Effect of Relative Volatilities on Inferential Temperature Control of

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Effect of Relative Volatilities on Inferential Temperature Control of Reactive Distillation Columns Devrim B. Kaymak,*,† Denizhan Yilmaz,† and Ahmet Z. G€urer† †

Department of Chemical Engineering, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey ABSTRACT: Several control structures for different types of reactive distillation columns have been studied in the literature. In general, these are the ternary systems with the chemistry A þ B T C and the quaternary systems with the chemistry A þ B T C þ D. In some cases, the ternary system may also contain an additional component I which is inert in terms of reaction. The growing literature includes both ideal generic and real chemical systems. However, all papers dealing with ideal generic systems assume constant relative volatilities between adjacent components. This paper explores the effect of temperature-dependent relative volatilities on inferential temperature control of reactive distillation columns. Three types of chemical systems are considered: a ternary system without inert component, a ternary system with inert component, and a quaternary system. Significant differences are observed between the results of these systems. The inferential control structure of the ternary system without inert provides effective control for any case of relative volatilities. However, there are some offset in bottoms and distillate compositions of quaternary system. The results illustrate that the magnitude of the offset in distillate composition increases significantly with the increase in the temperature-dependency of relative volatility.

1. INTRODUCTION The number of papers exploring reactive distillation columns and their application in the industry has increased during the past decade. Books dealing with this subject offer long lists of real chemistries.13 However, the own complexities of real chemical systems, such as azeotropes, reaction kinetics, and physical properties, may cloud the picture in understanding the effects of parameters on reactive distillation columns. Thus, there are several papers offering ideal generic reactive distillation columns with simple vaporliquid equilibrium, reaction kinetics, and physical properties to study the effects of parameters on design and control issues. A decade ago, Luyben offered an ideal quaternary system including two reactants and two products.4 In this system, the volatilities are such that the products are the lightest and heaviest components, while the reactants are intermediate boiling between the products. This quaternary system has been studied by several research groups. Luyben and co-workers examined alternative control structures for this quaternary system.57 This reactive column has been compared with a multiunit system in terms of design and control.8,9 Huang and co-workers studied the effects of internal heat integration on design and control.1012 The effects of steady-state multiplicity, feed tray location, catalyst loading, and number of reactive trays on controllability have been investigated by Kumar and Kaistha.1316 In addition, Olanrewaju and Al-Arfaj published a series of control papers using this generic reactive column.1719 The boiling point ranking of reactants and products has been changed by Tung and Yu20 meaning 24 possible boiling point rankings. It has been found that the relative volatility ranking plays an important role in the configuration of reactive distillation columns. Total annual cost of the configurations increases up to a factor of 7, when the products are selected as intermediate boiling components. r 2011 American Chemical Society

Another ideal system studied in the literature is the ternary system including two reactants and one product, where the reactants are lighter than the product.21,22 Two different cases have been considered for this ternary system. Although there are only three components in the first case, the second case includes a fourth component, which is inert in terms of reaction and has the same volatility as the lightest reactant. Chen and Yu investigated the design and control of reactive column systems for ternary decomposition reactions with one reactant and two products in the system.23 They studied two different relative volatility rankings. In the first case the reactant is the intermediate boiling component, while it is the heaviest one in the second case. In a recent paper, Sun and co-workers investigated the effectiveness of deepening internal mass integration using four different generic ideal systems mentioned above.24 The main assumption of all these papers is that the relative volatilities between adjacent components are equal and kept constant at 2. This means that the relative volatilities are independent of temperature, so they are constant throughout the column despite the differences in tray temperatures. This assumption is reasonable for hydrocarbon systems, where relative volatilities are fairly constant with similar chemical structures. However, the temperature dependences of vapor pressures are not the same in many systems. This results in a decrease in the relative volatilities as the temperature increases. If “neat” reactive distillation columns are used to separate systems with two reactants, composition analyzers could be used to achieve the desired conversion and product purity. However, it Received: November 8, 2010 Accepted: May 10, 2011 Revised: April 22, 2011 Published: May 10, 2011 8138

dx.doi.org/10.1021/ie102256j | Ind. Eng. Chem. Res. 2011, 50, 8138–8152

Industrial & Engineering Chemistry Research

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is preferable to use inferential temperature measurements instead of direct composition measurements whenever possible, because composition analyzers are expensive, have high maintenance requirement, and introduce deadtime into the control loop. At this point, the temperature profile should display enough information to enable the use of temperature based inferential control to provide the desirable composition purity. Here, the temperature dependency of relative volatilities may significantly affect the use of inferential temperature control structures, as it changes the temperature profile throughout the column. In their paper, Kaymak et al. illustrated how significantly the temperature dependency of relative volatilities affects the design of a generic quaternary ideal system.25 However, to the best of our knowledge, the effect of temperature-dependent relative volatilities on the controllability of reactive distillation columns has not been handled in the open literature until now. The novelty of this paper is to evaluate the effect of temperature-dependent relative volatility on inferential temperature control of reactive distillation columns. A literature survey shows that ∼65% of the total reaction systems using reactive distillation columns are ternary and quaternary systems.3 Thus, this study focuses on three different chemical systems: (i) ternary system without inert component, (ii) ternary system with inert component, and (iii) quaternary system. First, a brief introduction is given on how the relative volatilities are made temperature dependent. Then, studies on the optimum design of ternary system without inert component are performed for three different relative volatility cases. These optimum designs are used for the control studies of that chemical system. Next, the same procedure is applied to the ternary system with inert component and the quaternary system, respectively. Finally, some conclusion remarks are given in the last section of the paper.

2. TEMPERATURE-DEPENDENT RELATIVE VOLATILITY Relative volatility is a measure comparing the vapor pressures of the components in a liquid mixture of chemicals. A component having a higher vapor pressure than another component at a given temperature is more volatile. Relative volatility Rij for an ideal mixture is equal to the ratio of the vapor pressure of component i to the vapor pressure of component j Rij ¼

PiS PjS

ð1Þ

For ideal mixtures, Raoult’s law relates the partial pressure of a component in the vapor phase to its liquid phase composition and vapor pressure. The vapor pressure of components is a function of temperature and can be calculated from a twoparameter Antoine equation over a limited temperature range ln PS ¼ AVP

BVP T

ð2Þ

where AVP and BVP are constants. For the base case of any chemical system considered in this study, it is assumed that the vapor pressures of the ideal mixture have the same temperature dependency. Accordingly, the relative volatilities between adjacent components are not temperaturedependent, and they are assumed constant (Rij = 2.00) at any temperature. To make them temperature-dependent, the relative volatilities between adjacent components at 320 K are kept as 2.00 but reduced at some higher reference temperature. This reference temperature is selected as 390 K. By keeping the AVP

Table 1. Vapor Pressure Constants R390 = 2.00

R390 = 1.75

R390 = 1.50

AVP

BVP

AVP

BVP

AVP

BVP

A

12.34

3862.00

12.34

3862.00

12.34

3862.00

B

11.65

3862.00

12.40

4100.07

13.26

4374.90

C

10.96

3862.00

12.45

4338.13

14.17

4887.80

D

13.04

3862.00

12.30

3623.93

11.44

3349.10

I

12.34

3862.00

12.34

3862.00

12.34

3862.00

and BVP coefficients of the light reactant A constant, the coefficients of the two-parameter Antoine equation for other components are calculated for different temperature-dependent relative volatility cases. The details of the procedure have been given in Kaymak et al. for a quaternary system.25 Table 1 gives the AVP and BVP coefficients for three R390 cases. Since the relative volatilities are constant for the base case of R390 = 2.00, the BVP coefficients are the same for all components. The least volatile component C has the lowest AVP value among five components, and the value of this coefficient increases as the volatility of the components increases. With the reduction of R390, the values of AVP and BVP coefficients for less volatile components (B and C) increase, while those of the more volatile component (D) decrease. Vapor pressures for these relative volatility cases are given in Figure 1. Since the BVP coefficients are the same for all components in the case of R390 = 2.00, the slopes of all components are the same. Thus, the lines are arranged parallel to each other starting from component D with the highest AVP. The slope of the vapor pressure line for component A is the same for all R390 cases because of its constant coefficients. However, it is easily seen that the lines of other components are getting closer to each other as the temperature increases for smaller R390 cases.

3. RESULTS AND DISCUSSION 3.1. Ternary System without Inert Component. Figure 2 shows the flowsheet of the ternary reactive distillation column. The product C leaves the column from the bottoms stream as it is heavier than the reactants A and B. Since the system does not include any light product, there is no distillate stream. All the condensed vapor returns to the column by a reflux stream. Thus, the column only consists of a stripping section and a reactive zone. The pure light and heavy reactants are fed from the bottom and top trays of the reactive zone, respectively. The design objective is to obtain a product C with a purity of 98 mol % and a production rate of 12.6 mol/s. Thus, the bottoms flow rate is 12.86 mol/s. Since the impurity in the bottoms stream is mostly the heavier reactant B, the flow rates of the fresh feed streams FA0 and FB0 are slightly different. The liquid holdup in the reactive trays is 1000 mols. Kinetic parameters of the ternary system are given in Table 2. The relaxation method is used to find the steady-state conditions. The basic design procedure is based on an existing paper in the literature.21 The specifications and assumptions on locations of feed tray and holdup of reactive trays reduce the design of ternary system to an optimization problem including three variables: (1) the number of stripping trays NS, (2) the number of reactive trays NRX, and (3) the column pressure P. The optimum design is determined by minimizing the total annual cost, TAC. Total 8139

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Figure 1. Vapor pressures for temperature-dependent relative volatility cases.

Table 3. Optimum Design Parameters and Economics for Ternary System R390 2.00

Figure 2. Flowsheet of the ternary reactive distillation column.

Table 2. Kinetic Parameters for Ternary System parameter

value

1

activation energy (kcal mol ) forward

30

reverse

40

specific reaction rate at 366 K (kmol s1 kmol1) forward reverse

0.008 0.0004

chemical equilibrium constant at 366 K

20

heat of reaction (kcal mol1)

10

heat of vaporization (kcal mol1)

6.944

annual cost is the sum of energy and capital costs with a payback period of 3 years. The differential temperature driving force in the

1.75

1.50

NS

13

18

20

NRX

5

5

9

P (bar)

6

7

6

VS (mol/s)

33.45

41.12

51.77

R (mol/s)

51.60

59.26

69.92

B (mol/s)

12.86

12.86

12.86

F0A (mol/s)

12.60

12.60

12.60

F0B (mol/s) DC (m)

12.86 0.94

12.86 0.97

12.86 1.09

CC (103$)

395.36

459.75

559.71

CE (103$/yr)

144.15

177.18

223.10

TAC (103$/yr)

275.94

330.43

409.67

reboiler is 34.8 K. An overall heat-transfer coefficient of 0.568 kJ/ s.m2.K is used to size the reboiler area. The condenser area is calculated assuming an overall heat-transfer coefficient of 0.852 kJ/s.m2.K. The differential temperature driving force for condenser is 13.9 K. The cost of energy is 4.7 $/106 kJ. Table 3 gives the optimum design parameters and economics for three different temperature-dependent relative volatility cases. The column operates at an optimum pressure of 6 bar for the base case (R390 = 2.00). Using 13 stripping and 5 reactive trays with a vapor boilup of 33.45 mol/s yields a bottoms purity of 98 mol % component C. The column diameter is found 0.94 m. The total capital investment in the column and associated heat exchangers is $395.36 103. The cost of energy is $144.15 103/yr. The resulting TAC is $275.94 103/yr. Since the range of relative volatilities investigated in this paper is fairly narrow (R390 = 1.502.00), the results show no significant change in the operating pressure as the value of R390 decreases. However, a 8140



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Figure 3. Temperature profiles for ternary system.

visible pressure drop would be expected for lower values of R390. On the other hand, the decrease in the value of R390 results in a gradual increase of the optimum number of total trays. There is an increase in the number of stripping trays, when the value of R390 is reduced from 2.00 to 1.75. For the case of R390 = 1.50, the increase is observed in the numbers of both stripping and reactive trays. There is also a growth in the reboiler duty with the decrease of R390. This expands the diameter of the column and increases the heat exchanger areas. Thus, the capital and energy costs get higher as the value of R390 decreases. Figure 3 shows the temperature profiles for all temperaturedependent relative volatility cases. As the value of R390 decreases, the temperature difference between top and bottom of the column declines. The change of temperature profile is most clearly observed in the stripping section, while the hump in the reactive zone is obvious for any R390 cases. The figure shows lower base temperatures for the lower R390 cases despite having the same composition and essentially the same pressure. The temperatures satisfying these conditions are different for each R390 case as they have different vapor pressure constants. The decrease in the base temperature for the lower R390 cases prevents the actual relative volatilities from becoming too small. Optimum design of each relative volatility case is used to compare the effectiveness of two-temperature inferential control structure. Reflux drum and column base are sized to provide 5 min of liquid holdup when levels are at 50%. Figure 4 shows the control structure used for this system. The reflux drum level is controlled by manipulating the reflux flow rate. The base level is controlled by the bottoms flow rate. Proportional controllers with a gain of 2 are used for both liquid levels. The temperatures of two selected trays are controlled by manipulating the fresh feed streams. Locations of temperature control trays are selected using the sensitivity analysis and singular value decomposition (SVD) method. Both temperature loops have two 60-s first order lags. PI controllers are used for the temperature loops. Their ultimate gains and ultimate periods are determined using the relay feedback test. Then the Tyreus-Luyben tuning rules are applied to tune the controller parameters. Vapor boilup is the production rate handle. The first row of Figure 5 shows the steady-state gains between tray temperatures and two inputs of the column for three different relative volatility cases. The bottom row of Figure 5 gives the SVD results for the same cases. In this figure and

Figure 4. Control structure of reactive distillation column for ternary system.

following ones, the solid lines are used for the case of R390 = 2.00. The case of R390 = 1.75 is illustrated using dashed lines, while the dash-dot lines are used to illustrate the case of R390 = 1.50. These results show that the middle of stripping section is the most sensitive region to the changes in fresh feed stream FA0. On the other hand, the top tray of the column is the most sensitive region to the changes in fresh feed stream FB0. The SVD results are also coherent with the results of the sensitivity analysis. Table 4 gives controller pairings and parameters for the temperature loops of each case. Note that some loops are detuned for the ternary system. Figure 6 shows the response of the system to a 20% step increase in the production rate handle VS at time equal to 0.5 h. The increase in the vapor boilup ends up with coherent increments in the fresh feed streamflow rates. The controlled tray temperatures settle down to their set points easily for any case of R390. The purity of bottoms product recovers back to its specified value of 98 mol % C within 3 h. The case of R390 = 2.00 exhibits the shortest settling time compared to the other cases. The transient deviation of bottoms purity does not exceed 1 mol % in any case. In addition, the integral absolute error (IAE) for the case of R390 = 2.00 is smaller than that of other cases. The response of the system to a 20% step decrease in production rate handle is shown in Figure 7. The controlled tray temperatures return to their set points for any case of R390. For the case of R390 = 2.00, the product composition is maintained at its specified value of 98 mol % C in less than 2 h. Although it takes longer to settle down, the product compositions for other R390 cases are also held at their desired values. The transient deviation of bottoms purity is less than 1 mol % for any R390 case. It is observed that the case of R390 = 2.00 has the smallest IAE compared to that of other cases. In addition, the response of control structure to composition changes in the fresh feed streams is considered. Figure 8A shows the results for 3% of B impurity in FA0, while 3% of A impurity in FB0 is considered in Figure 8B. The results illustrate that controlled tray temperatures settle down to their set points easily for both of these impurities. It is seen that the settling time and the magnitude of the offset for the bottoms product purity 8141



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Figure 5. Sensitivity analysis and SVD results for ternary system.

Table 4. Controller Pairing and Parameters for Ternary System R390

loop

KC

τI

action

2.00

T4 - FA0

8.29

16.76

direct

T18 - FB0

19.54

22.57

reverse

1.75

T6 - FA0

5.70

39.6

direct

1.50

T23 - FB0 T9 - FA0

11.26 8.87

11.22 40.13

reverse direct

T29 - FB0

51.01

20.20

reverse

increase as the value of R390 decreases. However, the offset does not exceed 0.1 mol % even for the case of R390 = 1.50. The results of this ternary system differ significantly from those of the quaternary systems in terms of the actions of temperature controllers. The studies on quaternary systems concluded that the two temperature controllers had to have the same action.6 However, dynamically stable results are obtained for this ternary system with temperature controllers having opposite actions. Details of the physical explanation of this situation are given in a recent paper.26 3.2. Ternary System with Inert Component. The flowsheet of ternary system with inert is given in Figure 9. This system involves four components. However, one of them is an inert component I in terms of the reaction. Since its volatility is assumed the same as that of the light reactant A, it is fed from the fresh feed stream FA0 as a mixture with A. The other fresh feed stream FB0 contains pure reactant B. While the heavy product C leaves the column from the bottom, the low-boiling inert I is removed from the distillate without taking part in the reaction. Thus, the column has three zones: a stripping section, a reactive zone, and a rectifying section. It is assumed that the light and heavy fresh feed streams are fed from the bottom and top trays of the reactive zone, respectively. The main design objective is to obtain the purity of the bottoms product at 98 mol % C. On the other hand, the amount of reactants escaping from the distillate stream should also be considered. This is especially important for the light reactant A, which has an identical volatility with the inert component I.

Thus, a constraint of 3 mol % is defined for the maximum amount of reactants leaving the column from the top. The liquid holdups in the reactive trays are selected as 2000 mols to have reasonable liquid height. The composition of FA0 is 50 mol % A and 50 mol % I. Kinetic parameters of the ternary system are given in Table 5. The basic design procedure is based on the existing paper in the literature.21 The optimization problem for the ternary system with inert includes five design variables. These are (1) the number of stripping trays NS, (2) the number of reactive trays NRX, and (3) the number of rectifying trays NR, (4) the column pressure P, and (5) the reflux R. The objective function is TAC, and the same basis of economics is used as that has been used for the ternary system without inert component. Table 6 gives the optimum design parameters and economics for the relative volatility cases considered. For the base case of R390 = 2.00, the optimum operating pressure of the column is 9 bar. The bottoms purity of 98 mol % C is achieved with a column having 7 stripping, 12 reactive, and 7 rectifying trays. The column diameter is 1.00 m with a vapor boilup of 55.06 mol/s. The total capital investment for the column, reboiler, and condenser is $532.42 103, while the cost of energy is $237.33 103/yr. Assuming a payback period of 3 years, the TAC is $414.80 103/yr. There is a slight decrease in the operating pressure with the decrease of R390, but the change is not dramatic. This decrease is reasonable because lower pressure helps the VLE by reducing temperatures and increasing relative volatilities. There is an increase in the optimum number of total trays with the decrease of R390. Decrease of R390 from 2.00 to 1.75 results in a higher change in the number of separation trays. On the other hand, the increase in the number of reactive trays is more remarkable for the case of R390 = 1.50. The other design variable reflux R and the vapor boilup VS increase dramatically as R390 declines. Thus, the capital and energy costs get significantly higher as the value of R390 decreases. Figure 10 shows the temperature profiles of three relative volatility cases. There is a sharp temperature profile for the case of R390 = 2.00. This is especially true for the stripping section. The decrease of the value of R390 moderates the sharpness of the temperature profile. The size of the hump in the reactive zone also 8142



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Figure 6. Response of ternary system to þ20% step change in VS.

Figure 7. Response of ternary system to -20% step change in VS.

reduces as the value of R390 decreases. There is almost a flat temperature profile for the reactive zone of the case of R390 = 1.50. The optimum designs are used to compare the dynamics of three relative volatility cases. Sizing of reflux drum and base holdups is done as in the case of the ternary system without inert component. The inferential control structure used for this system is given in Figure 11. 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. Temperatures of two selected trays are controlled by manipulating the fresh feed streams FA0 and FB0. In this case, the vapor boilup VS is flow controlled and serves as the production rate handle. The sensitivity analysis and SVD results for three different relative volatility cases are given in Figure 12. The results show

that the stripping section is the most sensitive region to changes in both fresh feed streams FA0 and FB0. The SVD results are also coherent with the results of the sensitivity analysis. The fresh feed stream FA0 is paired with trays in the lower part of the stripping section, while the trays in the upper part of the stripping section are chosen for the fresh feed stream FB0. Table 7 provides controller pairings and their parameters for the temperature loops of each case. The results for a 20% step increase in the production rate handle VS are given in Figure 13. It is seen that the controlled temperatures increase rapidly when the disturbance is first applied at time equal to 1 h. In some cases, the temperature increase rises up to 20 K. However, the controlled variables turn back to their set points in all cases of R390. The pairings including FA0 as the manipulated variables settle down shortly, while 8143



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Figure 8. A. Response of ternary system to 3% of B impurity in FA0. B. Response of ternary system to 3% of A impurity in FB0.

settling down takes longer for the pairings including FB0. Although the first actions of manipulated variables in both loops are different, both fresh feed streams end up with positive changes. The increase in FA0 seems to be twice of the increase in FB0. However, it should be noted that the fresh feed stream contains 50 mol % inert component. Thus, there is a balance in the amount of reactants fed to the system. Although the controlled tray temperatures return to their set points easily, it takes longer for the product purities. For two cases of R390 (2.00 and 1.75), the purity of bottoms product recovers back to its specified value of 98 mol % C in around 12 h. On the other hand, the settling time takes ∼24 h for the case of R390 = 1.50 (out of the range given in Figure 13). Besides, it is seen that the

maximum transient deviation does not exceed ∼1.5 mol % in any R390 case. The purity of the inert component is maintained close to the desired value. However, the transient deviation and settling time of the inert component in distillate stream is worse than those of the product C in the bottoms stream. The response of the system to a 20% step decrease in production rate handle is given in Figure 14. Similar to the results of the positive disturbance, there is also a rapid change in the controlled tray temperatures once the disturbance is applied, and the temperature decreases up to 20 K in some cases. The controllers including FA0 as the manipulated variable response faster than the controllers including FB0. In the worst case, controlled temperature settles down in less than 10 h. This is 8144



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Table 6. Optimum Design Parameters and Economics for Ternary System with Inert Component R390 2.00

Figure 9. Flowsheet of the ternary reactive distillation column with inert component.

1.75

1.50

NS

7

10

NRX

12

13

18

NR P (bar)

7 9

9 9

10 8

VS (mol/s)

55.06

65.13

95.15

R (mol/s)

60

70

100

B (mol/s)

12.36

12.31

12.35

D (mol/s)

12.50

12.50

12.59

FA0 (mol/s)

24.37

24. 29

24.42

FB0 (mol/s)

12.60

12.60

12.60

DC (m) CC (103$)

1.00 532.42

1.07 617.44

1.28 813.84

11

CE (103$/yr)

237.33

280.56

409.90

TAC (103$/yr)

414.80

486.37

681.18

Table 5. Kinetic Parameters for Ternary System with Inert Component parameter

value

1


30

reverse

40

specific reaction rate at 366 K (kmol s1 kmol1) forward reverse

0.008 0.00016


50


10


6.944

achieved by coherent reduction of valve openings of the fresh feed streams. Although the bottoms purity decreases up to ∼95 mol % C during the transient for the case of R390 = 1.50, it recovers back to its specified value for all R390 cases. The performance is faster than that observed for the positive disturbance. The product purity settles down in ∼10 h for the worst case of R390 = 1.50, while it takes ∼5 h for other R390 cases (2.00 and 1.75). A similar performance is observed for the inert purity of the distillate stream. The purity of distillate stream settles down faster in the case of negative disturbance. The response of control structure to 3% of B impurity in FA0 is given in Figure 15A. The results illustrate that controlled tray temperatures settle down to their set points easily. Although the settling time is ∼5 h for the control loop including FA0 as the manipulated variable, it takes 10 h for the control loop including FB0. It is seen that the settling time and the magnitude of the offset for the bottoms product purity increase for the case of R390 = 1.50. However, the offset does not exceed 0.1 mol % even for this case. Figure 15B shows that mainly similar results are observed in the case of 3% of A impurity in FB0. 3.3. Quaternary System. The quaternary system involves two reactants and two products. The products are the lightest and heaviest components in the systems. The heaviest product C

Figure 10. Temperature profiles for ternary system with inert component.

leaves the column from the bottoms stream, while the lightest product D comes out from the top of the column. Thus, the basic flowsheet of quaternary system is similar to that of the ternary system with inert component given in Figure 9 despite the differences of both systems. It means that the column has three zones: a stripping section to keep lighter components from escaping out of the bottoms, a reactive zone where the reaction occurs, and a rectifying section to keep heavier components from escaping out of the top. The fresh feed streams of quaternary system consist of pure reactants, since there is no inert component in the system. It is assumed that the light and heavy fresh feed streams are fed from the bottom and top trays of the reactive zone, respectively. In this case, the design objective is to obtain 95% conversion for pure reactant fresh feed flow rates of 12.6 mol s1. Thus, the bottoms stream contains mostly heavy reactant B as impurity, while the distillate stream contains mostly light reactant A as impurity. Reactive tray holdup of 1000 mols is used yielding reasonable tray liquid heights. Table 8 gives the kinetic parameters 8145


Industrial & Engineering Chemistry Research of the quaternary system. The basic design procedure is given in Kaymak and Luyben.8 For the base case of R390 = 2.00, three design variables are used to find the optimum design of the quaternary system. These are (1) the number of separation trays NS/NR, (2) the number of reactive trays NRX, and (3) the column pressure P. However, once the relative volatilities are made temperature-dependent, the stripping trays NS and the rectifying trays NR have to be taken as two separate design variables resulting in four design variables in total. The aim is minimizing TAC. The basis of economics used in this case is the same as that of the ternary systems. Optimum design parameters and economics for three different relative volatility cases are given in Table 9. It is seen that a column operating at 8.5 bar with 5 stripping, 7 reactive, and 5 rectifying trays has the minimum TAC for the case of R390 = 2.00. The vapor boilup is 28.79 mol/s, and the column diameter is 0.80 m. The TAC is $240.10 103/yr as the sum of the capital cost and the energy cost with a payback period of 3 years. It is observed that the operating pressure decreases slightly with the decrease of R390.

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The decrease of R390 results in an increase of optimum number of total trays. The change is mainly in the number of stripping and reactive trays. This is reasonable because the relative volatilities at lower temperatures do not change significantly. There is a gradual increase in vapor boilup VS as R390 gets smaller. As a result of increasing number of trays and energy consumption, there is a rise in TAC as the value of R390 decreases. The temperature profiles of these three cases are illustrated in Figure 16. The sharpness of the temperature profile decreases as the value of R390 decreases. While the temperature difference between top and bottom of the column is ∼75 K for the case of R390 = 2.00, that difference decreases to ∼35 K for the case of R390 = 1.50. There is also a difference in the temperature profile of the reactive zone. As a result of reduction in R390, a decrease in the magnitude of hump is observed. Figure 17 shows the inferential control structure used for the quaternary system. The method used to size the reflux drum and base holdups is the same as that for ternary systems. Bottoms flow rate and reflux flow rate are manipulated to control base and reflux drum levels, respectively. There is also a ratio control by adjusting the distillate flow rate to keep the reflux ratio constant. For this system, the manipulated variables to control tray temperatures are different than those of the ternary systems. Two selected trays are controlled by manipulating the fresh feed streams FA0 and the vapor boilup VS. Thus, the heavy fresh feed stream FB0 serves as the production rate handle. Figure 18 shows the results of sensitivity analysis and SVD method for related inputs. The sensitivity analysis results illustrate that the most sensitive region to the changes in both inputs are in Table 7. Controller Pairing and Parameters for Ternary System with Inert Component R390

KC

τI

T3 - FA0

1.04

15.18

T6 - FB0

0.34

44.77

1.75

T4 - FA0 T9 - FB0

2.79 0.62

15.69 47.22

1.50

T7 - FA0

2.20

14.66

T10 - FB0

1.08

91.81

2.00

Figure 11. Control structure of reactive distillation column for ternary system with inert component.

loop

Figure 12. Sensitivity analysis and SVD results for ternary system with inert component. 8146



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Figure 13. Response of ternary system with inert component to þ20% step change in VS.

Figure 14. Response of ternary system with inert component to -20% step change in VS.

the middle of stripping section. However, it is easily observed that there are also other regions showing secondary sensitivity to both inputs in different sections of the column. It is observed that the magnitude of the sensitivities reduces as the value of R390 decreases. For input FA0, the locations suggested by SVD analysis are similar to those indicated by sensitivity analysis. On the other hand, SVD results suggest to pair the vapor boilup VS with trays in the reactive zone, which are the secondary sensitive trays according to the sensitivity analysis results. It is seen that the location of controlled trays slightly shifts to the upper parts of the column with the decrease of R390. Controller loops and their parameters are given in Table 10. Figure 19 shows the response of the system to a positive 20% step change in the production rate handle FB0 at time equal to 0.5 h.

It is seen that the controlled temperatures recover back to their set point in ∼4 h. The transient deviation does not exceed 2 K for any R390 cases. This is achieved by opening the valves of the manipulated variables. New steady-state of fresh feed stream FA0 is the same for all R390 cases, while the vapor boilup VS settles down to different steady-states for each R390. This control structure provides stable base-level regulatory control, and product purities settle down in ∼4 h. It is observed that the bottoms purity of product C is maintained close to the desired value for all R390 cases. However, the results for the distillate purity of product D illustrate that the deviation from the desired value increases significantly as the value of R390 decreases. As a result, the purity of the distillate product is ∼1 mol % far from its specification for the case of R390 = 1.50. 8147



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Table 9. Optimum Design Parameters and Economics for Quaternary System R390 2.00

1.75

1.50

NS

5

7

9

NRX

7

9

13

NR P (bar)

5 8.5

6 8

6 7

VS (mol/s)

28.79

36.07

48.80

R (mol/s)

33.43

40.70

53.43

B (mol/s)

12.60

12.60

12.60

D (mol/s)

12.60

12.60

12.60

FA0 (mol/s)

12.60

12.60

12.60

FB0 (mol/s)

12.60

12.60

12.60

DC (m) CC (103$/yr)

0.80 348.0

0.87 415.2

1.00 521.8

CE (103$/yr)

124.1

155.4

210.3

TAC (103$/yr)

240.1

293.8

384.2

Figure 15. A. Response of ternary system with inert component to 3% of B impurity in FA0. B. Response of ternary system with inert component to 3% of A impurity in FB0.

Table 8. Kinetic Parameters for Quaternary System parameter

value

Figure 16. Temperature profiles for quaternary system.

1


30

reverse

40

specific reaction rate at 366 K (kmol s1 kmol1) forward

0.008

reverse

0.004


2


10


6.944

Results for a negative 20% step change in FB0 are given in Figure 20. The control performance is essentially the same as that observed for the positive disturbance. The tray temperatures turn back to their set point easily in ∼4 h with transient deviation less than 2 K. This is done by reducing the flow rates of both manipulated variables. Fresh feed stream FA0 settles down to the same steady-state value for all R390 cases. This helps to satisfy the stoichiometric balance between the reactants fed to the

column. On the other hand, the decrease in the amount of VS rises as the value of R390 declines. Similar to the directly controlled tray temperatures, the inferentially controlled product purities settle down in ∼4 h. However, the offset in the distillate composition increases with the decrease of R390, while the bottoms purity of product C is maintained close to its specification for all R390 cases. Consequently, for the case of R390 = 1.50, the purity of the distillate product settles down to a new steadystate, which differs more than 1 mol % from its specification. 3.4. Comparison of Systems. As it is seen from the results given in the previous sections, there are specific differences between inferential controllability of reactive columns for different type of reactions. The first difference is the inferential control structure used for different chemical systems. While the fresh feed streams FA0 and FB0 are used as manipulated variables in both ternary systems, it is observed that this control structure does not work for the optimum designs of quaternary system in any R390 case. Thus, another two-temperature control 8148


Industrial & Engineering Chemistry Research structure is used for quaternary system, where FA0 and VS are the manipulated variables. The location of controlled trays is the second difference. For all R390 cases of the ternary system without inert, FA0 is paired with a tray in the stripping section, while FB0 is paired with a reactive tray. This reactive tray is located at the top of the column. On the other hand, both of the controlled trays are located in the stripping section of the column for the ternary system with inert. In the case of quaternary system, a stripping tray temperature is controlled by manipulating FA0, while a reactive tray temperature is controlled by manipulating VS. These trays are located in the middle of related zones. The final difference is in the dynamic responses of different reactive columns. Table 11 shows the steady-state deviation of

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the bottoms and distillate purities as a metric of quantitative comparison. It is observed that the bottoms purities of both ternary systems are maintained well with very slight deviations. The results show that the deviation in the bottoms purity of quaternary system is significantly higher than that of other chemical systems. In addition, it gradually increases with the decrease of R390 for both positive and negative changes. However, this increase is not gigantic, and the deviation does not exceed 1% for any case of R390 considered. For the ternary system with inert component, the change in the inert composition of the distillate is significantly bigger than the change in the product composition of the bottoms. However, no systematic change is observed with the decrease of R390, and the deviation is not more than 1% for disturbances in any direction. On the other hand, it is clearly seen that the distillate purity of quaternary system deviates significantly for both positive and negative disturbances. As the value of R390 decreases, the deviation from the desired purity increases gradually. Finally, a new steady-state with a deviation of more than 1% occurs for the case of R390 = 1.50. Although the purity of bottoms product recovers back to its specified value for both ternary systems, the time necessary for settling down differs significantly from one system to the other. While the ternary system without inert turns back to its desired Table 10. Controller Pairing and Parameters for Quaternary System R390

loop

KC

τI

2.00

T4 - FA0 T10 Vs

2.70 4.46

5.75 6.25

1.75 1.50

Figure 17. Control structure of reactive distillation column for quaternary system.

T5 - FA0

3.84

6.42

T13 Vs

8.04

6.50

T6 - FA0

7.06

6.67

T17 - Vs

15.72

5.92

Figure 18. Sensitivity analysis and SVD results for quaternary system. 8149



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Figure 19. Response of quaternary system to þ20% step change in FB0.

Figure 20. Response of quaternary system to -20% step change in FB0.

Table 11. Final Product Purity Deviations ternary

ternary with inert

disturbance R390 |ΔXB,C| |ΔXD,-| |ΔXB,C| þ20

20

quaternary

|ΔXD,I| |ΔXB,C| |ΔXD,D|

2.00 0.0002

---

0.0001

0.0037

0.0021 0.0052

1.75 0.0001

---

0.0001

0.0030

0.0022 0.0075

1.50 0.0001

---

0.0000

0.0082

0.0033 0.0105

2.00 0.0001

---

0.0001

0.0038

0.0022 0.0051

1.75 0.0001

---

0.0001

0.0030

0.0026 0.0073

1.50 0.0004

---

0.0003

0.0019

0.0037 0.0101

specifications within ∼3 h, it takes more than 12 h for the ternary system with inert. On the other hand, the quaternary system settles down to its new steady-state within ∼4 h.

4. CONCLUSIONS In this study, the effect of relative volatility on inferential temperature control of reactive distillation columns is explored. Three widely used chemical systems are considered. These are the ternary system without inert component, the ternary system with inert component, and the quaternary system. Instead of real chemicals, generic systems are used to enable a quantitative comparison. First, economically optimum steady-state designs are found for each case. These optimum designs are further studied in the control part of the paper. For each chemical system, the most proper two-temperature control structures are considered among the alternative structures. The results indicate that the effect of temperature-dependent relative volatilities on inferential controllability differs from one system to the other. For the ternary system without inert, the 8150


Industrial & Engineering Chemistry Research bottoms purity recovers back to its specified value within ∼3 h with a small transient deviation for any case of R390. The bottoms purity of the ternary system with inert is also maintained at its specified value. However, it takes more than ∼12 h even for the base case of R390 = 2.00. As the value of R390 decreases, the settling time and the magnitude of transient deviation are increased. This is related to the location of controlled trays. The quaternary system with two products gives the worst response among the three chemical systems studied in terms of final product purities. Although the bottoms purity is maintained close to the desired value, there is a significant offset in the distillate composition for both positive and negative disturbances. It is observed that the size of the offset increases with the decrease of R390. In the case of R390 = 1.50, the magnitude of the deviation exceeds 1%.

’ AUTHOR INFORMATION Corresponding Author

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

’ ACKNOWLEDGMENT Financial support from the Scientific and Technological Research Council of Turkey (TUBITAK) through Project Number 108M504 is gratefully acknowledged. ’ NOMENCLATURE AVP vapor pressure constant B bottoms flow rate in the column (mol/s) vapor pressure constant BVP D distillate flow rate in the column (mol/s) column diameter (m) DC fresh feed flow rate of reactant j (mol/s) Fj0 K steady-state gain controller gain KC number of the rectifying trays NR number of the reactive trays NRX number of the stripping trays NS P column pressure (bar) vapor pressure (bar) PS R reflux (mol/s) column temperature on tray i (K) Ti U left singular vector matrix vapor boilup (mol/s) VS bottoms composition of component j in liquid xB,j distillate composition of component j in liquid xD,j Greek Symbols

R R390 Δ τI

relative volatility relative volatility at 390 K difference reset time (min)

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