Effect of Feed Tray Location on Temperature-Based Inferential Control

Ind. Eng. Chem. Res. 2009, 48, 11071–11080

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Effect of Feed Tray Location on Temperature-Based Inferential Control of Double Feed Reactive Distillation Columns Gu¨ls¸ah Sunar and Devrim B. Kaymak* Department of Chemical Engineering, Istanbul Technical UniVersity, 34469, Maslak, Istanbul, Turkey

Adjusting the feed tray locations may result in significant energy savings for double feed reactive distillation columns. In the literature, composition-based control structures have been suggested for the internally heat integrated designs of ideal reactive distillation columns. This paper explores the effect of feed tray location on the controllability of ideal reactive distillation columns using temperature-based control structures. Two different types of two-temperature inferential control structures are considered. In the first structure, two fresh feed flow rates are manipulated to control the temperatures on two selected trays. In the second structure, the light fresh feed flow rate and the vapor boilup are manipulated. Alternative designs are created by moving the feed trays in and out of the reactive zone. It is demonstrated that significant energy savings are obtained by shifting the feed tray locations into the reactive zone. The results indicate that there are internally heat integrated designs, which can be effectively controlled by a two-temperature inferential control structure without using any direct composition measurement. For the case study examined, a 13.17% energy saving is obtained by shifting the fresh feed trays into the reactive zone by two trays from their conventional locations. The second temperature-based control structure considered can handle fairly large throughput changes without any system shut down or steady-state transition for this energy-efficient design. 1. Introduction The increasing interest in the process intensification results in a growth of the number of papers dealing with reactive distillation columns. The applications of reactive distillation for real chemical systems have been listed in the work of Doherty and Malone,1 Sharma and Mahajani,2 and Luyben and Yu.3 Moreover, there are several papers dealing with ideal generic chemical systems. The basic idea behind working with generic systems is investigating the design and control issues without being obscured by the complexities of specific chemical systems such as nonideal vapor-liquid equilibriums, reaction kinetics, and physical properties. The most frequently studied generic system is an ideal two-reactant and two-product system. A+BTC+D Optimization of a reactive distillation column involves a very large number of design variables. In the literature, several engineering assumptions are used to reduce the number of design optimization variables. A common assumption is feeding the light reactant to the bottom tray of the reactive section, while heavy reactant is introduced from the top tray of the reactive zone.4 This is the conventional design of a double feed reactive distillation column. However, recent work of Cheng and Yu5 has shown that extending this assumption by including the feed tray location as a design optimization variable can lead to significant energy savings. Huang and co-workers6 have also claimed that the internal heat integration between reaction and separation operations can be maximized by altering the feed tray locations. In the past decade, many papers have explored the controllability of the double feed ideal reactive distillation columns operated with the generic reaction given above. Most of these papers have used the conventional design as the test bed.7-14 * To whom correspondence should be addressed. E-mail: [email protected]. Phone: +90-212-285-3539. Fax: +90-212285-2925.

The conventional design is also used in several studies dealing with real chemical systems.15-19 There have been a few papers which examined the impact of altering feed tray locations on the controllability of these columns. Huang and co-workers used a three-composition control structure to control their heat integrated reactive distillation column.20 In this control structure, the concentration of reactant A on a reactive stage is controlled along with the composition of components C and D in the top and bottom products, respectively. Cheng and Yu proposed another three-composition control structure where the feed tray locations are arranged depending on the disturbance coming into the system.5 Recently, Kumar and Kaistha21 compared the controllability of a conventional reactive distillation column design with that of an internally heat integrated design. The optimized design has been obtained by altering the feed tray locations to achieve minimum reboiler duty. They claimed that the composition-based control structures must be used for internally heat integrated ideal reactive distillation columns. However, measuring compositions is more difficult and expensive compared to measuring temperatures. Besides, they introduce large dead times to the control loop. Therefore, a temperature-based inferential control structure providing effective control is desirable whenever it is possible. There is no detailed work in the literature that evaluates the impact of feed tray locations on the effectiveness of temperature-based inferential control structures. There might be some feed tray locations for a double feed reactive distillation system, which result in significant energy savings without requiring significant loss in controllability. On the other hand, there might be other feed tray locations improving the controllability with a small increase in the energy cost. In this work, the impact of feed tray locations on the temperature-based controllability of an ideal double feed distillation column is studied. First, the controllability of the conventional design, where feed trays are located at the bottom and top of the reactive zone, is discussed. Then, two alternative designs are explored by shifting the feed tray locations in and

10.1021/ie901142s CCC: $40.75  2009 American Chemical Society Published on Web 10/15/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009

Table 1. Kinetic and VLE Parameters for Quaternary System pressure activation energy of reaction forward reverse specific reaction rate at 366 K forward reverse chemical equilibrium constant at 366 K average heat of reaction average heat of vaporization molecular weight of the mixture relative volatilities: RC/RA/RB/RD

8.5 bar 125520 J/mol 167360 J/mol 0.008 mol/(s mol) 0.004 mol/(s mol) 2 -41840 J/mol 29053.7 J/mol 50 g/mol 8/4/2/1

vapor pressure constantsa

A

B

C

D

AVP BVP

12.34 3862

11.65 3862

13.04 3862

10.96 3862

a ln Pjs ) AVP,j - BVP,j/T with temperature in kelvin and vapor pressure in bar.

out of the reactive zone by two trays. Finally, the controllability of the optimum design is given. Two different types of two-temperature inferential control structures are considered. The robustness of the control structures for different designs is compared based on two metrics: the steady-state deviation of the product purities and the change in the throughput that can be handled. 2. Process Studied The ideal reversible liquid-phase reaction considered in this study is a quaternary system including two reactants and two products. The overall reaction rate on the nth tray depends on the molar holdup MRX, the specific forward and backward reaction rates kF and kR, and the liquid mole fractions xn,j. Rn ) MRX(kFxnAxnB - kRxnCxnD)

(1)

Table 1 gives kinetic and vapor-liquid phase equilibrium (VLE) parameters used in this paper. The equilibrium constant at 366 K, (KEQ)366, is 2. The relative volatilities between adjacent components are constant at 2. The product C is the low-boiling component, so it is removed from the top of the column. The product D is the heaviest component and leaves the column from the bottom. R C > R A > RB > R D

(2)

The double feed reactive distillation column is sketched in Figure 1. It consists of a reactive zone in the middle of the stripping and rectifying sections. The column has a total number of 20 trays including 5 stripping, 10 reactive, and 5 rectifying trays. A reactive tray holdup MRX of 1000 moles is used, which gives reasonable tray liquid heights. The column operates at a pressure of 8.5 bar. In the conventional design, the light reactant A is fed from the bottom of reactive zone (tray 6), while the heavy reactant B enters from the top of the reactive section (tray 15). The fresh feed streams are pure, and they have flow rates of 12.6 mol/s. For convenience, the conventional design is briefly referred to as the 6/15 design. For a systematic evaluation, other designs differ from the conventional design in the location of feed trays. The second design is obtained by shifting the feed trays out of the reactive zone by 2 trays. In the third design, the feed trays are moved into the reactive section by 2 trays. Thus, these designs are referred to as 4/17 and 8/13 designs, respectively.

Figure 1. Flowsheet for a double feed reactive distillation column. Table 2. Salient Differences in Operating Conditions and Costs 4/17 design

6/15 design

8/13 design

2.97 32.80 11.82 269.70 8.54

2.70 29.34 base case 248.48 base case

2.39 25.47 -13.17 224.58 -9.62

RR VS (mol/s) energy saving (%) TAC ($103/yr) cost saving (%)

All three columns are designed to obtain 95 mol % purity of the distillate and bottoms products C and D, respectively. The reflux flow rate is manipulated by a proportional and integral (PI) controller to drive the composition of distillate product C to its desired value. This also sets the purity of the bottoms product D and the conversion in the reactive zone to their desired values. The distinct differences in operating conditions and costs are summarized in Table 2. The details of sizing and cost calculations are given in the work of Kaymak and Luyben.4 The results show that there is an increase of 11.82% and 8.54% in vapor boilup and total annual cost (TAC) by moving the feed locations out of the reactive zone by two trays, respectively. On the other hand, moving the feed locations into the reactive zone by two trays corresponds to vapor boilup and TAC reduction of 13.17% and 9.62%, respectively. Thus, these results show that significant energy savings can be obtained by changing the feed tray locations. 3. Control Structures Figure 2 shows two different types of two-temperature inferential control structures evaluated in this paper. In the first control structure, the base level is controlled by manipulating the bottoms flow rate. As the reflux drum level strategy, there might be two alternatives; constant reflux flow rate or constant reflux ratio. The constant reflux ratio strategy is used in this control structure based on the results of Kaymak and Luyben.8 The reflux drum level is controlled by the reflux flow rate, and the distillate flow rate is adjusted to give a constant reflux ratio. Perfect flow and pressure control is assumed. At this stage, there are three valves available: the vapor boilup and two fresh feed streams. Two of them should be manipulated to control the temperatures of two selected trays, while the third one serves


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Figure 2. Schematic of control structures: (a) CS1, (b) CS2.

as the production rate handle. For the first control structure, the fresh feed flow rates are manipulated to control the tray temperatures. Thus, the throughput is set by flow controlling the vapor boilup VS. This control structure with VS as the throughput manipulator is labeled CS1 for convenient reference. The second control structure uses the same base and reflux drum level strategies as used in CS1. The main difference between these two control structures is in the selection of the input variables to be used to control the tray temperatures. In this case, the flow rate of heavy-reactant fresh feed F0B is flow controlled and sets the throughput. The selected tray temperatures are measured and controlled by manipulating the flow rate of light-reactant fresh feed F0A and the vapor boilup VS. For convenient reference, the control structure using F0B as the throughput manipulator is labeled CS2. 3.1. Tray Selection. The most important issue for temperature-based inferential control structures is to find the most suitable trays to control. The locations of these trays and the loop pairings are chosen using sensitivity analysis and singular value decomposition method (SVD).22 The steady-state gain matrix between the inputs (two fresh feed flow rates and vapor boilup) and the outputs (the tray temperatures) is calculated numerically. SVD expresses the matrix of steady-state gains as a product of three matrices. KP ) UΣVT

(3)

The U matrix is the left singular vector matrix, the Σ matrix is the diagonal matrix of singular values, and the V matrix is the right singular vector matrix. The biggest element in each column of the U matrix indicates which tray of the column is the most sensitive to the input change applied. The ratio of the largest singular value to the smallest one is called as the

condition number. The large condition number indicates that the system is difficult to control. 3.2. Controller Tuning. The liquid capacities in the base of the column and the reflux drum are simply being used as surge volumes. Thus, maintaining these liquid levels at certain values is not necessary, and P only controllers with KC ) 2 are used on these two levels. Since the residence times of liquids in the base of the column and the reflux drum are set to be small enough, the system controllability is not affected. Two first-order lags of 60 s each are used in the temperature control loops of both control structures. The temperature loops in both control structures are tuned using the relay feedback test. The controller settings are calculated by the Tyreus-Luyben tuning method using the ultimate gain Ku and ultimate period Pu obtained from relay feedback test. Ku 3.2

(4)

τI ) 2.2Pu

(5)

KC )

Controller gains are dimensionless using a temperature transmitter span of 50 K. A detuning factor f ) 2 is used to get a faster closed-loop response while avoiding large oscillations. It should be noticed that all the valves are designed to be half open at steady-state. 4. Closed-Loop Results For three different designs, two-temperature inferential control structures CS1 and CS2 are evaluated using closed-loop dynamic simulations, where the system is disturbed with positive and negative step changes in the production rate handles.

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Figure 3. Sensitivity and SVD analysis for 6/15 design. Figure 4. Results of control structure CS1 for 6/15 design. Table 3. Tuning Parameters for the 6/15 Design CS CS1 CS2

pairing

KC

τI (min)

T2-F0A T12-F0B T18-F0B T2-F0A T12-VS T18-VS

0.96 8.78 0.20 0.96 2.20 0.75

12.83 16.87 347.60 12.83 15.03 10.27

4.1. Conventional (6/15) Design. The temperature sensitivities with respect to fresh feed streams and related SVD results for the control structure CS1 are given in the first column of Figure 3. Note that the gains are dimensionless using a temperature transmitter span of 50 K. The sensitivity analysis results for CS1 indicate that tray 2 and tray 3 are the most sensitive trays to the light and heavy reactant feed streams, respectively. Since these trays are very close to each other, it is not a good idea to use them in the same control structure. In addition, the signs of their steadystate gains are opposite. In a previous work, Kaymak and Luyben pointed out for CS1 type control structures that locations of the temperature control trays must be selected such that both temperature controllers have the same action.10 Thus, the second controlled tray has to be chosen from another section of the column, where the sign of its steady-state gain is the same as the sign of the gain of the first tray. SVD results are used to find the best pairings. Looking at the U parameters given in the lower left graph in Figure 3, the SVD analysis for control structure CS1 suggests that the temperature of tray 2 in the stripping section should be controlled by manipulating the fresh feed flow rate F0A, while the temperature of tray 18 in the rectifying section should be controlled by manipulating the fresh feed flow rate F0B. The values of controller gain KC and reset time τI are given in Table 3. Although the controller tuning parameters are reasonable for the T2/F0A loop, the value of reset time for the T18/F0B loop is extremely large due to the inverse response. Therefore, the T18/ F0B pairing is not a good choice for the controllability of the conventional design. Looking back to the SVD results indicates another sensitive tray located in the reactive zone. The secondary candidate is tray 12 with a fairly large steady-state gain and reasonable controller settings given in Table 3. The sensitivity analysis and SVD results of control structure CS2 are shown in the second column of Figure 3. Accordingly, the temperature of tray 18 in the rectifying section should be controlled by manipulating the vapor boilup VS, while the

temperature of tray 2 in the stripping section should be controlled by manipulating the fresh feed flow rate F0A. The controller gains and reset times calculated from the relayfeedback tests have reasonable values for both tray 2 and tray 18. Note that the T12/VS pairing may be an alternative to the T18/VS pairing based on the magnitude and sign of its steadystate gain. The tuning parameters for CS2 are also reported in Table 3. Closed-loop response of the control structure CS1 is given in Figure 4. A positive 20% step throughput change in VS is handled well. The response is well-behaved for controlled tray temperatures T2 and T12. They turn back to their respective setpoints in less than 4 h. More fresh feed streams F0A and F0B are required due to the increase of production rate handle VS. In the temperature-based inferential control structures, the purpose is holding the temperatures constant. Since the product compositions are not directly measured, there is no guarantee that the composition of the products will be held exactly at their specifications. Thus, small offsets in the product compositions are seen in Figure 4. However, these drops in product purities are quite acceptable, and final purities of the products remain close to 95%. Although the control structure CS1 successfully handles a fairly large +20% disturbance, it fails for the same size of throughput decrease. After a -20% change in VS is applied, the temperatures on tray 2 and tray 12 begin to decrease. Since less vapor boilup enters into the system, the temperature controllers start to decrease the flow rates of fresh feed streams F0A and F0B. Although the temperature on tray 2 settles down to its set-point within less than 4 h, the temperature on tray 12 cannot succeed recovering back to its set-point. As a result, the F0B valve shuts down in ∼1 h. On the other hand, CS1 handles a -10% throughput change without failing. Both T2 and T12 recover back to their specified values in less than 4 h. The fresh feed flow rates F0A and F0B decrease as the production rate handle VS is reduced. The bottoms and distillate compositions settle down slightly over their design value 95%. Figure 5 shows the closed-loop response of the control structure CS2 using the T18/VS and T2/F0A loops. The control structure is tested for (20% step changes in the throughput F0B. A 20% throughput increase is handled well, but the settling time increases to ∼8 h. It should be also noted that the purity of the bottoms product drops to less than 90 mol % D during the transient. On the other hand, the -20% throughput change


Figure 5. Results of control structure CS2 for 6/15 design using the T18/VS loop.

Figure 6. Steady-state variation of T12 and T18 with respect to F0B for 6/15 design.

results in a new steady-state with lower conversion for the control structure CS2 using tray 18 as the controlled variable. In this case, both the vapor boilup VS and the light fresh feed flow rate F0A increase to maintain the corresponding tray temperatures at their set-points. Although the controlled tray temperatures T2 and T18 recover back to their respective setpoints, the steady-state product purities do not settle down to their desired values. As a result of the increase in the F0A, there is an excess of reactant A in the system. This light excess reactant A flows up the column and leaves it from the distillate stream. This leads to an undesirable steady-state transition, where the final purity of the distillate stream decreases to less than 80 mol % C. As a result, the product purities are not wellcontrolled, while the system is stable. Highly nonlinear behavior of reactive distillation columns may result in steady-state multiplicity. In case of input multiplicity, different input values can give the same output value. The steady-state transition given in Figure 5 may be a result of an input multiplicity. Thus, a bifurcation study is performed. Figure 6 shows the steady-state variations of T18 (primary candidate) and T12 (secondary candidate) with respect to F0B. It is seen that T18 indicates gain sign reversals for a decrease and an increase in F0B. For an increase in F0B, the

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Figure 7. Results of control structure CS2 for 6/15 design using the T12/VS loop.

input-output relation exposes a crossover at 23.8%. The gain sign reversal occurs twice for a decrease in F0B, and the input-output relation crosses the base-case temperature value at -22.1% and -31.5%. This means that the sign of the process gain after the second crossover is same as the sign of the process gain around the base-case. Therefore, it can be concluded that a steady-state transition occurs as a result of the large excess of reactant A, while the system is stable with respect to the controller action. On the other hand, the input-output relation for T12 exhibits no crossover for large increases in F0B, while there is a crossover point slightly beyond -35% for a decrease in F0B. Thus, Figure 6 indicates that T12 has a wider operating window, and CS2 may handle a larger throughput change by controlling T12 instead of T18. Thus, the possible improvement is investigated by pairing the vapor boilup VS with T12 from the secondary sensitive region instead of T18 of the rectifying section. Figure 7 shows the result of this control structure to the (20% step changes in the production rate handle. The positive 20% throughput change response is well-behaved for T12. The transient drop in the purity of the bottoms product is much smaller than the drop with T18. The controlled tray temperatures recover back to their set-points in ∼4 h. The product purities are also maintained close to the desired values within 6 h. The control performance for the negative disturbance is essentially the same as that observed for the positive disturbance. On the basis of these results, it can be clearly seen that controlling T12 instead of T18 significantly improves the controllability of the system for CS2. 4.2. 4/17 Design. The control structures CS1 and CS2 are evaluated for the 4/17 design, where feed tray locations are moved out of the reactive zone by 2 trays. Trays to be controlled are found using the sensitivity analysis and SVD method by keeping track of the inverse response and the steady-state transition problems related to conventional design. The sensitivity analysis and SVD results for CS1 suggest that the temperature of tray 2 in the stripping section should be controlled by manipulating the fresh feed flow rate F0A, while the temperature of tray 12 in rectifying section should be controlled by manipulating the fresh feed flow rate F0B. For the control structure CS2, the sensitivity analysis and SVD results indicate that the temperature of tray 11 in the reactive zone should be controlled by manipulating the vapor boilup VS, while the temperature of tray 2 in the stripping section should be controlled by manipulating the fresh feed flow rate F0A. The

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Table 4. Tuning Parameters for 4/17 and 8/13 Designs design 4/17 8/13

pairing

KC

τI (min)

T2-F0A T12-F0B T11-VS T3-F0A T12-F0B T12-VS

1.36 8.86 2.14 0.74 3.63 1.77

9.35 13.20 13.75 16.68 28.97 19.80

controller tuning parameters are given in Table 4. They look reasonable. There is an increase in the magnitude of controller gain KC and a decrease in the magnitude of the reset time τI compared to the controller tuning parameters of the conventional design. Results for the control structure CS1 are given in Figure 8. The straight line shows the response of CS1 to a positive 20% step change in the production rate handle VS, while the dashed line is the response of a negative 10% step change. A +20% change in the throughput is handled well. The response of tray temperatures to the increase of the production rate handle results in the increase of fresh feed streams. The temperatures settle down to their set-points in less than 2 h, but the response is complete within ∼4 h for the purity of bottoms product. On the other hand, CS1 fails for a 10% throughput decrease. Although the decrease in the fresh feed stream F0A turns the temperature on tray 2 back to its set-point, the T12/F0B loop does not work as illustrated in Figure 8. The temperature controller on tray 12 starts to decrease the heavy fresh feed flow rate F0B with the decrease of the production rate handle VS, but the temperature on tray 12 cannot recover back to its set-point. As a result, the F0B valve shuts down in less than 2 h. Figure 9 gives the response of control structure CS2 to the (20% change in the production rate handle F0B. CS2 provides an effective control for the positive 20% disturbance, but its response time is a little longer compared to that of CS1. It takes ∼6 h for the controlled tray temperatures and the product purities settling down to their desired values. For the negative 20% disturbance, the control performance is essentially the same as that observed for the positive disturbance. The response is complete within 6 h. Less vapor boilup VS and light fresh feed F0A are required. 4.3. 8/13 Design. The control structures CS1 and CS2 are evaluated for the 8/13 design, where feed tray locations are moved into the reactive zone by 2 trays. For CS1, the results of sensitivity analysis and SVD method show that the temper-

Figure 8. Results of control structure CS1 for 4/17 design.



atures of tray 3 in the stripping section and tray 12 in the rectifying section should be controlled by manipulating the fresh feed flow rates F0A and F0B, respectively. Same results suggest T3/F0A and T12/VS pairings for control structure CS2. The controller tuning parameters are given in Table 4. It should be noted that as the magnitude of controller gain KC decreases, the magnitude of the reset time τI increases compared to the tuning parameters of the conventional design. This is true for all control loops used in CS1 and CS2. The dynamic response of control structure CS1 is shown in Figure 10. CS1 handles a +20% step throughput change successfully. A 10% throughput decrease is also handled without failing. The controlled tray temperatures turn back to their steady-state values for both positive and negative disturbances. The response time of the T12/VS loop is longer than the response time of the T3/F0A loop. The product purities settle down close to their specifications with acceptable offsets. Transient drop of distillate product purity is more than the transient drop of bottoms product purity. Settling down takes longer time for xD,C compared to xB,D. The response of T12 and xD,C is ∼7 h. Figure 11 illustrates the response of CS2 to the (20% change in the throughput F0B. The temperatures on tray 3 and tray 12 turn back to their set-points in ∼5 h for positive and negative disturbances. However, the responses of product purities take


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Figure 12. Variation of controlled reactive zone temperatures with respect to F0B.

Table 5. Magnitude of Throughput Changes That Could Be Handled

Table 6. Final Steady-State Deviations

control structure

CS1

CS2

design

pairings

4/17

T2-F0A T12-F0B

6/15

T2-F0A T12-F0B

8/13

T3-F0A T12-F0B

4/17

T2-F0A T11-VS

6/15

T2-F0A T12-VS

8/13

T2-F0A T12-VS

performance shut down handled -10% change in change in VS -14% change in change in VS -18% change in change in VS -28% change in change in F0B -41% change in change in F0B -44% change in change in F0B

VS +70%

steady-state deviation control structure

VS +70% VS +70%

+20% in VS CS1 -10% in VS

F0B +70% F0B +70% F0B +70%

longer. The distillate purity is maintained close to the specified value of 95 mol % C, but the settling time takes around 10 h. The results given in Figure 10 demonstrate that the 8/13 design provides stable control and the purities of products are held close to their specifications in the face of fairly large disturbances. 4.4. Comparison of the Designs. Two metrics are used for a quantitative comparison of the controllability of three designs with different feed tray locations. The first metric is the magnitude of the throughput step change that could be handled. The magnitude of step change is increased in positive and negative directions until the system shuts down. The change is limited by (70% in the case of no system shut down. The second metric is the steady-state deviation of distillate and bottoms purities for fairly large disturbances. For CS1, the disturbances are a positive 20% and a negative 10% step changes in throughput handle VS. The reason of selecting a -10% step change instead of a -20% can be concluded from the results of the first metric. For CS2, the disturbances are (20% step changes in the production rate handle F0B. Table 5 reports the magnitude of throughput changes that could be handled using CS1 and CS2 for three designs. Both control structures handle a +70% throughput change for all three designs, while they fail for smaller step throughput decreases. The control structure CS2 handles a larger throughput decrease compared to the control structure CS1 without failing for all designs. None of the designs are able to handle a -20% throughput change using control structure CS1, while the same magnitude of step decrease can be easily handled by all designs using control structure CS2.

disturbance

+20% in F0B CS2 -20% in F0B

design

|∆XD,C|

4/17 6/15 8/13 4/17 6/15 8/13 4/17 6/15 8/13 4/17 6/15 8/13

0.0023 0.0002 0.0037 0.0011 0.0042 0.0016 system shuts down 0.0018 0.0005 0.0021 0.0007 0.0024 0.0002 0.0037 0.0011 0.0044 0.0017 0.0023 0.0002 0.0037 0.0011 0.0043 0.0016

|∆XB,D|

Results from Table 5 indicate that larger throughput decreases can be handled as the feed tray locations are moved closer to each other for both control structures. This result can be supported by a bifurcation study. Figure 12 shows the steadystate variations of controlled tray temperatures in the reactive zone (T11 for 4/17 design and T12 for 6/15 and 8/13 designs) with respect to F0B. It is seen that all designs indicate gain sign reversal for a decrease in F0B, while no crossover is seen for large increases. The input-output relation exposes a crossover at -12.5% for 4/17 design. For 6/15 design, the input-output relation crosses the base-case temperature value at -36%, while the crossover point is beyond -40% for 8/13 design. Figure 12 shows that moving the feed tray locations closer to each other broadens the operating window. Thus, control structures handle larger throughput decreases without system shut down in 8/13 design compared to other designs. The steady-state deviations of distillate and bottoms purities are summarized in Table 6. The deviations are given as absolute values. Since CS1 cannot handle -20% step throughput changes for any design as given in Table 5, a 10% step decrease is used as the metric of this control structure. The results of CS2 show that there is no noticeable difference in the magnitude of deviations regarding the direction of the disturbances for all of the designs. The responses of CS1 and CS2 to a +20% step throughput change show that there is no significant difference between the final product purities regarding the control structures either. However, the results indicate that the product purity deviations increase for both control structures as the feed tray

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Figure 13. TAC change with respect to feed tray locations.

Figure 14. Results of control structure CS1 for optimum design.

locations move closer to each other. In addition, the deviations in the purity of distillate product are larger than the deviations in the purity of bottoms product for all of the designs. Although the 8/13 design deviates a little more than the conventional design for the same size of disturbances, it is able to handle larger throughput changes without any system shut down. The results show that the control structure CS2 handles larger throughput changes than the control structure CS1. Thus, it is reported that the 8/13 design, where significant energy saving is obtained by moving the fresh feed trays into the reactive zone by 2 trays, can be controlled by CS2 without requiring any significant loss in controllability compared to the conventional 6/15 design. 5. Results for Optimum Design Once it has been seen that the 8/13 design results in energy savings without losing significantly in terms of controllability, it is decided to investigate the controllability of the design with economically optimum feed tray locations. The effect of feed tray locations on the total annual cost (TAC) is given in Figure 13. The results show that the optimum design is obtained by feeding the light reactant A from tray 10 and the heavy reactant B from tray 13. This optimum design reduces the vapor boilup and TAC by 22.54% and 16.55% compared to the conventional design, respectively. The sensitivity analysis and the corresponding SVD results for CS1 indicate that the temperature of tray 5 in stripping section should be controlled by manipulating the fresh feed flow rate F0A, while the temperature of tray 15 in the rectifying section should be controlled by manipulating the fresh feed flow rate F0B. Although the tuned controller parameters have reasonable values for the T5/F0A loop, the reset time for the T15/F0B loop is extremely large (τI ) 322.67). In contrast to the designs investigated in the previous section, no other sensitive region is observed for the optimum design. On the other hand, the sensitivity analysis suggests T5/F0A and T15/VS pairings for control structure CS2. SVD results show that the T12/VS pairing can be also used as an alternative to T15/VS in the case of steadystate transition. Figure 14 illustrates the closed-loop response of CS1 for (5% step changes in the throughput VS. As expected from the extremely large value of the reset time for the T15/F0B loop, the response of CS1 is very sluggish even for a small disturbance. The temperature of tray 15 cannot turn back to its set-point in

Figure 15. Results of control structure CS2 for optimum design.

20 h, while T5 recovers in less than 10 h. Therefore, the product purities do not settle down to their specified values in 20 h. These results demonstrate that CS1 does not provide an effective control for the optimum design. Control structure CS2 using the T15/VS loop handles a 20% throughput decrease, but fails for a +20% step change. However, CS2 using the secondary sensitive tray temperature T12 handles both (20% step throughput changes as given in Figure 15. Tray temperatures turn back to their set-points in ∼10 h. The purity of the bottoms product decreases to less than 90 mol % D during the transient, but the product purities are maintained close to their specified value within 10 h. On the other hand, the distillate purity deviates more from its specified steady-state condition compared to the conventional design. As another metric, the magnitude of the disturbance is increased in both directions. Although CS2 handles large negative disturbances that could be also handled by the conventional design, the system shuts down in the face of a +70% disturbance. In addition, as the magnitude of disturbance increases, the response takes significantly longer time for completion due to the larger reset time of the tray 12 controller. These results indicate that the optimum steady-state design does not show the best controllability. There is a trade-off between steady-state economics and dynamic controllability. It


appears that reactive distillation columns with higher energy costs may have long run advantages by providing more stable operations and achieving less variability in product quality. 6. Conclusion The impact of feed tray locations on the energy savings and temperature-based controllability of an ideal double feed reactive distillation column is explored. For the conventional reactive distillation column design, the light and heavy fresh feed streams are located at the bottom and top of the reactive zone, respectively. Alternative designs are created by moving the feed trays in and out of the reactive zone by two trays. Results indicate that adjusting the feed tray locations leads to significant energy savings. For the case study examined, moving the feed locations in the reactive zone by two trays decreases the vapor boilup by 13.17% compared to the conventional design. Two different types of two-temperature control structures are evaluated for each of the designs. In CS1, two fresh feed flow rates are manipulated to control the temperatures on two trays, while the reboiler heat-input serves as the production rate handle. In CS2, the heavy fresh feed stream serves as the production rate handle, and tray temperatures are controlled by manipulating the light fresh feed flow rate and the vapor boilup. Results show that CS2 is superior to CS1 by handling significantly larger throughput changes for any feed tray locations studied. For fairly large (20% throughput changes, CS2 provides an effective control of the energy-efficient 8/13 design. In addition, the 8/13 design handles a larger range of throughput changes without any system shut down or steadystate transition compared to the conventional 6/15 design. On the other hand, the optimum internally heat integrated 10/13 design reduces the vapor boilup by 22.54% compared to the conventional design. The (20% throughput changes can be handled for the optimum design using CS2, but it has a longer settling time and a higher steady-state deviation for the distillate purity compared to the other designs. In addition, the optimum 10/13 design is not able to handle the same range of throughput changes that can be handled by the conventional 6/15 design. These results indicate that significant energy savings can be obtained by adjusting the fresh feed trays. Thus, it is claimed that the feed tray location should be included as a design optimization variable. In addition, the effects of feed tray locations on the dynamic controllability should be considered at the design stage. The results demonstrate that there are energyefficient designs, which can be controlled by a temperaturebased inferential control structure without any significant loss in controllability. 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) f ) detuning factor F0j ) fresh feed flow rate of reactant j (mol/s) kF ) specific reaction rate of forward reaction (mol/(s mol)) kR ) specific reaction rate of reverse reaction (mol/(s mol)) KC ) controller gain KEQ ) equilibrium constant KP ) steady-state gain KU ) ultimate gain MRX ) liquid holdup on reactive trays (mol)

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NR ) number of the rectifying trays NRX ) number of the reactive trays NS ) number of the stripping trays NT ) number of trays in the column PU ) ultimate period (min) R ) reflux (mol/s) Rn ) rate of reaction on tray n (mol/s) RR ) reflux ratio Ti ) column temperature on tray i (K) U ) left singular vector matrix V ) right 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 Rj ) relative volatility of component j with respect to the heaviest component D ∆T ) temperature difference (K) Σ ) diagonal matrix of singular values τI ) reset time (min)

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ReceiVed for reView July 16, 2009 ReVised manuscript receiVed September 25, 2009 Accepted October 02, 2009 IE901142S

Effect of Feed Tray Location on Temperature-Based Inferential Control

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