Comparison of Two Types of Two-Temperature Control Structures for

Ind. Eng. Chem. Res. 2005, 44, 4625-4640

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Comparison of Two Types of Two-Temperature Control Structures for Reactive Distillation Columns Devrim B. Kaymak and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

The purpose of this paper is to compare the effectiveness of two different types of two-temperature control structures for reactive distillation columns. The first was a structure proposed by Roat et al. (Roat, S.; Downs, J.; Vogel, E.; Doss, J. Integration of rigorous dynamic modeling and control system synthesis for distillation columns. Chemical Process Control; CPC III; Elsevier: Amsterdam, The Netherlands, 1986) almost 2 decades ago in which two tray temperatures in the column are controlled by manipulating two fresh feed streams. The second was recently proposed by Yu and co-workers (Yu, C. C. Private communication, 2004) in which two tray temperatures are controlled by manipulating one of the fresh feed streams and vapor boilup. The second fresh feed stream is flow controlled. We discuss the issues of the selection of the fresh feed stream to be manipulated and the importance of the tuning procedure for the two interacting temperature controllers. Both an ideal reaction system and the methyl acetate system are studied. The effectiveness and robustness of these control structures are compared in the face of disturbances in the production rate and fresh feed compositions. One of the main conclusions is that the selection of the manipulated fresh feed stream in the second structure has an important role in the stability of the system. Sequential tuning of the interacting temperature controllers is sometimes necessary. 1. Introduction Control studies of reactive distillation columns have explored a variety of chemical reactions, flowsheets, and control structures. For a chemical reaction with two reactants, the type of flowsheet depends on whether we want to operate the reactive distillation column “neat”, i.e., no excess of either reactant. The “excess-reactant” flowsheet requires two columns and is therefore more expensive. However, it is easier to control. The “neat” flowsheet has better steady-state economics but presents challenging control problems because of the need to precisely balance the stoichiometry of the reaction. In some chemical systems, the only way this can be achieved is by the use of a composition analyzer. A composition analyzer may not be required if two products are produced. In this case, the temperature information may be rich enough to infer compositions with sufficient accuracy so that direct composition measurement is not absolutely necessary for “neat” operation. The ideal case considered by Roat et al.1 (A + B S C + D) and the production of methyl acetate (methanol + acetic acid S methyl acetate + water) are examples of this type of chemistry. In our present study, we consider a reactive distillation column, operated without an excess of one of the reactants, in which two reactants are fed to produce two products. We consider both an ideal system and the methyl acetate system. In the paper by Roat et al.,1 two proportional-integral (PI) temperature controllers are used to maintain two tray temperatures in the column by manipulating the two fresh feed streams. Production rate changes are achieved by changing the vapor boilup. In a previous * To whom correspondence should be addressed. Tel.: (610) 758-4256. Fax: (610) 758-5057. E-mail: [email protected].

paper,3 we studied this control structure, in particular looking at the tradeoff between design and control, i.e., the controllability effects of adding more reactive trays. We call this control structure CS7. In a recent paper by Yu and co-workers,2 an alternative two-temperature control structure is proposed, which we call CS8. One of the fresh feed streams is flow controlled, which provides a direct handle on the production rate. The second fresh feed is manipulated to control a tray temperature. A second tray temperature is controlled by manipulating the reboiler heat input. The purpose of this paper is to compare the effectiveness of these two alternative control structures. The dynamic responses for several disturbances are explored. Control effectiveness is evaluated in terms of the system not shutting down and not producing products that deviate significantly from their specifications. This paper also addresses the issues of the selection of the manipulated fresh feed stream for the second type of control structure and the tuning procedure. 2. Processes Studied 2.1. Ideal Process. The first process considered is taken from a previous paper4 in which an ideal reversible liquid-phase reaction occurs on the reactive trays.

A+BSC+D

(1)

Kinetic and physical properties and vapor-liquid equilibrium parameters are given in Table 1. Ideal vapor-liquid equilibrium is assumed with constant volatilities, such that the lightest component is one of the products (C) and the heaviest component is the other product (D). Reactant component A is lighter than the

10.1021/ie058012m CCC: $30.25 © 2005 American Chemical Society Published on Web 05/26/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005

Table 1. Physical Data for the Ideal Process activation energy of the reaction forward reverse specific reaction rate at 366 K forward reverse average heat of reaction average heat of vaporization molecular weight of the mixture ideal gas constant relative volatilities RA RB RC RD

Table 2. Kinetic Data for the Methyl Acetate Process activation energy of the reaction forward reverse preexponential factor for the reaction forward reverse

cal/mol 30 000 40 000 kmol/(s kmol) 0.008 0.004 -10 000 6944 50 1.987

cal/mol cal/mol g/mol cal/(mol K)

4 2 8 1

vapor-pressure constantsa

A

B

C

D

AVP BVP

12.34 3862

11.65 3862

13.04 3862

10.96 3862

a ln Ps ) A VP,j - BVP,j/T with temperature in K and vapor j pressure in bar.

other reactant B. Thus, the relative volatilities are

RC ) 8, RA ) 4, RB ) 2, RD ) 1

(2)

The design objective is to obtain 95% conversion for fixed fresh feed flow rates of pure reactants at 12.6 mol/ s. Light reactant A is fed from the bottom of the reactive zone, while heavy reactant B is introduced from the top of this zone. Essentially all of C leaves in the distillate and all of D leaves in the bottoms because of the relative volatilities. The specifications for the product impurities are assumed to be 5 mol % B in the bottoms and 5 mol % A in the distillate stream. Reactive tray holdup Mi is 1000 mol, which gives reasonable tray liquid heights. A previous paper3 showed that the optimum design for this system has a total of 17 trays including 7 reactive trays. However, this design does not provide good dynamic controllability. Therefore, in the present paper, we use a suboptimum design with 20 total trays, where 10 of them are reactive trays. The optimum column pressure is 8.5 bar. The tray numbers are counted from the bottom up, excluding the condenser and reboiler. 2.2. Methyl Acetate Process. The methyl acetate reactive distillation column is used as an example of a real two-reactant/two-product system with a reversible reaction. We use the optimum economic steady-state design presented by Yu and co-workers,5 which is a modified version of the design first proposed by Agreda et al.6 The reaction is the esterification of acetic acid with methanol:

HOAc + MeOH S MeAc + H2O

(3)

The reaction rate is expressed in the pseudohomogeneous model using component activities aj and is based on the catalyst weight (mcat.). In the application of this reaction kinetics to the reactive distillation, a catalyst density of 770 kg/m3 is used to convert the volume into the catalyst weight. The catalyst holdup on each tray is 29.5 kg.

r ) mcat.(kFaHOAcaMeOH - kRaMeAcaH2O)

(4)

Table 2 gives the reaction kinetics used. The UNIQUAC

kJ/kmol 49 190 69 230 kmol/(s kgcat.)

2.961 × 104 1.348 × 106

physical properties package is used with parameters given by Yu and co-workers.5 For this system, the heavy reactant is acetic acid (HOAc) and the light reactant is methanol (MeOH). Water (H2O), the heavy product of the reaction, is taken from the bottom, while the light product methyl acetate (MeAc) is removed from the top. The purity of both products is 98 mol %. The fresh feed rates of acetic acid and methanol are 50 kmol/h. The total number of trays is 38, with 3 in the stripping section and 1 in the rectifying section. Thus, this design has a very large number of reactive trays (34). Because Aspen Plus and Aspen Dynamics are used to simulate this case, the Aspen stage-numbering notation is used. Stages are counted from the top down and include the condenser and reboiler. The reactive zone runs from stage 3 to stage 36. The light reactant is fed to stage 27, which is about a third of the way up in the reactive zone. The heavy reactant is introduced on stage 4, which is near the top of the reactive zone. It is interesting that the design developed by Yu and co-workers5 does not feed the reactants at the ends of the reactive zone. 3. Control Structures Because composition analyzers are expensive, require high maintenance, and introduce deadtime into the control loop, it is desirable to use inferential temperature measurements instead of direct composition measurements whenever possible. When two products are produced in the reaction, it is sometimes possible to use the temperatures on two trays in the column instead of using a composition analyzer. Two different types of control structures are compared in this paper. Figure 1 shows the first type (called “CS7”), in which two temperature controllers manipulate the two fresh feed flow rates to maintain the temperatures on two trays in the column. The reboiler heat input is flow controlled and serves as the production rate handle. The base level is controlled by manipulating the bottoms flow rate. The reflux drum level is controlled by the reflux flow rate, and the distillate flow rate is adjusted to give a constant reflux ratio (RR). This structure is selected because of the fairly high RR. Figure 2 shows two versions of the second type of control structure, which we call “CS8”. This structure uses the same base and reflux drum level strategies as those used in CS7. In the first version (CS8A), the lightreactant fresh feed stream is flow controlled and serves as the production rate handle. The heavy-reactant fresh feed stream is ratioed to the light-reactant feed. The ratio is set by one of the tray temperature controllers. In the second version (CS8B), the heavy-reactant fresh feed stream is flow controlled and serves as the production rate handle. The light-reactant fresh feed stream is ratioed to the heavy-reactant feed, and the ratio is set by one of the tray temperature controllers. In both versions, the temperature of another tray in the column is controlled by manipulating the vapor boilup.

Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005 4627

Figure 1. Control structure CS7.

The discussion in this paper considers only conventional linear PI controllers in a decentralized (single input single output) environment. Because the dynamics

Figure 2. Control structures: (a) CS8A; (b) CS8B.

of flow measurement are fast, PI controllers with values of KC ) 0.5 and τI ) 0.3 min are used for all flow controllers in the Aspen Dynamics simulations. The liquid capacities in the base of the column and in the reflux drum are simply being used as surge volumes. Therefore, maintaining these liquid levels at certain values is not necessary, and P-only controllers with KC ) 2 are used on all levels. Temperature controllers are tuned using the TyreusLuyben tuning method. Two first-order measurement lags of 60 s each are used in all temperature loops. The relay-feedback method is used to obtain the ultimate gain and ultimate period. The valves are designed to be half open at steady state. The column pressure is controlled by manipulating the condenser heat duty. The selection of the trays to temperature control is one of the main issues in these structures. On the basis of the steady-state results, several interaction and stability measures can be developed to assist in the analysis of complex, multivariable control systems. From the rigorous steady-state simulations, the gain matrix between the inputs (the two fresh feed flow rates and/or the vapor boilup) and the outputs (the temperatures on all trays) is calculated numerically. Using the steady-state gains, the singular value decomposition (SVD) method is used to indicate the most sensitive trays to be controlled. The steady-state gains between the tray temperatures and inputs can be positive in some sections of the column and negative in others. This means that the action of the two temperature controllers (direct or reverse acting) can be different depending on what trays are selected. These control structures are evaluated using dynamic simulation and disturbing the system with step changes

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Figure 3. Gains and SVD results for the ideal process.

in the production rate handle (∆F0j or ∆VS) and in the compositions of the fresh feeds (∆z0A and ∆z0B).

Table 3. Tuning Parameters for the Ideal Processa CS CS7

4. Results for the Ideal Process 4.1. Steady-State Gains and SVD Analysis. The suboptimum economic steady-state design of the ideal process studied in this paper operates at 8.5 bar and has 5/10/5 trays (stripping/reactive/rectifying). Figure 3 gives values for steady-state gains (KF0A, KF0B, and KVS) and their related SVD results for CS7, CS8A, and CS8B control structures. Note that these gains are dimensionless. Temperature transmitter spans of 50 K are used, and valves are designed to be 50% open. The steady-state gains are to be used to design a linear controller, so they should correspond to a linearized version of the process. They were calculated numerically by making very small changes in the inputs (0.005%). In this ideal process, we are using trays numbered from the bottom, so the left side of the graphs shown in Figure 3 corresponds to the base of the column and the right side to the top. The trays in the stripping section of the column are the most sensitive region to the changes for all three input manipulated variables, as shown in the upper left graph in Figure 3. For a two-temperature control structure, only one of the trays can be chosen from the stripping section, and the second controlled tray should be chosen from another region. The SVD method is used to choose the best pairings. The steady-state gains between stripping tray temperatures and the fresh feed stream F0A are negative, and they have the biggest magnitudes of all of the steady-state gains. In addition, the gains KF0A for the rest of the trays are very small, which indicates very little sensitivity to changes in input F0A. However, for the other two inputs, F0B and VS, there are several sensitive regions (secondary peaks/valleys that are smaller than the largest peak/valley). Let us first look at the CS7 control structure. The SVD analysis, looking at the UCS7 parameters shown in the upper right graph in Figure 3, suggests that the temperature of tray 18 in the rectifying section should be controlled by manipulating the fresh feed flow rate F0B, while the temperature of tray 2 in the stripping

CS8A

CS8B

tuning

pairing

KC

τI (min)

individual individual sequential individual sequential individual individual sequential individual sequential individual individual sequential

F0A/T2 F0B/T12 F0B/T12 F0B/T18 F0B/T18 VS/T2 F0B/T12 F0B/T12 F0B/T18 F0B/T18 VS/T18 F0A/T2 F0A/T2

0.95 8.78 7.60 0.20 0.19 0.56 8.78 7.96 0.20 0.17 0.73 0.96 0.95

12.83 16.87 17.23 347.60 339.90 8.98 16.87 12.28 347.60 261.43 10.27 12.83 13.02

a Controller gains are dimensionless using temperature transmitter spans of 50 K and valve sizes of twice the steady-state flow rates.

section should be controlled by manipulating the fresh feed flow rate F0A for control structure CS7. As given in Table 3, the controller gains and reset times calculated from the relay-feedback tests have reasonable values for tray 2. However, the controller for tray 18 has a reset time of 347 min. This occurs because the T18/F0B open-loop transfer function has an inverse response. Therefore, the T18/F0B pairing is not a good choice for the controllability of the system. As mentioned above, there is another region sensitive to the changes in the fresh feed flow rate F0B. This is located at tray 12 in the reactive zone, where the steadystate gain is fairly large. Step tests show that there is no inverse response behavior for the T12/F0B pairing, and the reset time is 17 min. Thus, the closed-loop dynamics of the system should be improved by using tray 12 instead of tray 18. It is important to realize that the open-loop gains of both of the transfer functions for the temperature locations selected in the CS7 structure are negative. This means that the gains of both of the controllers are negative, i.e., have “direct action”. The SVD analysis for the CS8A structure is given in the lower left graph in Figure 3. The UCS8A parameters suggest that the temperature of tray 18 in the rectifying section should be controlled by manipulating the fresh feed flow rate F0B, while the temperature of tray 2 in


Figure 4. Response of CS8A: step changes in fresh feed stream ∆F0A.

the stripping section should be controlled by manipulating the vapor boilup VS for control structure CS8A. The control loops of the CS8A structure have opposite actions. The tray 18 temperature controller will be direct acting because the T18/F0B steady-state gain is negative, while the tray 2 temperature controller will be reverse acting because the T2/VS steady-state gain is positive. As can be seen from Table 3, the T18/F0B pairing is not a good choice because of the unrealistically large reset time, which is the result of the inverse response of the open-loop transfer function. Therefore, tray 12 in the reactive zone is selected as the secondary sensitive region. The SVD analysis for the CS8B structure is given in the lower right graph in Figure 3. The UCS8B parameters suggest a somewhat unusual control pairing. The temperature of tray 18 in the upper part of the column 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 tray 2 temperature controller is direct acting because the T2/F0A steady-state gain is negative, while the tray 18 temperature controller will be reverse acting because the T18/VS steady-state gain is positive. As given in Table 3, the controller gains and reset times calculated from the relay-feedback tests have reasonable values for both tray 2 and tray 18. There are two ways to tune controllers when using the relay-feedback method: individual and sequential tuning. By individual tuning, we mean that each temperature-control loop is tuned individually while the other temperature-control loop is on manual. For sequential tuning, one of the loops is tuned individually first, and then the second loop is tuned while the first loop is kept on automatic. As is seen in Table 3, there is almost no difference between the controller parameters calculated by individual and sequential relayfeedback tests. Thus, the controller parameters calculated by individual tuning are used for the ideal process

case. As discussed later, there is a significant difference between these tuning methods in the methyl acetate process. 4.2. Performance of the CS7 Structure. The details of robustness and rangeablity for CS7 have been presented in our previous paper,3 and only important results are summarized below. Very large positive step changes in the vapor boilup (the production rate handle) can be handled by this control structure. The purities of both products are maintained within 1% of the desired 95% specification. However, the robustness of the system is more limited for negative step changes in the vapor boilup. When the vapor boilup decreases, tray temperatures throughout the column decrease. This results in both of the temperature controllers decreasing the fresh feeds. A negative 10% step change can be handled with CS7. However, the system shuts down for a 12% decrease in the vapor boilup for the 5/10/5 design. The second type of disturbance is a step change in the composition of the fresh feeds. For a change in the composition of fresh feed F0A from pure component A to a mixture of components A and B, the system is dynamically stable for a wide range of disturbances. However, disturbances larger than 4% of B in this feed stream result in a decrease in the bottoms purity below 94%, while the distillate purity is above specification. For a change in the composition of fresh feed F0B from pure component B to a mixture of components A and B, the system is dynamically stable for changes up to 8% of A in this feed stream, but the system shuts down for 10% composition changes. In the stable region, both the distillate and bottoms purities are within the desired specification. 4.3. Performance of the CS8A Structure. Figure 4 shows the responses of the CS8A structure to +20% and -10% step changes in fresh feed stream F0A (the production rate handle). The 20% increase can be handled by the CS8A structure, and the system settles down to a steady state in about 2 h. The purities of both

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Figure 5. Response of CS8A: step changes in fresh feed compositions.

products are maintained close to the desired 95% specification. However, the robustness of the system for step decreases in F0A is not as good as its response to the step increases. When a negative step change is applied to fresh feed stream F0A, the tray 12 temperature starts to decrease. That decrease in temperature results also in a decrease of fresh feed stream F0B. As Figure 4 shows, a negative 10% change shuts off the F0B stream completely, and the system shuts down after a short time. Figure 5 gives the responses of CS8A to 3% step changes in the composition of either of the fresh feeds. These disturbances in the compositions of the fresh feed streams F0A and F0B make the system dynamically unstable. The system shows a very oscillatory response for about 3 h, and subsequently the F0B stream shuts off completely. The CS8A control structure does not provide stable dynamic controllability for some of the disturbances. Thus, we can conclude that it is not as good a control structure as the CS7 structure for the ideal process. 4.4. Performance of the CS8B Structure. The responses of the CS8B structure to +20% and -15% step changes in fresh feed stream F0B (the production rate handle) are given in Figure 6. Results show that a 20% increase can be handled without any oscillation, and the system achieves a steady state in about 3 h, with the purities of both products maintained close to the desired values. However, there is a large transient drop in the purity of the bottoms xB,D all the way to about 88%. The CS8B structure is also stable for a 15% step decrease in F0B, but it takes almost 10 h for the system to settle down and there is a large transient drop in the purity of the distillate xD,C. It should be noted that the system is dynamically stable even for bigger step decreases, but disturbances larger than a 15% decrease result in a new steady state in which products are below the 94% limit.

The second type of disturbance is a step change in the composition of the fresh feeds. Figure 7 gives the results for changes in the composition of fresh feed F0A from pure component A to a mixture of components A and B. The system is dynamically stable for large changes. However, disturbances larger than 2% of B result in a decrease in the distillate purity below 94%. Notice that the bottoms purity is above specification. Figure 8 gives the results for changes in the composition of fresh feed F0B from pure component B to a mixture of components A and B. Although the system is dynamically stable for larger disturbances, the distillate purity drops below the 94% limit even for a 1% disturbance. The CS8B structure has a narrow operability region with respect to maintaining product purities for changes in the composition of the fresh feeds. However, this control structure is able to provide dynamically stable control for all types of disturbances without shutting down. Thus, the CS8B control structure appears to be the better of the two CS8 alternatives for the ideal process. For control structure CS8A, the T12/F0B pairing, which has the third biggest peak in the KF0B curve, is used. For control structure CS8B, the T18/VS pairing, which has the second biggest peak in the KVS curve, is used. This means that the second pairing (T18/VS) of control structure CS8B is more sensitive to the disturbances compared to the pairing T12/F0B of control structure CS8A. This explains why the CS8B control structure gives better responses to disturbances than the CS8A control structure. As we will see later in this paper, the opposite is true for the methyl acetate process. 4.5. Comparison of the CS7 and CS8B Structures. The robustness and limits of the three different control structures are discussed in the previous sections. It has been seen that the CS8A control structure cannot handle most of the disturbances as well as the CS8B control structure. Therefore, only the CS7 and CS8B


Figure 6. Response of CS8B: step changes in fresh feed stream ∆F0B.

Figure 7. Response of CS8B: step changes in fresh feed composition ∆z0A.

structures are compared. The sizes of the disturbances are selected so that the process operates in dynamically stable conditions, which means that some product purities can go beyond the limits for some disturbances. The purpose of this section is to compare the effectiveness of these two control structures when the same size of disturbances is applied. Figure 9 gives a direct comparison of the performances of the CS7 and CS8B structures for a 10% increase in the production rate handles (vapor boilup VS or fresh feed stream F0B). The results show that a

10% increase can be handled by both control structures. The final steady-state purities of both products are maintained close to the desired 95% specification. The CS7 structure shows more aggressive behavior and settles out faster than the CS8B structure. The response of the CS8B structure is smoother, but it takes more than 5 h for this control structure to come to the new steady state and there is a large transient drop in bottoms purity, which lasts for about 2 h. Retuning the temperature controllers did not improve this response. A possible reason for the slower response of CS8B is

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Figure 8. Response of CS8B: step change in fresh feed composition ∆z0B.

Figure 9. Comparison of CS7 and CS8B: a 10% step increase in the production rate handle.

that the interaction between the two temperature controllers is more severe. The vapor boilup affects both temperatures quickly, while changes in the fresh feed flow rate take some time to affect the tray 2 temperature because of the liquid hydraulic lags. The comparison of the CS7 and CS8B structures for a 10% decrease in the production rate handles (vapor boilup VS or fresh feed stream F0B) is given in Figure 10. Both control structures can handle this disturbance, but the CS7 structure responds more quickly. There is

a large transient drop in the distillate purity with the CS8B structure that lasts about 3 h. Figure 11 gives a comparison for the change in the composition of the fresh feed F0A from pure component A to a mixture of 97% A and 3% B. Both control structures have smooth responses to this disturbance. However, the CS8B structure results in a decrease in the distillate purity to 93%. Figure 12 gives responses for the change in the composition of fresh feed F0B to a mixture of 99% B and 1% A. The changes in the steady-


Figure 10. Comparison of CS7 and CS8B: a 10% step decrease in the production rate handle.

Figure 11. Comparison of CS7 and CS8B: a 3% step change in fresh feed composition ∆z0A.

state product purities are larger for the CS8B structure. As mentioned in a previous section, for larger disturbances, the purity of the distillate drops below the 94% limit with the CS8B structure, while the CS7 structure keeps both product purities within the limit. 5. Results for the Methyl Acetate Process Figure 13 gives values for steady-state gains (KMeOH, KHOAc, and KQR) and their related SVD results for the

CS7 structure and the two CS8 control structures. Note that these gains are dimensionless. In the methyl acetate process, we use stages numbered from the top down, so the left side of the graphs shown in Figure 13 corresponds to the top of the column and the right side to the base. As given in the upper left graph of Figure 13, the stages in the stripping section of the column are the regions most sensitive to the changes in FMeOH and QR. The steady-state gains between stripping tray temper-

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Figure 12. Comparison of CS7 and CS8B: a 1% step change in fresh feed composition ∆z0B.

Figure 13. Gains and SVD results for the methyl acetate process.

atures and the fresh feed stream FMeOH are negative, and they have the largest magnitudes of all of the steady-state gains. In addition, the FMeOH gains for the rest of the trays are very small, which indicates very little sensitivity to changes in input FMeOH. The openloop steady-state gains for the input QR are positive in the stripping section. There is also another sensitive region (secondary peak that is smaller than the largest peak) for QR in the column, where steady-state gains are also positive. However, the tray most sensitive to the changes in FHOAc is in the reactive zone (stage 29), and the signs of open-loop steady-state gains in this region are negative. Let us first look at the CS7 control structure. The

SVD analysis parameters shown in the upper right graph in Figure 13 suggest that the temperature of stage 38 in the stripping section should be controlled by manipulating the fresh feed flow rate FMeOH, while the temperature of stage 29 in the reactive zone should be controlled by manipulating the fresh feed flow rate FHOAc for control structure CS7. It is important to realize that the open-loop gains of both of the transfer functions for the temperature locations selected in the CS7 structure are negative. This means that the gains of both of the controllers are negative; i.e., they have “direct action”. The SVD analysis for the CS8A structure is given in the lower left graph in Figure 13. The UCS8A parameters

Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005 4635 Table 4. Tuning Parameters for the Methyl Acetate Processa CS CS7 CS8A

tuning

pairing

KC

τI (min)

individual individual sequential individual individual sequential

FMEOH/T38 FHOAC/T29 FHOAC/T29 QR/T38 FHOAC/T29 FHOAC/T29

2.40 1.70 0.54 0.58 1.60 0.60

21.65 79.86 122.76 20.86 87.12 246.84

a Controller gains are dimensionless using temperature transmitter spans of 50 K and valve sizes of twice the steady-state flow rates.

suggest that the temperature of stage 29 in the reactive zone should be controlled by manipulating the fresh feed flow rate FHOAc, while the temperature of stage 38 in the stripping section should be controlled by manipulating the reboiler heat duty QR for control structure CS8A. The temperature controllers in the CS8A structure have opposite actions. The stage 29 temperature controller will be direct acting because the T29/FHOAc steady-state gain is negative, while the stage 38 temperature controller will be reverse acting because the T38/QR steadystate gain is positive. The SVD analysis for the CS8B structure is given in the lower right graph in Figure 13. The UCS8B parameters suggest that the temperature of stage 29 in the reactive zone should be controlled by manipulating the reboiler heat duty, while the temperature of stage 38 in the stripping section should be controlled by manipulating the fresh feed of methanol. The temperature controllers in the CS8B structure have opposite actions. The stage 38 temperature controller will be direct acting because the T38/FMeOH steady-state gain is negative, while the stage 29 temperature controller will be reverse acting because the T29/QR steady-state gain is positive. 5.1. Effect of the Tuning Procedure. One of the differences between the ideal process and the methyl acetate process is the tuning procedure used for finding

controller parameters. As mentioned in the previous section, there are two ways to tune the two interacting controllers. In the first, both temperature controllers are tuned individually. By individual tuning, we mean that each temperature controller is tuned with the other temperature controller on manual. In the second tuning method, the controllers are tuned sequentially. One of the controllers is tuned individually first, and then the second controller is tuned with the first controller on automatic. For the ideal process, there is almost no difference between controller parameters calculated by individual and sequential tuning. However, this is not true for the methyl acetate process. The tuning results given in Table 4 show that there are big differences between the tuning parameters calculated by the two procedures. The temperature controller in the T38/FMeOH or T38/QR pairing, depending on the control structure, is tuned first. Then the controller in the T29/FHOAc pairing is tuned sequentially, with the T38/FMeOH or T38/QR pairing temperature controller on automatic. Figure 14 shows the responses of control structure CS7 to a 10% step decrease in the reboiler heat duty QR using two different types of tuning parameters. The process settles down to the steady state smoothly with the sequentially tuned parameters (dashed lines). However, the system is dynamically unstable when individually tuned controller parameters are used. Figure 15 shows the same comparison for control structure CS8A with a 10% step decrease in fresh feed stream FMeOH. Similar to control structure CS7, the process settles down to the steady state smoothly with sequentially tuned parameters. With individual tuning parameters, the system has a very oscillatory response. The results of these tuning procedures for other types of disturbances show similar responses. Unlike the ideal process, there is an important difference between using

Figure 14. Comparison of tuning methods for CS7 (the methyl acetate case).

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Figure 15. Comparison of tuning methods for CS8A (the methyl acetate case).

Figure 16. Comparison of CS7 and CS8A: a 10% step increase in the production rate handle.

individually or sequentially tuned parameters in the methyl acetate process. 5.2. Comparison of the CS7 and CS8A Structures. Figure 16 gives a direct comparison of the CS7 and CS8A structures for a 10% increase in the production rate handle (vapor boilup VS or fresh feed stream FMeOH). Results show that the 10% increase can be handled by both control structures with the purities of both products maintained within 1% of the desired specification. The CS7 structure shows a more aggressive behavior to the step change compared to the response of control structure CS8A, but it takes longer

to come to a steady state. It takes almost 5 h for the CS7 structure, while the CS8A structure achieves a steady state in less than 2 h. Although both of the temperatures and product purities settle down to the same values for both control structures, the CS7 structure has larger transient deviations in both temperatures and product purities. Figure 17 compares the CS7 and CS8A structures for a 10% decrease in the production rate handle. Results are similar to the responses for a 10% step increase. Figures 18 and 19 give comparisons for 5% changes in compositions of fresh feeds FMeOH and FHOAc from


Figure 17. Comparison of CS7 and CS8A: a 10% step decrease in the production rate handle.

Figure 18. Comparison of CS7 and CS8A: a step change in fresh feed composition ∆zMEOH.

pure components to the mixtures of MeOH and HOAc. Both control structures have smooth responses to these disturbances, with the CS7 structure responding more quickly. For the disturbance in the composition of the fresh feed stream FMeOH, the purities of both products are maintained within 1% of the desired specification. However, for the disturbance in the fresh feed stream FHOAc, the distillate purity drops to about 94% for both control structures. Thus, the methyl acetate reactive column is more sensitive to disturbances in the acetic acid composition than in the methanol composition.

The results for control structure CS8B are not presented because this structure did not yield stable control for the methyl acetate process. Remember that the CS8B structure worked better in the ideal process, while in the methyl acetate process, it did not work. A possible explanation for this difference is the differences in the steady-state gains for the two processes. Control structure CS8A in the methyl acetate process is different from that in the ideal process because the most sensitive trays to pair with both inputs FHOAc and QR can be used because there is no inverse response

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Figure 19. Comparison of CS7 and CS8A: a step change in fresh feed composition ∆zHOAC.

problem. Therefore, the CS8A control structure gives reasonable responses to any type of disturbance in the methyl acetate process. However, control structure CS8B does not work as well in the methyl acetate process because stage 38 is the most sensitive tray for both inputs FMeOH and QR. Thus, one of these inputs has to be paired with a tray that is located in a region with less sensitivity. Because the input QR has a reasonably large secondary peak in the KQR curve, the T29/QR pairing is used for control structure CS8B. This means that this pairing of the CS8B structure is less sensitive to the disturbances compared to the pairings of the CS8A control structure. Therefore, the CS8B control structure is less effective than the CS8A structure for the methyl acetate case. 6. Modified Ideal Design As discussed in previous sections, the effectiveness of the alternative control structures is different for the ideal and methyl acetate cases. Although these two processes appear to be quite similar in terms of having two reactants and two products, they differ quite significantly in their design parameters. The most important difference is the number of reactive trays. The ideal process has 10 reactive, 5 stripping, and 5 rectifying trays. The methyl acetate process has a large reactive zone with 34 trays. The most sensitive regions for changes in all inputs (F0A, F0B, and VS) are concentrated in the stripping section of the column for the ideal process (see Figure 3). This means that one of the trays must be selected using a secondary peak. However, if we look to Figure 13 for the methyl acetate case, the stages in the stripping section of the column are the most sensitive region to the changes in FMeOH and QR, while the most sensitive tray to the changes in FHOAc is in the reactive zone (stage 29). In this case, the most sensitive trays can be selected because they are in different regions.

Figure 20. Gains and SVD results for the modified ideal process.

Having more reactive trays appears to spread out the sensitive regions for the various inputs. To see if this difference in the number of reactive stages is the explanation for the difference between the two superficially similar processes, we explore a modified ideal process. The number of reactive trays is increased from 10 to 25, while both the stripping and rectifying tray numbers are decreased from 5 to 3. The operating pressure is decreased from 8.5 to 4.5 bar to lower temperatures and reduce specific reaction rates so that more reactive trays are required. This new design is far from the economic optimal design. The total annual cost of the ideal process increases from $248 500 for the 5/10/5 design to $301 980 for the 3/25/3 design. Figure 20 gives the values of the steady-state gains (KF0A, KF0B, and KVS) and their related SVD results for the modified ideal process with 25 reactive trays. Now the trays in the stripping section of the column are not the region most sensitive to the changes for all three inputs (F0A, F0B, and VS), as was the case in the 5/10/5


Figure 21. Comparison of control structures: a 10% step increase in the production rate handle.

Figure 22. Comparison of control structures: a 3% step change in fresh feed composition ∆z0B.

design. The gain curves of the modified design look quite similar to the gain curves of the methyl acetate process. The stripping section is the region most sensitive to the changes in F0A and VS, while the tray most sensitive to the changes in F0B is in the lower part of the reactive zone (tray 12). The difference from the methyl acetate process is that there is another peak for the KVS curve in the reactive zone, which has an equal magnitude with the peak in the stripping section. Thus, the tray selection from any of these peak regions may give similar

sensitivity and should result in similar controllability. Using the most sensitive trays helps controllability. Figure 20 shows the SVD results for the three different control structures studied in this paper. A direct comparison of the CS7, CS8A, and CS8B structures for a 10% increase in the production rate handles (vapor boilup or fresh feed streams) is given in Figure 21. The responses of the CS7 and CS8A control structures look similar to the responses in the methyl acetate process. Both control structures can handle this distur-

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bance easily. In addition, the CS8B structure also gives good response in this modified process, which is not what was found for the methyl acetate process. This is because of the high sensitivity of tray 12 used to pair with VS in the ideal case. Figure 22 compares the responses of the three control structures with a 3% A impurity in fresh feed stream F0B. The product purities settle down to new steady states within the purity limits for all control structures. These results show that adding more reactive trays changes the shape of the steady-state gain curves. This permits us to select more sensitive trays for control because the regions of sensitivity are more widely spread throughout the column. Using more sensitive trays improves controllability and helps to get better responses with all control structures. 7. Conclusions Two different two-temperature control structures have been studied for the control of reactive distillation columns operated in the “neat” mode (no excess reactant) with chemical reactions that have two reactants and produce two products. An ideal chemical system is explored first, and then the real methyl acetate reactive distillation system is studied. The effectiveness of these control structures is compared in the face of disturbances in the production rate and fresh feed compositions. For both control structures, the process does not require the measurement of an internal composition. For the first control structure (CS7), two tray temperatures are controlled by manipulating the two fresh feed streams. The vapor boilup (or reboiler duty) is the production rate handle. For the second control structure (CS8), two tray temperatures are controlled by manipulating one of the two fresh feed streams and vapor boilup (or reboiler duty). The other fresh feed stream is the production rate handle. The selection of the fresh feed stream to be manipulated is a key issue in the design of the CS8 control structure. For the ideal process, the heavy-reactant feed works better. Manipulation of the light-reactant fresh feed stream gives unstable dynamic responses. However, for the methyl acetate process, manipulation of the light-reactant fresh feed stream handles the disturbances well and provides dynamically stable control. The reason for this difference is the shape of the steadystage gain curves for two different processes. The distribution of the regions with most sensitive trays affects the selection of the trays used for control. For the ideal process, control structure CS8B uses more sensitive trays compared to the trays used for control structure CS8A. However, for the methyl acetate process, control structure CS8A uses more sensitive trays compared to the trays used for control structure CS8B. This study has demonstrated that the sensitivity of controlled trays affects the selection of control structures and/or selection of the manipulated fresh feed stream. Using more sensitive trays for two-temperature control structures helps the controllability of the system. Another second key issue is the tuning procedure for the two interacting temperature controllers. Because both individual and sequential tuning procedures give similar controller parameters, individually tuned temperature controllers are used in the ideal process. However, big differences occur in the methyl acetate process by using individually or sequentially tuned

controllers. The use of sequentially tuned temperaturecontrol loops is necessary in the methyl acetate process. Acknowledgment We thank C. C. Yu and co-workers from National Taiwan University for providing the Aspen subroutine that uses activities in the kinetics. Nomenclature aj ) activity of component j AVP,j ) Antoine constant of component j BVP,j ) Antoine constant of component j B ) bottoms flow rate in the column (mol/s) D ) distillate flow rate for the ideal process (mol/s) D ) distillate flow rate for the MeAc process (kmol/h) FHOAc ) fresh feed flow rate of acetic acid (kmol/h) FMeOH ) fresh feed flow rate of methanol (kmol/h) F0j ) fresh feed flow rate of reactant j (mol/s) kF ) specific reaction rate of the forward reaction (kmol‚s-1‚kgcat.-1) kR ) specific reaction rate of the reverse reaction (kmol‚s-1‚kgcat.-1) KP ) steady-state gain related to the input KC ) controller gain mcat. ) catalyst weight (kgcat.) Mi ) tray holdup (mol) NT ) number of trays in the column QR ) reboiler duty (GJ/h) s Pj ) vapor pressure of component j on tray i (bar) r ) reaction rate (kmol/s) R ) reflux (mol/s) Ti ) column temperature on tray i (K) U ) SVD parameters VS ) vapor boilup (mol/s) xB,j ) bottoms composition of component j in liquid xD,j ) distillate composition of component j in liquid z0j ) fresh feed mole fraction of component j Greek Symbols Rj ) relative volatility of component j τI ) reset time (min)

Literature Cited (1) Roat, S.; Downs, J.; Vogel, E.; Doss, J. Integration of rigorous dynamic modeling and control system synthesis for distillation columns. Chemical Process Control; CPC III; Elsevier: Amsterdam, The Netherlands, 1986. (2) Yu, C. C. Private communication, 2004. (3) Kaymak, D. B.; Luyben, W. L. Evaluation of a TwoTemperature Control Structure for a Two-Reactant/Two-Product Type of Reactive Distillation Column. Submitted for publication. (4) Kaymak, D. B.; Luyben, W. L. A Quantitative Comparison of Reactive Distillation with Conventional Multi-Unit Reactor/ Column/Recycle Systems for Different Chemical Equilibrium Constants. Ind. Eng. Chem. Res. 2004, 43, 2493-2507. (5) Tang, Y. T.; Chen, Y. W.; Hung, S. B.; Huang, H. P.; Lee, M. J.; Yu, C. C. Design of Reactive Distillations for Acetic Acid Esterification. AIChE J. 2005, in press. (6) Agreda, V. H.; Partin, L. R.; Heise, W. H. High Purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, 86 (2), 40.

Received for review January 20, 2005 Revised manuscript received April 6, 2005 Accepted April 18, 2005 IE058012M

Comparison of Two Types of Two-Temperature Control Structures for

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