Design of Petlyuk Distillation Columns Aided with Collocation

Ind. Eng. Chem. Res. 2007, 46, 5365-5370

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PROCESS DESIGN AND CONTROL Design of Petlyuk Distillation Columns Aided with Collocation Techniques Miguel Vaca,†,‡ Arturo Jime´ nez-Gutie´ rrez,*,† and Rosendo Monroy-Loperena§ Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya, AV. Tecnolo´ gico y A. Garcı´a Cubas, Celaya, 38010, Gto, Me´ xico, DiVisio´ n de Ingenierı´a Quı´mica y Bioquı´mica, Tecnolo´ gico de Estudios Superiores de Ecatepec, AV. Tecnolo´ gico S/N, Ecatepec, 55210, Me´ xico, and ROMON, S.A., Paseo de los Pirules 124, Me´ xico, 04250, Me´ xico

The use of shortcut methods for the design of complex distillation systems provides preliminary structures that need to be updated, typically by a recursive use of simulations. Although this procedure is generally effective, it can be time-consuming. In this work, collocation techniques are used as part of the design of Petlyuk distillation columns for the separation of ternary mixtures. Once a preliminary design is generated by a shortcut method, collocation techniques are applied to refine the location of the interconnecting and sidestream product stages, as well as the flowrate values of interlinking streams. The proposed method gives a proper distribution of stages for the Petlyuk system. The final designs are easily obtained with the proposed methodology, with a significant lower effort than that required when preliminary designs are corrected using recursive simulations. 1. Introduction Because of their potential energy savings, thermally coupled distillation arrangements have received special attention in the past few years. Through a vapor-liquid interconnection between two columns, thermally coupled arrangements can be implemented. Studies at minimum reflux conditions indicated that thermally coupled structures could provide lower internal vapor flowrates and lower energy consumption levels than the conventional direct and indirect sequences for the separation of ternary mixtures (see for instance the works by Cerda and Westerberg1 and Fidkowski and Krolikowski2). These initial results on thermally coupled systems, however, did not provide the tray structure for the distillation columns. The three most-widely analyzed thermally coupled distillation systems are the sequence with a side rectifier, the sequence with a side stripper, and the Petlyuk arrangement,3-6 although other simpler structures have been recently proposed.7-11 Petlyuk columns provide in general higher energy savings than the sequences with side columns. Despite this incentive, it was not until recent times that Petlyuk systems gained some practical implementation in Europe, Japan, North America, and South Africa.12 We focus in this work on a design strategy for Petlyuk columns. Annakou and Mizsey13 have classified the design methods for Petlyuk systems as those following hierarchical approaches and those using algorithmic methods. In the hierarchical approach, a preliminary design based typically on the FenskeUnderwood-Gilliland-Kirkbride (FUGK) shortcut equations14 is developed first. In order to apply the FUGK equations, the Petlyuk column is decomposed into a sequence of three * To whom correspondence should be addressed. E-mail: arturo@ iqcelaya.itc.mx. Fax: (+52- 461) 611-7744. † Instituto Tecnolo ´ gico de Celaya. ‡ Tecnolo ´ gico de Estudios Superiores de Ecatepec. § ROMON, S.A.

conventional distillation columns; although this procedure provides some basic tray structure for the Petlyuk system, it is not entirely equivalent to the thermally coupled system except for minimum reflux calculations, as indicated by Carlberg and Westerberg.15 Then, rigorous simulations are carried out to validate the design, and some optimization criterion such as the minimization of the energy consumption can be used to obtain the final design. Works based on this approach include those by Triantafyllou and Smith,4 Annakou and Mizsey,13 Herna´ndez and Jime´nez,6 Muralikrishna et al.,16 and Jime´nez et al.10 A similar approach has been used to design thermally coupled distillation sequences for quaternary and multicomponent mixtures.17,18 The other approach is based on the use of nonlinear mixed integer programming models that solve superstructures that include all of the alternatives of interest for the system.19-22 As a result, the best sequence, according to the specified objective function, is identified, but typically at the expense of a significant computational effort. Shah and Kokossis23 have proposed the use of supertasks as opposed to superstructures for the synthesis of complex distillation systems. The optimization problem is formulated as a mixed integer linear programming model. Shah and Kokossis23 incorporated some results from previous research efforts, such as shortcut methods, new methods for the calculation of minimum reflux ratios, and strategies for feasibility analysis. Models based on tray-by-tray calculations have also been reported. Amminudin et al.24 have developed a rather complicated semirigorous method based on material balances and equilibrium calculations, which also uses the decomposition of the thermally coupled system into a sequence of three conventional distillation columns. Kim25 distinguishes the structural design of the Petlyuk column, which is done first from calculations at total reflux, from the operating design, in which the required values for the reflux ratios and interconnecting flowrates must be used. In both cases, rigorous equilibrium

10.1021/ie070281k CCC: $37.00 © 2007 American Chemical Society Published on Web 07/12/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007

Figure 1. (a) Cascade of N stages. (b) Generic separation stage.

calculations were used. Nonetheless, the resulting structure typically needs to be adjusted to provide a final design. In general, corrections are needed when shortcut or simplified tray-by-tray methods are used for the design of complex distillation systems. Adjustments for the preliminary design of Petlyuk columns are normally carried out on the values of the reflux and interconnecting streams flowrates. However, there are cases in which the number of stages of some sections of the Petlyuk column needs also some adjustment. The need for such modifications can be attributed to the assumptions of constant molar overflow and constant relative volatilities, implicit conditions on the use of the FUGK shortcut method. Amminudin et al.24 have mentioned such limitations. Also, the decomposition of the thermally coupled system into three conventional distillation columns affects the preliminary design. Corrections on values of the reflux and flowrates of the interconnecting streams can be easily implemented because of the continuous nature of such variables, as shown for instance in the works by Triantafyllou and Smith4 and Herna´ndez and Jime´nez.6 In contrast, the corrections on the number of stages are not as straightforward because the task involves discrete variables. Under these conditions, the correction to the number of stages for each section of the column is typically carried out through repetitive simulations when commercial simulators are used.25 Collocation methods provide an alternative tool for the design of separation systems.26-35 Although collocation techniques have been successfully applied to the analysis and design of conventional distillation columns, their use for the design of complex distillation systems, in which the number of discrete variables is higher, has not been reported. In this work, a methodology to correct the preliminary design of Petlyuk systems using orthogonal collocation techniques is reported. The methodology thus falls into the hierarchical approach for the design of complex distillation systems, as defined by Annakou

and Mizsey.13 Within this procedure, standard nonlinear optimization methods are applied to modify the location of the interconnecting streams, the location of the side product, and the values of internal flowrates. The problem is formulated so as to meet the specified compositions for the three product streams. 2. Use of Collocation Techniques in Simulation Processes When collocation techniques are applied, the profiles of the process variables are approximated through polynomial equations, generally with low order. The process variables are then treated as continuous functions with respect to their positions. This approach provides a significant reduction in the size of the problem, and the discrete variables can be handled as continuous since the polynomial equation applies for any noninteger number of stages. Standard nonlinear optimization techniques can then be applied to solve the design problem, as opposed to the use of mixed integer nonlinear programming models. The design task involves discrete variables such as the number of stages and the location of the feed stage or side product. A good description of the use of collocation techniques for the simulation of conventional distillation processes has been reported in the works by Stewart et al.28 and Seferlis and Hrymak.31 Here we show the basic ideas of such an approach when a cascade of N stages is considered to separate a mixture of NC components (see Figure 1a). Molar flowrates, component mole fractions, and saturated states are specified for the liquid and vapor feed streams L0 and VN+1. Consider the generic stage sj shown in Figure 1b. One can develop a simplified model by assuming that an average value of the relative volatility can be used and that the internal flowrates are constants. Under these conditions, the basic equations for the stage are as follows.

Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007 5367

Component material balances

n

n

li(sj - 1) + Vi(sj + 1) - li(sj) - Vi(sj) ) 0 i ) 1, NC - 1 (1)

li(s) )

∑

GLm(s)li(sm) GLm(s) )

m)0

k)0,k*m sm n+1

n+1

Summation constraints

Vi(s) )

∑lk (sj) ) L(sj)

(2)

k)1

Equilibrium relationships

Rili(sj)

Vi(sj) )

V(sj)

NC

(3)

∑Rklk(sj)

k)1

When one applies eqs 1-3 for sj ) j, with j ) 1 to N, a complete order model is obtained. The model therefore comprises a system of 2NNC nonlinear equations that provide the 2NNC molar flowrates in the liquid and vapor phases, li,j and Vi,j. Collocation techniques can be used to provide a reducedorder model. The profile of molar flowrates can be approximated as a function of the number of stages through a polynomial of NCP - 1 degree. Since the polynomial contains NCP unknown coefficients, one only has to apply the model, in principle, to NCP stages or collocation points. The location of the sj collocation points could be arbitrary. However, Stewart et al.28 have shown that, for conventional columns, it is convenient to locate n ) NCP - 1 of such points (s1, s2, ..., sn) as the roots of a certain class of orthogonal polynomials of n degree, plus the entry points s0 ) 0 for the liquid phase and sn+1 ) N + 1 for the vapor phase. Hahn’s discrete orthogonal polynomials of n degree, Ηn(s - 1;δ,λ,N - 1), provide the recommended choice; Hahn’s polynomials are defined by the ortogonality condition:28 M

w(s;δ,λ,M)Ηm(s;δ,λ,M)Ηn(s;δ,λ,M) ) 0 ∑ s)0

m * n (4)

The weight function is defined by

w(s;δ,λ,M) )

(δ + 1)s (λ + 1)M-s

(5)

s!(M - s)!

with (δ + 1)s or (λ + 1)M-s determined with

{

GVm(s)Vi(sm) GVm(s) )

m)1

NC

(a)k )

∑

a(a + 1) ... (a + k - 1) k > 0 1 k)0

}

(6)

The parameters (δ,λ) of the weight function w(s;δ,λ,M) must be greater than -1; their values give the distribution of the collocation points. The set (δ,λ) ) (0,0), suggested by Stewart et al.,28 gives uniform weights, w(s;0,0,M) ) 1. Some of the Hahn polynomials, their roots, and properties are also provided by Stewart et al.28 One important property of Hahn’s polynomials is that the complete order model is recovered when the degree of the polynomial is equivalent to the number of stages, i.e., when n ) N. As a result of the procedure, the collocation model comprises a set of 2nNC equations, whose solution provides the 2nNC unknown molar flowrates in the collocation points. Whenever needed, the molar flowrates at any other stage sj ∈ [1,N] are obtained by Lagrangean interpolation,

∏

s - sk

∏

- sk

(7)

s - sk

k)0,k*m sm

- sk

(8)

It is frequently necessary to consider several finite elements, i.e., more than one polynomial, to represent the profiles within one section or cascade. In such cases, continuity equations must be added for the flows in the frontiers of the finite elements.31 Complex columns, such as thermally coupled distillation sequences, can be seen as an arrangement of several sections or cascades, with additional blocks for feed or product stages (reboilers, condensers, feed, and sidestream trays). For such complex systems, it is necessary to couple the set of collocation equations for all the tray sections together with the material and equilibrium relations for the additional blocks. More complex specific stages can be considered if energy balances are added to the model. 3. Refined Design of Petlyuk Columns Figure 2 shows the basic Petlyuk design. One can identify six tray sections as part of the Petlyuk structure. We need a preliminary design to apply the proposed methodology, and it was obtained by using the FUGK method as described by Triantafyllou and Smith4 and Herna´ndez and Jime´nez.6 The structure of the Petlyuk column is approximated as a sequence of three conventional distillation columns from which the six sections of the Petlyuk system can be obtained. Each conventional column was designed assuming 98% of recovery of the key components. The preliminary design procedure provides the number of stages for each of the six section of the arrangement(Nk, k ) 1, ..., 6), the flowrates of the interconnecting streams (LS, VI), the reflux for the main column (R), and the flowrate of two of the product streams (D, S, B). Such variables define the number of stages in the prefractionator, the number of stages in the main column, the feed stage, the upper and lower interconnecting stages (NL, NV), and the stage for the side product (NS). Figure 3 shows the approach used to correct the preliminary design of the Petlyuk column. A basic simulation model is used as a first validation of the preliminary design. The simulation model at this stage is based on constant molar overflows and constant relative volatilities. If the design purities are met (or within a specified tolerance), then the design can be accepted. Otherwise, the design must be corrected. From the set of design variables, R, LS, VI, NL, NS, NV, a subset X of such variables will be subject to correction, but keeping the total number of trays for each column of the Petlyuk structure the same as provided by the shortcut method. An optimization module is used to treat the subset X as optimization variables. It can be noted that since collocation techniques are used, continuous values of NL, NS, and NV are allowed, so that a nonlinear programming method can be applied. The optimization method is set to minimize eq 9 as the objective function. 2 sp 2 sp 2 (9) f(X) ) (xA,D - xsp A,D) + (xB,S - xB,S) + (xC,B - xC,B)

In this work, the Simplex method by Nelder and Mead36 was used to perform the optimization step. The resulting refined design from this procedure is finally validated with a commercial software such as Aspen-Plus 11.1.

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Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007 Table 1. Data for Case of Study I component

Ri

zi

xi,D

xi,S

xi,B

i-pentane n-pentane n-hexane

3.04 2.44 1.00

0.33 0.33 0.34

0.98 0.02 0.00

0.02 0.96 0.02

0.00 0.02 0.98

Table 2. Data for Case of Study II component

Ri

zi

xi,D

xi,S

xi,B

n-pentane n-hexane n-heptane

5.06 2.20 1.00

0.15 0.70 0.15

0.91 0.09 0.00

0.005 0.990 0.005

0.00 0.09 0.91

Table 3. Results for Case of Study I simulationa design

NL

NS

NV

preliminary 39 72 77 refined 30.89 69.90 76.98 validated 31 70 77 a

Figure 2. Schematic representation of the Petlyuk column.

xA,D ) 0.98 xB,S ) 0.96 xC,B ) 0.98 0.98 0.98 0.98

0.94 0.96 0.97

0.96 0.97 0.99

Numbers in bold indicate design specifications.

in the products, sharp splits between B and C at the top and between A and B in the bottoms of the main column are assumed. Also, a symmetrical separation is assumed between A and C in the side product. Such assumptions are fairly standard for these types of design problems. For the application of the collocation method, one finite element was generally taken for each 7 to 10 discrete stages, and four collocation points were used for each element. Both the number of finite elements and the number of collocation points were the same for each iteration of the search procedure. To avoid the addition of continuity equations between two given sections, one collocation point was set in each border of the finite elements. 4. Results and Discussion

Figure 3. Overall strategy to provide an improved design.

It is important to note that at each step of the optimization algorithm, the simulation of the Peltyuk column is performed with specific values of the interconnecting and side product stages, NL, NS, NV. The model is defined with the number of finite elements for each section of the Petlyuk column and the number of collocation points per element. At each step of the optimization algorithm, the locations of the collocation points, provided by the roots of the Hahn’s polynomials, are needed. These can be readily determined with standard routines for polynomials roots. To illustrate the application of the method, the two cases given in Tables 1 and 2, taken from Jime´nez et al.,10 were considered. The feed flowrate was taken as 45.4 kmol/h (100 lbmol/h), and the design pressure was set so as to ensure the use of cooling water in the condenser. To complete the component distribution

Table 3 gives the results for case I. From the preliminary design, the main column had the following tray distribution, NL ) 39, NS ) 72, and NV ) 77 (see Figure 2). The total number of trays for the prefractionator was 36 and, for the main column, 80. Tray-by-tray simulations using the simplified model (eqs 1-3) for this configuration showed that the side and bottoms products did not reach the desired purities (row 1). The main deviations are observed for the intermediate and heavy components. The application of the methodology to refine the design, with X ) (NL,NS,NV) as independent optimization variables, provided NL ) 30.89, NS ) 69.90, and NV ) 76.98. These refined locations were obtained without changes in the total number of stages for either the prefracctionator or the main column; the internal flowrates determined from the preliminary design also remained unchanged. It is also relevant to note that the liquid interconnecting stage in the main column had a significant modification from the preliminary design of the shortcut method. This interconnecting stage was changed from stage 39 to stage 31. The vapor interconnecting stage, on the other hand, remained unchanged, while the stage for the side product changed from stage 72 to stage 70. When the modified design was simulated with the full-order model (assuming constant molar overflows and relative volatilities), the design specifications were practically met, as can be observed in row 2. Finally, the validation process using Aspen Plus indicated that the design purities were met (row 3). The results for the second case of study are given in Table 4. It can be seen how the preliminary design was not satisfactory. A preliminary reflux ratio R ) 6.3 was obtained with the shortcut method, and under these conditions, the design provided

Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007 5369 Table 4. Results for Case of Study II simulationa design

NL

NS

NV

LS/F

VI/F

xA,D ) 0.91

xB,S ) 0.99

xC,B ) 0.91

preliminary refined validated

10 10.94 11

18 15.94 16

23 23.44 24

0.3885 0.5025 0.5025

0.7429 1.1053 1.1053

0.88 0.92 0.92

0.92 0.99 0.99

0.65 0.91 0.90

a

Numbers in bold indicate design specifications.

product purities out of specifications (row 1). The deviation is particularly noticeable for the heavy component in the bottom product. In this case, we found that the correction to the set of independent variables X ) (NL,NS,NV) used for the previous case of study was not sufficient to correct the design. It was therefore needed to try a new set of variables, which now included the interconnection flowrates, such that X ) (NL,NS,NV,LS,VI). Different values of the reflux ratio for the main column were tried, and the refined design was obtained with R ) 14, as shown in row 2 of Table 4. The basic tray-by-tray simulation indicated that the design met the specifications for products purities. When the Aspen Plus simulator was used, a minor adjustment in the reflux value was still needed to provide the desired product purities (row 3). It can be noted how a major change in the reflux value was needed in this case. The total number of trays for this case (as provided by the preliminary design) was 12 for the prefractionator and 27 for the main column. In both cases of study, there were at least three discrete variables to be determined, which would have required a large number of combinations to repeatedly simulate the column if the design had been carried out only with the use of commercial simulators. In contrast, the refined design was found rapidly using the suggested methodology. It is also convenient to emphasize that the basic simulation module used to validate the preliminary design and the one provided by the optimization search is based on constant relative volatilities and overflows, which are implicit in the FUGK method. The validation process that was carried out indicates that such a simple model is satisfactory for the types of hydrocarbon mixtures here considered. Therefore, the need for corrections to the preliminary design (which are required as part of the hierarchical approach) can be attributed mainly to the use of a sequence of three conventional distillation columns to represent the Petlyuk system, and to the limitations of the empirical correlations (Gilliland or Kirkbride) rather than to the assumptions of constant molar overflow and constant relative volatilities. This observation is consistent with the remarks by Amminudin et al.24 5. Conclusions A methodology based on the application of orthogonal collocation techniques to improve a preliminary design of Petlyuk columns has been presented. Since preliminary designs are usually obtained through the application of shortcut methods, they require some adjustments in the tray structure. The application of the proposed methodology uses orthogonal collocation techniques to a model based on constant relative volatilities and overflows and provides the proper distribution of stages and/or interconnecting flowrates that meet a specified set of values for product purities. A relevant characteristic of the method is the possibility of using standard nonlinear optimization techniques, such as the simplex method by Nelder and Mead, as opposed to the need for a mixed integer nonlinear programming model. Given the simplicity of the shortcut methods and the characteristics of the collocation method, the

methodology is quite straightforward, and for ideal mixtures, it competes favorably with methods that use stage-by-stage calculations, such as those by Amminudin et al.24 and Kim.25 It is apparent that the need for corrections to the preliminary design is mostly due to the decomposition of the Petlyuk system into a sequence of conventional distillation columns and to the limitations of the FUGK shortcut method, rather than to the assumptions of constant molar overflows and constant relative volatilities. The proposed method reduces significantly the effort to provide improved designs if compared to the recursive use of simulation runs with commercial software. The approach can be extended to thermally coupled distillation systems for quaternary and multicomponent separations, whose preliminary designs must be also refined to meet specified product purities.17,18 Also, with the inclusion of relative volatilities independent of temperature but dependent on the liquid-phase composition, as shown recently by Gutie´rrez-Antonio et al.37 for the construction of reside curve maps for nonideal mixtures, one could extend the basic equations and the methodology presented here to other complex systems, such as azeotropic and reactive distillation systems. Acknowledgment Financial support from Conacyt and the Instituto Mexicano del Petro´leo, Me´xico, is gratefully acknowledged. Nomenclature B ) bottoms molar flowrate D ) distillate molar flowrate F ) feed molar flowrate li ) molar flowrate of component i in liquid phase L ) liquid molar flowrate LS ) liquid molar flowrate for top interconnection M ) parameter NC ) number of components n ) number of interior collocation points NCP ) total number of collocation points N ) number of stages NL ) top interconnecting stage in main column NS ) stage for side product in main column NV ) stage for lower interconnection in main column R ) reflux ratio in main column sj ) generic stage S ) side product molar flowrate Vi ) molar flowrate of component i in vapor phase V ) vapor molar flowrate VI ) vapor molar flowrate for lower interconnection x ) liquid mole fraction X ) vector of independent variables for refined design y ) vapor mole fraction z ) feed mole fraction Greek Symbols R ) relative volatility

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Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007

δ ) parameter λ ) parameter Subscripts A ) light component B ) intermediate component, bottoms product C ) heavy component D ) distillate S ) side product i ) component j ) stage or collocation point k ) section in Petlyuk column m ) parameter n ) parameter Superscripts sp ) specified Literature Cited (1) Cerda, J.; Westerberg, A. W. Shortcut Methods for Complex Distillation Columns. 1. Minimum Reflux. Ind. Eng. Chem., Process Des. DeV. 1981, 20, 546-557. (2) Fidkowski, Z.; Krolikowski, L. Thermally Coupled System of Distillation Columns: Optimization Procedure. AIChE J. 1986, 32, 537546. (3) Glinos, K.; Malone, M. F. Optimality Regions for Complex Column Alternatives in Distillation Systems. Trans. Inst. Chem. Eng. 1988, 66, Part A, 229-240. (4) Triantafyllou, C.; Smith, R. The Design and Optimisation of Fully Thermally Coupled Distillation Columns. Trans. Inst. Chem. Eng. 1992, 70, Part A, 118-132. (5) Herna´ndez, S.; Jime´nez, A. Design of Optimal Thermally-Coupled Distillation System Using a Dynamic Model. Trans. Inst. Chem. Eng. 1996, 74, Part A, 357-362. (6) Herna´ndez, S.; Jime´nez, A. Design of Energy-Efficient Petlyuk Systems. Comput. Chem. Eng. 1999, 23, 1005-1010. (7) Agrawal, R.; Fidkowski, Z. More operable arrangements of fully thermally coupled distillation columns. AIChE J. 1998, 44, 2565. (8) Agrawal, R.; Fidkowski, Z. Ternary Distillation Schemes with Partial Reboiler or Partial Condenser. Ind. Eng. Chem. Res. 1998, 37, 3455-3462. (9) Agrawal, R.; Fidkowski, Z. New thermally coupled schemes for ternary distillation. AIChE J. 1999, 45, 485-496. (10) Jime´nez, A.; Ramı´rez, N.; Castro, A.; Herna´ndez, S. Design and Energy Performance of Alternative Schemes to the Petlyuk Distillation System. Trans. Inst. Chem. Eng. 2003, 81, Part A, 518-524. (11) Ramı´rez, N.; Jime´nez, A. Two Alternatives to Thermally Coupled Distillation Systems with Side Columns. AIChE J. 2004, 50, 2971-2975. (12) Halvorsen, I. J.; Skogestad, S. Shortcut Analysis of Optimal Operation of Petlyuk Distillation. Ind. Eng. Chem. Res. 2004, 43, 39943999. (13) Annakou, O.; Mizsey, P. Rigorous Comparative Study of EnergyIntegrated Distillation Schemes. Ind. Eng. Chem. Res. 1996, 35, 18771885. (14) Seader, J. D.; Henley, E. J. Separation Process Principles; Wiley: New York, 1998. (15) Carlberg, N. A.; Westerberg, A. W. Temperature-Heat Diagrams for Complex Columns. 3. Underwood’s Method for the Petlyuk Configuration. Ind. Eng. Chem. Res. 1989, 28, 1386-1397. (16) Muralikrishna, K.; Madhavan, K. P.; Shah, S. S. Development of Dividing Wall Distillation Column Design Space for a Specified Separation. Trans. Inst. Chem. Eng. 2002, 80, Part A, 155-166.

(17) Blancarte-Palacios, J. L; Bautista-Valde´s, M. N.; Herna´ndez, S.; Rico-Ramı´rez, V.; Jime´nez, A. Energy-Efficient Designs for Thermally Coupled Distillation Sequences for Four-Component Mixtures. Ind. Eng. Chem. Res. 2003, 42, 5157-5164. (18) Calzon-McConville, C. J.; Rosales-Zamora, M. B.; SegoviaHerna´ndez, J. G.; Herna´ndez, S.; Rico-Ramı´rez, V. Design and Optimization of Thermally Coupled Distillation Schemes for the Separation of Multicomponent Mixtures. Ind. Eng. Chem. Res. 2006, 45, 724-732. (19) Dunnebier, G.; Pantelides, C. Optimal Design of Thermally Coupled Distillation Columns. Ind. Eng. Chem. Res. 1999, 38, 162-176. (20) Yeomans, H.; Grossmann, I. E. Disjuntive Programming Models for the Optimal Design of Distillation Columns and Separation Sequences. Ind. Eng. Chem. Res. 2000, 39, 1637-1648. (21) Yeomans, H.; Grossmann, I. E. Optimal Design of Complex Distillation Columns Using Rigorous Tray-by-tray Disjunctive Programming Models. Ind. Eng. Chem. Res. 2000, 39, 4326-4335. (22) Grossmann, I. E.; Aguirre, P.; Barttfeld, M. Optimal Synthesis of Complex Distillation Columns Using Rigorous Models. Comput. Chem. Eng. 2005, 29, 1203-1215. (23) Shah, P. B.; Kokossis, A. C. New synthesis framework for the optimization of complex distillation systems. AIChE J. 2002, 48, 527550. (24) Amminudin, K. A.; Smith, R.; Thong, D. Y. C.; Towler, G. P. Design and Optimization of Fully Thermally Coupled Distillation Columns. Part 1: Preliminary Design and Optimization Methodology. Trans. Inst. Chem. Eng. 2001, 79, Part A, 701-715. (25) Kim, Y. H. Structural Design and Operation of a Fully Thermally Coupled Distillation Column. Chem. Eng. J. 2002, 85, 289-301. (26) Cho, Y. S.; Joseph, B. Reduced-Order Steady-State and Dynamic Models for Separation Processes. Part I. Development of the Model Reduction Procedure. AIChE J. 1983, 29, 261-269. (27) Cho, Y. S.; Joseph, B. Reduced-Order Steady-State and Dynamic Models for Separation Processes. Part II. Applications to Nonlinear Multicomponent Systems. AIChE J. 1983, 29, 270-276. (28) Stewart, W. E.; Levien, K. L.; Morari, M. Simulation of Fractionation by Orthogonal Collocation. Chem. Eng. Sci. 1985, 40, 409-421. (29) Swartz, C. L. E.; Stewart, W. E. A Collocation Approach to Distillation Column Design. AIChE J. 1986, 32, 1832-1838. (30) Swartz, C. L. E.; Stewart, W. E. Finite-Element Steady State Simulation of Multiphase Distillation. AIChE J. 1987, 33, 1977-1985. (31) Seferlis, P.; Hrymak, A. N. Adaptive Collocation on Finite Elements Models for the Optimization of Multistage Distillation Units. Chem. Eng. Sci. 1994, 49, 1369-1382. (32) Seferlis, P.; Hrymak, A. N. Optimization of Distillation Units Using Collocation Models. AIChE J. 1994, 40, 813-825. (33) Huss, R. S.; Westerberg, A. W. Collocation Methods for Distillation Design. 1. Model Description and Testing. Ind. Eng. Chem. Res. 1996, 35, 1603-1610. (34) Huss, R. S.; Westerberg, A. W. Collocation Methods for Distillation Design. 2. Applications for Distillation. Ind. Eng. Chem. Res. 1996, 35, 1611-1623. (35) Seferlis, P.; Grievink, J. Optimal Design and Sensitivity Analysis of Reactive Distillation Units Using Collocation Models. Ind. Eng. Chem. Res. 2001, 40, 1673-1685. (36) Edgar, T. F.; Himmelblau, D. M. Optimization of Chemical Processes; McGraw-Hill: New York, 1988. (37) Gutie´rrez-Antonio, C.; Vaca, M.; Jime´nez, A. A Fast Method to Calculate Residue Curve Maps. Ind. Eng. Chem. Res. 2006, 45, 44294432.

ReceiVed for reView February 23, 2007 ReVised manuscript receiVed May 29, 2007 Accepted June 5, 2007 IE070281K