Separator System with

Aug 12, 2006 - A Hierarchical Controller Design for a Reactor/Separator System with Recycle. Hiroya Seki* andYuji Naka. Chemical Resources Laboratory,...
2 downloads 27 Views 268KB Size
Download PDF
6518

Ind. Eng. Chem. Res. 2006, 45, 6518-6524

A Hierarchical Controller Design for a Reactor/Separator System with Recycle Hiroya Seki* and Yuji Naka Chemical Resources Laboratory, Tokyo Institute of Technology, 4259-R1-19, Nagatsuta, Midori-ku, Yokohama 226-8503, Japan

A hierarchical control structure that comprises regulatory and coordination control layers is designed for a reactor/separator process with recycle. At the regulatory control layer, the process is divided into subunits on the basis of the controlled group unit approach. In defining inventory controls for the subunits, Luyben’s rule (fixing one of the flows on the recycle path) is employed to reduce dynamic interaction between subunits. The higher level coordination control layer is designed on the basis of the time-scale separation principle, which slowly manipulates the recycle flow between the subunits. A control structure that results in a wider operation range and small dynamic interaction is proposed, and its control performance is compared with that of a conventional control structure through simulations. Introduction Chemical processes consist of a number of subunits that interact with each other, and needs for control structure design from a plantwide perspective have been well recognized.1-4 Control strategies designed for isolated units may sometimes fail in a plantwide context, especially for a process with recycle, because the presence of recycle loops can have a significant effect on process dynamics, such as an increase in the time constant and reduction in damping.5 Theoretically, multivariable control, which handles the whole process as a single unit, may give the best performance, but a completely centralized controller has a number of disadvantages, such as a high cost of modeling and difficulties in controller design and tuning, maintenance, and modification.3 Especially from the plant operators’ point of view, a control structure that encompasses a wide range of different units may be beyond understanding; transparency and intuitiveness of the control structure are important factors for operator acceptance and safe plant operation. In practical implementation, it is a convention to break down the process into subgroups with a well-defined function for regulatory control purpose. At the same time, control tasks are broken down into regulatory tasks and coordination tasks, and a hierarchical structure may be employed by putting the higher level coordination control layer above the regulatory control layer (Figure 1). Morari et al.2,6 developed mathematical measures to guide the breakdown of control tasks and the partitioning of the process. At the regulatory control layer, the controller design for each subunit becomes easier if we can focus on the single subunit only. For this purpose, it would be desirable to break down the process so that there may be little dynamic interaction between the subunits. At the higher level coordination control layer, the controller design reflects the rationale from the process design phase, which is in most cases based on steady-state considerations. Hence, a time-scale separation is often possible between the regulatory control layer, which handles fast dynamics on the time scale of seconds to minutes, and the coordination control layer, which handles slow phenomena on the time scale of hours. * To whom correspondence should be addressed. Tel: +81-45-9245258. Fax: +81-45-924-5270. E-mail: [email protected].

Figure 1. Hierarchical control structure.

Figure 2. Reactor/separator process with recycle.

In this paper, we address a hierarchical control structure design for a reactor/separator process with recycle shown in Figure 2. First, we employ a controlled group unit (CGU) approach7 for breaking down the plant into subunits for the regulatory control design. A CGU is defined as a group of interconnected units surrounded by a number of control valves: each CGU has an internal structure comprising several control and processing units and is equipped with the capability of material inventory control. The CGU concept was originally introduced to design operational procedures for startup of chemical processes. It helps to break down the process system into relatively independent subsystems and provides a basis for regulatory control design. In breaking down the process with the CGU approach and designing regulatory control, we put the emphasis on minimizing the dynamic interaction between the subunits. Then the coordination control layer is designed

10.1021/ie060538z CCC: $33.50 © 2006 American Chemical Society Published on Web 08/12/2006

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006 6519

on the basis of the time-scale separation principle of the control tasks to improve the overall control performance. The resulting controller turns out to be “self-optimizing”,8 which achieves economically acceptable steady state performance without the need for explicit optimization. This paper is organized as follows. In the next section, the reactor/separator process is described, including a brief review of the previous works. Next, the regulatory control structure design based on the CGU approach is described, followed by the design of the higher level coordination controller. Last, simulation results to compare with a benchmark control scheme are shown. Finally, conclusions are drawn. Reactor/Separator System with Recycle The process studied is a reactor and a distillation column interconnected with a material recycle, shown in Figure 2, which has been studied by several authors.5,9-12 An irreversible firstorder reaction A f B occurs in the continuous stirred tank reactor, and the reactor effluent, a mixture of A and B, is fed to the distillation column. The distillation column has 22 stages, including reboiler and condenser; liquid feed at stage 13; constant relative volatility, R ) 2.0; and constant molar flows. In the distillation column, the mixture of A and B is separated, and product B is withdrawn from the bottom. Unreacted A is recycled back to the reactor from the separator top. The reactor temperature is tightly controlled, and we assume isothermal reaction. In addition, the separator pressure is assumed to be kept constant, controlled by the overhead condenser. As a process model, nonlinear differential equations similar to those of Luyben13 are used. The control objective is a stable plant operation that keeps the composition of A in the product flow B at a specified value (xB ) 0.0105). Available measurements are the reactor holdup, MR; the separator bottom holdup, MB; the separator bottom composition, xB; and the condenser drum holdup MD. Only the separator bottom composition, xB, is measured, whereas the feed composition, x0; the reactor composition, xR, and the separator top composition, xD, are assumed to be unavailable for control, since composition measurements are sometimes expensive and unreliable. The manipulated variables are the reactor effluent flow, F; the separator bottom flow, B; boilup, V; the reflux flow, L; and the distillate flow, D. The fresh feed, F0, is assumed to be given and manipulated according to a production schedule or some other reason and treated as a measured disturbance. The material balance equations can be described as follows.10 Overall,

F0 ) B

(1)

F0x0 ) BxB + MRkxR

(2)

F0 + D ) F

(3)

F0x0 + DxD ) FxR + MRkxR

(4)

For the reactor,

where k is the kinetic constant. Figure 3 shows the conventional control structure, in which the three inventories MR, MB, and MD are controlled by F, B, and D, respectively; the bottom composition xB is controlled by the boilup, V; and the reflux, L, is manipulated by the L/F ratio controller in response to changes in the reactor effluent F (we adopt the following convention for the ratio control: writing

Figure 3. Conventional control structure for the reactor/separator process with recycle.13

the ratio between L and F as L/F means that L is manipulated to keep L/F constant). Luyben10 pointed out the problem with the conventional control structure that the constant reactor holdup may cause “snowballing” in the recycle flow, D, under the increase in the fresh feed, F0. He showed that in the limiting case of xD ≈ 1 and xB ≈ 0 the mass balance eqs 1-4 yield

D)

x0F02 kMR-x0F0

(5)

which implies that increasing the fresh feed, F0, increases the recycle flow rate, D, sharply. He introduced the so-called Luyben’s rule that one flow rate in a liquid recycle loop should be flow-controlled. He proposed a control structure in which the reactor effluent, F, is held constant, and the reactor level is controlled by manipulating fresh feed flow; throughput changes are accomplished by changing the setpoint of the reactor level controller. Wu et al.5,11 introduced a “balanced structure” that adjusts both the reactor holdup and the recycle rate to distribute the effects of the load disturbances evenly to the reactor and the separator units. They pointed out that the Luyben’s structure handles the load disturbance by the reactor alone and may result in the snowball effect in the reactor holdup, whereas the conventional structure handles the disturbance at the separator, leading to the snowball effect in the recycle flow rate. Larsson et al.12 argued that from a steady-state economics point of view, a liquid-phase reactor should be operated at maximum holdup to minimize the separator load and used the control structure shown in Figure 3 with a constant maximum reactor holdup. This operation policy is advocated by Ward et al.,14-16 who proposed a control structure design methodology of plants with recycle for a wide variety of process chemistry, considering the tradeoff between selectivity losses at high reactant conversion and costs of recycle at low reactant conversion. They showed that process chemistry is classified into “bounded” and “nonbounded” chemistry in the plantwide control context. For “bounded” chemistry, reactors should be always operated at their maximum capacity for economically optimal operation. Processes are called “bounded” if the selectivity increases with conversion for any recycle species. The simple reaction A f B involves the general case of a “bounded” chemistry. In this study, we follow their arguments that the reactor holdup and temperature are kept constant at their maximum. The control structure shown in Figure 3 will be used as a benchmark to evaluate control structures in the subsequent study.

6520

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006

Table 1. Nominal Operating Condition feed rate F0 (kmol/h) feed composition, x0 (mol of A/mol) kinetic constant, k (1/h) reactor holdup, MR (kmol) reactor composition, xR (mol of A/mol) reactor effluent, F (kmol/h) vapor boilup, V (kmol/h) bottom composition, xB (mol of A/mol) bottom holdup, MB (kmol) reflux, L (kmol/h) reflux drum holdup, MD (kmol) recycle (distillate) flow rate, D (kmol/h) recycle composition, xD (mol of A/mol)

460 0.9 0.34 2800 0.43 942 1280 0.0105 275 798 185 482 0.83 Figure 5. Control structure candidate S1: the constant recycle flow.

Figure 4. CGUs for the reactor/separator process with recycle.

The nominal operating condition is shown in Table 1, which is the same as that employed by Larsson et al.12 Control Structure Design First, the whole plant is divided into subunits, and regulatory control is designed for each subunit. Then a coordination control layer is designed on the basis of the time-scale separation principle of the control tasks. Design of Regulatory Control Layer: CGU Approach. For the sake of simplicity, breaking down the process into subunits, we employ the CGU approach rather than the mathematical guideline proposed by Morari et al.,2 since the decomposition by CGU can be done solely from the topological information of the plant. A CGU is defined as a group of interconnected units surrounded by a number of control valves, and it serves as a basis for design of inventory control. It can be combined with a contiguous CGU to make a combined CGU, and a whole plant can be viewed as one large CGU. For each CGU, we can assume the following: • There are at least one inlet stream and outlet stream, which can be flow-controlled; by definition, a CGU is a closed region surrounded by control valves. • At least one control valve or one combination of control valves on the CGU border has to be used to control the inventory of the CGU; otherwise, stability is not ensured. It should be noted that the above rules do not uniquely determine control structure. They serve only as a basis for further control structure design. For the reactor/separator system, it is straightforward to define CGUs: the reactor CGU and the separator CGU, as shown in Figure 4. When we consider the combined CGU comprising the reactor CGU and the separator CGU, it has one inlet stream, F0, which

has been assumed unavailable for feedback control, and one outlet stream, B. Since at least one flow has to be used for inventory control, the separator bottom flow B is the choice for inventory control of the combined CGU. In defining streams used for inventory control of each CGU, we introduce another design rule: At least one flow on the recycle path is not used for inventory control. By keeping constant one of the flows on the recycle path, we can cut off feedback effects from other CGUs on the recycle path, making controller tuning for each subunit easier and relatively independent. Otherwise, controller tuning would become highly interactive and complicated, because such feedback effects may differ for different controller tunings in other CGUs. This requirement, which is equivalent to Luyben’s rule,10 is used only for the regulatory controller design and will be eliminated eventually in realizing coordination control. Applying Luyben’s rule, we find that either the reactor effluent, F, or the recycle flow, D, is put on flow control, and one of these two flows has to be used for inventory control of the reactor CGU. The above design procedure gives the following two candidates for control structure: Structure S1: The reactor effluent, F, is used for inventory control of the reactor CGU, and the recycle D is flow-controlled. Structure S2: The recycle flow, D, is used for inventory control of the reactor CGU, and the reactor effluent, F, is flowcontrolled. For the reactor CGU, the regulatory control layer simply consists of a reactor level controller, which regulates the reactor holdup, MR, either by F or D. For the separator CGU, the two inventories, namely, the separator bottom holdup MB and the reflux drum holdup MD, have to be controlled. In addition, the separator bottom composition xB should be controlled because it is designated as a product specification. It may be possible to design multivariable control, but in this study, we employ multiloop PI controllers for the sake of simplicity: the bottom holdup MB is controlled by B, the reflux drum holdup MD is controlled by L, and the bottom composition xB is controlled by the boilup V. The resulting control structure candidates are shown in Figures 5 and 6. In fact, these two control structures have been already suggested and analyzed by Wu et al.5 Around the nominal operating condition, these control structures work quite well with small dynamic interaction between the CGUs. However, as Wu et al. pointed out, these structures can accommodate only small changes in the fresh feed. We leave the task of improving control performance of these structures up to coordination control layer. Design of Coordination Control Layer. The problem with the proposed control structures is that they can accommodate

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006 6521

Figure 6. Control structure candidate S2: the constant reactor effluent.

value of 4% shown by Wu et al. because we use the larger reactor holdup, but still the operability is poor when compared with the conventional structure. In the case of structure S2, the situation is even worse because the recycle flow D ) F - F0 is decreased for a larger F0 with a constant F; the structure S2 can accommodate only a 9.5% increase in the fresh feed. To prevent the process from filling up with A and to widen the operational range, we have to increase consumption of A in the reactor. From a design rationale of reactor operation, this can be achieved by increasing the recycle flow. Increasing the recycle flow D, which is richer in the reactant A (xD > x0) when the column is filling up with A (xD ≈ 1), results in an increased overall feed composition, x′, and a reduced residence time, τ.

x′ )

F0 D xD + x F0+D F0 + D 0

(8)

MR F0 + D

(9)

τ)

Figure 7. Step responses of the structure S1 to (20% changes in the fresh feed. Solid: +20% change. Dashed: -20% change.

only a small increase in the fresh feed. The distillation column begins to fill up with component A, which can be observed through the phenomenon that the separator top composition xD approaches 1, and a breakthrough of component A in the product flow occurs, as shown in Figure 7. Figure 7 shows the simulation results of the step responses of the process controlled by structure S1 to (20% changes in the fresh feed. PI control parameters are shown in Table 2. The response to -20% change in the fresh feed is quite satisfactory, but for +20% change, the column filled up with the component A. Since the bottom composition is PI-controlled, the controller continues to increase boilup V, endlessly trying to achieve the product specification. More precisely, we can describe this situation in the following way. From the steady-state mass balances 1-4, we have

F0(x0 - xB) MRk

(6)

F0 (x - xB) + xR D R

(7)

xR ) xD )

Equation 6 implies that the reactor composition xR becomes larger as F0 or x0 increases for a constant xB, then eq 7 implies that increasing F0 while keeping D constant results in a larger xD. The physical constraint of xD ) 1 is reached for F0 ) 520.4 kmol/h, which is an increase of +13% over the nominal fresh feed flow, F0 ) 460 kmol/h. This allowance is larger than the

Since consumption rate of A in the reactor can be calculated as kMRx′/(1 + τk), an increased recycle flow increases consumption of A. In the plantwide perspective, this is described in eq 7, which implies that a larger D leads to a smaller xD. Now, the problem is how we change the recycle flow. It would be possible to increase D according to F0 in a feedforward manner, but this scheme is vulnerable to unmeasured disturbances, such as feed composition and reaction kinetics changes. One of the reasonable ways would be to manipulate D according to L or V in the case of structure S1, and F in the case of structure S2, since the flows L and V, which are increased by the bottom composition controller when the column begins to fill up with A directly reflect the column load. Following the well-known practice in distillation column control, we employ a ratio control such as D/L or D/V for the S1 structure, F/V or F/L for S2, in this study. Then through steady-state analysis of the nonlinear process model, it has been found that structures S1 and S2 can accommodate more than a 100% increase in the fresh feed by the introduction of such ratio controls. Particularly, structure S2 with a F/L ratio control has the same steady-state property as the conventional structure. Note that only the physical constraint xD e 1 is considered in this analysis. In practice, such constraints as the liquid and vapor rates in the column limit the actual operable range of the system. Implementation of the ratio control, however, violates Luyben’s rule, which has been introduced to cut off dynamic interaction of subunits at the regulatory control layer. Here, we regard the ratio control as the task of coordination control layer and apply the time-scale separation principle. In many cases, we can expect that composition responses are much slower than total material responses, so coordination of the two subunits does not have to be fast. Indeed, we change D or F in response to L or V, but only through a heavy low-pass filter. For example, in the case of manipulating D in response to V, the ratio control is described, using the Laplace transformation, as

D(s) )

D h /V h V(s) TF s + 1

(10)

where TF is the filter time constant, and D h and V h are the nominal values of the recycle flow and the boilup, respectively. We set TF sufficiently large so that it may not affect the dominant dynamics of each subunit.

6522

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006

Table 2. PI Control Parameters for the Regulatory Control Layer

reactor level control separator bottom level control separator bottom composition control reflux drum level control

proportional gain

integral time, h

6.0 (kmol/h)/kmol 1.0 (kmol/h)/kmol 25000 kmol/h/(-) 30 (kmol/h)/kmol

0.5 0.5 1.0 0.1

Table 3. Relative Gain Analysis for the Ratio Control Structures S1

S2

D/L

D/V

F/L

F/V

2.04

2.04

-2.23

-1.02

In fact, a similar structure, which employs a D/L ratio control, has already been proposed by Wu et al.,11 but our approach is different from theirs in that we regard the ratio control as the task of coordination control layer and insert the low-pass filter for the time-scale separation purpose, which results in reducing dynamic interactions between the subunits and at the same time achieves a wider operation range. With the conventional structure, the recycle flow is automatically increased under a column load increase; the bottom composition controller increases the boilup, V, which is reflected to the reflux drum holdup, MD, and the reflux column level controller increases the recycle flow. In this case, interaction of the two subunits may be expected, because the action by the composition controller is fed back through the inventory control of the reactor, which manipulates the feed to the distillation column. As the coordination control layer, a more sophisticated controller that is equipped with constraint handling and optimization capabilities may be utilized instead of the simple ratio control. Also in this case, slow manipulation of interconnecting flows between CGUs may be incorporated. Screening by Relative Gain Analysis. It has been found that the above design procedure does not yet guarantee feasible control structure. A further screening by the following analysis reveals that structure S2 has a stability problem. With the assumption that the inventory controls for MD and MB are closed by L and B, we compare open-loop gains of the composition controller, that is ∆xB/∆V, for the cases with and without ratio control (Figure 8). The analysis is performed using the linearized model of the separator, neglecting the reactor part. Table 3 shows the calculation results of the relative gain; the relative gain is defined as the ratio of the open-loop steadystate gains with and without ratio control. Note that the relative gain presented here is different from Bristol’s RGA.17 For the S2 structure, the relative gain is negative, which implies that the composition controller tuned with a constant F becomes unstable once the ratio control is switched on. Physical interpretation is that open-loop, steady-state gains ∆xB/∆V and ∆xB/∆F have different signs and that the effect of the column feed F is significant. For the S1 structure, the relative gain is rather large (≈2), but this may cause little problem as long as it is positive, because a heavy low-pass filter makes the gain at higher frequencies negligible. Figure 9 shows the control structure S1 with a low-pass filtered D/V ratio control, which is used for the subsequent simulation studies. Optimality of the Proposed Control Structure. Here, we assume that the operational cost of this process can be approximated by the energy cost related to the distillation

Figure 8. Relative gain analysis. Comparison of gains with and without ratio control (e.g., structure S2 with F/V ratio control).

Figure 9. Proposed control structure (ratio control D/V with time-scale separation by low-pass filter).

Figure 10. Relation between the recycle flow, D, and the vapor boilup, V. Well-defined minimum exists for V.

column, and the optimal operation for a given fresh feed F0 is to minimize the vapor boilup V. Figure 10 shows the relation between the recycle flow, D, and the vapor boilup, V; with the three inventories and the bottom product composition regulated by F, B, L, and V under the nominal fresh feed condition, the steady states for various values of D are calculated using the nonlinear process model. There exists a well-defined minimum for V so that the corresponding recycle flow rate is the optimal one. Such a minimum boilup is calculated for various values of the fresh feed, F0, and compared with the boilup under the D/V ratio control in Figure 11. They almost coincide, only with a slight deviation at higher values of F0. This implies that the proposed controller realizes an almost economically optimal

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006 6523

Figure 11. Comparison of the boilup, V, from the proposed control structure and the minimum boilup for various values of F0. Solid: proposed structure. Dashed: the minimum boilup.

Figure 13. Step responses to -20% change in the boilup, V. Solid: proposed structure. Dashed: conventional structure.

Figure 12. Step responses to +20% change in the fresh feed. Solid: proposed control structure. Dotted: conventional structure.

operation without the need for explicit optimization and can be regarded as a self-optimizing controller.8 The conventional structure with the L/F ratio control is also self-optimizing.12 It should be noted that S2 structures with a F/L or F/V ratio control, which have been eliminated because of the stability reason, have the same steady-state properties. Simulations In the simulation studies, we compare the control performances of the control structure S1 shown in Figure 9 and the conventional structure by Larsson et al.12 shown in Figure 3. The PI control parameters for the reactor level, the separator bottom level, the reflux drum level, and the composition controls are the same for the two control schemes and are shown in Table 2. A measurement delay of 8 min is assumed in the composition measurement. The low-pass filter time constant for the proposed structure is set to TF ) 5 h. The ratio setpoints in the ratio controllers are employed from the nominal operating condition. Step responses to a 20% increase in the fresh feed and a -20% decrease in the boilup are shown in Figures 12 and 13, respectively. For the fresh feed change, the response of the proposed structure is improved from the one shown in Figure 7, with the ratio controller successfully preventing the column from filling up with A. Compared with the conventional structure, overall responses are slower due to the low pass filter in the ratio

controller, but from a plant operation point of view, this may be acceptable. However, if the recycle flow increase is too slow and the column fills up with A on its way to steady state, then we may have to tune the low-pass filter time constant. In this simulation example, the boilup V in the column shows an ∼40% increase for the +20% feed change. In practice, distillation columns are not designed to deal with a vapor load change as large as 40% so that the example here may be somewhat contrived to distinguish the two control schemes. Since we assume that the reactor is operated at its maximum holdup and temperature, the only way to avoid the vapor flow constraint is to reduce the feed rate. For this purpose, another coordination controller, which manipulates the fresh feed rate, F0, looking at the vapor rate and other constraints could be incorporated. The characteristics of the proposed control structure can be observed in the responses to the disturbance in the boilup, V. In the conventional structure, the disturbance that occurs in the separator propagates to the reactor through D and comes back through F, resulting in the larger fluctuation in the product flow, B. On the other hand, in the proposed structure, such a disturbance is diminished because the recycle flow, D, does not respond immediately. Note that the time scale in Figure 13 is shorter than that of Figure 12, and the responses at t ) 4h are still changing. Although results are not shown, the simulation calculations have been actually performed for a longer time range to confirm that a stable steady state really exists. In addition, eigenvalue analysis has been performed for the linearized model to show that the closed loop is, indeed, stable. Conclusion A hierarchical control structure for the reactor/separator system with recycle has been proposed, and its control performance is compared with that of the conventional structure. The proposed control structure comprises a lower-level regulatory control layer and a higher-level coordination control layer. To design the regulatory control layer, the whole process is divided into the reactor and separator subunits on the basis of the controlled group unit (CGU) approach. Regulatory controllers are constructed assuming one of the interconnecting flows between the subunits; namely, the recycle flow or the reactor effluent, is kept constant. In this way, dynamic interaction between the two subunits can be made small, and controller

6524

Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006

tuning for each subunit can be done relatively independently. To solve the problem of a narrow operational range, the ratio control is introduced as the task of the higher-level coordination layer. On the basis of the time-scale separation principle, the coordination layer controller slowly manipulates the recycle flow, which is assumed to be constant at the regulatory layer design phase, looking at the column load. The proposed hierarchical controller realizes such advantages as smaller dynamic interaction between the subunits, easier controller tuning, and a sufficiently wide operation range. In addition, the proposed controller turns out to be self-optimizing, which achieves economically acceptable steady-state operation without the need for explicit optimization. Although the reaction A f B treated here is very simple and may seem to bear no practical significance, the controller design methodology presented in this paper, that is, the reactor operation at its maximum capacity and manipulation of the recycle flow as the degree of freedom for coordination of different subunits, is believed to be widely applicable to many complex process chemistries that fall into the “bounded” category, as implied by Ward et al.14-16 Literature Cited (1) Foss, A. S. Critique of chemical process control theory. AIChE J. 1973, 19, 209-214. (2) Morari, M.; Arkun, Y.; Stephanopoulos, G. Studies in the synthesis of control structures for chemical processes. part I: Formulation for the problem. process decomposition and the classification of the control tasks. analysis of the optimizing control structures. AIChE J. 1980, 26, 220232. (3) Skogestad, S.; Postlethwaite, I. MultiVarible Feedback Control; John Wiley & Sons: New York, 1996. (4) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999.

(5) Wu, K. L.; Yu, C. C. Reactor/separator processes with recycle-1. Candidates control structure for operability. Comput. Chem. Eng. 1996, 20, 1291-1316. (6) Morari, M.; Stephanopoulos, G. Studies in the synthesis of control structures for chemical processes. part II: Structural aspects and the synthesis of alternative feasible control schemes. AIChE J. 1980, 26, 232-246. (7) Naka, Y.; Lu, M. L.; Takiyama, H. Operational design for start-up of chemical processes. Comput. Chem. Eng. 1997, 21, 997-1007. (8) Skogestad, S. Plantwide control: the search for the self-optimizing control structure. J. Process Control 2000, 10, 487-507. (9) Papadourakis, A.; Doherty, M. F.; Douglas, J. M. Relative gain array for units in plants with recycle. Ind. Eng. Chem. Res. 1987, 26, 12591262. (10) Luyben, W. L. Snowball effects in reactor/separator processes with recycle. Ind. Eng. Chem. Res. 1994, 33, 299-305. (11) Wu, K. L.; Yu, C. C.; Luyben, W. L.; Skogestad, S. Reactor/ separator processes with recycles-2. Design for composition control. Comput. Chem. Eng. 2003, 27, 401-421. (12) Larsson, T.; Govatsmark, M. S.; Skogestad, S.; Yu, C. C. Control structure selection for reactor, separator and recycle processes. Ind. Eng. Chem. Res. 2003, 42, 1225-1234. (13) Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers, 2nd ed.; McGraw-Hill: New York, 1990. (14) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Importance of process chemistry in selecting the operating policy for plants with recycle. Ind. Eng. Chem. Res. 2004, 43, 3957-3971. (15) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Novel reactor temperature and recycle flow rate policies for optimal process operation in the plantwide context. Ind. Eng. Chem. Res. 2005, 44, 6729-6740. (16) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Insight from economically optimal steady-state operating policies for dynamic plantwide control. Ind. Eng. Chem. Res. 2006, 45, 1343-1353. (17) Bristol, E. H. On a new measure of interactions for multivariable process control. IEEE Trans. 1966, AC-11, 133-134.

ReceiVed for reView April 27, 2006 ReVised manuscript receiVed July 11, 2006 Accepted July 18, 2006 IE060538Z