Modeling and Model Predictive Control of Composition and

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Modeling and Model Predictive Control of Composition and Conversion in an ETBE Reactive Distillation Column R. Khaledi and B. R. Young* Department of Chemical and Petroleum Engineering, The University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4

Reactive distillation is a novel technology that has been successfully used in the production of ether fuel additives. This process integrates reaction and separation in a single unit-operation. The interaction of reaction and separation makes the process exhibit complex behavior such as process gain nonlinearity, significant interactions, process gain bidirectionality (i.e., process gain sign change), and steady-state multiplicity. These complex dynamics make process control of the reactive distillation column very difficult. In this work, the nonlinearity of an ETBE reactive distillation column was investigated, and a 2 × 2 unconstrained model predictive control scheme was developed for the product purity and reactant conversion control. The process dynamics were approximated by a first-order plus dead time model to estimate the process model for the model predictive controller. The model predictive controller was able to handle the process interactions well and was found to be very efficient for disturbance rejection and set-point tracking. This controller was stable and performed robustly in the presence of process measurement noise. Introduction Air pollution has become one of the major environmental problems of the century. The emission of carbon monoxide and unburned hydrocarbons from consumption of gasoline in vehicles is a main source of the air pollution. Pollutant emission can be reduced by increasing the octane number of the fuel by addition of oxygenate additives such as methyl tert-butyl ether (MTBE) and ethyl tert-butyl ether (ETBE) to the fuel. These ether additives are produced by etherification of C4 isoolefins with methanol for MTBE or ethanol for ETBE. MTBE and ETBE may be used as replacement for lead-based octane enhancers such as tetramethyllead (TML) and tertaethyllead (TEL). Recently, more attention has been given to ETBE production because of its higher octane enhancing properties, lower volatility, and concerns in the United State over MTBE contamination of groundwater due to leaking underground storage tanks.1 The low vapor pressure of ETBE reduces emission of volatile organic compounds of the fuel. Therefore, it is considered as an environmentally friendly fuel additive. Reactive distillation is a favorable process for carrying out equilibrium-limited chemical reactions. This process recently has attracted attention of many of researchers and has become very important in the production of ether fuel additives.2 Reactive distillation is a process in which the chemical reaction and separation take place continuously in a single distillation tower. This technology offers significant benefits over the conventional process consisting of a separate reactor and distillation column(s). Compared to the conventional process, reactive distillation has a simpler flow sheet with fewer recycle streams and a smaller number of separation units. Reactive distillation can increase the reactant conversion by continuous removal of products * To whom correspondence should be addressed. Tel: (403)220-8751. Fax: (403)284-4852. E-mail: [email protected].

and recycling of reactants. It can also have higher energy efficiency for exothermic reaction systems since the heat generated by the reactions reduces the reboiler heat input to the column. Although reactive distillation has advantages over the conventional process, its process control is much more difficult than the conventional process where the reaction and separation units are connected in series. In the conventional process, the reactant conversion can be controlled separately in the reaction vessel by maintaining the reactor process variables at desirable conditions. Consequently, the product purity is controlled independently in the separation unit. On the other hand, the interaction of reaction and separation makes the reactive distillation column exhibit complex behavior such as multiplicity in steady-state solutions, high process nonlinearity, process gain sign changes (bidirectionality), and strong interactions between process variables. These complexities make reactive distillation process control extremely difficult and may reduce the flexibility of the system. There has been a considerable amount of research work on simulation and control of the ETBE reactive distillation process published recently. Sneesby et al.3 simulated an ETBE reactive distillation column using Pro/II4 and SpeedUp.5 Using this steady-state simulation model, they studied the effects of design and operating variables on column performance and developed a design method for ETBE reactive distillation column. Following steady-state simulation, they studied the dynamic behavior and control aspects of the ETBE column.6 In this work, they tested several control configurations, and among them LV and LB configurations were found to be more effective for single-composition control. Sneesby et al.7 proposed an inferential control scheme for two-point control of ETBE reactive distillation column in which both bottom product purity and reactant conversion are controlled using conven-

10.1021/ie049274b CCC: $30.25 © 2005 American Chemical Society Published on Web 03/19/2005

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tional PI controllers. Al-Arfaj and Luyben8 studied an ideal, generic two-product reactive distillation column with simple thermodynamic and kinetic behavior. They proposed a variety of control structures for two-point control of product purity in which the concentrations were controlled with direct measurement of compositions in the system. A single-point control structure with the stripping section stage temperature used as an inferred variable to control the bottom product purity was also investigated. In another study, Al-Arfaj and Luyben9 applied the control structures developed in the previous work to an optimized double-feed ETBE reactive distillation column. They also proposed a singlepoint control structure controlling either the bottom product composition or a stripping section stage temperature for a single-feed ETBE reactive distillation column. Jhon and Lee10 developed an algorithm for dynamic simulation of an ETBE reactive distillation column. They studied the open-loop dynamic responses of the system for total and partial condenser configurations. Tian et al.11 proposed a pattern-based predictive control (PPC) scheme for single-point control of the bottom product ETBE purity using the reboiler duty as the manipulated variable. They introduced a nonlinear transformation to obtain a pseudolinear input-output relationship and incorporated the PPC with conventional PI control. Bisowarno et al.12,13 proposed two adaptive PI control strategies, model gain-scheduling and nonlinear PI, to improve the single-point bottom product ETBE purity control performance. All of the control studies discussed above employ either conventional PI controllers for two-point control or an advanced control strategy with a PI controller for single-point control of the system. For single-point control, the PI controller can be conveniently applied with either a conventional or an advanced control strategy. However, for two-point control, because of strong interactions between the process and manipulated variables, an advanced control strategy may be desirable to control both product purity and reactant conversion without being affected by process variable interactions. In this paper, we propose a model predictive control (MPC) scheme for two-point control (purity and conversion) of an ETBE reactive distillation column. Although the process exhibits nonlinear behavior, it was found that it can be adequately estimated by a simple linear model for use in the MPC controller. Model mismatch has been addressed in the following manner. The simulation is essentially a virtual plant. Simplified models have been identified for control (MPC). Noise has also been added to the process variables to simulate process measurement noise. As such, the practical result of model mismatch is reflected in the results. It is also important to note here that it is well-documented that MPC is often outperformed by PI control on a one for one loop comparison basis (e.g., Luyben et al.14). However, MPC affords advantages when it comes to linking control strategy design (e.g., Downs15). This process could well benefit from these advantages due to its particular dynamics. ETBE Reaction Kinetics and Thermodynamics ETBE is the product of the etherification reaction of isobutylene with ethanol over an acid-base catalyst, such as an acidic ion-exchange resin (e.g., Amberlyst 15):

(CH3)2CdCH2 + C2H5OH S (CH3)3COC2H5 (1) The reaction is an exothermic equilibrium limited reaction that takes place in the liquid phase. The reaction extent is only 84.7% from a stoichiometric mixture of reactants at 70 °C. The equilibrium conditions of this reaction were studied by Jensen and Datta.16 The reaction equilibrium constant for ETBE resulting from this study was estimated as follows:

4060.59 - 2.89055 ln T T 0.0191544T + 5.28586 × 10-5 T2 - 5.32977 × 10-8 T3 (2)

ln(KETBE) ) 10.387 +

The ETBE reaction system also includes an equilibriumlimited side reaction that is the dimerization of isobutylene to produce diisobutylene (DIB):

(CH3)2CdCH2 + (CH3)2CdCH2 S [(CH3)2CdCH2]2 (3) However, in practice this side reaction can be prevented by using some excess ethanol and the DIB production becomes negligible. In this work a 3 mol % excess ethanol is used in the feed, and the DIB reaction is considered to be negligible. The ETBE-ethanol-hydrocarbon system is a highly nonideal system. Therefore, the system thermodynamic properties are appropriately calculated using UNIQUAC with UNIFAC parameters for the liquid phase.17 The vapor phase properties are calculated using the PengRobinson equation of state. ETBE Reactive Distillation Column A simplified reactive distillation process for production of ETBE is shown in Figure 1. The hydrocarbon feed should be rich in isobutylene. The industrial source of such a feed can be the products of refinery units such as a fluidized catalytic cracking (FCC) unit (15-35% isobutylene), a steam cracking unit (40-55% isobutylene), or an isobutene dehydration unit (40-55% isobutylene). The ethanol feed needs to be essentially pure to prevent the isobutylene hydration side reactions. The hydrocarbon and ethanol are fed into a catalytic tubular isothermal reactor where the majority of reaction (approximately 80%) takes place at about 90 °C. The operating pressure is kept between 1500 and 2000 kPa to ensure that all components remain in the liquid state. The reason for using the tubular reactor is to remove the liberated heat of reaction which affects the reaction equilibrium. This reactor is used as a first stage for reaction since it is not possible to load the reactive distillation column with the total amount of catalyst necessary for the whole conversion. Therefore, the feed to the reactive distillation column contains unreacted isobutylene, ethanol, nonreactive hydrocarbons, and ETBE. The second stage of reaction takes place in the reactive distillation column at a lower temperature (around 70 °C) to improve the conversion. The reactive stages are located below the rectifying section and above the stripping section of the column. The feed is introduced immediately below the reaction zone. ETBE is the least volatile component and is continuously removed from the reaction zone. In the stripping section, ETBE is concentrated and the lighter components (isobutylene and nonreactive hydrocarbons) are recycled

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Figure 2. Multiple steady-state solution; high and low conversion temperature profiles. Figure 1. ETBE synthesis via reactive distillation.

Table 3. Process Conditions for Low and High Conversion Solutions

Table 1. Reactive Distillation Feed Conditions feed condition pressure (kPa) temperature (°C) flow rate (kmol/h) composition (mol %)

950 30 100 29.1 ETBE, 9.1 ethanol, 7.3 isobutylene, 54.5 n-butylenes

Table 2. Reactive Distillation Column Specifications rectification theoretical stages reactive theoretical stages stripping theoretical stages total theoretical stages feed stage number overhead pressure (kPa) reflux ratio condenser reboiler

2 stages 3-5 5 10 6 950 6 total partial

to the reaction zone. In the rectifying section, ethanol and isobutylene are recycled to the reaction zone. By continuous recycling of the reactants into the reactive stages of the distillation column a high conversion can be attained (higher than 98 mol %). The product from the bottom of the reactive column would be mainly ETBE with a small amount of ethanol. The distillate product would mainly consist of nonreactive hydrocarbons and a small amount of isobutylene and ethanol. Case Study The feed conditions and column specifications for the ETBE reactive distillation under study in this paper are given in Tables 1 and 2, respectively. The feed conditions and column specifications are similar to the work of Sneesby et al.,6 except for the feed flow rate that is chosen to be higher (100 kmol/h) to represent an industrial-scale column. The column consists of 10 theoretical stages: two rectifying (including a total condenser), three reactive, and five stripping stages (including a partial reboiler). The feed is introduced on stage 6, immediately after the reactive stages. Steady-State Analysis of ETBE Reactive Column Nonlinearity and Multiple Steady States. The HYSYS 3.1 simulator18 was used to simulate the reactive distillation column of this study. Due to the fast chemical reaction, the reaction can be considered to be at equilibrium and it is not necessary to implement rate-

reflux rate (kmol/h) reboiler duty (kW) bottom ETBE (wt %) IBa conversion (mol %) a

high conversion

low conversion

411.5 2830 88.7 98.1

411.5 2830 0.53 2.5

Isobutylene.

based reaction kinetics.3 The ETBE reactive distillation column integrates reaction and separation. The interaction of reaction and separation makes the process exhibit extreme nonlinear behavior such as bidirectionality (i.e., process gain sign change) and multiplicity of steady states. Figure 2 shows the temperature profile for the high conversion and low conversion conditions for the process conditions given in Table 3. As can be seen from Table 3, the process conditions are similar for both cases, but there are multiple solutions for these process conditions. The simulation gives two different values for isobutylene conversion: 98.1% and 2.5% for the high and low conversion cases, respectively. This multiplicity reveals the nonlinearity of the process and points to potential difficulties in practical control of the column. Selection of Controlled and Manipulated Variables. The ETBE reactive distillation column combines reaction and separation elements in a single unitoperation. The primary objective of the reaction element is the conversion of the reactants that can be measured via the isobutylene conversion. The primary objective of the separation element is to separate the mixture of the components. Therefore, two control objectives can be defined in an ETBE reactive distillation column: the isobutylene conversion and the bottom product ETBE purity. For control of these two variables, two manipulated variables are required. The LV configuration was found to be the most appropriate configuration to accomplish this objective.19 The LV control structure is an example of a direct (or energy balance) configuration since the composition is controlled by manipulating internal variables (LV). The inventory (liquid levels and pressure) can be controlled using bottom and distillate flow rates and condenser duty. The bottom ETBE purity can be measured directly using an online analyzer. However, due to a large measurement lag in the analyzer, the direct measurement of this process variable is not recommended in practice. Instead, an easy to measure process variable is sought to adequately infer the bottom ETBE purity. The selection of this inferential process variable needs to be investigated. Figure 3 shows the bottom product ETBE purity and


Figure 3. Effect of reboiler duty on bottom ETBE purity and isobutylene conversion at constant reflux ratio (R ) 5.3 and 6.0).

Figure 4. Stage temperature profile in ETBE column at constant reflux ratio (R ) 5.3).

Figure 5. Effect of reboiler duty on stage 7 temperature at constant reflux ratio (R ) 5.3 and 6.0).

isobutylene conversion at different reflux ratios with respect to reboiler duty. It can be seen that the purity and conversion behave in a similar manner with change in reboiler duty. The nonlinearity and bidirectionality (i.e., the process gain changes from positive to negative in the operating range) can be seen in the graph. The bidirectionality of the process gain is another reason that makes the direct control of the ETBE purity with the reboiler duty an inadvisable control choice. A stage temperature close to the bottom of the column may be a good candidate for the bottom composition control as an inferred variable. Figure 4 shows the stage temperature profile for different reboiler duties. It can be seen that the stage 7 temperature is more sensitive to reboiler duty changes than the other stages and may be an appropriate candidate inferred variable. Figure 5 shows the stage 7 temperature variation against the reboiler duty. This temperature has a one-to-one relationship with reboiler duty and always has a positive

Figure 6. Bottom product ETBE purity and the stage 7 temperature relationship at constant reflux ratio (R ) 5.3 and 6.0).

Figure 7. Isobutylene conversion relationship with the reaction zone temperature difference at constant reboiler duty.

gain. The graph shows a nonlinear relationship between temperature and reboiler duty. However, the process gain is almost constant over selected ranges of operating conditions (approximately 130-143 °C). Figure 6 shows the bottom product ETBE purity relationship with stage 7 temperature. For the operating range of interest (i.e., 130-143 °C), the relationship is linear and the ETBE purity increases with an increase in stage 7 temperature. Therefore, this temperature is an appropriate variable to infer the bottom product ETBE purity. The direct measurement of the isobutylene conversion is difficult and requires a substantial amount of data obtained by a number of composition analyzers. Sneesby et al.7 have shown that the reaction zone temperature difference can be used as a good estimate for isobutylene conversion. As it can be seen from Figure 7, the reaction zone temperature difference (∆T ) T4 - T2) has a maximum peak close to 2 °C. The isobutylene conversion drops very fast for ∆T less than this value. We are interested in the ∆T range where the there is almost a linear relationship between the isobutylene conversion and reaction zone ∆T for a wide range of operating conditions. Therefore, the high isobutylene conversion can be achieved by controlling the reaction zone ∆T in a range between 2 and 6 °C. With the LV configuration, V was already used for stage 7 temperature and the remaining manipulated variable was reflux flow (L) that was to be used for reaction zone ∆T. Figure 8 shows the variation of reaction zone ∆T with respect to reflux flow rate. It can be seen that the reaction zone ∆T has a one-to-one relationship with reflux flow. Although with different gain magnitude, it has always the same sign for the gain (negative). Moreover, it has nearly linear behavior at the desired operation conditions (∆T in a range between 2 and 6 °C) where the isobutylene conversion reaches a maximum. The completed control

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Figure 8. Reaction zone temperature difference relationship with the reflux rate at constant reboiler duty.

Figure 10. Open-loop response to step changes in manipulated variables.

Figure 11. Variation of process model parameters with respect to changes in reflux molar rate from steady-state value.

Figure 9. Two-point MPC control configuration for the ETBE reactive distillation column.

structure for ETBE reactive distillation column is given in Figure 9, which includes a MPC controller as further discussed in the controller configuration and MPC setup section. Process Model Identification The open-loop dynamic response of the ETBE reactive distillation column is obtained by performing a series of step changes in the manipulated variables (reflux rate and reboiler duty). The step changes are made in the desired operating region where high conversion solutions exist. Figure 10 shows the dynamic response of the ETBE reactive distillation column for a 1% increase in reflux molar flow and a 5% decrease in reboiler duty. It can be seen from the graph that even for a small change in manipulated variables, the process variables deviate significantly from initial steady-state values. Also it is noted that the system shows a significant interaction between the process and manipulated variables. The shape of process variable responses to step changes in manipulated variables reveals that the process exhibits essentially first-order behavior. For a

Figure 12. Variation of process model parameters with respect to changes in reboiler duty from steady-state value.

series of different values of step change in manipulated variables, the first-order process gains and process time constants were found and plotted in Figures 11 and 12. The nonlinearity of process can be seen from variation of process gains and process time constants. Figures 11 and 12 suggest that an adaptive controller with variable process model parameters could be appropriate for control of ETBE reactive distillation column. However, it is still possible to estimate the system with a firstorder plus dead time model given by eq 4 with constant, averaged model parameters:

gij )

Kpe-τds τps + 1

(4)

The region with negative change in reflux molar rate change in Figure 11 and also the region with positive change in reboiler duty in Figure 12 corresponded

Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3139 Table 4. First-Order plus Dead Time Model Parameters

Table 6. Feed Composition Conditions for Step Change

model parameters

feed composition (mol %)

transfer function

Kp

τp (min)

τd (s)

component

condition 1

condition 2

g1.1 g1.2 g2.1 g2.2

-11 10 -50 50

12 12 38 36

10 10 30 5

isobutylene 1-butene ethanol ETBE

7.3 54.5 9.1 29.1

15.0 45.5 19.5 20.0

Table 5. Inventory PI Controller Parameters controller

kc

τI (min)

LC-1 LC-2 PC

4.5 10.1 0.29

0.75 0.33 0.14

to high T7 temperatures and reaction zone ∆T values where the conversion is very low. These regions are not in our operating range of interest. Therefore, the approximate model parameters should be selected from the positive side of reflux rate change and negative side of reboiler duty change. The approximate selected process model parameters after experimental tuning are given in Table 4. These process gains, process time constants, and dead times were used for model estimation in the model predictive control (MPC) controller implemented in this work. Controller Configuration and MPC Setup Figure 9 shows the control configuration for the twopoint control of the ETBE reactive column. Three singleloop PI controllers are used to control the column inventory. The column pressure is controlled by adjusting the condenser duty, and the condenser accumulator and reboiler levels are controlled by the distillate and bottom flow rates, respectively. The PI controllers are tuned with the Auto Tune Variation (ATV) technique of A° stro¨m and Ha¨gglund.20 The tuning parameters of these controllers are given in Table 5. The two remaining controlled variables, namely, reaction zone ∆T and stage 7 temperature, are controlled with the reflux molar flow rate (L) and the reboiler duty (Qreb) using a 2 × 2 unconstrained MPC controller built in HYSYS 3.1.18 For the MPC controller, the first-order plus dead time model of eq 4 with parameters as given in Table 4 is implemented to estimate the process model. The number of sampling intervals in the MPC controller is set to 100. A value of 1.5 min (which is in the range of 1/5th to 1/10th of the smallest process time constant) is used as control interval. This control interval value allows the process model to reach its steady state after 100 time intervals. The control horizon is set to two control moves into the future to ensure a nonaggressive control action, and the prediction horizon is set to 20 steps (30 min). Dynamic Open-Loop Response and Controller Performance Test MPC Control System. In this section, the performance of the MPC controller for disturbance rejection and set-point tracking is investigated. An efficient controller should be able to reject unexpected disturbances introduced to the system. Feed composition and feed rate changes are the most probable unpredicted upstream disturbances that affect the ETBE reactive distillation column operation. The reaction zone ∆T and the stage 7 temperature set-points may often need to be changed by the operators to achieve the desired

reaction conversion and bottom product purity. Therefore, the control system should be also capable of fast set-point tracking. To examine the MPC controller performance, the following disturbances and set-point changes were applied to the ETBE reactive distillation column: (a) Feed composition change from condition 1 to condition 2 as shown in Table 6. (b) (20% step changes in feed molar flow rate. (c) Sinusoidal disturbance in feed molar rate with an amplitude of 10% and a period of 120 min. (d) (3 °C step changes in T7 set-point. (e) (1.5 °C step changes in reaction zone ∆T set-point. (f) -3 and +1.5 °C followed by +3 and -1.5 °C step changes in T7 and reaction zone ∆T set-points, respectively. To observe the effect of disturbances on process variables, the dynamic open-loop response for feed composition and molar flow rate changes were obtained. Figure 13a-d shows the responses to step changes in feed composition, and Figure 14a-d shows the responses to both a step increase and decrease in feed molar flow rate. It can be seen that feed condition changes can cause significant deviations in the process variables from their steady-state values. Figure 13a,b shows that for a step change in feed composition from condition 2 to condition 1 of Table 6, both T7 and reaction zone ∆T respond quickly and reach steady-state conditions but for a change from condition 1 to condition 2 the response is slower (particularly for reaction zone ∆T), and it takes a longer time for the process variables to reach their steady-state conditions. This difference in dynamic response of the system indicates the nonlinearity of the process. However, both T7 and reaction zone ∆T variations are monotonic for feed composition step changes. For isobutylene conversion, this step change in feed composition causes the isobutylene conversion to exhibit complex behavior by passing through maximums and minimums before reaching a steady-state condition (Figure 13c). The ETBE purity response also shows a nonlinear characteristic (Figure 13d). A similar complex process response can be seen in Figure 14a-d for feed molar flow rate changes. These complex behaviors indicate significant nonlinearity and bidirectionality of the process gain. The closed-loop dynamic responses for feed composition and feed flow rate changes (tests a and b) are shown in Figures 15a-d and 16a-d, respectively. As it can be seen from Figures 15a,b and 16a,b, the controller is successfully able to maintain the T7 and reaction zone ∆T close to their set-points and reject the introduced disturbances. The capability of the controller can be noticed clearly when the closed-loop responses are compared with the open-loop responses for the same applied disturbances. For the feed composition step changes, the inferential control of the T7 and reaction zone ∆T is able to control the isobutylene conversion and bottom product ETBE purity around their desired values. However, the steady-state values for these two

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Figure 13. Open-loop response to feed composition step change: (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

process variables are slightly different from their original steady-state values before the disturbance was applied. To eliminate this difference, the operator can manually adjust T7 and reaction zone ∆T set-points to achieve the desired reactant conversion and product purity. For the feed molar flow rate step change (test b), the isobutylene conversion and bottom product ETBE purity deviate slightly from their desired values (Figure 16c,d) and reach steady-state conditions after the T7 and reaction zone ∆T are stabilized. These steady-state values are equal to the original steady-state values

Figure 14. Open-loop response to (20% step change in feed molar rate: (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

before the feed rate step change was introduced. Therefore, the control system is able to maintain the conversion and purity for feed rate changes without any need for adjustment in the T7 and reaction zone ∆T setpoints. Test c is performed to evaluate the stability and robustness of the MPC controller. To observe the effectiveness and robustness of controller with the presence of common process disturbances as in actual cases, random measurement noise with a variance of 0.1 °C is embedded in the process variables (T7 and reaction zone ∆T) for tests c-f. The dynamic closed-loop re-


Figure 15. Closed-loop responses of MPC controller for step changes in feed composition (test a; solid line, condition 1 to condition 2; dashed line, condition 2 to condition 1): (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

Figure 16. Closed-loop responses of MPC controller for (20% step changes in feed molar rate (test b; solid line, +20% change; dashed line, -20% change): (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

sponses for test c are presented in Figure 17a,b. The results show that the controller is stable and can maintain the process variables around their desired values for the sinusoidal variation in feed molar flow rate and also with measurement noise present in the process variables. Figure 18a-d shows the closed-loop response of the process variables for (3 °C step changes in T7 set-point

(test d). It can be seen from Figure 18a that the controller tracks the new set-point for T7 very quickly for both an increase and decrease in set-point value. Even with the presence of measurement noise of 0.1 °C in process variables, a relatively large noise for temperature measurement devices, the set-point tracking is carried out smoothly without any erratic deviation in controlled variables and instability of the process.

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Figure 17. Closed-loop response of MPC controller for sinusoidal disturbance in feed rate in the presence of measurement noise in process variables (test c): (a) stage 7 temperature and reaction zone ∆T; (b) isobutylene conversion and bottom product ETBE purity.

Figure 18b shows the effect of T7 set-point change on the reaction zone ∆T. The controller is able to reject the disturbance caused by T7 set-point change and prevent a large deviation in ∆T from its set-point value and return it to its set-point quickly. From Figure 18c,d it can be seen that both bottom product ETBE purity and isobutylene conversion increase and decrease with an increase and decrease in T7 set-point, respectively. Therefore, the operator can use the T7 as a singleadjusting process variable for slight increases or decreases in both conversion and bottom product ETBE purity. Figure 19a-d shows the closed-loop response of the process variables for (1.5 °C step changes in reaction zone ∆T set-point (test e). The controller is able to track the new set-points quickly for both an increase and decrease in set-point value. However, the process variable approach speed to track the set-point for an increase in set-point is faster than the approach speed for a decrease in set-point (Figure 19b). This is due to difference between the process gain magnitudes (nonlinearity) in both sides of the steady-state value of 4 °C for reaction zone ∆T. Figure 19a shows that the controller reacts quickly to upset the effect of the reaction zone ∆T set-point change on T7. Figure 19c reveals that the isobutylene conversion increases with a decrease in reaction zone ∆T set-point and vice versa. The bottom product ETBE purity also shows similar behavior for the same set-point change as shown in Figure 19d. Test f is performed to evaluate the controller ability in handling the interaction of the two process variables when two opposite control actions are expected from the controller. In this test, for a decrease in the T7 set-point the controller action would be to decrease reboiler duty (Qreb) and increase reflux flow rate (L). At the same time, the reaction zone ∆T set-point is increased, which requires the controller to act in the opposite direction

Figure 18. Closed-loop responses of MPC controller for (3 °C step changes in T7 set-point in the presence of measurement noise in process variables (test d): (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

in manipulating the manipulated variables (increase in reboiler duty and decrease in reflux flow rate). For such an upset, the non-decoupled controller may become unstable by increasing and decreasing the manipulated variables. Figure 20a,b shows the controller performance for this test. It can be seen that the controller is able to track both set-points by smooth manipulation of the manipulated variables toward the target without


Figure 20. Closed-loop response of MPC controller for simultaneous step change in T7 and reaction zone ∆T set-points in the presence of measurement noise in process variables (test f): (a) stage 7 temperature and reaction zone ∆T; (b) manipulated variables. Table 7. Controller Parameters for Single-Point Control Structure

Figure 19. Closed-loop responses of MPC controller for (1.5 °C step changes in reaction zone ∆T set-point in the presence of measurement noise in process variables (test e): (a) stage 7 temperature; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity.

encountering fluctuation and instability due to process variables interactions. Comparison of PI and MPC Controllers. The MPC controller performance is compared with two PI control structures. The first control structure is a singlepoint control scheme as proposed by Al-Arfaj and Luyben.9 Figure 21 shows the control configurations for this structure. In this control scheme, the stage 7 temperature is controlled by manipulating the reboiler

controller

kc

τI (min)

FC TC

0.17 0.5

0.1 4.0

heat duty while the reflux flow rate is kept constant using a flow controller. The advantage of this singlepoint control structure is that it is simple and can be easily tuned. The second control structure is shown in Figure 22. In this 2 × 2 PI control system, the reaction zone ∆T is controlled using the reflux flow rate, and the stage 7 temperature is controlled using the reboiler heat duty. This control system is similar to the control structure proposed by Sneezby et al.19 The PI controllers in the single-point control structure are tuned using the ATV technique of A° stro¨m and Ha¨gglund.20 In tuning of the controllers for the 2 × 2 PI control structure, the interactions between the reaction zone ∆T and T7 control loops should be taken in account. These controllers are tuned using the multivariable “biggest log modulus tuning” (BLT) method of Luyben.21 The controller parameters for single-point and 2 × 2 PI structures are given in Tables 7 and 8, respectively. Figure 23a-d compares the single-point control, 2 × 2 PI, and MPC performance for a +3 °C step change in T7 setpoint. In the single-point control scheme, the controller can track the set-point change much faster than the other two controllers (Figure 23a). This is due to the fact that, in the single-point control scheme, there is less interaction between the T7 control loop and other control loops. The effect of interaction of the reaction zone ∆T control loop with the T7 control loop can be clearly seen in 2 × 2 PI control scheme. Although slightly slower than the 2 × 2 PI control scheme, the

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Figure 21. Single-point PI control configuration for the ETBE reactive distillation column.

Figure 22. Two-point PI control configuration for the ETBE reactive distillation column.

MPC controller shows a smoother move and is less affected by the loop interactions. In the single-point control scheme, unlike the 2 × 2 PI and MPC, the reflux flow is constant, and no adjustment is made to keep the reaction zone ∆T on a specific set-point value. Therefore, the reaction zone ∆T increases for a step change in T7 set-point as shown in Figure 23b. This reaction zone ∆T

Figure 23. Comparison of the single-point PI, 2 × 2 PI, and 2 × 2 MPC closed-loop responses for +3 °C step change in T7 setpoint: (a) stage 7 temperature [for better illustration of the T7 response at the time of set-point change, a smaller time scale is used]; (b) reaction zone ∆T; (c) isobutylene conversion; (d) bottom product ETBE purity. Table 8. Controller Parameters for 2 × 2 PI Control Structure controller

kc

τI (min)

DTC TC

4.6 9.1

11.2 5.7

change affects the steady-state values of isobutylene conversion and bottom product ETBE purity (Figure


23c,d). The steady-state values of isobutylene conversion and bottom product ETBE purity in the single-point control structure are less than for the 2 × 2 PI and MPC control structures.

Note Added after ASAP Publication. This article was released ASAP on March 19, 2005, with the incorrect artwork for Figure 14b. The correct version was posted on March 29, 2005.

Summary and Conclusions

Literature Cited

The ETBE reactive distillation column is a process with a high degree of nonlinearity, bidirectionality of process gains, significant interactions, and multiplicity of steady-state solutions. This nonlinearity was investigated, and for a selected set of control objectives (composition and conversion), appropriate inferred process variables and manipulated variables are used. In this control configuration (i.e., LV), the reflux flow rate is used to control the reaction zone temperature difference, which infers isobutylene conversion, and the reboiler duty is used to control the stage 7 temperature, which infers bottom product ETBE purity. A 2 × 2 unconstrained model predictive control system was developed for two-point control of the ETBE reactive distillation column. The process dynamics are approximated by a first-order plus dead time model to estimate the process model in model predictive controller. The model predictive controller is able to handle the process interactions perfectly. This controller was found to be very efficient for disturbance rejection and setpoint tracking, was stable, and performed robustly with process measurement noise present. The MPC controller performance was compared with a simple single-point control scheme and a 2 × 2 PI control structure. The simple control structure showed a faster response as compared to the MPC and 2 × 2 PI control structures. However, it is not as able as the 2 × 2 PI and MPC in maintaining the isobutylene conversion. The MPC controller is slightly slower than the conventional 2 × 2 PI controller. However, the MPC controller is not affected by the loop interactions. Nomenclature B ) bottom product molar flow rate (kmol/h) D ) distillate molar flow rate (kmol/h) gij ) transfer function given by eq 4 relating the process variable, PVi, to manipulated variable, MVj Kc ) proportional gain in PI controller KETBE ) ETBE equilibrium constant Kp ) process gain (% process variable change/% manipulated variable change) L ) reflux molar flow rate (kmol/h) MV ) manipulated variable PV ) process variable Qc ) condenser duty (kW) Qreb ) reboiler duty (kW) R ) reflux ratio s ) Laplace operator T ) temperature (K) V ) internal vapor molar flow rate (kmol/h) Greek Symbols τd ) process dead time (s) τI ) integral time in PI controller (min) τp ) process time constant (min) Subscripts i ) process variable index (PV1 ) reaction zone ∆T, PV2 ) T7) j ) manipulated variable index (MV1 ) L, MV2 ) Qreb)

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Received for review August 11, 2004 Revised manuscript received February 10, 2005 Accepted February 11, 2005 IE049274B

Modeling and Model Predictive Control of Composition and

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