Insight from Economically Optimal Steady-State Operating Policies for

Ind. Eng. Chem. Res. 2006, 45, 1343-1353

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PROCESS DESIGN AND CONTROL Insight from Economically Optimal Steady-State Operating Policies for Dynamic Plantwide Control Jeffrey D. Ward, Duncan A. Mellichamp, and Michael F. Doherty* Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106-5080

Shortcut methods for the rapid determination of the optimal steady-state operating policy for certain classes of chemical plants have recently been published [Ward et al., Ind. Eng. Chem. Res. 2004, 43, 3957; 2005, 44, 6729]. The purpose of this work is to investigate the implications of these optimal operating policies for the selection of control methodology and control structure design, and to demonstrate that the operating policies can be implemented dynamically. These methods can predict, using basic information about the process chemistry, whether a self-optimizing control structure will give good economic performance, or whether an explicit optimization layer will be required in the control hierarchy. When a self-optimizing control structure is sufficient, this methodology allows the engineer to anticipate which binary pairings between controlled and manipulated variables are optimal. When an optimizing control layer (e.g., MPC or real-time optimizer (RTO)) is required, the method suggests a low-order model that will capture the essential nonlinearities of the process and that can be used for the design of the optimizing controller. Dynamic simulations of two case studies confirm that the control methodologies and control structures, which are predicted best by the methods of Ward et al., do, in fact, give the economically best operation of the process, when implemented dynamically. 1. Introduction Page Buckley1 is generally considered to be the first author to address the problem of plantwide process control. He advocated breaking down the process control problem into separate systems for product quality and inventory control, with the assumption that the product quality control system will operate on a much different time scale than the material balance control system, and, therefore, the two systems will not interact significantly. Fisher et al.2-4 recommended breaking down the plantwide control structure design problem along the same lines as Douglas’ well-known hierarchy for process design.5 For a more thorough review of the field of plantwide control, the reader is referred to the review by Larsson and Skogestad.6 Luyben and co-workers7,8 identified a potential problem in plantwide operation and control: if the reactor level is controlled at a constant value, then a modest change in the production rate may cause a large change in the recycle flow rate. Luyben called this the “conventional” operating policy (or control structure). To alleviate this problem, he proposed that a stream somewhere in each recycle loop be controlled to a constant flow rate, and the reactor level be allowed to fluctuate. However, although Luyben’s method mitigates large swings in the recycle flow rate, it causes wide swings in the reactor holdup. As an alternative, Yu and co-workers9,10 recommend so-called balanced methods, which distribute the load from a production rate change evenly between the reactor and the separation system. It is unclear whether any of these methods (conventional, balanced, or Luyben’s) will be optimal, from an economic point of view. Skogestad and co-workers6,11 suggested that, in most cases, it was economically best to operate with the reactor completely full at all times (the conventional method). * To whom correspondence should be addressed. Tel.: (805) 8935309. Fax: (805) 893-4731. E-mail: [email protected].

Self-optimizing control is advocated by several authors12,13 as a way to design economically attractive control structures. The basic idea is to identify variables that, when controlled to constant values, give good economic performance over the expected range of disturbances and operating conditions. The term arises by analogy to a self-regulating process: the selfoptimizing controller optimizes the system automatically without the need for an explicit optimizing layer, just as a self-regulating process regulates itself without the need for an explicit feedback controller. Recently, Ward et al.14,15 derived heuristics for plantwide optimal operation based on analysis of a simplified steady-state cost model. Processes are classified on the basis of the process chemistry, and the optimal operating policy is predicted for each class of processes. In these previous papers, however, the emphasis was on the steady-state optimal operating point, rather than on the dynamics and control of the process. The purpose of this work is 2-fold. First, the implications of the optimal steady-state operating policy for the selection of a control methodology and the design of control structures are explored. When a self-optimizing control structure is appropriate, the methodology of Ward et al.14,15 can predict which pairing of controlled and manipulated variables will be best from the economic point of view. When a supervisory controller such as MPC or real-time optimizer (RTO) is most appropriate, the same methodology suggests a low-order process model that captures the essential nonlinearities in the process and can be used in the design of such a controller. Second, it is demonstrated through case studies that the operating policies advocated in previous papers by Ward et al.14,15 can, in fact, be implemented dynamically and that the method predicted to be the best does, in fact, give the best economic performance.

10.1021/ie050396t CCC: $33.50 © 2006 American Chemical Society Published on Web 01/18/2006

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Table 1. Summary of Derived Heuristics for Plantwide Operating Policy Based on Active/Inactive Constraints for Processes with One Recycle Stream Bounded Chemistry constraint

Vmax

RA, max

isothermal E′A,1 > 1 E′A,1 < 1

active activea active

inactive inactivea inactive

Nonbounded Chemistry

Tmin inactivea inactive

Tmax

Vmax

RA, max

Tmin

Tmax

activea active

may switch may switch inactive

may switch may switch may switch

may switch inactive

inactive active

a Note: Following the analysis in Ward et al.,15 there are certain cases where this heuristic is not valid, depending on the relative magnitude of the activation energies and the ratio of the exponents in the kinetic rate expressions.

2. Design of Control Structures Based on Heuristics for Economically Optimal Steady-State Operation Recently, Ward et al.14,15 advocated a new methodology whereby processes are classified and assigned to an economically optimal plantwide operating policy on the basis of the process chemistry. Processes are called bounded if the selectivity increases with conversion for any recycled species. If selectivity decreases with conversion for all recycled species, then the process chemistry is called nonbounded. Processes are further classified on the basis of the ratio of activation energies of the desired and undesired reactions. Table 1 gives a concise summary of the optimal operating policy as predicted by the derived heuristics of Ward et al.,14,15 in terms of whether certain process constraints are expected to be active. For example, if a bounded chemistry with one recycle stream is operated isothermally, it is now known that the reactor volume constraint will always be active, whereas the recycle capacity constraint will always be inactive. This operating policy makes sense for a bounded chemistry, because selectivity is improved by maximizing the per-pass conversion of the bounded species. At the same time, maximizing the per-pass conversion minimizes the recycle flow rate, which reduces the burden on the separation system. Surprisingly, the new method of analysis shows that, for bounded chemistries operated with variable reactor temperature, it is usually optimal to operate at the highest possible temperature, regardless of the relative magnitudes of the activation energies. The physical basis for this result is that the byproduct-producing reaction can be suppressed more effectively by minimizing the concentration of the bounded species than by manipulating the ratio of the kinetic rate constants via the temperature. More details about the analysis and intuition underlying the results presented in this table can be found in Ward et al.14,15 Note that the guidelines presented in Table 1 are not a substitute for sound engineering judgment, but rather should be used in conjunction with sound engineering judgment to determine the optimal operating policy. For example, the methodology requires that the engineer specify an upper bound on the reactor temperature. In so doing, the engineer must take into account several factors, including the materials of construction, the area available for heat transfer, the dynamic stability limitations, and the possibility of thermal runaway. Also, some recommended policies call for a significant decrease in the throughput of the separation system during a production rate decrease. The engineer should be mindful that it may be necessary to maintain the vapor flow rate through the column above some minimum level to avoid weeping. Table 1 is not only useful for understanding the optimal operating policy for a chemical plant, it can also guide the design of a control structure. Table 2 shows the optimal control structure corresponding to the operating policies given in Table 1. The entries in Table 2 are discussed below. The approach for determining the optimal dynamic control structure is similar to that of a self-optimizing control structure.

Table 2. Summary of Derived Heuristics for Control Structure Design for Processes with One Recycle Stream chemistry

bounded

nonbounded

isothermal E′A,1 > 1 E′A,1 < 1

conventional conventional conventional

balanced supervisory balanced

Skogestad identifies three classes of controlled variables: “constrained optimum” (where the optimum always lies at the constraint), “unconstrained flat optimum” (where the optimum lies away from constraints but process economics are insensitive to the exact value of the controlled variable), and “unconstrained sharp optimum” (where the optimum lies away from constraints and process economics are sensitive to the exact value of the controlled variable). The first two of these are identified as good choices for controlled variables, whereas the third is not. Returning to Table 1, if the process chemistry is bounded and the process is to be operated isothermally, then a selfoptimizing control structure can be implemented with one controlled variable at a constrained optimum: the reactor volume. (The reactor temperature is also held constant, but at a value that is not necessarily chosen for reasons of economic optimization.) Therefore, in Table 2, the optimal control structure is the conventional structure. If a bounded chemistry is to be operated nonisothermally, then a self-optimizing control structure can be developed with two constrained optimum controlled variables: reactor level and reactor temperature. As discussed previously, the reactor temperature is maintained at the highest possible value, regardless of the ratio of the activation energies. This operating policy also represents a conventional policy. If the process chemistry is nonbounded and the reactor is to be operated isothermally, then Table 1 indicates that both the reactor volume constraint and the recycle flow rate constraint may switch; that is, they are not guaranteed to be either active or inactive. Ward et al.14 have shown that, to a first approximation for such a process chemistry, it is optimal to scale both the reactor level and the recycle flow rate linearly with the production rate. In this case, a self-optimizing control structure can also be designed, where the ratio between the reactor holdup and the recycle flow rate is a controlled variable. Thus, for this class of process chemistries, the balanced control structure is recommended in Table 2. This policy describes an unconstrained optimization problem, and the control structure will be most effective if the economics are insensitive to errors in the implementation, i.e., if the unconstrained optimum is “flat”. If the process chemistry is nonbounded and the activation energy of the undesired reaction is smaller than that of the desired reaction, then a self-optimizing control structure can also be designed. In this case, the reactor temperature should be maintained constant at the upper limit (constrained optimum) and the ratio between the reactor holdup and the recycle flow rate should also be held constant (unconstrained flat optimum). On the other hand, if the process chemistry is nonbounded and the activation energy of the undesired reaction is greater

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than the activation energy of the desired reaction, and the process is to be operated nonisothermally, then there are no obvious candidates to use for controlled variables with self-optimizing control. The reason is that, for optimal operation, the reactor temperature setpoint must be coordinated with the recycle flow rate in a manner that proportional integral derivative (PID) controllers cannot achieve. In this case, a supervisory control structure such as model predictive control (MPC) or RTO is expected to give better economic performance than a selfoptimizing control structure. Supervisory optimizing controllers are not recommended for other classes of chemistries, because, in those other cases, it is always optimal to operate the process with the reactor temperature constant at the high constraint. With the self-optimizing control structures considered in this work, the reactor temperature is not changed, because there is no simple way to design a control structure using PID controllers that can manipulate it in a sensible way. Therefore, the supervisory control structure has an inherent advantage over the self-optimizing PID control structures, namely, that it can manipulate both the reactor temperature and the reactor holdup simultaneously. Therefore, it is not surprising that improved economic performance results. Exactly how much improvement the RTO provides will be dependent on the details of the process. Because the methodology of Ward et al.14,15 allows the engineer to anticipate what the best self-optimizing control structure will be, the best available self-optimizing control structure can be compared to the results from a real-time optimizer, to determine whether the additional cost and complication of an optimizing controller can be justified. Because these control structures are designed to accommodate a substantial change in the production rate (either up or down), and because the kinetics of the reactions and the temperature dependence of their rates are nonlinear, a supervisory controller must take into account nonlinearities in the process model, to give a good economic response. In such cases, it is often very helpful to have a low-order model of the process that captures the inherent nonlinearities. The analysis of Ward et al.14,15 provides exactly such a model. Thus, the plantwide operation methodology provides valuable insight and guidance to the engineer, regardless of whether an optimizing or self-optimizing control methodology will be used. A detailed discussion of how the methodology of Ward et al.14,15 can facilitate the design of an optimizing controller such as a real-time optimizer is given in Appendix A. The following sections present results from case studies which demonstrate the efficacy of the proposed method. The case studies fall into two different categories in Table 1: either the benzene chlorination process has a nonbounded chemistry with E′A,1 > 1, or the etherification process has a bounded chemistry with E′A,1 > 1. Therefore, on the basis of Table 1, it is expected that, for isothermal operation, it will be optimal to operate the benzene chlorination process with a balanced control structure, where the reactor holdup and recycle flow rates are scaled linearly with the production rate. In contrast, it is expected that it will be optimal to operate the etherification process using the conventional method, in which the reactor holdup is kept constant at the maximum value at all times. Furthermore, for the benzene chlorination process, a supervisory optimizing controller that can adjust the reactor temperature and reactor holdup simultaneously will give enhanced economic performance, whereas for the etherification process it will always be optimal to operate at the highest possible temperature, regardless of the production rate.

Table 3. Values of Kinetic Parameters for the Benzene Chlorination Process k value

activation energy

k0,0 ) 774 min-1 k1,0 ) 3400 min-1

EA,0 ) 33.9 kJ/mol EA,1 ) 42.7 kJ/mol

Textbooks by Douglas5 and Doherty and Malone16 were used as references in the steady-state conceptual design of these processes. The dynamic behavior of the processes was simulated in HYSYS. The recent book by Luyben17 was a valuable reference in constructing the dynamic simulations. 3. First Case Study: Benzene Chlorination 3.1. Description of the Process. This document presents a case study of a process to produce chlorobenzene that illustrates the use of reactor temperature as a degree of freedom in plantwide control. It is a generalization of a previous case study presented by Ward et al.,14,15 which also investigated a process to produce chlorobenzene but was limited to isothermal reactor operation. Chlorination reactions are used to introduce reactive sites on organic molecules. For example, one route in the production of phenol (C6H5OH) from benzene is via the intermediate chlorobenzene (C6H5Cl). In this case, care must be taken to minimize the production of higher chlorinated benzenes. The reactions are

benzene + Cl2 f chlorobenzene + HCl chlorobenzene + Cl2 f dichlorobenzene + HCl These reactions can be performed in a continuously stirred tank reactor (CSTR) in the liquid phase at 60 °C. Chlorine gas is introduced into the reactor and dissolved in the liquid by means of a sparger. We assume that dichlorobenzene has no value and must be disposed of safely, i.e., the second reaction is a representative byproduct-forming reaction. The reactor temperature and the per-pass conversion of benzene are kept low to suppress the undesired reaction. Unreacted chlorine, hydrogen chloride, and the catalyst are removed from the reactor effluent stream through stripping operations which are not included here. All three isomers of dichlorobenzene (ortho-, meta-, and para-) are expected to be formed; however, these species have very similar normal boiling points and, for practical purposes, will exit together when separation is accomplished by distillation. Therefore, we treat the isomers of dichlorobenzene as a single species. The development of this case study is similar to that described by Kokossis and Floudas.18 Silberstein et al.19 have reported kinetic data for the above reactions when catalyzed homogeneously by stannic chloride. The reactions are assumed to be third-order overall, being first order in the concentration of catalyst, chlorine, and benzene (or chlorobenzene). With the assumption that the catalyst concentration is constant at [SnCl4] ) 0.030 mol/L and the concentration of chlorine dissolved in the liquid is constant near the saturation point (which is assumed, for simplicity, not to be a function of temperature) [Cl2] ) 0.25 mol/L, the kinetic model for the process reduces to

r0 ) k0[benzene] r1 ) k1[chlorobenzene]

(k0 ) k0,0e-EA,0/(RT)) (k1 ) k1,0e-EA,1/(RT))

Values of the kinetic parameters are given in Table 3.

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Figure 1. Process flow diagram (PFD) for benzene chlorination.

With these simplifications, the process chemistry is, effectively, of the type A f B f C, which is nonbounded. Furthermore, the activation energy of the undesired reaction is greater than the activation energy of the desired reaction. On the basis of this information alone, it is possible using the methods of Ward et al.14,15 to anticipate the optimal operating policy (cf. Table 2): If the reactor is to be operated isothermally, then the best operating policy is a balanced policy that distributes the load of a production rate change between the reactor and the separation system. These results also suggest that, if the reactor temperature can be varied, then a supervisory controller such as a real-time optimizer will be required to achieve the most economical operation. Two distillation columns are used to separate the components of the reactor effluent. The first column separates benzene from the mixture, so that it can be recycled back to the reactor. The second column separates the desired product chlorobenzene from the undesired dichlorobenzene. The reactor considered in this case study is a single CSTR, although a plug-flow reactor (PFR) or a cascade of CSTRs would be expected to give a better selectivity-conversion profile. With the aforementioned assumptions, the process flowsheet corresponding to level 4 in the procedure devised by Douglas5 is shown in Figure 1. 3.2. Alternative Control Methodologies and Structures. The focus of this paper is on plantwide control methodologies.

Variables whose control (or manipulation) influences many or most of the unit operations in the plant, not merely the unit to which they are attached, will be denoted here as “plantwide” variables. The objective of this work is to investigate the influence that different pairings of the plantwide variables have on the operation of the plant. Therefore, for the purposes of comparison, the pairings between nonplantwide controlled and manipulated variables are not changed between control structures. This feature is shown diagrammatically in the piping and instrumentation diagram (P&ID) in Figure 2, where the nonplantwide variables (associated with the distillation columns) are depicted as already paired, with a suitable controller linking each controlled and manipulated variable. By contrast, the plantwide variables, depicted unpaired and without controllers, appear within the dashed box around the reactor. Thus, for comparison of alternate control structures, the only changes made to the control structure are made within the dashed box because these are the only variables that affect the plantwide operating policy. Control of the gas flow rate is not considered in detail in this case study, because the adjustment of the gas flow rate is not directly relevant to the implementation of the optimal operating policy. The gas flow rate must be adjusted to satisfy overall material balance. One possibility would be to implement a ratio controller, which adjusts the gas flow rate proportionally to the fresh feed flow rate of benzene. This could be augmented with a cascaded feedback controller, which measures the chlorine concentration in the liquid (or partial pressure in the vapor) and manipulates the setpoint of the ratio controller. It is assumed that the throughput manipulator for the process is the fresh feed of benzene into the reactor, i.e., a change in the production rate is accomplished by adjusting the fresh feed flow rate of benzene into the process. For specificity, two-point composition control is implemented on both distillation columns, although the plantwide control structure design method works equally well if one or both columns are operated with a constant

Figure 2. Partial piping and instrumentation diagram (P&ID) of benzene chlorination process before plantwide variables are paired.


Figure 3. Expanded diagram of the benzene chlorination process flow diagram showing how controlled and manipulated variables are paired to implement Luyben’s plantwide control method. Both the reactor effluent flow rate and the reactor temperature are kept constant.

ratio of reflux to distillate flow rate or reflux to feed flow rate, or if one or both columns use a control structure that infers the composition in the condenser and/or reboiler by measuring the temperature at one or more places in the column. Although not depicted in the diagram for clarity, 5-min and 1-min delays are associated with all composition measurements and temperature measurements, respectively. Also not shown is that the feed flow rate to the second column is fed-forward to the composition controllers, to provide enhanced disturbance rejection. Holdups in reboilers and condensers are controlled using proportionalonly control. The compositions at the top and bottom of each column are controlled with proportional-integral (PI) controllers; the tuning parameters were determined by sequential relay auto-tuning,20 using the auto-tuner that is built into HYSYS. Tyreus-Luyben tuning parameters21 were used to determine the controller gain and integral time constant from the ultimate gain and ultimate period. Figure 3 depicts only the dashed region in Figure 2, and it shows how controlled and manipulated variables are paired to implement Luyben’s recommended plantwide control method. The reactor temperature is maintained constant by manipulation of the cooling water flow rate; the reactor effluent flow rate is controlled to a constant value by a flow controller. Figure 4 shows how controlled and manipulated variables are paired to implement the balanced method. The reactor temperature is kept constant by manipulation of the cooling water flow rate. The reactor level is controlled by a proportionalonly controller that manipulates the reactor effluent flow rate. When both the controlled and manipulated variables are expressed as a percentage of their nominal values, the result is that both reactor level and reactor effluent feed flow rate change linearly with the production rate. This clever method of implementing the balanced method is due to Wu et al.10 Figure 5 shows how the real-time optimizer (RTO) was implemented. It accepts the current value of the fresh feed flow rate as input and determines the optimal steady-state values of the reactor temperature and the reactor level. The setpoint for the reactor level was fed through a 1-h lag to prevent abrupt changes in the reactor level setpoint. As discussed previously, the RTO has an inherent advantage over the self-optimizing proportional-integral-differential (PID) control structures, namely that it can manipulate both the reactor temperature and the

Figure 4. Expanded diagram of the benzene chlorination process flow diagram showing how controlled and manipulated variables are paired to implement the balanced method. The level controller is P-only, which causes the reactor level and reactor effluent flow rate to scale linearly with the production rate. The reactor temperature is kept constant.

Figure 5. Expanded diagram of the benzene chlorination process flow diagram showing how the real-time optimizer (RTO) is implemented. The optimizer accepts as input the fresh feed flow rate and determines the optimal steady-state values of the reactor level and temperature.

reactor holdup. Further details concerning the design of the RTO are available in Appendix A. 3.3. Dynamic Responses. Figure 6 illustrates the dynamic responses for the benzene chlorination process following a 50% decrease in the production rate using three plantwide control structures (operating policies): Luyben’s method (blue), a balanced method (red), and RTO (black). The conventional method (where reactor holdup is kept constant) is not shown on this diagram, because it is not operable for a production rate change this large. However, results from a smaller step change show that the conventional method is inferior to both the balanced method and Luyben’s method, in terms of economics. The decrease in fresh feed flow rate from 50 kmol/h to 25 kmol/h is ramped over a period of 6 h. For both the balanced method and Luyben’s method, the reactor temperature is kept constant. Compared with Luyben’s method, the balanced method causes a less-severe reduction in the CSTR holdup by decreasing the reactor effluent and recycle flow rates. The balanced method is predicted by our methodology as the economically best isothermal method, and this result is borne out by the plot of economic potential versus time. Luyben’s method produces the least amount of byproduct, because the per-pass conversion is

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Figure 6. Dynamic response of the benzene chlorination process to a 50% decrease in production rate using three alternative control methodologies. As anticipated, the balanced control structure gives the best economics for isothermal operation, and a supervisory optimizer can further improve performance under nonisothermal operation. [Note: A line that is not visible on the diagram is obscured by another line. For example, all three dynamic responses are forced by the same change in the fresh feed flow rate. Therefore, all three lines lie on top of each other.]

kept low but expends far too much energy within the separation system. All methods are able to facilitate a smooth transfer between steady states and to reject disturbances while keeping the product purity within specifications. The analysis also predicts that a supervisory, optimizing controller such as a RTO will give improved economic performance for a chemistry of this type if the reactor can be operated nonisothermally. The RTO simultaneously optimizes the reactor temperature and reactor level in a manner that a PID control structure cannot, operating the plant in a substantially different manner than either of the PID self-optimizing control structures. The production rate decrease is accommodated by decreasing the reactor temperature as well as the recycle flow rate. The result is that the reactor holdup does not change substantially. The RTO yields only a modest improvement in economic performance, compared to the best available self-

optimizing control structure. The engineer would have to use his or her judgment to determine whether this improvement justifies the expense and complication of designing, implementing, and operating an RTO. 4. Second Case Study: Etherification 4.1. Description of the Process. Alkyl ethers are used as fuel additives in gasoline to improve the combustibility and octane number and to meet legislative requirements for oxygenate content. In the early 1990s, methyl tert-butyl ether (MTBE) was thought to be a valuable additive for this purpose. However, one critical drawback of MTBE is that it is moderately soluble in water: 4.3 wt %. Since its introduction, MTBE has been detected in the groundwater in many communities. Consequently, MTBE is being phased out as a gasoline additive in California, and other states may follow.


Figure 7. 2-methoxy-2-methylheptane. Table 4. Values of Kinetic Parameters for the Etherification Process k value

activation energy

k0,0 ) 7.5 × 1011 mol/(kg cat s) k1,0 ) 2.1 × 1011 mol/(kg cat s)

EA,0 ) 90 kJ/mol EA,1 ) 102.6 kJ/mol

The solubility of ethers in water decreases with increasing molecular weight, so one possible way of overcoming this drawback is to use ethers with higher molecular weights. Recently, Krause and co-workers22-25 have investigated the kinetics of the production of high-molecular-weight ethers from C6 to C8 alkenes and C1 to C4 alcohols. In this work, we consider the production of 2-methoxy-2-methyl heptane (MMH, Figure 7) from 2-methyl-1-heptene (MH) and methanol (MeOH). In the development of this case study, commercial processes to produce MTBE and TAME (tert-amyl-methyl-ether) are considered as prototypes. The MTBE process is conducted in the liquid phase, typically with a heterogeneous acid-resin catalyst and at temperatures in the range of 60-80 °C. The selectivity of the MTBE process is typically quite high (99%) with the principle byproducts being dimethyl ether (DME) and tert-butyl alcohol, which is formed from the water produced in the dehydration of MeOH. The hydrocarbon feed to the MTBE process is typically a mixture of alkane and olefin. Therefore, it is assumed that the hydrocarbon feed to the etherification process under study is also a mixture of alkenes and alkanes. This feature is represented in the case study as a stream that is comprised of 80 mol % 2-methylheptane and 20 mol % MH. Because of the difficulty of separating the alkane from the alkene, unconverted alkene is not separated and recycled back to the reactor, but rather is fed, along with the alkane, to a downstream blending process. A requirement of the blending process is that the hydrocarbon stream contain no more than 2 mol % alkene. Therefore, it is necessary to achieve an alkene conversion of ∼90%, but there is no incentive to go beyond this value. Unfortunately, highermolecular-weight olefins are considerably less reactive than isobutene in etherification reactions. To achieve high conversion with a reasonable reactor residence time, it is necessary to operate the process at a higher temperature than the MTBE process, and with a molar excess of methanol. Both of these considerations have a tendency to promote the undesired reaction and make selectivity losses a greater concern.

Figure 8. Etherification PFD.

Krause and co-workers have developed sophisticated kinetic models for these types of reactions, based on the activity of the species in solution and assuming either a LangmuirHinshelwood or Eley-Rideal-type reaction mechanism. As is the case for the MTBE process, the reactions are typically equilibrium-limited and high conversion can be achieved using multiple reactors with intervening distillation columns between them to remove the desired product.26,27 However, such a reactor network is too complicated for the pedagogical purpose of this case study. Therefore, the reactions are assumed here to be kinetically limited over the range of conversion of interest. Data reported by Krause and co-workers and/or data generated from the models suggested by Krause and co-workers were fit to a pseudo-homogeneous model of the form

MH + MeOH f MMH (r0 ) k0xMHxMeOH; k0 ) k0,0e-EA,0/(RT)) 2MeOH f DME + H2O (r1 ) k1x2MeOH; k1 ) k1,0e-EA,1/(RT)) MH + H2O f MHOH

(r2 ) fast)

where x is the mole fraction of species. Values of the derived parameters are given in Table 4. Because the per-pass conversion of the alkene is fixed, the process chemistry is effectively of the type A f C, A + A f D, where A is methanol, C is the desired ether, and D represents undesired byproducts. Therefore, the etherification chemistry is a bounded chemistry. Furthermore, the activation energy of the undesired reaction is greater than the activation energy of the desired reaction; therefore (cf. Table 1), it is expected that the optimal isothermal operating policy will be with the reactor holdup and reactor temperature at their maximum possible values at all times. It is assumed that the acid-resin catalyst used in these reactions has a cost of $10/kg and will deactivate at temperatures >120 °C. These values are representative of typical commercial acid-resin catalysts, such as Amberlyst. Figure 8 shows the process flow diagram corresponding to level 4 of the Douglas procedure.5 DME is, by far, the lightest species in the reactor effluent and can be removed by a flash unit or short distillation column (column 1 on the process flow diagram). This unit is not expected to contribute significantly to either the economics or the dynamics of the process; thus, it is neglected in the economic potential calculations and represented by a split block in the dynamic simulations. Separation of the alkene from the alkane by distillation is not practical, as already noted; thus, in effect, a single hydrocarbon species enters the separation system. A single

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Figure 9. Partial P&ID etherification process before plantwide variables are paired.

Figure 10. Expanded diagram of etherification PFD for the conventional method and the balanced method. The conventional method uses PI reactor level control to keep the reactor level constant. The balanced method uses proportional-only control so that both the reactor level and reactor effluent flow rate scale linearly with the production rate.

azeotrope exists in the system: a binary, low-boiling azeotrope between MeOH and hydrocarbon, at a composition of 86.5 mol % methanol and a temperature of 61.4 °C (at 1 atm). The azeotrope is an unstable node. Thus, it is possible to design a column in which a mixture with the azeotropic composition is removed from the top, and with essentially no methanol present in the bottoms product (column 2 in Figure 8). The methanolrich azeotropic mixture is recycled back to the reactor; the bottoms product is sent downstream for further processing. The remaining species (hydrocarbon, ether, and octyl alcohol) form essentially ideal mixtures that can be approximated with constant relative volatility models. Column 3 separates the spent hydrocarbon from the ether and octyl alcohol. The distillate hydrocarbon stream is sent to a downstream blending process. The bottoms product is sent to column 4, where the desired

Figure 11. Expanded diagram of the etherification PFD for Luyben’s plantwide control method.

product is collected in the distillate, and the undesired octyl alcohol is recovered at the bottom of the column and diverted to the process fuel line. Columns 3 and 4 were designed and costed using the shortcut methods suggested by Doherty and Malone.16 Column 2 was designed by trial-and-error using HYSYS, and costed according to the methods suggested by Doherty and Malone.16 It was determined, for column 2, that a reflux ratio of 0.5 and eight ideal stages (16 real stages) was sufficient to achieve the desired separation over the expected range of inlet compositions. Column 4 is also small, compared to columns 2 and 3, and is not expected to influence the dynamics of the process substantially. Therefore, it is also represented as a split block in the dynamic simulation. 4.2. Alternative Control Methodologies and Structures. Figure 9 shows a partial P&ID for the etherification process, in which the non-plantwide variables are shown paired with a


Figure 12. Dynamic response of the etherification process to a 50% decrease in production rate using three alternative control methodologies. As anticipated, the conventional control structure gives the best economics. Because the optimal operating point is always located at the intersection of process constraints, very good economic operation can be achieved with only a self-optimizing feedback control structure.

suitable controller linking each controlled and manipulated variable. As with the benzene chlorination process, the plantwide variables are depicted unpaired and without controllers. Columns 1 and 4 are modeled as instantaneous split blocks. Columns 2 and 3 are modeled with rigorous dynamic models and utilize two-point composition control. For enhanced disturbance rejection, the bottoms composition controller is cascaded onto a faster-acting reboiler temperature controller. As with the benzene chlorination process, 5-min and 1-min delays are associated with all composition and temperature measurements, respectively. Figure 10 illustrates an expanded P&ID of the plantwide variables, showing the pairings used for both the conventional and balanced methods. In both cases, the reactor temperature is controlled to a constant value and the reactor effluent is manipulated to control the reactor level. With the conventional method, a PI controller is used and the reactor level is kept constant at its maximum value (zero offset at steady state). With the balanced method, the reactor level controller is proportional only with a gain of unity (the controlled and manipulated variables are expressed as a percentage of steady-state values),

so that both the reactor holdup and recycle flow rate scale linearly with the production rate. Figure 11 shows the expanded P&ID for the controller configuration which implements Luyben’s recommendation that a flow rate somewhere in the recycle loop is controlled to a constant value. The recycle flow rate (stream 6) is chosen rather than the reactor effluent (stream 3), because the process has a very limited range of operability if the reactor effluent is flowcontrolled, because the majority of the material flowing through the reactor effluent stream is inert alkane. If the production rate is decreased, the fresh feed flowrate of alkane must be decreased. Therefore, the flowrate of alkane in the reactor effluent must also be decreased, and it is impossible to compensate for this decrease by increasing the recycle flow rate of methanol unless the production rate change is very small. As a result, any control structure in which the reactor effluent flow rate is maintained constant will be operable only over a very small range of process conditions. Controlling the recycle flow rate instead of the reactor effluent flow rate is a valid implementation of Luyben’s method. It ensures that the load of a production rate change is

1352


borne by the reactor rather than the separation system, which is what Luyben’s method is intended to do. Note that no RTO is implemented in this case study, because it would provide no added benefit over the best self-optimizing control structure: the optimal operating policy holds the reactor temperature constant at the maximum possible value. Therefore, an RTO that has been designed and implemented correctly would implement exactly the same operating policy as the conventional method. Thus, its use is not considered in this case study. 4.3. Dynamic Responses. Figure 12 shows the dynamic responses of the etherification process to a 50% decrease in the production rate, starting from an optimal operating point, using three different control structures: the conventional method (green), the balanced method (red), and Luyben’s method (blue). All three structures are operated isothermally, because, for chemistries of this type, it is always best to operate at the highest possible temperature. All three control structures facilitate a smooth transition between steady states and keep the product quality within the required range. However, the use of different control structures leads to very different steady states, as anticipated. Luyben’s method results in the greatest recycle flow rate and the greatest production rate of byproduct, and, therefore, it is the least attractive economically. The balanced method leads to an intermediate value of both the recycle flow rate and byproduct production. The conventional method, which is predicted to be best for a chemistry of this type, produces the least byproduct and results in the lowest recycle flow rate; therefore, it also exhibits the best economic performance. 5. Conclusions This work has explored the dynamic implications of the steady-state operating policies published by Ward et al.14,15 The intuition developed in these papers can facilitate the design of control structures that are not only stable and operable over a wide range of process conditions, but also are (almost) economically optimal. The procedure is as follows: (1) Determine the optimal operating policy on the basis of the process chemistry. (2) Decide whether a self-optimizing control structure can implement the optimal operating policy, or whether a supervisory controller is necessary. (3) Design the simplest control structure that implements the optimal operating policy. This methodology has the advantage that it can provide valuable insight into the nature of the optimization problem and its solution, while, at the same time, it is simple enough that it could be taught to undergraduate students or to plant operators or engineers. The method allows the engineer to anticipate what the optimal control methodology and control structure will be with only minimal information about the process chemistry. The method has been demonstrated through application to realistic case studies that verify the anticipated results. Appendix A. Design of the Real-Time Optimizer Generally speaking, supervisory controllers such as the MPC and the real-time optimizer (RTO) require a mathematical model of the process that they are intended to optimize. Because a properly designed control structure should accommodate a substantial change in the production rate, and because the kinetics of the reactions and the temperature dependence of their rates are nonlinear, it is expected that a supervisory controller must utilize a nonlinear process model to give a good economic response.

Table 5. Process Variables Measured and Incorporated into the Steady-State Model for the Real-Time Optimizer (RTO) variable

meaning

units

V RA PC T PD Qreb Qcond FA

reactor holdup recycle flow rate production rate of desired product reactor temperature byproduct production rate reboiler energy consumption rate condenser energy consumption rate fresh feed flow rate

m3 kmol/h kmol/h K kmol/h W W kmol/h

Although, in principle, it is possible to develop a so-called black-box model of the process with no knowledge whatsoever about its internal workings (e.g., artificial intelligence techniques), the task generally will be easier (and the result safer to extrapolate) if the designer can incorporate engineering intuition and knowledge to develop a low-order mathematical model of the process that captures the essential nonlinearities. For the class of problems studied here, the analysis of Ward et al.14,15 provides such a model. The intention in this work is not to develop the “best” optimization scheme, but merely to demonstrate that a supervisory optimizing controller can give improved economic performance for this process. Therefore, the simplest optimizer that demonstrates this characteristic is designed and used. For this work, designing the RTO consisted of four steps: developing a low-order process model, fitting the model to plant data, optimizing the model with respect to an economic objective function, and expressing the results of the optimization in the form of a lookup table. Steady-state values of eight process variables (Table A1) over a range of different steady states were used to fit the following model:

V ) (p1RA + p2PC)ep3/T PD ) p4ep5/T

() PC RA

() PC RA

(A1)

2

(A2)

Qreb ) p6 + p7(PC + RA)

(A3)

Qcond ) p8 + p9(PC + RA)

(A4)

F A ) PC + PD

(A5)

Equations A1 and A2 are based on the expressions of Ward et al.15 for the reactor volume and byproduct production rate for a process with one recycle stream and with the reactor operated nonisothermally. Equations A3 and A4 reflect the assumption that the cost of operating the separation system scales linearly with the feed flow rate into the separation system. Equation A5 is a global material balance. The unknown parameters p1, ..., p9 were determined by minimizing the sum of the squares of the errors (mismatch) between data and model for a range of steady states spanning the expected operating region. The model has eight variables and five equations; therefore, it has three degrees of freedom. One of these degrees of freedom (FA) is used to set the production rate and is not available for optimization. The RTO is designed to predict the optimum values of volume V and temperature T for a given value of FA. Because the RTO uses a purely steady-state optimization, the optimization problem can be solved off-line in advance. The objective function C is given by

C ) PCCC + PDCD + QrebCstm + QcondCcw

(A6)


where C is the cost of operating the plant (smaller values of C imply larger profit); CC is the negative of the revenue obtained from producing one mole of desired product (CC < 0); CD is the cost of producing one mole of undesired byproduct (including raw materials costs and separations costs) (CD > 0), and Cstm and Ccw are the costs of steam and cooling water, respectively (Cstm > 0, Ccw > 0). Thus, the optimization problem, for a given value of FA, is

V h, T h ) arg min C(V,T)

(A7)

subject to the equality constraints in eqs A1-A5 and the inequality constraints:

V e Vmax

(A8)

Tmin e T e Tmax

(A9)

The RTO determines the optimal steady-state values of V and T, given a value of FA. These offline results are stored in a lookup table, which is accessed by the RTO during operation to determine the setpoints for the reactor temperature and reactor level controllers, as functions of the fresh feed flow rate. To identify parameters for the steady-state model of the process, the process was operated at (and transitioned between) more than twenty different steady states spanning the entire feasible operating region. When the controllers were tuned in a reasonable manner, the process was seen to be stable at a single steady state at all points within the feasible region, and at no time was there any evidence that a bifurcation point was encountered. This suggests that, at least for this process, bifurcation analysis would not provide further insight into the dynamics of the process. Appendix B. Definition of Instantaneous Profitability for the Dynamic Behavior of the Economic Potential To compare control methodologies in terms of their economic performance, it is useful to define a quantity that is representative of the “instantaneous” economic potential of the process, with units of $/h. However, if the process is not at steady state, it is important to account for material accumulation within the process; otherwise, the dynamic response of the economic potential may be misleading. For example, if there is an increase in the reactor holdup during a dynamic transient, it is necessary to add the value of the material accumulated in the reactor back into the expression for the economic potential. If the expression for the economic potential includes merely the flow rates into and out of the process, then the process will be unfairly penalized for this accumulation. Similarly, if the reactor holdup is depleted during a dynamic transient, then it is necessary to subtract the value of the material depleted; otherwise, the transient process will seem to be more profitable than it really is. The steady-state economic potential of the process is given in eq A6. To form the instantaneous dynamic economic potential, a term is added to compensate for inventory changes:

Cdyn ) PCCC + PDCD + QrebCstm + QcondCcw +

∑CiA˙ i

(B1)

where A˙ i is the rate of accumulation of species i and the summation is over all species and all locations where material may accumulate. In practice, the accumulation in many unit operations is so small that it may be neglected. In the case

studies considered here, the only significant holdup of material is in the reactor; therefore, only the reactor holdup is considered in the modified economic potential equation. Literature Cited (1) Buckley, P. S. Techniques of Process Control; Wiley: New York, 1964. (2) Fisher, W. R.; Doherty, M. F.; Douglas, J. M. The Interface between Design and Control. 1. Process Controllability. Ind. Eng. Chem. Res. 1988, 27, 597. (3) Fisher, W. R.; Doherty, M. F.; Douglas, J. M. The Interface between Design and Control. 2. Process Operability. Ind. Eng. Chem. Res. 1988, 27, 606. (4) Fisher, W. R.; Doherty, M. F.; Douglas, J. M. The Interface between Design and Control. 3. Selecting a Set of Controlled Variables. Ind. Eng. Chem. Res. 1988, 27, 611. (5) Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill: New York, 1988. (6) Larsson, T.; Skogestad, S. Plantwide controlsA review and a new design procedure. Model., Identif. Control 2000, 21, 209. (7) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (8) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999. (9) Wu, K. L.; Yu, C. C. Reactor/Separator Processes with Recycles1. Candidate Control Structures for Operability. Comput. Chem. Eng. 1996, 20, 1291. (10) Wu, K. L.; Yu, C. C.; Luyben, W. L.; Skogestad, S. Reactor/ Separator Processes with Recycles2. Design for Composition Control. Comput. Chem. Eng. 2003, 27, 401. (11) 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. (12) Skogestad, S. Plantwide control: the search for the self-optimizing control structure. J. Process Control 2000, 10, 487. (13) Zheng, A.; Mahajanam, R. V.; Douglas, J. M. Hierarchical Procedure for Plantwide Control System Synthesis. AIChE J. 1999, 45, 1255. (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. (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. (16) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (17) Luyben, W. L. Plantwide Dynamic Simulators in Chemical Processing and Control; Marcel Dekker: New York, 2002. (18) Kokossis, A C.; Floudas, C. A. Synthesis of Isothermal ReactorSeparator-Recycle Systems. Chem. Eng. Sci. 1991, 46, 1361. (19) Silberstein, B.; Bliss, H.; Butt, J. B. Kinetics of Homogeneously Catalyzed Gas-Liquid Reactions: Chlorination of Benzene with Stannic Chloride Catalyst. Ind. Eng. Chem. Fundam. 1969, 8, 366. (20) Yu, C.-C. Autotuning of PID Controllers; Springer: London, 1999. (21) Tyreus, B. D.; Luyben, W. L. Tuning PI Controllers for Integrator/ Dead Time Processes Ind. Eng. Chem. Res. 1992, 31, 2625-2628. (22) Karinen, R. S.; Krause, A. O. I. Reactivity of Some C8-alkenes in Etherification with Methanol. Appl. Catal., A 1999, 188, 247-256. (23) Kiviranta-Paakkonen, P. K.; Struckmann, L. K.; Linnekoski, J. A.; Krause, A. O. I. Dehydration of the Alcohol in the Etherification of Isoamylenes with Methanol and Ethanol. Ind. Eng. Chem. Res. 1998, 37, 18-24. (24) Karinen, R. Etherification of Some C8 Alkenes to Fuel Ethers, Sc. D. Dissertation, Helsinki University of Technology, Espoo, Finland, 2002. (25) Karinen, R. S.; Linnekoski, J. A.; Krause, A. O. I. Etherification of C5 and C8 alkenes with C1 to C4 alcohols. Catal. Lett. 2001, 76, 81-87. (26) Bitar, L. S.; Hazbun, E. A.; Piel, W. J. MTBE production and economics. Hydrocarb. Process. 1984, 63, 10, 63-66. (27) Scholz, B.; Butzert, H.; Neumeister, J.; Nierlich, F. Methyl Tert-Butyl Ether. In Ullmann’s Encyclopedia of Industrial Chemistry; Verlagsgesellschaft: Weinheim, Germany, 1990; pp 543-550.

ReceiVed for reView March 29, 2005 ReVised manuscript receiVed October 27, 2005 Accepted October 28, 2005 IE050396T

Insight from Economically Optimal Steady-State Operating Policies for

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