Aggregated Models for Integrated Distillation Systems - Industrial

2330

Ind. Eng. Chem. Res. 1999, 38, 2330-2344

Aggregated Models for Integrated Distillation Systems J. A. Caballero† and I. E. Grossmann*,‡ Department of Chemical Engineering, University of Alicante, Ap Correos 99, Alicante, Spain, and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

In this work we present an aggregated representation for distillation columns that can be used in the synthesis of separation sequences with heat integration. A new aggregated model is first presented for the stripping and rectifying sections of individual distillation columns. This model is based on mass balances and equilibrium feasibility, expressed in terms of flows, inlet concentrations, and recoveries. The energy balance can then be decoupled from the mass balance, and the utilities can be calculated for each separation task. The proposed model is applied to three different superstructures: state task network, state equipment network, and an intermediate representation. The proposed model yields a lower bound to the vapor flow or to the total cost of the utilities. Performance of the different superstructure representations in terms of robustness and computational time is illustrated with several examples. Introduction The goal of conceptual design (process synthesis) is the identification of the best flowsheet structure (process system) that must carry out a specific task, such as conversion of raw material into a product or separation of a multicomponent mixture. To accomplish this goal, many alternative designs must be considered. Usually the flowsheet is divided into subsystems such as reaction, separation, and heat integration. These subsystems can, in turn, be divided into smaller ones, at either the level of the paths between unit operations or choices of different technologies among them. Two alternative approaches to process synthesis include (a) total enumeration and (b) physical insights. The former is limited to problems with very few alternatives, while the second relies on graphical representations of the problem, for example, the separation of azeotropic mixtures of three components using ternary diagrams1,2 or the synthesis of reactor networks using the attainable region representation.3,4 Both approaches are only useful for relatively small problems. Two other approaches are (c) search with pruning5 and (d) heuristic evolutionary search.6 The former can be algorithmic in nature, for example, through a branch and bound search (possibly based on heuristics and knowledge of previous problems), while the latter uses heuristics to generate a good base case design and then modifies the flowsheet until no improvement is possible. The two more recent approaches are (e) hierarchical decomposition developed by Douglas7-9 and (f) superstructure optimization developed fundamentally by Grossmann.10-12 In the first, the problem is addressed through hierarchical decisions and short-cut models at various levels: batch versus continuous, input-output structure of the flowsheet, recycle structure of the flowsheet, general structure of the separation system (vapor and liquid recovery), and heat integration. If the process becomes unprofitable as the design proceeds, the search is terminated. In the case of superstructure optimization, a systematic representation is postulated * To whom correspondence should be addressed. † University of Alicante. ‡ Carnegie Mellon University.

in which all of the alternatives of interest are embedded. The problem is then modeled and solved as an MINLP process flowsheet. While hierarchical decomposition can address complex problems, it cannot guarantee one to obtain the best solution because it is a sequential decomposition strategy, and therefore it does not take into account the interactions between the different levels of decomposition. As an example, Duran and Grossmann13 and Lang et al.14 have shown that the simultaneous optimization and heat integration of process flowsheets generally produce improvements compared to the sequential approach. The MINLP techniques have shown to be powerful in the synthesis of subsystems: heat exchanger networks,15-18 mass exchanger networks,19-21 distillation sequencing,18,22-24 and utility systems.25,26 MINLP techniques are limited to problems of moderate size for solving process flowsheets. As was observed by Rippin,27 hierarchical decomposition and superstructure optimization can be considered to be complementary to each other. Daichendt and Grossmann28 developed an approach for flowsheet synthesis that combines hierarchical decomposition with the MINLP approach to exploit advantages of both methods. The main idea is to solve the entire flowsheet at each step of the decomposition using a multilevel tree search. Aggregated optimization models were used to account for the interactions between the different levels and subsystems. Thus, a simultaneous optimization of the entire process is performed by a combination of simple and more detailed models. At each step of the decomposition, improved designs are obtained with tighter bounds and a monotonic decrease in the profit. The key idea of Daichendt and Grossmann28 is that the representation of alternatives for optimization can be specified at various levels of detail, from aggregated to detailed models. In the superstructure a detailed representation of all the potential units and streams in a flowsheet is explicitly considered. Aggregated models are representations at higher levels of abstraction and lower dimensionality that have implicitly embedded all of the alternatives of interest in which the design model is simplified by the use of design targets such us

10.1021/ie980803j CCC: $18.00 © 1999 American Chemical Society Published on Web 05/01/1999

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2331

Figure 1. STN representation for a mixture of four components.

minimum utility or maximum yield.29 This approach simplifies the mathematical programming problem either by reducing the size of the original MINLP or by reducing it into a nonlinear program (NLP), mixedinteger linear program (MILP), or linear program (LP). One of the limitations of the approach of Daichendt and Grossmann is that the bounds of some of the aggregated models may be weak, with which the space of feasible alternatives is not significantly reduced, and the next step in the search will have to deal with an MINLP problem of increased complexity. Daichendt and Grossmann used models based on thermodynamic efficiency for the separation step, which tend to produce weak bounds. Therefore, a major motivation of this work is to develop aggregated models that improve these bounds. The specific goal of this paper is to develop an aggregated model for the synthesis of heat-integrated distillation systems. Models are presented for the state task network (STN) and the state equipment network (SEN) representations that can be used to systematically derive superstructures for these systems.30 We first present the general framework for systematically deriving process superstructures and its application to distillation problems. Next, we present the aggregated model for one conventional distillation column and its extension to the case of heat integration. Finally, several examples are presented with mixtures of four and five components to illustrate the proposed approach.

Problem Statement The problem addressed in this paper can be stated as follows: Given is a mixture of N components that is to be separated into pure components. The objective is to develop an aggregated optimization model that can include heat integration in order to predict a tight lower bound for the cost of the distillation sequence. The model uses target objectives related to the cost. For instance, the objective in the model without heat integration is to minimize the total vapor flow rate in the columns. In the model with heat integration, the objective function is to minimize the total cost of the utilities. The proposed aggregated model avoids solving the complex MINLP problem of a rigorous plate-by-plate optimization. The model can be used to perform preliminary screening to decide which configuration should be considered in more detail in the next step of the design process. The cost of the aggregated model also underestimates the costs and the feasible region of the model. Systematic Generation of Superstructures To formulate the aggregated model for synthesizing distillation sequences as an optimization problem, the first step it is to develop a representation of the alternatives that will be considered as candidates for

2332 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

Figure 2. SEN representation for a mixture of four components.

the optimal solution. Yeomans and Grossmann30 proposed a general framework for automatically generating superstructures. These authors considered two extreme representations, the STN (see also the work of Sargent31) and SEN (see also Smith and Pantelides32). The former is concerned with the selection of tasks, leaving the equipment assignment to a second stage. In the latter the equipment is selected, leaving the selection of tasks to a second stage. Application of the STN Representation to Optimal Distillation Sequences. According to the proposed methodology, the first step is to identify the states and the tasks. For conventional columns, the tasks are each one of the different possible separations. For example, for a mixture of four components (say ABCD, where A is the most volatile and D the least) we can identify 10 different tasks (A/BCD, AB/CD, ABC/D, A/BC, AB/C, B/CD, BC/D, A/B, B/C, and C/D). The states are each one of the streams that are related to the task. In this case we assume that they are all saturated liquid. The second stage is to assign the equipment to the tasks. In this case only the one task one equipment (OTOE) assignment is considered, because we are restricted to distillation with conventional columns. Figure 1 shows an example of the STN representation for a mixture of four components. Application of the SEN Representation to Optimal Distillation Sequences. The SEN representation was developed by Smith and Pantelides.32 However, these authors considered full connectivity between states and equipment. An important characteristic of the SEN representation is that the state definition may not be unique, because the stream properties and qualitative characteristics will depend on which task is performed in the equipment. The state definition will have to consider all of the possible states as a function of the different tasks in the equipment. Thus, there is a tradeoff between the smaller combinatorial problem for the selection of equipment and the increasing combi-

natorial problem complexity in the state definition. Figure 2 shows an example of the SEN representation for a mixture of four components. Note that the STN and SEN representations are extreme cases. In the STN representation the number of columns is equal to the number of tasks, and in the SEN representation the number of columns is the minimum necessary to perform the separation (in the case we are considering N - 1 columns). Therefore, it is possible to generate superstructures with any number of columns between these two extremes.31 An intermediate situation is shown in Figure 3 for a mixture with four components. Note that if the columns of Figure 3 were thermally linked, the task of column 1 would be to separate A from D, and we could get as the top product a mixture of ABC and as the bottom product a mixture of BCD. We can decide then that the next column must separate A from C, the next one B from D, etc. This is basically the approach of Sargent and Gaminibandara,24 including the modification of Agrawal33 for thermally linked columns. It should also be noted that the tasks for the mixers and splitters must also be specified. In this case, both mixers and splitters are “single choice”, and only one input stream (mixer) or one output stream (splitter) takes a value different from zero. The particular realization will depend on the logical relations between tasks, which are specified at the modeling level. Aggregated Model for One Distillation Column Bagajewicz and Manousiouthakis34 developed a model in which a distillation column was considered as a composite heat- and mass-exchanger operation. Assuming a constant countercurrent molar flow rate, the mass exchange inside the distillation columns can be treated as a pure mass-transfer operation. Therefore, in this case a distillation network could be treated as a separable heat/mass exchange network. Papalexandri


Figure 3. Intermediate representation for a mixture of four components.

Figure 5. Composition-mass load diagram for a binary mixture. Figure 4. Scheme of a distillation column.

and Pistikopoulos35 introduced a multipurpose mass/ heat-transfer module as a building block of a systematic representation of conventional and nonconventional process units and process structures. In this paper, we also represent a column as a separable mass- and heattransfer network. The equilibrium feasibility (pinch) is considered as a criterion for the design of a multicomponent separation column. For the sake of simplicity, we consider in this paper conventional columns. The representation, however, can

easily be extended to columns with multiple feeds and multiple products and columns without condenser or reboiler. The main assumptions for the proposed model are as follows: 1. Each column is divided into two sections (two mass exchange zones); see Figure 4. In each of the sections the molar flow rates of vapor and liquid are assumed to be constant. 2. The pinch point can be located only in the extreme points of the sections. This can be illustrated with a binary mixture through the composition-mass load (C-


ML) pinch diagram as seen in Figure 5 (Bagajewicz and Manousiouthakis34). Feasibility of mass exchange is established when both ends of a mass exchanger’s operating line lie below the equilibrium curve (or above if the equilibrium curve is based on the heavier component). Fortunately, because the liquid curvesF(x) in Figure 5sis concave, thermodynamic feasibility of mass exchange can be verified by examining the end points at each stream. In a multicomponent mixture the feasibility restrictions will depend on the task that the column performs. For example, in a column with three components, say A, B, and C, in which we want to perform the separation A/BC (A is the most volatile and C the least), the following constraints must hold at the ends of the streams:

i ) B, C

yA e KAxA; yi g Kixi

(1)

The model for a column is as follows: We define the following index sets, COM ) {i|components to be separated}, S ) {s ∈(top, mt, mb, bot)|pinchpoint candidates}, where mt is the bottom part of the top section and mb is the top of the bottom section. 1. Overall Mass Balance for Each Section.

Vini,mt + Lini,top ) Vini,top + Lini,mt Vini,bot + Lini,mb ) Vini,mb + Lini,bot Vs )

∑i Vin

Ls )

∑i Lin

i,s

i,s

}

Vtop ) Vmt; Vmb ) Vbot

}

i ∈ COM (2)

i ∈ COM, s ∈ S

where Vin Lin make reference to the flow rates of the individual components in the vapor and liquid, respectively, and L and V are the overall liquid and vapor molar flow rates. 2. Overall Mass Balance.

(5)

where F is the individual flow rate of component i in the feed and pt and pb are the individual flow rates of the top and bottom products, respectively. 3. Mass and Energy Balances in the Feed Section. We assume that the feed is introduced at its bubble point and it mixes with the liquid stream.

Fi + Lini,mt + Vini,mb ) Lini,mb + Vini,mt

∑i Fihi + ∑i Lin hi,mt + ∑i Vin ∑i Lin hi,mb + ∑i Vin i,mt

i,mb

i,mb

i,mt

i ∈ COM

(6)

Hi,mb )

Hi,mt

Lini,top ) (1 - η1)Vini,top Lini,bot ) pbi + Vini,bot pbi ) η2Lini,bot

Vini,bot ) (1 - η2)Lini,bot

i ∈ COM (7)

where H and h correspond to the specific enthalpies of the vapor and liquid, respectively.

} }

i ∈ COM

(8)

i ∈ COM

(9)

where η1 and η2 are split fractions to be determined. 5. Equilibrium Equations. The equations are not restricted to any particular equilibrium model. In general,

Ki,s ) f(x1s,x2s,...,xns,P,Ts)

i ∈ COM, s ∈ S

(10)

where K is the equilibrium constant, xj,s (j ) 1, 2, ..., n) is the molar fraction of component j in the liquid fraction at position s in the column, P is the pressure in the column, and T is the temperature in section s of the column. 6. Products Must Be at Their Bubble Points. It is assumed that a total condenser is used and that the bottom product is extracted from the reboiler as liquid, which leads to the equation

∑i ptiKi,con ) ∑i pti (4)

i ∈ COM

Vini,top ) pti + Lini,top pti ) η1Vini,top

(3)

Ltop ) Lmt; Lmb ) Lbot

Fi ) pti + pbi

4. Mass Balance in the Condenser and Reboiler. These are treated as splitters

∑i pbiKi,reb ) ∑i pbi

}

i ∈ COM

(11)

where reb and con make reference to the reboiler and condenser, respectively. 7. Temperature Increases from the Top to the Bottom of the Column.

Tcon e Ttop e Tmt e Tmb e Tbot e Treb

(12)

8. Feasibility Restrictions. These constraints have two functions. First they represent the pinch constraints, and second they distribute the non key components:

Vini,s Vs

Lini,s e Ki,s Ls

i ∈ COM, s ∈ S

(13)

if the product i is mostly present in the top of the column or

Linj,s g Kj,s Vs Ls

Vinj,s

j ∈ COM, s ∈ S

(14)

if the product j is mostly present in the bottom of the column.


9. Recovery. A recovery factor (f) can be fixed for each component,

Fifi e pti

or

Fifi e pbi

i ∈ COM

(15)

depending on whether the product is obtained as a top or bottom product. 10. Objective Function. The vapor flow rate in the column is minimized. Because we have two column sections, we choose the maximum of those two flows in the column,

Min Max (Vtop, Vbot)

(16)

Defining a new variable R, we can transform the previous equation as follows:

R g Vtop

COM ) {j|components to be separated} COL ) {c|c is a separation task with an associated column} ISi ) {k|stream k is an input into splitter i} OSi ) {l|stream l is an output from splitter i} IMi ) {k|stream k is an input into mixer i}

Min R s.t.

states are the streams of the problem characterized qualitatively by their phase (we assume that they are all liquid) and quantitatively for their flow and composition. Tasks are all of the possible cuts that can be performed on the mixtures together with the mixing and splitting tasks. Each one of the cuts is assigned to a potential distillation column. Splitter and mixers do not appear in the final configuration but are necessary at the modeling level. We define the following new sets:

(17)

R g Vbot Note that the proposed model given by eqs 1-17 only includes mass balance and an energy balance in the feed section. We will only consider the heat balances in the reboiler and condenser when we consider heat integration. It is worth noting that eqs 2-15 represent the aggregation of the equations of a tray-by-tray model. In particular, the mass balance equations (2), (4)-(6), (8), and (9) represent a linear combination of component mass balances in each tray with the assumption of equimolar flow. The enthalpy balances are relaxed because they are removed, except for the feed tray in eq 7. Finally the equilibrium equations are relaxed by two inequalities (13) and (14) which are imposed at the extremes of each section. Thus, if the same thermodynamic model is used, the aggregated model will yield a lower bound in the vapor flows with respect to a rigorous tray-by-tray model with equimolar flows. Furthermore, if the heat of vaporization decreases with relative volatility, the model also predicts a lower bound of the utilities (in this case energy balances are added to the reboiler and condenser). This is due to the relaxation of the equilibrium equations, which in turn will overpredict recoveries of lighter than key component.

STN Model for 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 or equipment that involve discrete decisions, it is necessary to use discrete mathematical programming models. In this case, we will use generalized disjunctive programming36 in which the conditional constraints are represented by disjunctions which have assigned a Boolean variable that represents their existence. As mentioned above, the first step to generate the STN model is to identify the tasks and the states. The

OMi ) {l|stream l is an output from mixer i} Pj ) {t|task t in which compound j is obtained as a pure product} The variables are the following: Fj,l is the molar flow rate of component j in stream l, ξli is the split fraction related to stream l in the splitter I, and Yc is a Boolean variable that takes the value of true if the task is selected or false if not; the equations and constraints that are activated when task c is performed are defined by hc(x) (eqs 2-11) and gc(z) (eqs 12-15) where x and z are the vectors of variables that appear in the equations and constraints, respectively. The final parameters are fj, the recovery fraction of component j, and Fej, the molar flow rate of component j that is processed in the separation system. The disjunctive formulation of the problem is given by:

(P1):

∑

Min

Rc

c∈COL

Rc g Vtop,c Rc g Vbot,c

s.t.

∑ Fj,k ) Fj,l

k∈IMi

Fj,k )

∑

Fj,l

Fj,l )

l∈OSi ξil Fj,k

∑

ξil ) 1

k∈OMi

}

c ∈ COL

(18)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M (19)

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S (20)

fjFej e

[

}

∑ Fj,t

j ∈ COM

(21)

t∈Pj

][

Yc ¬Y hc (x) ) 0 ∨ c c Bx)0 gc (z) e 0

Ω(Yc) ) True

]

c ∈ COL

(22)

Yc ∈ {True, False}

(23)


Equations 19 and 20 represent the mass balances on mixers and splitters, respectively. Equation 21 fixes the minimum recovery of each component. The conditional tasks are represented by eq 22. When a conditional task is selected, the equations of the first term of the disjunction are activated (eqs 2-15). When the conditional task is not selected, all of the variables related to this task are set to zero. Equation 23 represents the set of logic constraints that relate the existence of a given task with the other tasks. Note that the vector z is a subvector of the vector of variables x. Bc is a diagonal matrix whose elements are set to 1 if the variable in the same row must be set to zero when the column is not selected.

(P2):

∑

Min

Rc

c∈COL


s.t.

∑ Fj,k ) Fj,l

}

c ∈ COL

(24)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M

k∈IMi

(25)

∑ Fj,l

Fj,k )

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S

l∈OSi

(26) fjFej e

∑ Fj,t

j ∈ COM

(27)

t∈Pj

SEN Model for Distillation Sequences For the SEN representation it is necessary to specify the equipment that is available and the tasks that they can perform. The tasks are the same as those in the STN representation. In this case we use N - 1 columns (where N is the number of components). The assignment of the tasks to the columns is implicit in the representation. Figure 2 shows the SEN representation for a mixture of four components. Note that the task A/B is repeated in columns 2 and 3 in order to perform the separation AB/CD, A/B, and C/D. In the SEN representation, there is not a one-to-one correspondence between states and streams. Therefore, the particular realization of streams will depend on the tasks that the columns perform. Splitters are single choice (only one of the output streams has a value different from zero). However, because of the relation between the task, states, and streams, the disjunctions must explicitly include which one of the streams is different from zero. Another important difference is that the equations related to the columns (mass and energy balances and equilibrium relations) need not be included in the disjunctions because they are independent of the particular task that a given column performs. To formulate the disjunctive model representation for the SEN, we define the new sets:

∨

[

∀t∈TCc

hc (x) ) 0

c ∈ COL

(28)

g′c (z) e 0

c ∈ COL

(29)

Yt,c Fj,k ) Fj,m j ∈ COM, k ∈ ISi, m ∈ STt Vinj,s,c Linj,s,c j ∈ TOPt, s ∈ S e Kj,s,c Vs,c Ls,c Vinj,s,c Linj,s,c j ∈ BOTt, s ∈ S g Kj,s,c Vs,c Ls,c

]

c ∈ COL (30)

Ω(Yc) ) True

Yt,c ∈ {True, False}

(31)

Fj,k, Fj,m, Vinj,s,c, Linj,s,c, Vs,c, Ls,c, Kj,s,c g 0 Equation 24 is the objective function. Equations 25 and 26 correspond to the mass balances in mixers and splitters, respectively. Equation 27 defines the recovery of each component. Equations 28 and 29 are the mass balances, equilibrium relations, and constraints of each one of the columns. In the disjunction (eq 30), the first term ensures that the splitters are single choice; the output stream and its state are selected depending on the task that the column performs. The last two equations in the disjunction are the feasibility constraints (pinch). Equation 31 represents the logic relations for the existence of different tasks in different columns.

TCc ) {t|t is a task that can be performed by column c}

Intermediate Representation for Distillation Sequences

STt ) {m|m is a state active if task t is performed}

The third case that we consider is an intermediate situation between the STN and SEN representations that was proposed by Sargent.31 The tasks are the same as those in previous cases, but the assignment of tasks to equipment is different. We show this assignment through an example. For a mixture of four components (ABCD), we can specify the more general task to separate A from D. Then, we can obtain, in the most general case, a mixture of ABC and a mixture of BCD. Now we can specify the task of separating A from C (for the mixture ABC) and separating B from D (for the mixture BCD) and so forth. Because we are using conventional columns, the separation of A from D can

TOPt ) {j|product j is fundamentally distillate if task t is performed} BOTt ) {j|product j is a bottom product if task t is performed} It is necessary to introduce a new Boolean variable Yt,c to indicate that task t is performed by column c. The constraints associated with a given column are not included in the disjunctions represented by g′(z). Equations 13 and 14 must remain within the disjunction.


be performed through one of the three tasks A/BCD, AB/ CD, or ABC/D, the separation of ACD through A/BC or AB/C, etc. Figure 3 shows the resulting structure. The model has some of the characteristics of the STN and SEN models. Some of the streams have their state fixed. However, others must be specified according to the tasks performed by the columns. Tasks performed by the splitters must be specified as a function of the task that each column performs. Using the sets and variables previously defined, the model can be written as follows:

logic propositions YA|BCD ∨ YAB|CD ∨ YABC|D YA|BCD w YB|CD ∨ YBC|D YAB|CD w YC|D ∧ YA|B YABC|D w YA|BC ∨ YAB|C YB|CD w YA|BCD ∧ YA|B YBC|D w YA|BCD ∧ YB|C

(P3): Min

∑ Fj,k ) Fj,l

YA|BC w YABC|D ∧ YB|C

∑ Rc c∈COL


s.t.

}

YAB|C w YABC|D ∧ YA|B

c ∈ COL

(32)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M

k∈IMi

(33)

Fj,k )

∑ Fj,l

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S

l∈OSi

∨

Table 1. Logic Propositions and Their Equivalence Constraints for the Different Superstructure Representations

[

∀t∈TCc

(34)

fjFej e

∑ Fj,t

]

j ∈ COM

(35)

t∈Pj

Yt,c Fj,k ) Fj,m j ∈ COM, k ∈ ISi, m ∈ STt Vinj,s,c Linj,s,c j ∈ TOPt, s ∈ S e Kj,s,c Vs,c Ls,c ∨ Vinj,s,c Linj,s,c j ∈ BOT , s ∈ S g Kj,s,c t Vs,c Ls,c hc (x) ) 0 g′c (z) e 0

[

¬( ∨ Yt,c) t∈TCco

Fj,k ) 0 k ∈ ISi Fj,m ) 0 m ∈ STt Bcx ) 0

Ω(Yc) ) True

}

j∈C

]

c ∈ COL (36)

Yt,c ∈ {True, False}

YC|D w YAB|CD ∨ YB|CD YB|C w YBC|D ∨ YA|BC YA|B w YAB|CD ∨ YAB|C

algebraic constraints STN yA|BCD + yAB|CD + yABC|D ) 1 yB|CD + yBC|D - yA|BCD g 0 yAB|CD - yC|D e 0 yAB|CD - yA|B e 0 yA|BC + yAB|C - yABC|D g 0 yB|CD - yA|BCD e 0 yB|CD - yA|B e 0 yBC|D - yA|BCD e 0 yBC|D - yB|C e 0 yA|BC - yABC|D e 0 yA|BC - yB|C e 0 yAB|C - yABC|D e 0 yAB|C - yA|B e 0 yB|CD + yAB|CD - yC|D g 0 yBC|D + yA|BC - yB|C g 0 yAB|CD + yB|CD - yA|B g 0

SENa 1 1 1 1 1 1 YA|BCD ∨ YAB|CD ∨ YABC|D YA|BCD + yAB|CD + yABC|D )1 2 2 2 2 2 2 2 2 YAB|C ∨ YA|BC ∨ YBC|D ∨ YB|CD ∨ yAB|C + yA|BC + yBC|D + yB|CD + 2 2 )1 YA|B yA|B 3 3 3 3 3 3 YA|B ∨ YB|C ∨ YC|D yA|B + yB|C + yC|D )1 1 2 2 2 2 1 yB|CD + yBC|D - yA|BCD g0 YA|BCD w YB|CD ∨ YBC|D 1 2 3 2 1 w YA|B ∧ YC|D yA|B - yAB|CD g0 YAB|CD 2 1 - yAB|CD g0 yC|D 1 2 2 2 2 1 YABC|D w YAB|C ∨ YA|BC yAB|C + yA|BC - yABC|D g0 2 3 3 2 yC|D - yB|CD g 0 YB|CD w YC|D 2 3 3 2 w YB|C yB|C - yBC|D g0 YBC|D 2 3 3 2 YA|BC w YB|C yB|C - yA|BC g0 2 3 3 2 w YA|B yA|B - yAB|C g0 YAB|C Intermediate Representationa 1 1 1 1 1 1 YA|BCD ∨ YAB|CD ∨ YABC|D yA|BCD + yAB|CD + yABC|D )1 2 2 2 2 YA|BC ∨ YAB|C yA|BC + yAB|C )1 3 3 3 3 ∨ YBC|D yB|CD + yBC|D )1 YB|CD 1 3 3 3 3 1 w YB|CD ∨ YBC|D yB|CD + yBC|D - yA|BCD g0 YA|BCD 1 4 6 4 1 yA|B - yAB|CD g 0 YAB|CD w YA|B ∧ YC|D 6 1 yC|D - yAB|CD g0 1 2 2 2 2 1 yAB|C + yA|BC - yABC|D g0 YABC|D w YAB|C ∨ YA|BC 2 5 5 2 w YB|C yB|C - yA|BC g0 YA|BC 2 4 4 2 w YA|B yA|B - yAB|C g0 YAB|C 2 6 6 3 YBC|D w YC|D yC|D - yB|CD g0 3 5 5 3 w YB|C yB|C - yBC|D g0 YBC|D a

column Ytask .

(37)

Fj,k, Fj,m, Vinj,s,c, Linj,s,c, Vs,c, Ls,c, Kj,s,c g 0 Note that the vector z is a subset of the vector of variables x, and that the variables Lin, Vin, L, V, and K are also included in x. Equation 32 is the objective function, (33) and (34) correspond to the mass balances for mixers and splitters, respectively, and eq 35 defines the recovery of each component. In the disjunctions (eq 36), the terms in the first bracket are active if the task Yt,c (task t is selected in column c) is true. If it is not selected, the variables are set to zero. The first equation in the disjunction ensures that the splitters are single choice; the output stream and its state are selected depending on the task that the column performs. The next two equations in the disjunction are the pinch constraints. The last two equations correspond to the mass balances, equilibrium equations, and constraints if column c is selected. Equation 37 represents the logic relations for the existence of different tasks in different columns.

Transformation of the Disjunctive Problems into MINLP Problems Turkay and Grossmann36 developed a logic-based outer-approximation algorithm for solving nonlinear disjunctive problems. Because we are approaching the problem in aggregated form, we will convert the problem into an MINLP problem, which under certain circumstances reduces to an NLP problem. For the general procedure for the reformulation, consider the following disjunction:

[ ][

Y ¬Y h(x) ) 0 ∨ Bx ) 0 g(x) e 0

]

(38)

The first step to transform the disjunctive problem into an MINLP problem is to assign a binary variable to each one of the Boolean variables. If the boolean variable is set to true, the binary is set to 1; if the Boolean variable


is false, the binary variable is zero. In this way the disjunction in 38 can be written as an MINLP problem as follows:

h(x) + s1 - s2 ) 0

(39)

s1 + s2 e U1(1 - y)

(40)

g(x) e U2(1 - y)

(41)

xLi y e x e xU i y

for bii ) 1

s1, s2 g 0

(42) (43)

where s1 and s2 are positive slack variables. U1 and U2 are valid upper bounds for h(x) and g(x), respectively, xL and xU are lower and upper bounds for the variable x, and bii is the diagonal element in the matrix b. Note that if y ) 1, then s1 ) s2 ) 0, while if y ) 0, then the slack variables are allowed to vary in order to relax the model constraints, and the variable xi is set to zero. Note also that care should be taken to identify the tightest upper bounds U1 and U2 in order to avoid a poor relaxation. The same argument holds true for xL and xU. The logical relations Ω(Yi) ) True written in form of propositional expressions can be transformed into a set of independent linear constraints.37 Table 1 shows the logical expressions and their corresponding integer linear equations for the three superstructures considered in this work. MINLP Model for the STN Representation. Using eqs 39-43, the STN (problem P1) can be transformed into the MINLP problem:

(P4):

∑ Fj,k ) Fj,l k∈IM i

∑

Fj,l

l∈OSi ξliFj,k

ξli ) 1 ∑ k∈OM i

∑

Min

Rc

c∈COL


s.t.

∑

Fj,k ) Fj,l

}

c ∈ COL

}

}

(51)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M

k∈IMi

(52)

Fj,k )

∑

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S

Fj,l

∑ Rc c∈COL


s.t.

Fj,l )

(P5):

l∈OSi

Min

Fj,k )

Equations 45 and 46 correspond to the mass balances for the splitters and mixers, while eq 47 fixes the recovery fraction for each of the components. Equations 48 and 49 are the transformations of the disjunctions to the MINLP formulation. s1c and s2c are slack variables, and U1c and U2c are valid upper bounds for h(x) and g(z), respectively. Equation 50 corresponds to the logic relations between tasks expressed in algebraic form. In all of the cases studied, the STN representation can be simplified and the binary variables removed, because the splitters become single choice in the optimization process. In other words, the optimal solution of the problem written as an NLP is a solution with N - 1 columns and the flows associated to the other columns are zero. Thus, in the model P4, eqs 49 and 50 can be removed and the slack variables and the upper bounds can be set to zero. MINLP Model for the SEN Representation. When the same procedure is followed, the disjunctive SEN representation can be written as the MINLP:

(53) fjFej e

∑ Fj,t

j ∈ COM

(54)

t∈Pj

c ∈ COL

(44)

hc(x) ) 0

c ∈ COL

(55)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M

g′c(z) ) 0

c ∈ COL

(56)

(45)

Fj,k - Fj,m e U (1 - yt,c) j ∈ COM, k ∈ ISi, m ∈ STt, t ∈ TCc, c ∈ COL (57) Vinj,s,c

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S (46)

fjFej e

∑ Fj,t

t∈Pj

s1c

s2c

hc (x) + - ) 0 s1c + s2c e U1c (1 - yc) g(z) e U2c (1 - yc)

j ∈ COM

}

(BcxL)yc e Bcx e (BcxU)yc Ay e b y ) {0, 1}; x g 0; x )

[]

(47)

Linj,s,c

j ∈ TOPt e U(1 - yt,c) Vs,c Ls,c Linj,s,c Vinj,s,c + Kj,s,c e U(1 - yt,co) j ∈ BOT Vs,c Ls,c - Kj,s,c

Ay e b

c ∈ COL

(48)

(49) (50)

z , Fj,k, Fk,l, ξli, s1c , s2c g 0 .

Like in the previous cases eq 44 is the objective function.

(58) (59)

z , . Fj,k, Fj,m, Vinj,s,c, Vs,c, Kj,s,c, Linj,s,c, Ls,c g 0

y ) {0, 1}; x g 0, x ) c ∈ COL

[]

}

s∈S t ∈ TCc c ∈ COL

Note that in this case the variables are not set to zero if the task is not selected because of the SEN representation. Note also that the binary variables cannot be removed because of eqs 57-59. MINLP Model for the Intermediate Representation. The intermediate case, model P3, can be also transformed into a MINLP formulation, although the transformation is not as straightforward as in the previous cases. When the binary variable yt,c that takes the value


1 if the task t is performed by the column c is defined, the MINLP formulation can be written as follows:

(P6): Min

∑

Rc

c∈COL


s.t.

∑

Fj,k ) Fj,l

}

c ∈ COL

(60)

j ∈ COM, l ∈ OMi, i ) 1, 2, ..., M

k∈IMi

(61)

Fj,k )

∑

j ∈ COM, k ∈ ISi, i ) 1, 2, ..., S

Fj,l

l∈OSi

(62) fjFej e

∑ Fj,t

j ∈ COM

(63)

t∈Pj

Fj,k - Fj,m e U (1 - yt,c) j ∈ COM, k ∈ ISi, m ∈ STt, t ∈ TCc, c ∈ COL (64) Vinj,s,c

- Kj,s,c

Linj,s,c

j ∈ TOPt

e U(1 - yt,c)

Vs,c Ls,c Linj,s,c Vinj,s,c + Kj,s,c e U(1 - yt,c) j ∈ BOT Vs,c Ls,c

hc(x) +

s1c

s1c

e U(1 -

+

s2c

-

s2c

)0

∑ yt,c)

t∈TCco

g′(z) e U(1 -

∑ yt,c)

t∈TCco

∑ yt,c e 1

}

}

s∈S t ∈ TCc c ∈ COL

c ∈ COL

c ∈ COL

(65)

(66)

(67)

t∈TCc

Ay e b BcxL

∑

e Bcxc e BcxU

t∈TCc

y ) {0, 1};

∑ yt,c

(68) c ∈ COL

(69)

t∈TCc

[]

z , . Fj,k, Fk,l, Vinj,s,c, Vs,c, Kj,s,c, Linj,s,c, Ls,c g 0 x g 0,

x)

Equations 60-63 are identical with those in problem P3. Equations 64 and 65 are the single choice splitters and the pinch restrictions, respectively. If one of the tasks that the column c can perform is selected, then the slack variables are zero and the equations of the column c are active (eq 66). Note that column c can perform at most one task. This is mathematically expressed by eq 67. Equation 68 represents the logical constraints written in algebraic form. If no task is selected in a given column, the variables of this column are set to zero (eq 69). Distillation Sequences with Heat Integration We will consider only the heat integration between different separation tasks. Thus, multi-effect columns are excluded. For the sake of simplicity, we consider only one cold and one hot utility, and the objective function considers only the cost of the utilities. The approach

used in this work for heat integration was originally developed by Floudas and Paules.18 Raman and Grossmann38 developed a special version of their model in which all of the constraints and the objective function are linear. Given the parameters CHU and CCU for the costs of the heating and cooling utilities, respectively, and introducing also the variables QCc and QHc, which are the cooling and heating duties supplied to satisfy the load Qc of each column, the objective function can be written as follows:

Min (

∑

CHUQHc + CCUQCc) + f(x)

(70)

c∈COL

where f(x) is a function that can include the costs associated with the columns, heat exchangers, or other items. To calculate the heat duties of a column, energy balances are applied in the reboiler and in the condenser:

Qcon ) c

∆HviVin ∑ i∈C

Qreb c )

∆HviVin ∑ i∈C

i,con,c

i,reb,c

}

c ∈ COL

(71)

where ∆Hvi is the heat of vaporization of the component and Qcon are the heat load of the reboiler i, and Qreb c c and condenser. and Tcon are the reboiler and condenser temIf Treb c c peratures of column c, EMAT is the minimum exchanger approach temperature, and TS and TC are the temperatures of steam and cooling utility, the two following constraints apply:

Treb c e TS - EMAT g TC + EMAT Tcon c

}

c ∈ COL

(72)

To consider the potential exchanges of heat, we introduce the variable QEXkj that is the amount of heat exchanged between the condenser of column k and the reboiler of column j. We also define the binary variable wkj which is equal to 1 if the condenser of column k supplies heat to the reboiler of column j and zero otherwise. Thus, the following conditional constraints apply:

}

QEXk,j e Ukwk,j reb Tcon + EMAT - Uk,j(1 - wk,j) k g Tj k, j ∈ COL, k * j (73) If wk,j ) 1, the temperature of the condenser in column k must be larger than the temperature in the reboiler of column j. If wk,j ) 0, the heat exchanged between the condenser of column k and the reboiler of column j is forced to be zero. Heat balances must hold for cooling and heating utilities:

QEXc,j + QCc ) Qreb ∑ c j∈COL QEXj,c + QHc ) Qcon ∑ c j∈COL

}

c ∈ COL

(74)

2340 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 Table 2. Comparison between Rigorous and Aggregated Models (Feed: Propane, 125 k‚mol/h; n-Butane, 125 k‚mol/h; n-Pentane, 125 k‚mol/h; n-Hexane, 125 k‚mol/h)a temp (°C) tray 1C 2 3 4 5 6 7 8 9 10 11 12 13 14R 1C 2 3 4 5 6 7 8 9 10 11 12 13 14R a

R -38.3 -21.3 -16.7 -15.4 -14.4 -13.1 -11.9 -11.9 -11.8 -11.6 -11.1 -9.4 -4.3 9.7 -0.5 19.7 20.5 20.5 20.5 20.5 20.5 21.2 21.5 21.8 22.2 23.4 26.6 35.1

A -42.1

-12.0 -12.0

liquid flow (k‚mol/h) R

A

20.1 20.1

R

A

product or feed (k‚mol/h)

heat duties (kW)

R

A

R

A

119.0

119.0

772.0

656.5

125.5 119.5

500

500

119.5

381.0

381.0

1401.5

769.0

121.0

121.0

895.5

768.0

123.5 123.0

380.5

380.5

123.0

259.5

259.5

1014.0

926.0

a. Column 1 in Example 3 11.5 8.5 7.5 7.0 6.0 5.5 505.5 505.5 505.5 505.5 505.5 505.5 506.0

5.8

500.0

500.0

18.4 -0.4

vapor flow (k‚mol/h)

130.5 127.0 126.5 125.5 125.0 124.0 124.0 124.0 124.0 124.0 124.0 124.5 125.0

125.5

b. Column 2 in Example 3 2.5 2.0 2.0 2.0 2.0 2.0 386.0 387.5 388.0 388.0 387.5 387.0 384.5

37.7

2.35

2.35 382.5

382.5

124.0 123.5 123.5 123.0 123.0 123.0 127.0 128.0 128.5 128.5 128.5 127.5 125.5

123.5

R ) rigorous. A ) aggregated.

Finally, the logic constraints that related the existence of columns with the selection of matches can be added. The heat integration can be implemented independently of the representation used. Note, however, that the STN representation becomes an MINLP that includes the binary variables associated to the columns and those related to the heat integration. Examples In all of the examples GAMS39 was used as the equation modeling system. GAMS/DICOPT++ was the MINLP solver. This solver uses the Outer approximation with equality relaxation and augmented penalty algorithm (OA/ER/AP) by Viswanathan and Grossmann.40 The NLP solver was CONOPT2 and the MILP solver OSL. The computer used was a Pentium Pro 200 MHz. Example 1. In the first example we compare the results obtained with the model for one column and a rigorous simulation. Two cases are presented. In both we minimize the vapor flow rate in a column (first two columns in the best sequence of example 2). The data and the results of the proposed model, and the comparison with rigorous simulation with the simulator PROII,41 are shown in Table 2a,b. As can be seen, the results obtained from the simulation and with the aggregated model are reasonably close. As pointed out before, the model predicts bounding properties to the flows and heat loads. For example, the vapor flow rate in the first case is 125.5 kmol/h in the aggregated model vs the 130.5 kmol/h of the rigorous simulation. For the second case, it is 123.5 kmol/h vs 124 kmol/h. The heat loads are also a lower bound to the total

utilities: for the first one 769 kW vs 1401.5 kW; for the second 926 kW vs 1014 kW. It should be noted that some differences between the aggregated and the rigorous problems can be due to the thermodynamic properties used in both models. We used the correlations given by Reid et al.,42 and the simulation was carried out with the default values of the simulation software (ProII). (In both cases ideal behavior was assumed.) The differences can also be due to numerical limitations in the aggregated model. To avoid divisions by zero and large derivative values, the lower bound of some of the variables, in particular the flows, was fixed to a value different from zero. If this constraint becomes active, for example, in the flow rate of the heaviest compound in the top product, the effect is that the total flow must increase to satisfy the feasibility constraints. Note that the most important difference is the temperature in the reboiler of the first column. However, the rigorous simulation showed that this temperature is very sensitive to small perturbations of flows or heat loads, changing from 9 to almost 20 °C. Additionally, the relaxation of equilibrium equations produces an increase of light products in the bottom products, and because of the heat of vaporization decreases with volatility, the heat load in the aggregated model also decreases. The effect is more important in the first column than in the second, in which the most volatile compound has been almost removed. Example 2. In this example we present the separation of a mixture of four components without heat integration. The four components are propane, n-butane, npentane, and n-hexane. Ideal behavior is assumed in the liquid and vapor phases. Data for thermodynamic properties are from Reid et al.42 Table 3 shows the

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2341 Table 3. Objective Function for the Five Possible Separations of Example 2a separation sequence

Table 6. Objective Function for the Five Possible Separations of Example 2

objective function (k‚mol/h)

separation sequence

objective function (k‚mol/h)

372.35 492.5 490.5 611.5 730.0

A/BCDE, B/CDE, C/DE, D/E A/BCDE, B/CDE, CD/E, C/D A/BCDE, BC/DE, B/C, D/E A/BCDE, BCD/E, B/CD, C/D A/BCDE, BCD/E, BC/B, B/C AB/CDE, A/B, C/DE, D/E AB/CDE, A/B, CD/E, C/D ABC/DE, A/BC, B/C, D/E ABC/DE, AB/C, A/B, D/E ABCD/E, A/BCD, B/CD, C/D ABCD/E, A/BCD, B/CD, C/D ABCD/E, AB/CD, A/B, C/D ABCD/E, ABC/D, A/BC, C/D ABCD/E, ABC/D, AB/C, A/B

421.0 519.0 519.0 672.0 922.5 724.0 630.0 615.5 710.5 866.0 869.0 882.0 981.5 1064.5

A/BCD, B/CD, C/D A/BCD, BC/D, B/C AB/CD, A/D, C/D ABC/D, A/BC, B/C ABC/D, AB/C, A/B

a A ) propane; B ) n-butane; C ) n-pentane; D ) n-hexane. Feed: 500 kmol/units of time. Composition (molar fraction): A, 0.25; B, 0.25; C, 0.25; D, 0.25.

Table 4. Results and Statistics of Example 2: Separation of Four Components without Heat Integration representation

model

no. of equations

no. of variables

STN intermediate SEN

NLP MINLP MINLP

1700 1890 632

1699 1431 (6 binary) 512 (11 binary)

representation

best OF (k‚mol/h)

iterations

CPU time (s)


492.5 372.35 372.35

3 4

310.8 381.2 130.7

Table 5. Results and Statistics of Example 3: Separation of Four Components with Heat Integration representation

model

no. of equations

no. of variables


MINLP MINLP MINLP

2286 2097 662

1909 (45 binary) 1503 (21 binary) 530 (14 binary)

representation

best OF ($/year)

iterations

CPU time (s)


251 240 87 280 87 280

4 4 4

510.6 441.9 136.8

objective function (minimum vapor flows) for each one of the five possible configurations. Table 4 shows the statistics and the results obtained with each one of the representations. Note that the SEN representation yields the smallest model (the third part about the size of the other two). The reason is that the SEN representation uses the minimum number of columns. Although the variables associated with the nonactive columns are set to zero in the solution of the STN and intermediate representation, they cannot be removed from the model. As was previously mentioned, the STN representation gives an NLP model, while the other two representations yield an MINLP. Unfortunately, because of the nonconvexities in eq 46, the model is trapped in a suboptimal solution. The results obtained with the STN representation are very dependent on the initial point, and different solutions can be obtained if the initial values are changed. Table 4 shows the results when all of the representations are started from the same initial point. Both, the intermediate and the SEN representations reach the global optimum in three and four major iterations, respectively. Note also that the time needed to reach the optimum is considerably longer with the intermediate representation than with the SEN (around 3 times more), although the SEN carried out one more iteration. Example 3. In this case we solve the same separation problem as that in example 2 but with heat integration. Table 5 shows the statistics of the models and the results obtained.

a A ) propane; B ) methylacetylene; C ) n-butane; D ) n-pentane; E ) n-hexane. Feed: 500 kmol/units of time. Composition (molar fraction): A, 0.2; B, 0.2; C, 0.2; D, 0.2; E, 0.2.

Table 7. Results and Statistics of Example 4: Separation of 5 Components without Heat Integration representation

model

no. of equations

no. of variables

intermediate SEN

MINLP MINLP

4595 1156

2876 (10 binary) 956 (26 binary)

representation

best OF (k‚mol/h)

iterations

CPU time (s)

intermediate SEN

421.0 421.0

3 4

2477 445

The separation sequence is the same as that in example 2. However, the flows in the columns change to allow the heat exchange between columns. Figure 6 shows the optimal configuration together with the temperature enthalpy diagram. In the optimal solution, the condenser of the second column is heat integrated with the reboiler of the first one, and the condenser of the third is heat integrated with the reboiler of the second one. The heat that is integrated is a very important fraction of the total heat load of the columns. Around 75% of the cooling utility and 66% of the heating utility are reduced. Again the smallest model in number of equations and variables is the SEN representation (662 equations and 530 variables). The other two representations have about 3 times more equations and variables. In these problems the bottleneck is the time required for solving the NLPs. However, if the number of binary variables increases, the time consumed by the MILPs also increases. For example, in the SEN representation, the MILPs required about 20% of the total time, while in the intermediate representation with more binary variables, the MILPs require over 35% of the total time. The bests results are obtained with the SEN; the optimal solution is obtained in four iterations and 136 s versus the more than 440 s of the intermediate representation with also four iterations. Example 4. In this case a mixture of five components was considered. The components are the same as those in previous examples with the addition of methylacetylene. Heat integration is not considered. Table 6 shows the optimal solution for each one of the 14 possible separation sequences. Table 7 shows the results obtained and the model statistics. Because of the poor results obtained with the STN representation, it was not considered here. Both the intermediate and the SEN representation reach the optimal solution (direct sequence) in three and


Figure 6. Optimal configuration and a temperature-heat load diagram for example 6.

Figure 7. Optimal configuration and a temperature-head load diagram for example 5. Table 8. Results and Statistics of Example 5: Separation of Five Components with Heat Integration representation

model

no. of equations

no. of variables

intermediate SEN

MINLP MINLP

5285 1216

3076 (55 binary) 988 (32 binary)

representation

best OF ($/year)

iterations

CPU time (s)

intermediate SEN

637 600 301 200

3 5

4675 657

four iterations, respectively. However, while the SEN model reaches the solution in 445 s, the intermediate representation takes more than 2500 s. The reasons are the same as those in previous cases. The SEN representation has a smaller number of variables and equations than the other representations. Example 5. Here we present the same mixture as that in example 4 but with heat integration. Table 8 shows the results and the statistics. The comments about

model performance and statistics made in previous examples also apply here. The most important point in this example is that the best heat-integrated sequence and the best sequence without heat integration are different. The best sequence obtained with heat integration was A/BCDE, BCD/E, BC/D, and B/C where A is the most volatile product and E the heaviest. The optimal objective function for 8000 h/year of operation is $301 200/year. The best objective in the case when we use the direct sequence (optimal for no heat integration) is $461 200/ year, approximately 35% more expensive. However, the objective function only takes into account the costs of the utilities, and no other costs are associated with the conditions of operation or the investment cost. The optimal heat-integrated sequence has a total vapor flow rate of 1403 kmol/h versus the 1172.5 kmol/h of the direct separation sequence with heat integration. The increase of vapor flow rates in the heat-integrated columns allows one to reduce the utilities by increasing


the heat exchange between columns. The pressures of operation of the columns in the best heat integration sequence are 1, 2.05, 2.7, and 2.9 atm, respectively. The pressures in the direct separation are 1, 1, 1.3, and 1.3 atm, respectively. Therefore, there is a tradeoff between the cost of utilities and the cost of operation associated with the flows and the pressure. Figure 7 shows the optimal configuration predicted by the model. Conclusions This paper has proposed an aggregated model for the synthesis of heat-integrated distillation columns. The model assumes that the liquid and vapor molar flows can be considered constant in each one of the sections of the column and that the pinch-point candidates are placed in the end points of the sections of the column. Under these assumptions, the aggregated model yields lower bounds of vapor flows and utilities that are fairly close to those obtained in a rigorous simulation. It has been shown that the proposed model can be used in three different superstructure representations: STN, SEN, and an intermediate representation. The three representations were modeled as a generalized disjunctive program and transformed into MINLP problems. In all of the cases the STN representation had poor performance and was easily trapped in local solutions. The reasons for this behavior are that the STN model involves a larger number of equations and variables than the other representations and that bilinear terms are associated with the splitters while they do not appear in the other representations. The SEN and intermediate representations always reached the correct optimal solution, but the best results in terms of the size of the program and computer times were obtained with the SEN representation. This representation has shown very good performance in reaching the global optimum, although there is no guarantee given the nonconvexities that are present. The main usefulness of these aggregated models is that they can be used in a preliminary process design or as a part of a formal search strategy such as that presented by Daichendt and Grossmann.28 Acknowledgment The authors acknowledge financial support from the Fulbright (MEC-Fulbright 97). Partial support from the Computer Aided Process Design Consortium at Carnegie Mellon is also acknowledged. Literature Cited (1) Westerberg, A. W.; Wahnschafft, O. In Synthesis of Distillation Based Separation Processes; Anderson, J. L., Ed.; Advances in Chemical Engineering; Academic Press: New York, 1996. (2) Doherty, M. F.; Cardarola, G. A. Design and Synthesis of Homogeneous Azeotropic Distillations. 3. The Sequencing of Columns for Azeotropic Distillation. Ind. Eng. Chem. Fundam. 1985, 24, 474. (3) Glasser, D.; Crowe, C.; Hildelbrandt, D. Geometric Approach to Steady-State Flow Reactors: The Attainable Region and Optimization in Concentration Space. Ind. Eng. Chem. Res. 1987, 26 (9), 1803. (4) Hildebrandt, D.; Glasser, D.; Crowe, C. The Geometry of the Attainable Region Generated by Reaction and Mixing: With and Without Constraints. Ind. Eng. Chem. Res. 1990, 29 (1), 49. (5) 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.

(6) Stephanopoulos, G.; Westerberg, A. W. Studies in Process Synthesis II. Evolutionary Synthesis of Optimal Process Flowsheets. Chem. Eng. Sci. 1996, 31, 195. (7) Douglas, J. M. A Hierarchical Decision Procedure for Process Synthesis. AIChE J. 1985, 31, 353. (8) Douglas, J. M. Conceptual Design of Chemical Process; McGraw Hill: New York, 1988. (9) Douglas, J. M. Synthesis of Multistep Reaction Processes. In Foundations of Computer Aided Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; Cache-Elsevier: Amsterdam, The Netherlands, 1990. (10) Grossmann, I. E. MINLP Optimization Strategies and Algorithms for Process Synthesis. In Foundations of Computer Aided Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; Cache-Elsevier: Amsterdam, The Netherlands, 1990. (11) Grossmann, I. E. Mixed Integer Nonlinear Programming Techniques for the Synthesis of Engineering Systems. Res. Eng. Des. 1990. (12) Grossmann, I. E. Mixed-Integer Programming Approach for the Synthesis of Integrated Process Flowsheets. Comput. Chem. Eng. 1985, 9, 463. (13) Duran, M. A.; Grossmann, I. E. Simultaneous Optimization and Heat Integration of Chemical Processes. AIChE J. 1986, 32, 123. (14) Lang, Y. D.; Biegler, L. T.; Grossmann, I. E. Simultaneous Optimization and Heat Integration with Process Simulators. Comput. Chem. Eng. 1988, 12, 311. (15) Yee, T. F.; Grossmann, I. E.; Kravanja, Z. Simultaneous Optimization Models for Heat Integration: I. Energy and Area Targeting. Comput. Chem. Eng. 1990, 14, 1151. (16) Yee, T. F.; Grossmann, I. E. Simultaneous Optimization Models for Heat Integration: II. Heat Exchanger Network Synthesis. Comput. Chem. Eng. 1990, 14, 1165. (17) Yee, T. F.; Grossmann, I. E.; Kravanja, Z. Simultaneous Optimization Models for Heat Integration: III. Optimization of Process Flowsheets and Heat Exchanger Networks. Comput. Chem. Eng. 1990, 14, 1185. (18) Floudas, C. A.; Paules, G. E. A Mixed-Integer Nonlinear Programming Formulation for the Synthesis of Heat Integrated Distillation Sequences. Comput. Chem. Eng. 1988, 12, 531. (19) El-Halwagi, M.; Manousiouthakis, V. Automatic Synthesis of Mass Exchange Networks with Single Component Targets. Chem. Eng. Sci. 1990, 45 (9), 2813. (20) El-Halwagi, M.; Manousiouthakis, V. Design and Analysis of Mass Exchange networks with Multicomponent Targets, AIChE Meeting, San Francisco, 1989; Paper 137f. (21) El-Halwagi, M.; Manousiouthakis, V. Synthesis of Mass Exchange Networks. AIChE J. 1989, 35 (8). (22) Andrecovich, M. J.; Westerberg, A. W. A Simple Synthesis Method Based on Utility Bounding for Heat Integrated Distillation Sequences. AIChE J. 1985, 31, 363. (23) Andrecovich, M. J.; Westerberg, A. W. An MILP Formulation for Heat Integrated Distillation Sequence Synthesis. AIChE J. 1985, 31, 1461. (24) Sargent, R. W. H.; Gaminibandara, K. Introduction: Approaches to Chemical Process Synthesis. In Optimization in Action; Dixon, L. C. W., Ed.; Academic Press: London, 1976. (25) Papoulias, S. A.; Grossmann, I. E. A Structural Optimization Approach in Process Synthesis. Part I. Utility Systems. Comput. Chem. Eng. 1983, 7, 695. (26) Bruno, J. C.; Ferna´ndez, F.; Castells, F.; Grossmann, I. E. A Rigorous MINLP Model for the Optimal Synthesis and Operation of Utility Plants. Trans. Inst. Chem. Eng. 1998, 76A, 246. (27) Rippin, D. W. T. Introduction: Approaches to Chemical Process Synthesis. In Foundations of Computer Aided Process Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; Cache-Elsevier: Amsterdam, The Netherlands, 1990. (28) Daichendt, M. M.; Grossmann, I. E. Integration of Hierarchical Decomposition and Mathematical Programming for the Synthesis of Process Flowsheets. Comput. Chem. Eng. 1997, 22, 147. (29) Grossmann, I. E. In Mixed-Integer Optimization Techniques for Algorithmic Process Synthesis; Anderson, J. L., Ed.; In Advances in Chemical Engineering 23; Academic Press: New York, 1996; pp 171-245.

2344 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 (30) Yeomans, H. Grossmann, I. E. A Systematic Modeling Framework of Superstructure Optimization in Process Synthesis. Comput. Chem. Eng. 1999, in press. (31) Sargent, R. W. H. A Functional Approach to Process Synthesis and its Applications to Distillation Systems. Comput. Chem. Eng. 1998, 22, 31. (32) Smith, E. M. On the Optimal Design of Continuous Process. Ph.D. Dissertation under the supervision of C. Pantelides, Imperial College of Science, Technology and Medicine, London, U.K., 1996. (33) Agrawal, R. Synthesis of Distillation Column Configurations for a Multicomponent Separation. Ind. Eng. Chem. Res. 1996, 35, 1059. (34) Bagajewicz, M. J.; Manousiouthakis, V. Mass/Heat exchange Network representation of Distillation Networks. AIChE J. 1992, 38, 1769. (35) Papalexandri, K. P.; Pistikopoulos, E. N. Generalized Modular representation Framework for Process Synthesis. AIChE J. 1996, 42, 1010. (36) Turkay, M.; Grossmann, I. E. Logic-Based MINLP algorithms for the Optimal Synthesis of Process Networks. Comput. Chem. Eng. 1996, 29, 959.

(37) Raman, R.; Grossmann, I. E. Relation Between MILP Modeling and Logical Inference for Chemical Process Synthesis. Comput. Chem. Eng. 1993, 15, 73. (38) Raman, R.; Grossmann, I. E. Symbolic Integration of Logic in Mixed Integer linear Programming techniques for Process Synthesis. Comput. Chem. Eng. 1993, 17, 909. (39) Brooke, A.; Kendrik, D.; Meeraus, A. GAMS: A User’s Guide; Scientific Press: Palo Alto, CA, 1988. (40) Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990, 14, 769. (41) ProII keyword Input Manual, Simulation Science, 1996. (42) Reid, C. R.; Prausnitz, J. M.; Sherwood, T. The Properties of Gases and Liquids; McGraw Hill: New York, 1977.

Received for review December 30, 1998 Revised manuscript received March 22, 1999 Accepted March 25, 1999 IE980803J

Aggregated Models for Integrated Distillation Systems - Industrial

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