Inferential Temperature Control Structures for Different Types of Two

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Inferential Temperature Control Structures for Different Types of Two-Reactant Reactive Distillation Systems Devrim B. Kaymak,* Denizhan Yilmaz, and Ahmet Z. G€urer Department of Chemical Engineering, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey ABSTRACT: The control performance of three types of two-temperature inferential control structures for the economic optimum designs of two-reactant ideal reactive distillation systems is evaluated. Three commonly encountered reaction chemistries are studied, namely, a single-product ternary system without inert, a single-product ternary system with inert, and a two-product quaternary system. Closed loop results for the different reactive distillation systems demonstrate significant differences in terms of workable control structures and appropriate temperature control tray locations. A suitable two-temperature inferential control structure for the optimum designs of all the systems is found. The result for the ternary system with inert is in contrast to recent reports claiming the need for column internal composition control. The results also suggest that, with a properly designed twotemperature inferential control system, column overdesign for good controllability is not needed.

1. INTRODUCTION Since the emergence of the pioneering paper by Agreda et al.1 more than two decades ago, there has been a dramatic growth in the number of studies on reactive distillation columns. This growth is also evident in the increasing size of comprehensive lists of reaction systems given in the books dealing with this subject.24 A recent literature survey shows that 40% of all the reactive distillation systems belong to the quaternary class, while 25% of them are from the ternary class.4 Although most of the early literature was focused on the steady-state design of reactive distillation columns, the number of articles dealing with the issues of dynamics and controllability has shown rapid growth during the past decade. Challenging control problems have been presented for “neat” reactive distillation columns with two fresh feed streams. In addition to the papers focusing on real chemical systems, there are several studies in the literature using ideal generic reactive distillation systems. It is assumed that the control issues of reactive distillation columns can be explored without being distracted by the complexities of specific chemical systems, when ideal generic systems are used. In their first paper on the controllability of an ideal twoproduct quaternary reactive distillation system, Al-Arfaj and Luyben5 evaluated six alternative control structures. All of these control schemes have a composition analyzer in the reactive zone of the column to satisfy the stoichiometric balance between reactants. Control structures including at least one composition controller were used to investigate the performance of linear models,6 the effects of feed tray locations,7 and the internal heat integration.8 Whenever possible, inferential temperature control is preferred to direct composition control in the operation of distillation columns, since the composition analyzers are expensive, have high maintenance requirements, and introduce dead time into the control loops.9 Thus, Al-Arfaj and Luyben10 investigated an inferential temperature control structure for the quaternary system, which was first suggested by Roat et al.11 In this structure, r 2011 American Chemical Society

two tray temperatures are controlled by manipulating two fresh feed streams, while the vapor boilup is flow controlled and serves as the production rate handle. They claimed that this twotemperature control structure enables an effective control provided that the system is overdesigned in terms of the number of reactive trays, holdup, or catalyst load. Kaymak and Luyben12 extended this claim by quantitatively showing the impact of number of reactive trays on the effectiveness of the control structure. These overdesigned column configurations of quaternary systems were used to investigate the effectiveness of other inferential two-temperature control structures.13 Once levels and pressure are controlled, there are still three column inputs available for two-feed reactive distillation columns. Thus, any two of these three inputs can be used to control suitable tray temperatures, while the remaining input is chosen as the production rate handle. These alternative inferential control structures were also used to investigate the effect of feed tray locations,14,15 multiplicity,16 and design17 on controllability. Although there are several papers on the controllability of twoproduct quaternary systems, a few papers explored the control of single-product ternary systems, and only one of these papers deals with ideal generic reactive distillation systems. In that paper, Luyben studied the basic two-temperature control structure, where the fresh feed streams are manipulated to control two tray temperatures, for an ideal ternary system with and without inert components.18 Roat’s two-temperature inferential control structure provided an acceptable control for the ternary system without inert. On the other hand, the structure was found to be unworkable for a ternary system with inert and direct internal composition control was recommended for an effective control.

Received: November 18, 2010 Accepted: March 31, 2011 Revised: February 22, 2011 Published: April 27, 2011 6777

dx.doi.org/10.1021/ie102333q | Ind. Eng. Chem. Res. 2011, 50, 6777–6793

Industrial & Engineering Chemistry Research

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Table 1. Kinetic and VaporLiquid Phase Equilibrium Parameters for Systems ternary þ inert

ternary

quaternary

activation energy (kcal mol1) forward

30

30

30

reverse

40

40

40

forward

0.008

0.008

0.008

reverse

0.0004

0.000 16

0.004

chemical equilibrium constant at 366 K

20

50

2

heat of reaction (kcal mol1) heat of vaporization (kcal mol1)

10 6.944

10 6.944

10 6.944

specific reaction rate at 366 K (kmol s1 kmol1)

vapor pressure constantsa (AVP,j BVP,j)

a

A

12.34 3862

12.34 3862

12.34 3862

B

11.65 3862

11.65 3862

11.65 3862

C

10.96 3862

10.96 3862

10.96 3862

D

13.04 3862

I

12.34 3862

ln Psj = AVP,j BVP,j/T with temperature in kelvins and vapor pressure in bars.

Figure 1. Flow sheets of reactive distillation columns: (A) ternary system without inert component; (B) ternary system with inert component and quaternary system.

In the extant literature on the control of generic ideal reactive distillation systems, each system is studied individually and the recommendations are very specific to the particular column design used, so much so that temperature inferential control may or may not be feasible for a particular system depending on the column design. There thus exists the need to systematically evaluate the feasibility of temperature inferential control, which is

of much practical utility in industrial settings, of these generic ideal reactive distillation systems. In this work, such a comprehensive study is undertaken on economic optimum designs of three ideal reactive distillation systems. In the following, after briefly describing the reactive distillation systems, the dynamic control performance of three types of temperature inferential control structures is evaluated followed by a summary of workable 6778

dx.doi.org/10.1021/ie102333q |Ind. Eng. Chem. Res. 2011, 50, 6777–6793


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Figure 2. Control structures: (A) ternary system without inert component; (B) ternary system with inert component and quaternary system.

controlled and manipulated variable pairings for each system and interpretation of the dynamic results in light of steady-state multiplicity.

Table 2. Steady-State Conditions for Optimum Designs ternary

2. STEADY-STATE COLUMN DESIGNS 2.1. Kinetics and Phase Equilibrium Parameters. The

reactions considered in this study are ideal, reversible, and liquid phase. The overall reaction rates on the reactive trays depend on the molar holdup MRX, the specific forward and backward reaction rates kF and kR, and the liquid mole fractions xi,j. The 6779

ternary þ inert

quaternary

NS

13

7

5

NRX

5

12

7

NR

7

5

P (bar) R (mol/s)

6.0 51.60

9.0 60.00

8.5 33.45

VS (mol/s)

33.45

55.06

28.82

DC (m)

0.94

1.00

0.80



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Figure 3. Temperature and composition profiles: (A) ternary system without inert component; (B) ternary system with inert component; (C) quaternary system.

Figure 4. Sensitivity and SVD analysis for ternary system without inert component.

reaction kinetics and vaporliquid phase equilibrium parameters used are taken from the literature19,20 and summarized in Table 1. In this study, the notation for the components is altered slightly

for compatibility across the different systems. Component C is taken as the high-boiling component for all the systems, and component D is taken as the most volatile one in the quaternary 6780



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Table 3. Controller Tuning Parameters: Ternary without Inert Component KC

τI (min)

FA0T4

8.29

16.76

FB0T18 FA0T4

19.54 8.29

22.57 16.76

VST18

2.11

80.12

VST7

3.01

10.56

19.54

7.52

CS

control loop

CS1 CS2 CS3

FB0T18

system. The volatility of the inert component I is considered to be the same as the light reactant A in the ternary system with inert. 2.2. Column Configurations. Schematics of the double feed reactive distillation column configurations for each reaction system are given in Figure 1. For the ternary system without inert component, the column has only a bottoms product. Since there is no distillate product, the column operates at total reflux with no rectifying section. The light and heavy fresh feed streams are fed to the bottom and top trays of the reactive zone, respectively. For the ternary system with inert component, the lower feed stream is a mixture of the light reactant A and the inert component I, since both components have identical volatilities. The other feed stream contains pure B, and it is fed to the top tray of the reactive zone. In this case, the column has both distillate and bottoms streams. While the heavy product C leaves the column from the bottoms stream, the low-boiling inert I, which is not involved in the reaction, is removed by the distillate stream. Thus, there are three sections throughout the column: a stripping section, a reactive zone, and a rectifying section. Here, the main function of the rectifying section is to prevent the escape of heavy reactant B in the distillate. The quaternary system column configuration is similar to that of the ternary system with inert component. It contains a reactive zone between the stripping and rectifying sections. The fresh feed streams consisting of pure A and B are fed to the bottom and top of the reactive section, respectively. In this configuration, heavy product C and light product D are taken out of the column using the bottoms and distillate streams, respectively. 2.3. Design Specifications and Procedure. For both ternary systems, the design objective is to obtain a bottoms product C with 98 mol % purity. The rest of the bottoms stream consists of the impurities of other components. For the ternary system with inert component, the total mole fraction of reactants (A þ B) escaping in the distillate must be less than 3 mol %. The quaternary system column is designed for 95 mol % purities of products C and D in the bottoms and distillate streams, respectively. In this study, the relaxation method is used to find the steadystate conditions of all configurations. The liquid and vapor flow rates are constant in the nonreactive sections of the columns due to equimolal overflow assumption. However, the exothermic reaction taking place in the reactive zone results in vaporization of some of the liquid. Thus, the liquid flow rates decrease down the column, while the vapor flow rates increase up the column. In addition, the nonequimolar reaction of the ternary systems causes changes in the flow rates of this zone. For this study, the catalyst holdup per tray is assumed constant to decrease the number of optimization variables. The holdup on each reactive tray is assumed 1000 mol for the ternary without inert and quaternary

systems. The assumption has been increased to 2000 mol in the ternary system with inert. These values give reasonable liquid heights (in terms of hydraulics) on the reactive trays with calculated column diameters. Based on these specifications and assumptions, a grid-search optimization strategy is used to find the optimum design for each of these reactive column columns. The objective function is the total annual cost which involves the capital and energy costs of processes assuming a payback period of 3 years for capital. The details of design procedures and cost calculations used for these systems are given in the literature.19,20 The common optimization variables of these three columns are (i) the number of stripping trays NS, (ii) the number of reactive trays NRX, and (iii) the column pressure P. For the ternary system with inert component, there are two additional optimization variables such as (iv) the number of rectifying trays NR and (v) the reflux flow rate R. Although the number of rectifying trays NR could be an additional optimization variable for the quaternary system, it is assumed equal to the number of stripping trays NS. This assumption is based on the similarity of the nontemperature dependent relative volatilities between adjacent components.

3. CONTROL STRUCTURES Three alternative temperature-based inferential control structures are explored for three different types of reactive distillation systems. For all control structures of all reactive column systems, the base level is controlled by manipulating the bottoms flow rate, while the reflux drum level is controlled by manipulating the reflux flow rate. In addition, the distillate flow rate is adjusted to give a constant reflux ratio in the cases of the ternary system with inert component and the quaternary system. There are still three column inputs available, namely, the fresh feed streams and the vapor boilup. While one of these three inputs is chosen as the production rate handle, the other two inputs are used to control the temperatures of two trays. Accordingly, there are three basic inferential control structure alternatives. The first structure evaluated in this study is a well-known control structure, which has been first proposed by Roat et al.11 In this control structure, the vapor boilup is chosen as the production rate handle and two fresh feed streams are paired with two temperature control trays. In the second control structure, the heavy reactant fresh feed stream is flow controlled and serves as the production rate handle. In this case, the temperatures of two suitable trays are controlled by manipulating the vapor boilup and the light fresh feed stream. The third control structure sets the production rate by controlling the flow rate of the lower feed stream. Thus, the tray temperatures are controlled by manipulating the vapor boilup and the heavy fresh feed stream. Besides, perfect pressure and flow controls are assumed for all control structures. These three basic control structures are given in Figure 2, and they are labeled in this paper as CS1, CS2, and CS3, respectively. The temperature control trays and loop pairings are selected using the sensitivity analysis and singular value decomposition (SVD) methods. The level controllers are P only with a gain of 2, while the temperature loops are tuned using the relay-feedback test and the TyreusLuyben tuning method. However, in some cases, the loops are detuned by a factor f to prevent the sustained oscillations. Two 60-s first-order measurement lags in series are used for all temperature control loops. The temperature transmitter spans are 50 K, and all valves are 50% open at steady-state conditions. 6781



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Figure 5. Results of CS1 for ternary system without inert component: (A) (20% VS; (B) z0A,B = 0.03, z0B,A = 0.03.

4. RESULTS AND DISCUSSION 4.1. Optimum Steady-State Designs. Optimization results of three reactive distillation systems including the optimum design variables and parameters are summarized in Table 2. Figure 3 plots the steady-state temperature and composition

profiles for the economic optimum designs. It is observed that the temperature profiles resemble the heavy product profiles throughout the columns. There are humps in the reactive zones of all columns. 4.2. Closed Loop Dynamics. 4.2.1. Ternary System without Inert Component. The steady-state gains between the output 6782



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Figure 6. Results of CS2 for ternary system without inert component: þ20% FB0.

(tray temperatures) and input (two fresh feed stream flow rates and vapor boilup) variables are calculated numerically. SVD is performed on the gain matrix to obtain the two U (left singular) vectors. Figure 4 shows the steady-state gains and U vectors for three different control structures. It should be noted that these gains are normalized to be dimensionless. From the plot, the most sensitive trays to the fresh feed stream FA0 are in the stripping section with negative steady-state gains. On the other hand, the top reactive trays of the column have high sensitivity to the fresh feed stream FB0 and the vapor boilup VS. These trays have positive steady-state gains. It is easily seen that the changes in the tray temperatures throughout the column with respect to inputs FB0 and VS are very similar in terms of both sign and magnitude so that the U vectors for both CS1 and CS2 are similar. According to the SVD results for CS1, the temperature of tray 4 should be controlled by manipulating the fresh feed stream FA0, while the temperature of tray 18 should be controlled by manipulating the fresh feed stream FB0. The T4/FA0 controller will be direct acting because of the negative steady-state gain, while the T18/FB0 controller will be reverse acting because of the positive steady-state gain. The U vectors for CS2 suggest that tray 4 should be paired with fresh feed stream FA0, while tray 18 is paired with vapor boilup VS. These results are consistent with the results of sensitivity analysis. On the other hand, results of SVD analysis differ from those of the sensitivity analysis for CS3. Since the gain curves for FB0 and VS are the same, the sensitivity analysis suggests pairing both inputs with the trays in the top section of the column. However, the SVD results for CS3 indicate that tray 7 and tray 18 are the most sensitive locations to the changes in vapor boilup VS and fresh feed stream FB0, respectively. In this case, both the T7/VS and T18/FB0 controllers are reverse acting because of their positive steady-state gains. Table 3

shows the values of controller gains and reset times calculated by the relay-feedback test and TyreusLuyben tuning. Figure 5A plots the closed loop response of CS1 to (20% step changes in the production rate handle VS. The disturbance is applied at minute 30. The closed loop response is dynamically stable, and the controlled tray temperatures settle to their set points in less than 2 h. Also, the bottoms purity recovers to its desired value (98 mol %) within 2 h with a maximum of 0.5 mol % transient deviation. The result is in contrast to the need for direct action in both temperature loops in quaternary systems12 and merits explanation. Initially, an increase in the vapor boilup causes an increase in both of the temperature control trays. The temperature controller of tray 4 has a direct action, so it increases the feed flow rate FA0. This change corresponds to an increase in the amount of reactant A. This excess reactant A moves up through the column. Since there is no distillate stream in the column configuration of the ternary system, the light reactant A moving up cannot escape and it turns back to the column by the reflux. That results in a quick decrease of T18, which forces the feed flow rate FB0 to increase. Finally, the fresh feed streams settle down to their new steady-state values providing a precise balance of the stoichiometry. The responses of CS1 for composition changes in the fresh feed streams are also considered. Figure 5B gives the results for 3% B impurity in FA0 and 3% A impurity in FB0. The results show that both of these impurities can be handled easily by CS1 within less than 2 h. For the case where FA0 contains a mixture of A and B, there is an increase in the flow rate of FA0 and a decrease in the flow rate of FB0 to balance the stoichiometry. The reverse case is also correct when the FB0 flow rate contains a mixture of A and B. Here, the flow rate of FB0 increases, while there is a decrease in the flow rate of FA0. 6783



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Figure 7. Results of CS3 for ternary system without inert component: (A) (20% FA0; (B) z0A,B = 0.03, z0B,A = 0.03.

The closed loop response of CS2 to þ20% step change in the production rate handle FB0 is given in Figure 6. In this case, the FA0 valve shuts down within less than 1 h, while it takes a little longer for the VS valve. Thus, the control structure fails in providing a stable base-level regulatory control. Figure 7A shows the performance of CS3 to (20% step changes in the production rate handle FA0. These disturbances are handled

by CS3 successfully. The tray temperatures return to their steadystate values in about 1 h. The bottoms purity is also maintained very close to its specification by settling down within 1 h. It is also observed that the maximum transient deviation for CS3 is quite small. Figure 7B gives the response of CS3 to composition changes in the fresh feed streams FA0 and FB0. Results demonstrate that these impurities can be easily handled within 1 h. 6784



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Figure 8. Sensitivity and SVD analysis for ternary system with inert component.

The closed loop results for the ternary system without inert component suggest that CS1 and CS3 provide effective column regulation with CS3 giving tighter control. It is also highlighted that the economic optimum design rejects the principal disturbances in a shorter time than the design studied in the literature.18 4.2.2. Ternary System with Inert Component. The sensitivity of tray temperatures for the ternary system with inert component is given in Figure 8 with respect to inputs FA0, FB0, and VS. Similar to the ternary system without inert component, the most sensitive trays to the fresh feed stream FA0 are in the stripping section with negative steady-state gains. However, the locations of the most sensitive trays to fresh feed stream FB0 and vapor boilup VS differ from those of the ternary system without inert component. In this case, the sensitivity analysis results show the top stripping trays as the most sensitive. The steady-state gains of these trays are positive. It should also be noted that there is a remarkable similarity between the steady-state gains related to FB0 and VS. Thus, SVD analysis results for control structures CS1 and CS2 are also similar. The U vectors for CS1 show that the most sensitive trays to the changes in fresh feed streams FA0 and FB0 are tray 3 and tray 6, respectively. These control loops have opposite actions. Although T3/FA0 pairing is direct acting due to the negative steady-state gain, the positive steady-state gain of the T6/FB0 pairing results in reverse acting. The SVD results for CS2 suggest that the temperature of tray 3 is controlled by manipulating the fresh feed stream FA0, while the temperature of tray 6 is controlled by manipulating the boilup VS. These results are well matched with the results of the sensitivity analysis. In the case of CS3, results of SVD analysis differ from the results of the sensitivity analysis. The U vector for FB0 suggests tray 6 to be controlled. This is the tray with the highest sensitivity. On the other hand, the second U vector indicates that more than one tray is controllable by manipulating vapor boilup VS. These are trays

Table 4. Controller Tuning Parameters: Ternary System with Inert Component CS

control loop

KC

τI (min)

CS1

FA0T3

1.04

15.18

FB0T6

0.34

44.77

CS2

FA0T3

1.00

14.74

CS3

VST6 VST4

0.05 0.31

51.59 11.40

FB0T6

0.36

94.31

4, 7, and 13. Since the T7/VS pairing is too close to the T6/FB0 pairing for providing independent temperature measurements, this option is eliminated. Thus, two different versions of this control structure can be evaluated in which the temperature at tray 4 or tray 13 is controlled by manipulating VS, while tray 6 is controlled by manipulating FB0. In these pairings, both controllers have the same action. The values of controller gain and reset time are given in Table 4. Figure 9A illustrates the response of CS1 to (20% step changes in the production rate handle VS. The disturbance is applied at minute 60. Here, CS1 provides a stable control for both positive and negative 20% throughput changes. Although the tray temperature T3 settles quickly to its set point, it takes ∼5 h for the tray temperature T6. As a result, the bottoms purity recovers back to its specified value of 98 mol % C within 10 h. The maximum transient deviation is observed as 2 mol %. On the other hand, settling down of the inert component I takes a longer time. The responses of CS1 to the disturbances in the feed compositions are given in Figure 9B. The disturbances applied are the same as those of the ternary system without inert component. The results illustrate that CS1 handles both of these disturbances and 6785



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Figure 9. Results of CS1 for ternary system with inert component: (A) (20% VS; (B) z0A,B = 0.03, z0B,A = 0.03.

returns the product purity to its desired value in around 10 h. It is observed that the transient deviation in the case of z0A,B = 0.03 is

higher compared to the case of z0B,A = 0.03. Besides, the inert composition of the distillate settles to new steady-state values 6786



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Figure 10. Results of CS2 for ternary system with inert component: (20% FB0.

Figure 11. Results of CS3 for ternary system with inert component: (20% FA0.

instead of turning back to its specific value. This is due to the stoichiometric imbalance between the two fresh feed streams. Figure 10 gives the response of CS2 to (20% step changes in the production rate handle FB0. In the case of a positive disturbance, despite the continuous decrease of FA0 flow rate, tray temperature T3 cannot recover. Thus, the FA0 valve shuts down after a short while. In the case of a negative disturbance, temperature control tray T6 tends to settle down to a new steady state. As a result, large amounts of the reactants are being lost from the top of the column. Hence, it is concluded that control structure CS2 fails to provide a stable base-level regulatory control.

Figure 11 demonstrates the performance of CS3 to (20% step changes in the production rate handle FA0. The results show that the tray temperature T4 returns to its steady-state value within ∼3 h. However, the settling time is too long for tray temperature T6 as it takes ∼15 h to settle down for the bottoms product purity. Besides, the transient deviation of the bottoms product purity exceeds 2 mol % in the case of a negative 20% step change. In addition, the recovery time for the inert composition of the distillate is longer than 15 h. Based on these results, the large production rate changes can be handled using a two-temperature control structure for the ternary system with inert. This conclusion significantly differs 6787



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Figure 12. Sensitivity and SVD analysis for quaternary system.

Table 5. Controller Tuning Parameters: Quaternary System CS

control loop

KC

τI (min)

CS1

FA0T4 FB0T10

0.85 41.77

12.65 8.25

CS2

FA0T4

0.85

12.65

VST14

1.13

9.17

VST10

1.39

13.75

0.53

9.53

41.77

8.25

CS3

VST4 FB0T10

from the results in the literature claiming the need for a direct composition measurement of the column internal composition.18,2124 4.2.3. Quaternary System. The sensitivity analysis results given in Figure 12 show that the most sensitive trays to any of the inputs are in the stripping section. The most sensitive trays to FA0 have negative steady-state gains, while the most sensitive trays to FB0 and VS have positive steady-state gains. There are other sensitive regions in the sensitivity figure of the quaternary system in contradistinction to those of the ternary systems. The SVD results of CS1 suggest T4/FA0 and T10/FB0 pairings. These results are consistent with the results of sensitivity analysis. Tray 10 is one of the secondary sensitive regions with a negative steadystate gain. Both the T4/FA0 and T10/FB0 loops of CS1 have direct action. The main concern here is the magnitude of T10/FB0 steadystate gain, which is very small compared to T4/FA0 steady-state gain. The U vectors for CS2 suggest tray 4 and tray 14 to be controlled by manipulating the fresh feed stream FA0 and vapor boilup VS, respectively. In addition, a second location at tray 10 is indicated as being sensitive to the changes in vapor boilup VS. Therefore, two different versions of this control structure are evaluated in which the temperature at tray 14 or tray 10 is controlled by manipulating VS. By looking to the SVD results of CS3, the most sensitive locations to the changes in vapor boilup VS and fresh feed stream

FB0 are indicated as tray 4 and tray 10, respectively. The singular values for this control structure are σ1 = 27.61 and σ2 = 2.37. The condition number of 11.7 indicates that this control structure should perform adequately. These locations are also validated by the results of sensitivity analysis. However, the magnitude of steady-state gain of T10/FB0 pairing is very small compared to that of T4/VS pairing. Controller gains and reset times of these control structures are given in Table 5. The closed loop responses of CS1 to (20% step changes in the production rate handle VS are given in Figure 13. The results show that a positive throughput change is handled well with CS1. Although there are transient deviations up to 3 mol %, the production purities settle down to their steady-state values within less than 2 h. However, the system is dynamically unstable for a negative 20% throughput change. The T10/FB0 loop does not work properly, while the tray temperature T4 returns to its respective set point by increasing FA0. Thus, the system shuts down within 1 h, and the control structure CS1 fails. The closed loop performance of CS2 using T4/FA0 and T14/VS temperature control loops is presented in Figure 14. The results show that the control structure fails for a 20% step increase in the production rate handle FB0. In this case, the FA0 valve shuts down within less than 2 h. Based on the SVD results, another version of CS2 is evaluated where temperature of tray 10 is controlled by manipulating VS. The results given in Figure 15A demonstrate that CS2 provides a stable control using the T10/VS pairing. For both positive and negative 20% step changes in the production rate handle FB0, the temperatures turn back to their set points within 4 h. Although the product purities cannot recover back to their specifications, they are maintained close to these desired values. This is especially true for the heavy product leaving the column with the bottoms stream. Figure 15B shows the response of CS2 for composition changes in the fresh feed streams. Two different cases of 3% B impurity in FA0 and 3% A impurity in FB0 are considered. The results demonstrate that CS2 provides a stable control for both 6788



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Figure 13. Results of CS1 for quaternary system: (20% VS.

Figure 14. Results of CS2 for quaternary system with T14/VS pairing: þ20% FB0.

types of disturbances. The temperatures settle down to their set points. However, the purities of products cannot be held very close to their specifications. For the case where FA0 contains 3% B impurity, the purities are maintained within (1 mol % of the specification. However, 3% A impurity in FB0 results in an increase of the bottoms

purity over 96 mol % and a decrease of the distillate purity below 94 mol %. The dynamic response of CS3 to (20% step changes in the production rate handle FB0 is given in Figure 16. The results show that the control structure fails for disturbances in all directions. In 6789



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Figure 15. Results of CS2 for quaternary system with T10/VS pairing: (A) (20% FB0; (B) z0A,B = 0.03, z0B,A = 0.03.

this case, the FB0 valve shuts down within less than 1 h, while the VS valve saturates at fully open. The results of the quaternary system demonstrate that there is no need for an overdesigned column to handle large production rate changes, if the correct two-temperature control structure is selected.

4.3. Comparison of Systems. The closed loop results of three different reactive column systems show that there are distinct differences between their inferential controllabilities. The first difference is in the selection of the proper inferential control structure. For both ternary systems, fresh feed streams (FA0 and FB0) are suggested to be used as manipulated variables. In 6790



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Figure 16. Results of CS3 for quaternary system: (20% FA0.

addition, another control structure, where the vapor boilup VS and the heavy feed stream FB0 are the manipulated variables, is also found suitable for the ternary system without inert component. However, the optimum design of the quaternary column cannot be controlled using any of these control structures. A stable regulatory control for the quaternary column is achieved using a control structure in which the vapor boilup VS and the light feed stream FA0 are the manipulated variables. The second difference is in the location of the temperature control trays. For the ternary system without inert component, a stripping tray temperature is controlled using FA0 or VS, while the top reactive tray is controlled by manipulating FB0. For the ternary system with inert component, both temperature control trays are located in the stripping section of the column. Here, the location of the tray paired with FA0 should be in a lower part of the stripping section compared to the location of the tray paired with FB0. For the quaternary system, one temperature control tray is located in the stripping section, while the other one is located in the reactive zone, and both of these trays should be in the middle parts of the related zones. These suggestions are summarized in Table 6. The third difference is in the dynamic response of these systems. The ternary system without inert component and the quaternary system respond quickly to the changes in the production rate handles. Thus, both the controlled temperatures and the product purities settle down to the steady-state values easily. However, the response of the ternary system with inert component takes a longer time due to the location of the controlled trays. The final difference is in the steady-state deviations of the product compositions in response to disturbances. The bottoms products of both ternary systems recover back to their specified values. However, the product purities of the quaternary system cannot be held as close to the desired values as with the ternary systems.

Table 6. Summary for the Selection of Control Structure and Tray Location loop 1

loop 2

location of system

location of

manipulated

controlled

manipulated

controlled

variable

variable

variable

variable

ternary

FA0/VS

stripping section

FB0

reactive zone

ternary þ inert

FA0

stripping section

FB0

stripping section

quaternary

FA0

stripping section

VS

reactive zone

4.4. Bifurcation Analysis. In reactive distillation columns, steady-state multiplicities can occur as a consequence of the high process nonlinearities. As investigated in a recent paper of Kumar and Kaistha, input multiplicities may lead to wrong control action under feedback control.25 Closed loop results of all chemical systems studied here indicate that the failure of the control structures might be due to the wrong control action. To investigate this phenomena and the failure of control structures, a bifurcation study is performed of both the ternary system without inert component and the quaternary system. For the ternary system without inert, the steady-state variations of the controlled tray temperature T18 with respect to corresponding inputs FB0 and VS are given in Figure 17A. Here, the inputoutput relation of T18 with respect to FB0 is wellbehaved, and there is no sign reversal. A decrease in FB0 causes a gradual decrease in T18, while an increase in FB0 leads to a temperature increase. Thus, control structures CS1 and CS3 using the T18/FB0 pairing results in stable controllability. On the other hand, T18 exhibits a gain sign reversal for a decrease in VS, while it behaves well for the changes in the positive direction. The crossover point is around 13.5% in VS. For CS2, a þ20% 6791



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Figure 17. Bifurcation analysis: (A) ternary system without inert component; (B) quaternary system.

change in FB0 results in an increase of T18. Based on the sensitivity analysis, the decrease in VS flow rate should recover T18 back to its set point, since this loop is reverse acting. However, the large decrease in VS causes an increase in T18 due to the wrong control action. Thus, T18 increases as the VS valve turns down, and the VS valve eventually ends up shutting down. For the quaternary system, Figure 17B illustrates the temperature change of tray 10 with respect to corresponding inputs. It is demonstrated that the inputoutput relation of T10 exhibits a gain sign reversal for a decrease in FB0. The crossover point is slightly beyond 30%. On the other hand, the inputoutput relationship of T10 with respect to VS is well-behaved without any sign reversal in any directions. For CS1, a negative 20% change in VS results in a decrease of T10. Based on the sensitivity analysis, T10 should be recovered back to its set point by decreasing the FB0 flow rate, since this loop is direct acting. However, the bifurcation analysis shows that the large decrease in FB0 results in a decrease in T10 because of the wrong control action. Finally, the VS valve shuts down, and CS2 fails. Similarly, the stability of CS3 is affected by the wrong control action of FB0.

5. CONCLUSIONS In conclusion, this comprehensive study on the control of three generic ideal two-reactant reactive distillation systems shows that inferential temperature control with proper choice of the control structure and control tray locations provides acceptable control of the optimal design for each system. Three two-temperature control structures with FA0 and FB0 (CS1), FA0 and VS (CS2), and FB0 and VS (CS3) as the manipulated variables have been evaluated. For the ternary system without inert, CS1 and CS3 effectively reject large throughput and feed composition disturbances. For the ternary system with inert, CS1 is found to be effective in handling these disturbances. For the quaternary system, CS2 effectively handles large throughput changes, but feed composition changes result in large steady-state product purity deviations. In a future work, control structures including a third tray temperature might be considered to mitigate these deviations. For each system, the reason for the failure of a particular control structure can be traced to input multiplicity in the steady-state inputoutput relations.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: þ90-212-285-3539. Fax: þ90-212-285-2925.

’ ACKNOWLEDGMENT Financial support from the Scientific and Technological Research Council of Turkey (TUBITAK) through Project No. 108M504 is gratefully acknowledged. ’ NOMENCLATURE AVP = vapor pressure constant B = bottoms flow rate in the column (mol/s) BVP = vapor pressure constant D = distillate flow rate in the column (mol/s) DC = column diameter (m) f = detuning factor Fj0 = fresh feed flow rate of reactant j (mol/s) kF = specific reaction rate of forward reaction (mol 3 s1 3 mol1) kR = specific reaction rate of reverse reaction (mol 3 s1 3 mol1) K = steady-state gain KU = ultimate gain MRX = liquid holdup on reactive trays (mol) NR = number of rectifying trays NRX = number of reactive trays NS = number of stripping trays P = column pressure (bar) PU = ultimate period (min) R = reflux (mol/s) Ti = column temperature on tray i (K) ΔT = temperature difference (K) U = left singular vector matrix VS = vapor boilup (mol/s) xn,j = composition of component j in liquid on tray n xB,j = bottoms composition of component j in liquid xD,j = distillate composition of component j in liquid Greek Symbols

σ = singular value τI = reset time (min) 6792



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Inferential Temperature Control Structures for Different Types of Two

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