratio needed to achieve the desired separation. As shown in the fi gure, the desired refl ux ratio is readily calculated using Solver to determine the refl ux 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 non-optimum 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 refl ux 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 refl ux ratio and column fl ows and hence a smaller column diameter), as well as a decrease in operating costs (mainly a lower heating duty, which is usually the

major operating cost). The value of this analysis is that the design engineer can easily understand the benefi ts 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-coeffi cient model), but rigorously deals with heat effects. It does not make the Lewis approximations (con-stant molal overfl ow or con(con-stant heat of vaporization even as the liquid composition changes, and negligible sensible

Table 1. The thermal state of the feed affects the refl ux 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 Refl ux Ratio Optimum Tray QR, GJ/h QC, GJ/h QR + QF, 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 3 4 5 6 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 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y

= Mole Fraction of Acetone in Vapor

1

2 XD

XB

S Figure 2. Acetone-ethanol separation at 1 atm and total refl ux requires between six and seven stages to achieve xD = 0.09 and xB = 0.03.

S Figure 3. The McCabe-Thiele diagram for acetone-ethanol at 1 atm shows the input variables (design specifi cations) 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 refl ux ratio to drive xB-error to a minimum value (effectively zero).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Top Operating Line Bottom Operating Line Feed Line 1 2 3 4 5 6 7 8 9 10 11

x = Mole Fraction of Acetone in Liquid

y

= Mole Fraction of Acetone in Vapor

Design Specifi cations

xF 0.5 Feed Stage 5 q 0.8 xD 0.90 EML 1.00 xB 0.03 Refl ux Ratio 2.025 Calculated xB Stage 10 0.0300 QC, GJ/h –4.984 QR, GJ/h 4.299 xB-error 8.9 E–23

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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 approxima-tions cause errors of about 10% in the refl ux ratio and up to 15% in the reboiler duty. The Lewis approximations are in excellent agreement with the rigorous calculation for the liquid composition profi le (Figure 6), but poor agreement with the rigorous calculation for the fl ow profi les (Figure 7).

Today, rigorous calculations rather than shortcut meth-ods 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.

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-1.6 1.8 2 2.2 2.4 2.6 0 0.2 0.4 0.6 0.8 1 Reflux Ratio

q = Thermal Quality of Feed McCabe-Thiele

Rigorous

S Figure 4. The required refl ux ratio is a function of the thermal quality of the feed. 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 Reboiler Duty, GJ/h

Error bars show ±15% variance from rigorous calculation McCabe-Thiele

Rigorous

q = Thermal Quality of Feed

S Figure 5. The reboiler duty is a function of the thermal quality of the feed. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 Tray Number McCabe-Thiele Rigorous x

= Mole Fraction of Acetone in Liquid

S Figure 6. The liquid composition profi les obtained using the McCabe-Thiele techinique and rigorous calculations are similar.

75 100 125 150 175 200 0 2 4 6 8 10 Flow, kmol/h Tray Number Liquid Vapor

Rigorous: Solid lines McCabe-Thiele: Dashed lines

S Figure 7. There is poor agreement between the Lewis approximations and rigorous calculations for the column fl ow profi les.

3 4 5 6 7 8 9 3 4 5 6 7 8 9 Reboiler Duty, GJ/h 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.

1 2 3 4 5 6 7 8 9 10 11 Top Operating Line Bottom Operating Line Feed Line 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

x = Mole Fraction of Acetone in Liquid

y

= Mole Fraction of Acetone in Vapor

S Figure 9. A pinch point occurs when a feed point is too low, such as tray 9 rather than the optimum tray 6.

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Reactions and Separations

totic value of about 0.09 at tray 9, and the addition of more stages above the feed will have no effect on improv-ing product purities.

The Excel fi le can be used to further study the analogous detrimental effect that occurs if the feed point is too high.

Effect of reduced tray effi ciency

It is usually recommended that the sectional column effi ciency be defi ned as the ratio of the theoretical num-ber of stages to the actual numnum-ber of stages to achieve a particular separation (8). Since effi ciencies vary from one section to another, it is best to apply the effi ciency separately for each section (i.e., rectifying and stripping).

The concept of Murphree effi ciency (EML, Eq. 4) has been used to investigate the effect of tray effi ciency. Table 2 and Figure 11 show that the required refl ux ratio and the resulting reboiler duty increase substantially as EML decreases below unity. Note also that the optimum feed location changes as effi ciency changes, although the results are insensitive to the feed location when the refl ux ratio is high.

Figure 12 presents the McCabe-Thiele diagram for the case where EML = 0.7. The reduced Murphree effi ciency effectively reduces the relative volatility, which makes the separation more diffi cult, requiring a higher refl ux ratio. In fact, the desired separation is barely possible for EML = 0.67, and further reduction in the effi ciency will make the desired separation impossible, even at total refl ux.

It should be emphasized that the Murphree effi ciency 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 effi ciency.

Solution to the acetone-ethanol example problem

a. Constant molal overfl ow 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

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 refl ux is between six and seven. Thus, the available 10-stage col-umn is likely to be adequate for the desired separation.

c. The refl ux ratio needed for the specifi ed separation is 2.0. The refl ux 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 refl ux 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 diffi cult as EML

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 Tray Number McCabe-Thiele Rigorous x

= Mole Fraction of Acetone in Liquid

S Figure 10. When the feed point is too low (tray 9 rather than the opti-mum tray 6), the liquid composition reaches an asymptotic value.

0 5 10 15 20 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Reflux Ratio Efficiency

S Figure 11. The Murphree effi ciency (EML) substantially decreases with a higher refl ux ratio.

Table 2. The Murphree effi ciency (EML) affects the refl ux ratio and reboiler duty.

xF = 0.5, xD = 0.9, xB = 0.03, q = 1.0

EML Refl ux 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 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 0.68 18.2 5 30.9 0.67 28.8 5 48.1

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decreases below unity (Figure 11). Note also that low values of EML may cause pinch points that do not exist at higher effi ciencies.

Other binary separations

Other binary systems may easily be studied by replac-ing the x-y table used here for the acetone-ethanol binary mixture. The Excel fi le supplied online provides two additional examples of x-y diagrams for binary systems. The fi rst example is representative of the benzene-toluene system and assumes constant relative volatility, α:

( )/( ) / y x y x 1 1 13 / a - - ^ h ( ) y=1+ axa-1 x ^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 refl ux 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 diffi cult 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 refl ux ratio almost doubles (to 10.2), which

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Top Operating Line Line Feed Line 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 6 7 8 9 10 11

x = Mole Fraction of Acetone in Liquid

y

= Mole Fraction of Acetone in Vapor

S Figure 12. The effective equilibrium curve (green) at the reduced effi ciency of EML = 0.7 on the McCabe-Thiele diagram reveals that the required refl ux ratio increases from 1.89 (EML = 1.0, optimum feed location) to 10.6.

A12, A21, B12, B21 = temperature-dependence parameters in NRTL model (Eq. 12)

B = bottom fl owrate, kmol/h

D = distillate fl owrate, kmol/h

EML = Murphree effi ciency, applied to the liquid compositions (Eq. 4)

GE = excess Gibbs energy (Eq. 10)

G12, G21 = terms in NRTL model (Eqs. 10 and 11)

ΔH1vap, ΔH

2vap

= enthalpy of vaporization of components 1 and 2, kJ/mol

L = liquid fl owrate in rectifying section, kmol/h

= liquid fl owrate in stripping section, kmol/h

P = pressure, kPa

P1sat, P

2sat

= vapor pressures of components 1 and 2, kPa

q = liquid fraction or thermal state of the feed; q = 1 corresponds to saturated liquid

Q = heat rate, GJ/h

R = gas constant

R = refl ux ratio

T = temperature, K

V = vapor fl owrate in rectifying section, kmol/h

= vapor fl owrate in stripping section, kmol/h

x = liquid mole fraction (of the more-volatile component)

x* = liquid composition in equilibrium with y (Eq. 4)

y = vapor mole fraction (of the more-volatile component)

Greek Letters

α = nonrandomness parameter in NRTL model (Eq. 11)

α = relative volatility (Eq. 13)

γ1, γ2 = activity coeffi cients of components 1 and 2 τ = interaction-energy parameter in NRTL model (Eq. 10) Subscripts 1, 2 = components 1 and 2 B = bottom C = condenser D = distillate F = feed j = stage number R = reboiler V L

(9)

Reactions and Separations

highlights the diffi culty 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 purifi cation of a mixture that forms an azeotrope.

Closing thoughts

The ability to easily vary input specifi cations in the Excel spreadsheet and to visualize the effects on column performance has signifi cant value in teaching distilla-tion 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 fi rst 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 confi gurations, 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 fl exible 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 benefi t from both the accuracy and fl exibility of rigorous calculations and the insight and understanding gained by visualization of the venerable McCabe-Thiele diagram.

Literature Cited

1. McCabe, W. L., and E. W. Thiele, “Graphical Design of Fractionating Columns,” Ind. Eng. Chem., 17, pp. 605–611 (1925).

2. Taylor, R., et al., “Real-World Modeling of Distillation,”

Chem. Eng. Progress, 99 (7), pp. 28–39 (2003).

3. Wankat, P. C., “Teaching Separations: Why, What and When,” Chem. Eng. Education, 35 (3), pp. 168–171 (2001).

4. Kister, H. Z., “Distillation Design,” McGraw-Hill, New York, NY (1992).

5. Burns, M. A., and J. C. Sung, “Design of Separation Units Using Spreadsheets," Chem. Eng. Education, 30, pp. 62–69 (1996).

6. Lee, M.-J., and C.-H. Hu, “Isothermal Vapor-Liquid Equilib-ria for Mixtures of Ethanol, Acetone, and Diisopropyl Ether,”

Fluid Phase Equilibria, 109, pp. 83–98 (1995).

7. Chaudhry, M. M., et al., “Excess Thermodynamic Func-tions for Ternary Systems. 6. Total-Pressure Data and GE for

Acetone-Ethanol-Water at 50°C,” J. Chem. Eng. Data, 25, pp. 254–257 (1980).

8. Kister, H. Z., “Effects of Design on Tray Effi ciency in Com-mercial Towers,” Chem. Eng. Progress, 104 (6), pp. 39–47 (2008).

9. Egloff, G., and C. D. Lowry, Jr., “Distillation Methods, Ancient and Modern,” Ind. Eng. Chem., 21, pp. 920–923 (1929).

10. Sorel, E., “Sur la Rectifi caton de l’alcool,” Comptes Rendus,

58, p. 1128 (1889).

11. Seader, J. D., “The B. C. (Before Computers) and A. D. of Equilibrium-Stage Operations,” Chem. Eng. Education,

19 (2), pp. 88–103 (1985).

12. Kister, H. Z., “What Caused Tower Malfunctions in the Last 50 Years?,” Trans. IChemE., 81A, pp. 5–26 (2003).

13. Carlson, E. C., “Don’t Gamble With Physical Properties for Simulations,” Chem. Eng. Progress, 97 (10), pp. 42–46 (1996).

14. Doherty, M. F., and M. F. Malone, “Conceptual Design of Distillation Systems,” McGraw-Hill, New York, NY (2001).

15. Kister, H. Z., “Can We Believe the Simulation Results?,”

Chem. Eng. Progress, 103 (10), pp. 52–58 (2002).

16. Ohe, S., “Energy-Saving Distillation Through Internal Heat Exchange (HiDiC),” in “Distillation 2007,” Topical Confer-ence Proceedings, AIChE Spring National Meeting, Houston, TX, p. 13 (Apr. 22–26, 2007).

17. Zygula, T. M., “A Design Review of Steam Stripping Col-umns for Wastewater Service,” in “Distillation 2007,” Topical Conference Proceedings, AIChE Spring National Meeting, Houston, Texas, p. 609 (Apr. 22–26, 2007).

18. Kister, H. Z, “Distillation Troubleshooting,” Wiley, Hoboken, NJ (2006).

19. Love, D. L., et al., “Rethink Column Internals for Improved Product Separation,” Hydrocarbon Processing, pp. 97–105 (May 2007).

20. Bellner, S. P., et al., “Hydraulic Analysis is Key to Effective, Low-Cost Demethanizer Debottleneck,” Oil & Gas Journal,

102 (44), pp. 56–61, (2004).

21. Green, D. W., and R. H. Perry, “Perry’s Chemical Engineers’ Handbook,” 8th ed., p. 13–17, McGraw-Hill, New York, NY (2008).

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 publica-tions and 75 presentapublica-tions 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.

References

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