Energy-efficient complex distillation sequences: control properties

Energy-Efficient Complex Distillation Sequences: Control Properties Salvador Robles-Zapiain,1 Juan Gabriel Segovia-Hern´andez,1 * Adri´an Bonilla-Petriciolet2 and Rafael Maya-Yescas3 1. Universidad de Guanajuato, Facultad de Qu´ımica, Noria Alta s/n, Guanajuato, Guanajuato, 36050, M´exico 2. Instituto Tecnol´ogico de Aguascalientes, Departamento de Ingenier´ıa Qu´ımica, Av. L´opez Mateos 1801, Aguascalientes, Aguascalientes, 20256, M´exico 3. Universidad Michoacana de San Nicol´as de Hidalgo, Facultad de Ingenier´ıa Qu´ımica, Ciudad Universitaria, Edif. M, Col. Fel´ıcitas del R´ıo, 58060 Morelia, Michoac´an, M´exico

Four thermally coupled distillation systems were designed for the separation of five-component mixtures (the light-ends separation section of a crude distillation plant); their steady-state design was obtained by starting from a conventional distillation sequence and then optimizing for minimum energy consumption. The thermally coupled distillation systems were compared to sequence based on conventional columns design. Comparison was based on controllability properties under open and closed loop operation, following the dynamic behaviour after common industrial operating disturbances. Simulation results were analyzed by the singular value decomposition technique and with the performance examination of elimination of feed disturbances using PI controllers. It was found that thermally coupled distillation systems are controllable and, sometimes, they exhibit dynamic responses that are easier to manage than in the case of conventional distillation sequences; this result is innovative in the study of this kind of systems. Quatre syst`emes de distillation coupl´es thermiquement ont e´ t´e conc¸us pour la s´eparation de m´elanges a` cinq composantes (la section de s´eparation de l´egers d’une installation de distillation de brut); leur conception en r´egime permanent a e´ t´e obtenue en d´ebutant par une s´equence de distillation conventionnelle qui a ensuite e´ t´e optimis´ee quant a` la consommation d’´energie minimale. Ces syst`emes de distillation coupl´es thermiquement ont e´ t´e compar´es a` une s´equence bas´ee sur la conception de colonne conventionnelle. La comparaison s’appuie sur les propri´et´es de contrˆolabilit´e pour un fonctionnement en boucle ouverte et ferm´ee, en suivant le comportement dynamique a` la suite de perturbations op´eratoires industrielles communes. Les r´esultats de simulation sont analys´es par la technique de d´ecomposition des valeurs singuli`eres et l’examen des performances pour e´ liminer des perturbations sur l’alimentation a` l’aide de contrˆoleurs PI. On a trouv´e que les syst`emes de distillation coupl´es thermiquement sont contrˆolables, et parfois, ceux-ci montrent des r´eponses dynamiques qui sont plus faciles a` g´erer que dans le cas des s´equences de distillation conventionnelles; c’est un r´esultat novateur dans l’´etude de ce type de syst`emes. Keywords: thermally coupled distillation schemes, controllability, energy consumption, multi-component mixtures, open loop dynamics, closed loop dynamics

INTRODUCTION

D

istillation column configurations used to separate mixtures containing three or more components into pure product streams have been studied for a long. For example, Seader and Henley (1998) discuss algorithms to draw all the possible configurations looking for sharp splits between components of adjacent volatilities. Such schemes are generally referred as having direct or indirect splits. Some other known configurations include schemes similar to the prefractionator configuration for ternary feed mixtures. Additionally, thermally coupled configurations with reduced numbers of reboilers and condensers have been

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proposed (Petlyuk et al., 1965; Tedder and Rudd, 1978; Agrawal, 1996; Agrawal and Fidkowski, 1999). The thermally coupled distillation schemes are considered to be one of the most promising systems because of its savings on both energy and capital costs.

∗ Author to whom correspondence may be addressed. E-mail address: [email protected] Can. J. Chem. Eng. 86:249–259, 2008 © 2008 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20021

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Figure 1. Distillation sequence for five-component mixtures: (a) conventional sequence, (b) complex distillation column (TCDS-1).

Numerous research works on thermally coupled distillation schemes have been conducted to analyze steady-state and dynamic performance, especially for ternary mixtures (Tedder and Rudd, 1978; Triantafyllou and Smith, 1992; Wolff and Skogestad, 1995; Abdul Mutalib and Smith, 1998; Hern´andez and Jim´enez, 1999; Kim, 2000; Jim´enez et al., 2001; Kim et al., 2002, 2003; Segovia-Hern´andez et al., 2002, 2004, 2006, 2007a). As consequence, some interesting observations have been built: for example, for ternary mixtures the energy savings in the fully thermally coupled configuration can be less than 30% compared to those of traditional schemes. Among the three possible thermally coupled systems (the side rectifier, the side stripper configurations, and the Petlyuk column) for ternary mixtures, over a wide range of relative volatilities and feed compositions, the Petlyuk column tends to provide the most efficient designs, thermodynamically, compared to traditional configurations (Flores et al., 2003). There are few works on extensions towards design and control properties of integrated systems for mixtures of more than three components (Chrsitiansen et al., 1997; Blancarte-Palacios et al., 2003; Esparza-Herm´andez et al., 2005; Hern´andez et al., 2005). The reduced number of works in this field is due to the combinatorial problem of the possible configurations for multi-component separations and, mainly, the lack of experience and knowledge on thermally coupled distillation schemes for four or more components (Rong et al., 2000, 2001, 2003). This lack of knowledge has caused industry people to expect that dynamic properties of thermally coupled systems may cause more operational problems than conventional sequences, which lead to the lack of industrial implementation.

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There are only few studies about dynamic properties of thermally coupled configurations for the separation of multicomponent mixtures. Recently, a design procedure has been reported for the design and optimization of multi-component thermally coupled distillation columns (Calzon-McConville et al., 2006), which also developed a methodology for the study of control properties. Taking as starting point that methodology, in this work we developed a comparative study of control properties of the four thermally coupled distillation sequences for the separations of five-component mixtures and those of conventional sequences (Figures 1 to 4); dynamic operating performance is evaluated by using the singular value decomposition technique and rigorous dynamic simulations.

ENERGY-EFFICIENT DESIGN The design of the thermally coupled distillation arrangements could be performed through a sequence of superstructures suitable for optimization procedures. This task is complicated and, frequently, may fail to achieve convergence. There have been three important papers that deal with the optimal design of complex distillation columns, Dunnebier and Pantelides (1999), Yeomans and Grossmann (2000), and Caballero and Grossmann (2001). Nevertheless, for five-component mixtures, the problem is clearly more complicated than those solved by the mentioned authors, as consequence of the combinatorial nature of the system that gives rise to a superstructure significantly more complex to solve. To deal with this problem, Calzon-McConville et al. (2006) have proposed a method to overcome the complexity of the

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Figure 2. Distillation sequence for five-component mixtures: (a) conventional sequence, (b) complex distillation column (TCDS-2).

Figure 3. Distillation sequence for five-component mixtures: (a) conventional sequence, (b) complex distillation column (TCDS-3).

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Figure 4. Distillation sequence for five-component mixtures: (a) conventional sequence, (b) complex distillation column (TCDS-4).

simultaneous solution of the tray arrangement and energy consumption within a formal optimization algorithm. They decoupled the design problem into two stages: tray configuration and optimal energy consumption. The first stage of their approach is the development of preliminary designs for complex systems based on conventional distillation columns (see Figures 1 to 4). For this case study, conventional sequences consist of eight different tray sections, which are used as a basis for the arrangement of the tray structure of the thermally coupled schemes through a section analogy procedure (see Calzon-McConville et al., 2006, for details). After the tray arrangement for the integrated designs have been obtained, an optimization procedure (second stage) is used to minimize the heat duty supplied to reboilers of each coupled scheme, using as constrains the required purity of the five product streams, which yields operating conditions for each set of design specifications and tray arrangements. Then, remaining degrees of freedom are used to design integrated designs that provide minimum energy consumption. More details about the optimization procedure are shown in Calzon-McConville et al. (2006).

SINGULAR VALUE DECOMPOSITION One of the basic and most important tools of modern numerical analysis is the singular value decomposition (SVD). There are numerous important applications of the SVD when quantitative and qualitative information is desired about linear maps. One important use of the SVD is in the study of the theoretical control properties in chemical process. One definition of SVD is: G = V
WH

(1)

Here, G is the matrix target for SVD analysis,
is a diagonal matrix which consists of the singular values of G, V is a matrix

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which contains the left-singular vector of G and W is the matrix composed by the left-singular vectors of G (more details about mathematic fundaments are shown in Klema and Laub, 1980). In the case where the SVD is used for the study of the theoretical control properties, two parameters are of interest: the minimum singular value (␴∗ ) the maximum singular value (␴* ), and its ratio known as condition number (␥): ␥=

␴∗ ␴∗

(2)

The minimum singular value is a measure of the invertibility of the system and represents a clue of potential problems of the system under feedback control. The condition number reflects the sensitivity of the system to uncertainties in process parameters and modelling errors. These parameters provide a qualitative assessment of theoretical control properties of the alternate designs. The systems with higher minimum singular values and lower condition numbers are expected to show the best dynamic performance under feedback control (Klema and Laub, 1980; Papastathopoulou and Luyben, 1991). Also, it is important to note that a full SVD analysis should cover a wide range of frequencies. The SVD technique requires a transfer function matrix (G in Equation (1)) around the optimum design of the distillation sequences, and registering the dynamic responses of products composition. For the distillation sequences presented in this work, five controlled variables were considered, the products composition A, B, C, D, E. Similarly, five manipulated variables were defined, reflux ratios (Rj ) and heat duties supplied to reboilers (Qj ), depending on the structure. For example in the case of CS-1 and TCDS-1, we use the reflux ratio, where A is obtained and heat

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Table 1. Transfer function matrix for CS-1 (M1F1)

Table 2. Transfer function matrix for TCDS-1 (M1F1)

duty in the other components. Tables 1 and 2 show the transfer function matrix (G) for CS-1 and TCDS-1. It can be noted that the dynamic responses can be adjusted to first- or second-order models. Similar transfer function matrix can be obtained for TCDS-2, TCDS-3, and TCDS-4 and for the all cases of study.

Table 3. Design variables for the TCDS-4, case M1F1 Column

Variables

Main column

Stages = 41 Feed stage = 9 Reflux ratio = 1.72 FV1 = 15.5 kmol/h FV2 = 19.8 kmol/h FV3 = 22.75 kmol/h Pressure = 4.52 atm Stages = 10 Distillate flow rate = 4.47 kmol/h Stages = 11 Distillate flow rate = 4.41 kmol/h Stages = 11 Distillate flow rate = 4.39 kmol/h

CASE STUDY To compare the dynamic behaviour of the integrated arrangements with the conventional sequence, two five-component mixture of n-butane, n-pentane, n-hexane, n-heptane, and n-octane (mixture 1) and n-butane, isopentane, n-pentane, n-hexane, and n-heptane (mixture 2) were considered (examples of the light ends separation section of a crude distillation plant), with a feed flow rate of 45.5 kmol/h. We selected those mixtures to reflect different separation difficulties. It is important to establish that studying a five-component mixture of hydrocarbons is a suitable example, given the applications of the hydrocarbon mixtures in the petrochemical industry (Harmsen, 2004). The specified molar product purity for components A, B, C, D, and E were 98, 94, 94, 94, and 97%, respectively. Two different molar compositions (A, B, C, D, E) equal to (0.35, 0.10, 0.10, 0.10, 0.35; F1) and (0.125, 0.25, 0.25, 0.25, 0.125; F2) to examine the effect of the content of the intermediate components were studied (similar to the case of ternary and quaternary mixtures). Since the feed involves a hydrocarbon mixture, the Chao-Seader correlation was used for the prediction of thermodynamic properties. The tray arrangements and some important design variables for a representative sequence, after the optimization task, are given in Table 3. As far as energy consumption is concerned, the optimized steadystate design provides energy savings of ∼35% with respect to the best energy-efficient sequence based on conventional distillation columns (Table 4; more details in Calzon-McConville et al., 2006).

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Side rectifier 1 (where component B is purified) Side rectifier 2 (where component C is purified) Side rectifier 3 (where component D is purified)

Table 4. Optimum reboiler duty (kW) for complex distillation sequences (case M1) Feed

Sequence

Total reboiler duty (kW)

F1

CS-1 TCDS-1 CS-2 TCDS-2 CS-3 TCDS–3

1181 793 1062 931 1168 708

F2

CS-1 TCDS-1 CS-2 TCDS-2 CS-3 TCDS–3

2039 1471 1213 1123 1332 960

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RESULTS

1.E+32

1.E+28

1.E+24

Condition number, γ

The theoretical control properties of conventional and thermally coupled distillation sequences were obtained. The SVD technique requires transfer function matrices, which are generated by implementing step changes in the manipulated variables of the optimum design of the distillation sequences. In the second part of the study, dynamic responses under the action of proportional integral controllers (PI) were obtained for changes in the feed composition.

1.E+20

1.E+16

TCDS-1, 1.E+12

1.E+08

1.E+04

CS-1,

1.E+00

SVD Analysis

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

Frequency, ω [ rad/h ]

For the case study of M1F1, we got the next results: TCDS-1 and CS-1 arrangements (Figures 5 and 6) present similar condition number and values of the minimum singular value for the whole frequency range (specially in higher frequencies); therefore, it can be expected that the conventional and complex systems exhibit similar control properties under feedback control and they are better conditioned to the effect of disturbances. Similar results can be showed in the case of M1F2 (Figures 7 and 8) and with the mixture M2. The results indicate that a conventional scheme does not necessarily provide an improvement of its controllability properties in comparison with a complex scheme (thermally coupled).

Figure 8. Condition number TCDS-1 and CS-1 (M1F2).

Figures 9 and 10 show the minimum singular value and condition number for the case study of M1F1. The CS-2 presents higher values of ␴∗ and lower values of condition number for the whole frequency range. Therefore, the CS-2 is expected to require less effort control under feedback operation and it is better conditioned to the effect of disturbances than the TCDS-2 scheme. According to 1.E-14 1.E-15

1.E+02 1.E+01

1.E-16

CS-1,

1.E-17

1.E+00 1.E-01

1.E-18

Minimum singular value, σ *

1.E-02 1.E-03

Minimum singular value, σ *

1.E-04 1.E-05

TCDS-1,

1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12

1.E-19 1.E-20 1.E-21

CS-2,

1.E-22

TCDS-2,

1.E-23 1.E-24 1.E-25 1.E-26

1.E-13 1.E-14

1.E-27

1.E-15 1.E-16

1.E-28

1.E-17

1.E-29

1.E-18 1.E-30 1.E-04

1.E-19 1.E-20

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Frequency, ω [ rad/h ]

1.E-21 1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

1.E+22

1.E+24

Frequency, ω [ rad/h ]

Figure 9. Minimum singular value TCDS-2 and CS-2 (M1F1).

Figure 5. Minimum singular value TCDS-1 and CS-1 (M1F1). 1.E+30

1.E+03

Condition number, γ

Condition number, γ

1.E+04

TCDS-1, F1

1.E+23

TCDS-2,

1.E+16

CS-1, F1

CS-2,

1.E+02

1.E+09 0.0001

1.E+01 0.0001

0.001

0.01

0.1

1

100

10

1000

10000

100000

1000000

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000 1000000000 10000000000

10000000

Frequency, ω [ rad/h ]

Frequency, ω [ rad/h ]

Figure 10. Condition number TCDS-2 and CS-2 (M1F1).

Figure 6. Condition number TCDS-1 and CS-1 (M1F1). 1.E+01

1.E+00 1.E+00 1.E-01

1.E-01

TCDS-1,

1.E-02

1.E-02

CS-1,

1.E-03 1.E-04

1.E-04

Minimum singular value, σ *

Minimum singular value, σ *

1.E-03

1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11

1.E-05

CS-2, 1.E-06 1.E-07 1.E-08

TCDS-2, 1.E-09 1.E-10 1.E-11

1.E-12

1.E-12 1.E-13

1.E-13 1.E-14

1.E-14 1.E-15

1.E-15 1.E-16 1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

Frequency, ω [ rad/h ]

1.E+20

1.E-16 1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Frequency, ω [ rad/h ]

Figure 7. Minimum singular value TCDS-1 and CS-1 (M1F2).

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Figure 11. Minimum singular value TCDS-2 and CS-2 (M1F2).

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1.E+12

1.E+08

Condition number, γ

Condition number, γ

1.E+12

TCDS-2,

1.E+04

1.E+08

CS-3,

1.E+04

CS-2,

TCDS-3,

1.E+00 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

1.E+00

Frequency, ω [ rad/h ]

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

Frequency ω [ rad/h ]

Figure 12. Condition number TCDS-2 and CS-2 (M1F2).

Figure 16. Condition number TCDS-3 and CS-3 (M1F2).

1.E+00 1.E-01 1.E-02

1.E+00

1.E-02

1.E-04

CS-4 Minimum Singular Value σ *

SVD (Figures 11 and 12) for the case of M1F2 and CS-2 shows the better control properties than the TCDS-2 because that scheme presents lower values of the condition number and higher values of the minimum singular value in comparison with TCDS-2 arrangement. Similar results can be obtained for the case study with mixture M2.

1.E-03

1.E-06

1.E-08

TCDS-4 1.E-10

1.E-12

1.E-04

1.E-14

1.E-06 1.E-07

CS-3,

1.E-16

1.E-08 1.E-09

TCDS-3,

1.E-12

1.E +1

6

1. E +1 4

1.E +1

1.E +1

2

0

1.E + 08

6

4 1.E +0

1.E +0

2 1. E +0

1 .E -0 2

1.E-11

1.E + 00

1.E-18

1.E-10

1 .E -0 4

Minimum singular value, σ *

1.E-05

Frequency,ω [rad/h]

1.E-13 1.E-14 1.E-15

Figure 17. Minimum singular value TCDS-4 and CS-4 (M1F1).

1.E-16 1.E-17 1.E-18 1.E-19 1.E-20

1.E+06

1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Frequency, ω [ rad/h ]

Figure 13. Minimum singular value TCDS-3 and CS-3 (M1F1). Condition number, γ

1.E+29

Condition number, γ

TCDS-4

1.E+05

1.E+36

1.E+22

1.E+04

TCDS-3,

1.E+03

CS-4

1.E+15

1.E+08

18 1. E +

1. E +

1.E +1

6

14

2 1.E +1

1. E + 10

8 1. E +0

4

1. E + 06

1.E +0

1.E +0

2

0 1.E +0

CS-3,

1 .E

1 .E

-0 2

-0 4

1.E+02 1.E+01

Frequency, ω[rad/hr] 1.E-06 0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000 1000000000 10000000000

Frequency, ω [ rad/h ]

Figure 18. Condition number TCDS-4 and CS-4 (M1F1).

Figure 14. Condition number TCDS-3 and CS-3 (M1F1).

1.E+01

1.E-01 1.E+00 1.E-01

1.E-03 1.E-02

Minimum Singular Value σ *

1.E-03

Minimum singular value, σ *

1.E-04 1.E-05 1.E-06

TCDS-3,

1.E-07 1.E-08

CS-3,

1.E-05

CS-4

1.E-07

1.E-09

TCDS-4

1.E-11 1.E-09 1.E-10

1.E-13

1.E-11 1.E-12

1.E-15

1.E-13

1.E-16 1.E-04

12

10 1.E+

1.E+

08

6 1.E +0

1.E+

4

02 1.E+

1.E +0

00 1.E+

1.E-15

1.E -0

1.E -04

2

1.E-17

1.E-14

Frequency, ω [rad/h] 1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12

1.E+14

1.E+16

1.E+18

1.E+20

Frequency, ω [ rad/h ]

Figure 19. Minimum singular value TCDS-4 and CS-4 (M1F2). Figure 15. Minimum singular value TCDS-3 and CS-3 (M1F2).

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1.E+06

Condition number γ*

1.E+05

CS-4 1.E+04

TCDS-4

08 1.E+

7

06 1.E+

1.E +0

05 1.E+

4 1.E +0

03 1.E+

2

01 1.E+

1.E +0

00

2 1.E -0

1.E+

3 1.E-0

1.E01

4 1.E -0

1.E+03

Frequency,ω [rad/h]

Figure 20. Condition number TCDS-4 and CS-4 (M1F2).

Results for TCDS-3 and CS-3 are displayed in Figures 13 and 14 (case M1F1). TCDS-3 exhibit similar values of ␴∗ than the conventional scheme (CS-3), and the complex arrangement presents similar values of condition number in comparison with the CS-3 scheme. In the case of M1F2, the TCDS-3 shows higher values of ␴∗ and lower values of condition number for the whole frequency range. Therefore, the coupled scheme is expected to require less effort control under feedback operation and it is better conditioned to the effect of disturbances than CS-3 sequence (Figures 15 and 16). One more time, we can say that complex distillation sequence offers better conditioning properties against model uncertainties and process disturbances than the conventional columns. Similar results can be showed for the mixture M2. In the case of TCDS-4 and CS-4, for the case M1F1, results are shown in Figures 17 and 18. CS-4 arrangement present higher values of the minimum singular value and lower condition number for the whole frequency range; therefore, it can be expected that CS-4 system exhibit better control properties than the other sequence under feedback control and it is better conditioned to the effect of disturbances than the other distillation scheme. For the case M2F1, Figures 19 and 20 show that at low frequencies TCDS-4 exhibit higher values of ␴∗ than the other scheme, but as the frequency increases, the minimum singular value decreases drastically, and the CS-4 offer the best values of this parameter. In the case of the number condition, TCDS-4 shows the lowest values at low frequencies.

Based on the trends observed, a distinction is given between the best control option for coupled scheme and conventional sequence. In the case of complex schemes, they have better control properties in comparison with conventional sequence, when coupled arrangement preferentially has side strippers (case TCDS-1 and TCDS-3). In this case, the arrangement with thermal coupling is expected to require less control efforts under feedback operation. However, if the complex scheme preferentially has side rectifiers, its control properties are worst in comparison with the conventional sequence (for example, TCDS-2 and TCDS-4). In general, the structure of the sequence affects the dynamic behaviour of the TCDS for the separation of multi-component mixtures. This result is similar to that reported by Segovia-Hern´andez et al. (2005) in the case of control properties of alternative sequences to the Petlyuk column (the dynamic behaviour depends on the topology of the scheme).

Closed-loop analysis According to previous studies in thermally coupled distillation sequences (Cardenas et al., 2005; Esparza-Herm´andez et al., 2005; Segovia-Hern´andez et al., 2007b, among others) TCDS options can have good dynamic responses in comparison to those obtained in the conventional distillation sequences considering heuristic pairings in the control loops, that is, distillation composition–reflux rate and bottoms composition–reboiler heat duty (Haggblom and Waller, 1992). The PI controllers were tuned by minimizing the integral absolute of error (IAE) (Stephanopoulos, 1984). Therefore, for each loop, an initial value of the proportional gain was set; a search over the values of the integral reset time was conducted until a local optimum value of the IAE was obtained. The process was repeated for other values of the proportional gain. The selected set of controller parameters was the one that provided a global minimum value of the IAE. Although the tuning procedure is fairly elaborated, the control analysis is conducted based on a common tuning method for the controller parameters. For the distillation sequences of Figure 2, the dynamic responses are showed in Figures 16 to 20 for the case of feed disturbance (when a 5% change in the feed composition of component A was implemented). For the case of component A, Figure 21 shows that both the CS and the TCDS successfully rejected the disturbance to bring the product composition back to its design value. However, the response of the TCDS was remarkably better. The integrated dis-

Figure 21. Dynamic responses of component A for feed disturbance (M1F1; Figure 2). Table 5. IAE results for CS-2 and TCDS-2 (M1F1)

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Sequence

Component A

Component B

Component C

Component D

Component E

CS-2 TCDS-2

7.13065 × 10−4 2.16125 × 10−4

2.70389 × 10−4 5.97292 × 10−5

2.92806 × 10−4 2.0778 × 10−5

4.77071 × 10−4 0.00269654

2.16417 × 10−4 2.8497 × 10−5

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Figure 22. Dynamic responses of component B for feed disturbance (M1F1; Figure 2).

Figure 23. Dynamic responses of component C for feed disturbance (M1F1; Figure 2).

Figure 24. Dynamic responses of component D for feed disturbance (M1F1; Figure 2).

Figure 25. Dynamic responses of component E for feed disturbance (M1F1; Figure 2).

tillation sequence gives an IAE value of 2.16125 × 10−4 which is lower than that of the conventional distillation sequence (IAE = 7.13065 × 10−4 ) (see Table 5). As a result, the integrated distillation sequence presents a better dynamic response. When a feed disturbance was implemented in both distillation sequences, the dynamic response of the component B in the conventional distillation sequence reaches the new steady-state faster, but it presents more oscillations. TCDS present better IAE value and dynamic behaviour (Table 5 and Figure 22). Figure 23 shows the responses of the component C in the TCDS and conventional con-

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figuration. The TCDS provided smooth response, while CS yielded high deviation. The IAE values showed TCDS = 2.0778 × 10−5 and CS = 2.92806 × 10−4 (see Table 5). For component D (Figure 24), the dynamic response of the integrated distillation sequence presents more oscillations and a rather poor behaviour with large settling times. The conventional sequence present better control properties and better IAE value (Table 5). Figure 25 presents a similar behaviour for composition E of the heat integrated distillation sequence and conventional arrangement. The new steady-state is obtained in a very short time.

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However, IAE value is better in the case of complex column (Table 5). In general, the dynamic responses of TCDS options can be better in comparison to the conventional distillation sequences. When other cases of study were subjected to the same test, similar trends on the dynamic responses of TCDS were obtained.

CONCLUSIONS An analysis on control properties of four-coupled distillation sequences that arise from modifications to conventional sequences for the distillation of five-component mixtures has been presented. Results from singular value decomposition indicate, in general, that thermally coupled distillation systems exhibit better control properties than conventional schemes. The results from the theoretical control properties indicate that the presence of interconnections does not necessarily provide operational disadvantages, as originally expected due to the resulting complex structural design. Also, in general, the thermally coupled distillation sequences outperformed the dynamic responses of the conventional distillation sequences under closed-loop fashion. The results also suggest that control properties are ruled by the number of side strippers; when the coupled scheme preferentially has side strippers, their control properties are better than the conventional arrangement. If the complex scheme preferentially has side rectifiers, its control properties are worse in comparison with the conventional sequence. In general, it is apparent that the presence of recycle streams instead of deteriorating the dynamic behaviour of thermally coupled distillation sequences may contribute positively to their dynamic properties.

ACKNOWLEDGEMENTS The authors acknowledge financial support received from Universidad de Guanajuato, Universidad Michoacana, Instituto ´ Technologico de Aguascalientes and CONCyTEG, M´exico.

NOMENCLATURE G V W

transfer function matrix matrix of left eigenvectors matrix of right eigenvectors

Greek Symbols ␥ ␴* ␴∗ ω

condition number maximum singular value minimum singular value frequency

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Manuscript received July 6, 2007; revised manuscript received September 26, 2007; accepted for publication October 2, 2007

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