Generalized Disjunctive Programming Model for the Optimal Synthesis

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Ind. Eng. Chem. Res. 2001, 40, 2260-2274

Generalized Disjunctive Programming Model for the Optimal Synthesis of Thermally Linked Distillation Columns Jose´ A. Caballero† and Ignacio E. Grossmann*,‡ Department of Chemical Engineering, University of Alicante, Ap. correos 99, Alicante, Spain, and Department of Chemical Engineering, Carnegie Mellon Universtity, Pittsburgh, Pennsylvania 15213

This paper addresses the synthesis of distillation column configurations to separate nonazeotropic multicomponent mixtures containing N components. It is shown that, for sharp separations of an N-component mixture, it is possible to develop a superstructure that takes into account all of the possibilities, from thermally linked systems with only one reboiler and one condenser to sequences with only conventional columns [2(N - 1) condensers and reboilers]. All of the partially thermally linked superstructures are included. The superstructure is systematically generated using the state task network (STN) formalism, in which only the tasks that can be performed are specified, but not equipment. A set of logical relationships between tasks is proposed that allows only feasible configurations that use the minimum number of column sections and that takes into account the fact that the minimum number of column sections and the number of heat exchangers are not independent. The superstructure is modeled using generalized disjunctive programming independent of the equations that represent each of the tasks (shortcut, aggregated, or rigorous models). In this paper, the procedure is illustrated using a modified version of Underwood’s equations proposed by Carlberg and Westerberg (Carlberg, N. A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 2. Underwood’s Method for Side Stripers and Enrichers. Ind. Eng. Chem. Res. 1989, 28, 1379-1386. Carlberg, N. A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 3. Underwood’s Method for the Petlyuk Configuration. Ind. Eng. Chem. Res. 1989, 28, 1386-1397). A modified version of the logic-based outer approximation algorithm is used to solve the resulting model. Introduction The objective of a distillation-based synthesis problem is to find the column arrangement that provides the best scheme in terms of investment and operational costs. Column sequences to separate a nearly ideal multicomponent feed using conventional distillation columns (one feed to each column with one reboiler and one condenser) have been widely studied. Unfortunately, even in this simple case, the number of alternatives increases in a combinatorial way with the number of components to be separated.3 For a mixture of N components, N - 1 conventional distillation columns are needed, but each one of these distillation columns needs one reboiler and one condenser with their own contributions to the total investment costs and energy (utilities) consumption. Reductions in energy consumption can be achieved through energy integration either by synthesizing multieffect columns or by exchanging heat between different separation tasks (columns) in order to minimize the consumption of utilities. One method for reducing the number of reboilers and condensers is to use thermally linked columns. One particular arrangement was proposed by Petlyuk et al.4 In a Petlyuk configuration, the entire distillation can be performed using only one reboiler and one condenser, independent of the number of components to be separated.2,5 Several approaches have been proposed for the design of efficient separation systems, including heuristic * Author to whom correspondence should be addressed. † University of Alicante. ‡ Carnegie Mellon Universtity.

methods,6 evolutionary techniques,7 hierarchical decomposition,8 implicit enumeration,9 dynamic programming,10 and stochastic methods.11 General reviews of distillation can be found by Westerberg,12 Flouquet et al.,13,14 Gert-Jan et al.,15 Juergen et al.,16 and Westerberg and Wahnschafft.17 As an alternative to the above methods, the use of “superstructures” has been suggested, in which all of the alternatives of interest for the system must be included. The pioneering work in superstructure optimization was done by Sargent and Gaminibandara.18 These authors proposed a superstructure of linked columns that included not only simple sharp splits, but also complex columns such as the Petluyk configuration. That superstructure has a large number of possible configurations embedded in it, and any particular configuration can be obtained by removing some column sections, condensers, and reboilers and the interconnections between them. Andrecovich and Westerberg19 presented a network superstructure for the separation of near-ideal mixtures into pure components. The model considers only sharp splits and is based on shortcut methods, but its major contribution is the formulation of a mixed integer linear programming (MILP) problem associated with the superstructure representation. The model is suitable for the design of heat-integrated distillation sequences and uses a network representation instead of a tree representation. Since the pioneering work cited above, the use of superstructures has received renewed attention, and a large number of papers on it have been published. A recent review can be found in Grossmann et al.20 It is not until recently, however, that greater attention has been paid to the systematic generation and modeling of superstructures for distillation systems. Bagajewicz

10.1021/ie000761a CCC: $20.00 © 2001 American Chemical Society Published on Web 04/13/2001

Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2261

Figure 1. Example of sequences for separation of three components: (a) conventional columns (four column sections), (b) thermally linked columns (four column sections), (c) only two heat exchangers (heat exchanger of compound B is removed, but the structure has six column sections).

and Manousiouthakis21 proposed the state space representation for separation network synthesis. This approach allows for a specific number of process operators and all possible interconnections among them, providing a framework for network design with minimal assumptions on process structure. Sargent22 proposed a superstructure that is based on the state task network representation. He proposed a representation that is able to include all possible separations, including azeotropic mixtures, derived from a known equilibrium system. Yeomans and Grossmann23 proposed a systematic modeling framework based on the state task network (STN) and the state equipment network (SEN) formalisms and studied the influence of these representations on the resulting optimization model. Most of these works were applied to systems that use only conventional columns, and only some of them were applied to to thermally linked distillation systems.18,24 In particular, Agrawal24 showed that the widely accepted superstructure of Sargent and Gaminibandara18 does not include some alternatives and does not use the minimum number of distillation sections for some separations. Agrawal24 gives a set of heuristics to generate superstructures. In this work, we first present a systematic procedure for generating superstructures for thermally linked distillation system based on the STN formalism of Yeomans and Grossmann.23 We integrate this systematic procedure with some of the heuristics proposed by Agrawal24 in order to obtain superstructures that include all of the separation options using the minimum number of separation sections. Next, we show how this superstructure can be modeled using a disjunctive representation. Later, we also show how the system can be extended to mixed systems in which combinations of complex and conventional columns are included. The goal is to allow for the introduction of more than two heat exchangers so that all of the possibilities, from thermally linked systems with only one rebolier and one condenser to sequences with only conventional columns, can be included in a single representation. Finally, the optimization of this mixed system is modeled using generalized disjunctive programming (GDP). Superstructures for Thermally Linked Columns (One Reboiler and One Condenser) In this section, we consider the separation of an N-component mixture that does not form azeotropes. Only thermally linked configurations with one reboiler and one condenser will be considered.

The first aspect to take into account when generating superstructures for thermally linked columns is that there is a relationship between the number of heat exchangers and the minimum number of columns (or better, column sections) for a given separation. Whereas for conventional columns this relation is fixed to N - 1 columns and 2(N - 1) heat exchangers, this is not true for thermally linked distillation systems. A column (separation) section is defined to be a portion of a distillation column that is not interrupted by entering or exiting streams or heat flows.25 For the separation of a mixture of N components, N - 1 conventional columns are needed or, equivalently, 2(N - 1) separation sections. This is also the minimum number of column sections required to perform that separation. Using only conventional columns, 2(N - 1) heat exchangers are needed (one for each section). In thermally linked columns (side stripers or side rectifiers), the vapor and liquid flows from a given section in a column are shared with another column, i.e., the rectifying section of a column is shared with a side stripper and the enriching section of a column is shared with a side rectifier. Therefore, the condenser is able to provide the condensing duty for the side stripping column, and the rebolier is able to provide the boiling duty for the side rectifying column. Therefore, the minimum number of reboilers and condensers that uses the minimum number of distillation sections is equal to the number of components in the feed mixture. However, it is possible to eliminate a reboiler or a condenser associated with a component of intermediate volatility by adding two more distillation sections. See, for example, Figure 1 in which the reboiler associated with compound B in scheme 1b is removed in scheme 1c. Note that the condenser associated with the lightest compound and the reboiler associated with the heaviest one cannot be removed.24 Figure 1 shows graphically some of the aspects previously mentioned. Summarizing, the minimum number of distillation sections for thermally linked systems is 2(N - 1), and the minimum number of heat exchangers (reboilers and condensers) is N (see, for example, Figure 1b). It is possible to eliminate N - 2 of those condensers or reboilers, but this implies the addition of two new column sections for each of the exchangers removed. Thus, the minimum number of column sections in a thermally linked system is 2(N - 1) + 2(N - E), where E is the number of heat exchangers (2 e E e N). If only one condenser and one reboiler are present, then the

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Figure 2. General STN superstructure for a mixture of four components.

minimum number of column sections is 2(N - 1) + 2(N - 2) ) 4N - 6. As pointed out by Agrawal,24 the superstructure proposed by Sargent and Gaminibandara18 implies the use of N(N - 1) separation sections. Therefore, for systems with N g 4, N(N - 1) is always greater than 4N - 6, and there are some configurations that cannot be derived from the Sargent and Gaminibandara18 superstructure. Further details can be found in Agrawal.24 Another aspect to take into account is that, in a thermally linked (or partially linked) separation system, it is not possible to prespecify whether a given separation task is going to be carried out by the main column or by a side stripper, a side rectifier, etc. It is possible to use configurations in which more than one column is integrated in the same column shell. Hence, especially in the early stages of design, we cannot prespecify the equipment in which a given separation task will be performed. However, we know the separation possibilities (tasks). Yeomans and Grossmann23 proposed a general framework for systematically generating superstructures. These authors considered two extreme cases: the state task network (STN) and the state equipment network (SEN). The former is concerned with the selection of tasks, leaving the selection of equipment for a later stage. In the second, the equipment is selected, leaving the selection of tasks for a later stage. The STN formalism is especially well-suited for the case studied here. We will illustrate the methodology with a mixture of four components, say, A, B, C, and D that are ranked according to their volatilities, with A being the most volatile and D the least. The states are all of the possible

mixtures that can be produced after a separation. In our example, the possible states are ABCD, ABC, BCD, AB, BC, CD, A, B, C, and D (defined, in this case, only as a function of their composition). Identification of tasks is not always trivial. The reason is that the simple enumeration of tasks can include tasks that cannot be performed by a thermally linked distillation system. However, this difficulty can be circumvented if a set of logical relationships between tasks and states is included to yield only correct separation sequences. Therefore, for a four-component mixture, the tasks are A|BCD, AB|CD, AB|BCD, ABC|BCD, ABC|CD, ABC|D, A|BC, AB|BC, AB|C, B|CD, BC|CD, BC|D, A|B, B|C, and C|D. Deriving the superstructure is now a simple task. We only need to join the different states with the tasks that give rise to a particular state, i.e., the state ABCD will be connected to the tasks A|BCD, AB|CD, AB|BCD, ABC|BCD, ABC|CD, and ABC|D. Also, a given task is connected to two states, i.e., the task AB|BCD is connected to the states AB and BCD. Figure 2 shows the resulting superstructure. We must include some logical relations between tasks to avoid forbidden configurations or sequences with more than 4N - 6 separation sections. The following fives rules are sufficient to completely specify a feasible separation system (some of these rules were previously proposed by Agrawal):24 (1) A given state can give rise to at most one task. (2) A given state can only be produced by at most one task (except products). (3) For the products, the lightest and the heaviest can be produced by only one task. However, the intermediate products are produced by two sections, one by a stripping section of a task (considering the task as a pseudocolumn) and another by a rectifying section. (4)


Figure 3. Example of fully thermally linked superstructure with sharp splits AB|CD and BC|DE. The sequence has 14 columns sections.

The property of connectivity holds, i.e., task AB|BCD implies task A|B and task AB|BCD implies task B|CD or task BC|CD or task BC|D. (5) The number of tasks is equal to (4N - 6)/2. Note that each task produces two separation sections. Agrawal24 gave a set of nine heuristics for generating configurations for thermally linked distillation sequences. These nine heuristics are implicitly included in the set of logical relationships presented above. However, observation 2 by Agrawal’s work is incomplete. The original rule states that “Any subgroup (state) containing more than three components should not be

Figure 4. STN superstructure after simplification.

sharp split into two subgroups each containing two or more components. This would require the use of an additional reboiler or condenser”. This rule is only true if the state is not coming from a previous unsharp separation involving the compounds to be sharp separated. Figure 3 shows an example with splits for AB|CD and BC|DE that violates this rule and corresponds to a valid separation of a mixture of five compounds with the minimum number of tasks and only two heat exchangers. Before translating these rules to logical or algebraic expressions, we can use them to simplify the superstructure for components A, B, C, and D as follows: (a) The only task that produces pure B (the second compound in relative volatility) from a stripping section is A|B. Therefore, from rule 3, task A|B always exists. The same happens with the second heaviest compound (i.e., in a mixture of the four compounds A, B, C, and D, task C|D appears in all of the configurations). (b) As a consequence of point a and rule 3, tasks that produce pure A (except for A|B) can be removed. The same is true for tasks that produce pure D (for the mixture ABCD) (except for C|D). (c) Superstructures with more than 4N - 6 separation sections can be generated by modifying rule 2 and allowing that states with no extreme compounds (no A or D, in our example) can be produced by at most two states. (d) From Agrawal’s rule 2, task AB|CD can be removed. With the previous considerations taken into account, the simplified superstructure is as shown in Figure 4. Note that it is not necessary to remove the separations AB|CD, A|BCD, A|BC, ABC|D, and BC|D because the logical relationships between tasks (rules 1-5) avoid the appearance of those tasks. However, the previous

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elimination simplifies the superstructure and reduces the number of logical relationships. STN Model for the Thermally Linked Distillation Sequences After the superstructure has been generated, the second step involves the modeling of the representation as a mathematical programming problem. Because there are conditional tasks that involve discrete decisions, it is necessary to use discrete mathematical programming models. In this case, we use generalized disjunctive programming26 (GDP) in which the conditional constraints are represented by disjunctions that have assigned Boolean variables to represent enforcement of the constraints. The first stage consists of developing the logical relations in terms of Boolean variables. We illustrate the procedure for a mixture of four components. The extension to five or more components is straightforward. Let us define the following index sets:

ration and false otherwise. The five previous rules can be expressed as logic propositions or translated into equivalent algebraic expressions as a function of binary variables. (1) A given state s can give rise to at most one task.

∑ Yt e 1

∑ Yt e 1

∨

TSBCD ) {(B/CD), (BC/CD)} (4) STs ) {tasks t that are able to produce state s} e.g., STABC ) {(ABC/CD), (ABC/BCD)} STBCD ) {(AB/BCD), (ABC/BCD)} STAB ) {(AB/BCD), (AB/BC), (AB/C)} STBC ) {(AB/BC), (BC/CD)} STCD ) {(ABC/CD), (B/CD), (BC/CD) } (5) PREi ) {tasks t that produce pure product i through a rectifying section} e.g., PREA ) {(A/B)} PREB ) {(B/CD), (B/C)} PREC ) {(C/D)} (6) PSTi ) {tasks t that produce pure product i through a stripping section} e.g., PSTB ) {(A/B)}

Yt ;

t∈PREi

∨

Yt

t∈PSTi

Yt w

∨

Yk ∀ t ∈ STs, s ∈ STATES

(5)

∨

Yk ∀ t ∈ TSs, s ∈ STATES

(6)

k∈TSs Yt w

k∈STs (5) The number of tasks is equal to (4N - 6)/2.

∑

Yt ) (4N - 6)/2

(7)

t∈TASK

Note that each task produces two separation sections. The previous equations (1-7) can be translated into algebraic expressions as a function of the binary variables. In this case, there is a one-to-one correspondence between the binary and Boolean variables. If the Boolean variable is true, the binary variable takes the value 1, and it takes the value zero otherwise. See Raman and Grossmann27-29 for a systematic method for carrying out this transformation. The next step consists of writing the equations of the model. The advantage of using the STN formalism together with generalize disjunctive programming (GDP) is that the formal structure of the model is always the same (see, for example, Yeomans and Grossmann23 and Caballero and Grossmann30). Conceptually, the model can be written as follows:

(P1) min f(x) s.t. r(x) ) 0 connectivity equations Yt ¬Yt ht(x) ) 0 ∨ Btx ) 0 ∀ t ∈ TASK gt(x) e 0

[ ][ ]

PSTC ) {(AB/C), (B/C)}

Ω(Yt) ) true; Yt ) {true, false}

PSTD ) {(C/D)}

x∈Rn

We next define the Boolean variable Yt such that the variable is true if the task t is selected in the configu-

(3, 4)

(4) Connectivity

e.g., TSABCD ) {(AB/BCD), (ABC/BCD), (ABC/CD)} TSABC ) {(AB/BC), (AB/C)}

(2)

(3) The lightest and the heaviest products can be produced by only one task. However, the intermediate products i are produced by two contributions, one by a stripping section of a task (considering the task as a pseudo column), and another by a rectifying section.

e.g., STATES ) {(ABCD), (ABC), (BCD), (AB), (BC), (CD), (A), (B), (C), (D)} to produce}

s ∈ STATES

t∈STs

(2) STATES ) {s | s is a state}

(3) TSs ) {tasks t that the state s is able

(1)

(2) A given state can only be produced at most by one task (except for products).

(1) TASK ) {t | t is a given task} e.g., TASK ) {(ABC/BCD), (AB/BCD), (ABC/CD), (AB/BC), (AB/C) (B/CD), (BC/CD), (A/B), (B/C), (C/D)}

s ∈ STATES

t∈STs

where x is a vector of continuous variables. f(x) is the objective function, and the equations and constraints


that are activated when task t is performed are given by ht(x) ) 0 and gt(x) e 0, respectively. The equations Ω(Yt) ) true are the logical relationships among the tasks in the superstructure. The model is valid for shortcut as well as rigorous calculations. The connectivity equations r(x) ) 0 simply indicate the flow relations between tasks. They can be modeled by admitting that a state generates a mixer and a splitter. To write the connectivity equations, we define the following new index sets: COMP ) {i | components to be separated} RECTs ) {task t that produces state s by a rectifying section} e.g., RECTABC ) {(ABC/CD), (ABC/BCD)} RECTAB ) {(AB/BCD), (AB/BC), (AB/C)} RECTBC ) {(BC/CD)} STRIPs ) {task t that produces state s by a stripping section} e.g., STRIPBCD ) {(AB/BCD), (ABC/BCD)} STRIPBC ) {(AB/BC)}

A conceptual objective will be considered with the contributions of fixed costs of the heat exchangers and tasks (each task will produce two sections of final column) and a linear variation with the total vapor flow rate and energy consumption (utilities). These cost coefficients are “conceptual” in the sense that they can easily be changed to study their effects on the final optimal configuration. These costs should be sufficient for a preliminary design. Note also that the optimum corresponds not to a single column configuration, but rather to a sequence of tasks that can be rearranged in N - 1 columns in very different forms. Therefore, it is not easy to develop a more rigorous cost estimation. However, at this level of design, this is not necessary. A two-step recursive procedure is currently under development in which, first, the optimal sequence of tasks is obtained (the model under discussion in this paper), providing a rigorous lower bound with the dominant costs. Then, a second stage is performed in which the optimal rearrangement of the predicted tasks in the actual N - 1 columns is obtained with more rigorous cost estimation models (providing an upper bound). The disjunctive part of the model can be then written as follows:

STRIPCD ) {(ABC/CD), (B/CD), (BC/CD)} The connectivity equations for each state s are then given as follows:

∑

Ft -

t∈TSs

∑

Dt -

t∈RECTs

∑

Bt ) 0 s ∈ STATES

t∈STRIPs

(8)

∑

FIt,i -

t∈TSs

∑

∑

(V1t - V2t) -

t∈TSs

∑ (L

t∈TSs

DIt,i -

t∈RECTs

∑

∀ i∈COMP, s ∈ STATES (9)

∑

VIt +

t∈RECTs

1

- L2)t -

BIt,i ) 0

t∈STRIPs

∑

L1t +

t∈RECTs

∑

V2t ) 0

t∈STRIPs

∑

t∈STRIPs

s ∈ STATES (10)

L2t ) 0 s ∈ STATES (11)

where Ft is the molar flow rate of feed in task t; V1t and L1t are, respectively, the molar liquid and vapor flow rates leaving a task through a rectifying section; and V2t and L2t are, respectively, the molar vapor and liquid flow rates leaving a task through a stripping section. Dt and Bt are the net distillation and bottom flow rates, respectively, of the task t, defined as Dt ) V1t - L1t

(12)

Bt ) L2t - V2t ∀ t ∈ TASK

(13)

FIi,t, BIi,t, and DIi,t are the individual flow rates of component i in streams F, B, and D, respectively. Note that all of the connectivity equations are linear. Therefore, the only nonlinearities in the model are those due to equations in the disjunctions or those included in the objective function. We will illustrate the model with an example using Underwood’s equations for thermally linked systems proposed by Carlberg and Westerberg.1,2

The first six equations in the left part of the disjunction are only mass balances. The next two correspond to Underwood equations. Superstructures for Thermally Linked Separation Systems (from 2 to N Heat Exchangers) The STN superstructure for thermally linked systems using only one reboiler and one condenser can be generalized to thermally linked systems with up to N heat exchangers. The derivation of the superstructure is exactly the same as in the previous cases. However, in this case, associated with each of the final states (pure

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products), there is the possibility of including a heat exchanger. In this case, a disjunction appears associated with these states. In fact, this is equivalent to defining two different states (e.g., B and B′) that are characterized not only by the composition, but also by the thermal state. The logical relationships between tasks change slightly. The logical relationships for this case are as follows: (1) A given state can give rise to at most one task. (2) A given state can be produced by at most one task (except for products). (3) The lightest and heaviest products are produced by only one task, and the heat exchanger associated with these components will always appear in the superstructure. However, the intermediate products are produced by one or two contributions. (3.1) If the intermediate product is produced by two contributions, one must come from a stripping section of a task and the other from a rectifying section. There is no heat exchanger associated with this product. (3.2) If the intermediate product is produced by only one contribution, the heat exchanger associated with this product must be selected. (4) Connectivity holds. (5) The number of tasks is equal to N - 1 + N - E ) 2N - 1 - E (E is the number of heat exchangers), and 2 e E e N. Note that logical relations (rules) 1, 2, and 4 are exactly the same as in the case in which only one reboiler and one condenser are included. The important differences arise from the introduction of the possibility of new heat exchangers. Rules 3.1 and 3.2 implicitly include the following logical relationships. Let Wi be a Boolean variable that takes the value true if the heat exchanger associated with product i is selected and false otherwise. The new logical expressions are then as follows: (3) An intermediate product i can be produced by one or two contributions

∨

t∈(PREi∪PSTi)

Yt i ∈ COMP

(4) The number of tasks is equal to N - 1 + N - E ) 2N - 1 - E (E is the number of heat exchangers), and 2 e E e N.

∑

t∈TASK

Yt ) N - 1 -

∑

Wi

(21)

i∈COMP

The disjunctive formulation for this problem is then as follows:

(15)

∑

Yt e 1 i ∈ COMP

(16)

∑

Yt e 1 i ∈ COMP

(17)

t∈PREi

t∈PSTi

Equation 15 ensures that product i is produced by at least one task, while eqs 16 and 17 ensure that product i is produced by the contributions of at most one rectifying section and at most one stripping section. (3.1) If the intermediate product is produced by two contributions, one must come from a stripping section of a task and the other from a rectifying section, and no heat exchanger is associated with this product.

(

∨

t∈PREi

)∧(

Yt

)

∨

Yt w ¬W i ∈ COMP i

t∈PSTi

(18)

(3.2) If the intermediate product is produced by only one contribution, the heat exchanger associated with this product must be selected.

( (

) )

¬ t∈PRE i

∨

Yt

w Wi i ∈ COMP

(19)

¬ t∈PST i

∨

Yt

w Wi i ∈ COMP

(20)

In the above model (P2), Qc and Qr are the utilities needed in the condenser and reboiler, respectively; Hj


is the enthalpy of vaporization of component j; Ct is the fixed cost of task t; Cv is a cost coefficient related to the vapor flow rate in task t; CWi is the fixed cost of installing a heat exchanger; and CC and CH are the cost coefficients of the utilities. It should be noted that the model with only one condenser and one reboiler is a particular case of model P2. Complex columns are more energy efficient but have larger temperature ranges than simple column sequences. More specifically, complex columns are more efficient with respect to first law effects, but less efficient with respect to second law effects. Therefore, from an energy point of view, complex columns are favored if there is an adequate temperature driving force.1 From the point of view of investment, complex columns use more column sections but fewer heat exchangers than conventional columns. As a consequence, the optimal sequence of columns could be an intermediate design. In the next section, we extend the model to take into account all of the possible configurations from a system with 2(N + 1) heat exchangers to only 2 heat exchangers. Disjunctive Model for Generalized Distillation Systems. Partially Thermally Linked Distillation Systems Superstructures generated for thermally linked systems are also valid for systems that take into account all of the combinations of nonconventional and conventional columns. Note that, at the level of the STN representation, the only difference between thermally linked columns and conventional ones is the presence of intermediate condensers and reboilers. Again, the thermal state of intermediate states is important. Note also that, if we introduce the possibility of a heat exchanger associated with each one of the intermediate states, we obtain a superstructure in which all of the possibilities are taken into account. However, although there are no important changes at the level of the superstructure, there are important modifications at the modeling level. In this case, the connectivity equations cannot be treated as simple mixers and splitters, because the presence or absence of an intermediate heat exchanger changes the flows between columns. Whereas, in thermally linked systems, separation tasks exchange vapor and liquid flows, if an intermediate heat exchanger is selected, only liquid is fed to the tasks. To model this last case, it is necessary to introduce a Boolean variable to indicate the existence of these intermediate heat exchangers. (Here, only total condensers are considered. The extension to systems with partial condensers can also be carried out.) It is important to point out that we differentiate between heat exchangers associated with pure products and intermediate heat exchangers (related to each one of the intermediate states). The reason is that the number of intermediate heat exchangers does not change the minimum number of column sections (or tasks) for performing the separation. The heat exchangers only break the thermal linkages between tasks, although the heat exchangers related to the final products are related to the minimum number of separation sections. To model the general case, let us to introduce the following new subsets:

ISTATE ){m | m is an intermediate state} e.g., ISTATE ) {ABC, BCD, AB, BC, CD} IRECm ) {task t that produces intermediate state m from a rectifying section} e.g., IRECABC ) {(ABC/D, ABC/CD, ABC/BCD} IRECAB ) {(AB/CD, AB/C, AB/BC, AB/BCD)} IRECBC ) {(BC/D, BC/CD)} ISTRIPm ) {task t that produces intermediate state m from a stripping section} e.g., ISTRIPBCD ) {(A/BCD, ABC/BCD, AB/BCD)} ISTRIPBC ) {(A/BC, AB/BC)} ISTRIPCD )

{(AB/CD, B/CD, ABC/CD, BC/CD)}

Let Zm be a Boolean variable that takes the value true if the heat exchanger associated with the intermediate state m appears in the final structure and false otherwise. In this case, no new logical relationships are needed, but a new disjunction is required in the model. The disjunctive formulation of the generalized distillation system is as follows:

(P3) min Z )

∑

(Ct + CvV1t + CvV2t + CCQrt +

t∈TASK

∑

CHQct) +

CWi +

i∈COMP

∑

CZm

m∈ISTATE

s.t. common connectivity equations

∑

Ft -

t∈TSs

∑

t∈TSs

FIt,i -

∑

Dt -

∑

DIt,i -

t∈RECTs

t∈RECTs

∑

Bt ) 0 s ∈ STATES

t∈STRIPs

∑

BIt,i ) 0

t∈STRIPs

∀ i ∈ COMP, s ∈ STATES

Disjunctions due to heat exchangers associated with intermediate states (including connectivity and heat balances)

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[


]

Disjunctions due to heat exchangers related to products

Wi QctDt )

∑ HjDj,tV1t j∈COMP

t ∈ PREi

QrtBt )

∑ HjBj,tV2t j∈COMP

t ∈ PSTi

CWi ) γi

[

∨

¬Wi Qct ) 0 t ∈ PREi Qht ) 0 t ∈ PSTi CWi ) 0

]

∀ i ∈ COMP

Yt e 1 s ∈ STATES

∑

Yt e 1 s ∈ STATES

∨

Yk ∀ t ∈ STs, s ∈ STATES

t∈TSs

∨

Yk t ∈ TSs, s ∈ STATES

∨

t∈(PREi∪PSTi)

Yt i ∈ COMP

∑

Yt e 1 i ∈ COMP

∑

Yt e 1 i ∈ COMP

t∈PREi

t∈PSTi

∨ ∨

∨

t∈PSTi

t∈PREi t∈PSTi

∑

∑

k∈STs

( (

)∧(

t∈TASK

t∈STs

Yt w

Yt

¬

Logical relationships

k∈TSs

∨

t∈PREi

¬

Disjunctions related to the diferent tasks

Yt w

(

) )

)

Yt w ¬W i ∈ COMP i

Yt w W i ∈ COMP i Yt w W i ∈ COMP i

Yt ) N - 1 -

∑

Wi

i∈COMP

This model (P3) contains all of the possible sequences, including those that only use conventional columns. Examples To illustrate the proposed models, three different examples have been studied. All data and parameters of the example systems are included in Tables 1, 3, and 5. Thermodynamic properties were obtained from the ChemCad V database.31 The most straightforward way of solving the models is transforming them into a MINLP. In this approach, the logical relationships are transformed to linear algebraic expression functions of only binary variables.27-29 The disjunctions can be transformed using either a big M formulation or the convex hull formulation. The model can then be solved using GAMS/ DICOPT. However, three important problems arise. First, in most cases, the NLP solver was even unable of finding a feasible solution because of the singularities that arise with disappearing sections. Second, the NLPs and the master MIPs are very time-consuming to solve. Third, even if a solution is found, it often corresponds to a poor local optimum. The following observation is useful for developing an alternative algorithm: For a fixed topology (all of the Boolean variables fixed to given feasible values), the problem was found to always converge to the same solution. This is only a heuristic from experience because there are nonconvexities in the model. If anything, the problem is that the solver often cannot find a feasible solution for an arbitrary starting point. However, the problem always converges if the initial values of the variables are relatively close to the solution, and it usually finds what appears to be the global optimum. Based on the above observation a modified version of the logic-based outer approximation algorithm of Turkay and Grossmann26 was used. If the algorithm is used directly as described in the original paper of these authors, the following problem arises. Because of the nonconvexities, the slacks of the master problem are active even in the first major iteration. This causes the master to lose sensitivity with respect to the best set of binary variables in the following iterations and predicts poor initial values for the continuous variables. In this paper, we propose to modify the original master problem (obtained by accumulation of linearizations) by a master in which the nonlinear equations are changed by their convex (or linear) envelope. Note that all of the nonlinear equations in the model are combinations of bilinear and linear fractional terms, and therefore, it is possible to systematically generate the linear envelopes.32,33 The proposed master has the following advantages. First, it avoids the accumulation of linearizations.

Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2269 Table 1. Summary of Example 1 components composition (molar fraction) feed flow rate ) 100 kmol/h pressure cost coefficients

A ) benzene B ) toluene C ) o-xylene A ) 0.2, B ) 0.3, C ) 0.5 1 atm Ct ) 3500 Cv ) 10 CWi ) 500 CC ) 0.5 CH ) 1 CZm ) 500

cost unit/task cost unit/kmol cost unit/heat exchanger cost units/MJ cost units/MJ cost unit/heat exchanger

(a) objective ) 17745 best sequence ) AB/C, A/B heat exchangers in states A, B, C (b) same coefficients as in previous case except Ct ) 2000 cost unit/task objective ) 14356 best sequence ) AB/BC, A/B, B/C

Figure 5. Schematic of the algorithm.

Second, it does not require that the search be started from scratch in each iteration. We can update the tree search with the binary cut and continue from that point, producing better initial values for the NLPs. It should be noted that the master has low sensitivity to the presence, or lack thereof, of heat exchangers only in the most general case (when heat exchangers associated with intermediate states are allowed). In this case, the NLPs are substituted by a small MINLP (big M formulation) with a fixed configuration of tasks. A schematic of the proposed algorithm is presented in Figure 5.

Figure 6. Separation alternatives for a mixture of three components.

Example 1. The first example consists of separating a mixture of three hydrocarbons (benzene, toluene, and o-xylene). Figure 6 shows the alternatives included in this simple example. Note that, with only one condenser and only one reboiler, only the Petlyuk configuration is possible. However, with the generalized distillation superstructure, up to eight configurations are possible (only two of which involve only conventional columns). This simple example is useful for understanding the trade-offs in the system. (a) The optimal structure (using the costs coefficients given in Table 1) is presented in Figure 6f, and the details of the flows can be found in Table 2. In this example, a configuration with the minimum number of column sections is obtained (four sections). It is well-

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Table 2. Flows and Heat Loads for Example 1 (a) Example 1a task

F (kmol/h)

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

AB/C A/B

100 50

50 20

50 30

122.45 149.070

63.897 26.620

72.450 129.070

113.897 56.620

task

Qc (MJ/h)

Qr (MJ/h)

AB/C A/B

4585.541 -

2352.67 829.995

(b) Example 1b task

F (kmol/h)

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

AB/BC A/B B/C

100 36.469 63.531

36.469 20.000 13.531

63.531 16.469 50.000

74.528 92.342 41.460

28.747 17.815 70.207

38.058 72.342 27.929

92.278 34.284 120.207

task

Qc (MJ/h)

Qr (MJ/h)

AB/BC A/B B/C

2840.538 -

2585.027

known that, for a three-component system, the configuration with the minimum vapor flow is that proposed by Petlyuk.5,34 In this case, we are assuming that each section has a cost. Then, there is a trade-off between the total number of column sections and the cost of energy. Therefore, if the cost coefficients of the column sections decrease, it is expected that the optimal configuration is the fully thermally coupled configuration. (b) In this example, the cost coefficients of the column sections are decreased from 3500 units/year to 2000 units/year. As expected, the fully thermally coupled configuration with six sections shown in Figure 6a is obtained. Details of the solution are given in Table 2. Notice that heat exchangers associated with intermediate states do not appear in either example 1a or example 1b. Agrawal and Fidkowski35 showed that, in some cases, sequences involving three components with intermediate heat exchangers have energy demands similar to those of the Petlyuk configuration, but always larger. The interest in these configurations is that these heat exchangers could be important if heat integration is taken into account and, in some cases, could improve the controllability of the system with small energy penalties.35 Example 2. This example consists of separating a mixture of five hydrocarbons (n-pentane, n-hexane, n-heptane, n-octane, and n-nonane). All of the alternatives discussed in the paper were tested. Cost coefficients and other parameters are shown in Table 3. (a) If only one reboiler and one condenser are allowed in the system, the optimal sequence shown in Figure 7a is obtained. (Figure 7b shows a possible rearrangement of this solution.) (b) If the problem is solved allowing up to N heat exchangers, the optimal configuration will depend on the fixed costs of the column sections (tasks), the costs of the utilities, and the fixed costs of the new heat exchangers. If the utilities are expensive and no constraints are imposed on the system, then it is likely that one would obtain a configuration with a minimum number of new heat exchangers. However, if the energy and heat exchangers are cheap compared to the columns, then a system with a reduced number of tasks

Table 3. Summary of the Example 2 A ) n-pentane B ) n-hexane C ) n-heptane D ) n-octane E ) n-nonane composition (molar fraction) A ) 0.1, B ) 0.1, C ) 0.2, D ) 0.3, E ) 0.3 feed flow rate ) 100 kmol/h pressure ) 1 atm cost unit/task cost coefficients Ct ) 3500 cost unit/kmol Cv ) 10 cost unit/heat exchanger CWi ) 500 CC ) 0.025 cost unit/MJ CH ) 0.05 cost//MJ cost unit/heat exchanger CZm ) 500 (a) fully thermally linked objective ) 31165 best sequence ) AB|BCDE, BC|CDE, CD|DE, A|B, B|C, C|D, D|E (b) up to N heat exchangers (b1) same conceptual coefficients as in previous case objective ) 22508 best sequence ) A/BCDE, B/CDE, C/DE, D/E heat exchangers in states A, B, C, D, E (b2) increasing cost of energy to CC ) 0.5, CH ) 1 objective ) 32153 best sequence ) ABCD|DE, D|E, AB|CD, A|B, C|D heat exchangers in states A, B, C, E (c) up to 2(N - 1) heat exchangers general distillation sequence same results as case b

components

could be interesting. (Remember that the numbers of tasks and heat exchangers are not independent.) Figure 8a is obtained using the same parameters as in example 2a. In Figure 8c, the cost of the utilities is increased, and as expected, the number of heat exchangers decreases. In any case, the optimal configurations are not intuitive. Details about this configurations are shown in Table 4(c) For the sake of comparison, a general distillation system with the same cost coefficients as in the previous examples was examined. The optimal configurations are the same that those obtained in example 2b. It has not been shown that, for systems with more than three components, intermediate heat exchangers cannot improve the efficiency, but the results obtained follow the behavior obtained for three components, and no intermediate heat exchangers appear. Note that, in systems with partially or thermally linked columns, there are different possible rearrange-


Figure 7. Optimal configuration for example 2a, five-component separation using only two heat exchangers: (a) keeping the tasks separated, (b) one of the possible rearrangements.

Figure 8. Optimal configuration for sequences with up to N heat exchangers for example 2: (a) and (b) different possible rearrangements for the same tasks, (c) one possible rearrangement for the optimal result increasing the cost of energy.

ments corresponding to the same sequence.36-38 The advantage of the STN representation is that no particular arrangement of columns is assumed a priori,

which is particularly useful in the early stages of design where decisions have a large influence on the final performance of the system.

2272


Table 4. Flows and Heat Loads for Example 2 Example 2 a task

F (kmol/h)

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

AB/BCDE BC/CDE CD/DE A/B B/C C/D D/E

100 85.555 75.968 14.445 9.587 21.823 54.145

14.445 9.587 21.823 10.000 5.555 15.968 24.145

85.555 75.968 54.145 4.445 4.032 5.855 30.000

29.677 19.121 36.725 33.782 23.672 49.096 62.718

15.187 34.308 71.033 4.104 4.551 12.372 133.751

12.232 9.535 14.902 23.785 18.117 33.128 38.573

100.742 110.276 125.178 8.550 8.538 18.227 163.751

task

Qc (MJ/h)

Qr (MJ/h)

A/B D/E

1164.5 -

6715.2 Example 2 b1

task

F (kmol/h)

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

A/BCDE B/CDE C/DE D/E

100 90 80 60

10 10 20 30

90 80 60 30

31.993 29.827 53.468 72.809

19.918 49.754 103.213 176.023

21.993 19.827 33.468 42.809

109.918 129.745 163.213 206.023

task

Qc (MJ/h)

Qr (MJ/h)

A/BCDE B/CDE C/DE D/E

1102.853 1144.186 2267.620 3354.941

8837.483 Example 2 b2

task

F (kmol/h)

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

ABCD/DE A/BCD B/CD C/D D/E

100 58.876 48.876 38.876 41.124

58.876 10.000 10.000 20.000 11.124

41.124 48.876 38.876 18.876 30.000

110.855 121.824 29.495 45.328 41.723

23.509 10.969 40.464 85.792 65.232

51.979 111.824 19.495 25.328 30.599

64.634 59.844 79.340 104.667 95.232

task

Qc (MJ/h)

Qr (MJ/h)

A/BCD B/CD C/D D/E

4199.478 1131.467 1922.363 -

3275.094

Table 5. Summary of the Example 3 A ) i-butane B ) 1-butene C ) n-butane D ) trans-2-butene E ) cis-2-butene composition A ) 0.1, B ) 0.1, C ) 0.2, D ) 0.3, E ) 0.3 (molar fraction) feed flow rate ) 100 kmol/h pressure ) 1 atm cost unit/task cost coefficients Ct ) 3500 Cv ) 10 cost unit/kmol cost unit/heat exchanger CWi ) 500 CC ) 0.5 cost units/MJ CH ) 1 cost units/MJ cost unit/heat exchanger CZm ) 500 objective ) 91996 best sequence ) ABC/BCDE, BCD/DE, B/CD, C/D, D/E, AB/C, A/B heat exchangers in states A, E components

Example 3. In this case, a mixture of five hydrocarbons is again considered, but whereas in example 2, the differences in relative volatilities are in some cases greater than 40, in this case, the maximum difference is around 2, which makes the separation much more difficult. The components to be separated are i-butane, 1-butene, n-butane, trans-2-butene, and cis-2-butene. The data are given in Table 5.

Figure 9. Optimal configuration for the general separation system of example 3.

For the sake of comparison, we use the same cost coefficients as in example 2. In this case, energy is the dominant cost, and the optimal configuration is a completely thermally linked structure, as shown in

Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2273 Table 6. Flows and Heat Loads for Example 3 task ABC/BCDE BCD/DE AB/C B/CD A/B C/D D/E task A/B D/E

F (kmol/h) 100 84.822 15.178 41.239 13.885 35.124 43.583 Qc (MJ/h) 2945.8 -

D (kmol/h)

B (kmol/h)

V1 (kmol/h)

V2 (kmol/h)

L1 (kmol/h)

L2 (kmol/h)

15.178 41.239 13.885 6.115 10.000 18.707 13.583

84.822 43.583 1.293 35.124 3.885 16.417 30.000

105.146 414.135 112.265 547.091 138.614 597.558 198.492

194.698 608.833 7.118 132.965 26.350 730.514 807.325

89.969 372.896 98.380 540.976 128.614 578.851 184.908

279.520 652.416 8.411 168.080 30.235 746.931 837.325

Qr (MJ/h) 18812

Figure 9. If the cost of the columns increases with the cost of energy, the optimal configuration would change to a configuration with a smaller number of column sections. Table 6 shows the results of this case. The cost is greater than in example 2 because of the larger flows through the columns and, therefore, the increase in operation and energy costs. However, although there are quantitative differences from the results of example 2, the optimal configurations are qualitatively equivalent. This example illustrates the versatility of the proposed approach. Conclusions The synthesis of distillation column configurations for the separation of near-ideal mixtures into their constituent product streams has been considered in this paper. The proposed approach considers both conventional columns and thermally linked systems in the same model. For partially thermally linked systems, the minimum number of column sections varies from 4N - 6 to 2(N - 1) + 2(N - E), where E is the number of heat exchangers (2 e E e N). If all of the heat exchangers are included, then the number of column sections is a minimum. The introduction of more heat exchangers, up to a maximum of 2(N - 1), does not reduce the number of column sections. It was shown that it is possible to systematically generate a superstructure based on the state task network (STN) formalism. Once all of the tasks and states are identified, the superstructure is systematically generated by connecting each task with the possible states to which this task could give rise and connecting each state with the tasks that this state could produce. If all of the tasks are included, then the superstructure is valid for thermally linked, partially thermally linked, and nonthermally linked systems. The only difference arises from the logical relationships between the tasks and states. The logical relationships can be generated systematically using a set of simple rules based on topological considerations. The superstructure can also be systematically modeled as a generalized disjunctive program with a mathematical structure that is independent of the particular model used (rigorous aggregated or shortcut). In this paper, the proposed methodology was illustrated with a variation of Underwood’s method.1,2 The results show that configurations of partially thermally linked configurations can be optimal if there is not a strong dominant cost in the process. These results are not intuitive, showing that these design alternatives should be taken into account in the early stages of design.

Finally, the heat integration between columns, which was not considered in this paper, is currently the subject of our research. Literature Cited (1) Carlberg, N. A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 2. Underwood’s Method for Side Stripers and Enrichers. Ind. Eng. Chem. Res. 1989, 28, 13791386. (2) Carlberg, N. A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 3. Underwood’s Method for the Petlyuk Configuration. Ind. Eng. Chem. Res. 1989, 28, 1386-1397. (3) Thomsom, R. V.; King, C. J. Systhematic Synthesis of Separation Schemes. AIChE J. 1972, 18, 941. (4) Peltlyuk, F. B.; Platonov, V. M.; Slavinskii, D. M. Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Int. Chem. Eng. 1965, 5 (3), 555. (5) Triantafyllou, C.; Smith, R. The Design and Optimization of Fully Thermally Coupled Distillation Columns. Trans. Inst. Chem. Eng. 1992, 70A, 118-132. (6) Seader, J. D.; Westerberg, A. W. A Combined Heuristic and Evolutionary Strategy for Synthesis of Simple Separation Sequences. AIChE J. 1977, 23, 951. (7) Stephanopoulos, G.; Westerberg, A. W. Studies in Process Synthesis. II. Evolutionary Synthesis of Chemical Process Flowsheets. Chem. Eng. Sci. 1976, 31, 195. (8) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (9) Johns, W. D.; Romero, D. Automated Generation and Evaluation of Process Flowsheet. Comput. Chem. Eng. 1979, 3, 251. (10) Fraga, E. S.; McKinnon, K. I. M. Portable Code for Process Synthesis Using Workstation Clusters and Distributed Memory Multicomputers. Comput. Chem. Eng. 1995, 19, 759. (11) Fraga, E. S.; Matias, T. R. S. Synthesis and Optimization of a Nonideal Distillation System Using Parallel Genetics Algorithms. Comput. Chem Eng., Suppl. 1996, 20, S79. (12) Westerberg, A. W. The Synthesis of Distillated Based Separations. Comput. Chem. Eng. 1985, 9, 421. (13) Floquet, P.; Pibouleau, L. Domoenech, S. Mathematical Programming Tools for Chemical Engineering Process Design Synthesis. Comput. Chem. Eng. 1988, 23, 1. (14) Floquet, P.; Pibouleau, L. Domoenech, S. Separation Sequence Synthesis: How to Use Simulated Annealing Procedures. Comput. Chem. Eng. 1994, 18, 1141. (15) Gert-Jan, A. F.; Liu, Y. A. Heuristic Synthesis and Shortcut Design of Separation Processes Using Residue Curve Maps: A Review. Ind. Eng. Chem. Res. 1994, 33, 2505. (16) Juergen, K.; Poellmann, P.; Blass, E. A Review of Minimum Energy Calculations for Ideal and Nonideal Distillation. Ind. Eng. Chem. Res. 1995, 34, 1003. (17) Westerberg, A. W.; Wahnschafft, O. Synthesis of Distillation Based Separation Systems. In Advances in Chemical Engineering; Academic Press: New York, 1996; Vol. 23, p 63. (18) Sargent, R. W. H.; Gaminibandara, K. Introduction: Approaches to Chemical Process Synthesis. In Optimization in Action; Dixon, L. C., Ed.; Academic Press: New York, 1976. (19) Andrecovich, M. J.; Westerberg, A. W. A MILP formulation for Heat Integrated Distillation Sequence Synthesis. AIChE J. 1985, 31 (9), 1461.

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(20) Grossmann, I. E.; Caballero, J. A.; Yeomans, H. Mathematical Programming Approaches to the Synthesis of Chemical Process Systems. Korean J. Chem. Eng. 1999, 16 (4), 407-426. (21) Bagajewitz, M. J.; Manousiouthakis, V. Mass/Heat Exchange Network Representation of Distillation Networks. AIChE J. 1992, 38, 1769. (22) Sargent, R. W. H. A Functional Approach to Process Synthesis and Its Applications to Distillation Systems. Comput. Chem. Eng. 1998, 22, 31. (23) Yeomans, H.; Grossmann, I. E. A Systematic Modeling Framework of Superstructure Optimization in Process Synthesis. Comput. Chem. Eng. 1999, 23, 709. (24) Agrawal, R. Synthesis of Distillation Column Configurations for a Multicomponent Separation. Ind. Eng. Chem. Res. 1996, 35, 1059-1071. (25) Hohmann, E. C.; Sander, M. T.; Dunford, H. A New Approach to the Synthesis of Multicomponent Separation Schemes. Chem. Eng. Commun. 1982, 17, 273-284. (26) Turkay, M.; Grossmann, I. E. A Logic Based Outer Approximation Algorithm for MINLP Optimization of Process Flowsheets. Comput. Chem. Eng. 1996, 20, 959. (27) Raman, R.; Grossmann, I. E. Relation Between MILP Modeling and Logical Inference for Chemical Process Synthesis. Comput. Chem. Eng. 1991, 15, 73. (28) Raman, R.; Grossmann, I. E. Symbolic Integration of Logic in Mixed Integer Linear Programming Techniques for Process Synthesis. Comput. Chem. Eng. 1993, 17, 909. (29) Raman, R.; Grossmann I. E. Modeling and Computational Techniques for Logic Based Integer Programming. Comput. Chem. Eng. 1994, 18, 563. (30) Caballero, J. A.; Grossmann I. E. Aggregated Models for

Integrated Distillation Systems. Ind. Eng. Chem. Res. 1999, 38, 2330. (31) ChemCad V Users Guide and Tutorial; Chemstations, Inc.: Houston, TX, 1999. (32) Quesada, I.; Grossmann, I. E. A Global Optimization Algorithm for Linear Fractional and Bilinear Programs. J. Global Optim. 1995, 6, 39-76. (33) Smith, E. M. On the Optimal Design of Continuous Processes. Ph.D. Dissertation, Imperial College of Science, Technology and Medicine, London, U.K., 1996. (34) Rudd, H. Thermal Coupling for Engineering Efficiency. Chem Eng., Suppl. 1992, S14. (35) Agrawal, R.; Fidkowski, Z. T. New Thermally Coupled Schemes for Ternary Distillation. AIChE J. 1999, 45 (3), 485496. (36) Agrawal, R. A Method to Draw Fully Thermally Coupled Distillation Column Configurations for Multicomponent Distillation. Trans. Inst. Chem. Eng. 2000, 78A, 454-464. (37) Agrawal, R.; Fidkowski, Z. T. More Operable Arrangements of Fully Thermally Coupled Distillation Columns. AIChE J. 1998, 44, 4 (11) 2565-2568. (38) Agrawal, R. More Operable Fully Thermally Coupled Distillation Column Configurations for Multicomponent Distillation. Trans. Inst. Chem. Eng. 1999, 543-553.

Received for review August 16, 2000 Revised manuscript received December 29, 2000 Accepted February 26, 2001 IE000761A

Generalized Disjunctive Programming Model for the Optimal Synthesis

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