Decomposability of the Control Structure Design Problem of Recycle

Jun 8, 2009 - The control structure design is an important part of process synthesis tasks. The design becomes more difficult if recycle is present in...
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Ind. Eng. Chem. Res. 2009, 48, 6339–6345

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Decomposability of the Control Structure Design Problem of Recycle Systems Marcell Horvath* and Peter Mizsey Department of Chemical and EnVironmental Process Engineering, Budapest UniVersity of Technology and Economics, Mu¨egyetem rkp. 3, H-1111 Budapest, Hungary

The control structure design is an important part of process synthesis tasks. The design becomes more difficult if recycle is present in the process to be controlled. The objective of this work is to clear up if the control structure design should be completed for the whole process simultaneously or it can be decomposed into subproblems involving only one unit of the whole system. Investigations show that the recycle can have an effect on the control structure that complicates the design problem. Simple case studies and exhaustive investigation of the industrial control problem are completed. Our investigations prove that decomposability can be applied simplifying the proper pairing of the controlled and manipulated variables, and this can facilitate the control structure design problem. 1. Introduction Recycling is a frequently used solution in different technologies to utilize material and energy more efficiently. The recycling in a technology means, however, not only process design problems but also special controllability considerations. From a controllability point of view, recycling is a positive feedback (Figure 1), and in several cases it might cause special, sometimes catastrophic problems. With the increasing need for efficient use of energy and material and, on the other hand, more severe environmental regulations, the different industries tend to apply recycling more and more in existing and new technologies. These incentives urge the comprehensive investigation of the effects and problems of recycling, and among them is the investigation of the controllability problems. Preliminary conceptions for our work are based on books,1,2,7 software,33 and articles.9,13,15,19,20,22,23,27-29 Several authors3,4 already drew attention to the interaction between process and control structure designs. The control structure design problem is a complex task for multivariable systems, and this activity is, however, properly supported with different control structure design tools, for example, the work of Luyben.8 As a consequence, the control structure can be designed in the early stage of the process design. The controllability problems associated with the recycle were investigated by Luyben.10-12 He exhaustively studied the problem of the controllability of recycle systems and showed that the loop gain and the dynamic behavior of the units in the recycle loop basically determine the stability of the recycle system. He investigated also a reactor-separator system and demonstrated the so-called “snow-ball” effect that is due to the recycle. Mizsey et al.16 also analyzed transfer function models and also showed that the recycle loop gain has a primary responsibility for the stability of the whole system. They pointed out also the importance of the control of the recycle unit, because if the recycle stream is controlled, its time constant has a smaller effect on the range of the stability of the system. The authors applied the results of the transfer function models to a reactor/ distillation column system, and the theory was found to be applicable: if the recycle stream was controlled, such a control allowed handling the so-called snowball effect. * To whom correspondence should be addressed. Phone: +36-14633104. E-mail: [email protected].

Morud and Skogestad14 declared the recycle a positive feedback and investigated a recycle-separator system from the point of view of the stability and the poles of the plant. They recommended a systematic classification of the different effects of the recycle. Scali and Ferrari17 exhaustively analyzed the possibilities and the effect of the application of recycle compensators, showing the drastically changed properties of processes with recycle. The basic issue of these investigations was the possibility of decomposing the global system in a part without recycle and the recycle itself. Their simulation results showed that by applying a compensator the performance could be significantly improved and the interactions could be effectively reduced. Daoutidis and Kumar25 investigated the correlation between the recycle flow rates and the dynamic properties of the recycle system. It was found that when the recycle flow rate is significantly larger than the feed flow rate, the recycle network exhibits a time-scale separation: in the fast time scale the network shows fast dynamics with a weak coupling; in the long scale the weak interactions became significant, and the overall recycle network shows a slow core dynamics. The authors

Figure 1. Scheme of the recycle.

Figure 2. Investigated 2 × 2 MIMO system without recycle.

Figure 3. Investigated 2 × 2 MIMO system with recycle.

10.1021/ie800976v CCC: $40.75  2009 American Chemical Society Published on Web 06/08/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009 Table 2. IAE Values of a Single 2 × 2 System (Figures 4 and 5) IAE

Figure 4. 2 × 2 MIMO unit without recycle.

structure

without recycle

with recycle

G1 (DP) G1 (CP) G2 (DP) G2 (CP) G3 (DP) G3 (CP)

0.362 0.37 0.432 0.44 0.54 0.55

0.892 0.88 1.03 1.1 1.34 1.35

Table 3. Integral Absolute Errors of the Whole Connected System (Figures 6 and 7)

Figure 5. 2 × 2 MIMO unit with recycle.

Figure 6. Two connected 2 × 2 MIMO units without recycle.

structure

IAE without recycle

IAE with recycle

G1 (DP), G2 (DP) G1 (DP), G2 (CP) G1 (CP), G2 (DP) G1 (CP), G2 (CP) G1 (DP), G3 (DP) G1 (DP), G3 (CP) G1 (CP), G3 (DP) G1 (CP), G3 (CP) G2 (DP), G3 (DP) G2 (DP), G3 (CP) G2 (CP), G3 (DP) G2 (CP), G3 (CP)

0.552 0.569 0.573 0.58 0.672 0.68 0.682 0.688 0.735 0.742 0.75 0.759

1.595 1.587 1.583 1.588 1.762 1.82 1.712 1.755 1.95 1.977 1.912 2.015

but it can be supported with the modified analytical hierarchical approach (AHP). The plantwide decomposition method allows the prioritizing the design objectives, the operability constraints, and the alternative decompositions. This method does not indicate, however, the effect of the recycle between two large scale units; namely, how the control structure design tools can be used and the results they deliver may be evaluated. 2. Discussion

Figure 7. Two connected 2 × 2 MIMO systems with recycle.

developed a nonlinear supervisory controller based on low-order state-space realizations of the system that describe the slow dynamics. Dimian et al.18 investigated the possibilities of the integration of process design and controllability analysis in the case of a large plant, where recycle effects are significant. It was demonstrated how to create flowsheet alternatives with feasible plantwide control structures exploiting the connections and the interaction between the recycles. Open-loop and closed-loop analysis was carried out, and the properties of the investigated main and side columns were studied. Different alternative structures of the investigated system are possible, and the case of a flowsheet alternative with a shorter recycle path showed significant advantages. Vasbinder and Hoo24 studied the opportunities of the integration of the control structure synthesis and plant design, and they recommended a novel plantwide decomposition method. The automatic decomposition is the selection of the unit operations,

Since in complicated systems recycle exists and there are several operating units in the recycle loop, each unit can influence the operation of every unit in the recycle loop. As a consequence, it should be clarified for the sake of accurate control structure design if (i) the whole recycle loop should be investigated at the same time or (ii) there might be some decomposition possibilities of the control structure design problem. This question has not been clarified yet. The objective of this work is to clear it up if there might be any decomposition possibility, and this would make the control structure design problem much easier and simpler. Such a concept is followed during process design as well, so the same philosophy might be followed. 2.1. Effect of Recycle on the Decomposability. To clear up the effect of recycle on the decomposition option, first the effect of recycle on the control structure is to be studied. It has been already investigated by several authors (e.g., Luyben et al.), but its effect can be easily understood by a simple case study problem, that is, the investigation of control structure design tools in the presence and absence of recycle. The simplest

Table 1. Investigated Transfer Function Matrices for the Hypothetical System

[

]

1 10 10s + 1 1s + 1 -0.1s G1 ) e 10 1 1s + 1 10s + 1

[

]

1 10 1s + 1 100s + 1 -0.1s G2 ) e 10 1 100s + 1 1s + 1

[

10 1 e-1s e-0.1s 0.1s + 1 10s + 1 G3 ) 1 10 e-0.1s e-1s 10s + 1 0.1s + 1

]

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009

( )

Table 4. The Material Flow Rates of the System stream

main components

flow rate [kg/h]

1 2 3 4 5 6 7 8 9

benzene ethylene B, EB, DEB, TEB EB, DEB, TEB DEB, TEB rec. benzene rec. DEB EB TEB

6600 3200 50450 13950 8150 36500 4150 7350 2450

∂y1 ∂x1

( ) ∂y1 ∂x1

λ11 )

∂y1 ∂x1

x2

∂y1 ∂x1

y2

RGA ) Λ )

[

λ11 λ12 λ21 λ22

(1)

]

(2)

RGA can be determined directly from the steady state transfer gains: λ11,without recycle ) a11

a11 a11a22 ) a12a21 a11a22 - a12a21 a22

(3)

If recycle exists (Figure 3), the calculation of the RGA is different as follows:

Figure 8. Iinvestigated process.

)

y2

1 - a11

a11 1 - a11

(4)

a12a21 a11a22 - a12a21 a22 ) a12a21 a22 - a11a22 + a12a21 + a22 (5)

cases are selected for study, that is, 2 × 2 MIMO systems with one recycle loop. Figures 2 and 3 show the systems investigated. 2.1.1. Effect of Recycle on the Control Structure. Since the control structure design is supported with different design tools, if it is detected that the presence of recycle influences the value of such a design tools, it can be concluded that the recycle can change the control structure, for example, a different pairing of the controlled and manipulated variables is recommended. A simple design tool is selected for more detailed study. This is the relative gain array (RGA) that shows the steady state interaction between control loops.26 RGA values are compared for the nonrecycle (Figure 2) and recycle cases (Figure 3). If the two values differ from each other, the recycle has an effect on the controllability features and, as a consequence, on the control structure design. So, the RGA is determined for the nonrecycle and the recycle system, generally. The RGA (eqs 1 and 2) can be determined from the steady state transfer gains (aii) of the system. Equation 3 shows the calculation for the nonrecycle case:

( ) ( )

a11 -

x2

)

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)

a11 1 - a11

(

)

a11 a11a22 a22 + a12a21 + 1 - a11 a11a22 - a12a21 a11a22 - a12a21 a22 + a12a21 -λ11,withoutrecycle + a11a22 - a12a21 (6)

λ11,withrecycle )

(

)

It can be seen that in the case of recycle this controllability index differs from that of the nonrecycle system (eqs 6 and 3). This shows that the recycle has an effect on the value of this controllability tool, and it can have an effect on the control structure design. So, it can be misleading if the control structure is designed without consideration of the effect of the recycle. 2.1.2. Study of Decomposability. The decomposition option can be studied on case studies with known transfer functions. Several 2 × 2 units can be selected and connected in recycle to study its effect. Now, the effect of the recycle is studied in the dynamic domain. The systems selected for the investigation are shown in Figures 4-7.The following case studies are considered: (1) 2 × 2 unit without recycle (Figure 4), (2) 2 × 2 unit with recycle (Figure 5), (3) two connected 2 × 2 units without recycle (Figure 6), and (4) two connected 2 × 2 units with recycle (Figure 7). In the case of the two connected units, the simplest connection is selected, that is, one interconnecting stream. However, the recycle overlaps more than one unit. Such a recycle system is selected since these cases are quite typical for industrial systems, like reactor and separator units where one stream from the separator, usually the unreacted material, is fed back to the reactor.10 For such systems it should be investigated if the recycle changes the control structure of the units. This question is important since it decides if the control structure design problem can be decomposed or not. The load rejection is selected for the measure and classification of the different control structures. The load rejection is measured with the integral absolute error (IAE), and this is investigated. The different cases are ranked according to their IAE values. Figures 4-7 show the structure of the investigated MIMO systems.

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Table 5. Recycle Ratios and the Time Constants of the Industrial System

recycle ratio

characteristic time constant, first column [h]

characteristic steady-state gain, first column [%h/kmol]

characteristic time constant, second column [hl]

characteristic steady-state gain, second column [%h/kmol]

0.42 3.72 4.14

1.5 1.6 3 3.5

0.2 0.3 1.2 1.5

2 2.4 4 4.3

0.3 0.4 1.4 1.7

no recycle only DEB recycle only B recycle both recycles

Table 6. IAE Values of the Composition Control Alternatives IAE without recycle structure L-Q L-B R-B D-BR L-BR R-Q

first column 0.082 0.085 0.091 0.115 0.121 0.135

second column 0.850 0.075 0.072 0.970 0.113 0.069

third column 0.933 0.108 0.142 0.732 0.157 0.095

IAE with recycle first column 1.142 1.167 1.150 1.850 2.682 2.151

second column

third column

1.162 1.588 0.952 1.945 2.130 1.312

2.175 1.392 1.415 2.011 1.832 1.210

Table 7. Effect of the Structure of the First Column on the Global System, while the Second and the Third Columns Have Their Optimal Control Structures structure of the first column

IAE of the first column (recycle)

IAE of the global system

L-Q R-B L-B D-BR L-BR R-Q

1.142 1.150 1.167 1.850 2.682 2.151

2.528 2.573 2.657 2.663 2.791 2.830

Table 10. IAE Values of the Composition Control Loops at Different Control Structuresa IAE values of the global system with recycle second column

first column

best

2nd

3rd

4th

5th

worst

best 2nd 3rd 4th 5th worst

2.528 2.573 2.657 2.663 2.791 2.83

2.552 2.595 2.668 2.672 2.811 2.95

2.572 2.632 2.67 2.8 2.85 2.981

2.602 2.663 2.758 2.812 2.864 2.997

2.789 2.811 2.852 2.887 2.91 3.16

2.824 2.843 2.91 3.03 3.12 3.374

a

The third column is controlled with its best control structure.

Table 8. Effect of the Structure of the Second Column on the Global System, while the First and the Third Columns Have Their Optimal Control Structures structure of the second column

IAE of the second column (recycle)

IAE of the global system

R-B L-Q R-Q L-B D-BR L-BR

0.952 1.162 1.312 1.588 1.945 2.130

2.528 2.552 2.572 2.602 2.789 2.824

Table 9. Effect of the Structure of the Third Column on the Global System, while the First and the Second Columns Have Their Optimal Control Structures structure of the third column

IAE of the third column (recycle)

IAE of the global system

R-Q L-B R-B L-BR D-BR L-Q

1.210 1.392 1.415 1.832 2.011 2.175

2.528 2.542 2.559 2.592 3.655 2.712

Table 1 shows the transfer functions of the simple 2 × 2 units selected for the investigations. As far as the pairing is considered, in these 2 × 2 units two kinds of pairing can be considered: direct pairing (DP) when the first controlled variable is controlled with the first manipulated variable and the cross pairing. PID controllers are selected, and the Ziegler-Nichols8 cycling tuning method is used. Step disturbances in the inputs are applied and the integral absolute errors of the different pairings are measured. The calculations are carried out for all three transfer function matrices. Table 2 shows the results of the dynamic investigations, that is, the IAE values for Figures 4 and 5 considering all the three dynamic unit models shown in Table 1. The optimal control structures are chosen based on the minimal IAE values (shaded

Figure 9. IAE values of the composition control loops at different control structures (the third column is controlled with its best control structure). Table 11. IAE Values of the Composition Control Loops at Different Control Structuresa IAE values of the global system with recycle second column

first column

best

2nd

3rd

4th

5th

worst

best 2nd 3rd 4th 5th worst

2.542 2.594 2.671 2.715 2.88 3.125

2.568 2.612 2.71 2.834 2.922 3.243

2.585 2.654 2.731 2.912 3.093 3.311

2.681 2.72 2.858 3.056 3.113 3.426

2.91 3.073 3.167 3.231 3.295 3.661

3.052 3.095 3.267 3.481 3.588 3.825

a

The third column is controlled with its best control structure.

cells), although the similar values indicates that the investigated theoretical systems are highly decoupled and essentially noninteracting systems. It can be seen that the recycle changes the optimal control structure in G1, but in the other two cases it is the same both for nonrecycle and recycle cases. The same investigation is carried out for the two connected 2 × 2 units (Figures 6 and 7). Every combination of the unit models (Table 1) is considered. Also investigated is whether the positioning of the units has an effect on the IAE values.

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Figure 10. IAE values of the composition control loops at different control structures (the third column is controlled with its best control structure).

Table 3 shows the IAE values for the connected units in the case of step disturbance. Here the recycle changes the best pairing in every combination. On the other hand, it is also found that the units’ positioning has no influence on the control structure; it is always the same. That is, for the best control structure it has no influence on which one from the selected two simple units is positioned in the first place or in the second place. Therefore, in Table 3 G1 and G2 mean also G2 and G1 since they are commutable. The global optimal control structure for the system built from the connected blocks depends only on the local control structures of the different blocks, and it does not depend on the placement of the separate units. So the design task of the control structure for the whole system can be carried out by designing the local optimal structures for the different blocks individually. That is, the control structure design problem of the recycle system can be decomposed into the control structure design problem of individual units being in the recycle loop. 3. Decomposability Study of an Industrial Problem The ethylbenzene production technology is a good industrial case study for the investigation of the recycle systems since the technology consists of a continuously stirred tank reactor and three distillation columns with two recycle loops (Figure 8). The flow rates of the material streams of the system can be seen in Table 4. The investigations are carried out on the Aspen Plus30 and Aspen Dynamics31 models of the technology, respectively. Matlab32 is also applied for the identification of the system elements in the form of transfer functions. The temperature of the reactor is 180 °C, and the pressure is 10 bar. The applied catalyst is aluminum-trichloride. From controllability aspects the aim is to hold the output compositions of the system on their constant, prescribed values, set points. To design control loops, manipulated and controlled variables are selected. On the continuously stirred tank reactor (block B1) two control loops are considered: pressure and temperature controls. These control loops are automatically set up, and these are not the subject of further investigation. The subject of the detailed dynamic investigation is the three column separation system (blocks B2, B3, and B4). Table 5 contains the recyclerelated properties of the system, like the recycle ratios and the characteristic time constants in different recycle cases. Table 5 shows the relationship between the recycled material flows and the different parameters. As it can be expected the higher the

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recycle flow, the higher the time constants. During the investigation, the decomposability is selected as a major aim. 3.1. Pairing of Manipulated and Controlled Variables. The dynamic behavior of the system is modeled and investigated. Since the control structure of the reactor is given, the control structures of the three distillation columns are studied with dynamic investigations. The top and bottom product compositions are the controlled variables, and six frequently applied manipulated variable sets are considered (Table 6). In the course of the dynamic investigations, all six pairing possibilities are investigated with the dynamic tools of the ASPEN. PIDIncr. controllers are applied that are tuned according to the internal tuning method of the Aspen Dynamics Simulator,31 which is based on the identified transfer functions of the system. Step disturbances (1% of the feed flow rates) are applied, and the responses are registered. The responses of the output compositions are measured as a function of the time. The integral absolute errors of the control loops are calculated and can be seen in Table 6 for all three columns. The IAE values of the non-recycle systems and the recycle systems are compared, or all the six control structure versions investigated. The shaded cells show the minimal IAE values which represent the optimal control structures. The optimal control structure is changing in the case of recycle. This column is the most important block of the separation system, because the sharpest separation takes place there: the ethylbenzene product needs to be produced with a purity of 99.9%; therefore, this recycle-caused change of the optimal control structure is important. Another important fact is also concluded from the values of the IAE of the optimal control structures. It can be clearly seen that the IAE values are significantly higher in the case of recycle. This is in agreement with the general conclusion by several other authors that the recycle has a negative effect on the quality of the control.17 3.2. Investigation of the Decomposability. To further study of the decomposability, it is necessary to investigate if the individually determined optimal control structures for each column can form the overall optimum or not. If they form the overall optimum, it is possible to determine it for the recycle system with the application of the optimal control structure of the individual units. In such a case it results that the control structure design can be decomposed. To determine such an option further study is carried out, where feed flow rate disturbances of 1% are applied on each column and the IAE values of the composition control loops are investigated. The overall optimum is also determined. The objective function is the sum of the IAE values of each column. If the overall IAE value is minimal, the control structure is considered as the optimal one. The results of the investigation are shown in Table 7. The IAE values of the first column in the cases of the different control structures and the sum of the IAE values of each column are presented. During the investigation, if the control structure of column 1 is changed, the global optimum, that is, the sum of the IAE values of each column response, is newly determined. This also means that columns 2 and 3 always have their optimal control structure. The same investigation is carried out for columns 2 and 3 (Tables 8 and 9). Tables 7-9 show that replacing a control structure with a more unfavorable one results in a more unfavorable global system. The control loops of the first and the second have significantly stronger effect on the global system than the control

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loops in the third column. The global system is less sensitive to the control structure changes of the third column, since the third column operates with the smallest flow rates. The results emphasize the direct correlation between the optimality of the local and the global control structures. Additionally, to certify that the best control structure for the global system is built from the optimal control structures of the individual columns, the interactions between the control structures of the different columns have to be investigated. It means that a systematic measurement is required which simultaneously investigates the nonoptimal control structures on the different columns and checks whether it is possible to create a better control structure than the previously determined optimal one by pairing two worse control structures. For such an investigation columns 1 and 2 are chosen, and simultaneous measurements of the IAE values of the global system in the case of different control structures are determined. The effect of column 3 is investigated as a fixed parameter. Table 10 shows the IAE values at different control structure combinations for columns 2 and 3, while the control structure of column 3 is the best one determined in the previous steps. The results are also demonstrated in Figure 9. The same investigation is carried out for that case where column 3 is controlled with its second best controller pairing (Table 11, Figure 10). It can be seen that the global IAE values are always higher if column 3 is operated with its second best control structure. The IAE surfaces presented in Figures 9 and 10 are monotonously increasing. If a control structure is replaced with that one that has a higher IAE, that is, a control structure with worse performance, the point representing the IAE of the global recycle system has a higher value in the 3D plot. The control structure of the recycle system has the best performance if each unit being in the recycle loop is controlled with its best control structure, that is, that the control structure that is applied has the lowest IAE. 4. Results and Conclusions The complex investigation of the control structure design task is demonstrated and investigated for recycle systems. The investigation certifies that the control structure design task can be decomposed in the case of recycle. This means that the control structure design for the different units being in a recycle loop can be completed for each unit individually and the effect of the recycle on the control structure can be neglected. Such a decomposability makes the complex process design task more simple and facilitates the work of the design engineer. Acknowledgment The authors are grateful for the Grants OTKA 49849 and OTKA 46218 of the Hungarian Scientific Foundation and to Sanofi-Aventis Hungary. Nomenclature 2 × 2 ) two manipulated and two controlled variables B ) benzene BR ) boilup ratio CP ) cross pairing D ) distillate DEB ) diethyl-benzene DP ) direct pairing

EB ) ethylbenzene ETH ) ethylene F ) feed G ) transfer function (or transfer function martix) IAE ) integral absolute errors L ) reflux stream Q ) reboiler heat duty R ) reflux ratio RGA ) relative gain array TEB ) triethyl-benzene x ) input signal xF ) composition of the feed y ) output signal

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ReceiVed for reView June 23, 2008 ReVised manuscript receiVed May 9, 2009 Accepted May 18, 2009 IE800976V