Structural Design of Extended Fully Thermally Coupled Distillation

Apr 27, 2001 - Because of extra degrees of freedom in the design of the columns, various designs can yield the same products for a given specification...
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Ind. Eng. Chem. Res. 2001, 40, 2460-2466

Structural Design of Extended Fully Thermally Coupled Distillation Columns Young Han Kim† Department of Chemical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu, Pusan, 604-714 Korea

A structural design procedure for extended fully thermally coupled distillation columns is proposed and applied to the example quaternary systems of several combinations of relative volatility in order to show the design performance. The procedure gives the optimum structure for a given quaternary separation, and therefore iterative computation encountered in the design using commercial packages is not required. Because of extra degrees of freedom in the design of the columns, various designs can yield the same products for a given specification. The structural design of this study leads to the most efficient design thermodynamically. With the outcome of the structural design, thermodynamic efficiency, arrangement of main and satellite columns, and location of interlinking streams are also examined. Introduction An extension of a fully thermally coupled distillation column (FTCDC) for the separation of a quaternary mixture is conceptually arranged as shown in Figure 1, which is proposed by Sargent and Gaminibandara.1 Though the design and control of FTCDCs, the Petlyuk columns, have been investigated for many years, the design of an extended FTCDC was rarely studied. Moreover, the industrial application of the Petlyuk column is reported in several publications,2-4 and many more implementations are expected soon. Most of chemical and petrochemical processes handle more than four components to be separated. The extension of the Petlyuk column is suitable for the separation of four or more components, because its thermodynamic efficiency is higher than that of a conventional distillation system or the original Petlyuk column. The conceptual study of the structure of the extended FTCDCs has been conducted by Sargent and Gaminibandara,1 Kaibel,5 and Agrawal.6 Their proposed three different arrangements of a main column and one or two satellite columns are evaluated by comparing the degrees of freedom for the difficulty of column operation and the minimum energy requirement for the thermodynamic efficiency.7 The study indicates that Agrawal’s structure is the most efficient. For the easy vapor transfer between a main column and satellite columns, the rearrangement of sections of the columns is proposed by Agrawal.8 The interchange of sections of the columns controls the pressure of the sections to make vapor flow without an external compression between the columns. Because the compression of vapor is complex and costly, the proper arrangement of the sections alleviates the operational difficulty of the extended FTCDC. An optimal design using a rigorous process model for the FTCDCs and the extended FTCDCs is conducted by Du¨nnebier and Panteides.9 More design variables than in previous studies are included in the design, and a rigorous simulation with nonlinear equilibrium is conducted in order to minimize the total of investment and operational costs. †

E-mail: [email protected]. Fax: 82-51-200-7728.

Figure 1. Schematic diagram of an extended fully thermally coupled distillation column. (Sargent and Gaminibandara arrangement)

Though a general structure of an extended FTCDC was presented in the previous studies, its structural design has not been addressed yet. The structural information, such as the numbers of trays and locations of interlinking, feed, and side product, helps to eliminate the iterative computation involved in the design using the conventional design method and design software. Also, the optimum structure ensures the highest thermodynamic efficiency of the column. In this study, the procedure of the structural design of the extended FTCDC is addressed and its structural analysis related to the design and operation of the column is conducted. Degrees of Freedom Analysis An extended FTCDC has a main column and two satellite columns along with a single reboiler and a condenser unlike a conventional three-column system for the separation of a quaternary mixture, which has three sets of reboilers and condensers. Among many arrangements of the sections of the extended FTCDC,

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groups: structural design related and operation related variables. Ten structural decisions are the numbers of stages of a main column and two satellite columns, feed and two side product locations, and four interlinking stages between the main column and satellite columns. The six operational decisions are reflux flow rate, vapor boilup rate, two liquid split ratios from the highest column to the top of the next column in two steps, and two vapor split ratios from the lowest column to the bottom of the next higher column. In the design of steady-state operation, the vapor split ratio is found from the optimum split ratio of the middle components to the top of the satellite columns and liquid split ratio. The computation detail of the design is given below. Thermodynamic Efficiency

Figure 2. Schematic diagram of a modified extended FTCDC.

the cascade arrangement shown in Figure 2 is employed for the analysis. The arrangement is the same as the Agrawal configuration6 except the top and bottom sections are combined with satellite columns. This is also shown in one or more operable configurations in Agrawal.8 The analysis of the degrees of freedom gives the information of design and selection of manipulated and controlled variables for the column operation. In Table 1, all of the variables associated with design and operation are listed, and the numbers of balance equations, equilibrium relations, and weir equations for liquid flow are counted. Liquid and vapor compositions are given for all of the trays of a main column and two satellite columns. While equilibrium is obtained in a reboiler, no equilibrium is formed in a total condenser. The additional number of 2 in material balance denotes a reboiler and a total condenser and 1 in equilibrium indicates the equilibrium of the reboiler. As a result, 18 degrees of freedom are found. When tight inventory control in a reboiler and a reflux drum is assumed, 2 degrees of freedom are reduced from the 18 degrees. The rest of them are divided into two

In binary separation, feed composition is located in the composition profile between top and bottom trays, and no significant mixing is produced in the feed tray. In a conventional two-column system of direct sequence for ternary separation, feed composition and the bottom product composition of the first column (feed to the second column) have to be the same distillation line, because two compositions are from one column. However, any distillation line of the ternary system hardly passes these two compositions at the same time, which leads to large mixing at the feed tray and the reduction of thermodynamic efficiency owing to the irreversibility of the mixing. On the other hand, the distillation lines of a main column and a prefractionator of the Petlyuk column for ternary separation include the compositions of the feed and interlinking trays. Therefore, no significant mixing occurs at the trays, and a higher thermodynamic efficiency than that of the conventional system is yielded. This explanation applies to the quaternary separation using an extended Petlyuk column. When a suitable distillation line is selected for the feed composition and two end compositions of the line and the middle of two other distillation lines of two satellite columns are matched, no significant mixing resulted. Because the shapes of the distillation lines and residue curves are similar, an illustration of residue curves is shown in Figure 3. In the figure, it is noticed that a single column

Table 1. Degrees of Freedom Analysis unknowns

number

nos. in trays flow splits in operation liquid composition vapor composition liquid flow rate in the tray vapor flow rate in the tray tray holdup reboiler holdup reflux drum holdup vapor boilup rate reflux flow rate

10 (NT1, NT2, NT3, NF2, NP1, NP3, NR2, NR3, NS1, NS2) 4 (VS1, VS2, LS2, LS3) (NT1 + NT2 + NT3 + 2)NC (NT1 + NT2 + NT3 + 1)NC NT1 + NT2 + NT3 NT1 + NT2 + NT3 NT1 + NT2 + NT3 1 1 1 1

total

(2NT + 2NT2 + 2NT3 + 3)NC + 3NT + 3NT2 + 3NT3 + 18 equations

component material balance equilibrium relation total material balance in the trays energy balance Francis weir equation

number (NT1 + NT2 + NT3 + 2)NC (NT1 + NT2 + NT3 + 1)NC NT1 + NT2 + NT3 NT1 + NT2 + NT3 NT1 + NT2 + NT3

total

(2NT + 2NT2 + 2NT3 + 3)NC + 3NT + 3NT2 + 3NT3

degrees of freedom

18

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Figure 3. Residue curves of quaternary systems.

can produce all four products, but a high liquid flow is required to maintain a high composition of the key component in side products. That is why the structures of Sargent and Gaminibandara1 and Kaibel5 are less efficient than that of Agrawal.6 The detailed design procedure is given in the next section. Structural Design In the design of a conventional distillation column, the number of trays is calculated by setting the reflux flow rate at between 1.2 and 2.0 times the minimum reflux flow rate.10 Otherwise, optimization for the minimization of the total annual expense including capital cost and energy expense is conducted to find the optimum tray number. When this procedure is applied to the design of an extended FTCDC, not only the total number of trays but also the locations of interlinking have to be found. Though the short-cut design equations for multicomponent distillation can be implemented by dividing the column into sections,11 the outcome is not accurate enough to be used for the rigorous simulation. Otherwise, time-consuming iterative computation is necessary. This problem applies to the design using commercial software. Moreover, many combinations of tray number and interlinking location do not give a convergent solution, because the composition match between two columns of interlinking cannot be obtained. Therefore, the minimum numbers of trays of a main column and satellite columns are computed first using a stage-to-stage computation in this study. The stage location of feed, side product, and interlinking between the main column and satellite columns is also obtained from the computation. Then, all of the tray-related numbers are proportionally increased for a practical distillation column. Because the column structure is derived from the structure of the minimum tray, the highest thermodynamic efficiency is obtained with the structure. The minimum tray system has no mixing in the feed tray and no remixing of intermediate componentssno irreversibility. From the residue curves of simple distillation of a quaternary mixture as seen in Figure 3, it is noticed that the curve varies widely according to the operational variables, such as liquid and vapor flow rates. Though the residue curves represent the composition profile of a packed column operated in total reflux and the distillation lines follow that of a tray column, both curves have similar patterns and the residue curves are used here for the simplicity of explanation.

The increased liquid flow moves the curve close to the intermediate components, but the lightest and heaviest products have little change of composition. In other words, the composition of side products determines the path of a residue curve. The paths of distillation lines of two satellite columns producing side products are determined by the composition of the products. If the feed composition is not far from the feed stage composition, it also decides the path for the main column. Unless the compositions of intermediate components in the feed and side product are close, the distillation lines of a main column and satellite columns are distant. That is, the numbers of trays in the sections of the main and satellite columns are different, and the dividing wall structure is not suitable because irreversible remixing of middle components occurs from the intentional matching of tray numbers. The minimum number of trays of a distillation column is computed in a total reflux operation in which the vapor composition of a stage is the same as the liquid composition of one stage above the stage. Assuming that the feed stage has equal liquid composition to the composition of a saturated liquid feed and the tray efficiency is ideal, one can calculate the liquid composition of the stages above the feed tray from eq 1

xn+1,i ) Rixn,i/

∑j Rjxn,j

(1)

where the subscript n denotes the nth tray counted from the bottom and R is the relative volatility. For a nonideal system, an equilibrium constant computed from the equilibrium relation can replace the relative volatility with iterative updating of the constant. Similarly, the liquid composition of the stages below the feed tray is given as

xn-1,i ) xn,i/[Ri

∑j (xn,j/Rj)]

(2)

This stage-to-stage design procedure is applied to satellite columns beginning with the composition of side products. The locations of interlinking between the main column, column II in Figure 2, and the satellite columns, columns I and III, are determined by comparing the profiles of the liquid composition of the main column and the satellite columns. The exact matches between them are impossible, and therefore the closest match is adopted to determine the location of interlinking. Because the composition of overhead and bottom products is given, the minimum number of the trays in the satellite columns, columns I and III in Figure 2, is obtained from the stage-to-stage computation. Then, the location of side product trays is also given from the computation. For a better understanding of the procedure, the result of stage-to-stage computation is given in Figure 4. This is the outcome of an ideal example system having 3.375, 2.25, 1.5, and 1 of relative volatility and equimolar feed. The specification of the key component in overhead and bottom products is 0.975 mole fraction, and that of two side products is 0.95 mole fraction. The circles indicate the liquid composition of column II, where F is the composition of the saturated liquid feed. The plus symbols represent the composition of column I, which produces the bottom product and the second side product. The circle and plus symbols meet

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the addition of the irreversible mixing in the arrangement of sections of an extended FTCDC. The arrangement of Figure 2 is more operable than the original Agrawal arrangement. Placing column III at the highest position and column II at the next highest makes all of the liquid flow in order without external pumping. Obviously, the dynamic head difference has to be greater than the difference of column pressure. Now maintaining the pressure of column I highest and column II next highest allows all of the vapor to be transported in order without external compression. Operational Decisions There are six operational variablessreflux flow rate, vapor boilup rate, two liquid split ratios, and two vapor split ratiossbetween a main column and two satellite columns. Among them, three are eliminated in the steady-state calculation with the equimolal overflow assumption. Namely, the vapor boilup rate is found from the reflux flow rate and overhead product rate. The vapor split ratio is also derived from the liquid split ratio and the flow rate of the side product. Therefore, three variablessreflux flow rate and two liquid split ratioss have to be determined, and the minimum liquid flow requirement provides an initial guideline to find the variables. For the simple analysis to calculate the minimum liquid flow, the arrangement of an extended FTCDC of Figure 1 is utilized, which is the structure of Sargent and Gaminibandara.1 Though the structures of Figures 1 and 2 are different, liquid flows in Figure 2 are computed from liquid flows in Figure 1 by relocating sections of the column. The minimum liquid flow13 for column I is given as

Figure 4. Liquid composition in an extended FTCDC system with a minimum number of trays.

at two places of their ends: one close to the vertex and the other on the base of the right quadrilateral in Figure 4. The former interlinking denotes lower interlinking between columns I and II, and the latter is upper interlinking. Similarly, the interlinking between columns II and III is illustrated with two close places of circle and times sign symbols. The minimum numbers of trays of three columns and the location of feed, side products, and interlinking are obtained by counting the number of symbols, and the actual numbers are taken as twice the minimum numbers, which is of common practice.12 The numbers are listed in Table 2. The first part of the structural design, the minimum tray numbers, of this study is based on the assumption of total reflux operation and ideal tray efficiency, which indicate an equilibrium process of no irreversible mixing. In other words, the maximum thermodynamic efficiency is yielded from the designed structure of a distillation system. Though many different arrangements of sections of the extended FTCDC are available, the structural design of this study adopts the arrangement of Figure 2, a modification of Agrawal arrangement.6 Christiansen et al.7 compared the thermodynamic efficiency for three different systems and found that the arrangement shows the highest efficiency. This result supports that the structural design of this study gives the most efficient arrangement for an extended FTCDC. For a more operable arrangement of the column, various alternatives are suggested in Agrawal.8 Figure 4e in the study is similar to Figure 2 of this study except the interlinking between columns I and III composed of components of B and C. However, this interlinking may reduce the thermodynamic efficiency of the column. Because column I has C-rich streams and column III has B-rich streams, their interlinking produces an irreversible mixing to deteriorate the efficiency. The utilization of the structural design prevents

L1M )

RDF RA - RD

(3)

From column I, components B and C leave from either the top or bottom of the column. Let the portion of component B from the top of the column be γB and that of component C be γC. With the minimum liquid flow, the number of trays of column I is infinite and the following relation is formulated.13

RB dxB,d/L1MxB + 1 ) RD dx /L Mx + 1 D,d

1

(4)

D

No component D is contained in the overhead product, and replacing L1M with eq 3 gives

γB )

RB - RD RA - RD

(5)

Table 2. Design Result of the Example Systems system

feed

NT1

NT2

NT3

NS1

NP1

NF2

NR2

NS2

NP3

NR3

L1

L2

L3

S1

F1 F2 F3 F1 F2 F3 F1 F2 F3

57 61 59 54 60 54 49 49 47

22 18 24 22 16 22 16 16 18

58 62 60 55 55 51 (55) 40 40 40

7 7 26 (7) 8 8 8 9 9 11 (7)

37 37 48 (37) 34 34 34 37 37 31 (37)

12 14 14 12 12 12 12 12 14

16 16 20 14 14 14 14 14 16

8 8 6 9 7 7 7 7 7

21 25 23 21 21 21 32 (21) 32 (21) 32 (21)

49 53 51 45 47 45 39 (33) 39 (37) 39 (35)

910 910 1160 1100 440 610 1560 1650 860

1240 1320 1220 1190 1060 890 1670 1710 1080

2190 1750 2840 2390 1980 2280 2430 2330 2370

S2 S3

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Similarly,

yn,i ) Rixn,i/ γC )

RC - RD RA - RD

(6)

The vapor flow in column I is liquid flow and net material flow from column I to column II; therefore,

V1M ) L1 + FzA + FzBγB + FzCγC

(7)

The vapor flow in column II is obtained from

V2M )

RCFzBβ RACFzA + RAC - φ RBC - φ

(8)

where φ is a solution of the following Underwood equation:14

RBCFzBγB FzAγC RACFzA + + ) V1M RAC - φ RBC - φ 1-φ

(9)

The ratio β is of component B from the top of column II and is given as

β)

RB - RC RA - RC

(10)

Then,

L2M ) V2M - FzA - FzBγBβ

(11)

The minimum vapor flow in column III is obtained from

V3M )

RABFzA RAB - φ

(16)

which is simplified as

yn,i ) Kn,ixn,i

(17)

In a matrix form, the material balance is

[

][ ] [ ]

A11 A12 A13 X1 C1 A21 A22 A23 X2 ) C2 A31 A32 A33 X3 C3

(18)

where the liquid composition is separated into three groups for a main column and two satellite columns. Because two coefficient matrices are known for a given liquid composition, the new liquid composition is computed from eq 18. However, an iterative renewal is necessary because a nonlinear equilibrium relation contains liquid composition, and a direct computation of the liquid composition is not available. To improve the convergence of the iterative composition calculation, a relaxation is added to the renewal of liquid composition.

Xk+1 ) rXnew + (1 - r)Xk

(19)

where r is a relaxation factor of the value between 0 and 1, and 0.3 is implemented in this study because fast convergence is obtained with the value. The convergence limit is set to 1.0 × 10-4 times the number of components times the number of trays. When the sum of the absolute value of composition variation is less than the limit, the iteration stops.

(12) Example Systems

where φ is a solution of the following Underwood equation:

FzAγBβ RABFzA + ) V2M RAB - φ 1-φ

(13)

L3M ) V3M - FzA

(14)

and

The actual liquid flow rate is obtained from the steady-state simulation by examining the specification of products, in which the minimum flow shows the initial values for the simulation. Steady-State Simulation A procedure15 for the composition calculation of a multicomponent distillation is modified for an extended Petlyuk column. An unsteady-state material balance for component i at the nth stage of a column is

Mn

∑j Rjxn,j

dxn,i ) Ln+1 xn+1,i + Vn-1 yn-1,i - Ln xn,i - Vn yn,i dt (15)

where Mn is liquid holdup on the tray. In the steadystate operation, the left-hand side of the equation is eliminated. The vapor composition is replaced with a liquid composition and an equilibrium relation given with a constant relative volatility.

Because the design of this study does not utilize shortcut design equations, a generalized design using a wide range of relative volatilities is not conducted here. Instead, three systems of a typical combination of relative volatilities are examined with three different feed compositions. The systems are selected for the combination of relative volatilities of even (S1; R’s ) 3.375, 2.25, 1.5, and 1), 2-2 (S2; R’s ) 6, 4, 1.5, and 1), and 1-3 distributions (S3; R’s ) 6, 2.25, 1.5, and 1). The system having a large difference in relative volatility is not examined here because the separation of the system is relatively easy. Also, because the system with very close relative volatility needs an unusually large number of trays, it is not included in the example systems. Three combinations of feed composition are equimolar (F1), 0.7-0.1-0.1-0.1 (F2), and 0.4-0.4-0.1-0.1 (F3). In the selection of feed composition, it is assumed that the effect of a high concentration of components C or D is similar to that of A or B. The design specification of products for all systems is set to 0.975 mole fraction of the key component in overhead and bottom products and 0.95 mole fraction of the key component in two side products. Results and Discussion As a base example, the system of even distribution of relative volatility (S1) and equimolar feed (F1) is selected and the explanation of design outcome is given for the system. Using the stage-to-stage design equa-

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between interlinking trays and between the feed and the feed tray is responsible. Because the design of this study is based on the structure of the minimum trays, the design outcome contains an optimum structure. On the other hand, the design using commercial packages surveys a proper structure until the liquid composition matches at interlinking trays. This procedure requires a lot of trialand-error computation and does not guarantee the optimum structure. Conclusion

Figure 5. Liquid composition in an extended FTCDC system with a practical number of trays.

tions, eqs 1 and 2, liquid composition profiles of main and two satellite columns are evaluated. The profiles of satellite columns are computed until two end products, overhead and bottom products, are obtained and there is matching between a main column and satellite columns for the interlinking streams. For the minimum number of trays with ideal tray efficiency, the liquid composition profile is demonstrated in Figure 4. The circles indicate the composition of a main column (column II in Figure 2) and so does the plus symbols for column I. The times signs are for column III. The F is the feed composition. The close plus or times symbol to a circle shows the interlinking tray composition. To examine the performance of the proposed design of this study, the design procedure is applied to the example systems and the design outcome is listed in Table 2. The system number and feed specification are given above. The symbols of the numbers of trays and location of interlinking, feed and side product are indicated in Figure 2 and so are the liquid flow rates. In the design of structural decisions, several systems have a large discrepancy in composition between main and satellite columns even for the closest connection of the compositions when the interlinking location is searched. The simulation in these cases gives a convergence problem or a quite distant product composition from the specification, and therefore the location of interlinking and product stream is modified to compensate the discrepancy. The parentheses in Table 2 contain the original location computed from the minimum number of trays. For a real column, the minimum numbers of trays and locations of interlinking, feed, and side products are multiplied by 2, which is common in practical design.12 Because the numbers are scaled proportionally, the structure of the minimum design is maintained and the high efficiency of the extended Petlyuk column is sustained. Using the minimum liquid flow calculated using eqs 3, 11, and 14, the initial distribution of liquid flow is employed in the steady-state simulation to compute a liquid composition profile. Because the minimum liquid flow is conservative,13 the modified liquid flow from the initial distribution is applied to find the optimum liquid flow. Also, the liquid flow rates are adjusted for the satisfaction of specification for all of four products. The profile of liquid composition for a real system of system S1 with feed F1 is shown in Figure 5. The usage of symbols is the same as that in Figure 4. Unlike the system of minimum trays, mixing in the feed tray and remixing of intermediate components in a main column are observed. The difference of compositions

A structural design procedure for extended FTCDCs is proposed and exercised to three different systems having different compositions of feed. The proposed design is based on the minimum tray structure of the columns, which is the optimum structure found from the equilibrium of the system. While most of field designs utilizing commercial software do not give any information on the optimum column structure, the structural design ensures the optimum structure of the column. The design outcome of example systems proves the effectiveness of the proposed design. Unless the structural information is given, numerous iterative computations are required to find a column system for the given separation and an optimization has to be followed to determine the optimum structure. In addition, the thermodynamic efficiency of the extended FTCDC is investigated using the structural design. Acknowledgment Financial support from the Korea Science and Engineering Foundation through Grant 2000-1-30700-001-2 is gratefully acknowledged. Nomenclature A ) coefficient matrix in eq 18 C ) coefficient vector in eq 18 d ) overhead product flow rate [mol/h] F ) feed flow rate [mol/h] F1 ) feed number 1 K ) equilibrium constant L ) reflux flow rate [mol/h] LS ) liquid split ratio M ) liquid holdup [mol] NC ) number of components NF ) feed tray number NP ) side draw tray number NR ) location of the upper side stream NS ) location of the lower side stream NT ) number of trays r ) relaxation factor S1 ) system number 1 V ) vapor flow rate [mol/h] VS ) vapor split ratio X ) liquid composition matrix in eq 18 x ) liquid composition [mole fraction] y ) vapor composition [mole fraction] z ) feed composition [mole fraction] y ) vapor composition [mole fraction] Greek Letters R ) relative volatility β ) split ratio of component B to the top of column II γ ) split ratio of a component to the top of column I φ ) solution of the Underwood equation

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Superscript M ) minimum Subscripts A ) component A B ) component B C ) component C D ) component D d ) overhead product i ) component i n ) tray number 1 ) column I 2 ) column II 3 ) column III

Literature Cited (1) Sargent, R. W. H.; Gaminibandara, K. Optimum Design of Plate Distillation Columns. In Optimization in Action; Dixon, L. W. C., Ed.; Academic Press: London, 1976; pp 267-314. (2) Wolff, E. A.; Skogestad, S. Operation of Integrated ThreeProduct (Petlyuk) Distillation Columns. Ind. Eng. Chem. Res. 1995, 34, 2094-2103. (3) Midori, S.; Nakahashi, A. Industrial Application of Continuous Distillation Columns with Vertical Partition. Proc. 5th Int. Symp. Sep. Technol.sKorea Jpn. 1999, 5, 221-224. (4) Lestak, F.; Egenes, D.; Yoda, H.; Hamnett, C. Kellogg Divided Wall Column Technology for Ternary Separation. Proc. 5th Int. Symp. Sep. Technol.sKorea Jpn. 1999, 5, 233-236. (5) Kaibel, G. Distillation Columns with Vertical Partitions. Chem. Eng. Technol. 1987, 10, 92-98.

(6) Agrawal, R. Synthesis of Distillation Column Configurations for a Multicomponent Separation. Ind. Eng. Chem. Res. 1996, 35, 1059-1071. (7) Christiansen, A. C.; Skogestad, S.; Lien, K. Complex Distillation Arrangements: Extending the Petlyuk Ideas. Comput. Chem. Eng. 1997, 21, S237-S242. (8) Agrawal, R. More Operatable Fully Thermally Coupled Distillation Column Configurations for Multicomponent Distillation. Trans. Inst. Chem. Eng. 1999, 77, Part A, 543-553. (9) Du¨nnebier, G.; Pantelides, C. C. Optimal Design of Thermally Coupled Distillation Columns. Ind. Eng. Chem. Res. 1999, 38, 162-176. (10) McCabe, W. L.; Smith, J. C. Unit Operations of Chemical Engineering, 3rd ed.; McGraw-Hill Book Co.: New York, 1976; p 568. (11) Triantafyllou, C.; Smith, R. The Design and Optimisation of Fully Thermally Coupled Distillation Columns. Trans. Inst. Chem. Eng. 1992, 70, Part A, 118-132. (12) Glinos, K. N.; Malone, M. F. Design of Sidestream Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 822828. (13) King, C. J. Separation Processes, 2nd ed.; McGraw-Hill Book Co.: New York, 1980; pp 416-421. (14) Glinos, K. N.; Malone, M. F. Minimum Vapor Flows in a Distillation Column with a Sidestream Stripper. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 1087-1090. (15) Wang, J. C.; Henke, G. E. Tridiagonal Matrix for Distillation. Hydrocarbon Process. 1966, 45, 155-163.

Received for review August 31, 2000 Accepted March 15, 2001 IE000791D