Optimal Design of Complex Distillation Columns Using Rigorous Tray

The GDP models use rigorous design equations, where the trays in the ..... The optimal solution will then include the values of the operating variable...
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Optimal Design of Complex Distillation Columns Using Rigorous Tray-by-Tray Disjunctive Programming Models Hector Yeomans and Ignacio E. Grossmann* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

This paper presents a modeling procedure for the derivation of generalized disjunctive programming (GDP) models for the optimal design of complex or thermally coupled distillation columns. Optimization models for the separation of ideal and azeotropic mixtures are derived on the basis of superstructures reported previously in the literature. The GDP models use rigorous design equations, where the trays in the column can be considered permanent or conditional, depending on the functions they perform (i.e., heat supply/removal, draw streams and feeds). The conditional trays are modeled with disjunctions to decide whether or not vapor-liquid equilibrium mass transfer should be applied in each potential tray. The GDP models derived are solved with a logic-based outer approximation algorithm. The performance of the proposed procedure is evaluated with three examples: (a) the separation of an ideal mixture, (b) the separation of an azeotropic mixture, and (c) an industrial problem involving ideal mixtures. These examples show that the proposed method produces fairly robust and computationally efficient models. 1. Introduction The design of complex distillation column configurations is an industrially relevant problem. Theoretical work under the assumption of minimum reflux has shown that these configurations can save up to 50% more utility than conventional column arrays (Fidkowski and Krolikowski,1 Glinos and Malone,2 Triantafyllou and Smith,3 Annakou and Mizsey4). However, the complex column configurations that can potentially produce larger energy savings, such as the fully thermally coupled columns (Petlyuk5), are not commonly used in industrial practice, largely because of control concerns (Dunnebier and Pantelides6). Another reason complex columns are not widely used is the fact that the design of such systems involves strong interactions, and requires detailed simulation models that are often difficult to converge. The works of Sargent and Gaminibandara,7 Fidkowski and Agrawal,8,9 and Agrawal15 have pioneered the development of superstructures for complex separations, but the problem of obtaining the best designs is still unsolved. Mathematical programming techniques can help to solve this design problem, but most of the efforts in the field have used only shortcut and simplified methods, mainly because of the convergence difficulties encountered in the optimization of rigorous distillation columns (Bauer and Stichlmair10). The recent developments of mathematical programming models in the form of logic-based disjunctive programs, which exhibit improved robustness compared to conventional mixed integer nonlinear programming (MINLP) models, has recaptured the interest in the performance of optimal rigorous distillation design (Yeomans and Grossmann11). With rigorous models, it is now conceivable to solve complex column configurations, not only for ideal systems (Dunnebier and Pantelides6), but also for azeotropic separations (Sargent12). * Author to whom correspondence should be addressed. E-mail: [email protected]. Tel.: 412-268-2230. Fax: 412268-7139.

This paper presents a procedure to model complex column configurations as generalized disjunctive programs (GDPs). On the basis of previous work reported in the literature, special superstructures for complex columns that use a state-task network (STN) representation are considered for determination of the number and location of potential column sections, as well as the number of trays and feed locations. This superstructure is modeled as a GDP based on the work of Yeomans and Grossmann11 and then solved with a modified logic-based outer approximation algorithm (Turkay and Grossmann13). Examples for an ideal system, an azeotropic separation, and an industrial problem are presented to illustrate the performance of the proposed approach for the separation of ideal and nonideal mixtures using complex column systems. 2. Problem Statement The objective of this paper is to systematically derive generalized disjunctive programming (GDP) models (Raman and Grossmann14) for the optimal design of complex distillation column systems. The proposed models can be used for design problems that require the separation of an ideal or nonideal mixture into relatively pure components. The GDP models will be derived starting with a preexisting superstructure for complex columns such as those derived by Sargent and Gaminibandara,7 Agrawal,15 and Sargent.12 As will be shown, the GDP models for complex columns are a generalization of the models derived by Yeomans and Grossmann11 and rely on the assumption that the heating, cooling, feed, and draw of material can take place only in prespecified locations of the column array, not in every tray. This assumption was made for practical reasons and to limit the alternatives for heat exchange. The proposed models also require the specification of an upper bound for the number of trays in order to define the search space ahead of time. Good estimates of this upper bound may require preliminary calculations (e.g., using shortcut methods).

10.1021/ie0001974 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/12/2000

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Figure 1. Relation between STN representation and equipment superstructure.

The GDP models for complex columns will be solved with the modified logic-based outer approximation algorithm (Turkay and Grossmann;13 Yeomans and Grossmann11). 3. Complex Column Superstructures for Distillation Systems The construction of a superstructure for complex systems of distillation columns is a nontrivial problem because of the large number of alternative designs that are possible (Dunnebier and Pantelides6). Major structural choices include the selection and interconnection of column sections, feed locations, and number of trays. There are several superstructure representations available from the literature for covering at least a portion of the combinatorial space. Some of the most important are those developed by Sargent and Gaminibandara7 and Agrawal.15 These superstructures can produce not only simple column configurations, but also thermally coupled columns (Petlyuk5) and columns with a siderectifier or side-stripper among others. It is worth noting that these superstructures apply to the case of ideal mixtures or near-ideal mixtures where no distillation boundaries exist. For the case of azeotropic mixtures, Bauer and Stichlmair10 have derived complex column configurations in which the interconnections among columns are based on a “preferred separation” scheme that makes use of a geometric analysis in a ternary diagram. Another systematic approach for deriving superstructures for distillation columns is that proposed by Sargent.12 This approach, which is a modification of the Sargent and Gaminibandara7 superstructure, considers that not all of the products that can potentially appear

in the original superstructure are reachable because of equilibrium limitations. To construct the superstructure for azeotropic systems, it is necessary to have a graphical or mathematical representation of the distillation boundaries and the residue curve maps of the system. The derivation of GDP models for complex columns in this paper is based on the Sargent and Gaminibandara7 superstructure for ideal systems and on the Sargent12 superstructure for nonideal systems. However, the proposed methodology can be applied to model other superstructures (Agrawal,15 Smith and Pantelides16). The superstructures for complex columns can be interpreted as a state-task network (Kondili et al.,17 Yeomans and Grossmann18). The states are all of the reachable products of the superstructure, and the tasks are the transformations (separation tasks) that join two different states. Figure 1a shows the Sargent and Gaminibandara7 superstructure in STN format. Figure 1a corresponds to the state-task network in which the states given by the ellipses are transformed by the separation tasks given by the rectangles. Each task in the STN is assigned to a particular equipment unit, which, in the case of distillation, is a separation section, or a group of trays that removes a given species from the bulk of the liquid or vapor flow. Figure 1b shows the interpretation of tasks as distillation sections. It can be seen that the representation also includes the location of heat exchangers for reboilers and condensers. It is possible that not all of the separation tasks on the STN or equipment superstructures will be needed to separate a given mixture. Figure 2 shows how the elimination of certain tasks from the superstructure, along with some mass and energy flows, transforms the

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Figure 2. Some separation sequences and equivalent complex superstructures.

Sargent and Gaminibandara7 superstructure into conventional or complex column configurations. 4. Modeling and Solution The main problem in optimizing the complex column superstructures is that finding the optimal configuration

Figure 3. Tray structure for complex columns.

requires the elimination of some of the sections in the column. Each section in the superstructure can, in principle, be represented with the MINLP model proposed by Viswanathan and Grossmann22 and modified by Dunnebier and Pantelides6 for its use in a stateoperator network (Smith and Pantelides16). For these models the location of the reflux and boil-up ratios in the column determines the use or disappearance of trays. However, the MINLP model has two important drawbacks. First, many constraints in the model that are redundant have to be solved at every iteration of the solution algorithm, which increases the computational requirements. Second, when the trays disappear from the superstructure, by setting the flows to zero, the equations of the MINLP can become discontinuous or undefined for certain values of the design or operation variables, which decreases the robustness of the computations. Yeomans and Grossmann11 have proposed a GDP modeling framework that overcomes these difficulties by allowing a “tray bypass” model to be in effect when a given tray is not needed. This modeling framework was applied for single-column configurations and will be extended in this paper for complex column arrays. It will be assumed that the reader is familiar with the work of Yeomans and Grossmann.11 The two main elements of the model are the permanent and the conditional trays. The former are fixed in the superstructure and correspond to the feed, reflux, and boil-up trays. The conditional trays are grouped in stacks between pairs of permanent trays and either perform mass transfer through vapor-liquid equilibrium (VLE) equations or simply let the vapor and liquid get trough the tray without any exchange of mass. Through a careful analysis of the superstructure in Figure 1a, it can be seen that the states are the elements that define the structure of the STN. The states represent the potential products that can be achieved by the system, and they, in turn, differentiate the task of one separation section from that of another. These characteristics of the states are the same as those required for the permanent trays in the Yeomans and Grossmann11 model. Therefore, the states can be considered permanent trays of the superstructure. The permanent tray structure, however, must be modified to perform multiple functions. In a complex column, there is no difference between a feed tray, a

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reflux tray, or a boil-up tray. Any of these trays can have a different function in different solutions. Figure 3 shows the structure of the permanent tray proposed for complex column configurations. With the new permanent tray, it is possible to feed liquid and/or vapor to the tray, draw liquid or vapor from the tray, and locate a reboiler or a condenser. With respect to the conditional trays, no modifications are required, as the problem of disappearing sections and trays is identical in simple and complex column problems when the trays are conventional (only one liquid and one vapor stream passing through the tray). Let C be the set of components i present in the mixture, and let p ∈ PT be the set of permanent trays in the superstructure. Then, for every permanent tray p, the following equations apply: out in cond LFpi + VFpi + Lin - Lout pi + Vpi + Lpi pi - Vpi ) 0

Lout pi ) LIQpxpi Vout pi ) VAPpypi

k k Lkji + Vkji - Lj+1,i - Vj-1,i )0

Lout pi ) LDpi + LS

LIQkj )

cond Vout pi ) VDpi + VSpi + Lpi

LDpi )

in in in [LFpihLFpi + VFpihVFpi + Lin ∑ pihLpi + VpihVpi + i∈C cond out out out reb - Lout Lcond pi hLpi pi hLpi - Vpi hVpi ] ) -Qp

- VDpihVDpi - VSpihVSpi cond Lcond pi hLpi ]

xpi ) 1 ∑ i∈C

)

Qcon p

ypi ) 1 ∑ i∈C

ypiPp ) γpiPvap pi xpi

VAPkj )

[ ][ xkji ) 1 ∑ i∈C

Lcond ) µpVout pi pi

∑ i∈C

Lkji ∑ i∈C

Vkji ∑ i∈C

k k k k [LkjihLkji + VkjihVkji - Lj+1,i hLj+1,i - Vj-1,i hVj-1,i ])0 ∑ i∈C

Rp Lout pi

VDpi ) βpVout pi

out [Vout pi hVpi

heat to the tray. This representation allows one to more directly relate the cooling operation to a condenser and the heating operation to the reboiler. In addition, tighter lower and upper bounds for the heat loads can be specified when two heat exchange points are considered. The equation set 1 represents the MESH equations for the permanent trays. For the conditional trays, the mass and energy balances for each tray are always applied, regardless of the configuration.11 The VLE constraints, however, are applied or not depending on whether the tray is needed or not in a particular solution. To avoid confusion, the variable names from eq 1 will be reused in the definition of the general constraints for the conditional trays. Consider that the variables will now correspond to the set j ∈ J that represents the conditional trays and the set k ∈ S that represents a given section of column. The variable cj represents the cost per tray based on vapor flow, and β is a cost parameter. The corresponding equations are as follows:

(1)

where, as seen in Figure 3, LF and VF represent the component liquid and vapor feeds to tray p, Lin and Vin represent incoming flows from sections above and below tray p, and Lout and V out represent the outgoing flows to sections below and above the permanent tray. LIQ and VAP represent the total component flows, LD and VD represent liquid and vapor draws, and Lcond represents the vapor draw to be condensed and recycled. R, β, and µ represent fractions between 0 and 1, and γ represents the activity coefficient for the equilibrium equations for a given component i. Note that the equilibrium equation assumes an ideal vapor phase and a nonideal liquid phase from an activity coefficient model. The variable h and the different flow variables indicate the enthalpies of each component. Finally, P represents the stage pressure, T represents the temperature, Pvap represents the vapor pressure of a component i, and x and y represent the mole fractions of each component. Two different heat exchange points are considered. One is for removing heat from a fraction of the outgoing vapor stream, and the other is for adding

ykji ) 1 ∑ i∈C

Wj k ykjiPk ) γjiPvap ji xji Lkji ) LIQjxkji ∨ Vkji ) VAPjykji TLkj ) TVkj cj ) VAPβj

¬Wj k k yji ) yj-1,i k xkji ) xj+1,i k k Lji ) Lj+1,i k Vkji ) Vj-1,i k TLkj ) TLj+1 k k TVj ) TVj-1 cj ) 0

]

(2)

The first four equations represent material and energy balances. The disjunction in (2) indicates which constraints define the existence of VLE in a conditional tray j, and this selection is controlled with the boolean variable Wj. If Wj is true phase equilibrium is imposed. If it is false equality of flows, compositions and temperatures are imposed to trivially reduce the mass and energy balance so that no mass transfer takes place between the vapor and liquid phases. Notice that it is necessary to differentiate from liquid (TL) and vapor (TV) temperatures in the conditional trays, since the streams might be bypassed. The remaining constraints for the GDP model correspond to the interconnection equations of conditional trays with each other and with the permanent trays. The permanent trays will be connected to conditional trays only at the top or bottom of a determined stack, and these constraints represent mass and energy balances. Equation block 3 shows the connection equations for a permanent tray p and the first tray (j ) 1) of a section k above it. The variable Tliq p represents the inlet temperature of the liquid above permanent tray p. For a detailed explanation of the GDP model structure, the reader is referred to Yeomans and Grossmann.11

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heptane (30% C) into relatively pure components at a pressure of 1 atm. The minimum purity specified is 90% for products A and B and 80% for C. The objective function considered for the superstructure optimization is the maximization of profit given by eq 4.

k Lin pi ) Lji k VSpi ) Vj-1,i

Tvap ) TVkj p k Tliq p ) TLj

(3)

Finally, the objective function can be economic in nature, such as the minimization of total cost (investment and operation) or the maximization of profit, or it can be a target objective, such as total energy use. The optimal solution will then include the values of the operating variables and indicate which trays are selected in every section and which are bypassed. 5. Numerical Examples Three examples will be presented to test the performance of the models for complex column separations. The first involves the separation of an ideal mixture of components, and the second involves the separation of an azeotropic mixture into (relatively) pure components. Finally, we present an actual industrial example that requires the separation of an ideal mixture with a somewhat less complicated superstructure. The first two examples were carried out at a fixed pressure, whereas the last example optimizes the pressure as well. The superstructures were modeled as a GDP and were solved using the modified logic-based outer approximation algorithm.11 The algorithm was coded in the GAMS19 modeling environment, using CONOPT 2.0 and CPLEX 4.5 as the solvers for the NLP and MILP, respectively. Because of the large size of the models, we only report the main input data and the overall results. Readers interested in the detailed models can obtain the GAMS files from the authors of this paper. 5.1. Separation of a Light Hydrocarbon Mixture. Consider the separation of a 1000 kmol/h mixture of n-pentane (20% A mole), n-hexane (50% B), and n-

Figure 4. STN and superstructure representation for problem 1.

profit ) cAFA + cBFB + cCFC - R λ(ARη +

∑

cj j∈NT ACη) - γQR

- µQC (4)

The parameters cA, cB, and cC correspond to the sales price of each product. The variables FA, FB, and FC represent the flows of a given component at the specific draw points where the product should be obtained. The sales terms are followed by the normalized cost for each existing tray j ∈ NT, where NT is the set of all trays in the column (conditional and permanent). Because the costing of complex columns is different from that of conventional columns, the cost of a column was normalized on a “per tray” basis, so that valid comparisons can be made between both configurations that depend on the total vapor flow on the tray. The remaining terms in eq 4 account for exchanger area (AC, AR) costs for condensers and reboilers and cooling and heating utility (QC, QR) costs. R, β, η, λ, and µ are cost parameters, obtained from Peters and Timmerhaus.19 To solve this problem, we used the superstructure shown in Figure 4. This representation is capable of reproducing simple separation sequences (direct and indirect), as well as the fully thermally coupled columns, and main column plus side-rectifier, or side-stripper configurations. The maximum number of trays allowed in each separation section is 20, the equilibrium is modeled with Raoult’s law, and the thermodynamic properties (enthalpies, vapor pressures) were obtained from Reid et al.20 The superstructure presented in Figure 4 includes a number of different design alternatives. Five of these alternatives, which were optimized individually, are

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Figure 5. Optimal solutions for separation of light hydrocarbons.

shown in Figure 5, in which the design parameters of the optimal solution for each configuration (number of conditional trays in each section, purity of products, heat loads) are also shown. The percentages reported in this

figure correspond to the purity of the products, and the total number of trays reported includes all conditional and permanent trays in the superstructure. The most profitable configuration for this problem was the fully

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Figure 6. Attainable products and superstructure for EtOH/MeOH/Water system separation with feed in the ABD region. Table 1. Computational Results for the Separation of Hydrocarbons parameter

thermally coupled

NLP equations NLP variables MILP equations MILP variables discrete variables LB OA iterations CPU min objective (k$/year)

4748 4565 14 989 19 745 132 3 21.8 1406.2

integrated (Petlyuk) column5 in Figure 5e, with earnings of at least 13% higher than the closest solution and with a relatively high-product purity. In terms of energy utilization, this configuration consumes 21.6% less energy than the best standard column configuration (direct sequence), with about the same number of trays (49). The superstructure in Figure 4 was also optimized simultaneously as a GDP model. As shown in Table 1, this model involves 4748 equations and 4565 variables in the optimal NLP subproblem, and 132 discrete variables, 19 745 continuous variables, and 14 989 constraints in the final master problem. The optimal solution was found after 21 min and 50 s (Pentium II

PC, 300 MHz). The optimal solution obtained is that of the Petlyuk column (Figure 5e). If this solution is eliminated with an integer cut, the following solution has a Petlyuk column structure, with a small condenser in the top of section 5. The operating variables, however, do not change significantly with the structure. 5.2. Separation of Methanol, Ethanol, and Water. As shown by Sargent,12 it is possible to extend the superstructure of Sargent and Gaminibandara7 to complex column configurations for the separation of azeotropic mixtures. As mentioned before, to identify the reachable states for the system, it is necessary to have a graphical or mathematical representation of the distillation space and residue curve map. Also, because of the presence of azeotropes, pseudocomponents consisting of binary pairs are introduced in the reachable space in order to identify all possible distillation paths.12 Because of the definition of these pseudocomponents and their relationship, the superstructure is problemdependent, and the number of sections that appear in it, as well as their interconnections, varies with the characteristics of the feed. The superstructure and recycle streams required to separate a mixture are dependent on the relative position of the feed with respect to the distillation boundaries of the system.

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4333 Table 2. Computational Results for Azeotropic Separation Problem

a

parameter

complex sequence

NLP equations NLP variables MILP equations MILP variables Discrete variables LB OA iterations CPU mina objective (k$/year)

6195 6093 21 633 25 532 176 3 59 4031.16

CPU minutes on a Pentium II 300 MhZ PC.

Figure 7. Optimal solution for azeotropic separation.

Consider the separation of a mixture 1000 kmol/h with 50 mol % methanol, 30 mol % ethanol, and 20 mol % water at a pressure of 1 atm, and for which a minimum purity of 90% is specified for all products. Figure 6b shows the reachable products for the methanol (A)-ethanol (C)-water (D) system (in order of decreasing relative volatility), where B is the ethanolwater azeotrope. For this example, the feed is located in the ABD region. Because relatively pure products are desired, sections 1-4 in Figure 6c are needed to cover the reachable space of products in the feed zone ABD. Because pure C is a desirable product, it is necessary to cross the distillation boundary of the system. The product at the top of section 2 is close to this boundary, and stripping sections 5 and 6 are able to reach mixed product ABC and pure C, respectively. Rectification sections 7 and 8) are needed to recover the azeotrope from mixtures of BC and BD, respectively. Note that both sections might not be necessary. The feed is located in the ABD region. If other feeds are available, superstructures similar to the one in Figure 6e can be generated, as explained by Sargent.12 The problem was modeled using the Wilson model for the calculation the activity coefficients for an ideal vapor phase.20 The objective considered was the minimization of investment and operation. Selection of the upper bound for the number of trays for each section is nontrivial and depends on the difficulty of the separation. In this example, that number was set arbitrarily to 20 trays per section. The results of this problem were validated with a commercial simulator (PRO II) to verify the behavior of the optimization model close to the distillation boundary. Table 2 shows the computational results for the complex azeotropic separation. The large computational time requirements for this problem arise for two reasons. A large part of the expense can be attributed to the use of nonideal equilibrium models,

and another part to the large number of sections (and therefore conditional trays) required to solve the superstructure. The results of the optimal design can be found in Figure 7, where it can be seen that it was not possible to achieve the required purity of 90% for component C (ethanol), even though the composition of the ethanolrich product is above the distillation boundary. To avoid infeasible solutions in the NLP due to violations in the purity, we used a simple penalty function. It is, of course, possible that we may have obtained a local solution that is suboptimal. We were unable, however, to find another solution that would meet the purity specification for C. 5.3. An Industrial Problem. This industrial example shows the advantages of using the modified permanent tray model presented in this paper in the construction of nonconventional column superstructures. The problem requires the separation of two components (A and E) into pure product streams (over 99% purity). The feed composition is 60.6% A, 0.1% B, 0.05% C, 0.15% D, and 39.1% E. What makes the separation nonconventional is the existence of impurities (components B, C, and D with intermediate relative volatility) that are present in amounts less than 1% mole in the feed and are not desired in the final products. The difference in feed composition renders conventional distillation sequences an expensive option. The objective function of the problem is the minimization of investment and operating costs of the design. Also, the pressure was considered to be an optimization variable for this problem. The approach taken to solve this problem is as follows. Consider the separation of a binary mixture, where a side draw stream will be located in the binary separation column to reduce the undesired contaminants to acceptable levels in the product streams. The superstructure representation for this problem is shown in Figure 8. The proposed modeling framework of this paper allows consideration of the potential location of the side draw in both the stripping and rectification sections, through the division of these two sections into four and the location of permanent trays between each of them. The permanent trays located in the middle of the original rectification and stripping sections have the ability to behave like conventional trays if a side draw is needed. If the permanent tray is required to work as such, it will allow for stream splitting, feeding, heating, or cooling. Table 3 shows the computational results for this example. As can be seen, the final NLP had 6519 variables and 6805 constraints, whereas the master problem had 120 0-1 variables, 33 730 continuous variables, and 31 896 constraints. This problem did exhibit convergence difficulties during solution of the NLP subproblems because of the difference of 4 orders of magnitude between product compositions in the column. This difference did not allow for the provision of appropriate bounding for the liquid and vapor flows inside the column. The equilibrium calculations are not an issue in this problem, because we considered the system to be ideal. The large CPU time (close to 3 h) was due to the convergence difficulties mentioned above and also to the maximum number of trays considered in each section (30 trays).

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Figure 8. Superstructure and optimal solution for purge location problem. Table 3. Computational Results for Purge Location Problem

a

parameter

complex sequence

NLP equations NLP variables MILP equations MILP variables Discrete variables LB OA iterations CPU mina objective (k$/year)

6805 6519 31 869 33 730 120 8 ∼180 232.3

CPU minutes on a Pentium II 300 MhZ PC.

problems directly as MINLP problems, the proposed method was significantly more robust and efficient. Acknowledgment The authors acknowledge the financial support for this project provided by Consejo Nacional de Ciencia y Tecnologia (CONACYT, Mexico) and the National Science Foundation, under Grant CTS-9710303. The collaboration of Bayer AG, Germany, in the design of some of the problems presented in the paper is also greatly appreciated.

6. Conclusions

Literature Cited

In this paper, we have introduced a procedure for the construction and solution of GDP models for optimizing the design of complex column configuration problems for distillation sequences. The optimization models are derived from the superstructures by Sargent and Gaminibandara7 and Sargent12 and are solved with the modified logic-based OA algorithm (Yeomans and Grossmann11). A modification to the permanent tray structure of the Yeomans and Grossmann11 GDP model was introduced to allow side draws and heat exchangers. It was shown that these permanent trays serve as the backbone for the column superstructure, and when they are not needed, they perform as conventional trays. Three examples were presented. The results for the separation of ideal mixtures show that complex column arrays can be more cost-effective than simple configurations. The azeotropic separation example illustrated the modeling and solution of a superstructure derived from Sargent.12 Finally, the industrial problem illustrated an application with components with low concentrations and the use of side draws. Although the solution of the proposed GDP models was not straightforward compared to our own experience in solving these type of

(1) Fidkowski, Z.; Krolikowski, L. Thermally Coupled System of Distillation Columns: Optimization Procedure. AIChE J. 1986, 32, 537. (2) Glinos, K.; Malone, F. Optimallity Regions for Complex Column Alternatives in Distillation Systems. Chem. Eng. Res. Des. 1988, 66, 229. (3) Triantafyllou, C.; Smith, R. The Design and Optimization of Fully Thermally Coupled Distillation Columns. Trans. Inst. Chem. Eng. 1992, 70, 118. (4) Annakou, O.; Mizsey, P. Rigorous Comparative Study of Energy-Integrated Distillation Schemes. Ind. Eng. Chem. Res. 1996, 35, 1877. (5) Petlyuk, F. B.; Platanov, V. M.; Slavinskii, D. M. Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Int. Chem. Eng. 1965, 5, 555. (6) Dunnebier, G.; Pantelides, C. C. Optimal Design of Thermally Coupled Distillation Columns. Ind. Eng. Chem. Res. 1999, 38, 162. (7) Sargent, R. W. H.; Gaminibandara, K. Optimal Design of Plate Distillation Columns. In Optimization in Action; Dixon, L. C. W., Ed.; Academic Press: New York, 1976. (8) Fidkowski, Z.; Agrawal, R. Utilization of Waste Heat Stream in Distillation. Ind. Eng. Chem. Res. 1995, 34, 1287. (9) Fidkowski, Z.; Agrawal, R. On the Use of Intermediate Reboilers in the Rectifying Section and Condensers in the Stripping Section of a Distillation Column. Ind. Eng. Chem. Res. 1996, 35, 2801.

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4335 (10) Bauer, M. H.; Stichlmair, J. Design and Economic Optimization of Azeotropic Distillation Processes Using Mixed-Integer Nonlinear Programming. Comput. Chem. Eng. 1998, 22, 1271. (11) Yeomans, H.; Grossmann, I. E. Disjunctive Programming Models for the Optimal Design of Distillation Columns and Separation Sequences. Ind. Eng. Chem. Res. 2000, 39, 1637-1648. (12) Sargent, R. W. H. A Functional Approach to Process Synthesis and its Application to Distillation Systems. Comput. Chem. Eng. 1998, 22, 31. (13) Turkay, M.; Grossmann, I. E. Logic-Based MINLP Algorithms for the Optimal Synthesis of Process Networks. Comput. Chem. Eng. 1996, 20, 959. (14) Raman, R.; Grossmann, I. E. Modeling and Computational Techniques for Logic Based Integer Programming. Comput. Chem. Eng. 1994, 18, 563. (15) Agrawal, R. Synthesis of Distillation Column Configurations for Multicomponent Separation. Ind. Eng. Chem. Res. 1996, 35, 1059. (16) Smith, E. M. B.; Pantelides, C. C. Design of Reaction/ Separation Networks using Detailed Models. Suppl. Comput. Chem. Eng. 1995, 19, S83. (17) Kondili, E.; Pantelides C. C.; Sargent R. W. H. A General

Algorithm for Short-Term Scheduling of Batch Operations. I. MILP Formulation. Comput. Chem. Eng. 1993, 17, 211. (18) Yeomans, H.; Grossmann, I. E. A Systematic Modeling Framework of Superstructure Optimization in Process Synthesis. Comput. Chem. Eng. 1999, 23, 709. (19) Peters, M. S.; Timmerhaus, K. D. Plant Design and Economics for Chemical Engineers. 4th ed.; McGraw-Hill: New York, 1991; pp 617, 712, 815. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (21) Brooke, A.; Kendrick, D.; Meeraus, A. GAMSsA User’s Guide; Scientific Press: New York, 1988. (22) Viswanathan, J.; Grossmann, I. E. Optimal Feed Locations and Number of Trays for Distillation Columns with Multiple Feeds. Ind. Eng. Chem. Res. 1993, 32, 2942.

Received for review February 7, 2000 Revised manuscript received July 10, 2000 Accepted August 21, 2000 IE0001974