Pareto front of ideal Petlyuk sequences using a multiobjective genetic algorithm with constraints

Computers and Chemical Engineering 33 (2009) 454–464

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Pareto front of ideal Petlyuk sequences using a multiobjective genetic algorithm with constraints Claudia Gutiérrez-Antonio a , Abel Briones-Ramírez b,c,∗ a b c

CIATEQ, A.C., Av. del Retablo 150, Col. Fovissste, 76150 Querétaro, Querétaro, Mexico Instituto Tecnológico de Aguascalientes, Departamento de Ingeniería Química, Av. Adolfo López Mateos #1801 Ote. Fracc. Bonagens, 20256 Aguascalientes, Aguascalientes, Mexico Innovación Integral de Sistemas S.A. de C.V., Calle Número 2 #125 Interior 13, Parque Industrial Jurica, 76120 Querétaro, Querétaro, Mexico

a r t i c l e

i n f o

Article history: Received 12 December 2007 Received in revised form 3 November 2008 Accepted 4 November 2008 Available online 21 November 2008 Keywords: Multiobjective optimization Petlyuk sequence Distillation Genetic algorithms

a b s t r a c t Petlyuk sequences are a very promissory option to reduce energy consumption and capital costs in distillation. The optimal design of Petlyuk sequence implies determining 8 integers and 3 continuous variables with the minor heat duty and number of stages possible, but reaching the specified purities. Note that the heat duty and the number of stages are variable in competition, since we cannot decrease indefinitely one without increasing the other. In other words, we have a multiobjective problem with constraints. In contrast with other numerical strategies proposed, we consider the search of optimal designs set, from minimum reflux ratio to minimum number of stages and all designs between them. This set of optimal designs can be achieved by means of Pareto front, which represents a set of optimal solutions not dominated for a multiobjective problem. In this work, we implemented a multiobjective genetic algorithm with constraints to obtain the Pareto front of Petlyuk sequences. This algorithm is coupled to Aspen Plus simulator, so the complete MESH equations and rigorous phase equilibrium calculations are used. Results show clear tendencies in the design of Petlyuk sequence, and can be used to develop a short design method. In addition, the tool developed can be used to optimize not only the distillation columns but also complete chemical and petrochemical plants. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Petlyuk sequence is a thermally coupled distillation system, and it consists of prefractionator and main column linked between them by liquid and vapor streams (Fig. 1). The liquid and vapor streams replace condensers and reboilers in the Petlyuk sequence, decreasing its energy consumption and capital costs. The energy consumption of Petlyuk sequence is until 30% less than the energy required by conventional sequences (Fonyó, Rév, Emtir, Szitkai, & Mizsey, 2001); this is why Petlyuk sequence has been widely studied in the past decades in the areas of design, optimization and control. Particularly, the optimization problem of Petlyuk sequence is very interesting and complex. Fig. 1 shows all the variables that have to be determined, both continuous and discrete, for a ternary separation. The discrete variables are the total number of stages, location of feed stage, side stream stage, and interconnection flows

∗ Corresponding author at: Innovación Integral de Sistemas S.A. de C.V., Calle Número 2 #125 Interior 13, Parque Industrial Jurica, 76120 Querétaro, Querétaro, Mexico. Tel.: +52 442 220 3311; fax: +52 442 220 3312. E-mail addresses: [email protected] (C. Gutiérrez-Antonio), [email protected] (A. Briones-Ramírez). 0098-1354/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2008.11.004

stages as in prefractionator as in the main column. On the other hand, interconnections flows, reflux ratio, and heat duty are continuous variables. In this way, the optimization problem of Petlyuk sequence has an interesting combination of 8 discrete variables and 3 continuous variables, subject to achieve the required purities or recoveries in each product stream. The first approach to obtain an optimal design of Petlyuk sequence was developed by Fidkowski and Królikowski (1986, 1987, 1990). They developed an analytical solution for thermally coupled systems with ternary mixtures, including Petlyuk sequence, in order to calculate the optimal internal vapor flow using Underwood equation. This analytical solution was very important; however, the use of Underwood equation restricts the application of their procedure to ideal mixtures with saturated liquid as thermal condition. In order to present a more formal optimization environment, Yeomans and Grossmann (2000) presented a procedure based on generalized disjunctive programming models to design optimal complex distillation columns. The procedure employs rigorous design equations, and disjunctions to model the trays; it was tested with ideal and azeotropic mixtures, and the objective function could be selected: profit, total cost, or total energy. However, the number of stages in each section was bounded apparently arbitrarily in order to obtain the solution of the problem, reducing the searching space at the same time.

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Nomenclature R Q Ni xk yk Nj NS Fj XF,i Recj,i

reflux ratio in the main column of the Petlyuk sequence heat duty of the sequence (BTU/h) total number of stages in the column distillation i required recovery or purity of component k recovery or liquid-phase molar composition of component k number of stage of the j interconnection stream number of stage of the side stream interconnection flow stream j liquid phase composition of component i at feed stream recoveries of product i at stream j

Subscripts D distillate stream S side stream B bottom stream A light component B intermediate component C heavy component min minimum FV1 vapor interconnection flow at superior interconnection FL1 liquid interconnection flow at superior interconnection FV2 vapor interconnection flow at inferior interconnection FL2 liquid interconnection flow at inferior interconnection main main column prefr prefractionator F feed stream

Later, Caballero and Grossmann (2001) developed a programming model for the optimal synthesis of thermally coupled distillation columns, in terms of investment and operational costs. They used a state task network to design the sequence, simplifying a previous procedure (Yeomans & Grossmann, 2000), and applying it

Fig. 1. Variables involved in the structure of Petlyuk sequence.

455

to multicomponent hydrocarbon mixtures. In spite of the improvement in the optimization procedure, it was implemented with Underwood equation’s modification of Carlberg and Westerberg (1989); in other words, using a reduced model. On the other hand, Kim (2001) presented a structural design procedure for thermally coupled distillation columns to obtain the design with the greater thermodynamic efficiency, which is the optimization criterion. The procedure was tested in quaternary systems with typical combinations of relative volatilities and three feed compositions. Kim comments that resulting designs do not require iterative computation in commercial simulators; however, the procedure employs a simplified model based on a modified Underwood equation and relative volatilities. A year later, Shah and Kokossis (2002) proposed a new synthesis framework for the optimization of complex distillation systems. They optimized the operating conditions by combining discrete instances of simple tasks with hybrid transformations. The problem was formulated as a simple MILP problem, and the procedure was tested with non-azeotropic systems. Since the objective of the work was determined the appropriate separation schemes before detailed simulations, just basic information of the components was given. In 2003, Jiménez, Ramírez, Castro, and Hernández (2003) optimized Petlyuk and alternative sequences of ternary ideal mixtures. They minimized the heat duty of the sequence, keeping fixed its structure. In order to find the minimum heat duty of the sequence, they used recursive rigorous simulations in which they varied the interconnection flows and the reflux ratio of the main column. Ideal mixtures with different easy separation indexes, ESI, were used in this work. So, they considered material and energy balances rigorously, but without a formal optimization procedure. Leboreiro and Acevedo (2004) used a genetic algorithm coupled with the simulator Aspen Plus to optimize distillation sequences, including the Petlyuk sequence, based in the implementation of Carroll (1996). Their code includes alternatives as jump and creep mutation, single point and uniform crossover, and elitism. In order to reduce the search space, they imposed restrictions in the codification to delimit the values of some variables in a desired range; these kinds of restrictions were set using the results of previous simulations, so a priori knowledge of the problem is required. To manage the constraints they used two sets of penalties. A major penalty was applied if the simulation converges with errors, while a successful simulation was just slightly penalized. However, penalty functions have some well-known limitations (Richardson, Palmer, Liepins, & Hilliard, 1989), from which the most remarkable is the difficult to define good penalty factors. These penalty factors are normally generated by trial and error, although their definition may severely affect the results produced by the genetic algorithm (Coello-Coello, 2000; Richardson et al., 1989). In 2005, Kim (2005) proposed a new structural design of fully thermally coupled distillation columns. The obtained results were very similar to those obtained with rigorous structural design method. The procedure was tested with organic mixtures with three different feed compositions. In spite of the interesting scheme proposed, Kim used a semi-rigorous model for material balances. Also in 2005, Pistikopoulos and Proios (2005) presented the Generalized Modular Framework based on formal superstructure optimization techniques, where sequences were evaluated with respect to energy efficiency. They worked with several cases, and found substantial energy savings. Basically, this approach can be viewed as an intermediate between rigorous and shortcut models for the systematic generation and identification of energy-efficient column sequences (Pistikopoulos & Proios, 2005). Recently, Vaca, Monroy-Loperena and Jiménez-Gutiérrez (2007) used collocation techniques to refine a design obtained with a shortcut design method. The refinement of the design just manip-

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ulates the location of the interconnection and side stream stages, and the interconnection flows, while the rest of the variables remained unchanged. Nevertheless, orthogonal collocation technique employs a model based in constant as overflows as relative volatilities to make the refinement of the design. In general, all the works mentioned before presented new procedures and frameworks to obtain optimal designs of Petlyuk sequence using rigorous, semi-rigorous or short models for material and energy balances, with mathematical programming, stochastic methods or refinement techniques. Also, in all the cases they just obtained one optimal design for a given optimization criterion, such as total cost, thermodynamic efficiency, energy efficiency, among others. However, it is well known that the number of stages and the reflux ratio are variables in competition in a distillation column, since the first one is associated to capital costs and controllability, and the second one impacts directly the energy consumption. So, a first alternative to solve the problem is using a heuristic rule, for instance the well known heuristic rule of 1.3 times the minimum reflux ratio for the operating reflux ratio. Heuristic rules are just recommendations based on experience, and they reflected the best compromise between the size of the column and its energy consumption, principally for conventional distillation sequences. Then, perhaps for Petlyuk sequence the heuristic rules that we know are not the best compromise between capital and operating costs. A second alternative is the use of total annualized cost function as an optimization criterion, which includes capital and operating costs: number of stages and heat duty. However, this function considers the equipment cost, energy, parameters of actualization and projection that also depends of time and location. In Petlyuk sequence, the total number of stages, in prefractionator and main column, and the heat duty are variables in competition, since we cannot decrease indefinitely the number of stages without an increasing in the heat duty and vice versa. So, if we could know for different total numbers of stages the design that consumes the lowest amount of energy, and meet the purities or recoveries required, then we can easily calculate the minimum total annualized cost. In this way, we calculate the total annualized cost just for a reduced set of designs that we know beforehand that meet the desired purities or recoveries (constraints) with the lowest heat duty possible for each number of stages (variables in competition). A possible approach would be to apply an algorithm that minimizes the heat duty for each total number of stages selected. This approach would require high computing times, and also an elevated number of runs to cover the total searching space. Besides, we need to know beforehand the adequate range of values of the number of stages that must to be covered, as in the prefractionator as in the main column. The approach presented in this paper considers the search for the optimal set of designs, sequences with different total number of stages and reduced energy consumption that meet the required recoveries or purities, using a multiobjective genetic algorithm that conducts the search with a reduced number of runs, and of course less computing time. So, the result will be a set of optimal designs from minimal reflux ratio to minimum number of stages, along with all designs include between these extremes. This set of optimal designs can be achieved by means of Pareto front, which represents a set of optimal solutions not dominated for a multiobjective problem. Multiobjective optimization approach using genetic algorithms has been used to solve other problems in Chemical Engineering such as optimization of membrane separation modules (Gupta, Yuen, Aatmeeyata, & Ray, 2000), adiabatic styrene reactor (Syed Mubeen, Chakole, & Babu, 2005), Williams and Otto Chemical Plant (Mukherjee, Chakraborti, Mishra, Aggarwal, & Banerjee, 2006), selection and recycling of solvents (Xu & Diwekar, 2007), among others.

In this work, we implemented a multiobjective genetic algorithm with constraints to obtain the Pareto front of Petlyuk sequences. The genetic algorithm is coupled to a commercial simulator, Aspen Plus, so all the results presented here are rigorous. The implemented genetic algorithm is based on the work of Deb, Agrawal, Pratap and Meyarivan (2000), NSGA-II, and we handle the constraints using a slightly modification of the work of CoelloCoello (2000). The NSGA-II has been used in some problems of Chemical Engineering such as optimization of reaction processes (Modak & Sarkar, 2005; Raha, Majumdar, & Mitra, 2004), principally. 2. Pareto front The concept of Pareto optimum or Pareto front was formulated by Vilfredo Pareto in XIX century, and it is considered the origin of all research in multiobjective optimization (Mezura-Montes, 2001; Pareto, 1896). We can say that one point z¯ ∗ ∈  is a Pareto optimum if for each z¯ ∈ : ∧ (fi (¯z ) ≥ fi (¯z ∗ ))

i∈I

(1)

Or at least there is some i ∈ I, where I represents the set of objective functions to optimize, that: fi (¯z ) > fi (¯z ∗ )

(2)

In the searching space, one point is considered as Pareto optimum if there is not a feasible vector that can decrease some value of one objective without increasing at the same time the value of another objective, if minimization is the case. Moreover, we define  when f (z) < f (w),  if W ⊆  and w  ∈ W if none that z dominates w  we say that w  is not dominated with respect to z ∈ W dominates w, W. The set of solutions which are not dominated and optimums of Pareto integrates the Pareto front (Mezura-Montes, 2001). For Petlyuk sequences, the Pareto front includes all optimal solutions from minimum reflux ratio to minimum number of stages, passing through all the good compromises designs. In this way, the engineer can choose an acceptable tradeoff between the two goals by picking some point along the Pareto front. The principal advantage of using a multiobjective optimization technique is that just one run is required to obtain the Pareto front. 3. Strategy solution Genetic algorithms are a stochastic optimization technique based on natural evolution (Goldberg, 1989). The genetic algorithm begins with an initial design from which it generates the first population of individual (solutions to the problem). All individuals included in this first population will be evaluated in terms of fitness function, which measures the capacity of an individual to survive in its environment; then, a greater ability to survive represents a greater value of fitness function. Once the population has been evaluated, the algorithm selects the individuals with the greatest fitness function values to be the parents of the new population. Thus, selecting the best individuals to be the parents of the new generation, it is expecting that the new generation will be better than the first one. The selected parents are reproduced and muted, looking to generate diversity with this last operator. Also, with the aim to preserve the best individuals of each generation to the next generations, the operator elitism is used. Now, the new generation is evaluated in terms of fitness function, and the best individuals are selected to be the parents of the new generation. The process is repeated until a stop criterion is satisfied, as the maximum number of generations, maximum computing time, or fitness function value unchanged during the last n generations.

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In the optimization problem of Petlyuk sequence we have to minimize simultaneously the heat duty, Q, and the number of stages in prefractionator and main column, Ni , subject to meet the required recoveries or purities in each product stream. This minimization can be expressed as: min(Q, Ni ) = f (Q, R, Ni , Nj , NS , NF , Fj ) s.t. y k ≥ xk

(3)

where R is the reflux ratio, Ni is the total number of stages of the column i, Nj is the stage number of the interconnection flow j, NS is the side stream stage, NF is the feed stage number in the prefractionator, Fj is the interconnection flow j, xk and y k are the vector of required and obtained purities or recoveries. As can be noted, in the optimization problem the three variables in competition are considered for optimization: heat duty of sequence, number of stages in main column, and number of stages in prefractionator. In terms of the genetic algorithm, the fitness function includes the variables in competition and the rest of the variables of the Petlyuk sequence, and it can be written as: fitness = f (Q, R, Ni , Nj , NS , NF , Fj , y k )

(4)

The multiobjective algorithm implemented to obtain the Pareto front is based on NSGA-II (Deb, Agrawal, Pratap and Meyarivan (2000)), which is a multiobjective genetic algorithm with low computational requirements, elitist approach, and parameters-less sharing approach with respect to NSGA (Srinivas & Deb, 1995). In order to evaluate the fitness function, the genetic algorithm takes the results of the rigorous simulation of the design, which is realized in Aspen Plus. The term rigorous means that the complete set of MESH equations are used to the simulation of the Petlyuk sequence. The process is as follows. The genetic algorithm sends the input data to Aspen Plus (variables of the design generated by the algorithm). Then, the simulator makes the run, and the results of interest are taken back by the optimization procedure. So, the fitness function is calculated according to the concept of non-dominance, as can be seen in Fig. 2. It is worth to mention that the genetic algorithm allocates an infinite heat duty to cases that are not physically feasible and those where the simulation converges with errors. The link between Aspen Plus and the optimization procedure was realized with ActiveX Control Technology, which allows the manipulation and information exchange between applications. The constraints are handling with a multiobjective optimization technique, which guides the search of the genetic algorithm using as base the concept of non-dominance proposed by Coello-Coello (2000). The approaching of Coello-Coello (2000) is slightly modified just to simplify the computational implementation, and it is as follows. The entire population is divided into sub-populations according to the criterion of total number of constraints satisfied: the best individuals of the generation are those that satisfied the n constraints, and they are followed by the individuals those just satisfying n − 1 constraints, and so. Inside each sub-population the individuals are ranked using the fitness function value from the algorithm based on NSGA-II, considering as other objective function minimizing the difference between the required and obtained purities or recoveries; in other words, the degree of dissatisfaction constraints. The calculation of dominance within each subgroup is made as follows: dominance{Q, Ni , min[0, (xk − y k )]

(5)

As can be noted in Eq. (5) the heat duty, the number of stages of the sequence and the degree of dissatisfaction constraints are considered at the dominance calculation. Thus, we are trying to minimize inside each subgroup as the original objectives functions as the value of restrictions unfulfilled. We chose to handle the constraints using a multiobjective technique, since it was very difficult

Fig. 2. Flow diagram of the optimization procedure.

to obtain a good solution if the constraints are handled with penalties. If we implemented soft penalties, it could not be achieved a solution that satisfied all the constraints; but if the penalties were so strong, then it was easy to reach a suboptimal solution since we were putting a lot of pressure in the algorithm. Also, we tested adaptive penalties but such penalties are difficult to tune. On the other hand, with the multiobjective technique used to handle the constraints we do not have these problems, since we do not have to do any additional tuning of the algorithm. In other words, there are not problems associated with penalties tuning or aggregation functions. 4. Cases of study In order to test the procedure implemented we select five ternary mixtures (Table 1) with three different feed compositions and their respective recoveries required (Table 2). Due to we consider two different operation pressures for mixtures M2 and M3, we decided to called them as a different mixture since their easy separation index, ESI, is also different as can be noted in Table 1. The mixtures were selected with different ESI to study

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Table 1 Ternary ideal mixtures selected. Mixture

Components (A–B–C)

ESIa

M1 M2 M3 M4 M5 M6 M7

n-Butane–n-hexane–n-heptane n-Butane–i-pentane–n-pentane i-Pentane–n-pentane–n-hexane Benzene–toluene–ethylbenzene n-Pentane–n-hexane–n-heptane n-Butane–i-pentane–n-pentane i-Pentane–n-pentane–n-hexane

1.08 1.66 0.49 1.12 1.04 1.86 0.47

a

Easy separation index.

Table 2 Different feed compositions and desired recoveries in each product stream. Feed

XF,A

XF,B

XF,C

RecD,A

RecS,B

RecB,C

F1 F2 F3

0.40 0.33 0.15

0.20 0.33 0.70

0.40 0.34 0.15

0.98 0.98 0.98

0.98 0.98 0.98

0.98 0.98 0.98

the effect of the nature of the mixture in the design of Petlyuk sequence. The liquid–vapor equilibrium for all mixtures is modeled with Chao-Seader correlation, and the Petlyuk sequence is simulated in the module Radfrac of Aspen Plus. Saturated liquid is the thermal condition of the feed for all mixtures and feeds considered in this work. 5. Analysis of results For the 21 cases resulting from combining all mixtures and feeds, the genetic algorithm uses the parameters showed in Table 3. The set of parameters was obtained through a tuning process, with the aim to ensure the convergence in the lowest number of generations, avoiding premature convergence, and seeking diversity in the Pareto front. The analysis of the results is going to be presented as follows: first we are going to give detailed explanation of case M1F1, and after that the results for the rest of the cases. 5.1. Case M1F1 The Pareto front of Petlyuk sequence for case M1F1 is presented in Fig. 3. We can observe that the three variables in competition are considered in this graph: heat duty of the sequence, number of stages in main column, and number of stages in prefractionator. The optimal points include the observed minimum reflux ratio (minimum energy consumption) marked in a green square (1.614), and the observed minimum number of stages (maximum energy consumption) marked in a pink circle (39 from main column and 15 from prefractionator). The rest of the points are optimal designs including between these values. In Fig. 3, we have added two dotted lines to visualize easily the region that contains all the optimal designs in Pareto front. We observe that the variation of energy with the number of stages is abrupt, since a zone Table 3 Parameters of the genetic algorithm for all mixtures and feeds. Parameters

Value

Population size Generation number Reproduction Reproduction fraction Mutation Elitism Migration Migration fraction Selection

600 40 Disseminate 0.6 Gaussian 4 Forward 0.2 Stochastic uniform

Fig. 3. Pareto front of Petlyuk sequence for case M1F1. (For interpretation of the references to color in text related to the figure, the reader is referred to the web version of the article.)

almost flat begins approximately at 1.08 times the minimum reflux ratio, Rmin . Thus, for case M1F1 fix the reflux operating ratio with a heuristic rule like 1.3Rmin will conduct to obtain a design with the same stages required at 1.08Rmin but with higher energy consumption. The Pareto front for case M1F1 includes 43 optimal designs of Petlyuk schemes, which satisfy the recoveries specified in Table 2 with the lowest values for the number of stages in prefractionator, main column, and, of course, heat duty. At the end, the total annualized cost could be calculated with specific factors for location, costs and time, but just for the optimal designs included in Pareto front instead of doing this calculation during all the iterative process. Fig. 4 presents other interesting graphics derived from the Pareto front of case M1F1. From Fig. 4a we observe that the prefractionator designs are narrow to 15 and 19 stages, and also the location of the feed stage is almost the same. It seems that in optimal Petlyuk sequences, the prefractionator structure remains almost constant to performance the same distribution of components, for a same feed, while the main column makes the fine separation with different reflux ratios and stages. The values of interconnection flow FL1 are also in a small range, between 15 and 19 lbmol/h, and it is just in this interval were the heat duty is almost constant (Fig. 4b). A similar behavior is presented in the interconnection flow FV2 (Fig. 4c). So, there is a zone in which the interconnection flows vary without an increasing in the heat duty; in other words, there is a set of structures that can achieve the same specifications of recoveries or purities with different operating conditions, but keeping constant the heat duty. This behavior has as effect the presence of a zone where the recoveries of the components can be slightly greater than the specification, without this requires more energy consumption (Fig. 4d). Now, Fig. 4e shows that there is a relation between the interconnection flows of liquid, FL1, and vapor, FV2; being the vapor flow, in average, around 3.5 times greater than the liquid flow for this case. At last, there is a linear relation between the location of side and interconnection flows stages and the total number of stages in the main column (Fig. 4f). Observe how easy could be draw some straight lines in the variation of side stream stage with the total number of stages in main column, for instance. This observation applies to the rest of the cases analyzed in this work. In spite of the high non-linearity and complexity of the Petlyuk sequence, there are linear relations in the structure of the main column and prefractionator in the optimal set of designs, and zones where we can increase the interconnection flows or the recoveries of the components while the heat duty remains almost constant. Basically, with the same energy consumption we can select the best structure of the sequence according to our particular needs.

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Fig. 4. Other interesting graphics of case M1F1: (a) location of the feed stage in prefractionator, (b) variation of interconnection stream flow FL1 with heat duty, (c) variation of interconnection stream flow FV2 with heat duty, (d) variation of recovery of light component with heat duty, (e) relation between interconnection flow streams FL1 and FV2.

Pareto front allows the engineer has the complete set of optimal sequences that satisfy the design specifications, and he would choose the better for his own particular situation of materials, services costs, and geographic location, among others. The computing time required to obtain the Pareto front of each case was 3 h, in a Laptop Centrino at 1.5 GHz and 0.5GB of RAM. The rigorous simulation performed in Aspen Plus is the activity that consumes more computation time, almost 95%. Fig. 5 presents the Pareto fronts for cases M1F1, M1F2 and M1F3, and there we can observe the effect of feed composition in the design of Petlyuk sequence. If the intermediate component composition in the feed is increased (F1 < F2 < F3) the reflux ratio rises in the main column (energy consumption of the sequence), along with the interconnection flows of liquid (Fig. 6). In general, the lowest heat duties are in case M1F1, since it is the case with the lower

amount of intermediate component in the feed; this is the case where Petlyuk sequence has the better performance (Jiménez et al., 2003). 5.2. All study cases Table 4 presents 3 designs extracted from Pareto front for each one of the 21 cases. In order to analyze the effect of the feed composition in the design of Petlyuk sequence we take again the cases M1F1, M1F2 and M1F3. From Table 4, we can observe that in the prefractionator the feed stage is located almost in the middle since its ESI is 1.08, for all feeds. Particularly, for equimolar feed (F2), the feed stage is almost exactly in the middle of the prefractionator, due to be the more symmetric case as in the ESI value as in the feed composition. Moreover, in the main column the location of input and output

460

Table 4 Some designs taken from Pareto front for all study cases. NFV1

NFL2

NFL1

NFV2

NSide

NPrefr

NF,Pref

FFL1 (lbmol/h)

M1F1 1.614722837 1.670591147 2.399112052

21 13 11

55 32 30

14 13 10

46 30 33

26 19 16

15 16 15

9 9 9

18.35328268 16.18376924 30.61599231

M1F2 2.652374344 2.830995796 3.384201041

12 10 12

34 31 31

11 10 13

32 29 33

18 16 18

17 13 13

7 8 9

M1F3 9.440556859 10.23285576 11.76827565

10 12 10

36 31 28

10 14 10

35 34 23

18 16 16

26 17 15

M2F1 9.657312354 11.71978748 15.68769846

21 18 16

79 70 69

27 26 28

82 74 72

32 31 29

M2F2 13.85819677 17.93489633 23.74662707

17 24 22

100 77 69

7 10 16

105 81 70

M2F3 46.17200964 50.93316863 66.60287255

19 4 16

85 88 87

10 10 10

M3F1 7.23184268 9.9391933 14.25950753

38 31 26

88 89 67

M3F2 8.95740928 11.63676951 17.54468949

35 35 36

M3F3 26.24394314 30.73358266 77.34634929

FFV2 (lbmol/h)

Distillate (lbmol/h)

Side stream (lbmol/h)

Heat duty (BTU/h)

RecCOMP1

RecCOMP2

RecCOMP3

Nmain

62.40830248 60.48760739 72.3871606

39.24038696 39.59999847 39.9204216

21.14372826 20.80699921 20.23406029

1172172.5 1205024.21 1489272.34

0.981008225 0.989947279 0.996841928

0.985994744 0.982864694 0.98844643

0.983397112 0.981326552 0.991504644

78 42 39

23.31987438 18.58335993 22.71227118

70.99823946 59.39948046 65.68702548

33 32.44229507 33

33.02999878 33.71466446 33.02999878

1316291.88 1355134.95 1553872.41

0.998437798 0.982977598 0.995442906

0.980640842 0.98836109 0.98260762

0.981844113 0.984182345 0.986663711

70 41 38

12 7 7

26.10504687 25.49018099 21.49003495

67.05041555 61.68180124 54.9162712

15.23154259 14.85892582 14.94857025

69.34081268 69.14669037 70.03165436

1614066.01 1681638.94 1915259.03

0.999341459 0.987414588 0.991855508

0.98686397 0.982287254 0.995107674

0.983312605 0.986839788 0.983176155

52 42 35

32 29 26

20 20 17

62.59332416 59.10356698 58.82905353

94.37448234 88.39780905 89.4945151

39.59999847 39.59999847 39.59999847

20.80699921 20.80699921 20.80699921

3603456.66 4289837.58 5605832.1

0.988724276 0.988591393 0.989993132

0.991504832 0.980239429 0.982205967

0.986877135 0.981330406 0.980957299

181 112 97

34 32 29

36 35 46

21 21 29

74.0276783 64.92649145 71.39444564

101.1214856 93.9310722 92.98910726

32.49271774 33 32.69385147

33.46676254 33.02999878 33.50622177

4105640.18 5312604.87 6841087.61

0.984626926 0.98966815 0.990704413

0.986465057 0.981362176 0.990707037

0.988059042 0.991211998 0.987597301

173 127 104

86 86 97

25 26 25

44 35 35

26 28 25

82.69708738 88.38416523 90.25838488

103.7270621 104.2282779 101.855924

15.20128155 15.22493076 15.40127468

69.02986145 69.23899841 69.37637329

6079258.13 6718070.74 8848705.17

0.993213755 0.985077823 0.990589494

0.98051616 0.982474449 0.985223803

0.9805634 0.984295935 0.982166664

141 120 113

41 32 20

99 89 77

77 68 58

56 32 93

13 5 14

128.919031 116.0592405 113.4418432

144.2737249 145.9215098 138.5349162

39.61855316 39.59999847 39.59999847

20.87293434 20.80699921 20.80699921

3405638.59 4500141.15 6231395.86

0.981061425 0.981106823 0.984282094

0.998593995 0.982127059 0.988543223

0.987712855 0.989763819 0.989823579

126 96 84

102 91 78

37 33 43

93 81 75

74 84 73

59 63 75

13 18 11

119.6896828 106.2754841 154.5196538

142.9051331 145.3372748 187.0156329

33.48455048 33.11178207 33

32.80270004 33.04566193 33.02999878

3504520.14 4346222.3 6304616.39

0.983115111 0.997406536 0.99714762

0.980988044 0.983286859 0.995898485

0.991550627 0.984948046 0.997914923

119 99 91

39 34 35

114 86 74

40 28 35

90 76 75

84 75 69

79 70 151

16 18 16

143.096997 116.4541347 134.5351246

179.6766545 167.3097783 179.9681168

15.04354095 15.01971436 15.00526524

69.65705872 69.87612915 70.20599365

4233127.06 4918003.72 11984477.4

0.984817365 0.980281174 0.997735004

0.991838344 0.992258638 0.998884809

0.999848944 0.991876754 0.985868775

123 93 88

M4F1 2.264979458 2.412600152 3.445439108

11 11 12

42 35 37

12 11 13

43 35 38

20 17 18

21 17 16

11 8 8

31.01382386 33.58863953 34.79506599

73.41166174 73.39362864 73.03805704

39.42544937 39.59999847 39.59999847

21.20441628 20.80699921 20.80699921

1886004.86 1973573.83 2513022.77

0.983844811 0.986587122 0.989281748

0.988333569 0.981114865 0.982901066

0.980204968 0.983824086 0.982012176

73 55 46

M4F2 3.548455185 3.78091699 4.3200781

12 15 12

49 43 39

12 12 12

43 46 37

22 22 21

26 35 18

11 16 9

33.68676557 35.35498398 43.4899685

72.94161678 70.91684292 81.68281772

33.01390457 32.68009567 33

33.02692413 33.15010071 33.02999878

2149199.33 2232749.11 2489278.92

0.998332044 0.990175444 0.997019965

0.980284392 0.98762555 0.98518551

0.981678819 0.993106888 0.987687074

70 60 52

M4F3 12.00877598 13.63826691 49.58114564

10 11 11

49 48 42

13 16 16

53 51 42

19 22 21

27 27 21

12 12 9

41.68416739 42.53253927 35.89941265

77.51695134 73.50503985 77.0912212

15.36069298 15.3228159 14.76211929

69.31826019 69.25997925 70.3356781

2745127.6 3079209.31 9927728.25

0.99612616 0.998430524 0.984083297

0.985465796 0.984895356 0.997368791

0.981495621 0.980463665 0.981313027

67 56 48

C. Gutiérrez-Antonio, A. Briones-Ramírez / Computers and Chemical Engineering 33 (2009) 454–464

Reflux ratio

Table 4 (Continued ) NFV1

NFL2

NFL1

NFV2

NSide

NPrefr

NF,Pref

FFL1 (lbmol/h)

M5F1 2.121434802 2.443743069 2.926932089

12 13 13

36 29 30

12 14 15

35 29 34

20 21 21

17 17 19

11 10 9

30.84358431 35.42568457 35.25202593

M5F2 3.465203831 4.165590762 5.147153485

12 12 9

35 31 36

12 12 11

34 32 32

20 20 21

27 18 34

9 9 10

29.15127602 32.46125853 44.2538787

M5F3 12.55966464 13.58108778 15.3164699

9 8 9

27 24 23

11 10 11

27 25 26

18 17 15

28 25 27

13 11 13

56.12422133 61.52411353 56.12422133

M6F1 8.889103322 10.41909209 13.24476189

32 26 20

114 96 89

15 24 23

107 110 93

44 44 42

33 31 28

13 17 13

M6F2 13.60061519 14.7775467 16.54793221

30 25 26

108 109 109

20 24 24

111 116 111

40 35 35

38 38 42

M6F3 44.54451871 47.85959981 50.61039112

13 22 22

116 106 101

9 21 22

113 100 101

47 36 37

M7F1 7.673200609 8.853270669 15.60864844

42 39 42

88 78 75

41 32 37

81 76 69

M7F2 9.992496635 11.2089348 12.1745918

31 39 40

85 83 78

31 35 37

M7F3 28.41710899 33.5430024 37.03444207

37 32 32

99 87 83

35 35 29

FFV2 (lbmol/h)

Distillate (lbmol/h)

Side stream (lbmol/h)

Heat duty (BTU/h)

RecCOMP1

RecCOMP2

RecCOMP3

Nmain

77.2821477 81.86885815 86.37689128

39.4392395 39.59999847 39.59999847

21.07221031 20.80699921 20.80699921

1420034.32 1558022.58 1757582.13

0.983713955 0.989569248 0.989796463

0.98359216 0.98197971 0.982583991

0.981293697 0.981283021 0.981310241

54 42 40

74.00705118 79.08575486 83.60990541

32.54239273 33 32.54239273

33.65611267 33.02999878 33.4923172

1630624 1894904.78 2200946.88

0.985931752 0.997934168 0.985815248

0.98960376 0.98427158 0.980260792

0.984265486 0.985894637 0.980128548

52 41 42

109.1241737 117.3648592 103.8850701

15.1256218 15.35368633 15.3072567

69.56587982 69.25526428 69.56451416

2204776.52 2416654.4 2686077.7

0.990092909 0.984682561 0.987163728

0.987699671 0.982740245 0.986824649

0.981465216 0.984415374 0.980405672

41 36 32

66.4983713 72.53910119 63.96446656

98.39420931 103.9140705 86.50444578

39.59999847 39.59999847 39.59999847

20.80699921 20.80699921 20.80699921

3300851.97 3804855.49 4742256.56

0.989999997 0.99 0.984733233

0.982598239 0.982691319 0.982061043

0.98114049 0.981166031 0.986080053

186 148 134

24 24 24

95.39552866 95.39552866 95.39552866

125.0373126 125.0373126 126.0131571

32.55973816 32.52771378 32.52771378

33.64404678 33.3952446 33.58218002

3993301.51 4307805.45 4786874.82

0.986658677 0.985688346 0.985688346

0.987107099 0.980603975 0.986935097

0.981480398 0.983427272 0.98406764

159 147 140

35 45 35

20 18 21

92.47697299 95.78127368 100.2242385

112.1453294 107.3905049 109.6098019

14.9770689 14.80394363 14.79608631

69.61904144 69.17715454 69.17715454

5696224.09 6038164.15 6373536.15

0.998059554 0.986929593 0.986405739

0.987620358 0.981382889 0.981373813

0.981282244 0.980914668 0.981387402

179 141 132

75 67 73

47 65 18

18 11 5

90.72470886 116.6406433 32.72610371

125.424926 144.2100591 97.51614909

39.51995468 39.73983765 39.59999847

21.13442612 20.8531456 20.80699921

3494514.2 3984725.76 6651431.8

0.981253822 0.984247909 0.989453579

0.986653315 0.981022627 0.982707751

0.983639827 0.984829749 0.981701406

105 85 84

77 86 81

77 71 70

49 44 49

10 9 10

95.29129332 104.4103584 94.0989332

143.6035744 139.5498693 138.691567

32.92714691 32.69269562 33.0178794

33.43683243 33.18371964 32.9289474

3681374.63 4053162.78 4413812.1

0.982688081 0.983732403 0.988993113

0.984897319 0.988421031 0.983636262

0.98929321 0.999082368 0.996851292

106 88 84

82 84 71

85 73 64

61 57 67

10 9 8

95.6969959 96.98935662 91.73007343

149.4923147 137.7301461 148.0026638

15.29521465 15.07734108 15.44358444

69.25338745 69.9169693 69.73553467

4543707.82 5250548.25 5926381.13

0.980703112 0.990275487 0.987051441

0.984988981 0.995098688 0.989538746

0.998999236 0.992088495 0.981829774

112 90 87

C. Gutiérrez-Antonio, A. Briones-Ramírez / Computers and Chemical Engineering 33 (2009) 454–464

Reflux ratio

461

462

C. Gutiérrez-Antonio, A. Briones-Ramírez / Computers and Chemical Engineering 33 (2009) 454–464

flows in the structure is moving down, in direction to the reboiler, when the amount of intermediate component is increased. Again, the more symmetric structure is observed in case M1F2. Also, bigger interconnection flows of vapor and liquid are required when there is a bigger amount of intermediate component in the feed. Also from Table 4, we note that in the designs of mixtures M7 and M3, ESI < 1, the feed stage in the prefractionator is close to the top of it. For these mixtures A/B is the hard cut, so the feed stage in the prefractionator is located close to the top in order to send the hard cut to the main column, resulting this in a high recovery of B in the top of the prefractionator (at least 80%). Moreover, in mixtures M2, M4, and M6, ESI > 1, the feed stage is located close to the bottom of the prefractionator, since in this way the difficult cut, B/C, is sent to the main column; for this mixtures the recovery of B in the bottom of the prefractionator is at least 80%. In mixtures more symmetric, M1 and M5, the feed stage is located almost in the middle, since it is easy to separate as A from B as B from C. So, in these cases the intermediate component is distributed around 50% in the top of the prefractionator. Continuing with the analysis of optimal designs from Table 4, we observe that in mixtures with ESI ≥ 1, the relation of vapor and liquid interconnection flows decreases when the composition of the intermediate component in feed increases. On the other hand, in mixtures with ESI < 1, that relation is increased when the amount of intermediate component in feed is major. In general, the relation between interconnection flows and the ESI has a maximum point

Fig. 6. Interconnection flow streams FL1 and FV2 for cases M1F1, M1F2 and M1F3.

Fig. 5. Pareto front of cases M1F1, M1F2 and M1F3.

where the ESI = 1, as can be noted in Fig. 7. Is it worth to mention that for mixtures deviated from ideality the relation of interconnection flows is lower than that for ideal mixtures; however, the values of the interconnection flows are greater for mixtures deviated from ideality than those found in ideal mixtures. In the main column, the location of the side stream stage depends of the nature of the mixture, as can be noted in Table 4. If the ESI of the mixture is greater than one, hard cut B/C, the side stream is located near to the top of the main column, and also near to stages of superior interconnection; in this way, they are a lot of

C. Gutiérrez-Antonio, A. Briones-Ramírez / Computers and Chemical Engineering 33 (2009) 454–464

463

Table 5 Material balances in prefractionator for some splits. Component Case M5F1 n-Pentane n-Hexane n-Heptane

Feed flow (lbmol/h)

Interconnection liquid flow, FL1 (lbmol/h)

Interconnection vapor flow, FV2 (lbmol/h)

Interconnection vapor flow, FV1 (lbmol/h)

Interconnection liquid flow, FL2 (lbmol/h)

40 20 40

19.31608 18.74867 0.2364782

0.0169335 51.51504 36.66856

59.30436 28.67522 0.8160373

0.028649 61.58849 76.089

Total flow

100

38.3012282

88.2005335

88.7956173

137.706139

Case M5F2 i-Pentane n-Pentane n-Hexane

33 33 34

24.13023 29.95756 0.2611462

2.855007 110.4212 0.8103858

57.16589 55.11917 2.547248

2.819351 118.2595 32.52428

114.832308

153.603131

Total flow

100

54.3489362

114.086593

Case M7F3 n-Butane i-Pentane n-Pentane

15 70 15

60.83036 41.43027 0.0534801

0.1135027 112.4811 15.26556

Total flow

100

102.31411

stages to make the separation of the hard cut. If the difficult cut is A/B, ESI < 1, then the side stream stage is located near to the bottom of the main column, in order to have a major number of stages to performance the hard cut. Also, it is important to mention that for the most of the cases there are four interconnection streams stages instead of the two classics defined in the work of Petlyuk, Platonov, and Slavinskii (1965). In the superior interconnection flows, generally the liquid interconnection flow, FL1, is extracted in previous stages to the feeding stage of vapor interconnection flow, FV1, counting from top to bottom. In mixtures with ESI < 1, M3 and M5, the side stream is located between the liquid and vapor streams, due to the presence of isomers in the mixtures: n-pentane–i-pentane. In the inferior interconnection, in most of the cases the vapor interconnection flow is extracted from previous stages than the stage where the liquid interconnection flow is fed to the main column, counting from top to bottom. The presence of four interconnection flows instead of the original two is due to the significant difference in the composition of the flows. For instance, take the case M5F1 from Table 5 and observe the compositions of interconnection flows FV2 and FL2. Note that these flows contain principally the intermediate and heavy com-

Fig. 7. Variation of vapor and liquid interconnection flows ratio with the ESI, for all cases.

127.860163

75.572 52.27646 0.2466264 128.095086

0.3718629 171.6349 30.07242 202.079183

ponents and just traces of the light. The interconnection flow FL2 has 60% of the intermediate and 40% of the heavy component. If we look in the composition of interconnection flow FV2 we see that the percents are exactly the inverse: 40% of the intermediate and 60% of the heavy. That is the reason why there are two interconnection stages for FL2 and FV2: the difference in compositions is so big due to it is not possible to locate the flows in the same stage. In general, the distribution of interconnection flows is: FL1, FV1, FV2, and FL2. As can be noted, the vapor flows are located between the liquid flows. Besides, we found that both optimal vapor flows, FV1 and FV2, have almost identical numerical values; as consequence, the interconnection flow FL2 is the sum of the interconnection flow FL1 plus the feed flow F (Table 5). Also, the vapor flows are pseudo binary: FV1 is composed principally of light and intermediate components, while the intermediate and heavy components are present mostly in flow FV2. 6. Concluding remarks Based on NSGA-II (Deb, Agrawal, Pratap and Meyarivan (2000)) and the non-dominance concept (Coello-Coello, 2000) we implemented a multiobjective genetic algorithm with constraints to obtain rigorous Pareto fronts of Petlyuk sequences, since the genetic algorithm was coupled to processes simulator Aspen Plus. Pareto front resulting allows identifying the observed minimum reflux ratio and minimum number of stages in the prefractionator and in the main column, along with the rest of optimal designs that meet the specified constraints. In the set of optimal designs, the structure of the prefractionator remains almost constant, and the principal differences are observed in the size of the main column. It seems that the prefractionator structure remains unchanged, and the change in the structure of main column conducts to the creation of Pareto front. The location of the feed stage in prefractionator depends on the composition of the feed and the nature of the mixture. If the mixture has an ESI < 1 the feed stage in the prefractionator is close to the top of it, where the recovery of intermediate component is at least 80%. In mixtures with an ESI > 1, the feed stage is located near to the bottom, where the recovery of the intermediate component is at least 80%. The feed stage is located in the middle of the prefractionator is the ESI of the mixture is 1, and in this case the intermediate component is distributed around 50% in the top. The objective is sent the hard cut to the main column, where we have more stages to performance the separation.

464

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The location of the side stream in the main column follows the same logic. If the difficult cut is A/B, ESI < 1, then side stream stage is located near to the bottom of the main column, in order to have a major number of stages to performance the hard cut. For an ESI > 1 the side stream is located close to the top of the column, while with an ESI = 1 the side stream is in the middle of the column. Another aspect of great importance is that for the optimal set of designs there is a zone where the interconnection flows can vary without an increasing in the heat duty, this situation also helps to have higher recoveries without more energy consumption. The analysis of the cases suggests that there is a linear relation between the interconnection flows of liquid and vapor, FV2 and FL1, and in function of the nature of the mixture the vapor flow is a certain times higher than the liquid flow. An interest fact is that in spite of the high non-linearity of Petlyuk sequence, there are simple linear relations in the structure of the main column. Practically, having a given design one can expand the design just keeping the proportion in location of interconnection and side streams. In general, the idea behind is give more stages to the harder separation, and avoid do it in the prefractionator. Also, the structure of the prefractionator is defined for this hard cut. References Caballero, J., & Grossmann, I. (2001). Generalized disjunctive programming model for the optimal synthesis of thermally linked distillation columns. Industrial & Engineering Chemistry Research, 40(10), 2260–2274. Carlberg, N. A., & Westerberg, A. (1989). Temperature heat diagrams for complex columns. 3: Underwood’s method for the Petlyuk configuration. Industrial & Engineering Chemistry Research, 28, 1386–1397. Carroll, D. L. (1996). Chemical laser modeling with genetic algorithms. AIAA Journal, 34, 338–346. Coello-Coello, C. A. (2000). Constraint-handling using and evolutionary multiobjective optimization technique. Civil Engineering and Environmental Systems, 17, 319–346. Deb, K., Agrawal, S., Pratap, A., & Meyarivan, T. (2000). A fast elitist non-dominated sorting genetic algorithm for multiobjective-optimization: NSGA-II, KanGAL report 200001. Kanpur, India: Indian Institute of Technology. Fidkowski, Z., & Królikowski, L. (1986). Thermally coupled system of distillation columns: Optimization procedure. AIChE Journal, 32, 537–546. Fidkowski, Z., & Królikowski, L. (1987). Minimum requirements of thermally coupled distillation systems. AIChE Journal, 33, 643–653. Fidkowski, Z., & Królikowski, L. (1990). Energy requirements of nonconventional distillation systems. AIChE Journal, 36, 1275–1277. Fonyó, Z., Rév, E., Emtir, M., Szitkai, Z., & Mizsey, P. (2001). Energy savings of integrated and coupled distillation systems. Computers & Chemical Engineering, 25, 119–140.

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