Steady-State Multiplicity and Its Implications on the Control of an Ideal

2778

Ind. Eng. Chem. Res. 2008, 47, 2778-2787

Steady-State Multiplicity and Its Implications on the Control of an Ideal Reactive Distillation Column M. V. Pavan Kumar and Nitin Kaistha* Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

A two-temperature-control structure for the generic ideal reactive distillation (RD) column is systematically evaluated for nonlinear dynamic phenomena. The steady-state open- and closed-loop input-output (IO) relations are presented to show the existence of severe input multiplicity in the most sensitive controlled variable, namely, a rectifying tray temperature. The bifurcation study shows that, for the studied control structure, steady-state transition and “wrong” control action can occur for a large throughput decrease and increase, respectively, when controlling the same. The severity of the input multiplicity is significantly mitigated when a temperature in the reactive section is controlled instead. Wrong control action can still occur for a large throughput change in either direction (increase or decrease). Closed-loop dynamic simulation results confirm the occurrence of these nonlinear dynamic phenomena. The results also show that the magnitude of throughput change that can be handled without the decentralized control system failing is much larger when the reactive tray temperature is controlled instead of the rectifying tray temperature. The key role of steady-state multiplicity in the design of a robust control system for RD systems is highlighted. Introduction Reactive distillation1,2 (RD), combining reaction and separation in a single column, has emerged as a promising technology for process integration with significant economic benefits over conventional ‘reaction followed by separation’ processes. When feasible, RD can reduce the capital and operating costs by a factor of 2-5 over conventional processes.3,4 Commercial RD processes include esterification systems for producing methyl acetate,5 ethyl acetate,6 and butyl acetate7 as well as etherification systems for producing methyl tert-butyl ether (MTBE),8 ethyl tert-butyl ether (ETBE),9 and tert-amyl methyl ether (TAME).10 Sharma and Mahajani11 provide a comprehensive review of possible RD applications. One of the key challenges in the successful commercialization of RD technology is the design of a robust control system that effectively rejects load disturbances such as production rate or feed composition changes. The coupling of reaction and separation in a single unit causes high nonlinearity and also reduces the control degrees of freedom for regulating both the reaction and the separation.12 The occurrence of multiple steady states is a direct consequence of the high process nonlinearity and has been experimentally verified for a laboratory-scale TAME column.13 This is in addition to the simulation reports in the literature on the existence of multiple steady states in RD systems.14-18 In view of the high process nonlinearity, the application of advanced control techniques such as pattern based predictive control,19 gain scheduling,20 and model predictive control21 has been propounded in the RD control literature. Gru¨ner et al.12 report the application of nonlinear control to an industrial RD column operated by Bayer AG. In contrast to advanced control, pioneering work by Luyben and co-workers22-28 on a variety of RD systems has demonstrated that, for an appropriately chosen control structure, traditional decentralized control provides effective column regulation. More recently, Hung et al.29 * Corresponding author. E-mail: [email protected]. Phone: +91512-259-7513. Fax: +91-512-25890104.

Figure 1. Typical steady-state input-output relation with input multiplicity between manipulated and controlled variables.

Figure 2. Schematic of control structure.

show that simple decentralized control can provide good control for nonlinear RD columns designed for the esterification of acetic acid. In order to facilitate the understanding of the key issues in RD control, Luyben3 proposed an ideal double-feed RD column with two reactants and two products (A + B T C + D) as a test-bed problem. Constant relative volatilities (RC/RA/RB/RD ) 8:4:2:1) for all components are assumed along with constant

10.1021/ie701720r CCC: $40.75 © 2008 American Chemical Society Published on Web 03/22/2008

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2779 Table 1. Physical and Operating Design Parameters of Generic Ideal RD Column pressure no. of stripping trays no. of enriching trays no. of reactive trays tray holdup flow rate of reactant A, FA flow rate of reactant B, FB feed tray location (feed-tray number) relative volatility (RD/RB/RA/RC) average latent heat of vaporization, (kJ kmol-1) reaction stoichiometry average heat of reaction, (kJ kmol-1) specific reaction rate at 366 K, (kmol s-1 kmol-1) reaction rate, kmol kmol-1 s-1 specific reaction rate, forward specific reaction rate, backward products, mol % vapor pressure constantsa

a

xC, distillate xD, bottoms C B A D

8.5 bar 5 5 10 1 kmol 12.6 mol s-1 12.6 mol s-1 FA-6, FB-15 1:2:4:8 29073.14 A+BTC+D -41840 forward, 0.008 backward, 0.004 r ) kfxAxB - kbxCxD, kf/kb ) 2 at 366 K kf ) af e-Ef/RT Ef ) 125 520 kJ kmol-1 kb ) ab e-Eb/RT Eb ) 167 360 kJ kmol-1 95.00 95.00 Avp/Bvp Avp/Bvp Avp/Bvp Avp/Bvp

13.04/3862 11.65/3862 12.34/3862 10.96/3862

ln Psi ) Avp,i - Bvp,i/T, i ) component, vapor pressure in bar and T in

K.

heat of vaporization/reaction and elementary reaction kinetics. The hypothetical system is thus stripped of nonlinearities due to nonideal VLE and temperature/composition dependence of heat of reaction/vaporization and is well-suited for understanding the nonlinearity due to interaction between reaction and separation. Al-Arfaj and Luyben22 studied various control structures for the generic ideal RD column. A key insight from their work is the need for stoichiometrically balancing the two fresh feeds. This is accomplished by controlling a reactive tray composition using one of the fresh feeds as the manipulated variable in a feedback loop. Kaymak and Luyben27,28 extended the work to study temperature-based inferential control, the same being desirable considering the large lags, poor reliability, and expensiveness of analytical composition measurements. Twopoint temperature-control structures were studied with some of the structures exhibiting controllability problems in handling large production rate changes in a direction27 (either increase or decrease). The dependence of the RD column output response on both the direction and the magnitude of the input change

was reported by Olanrewaju and Al-Arfaj.30 Kaymak and Luyben28 also showed that distributing the catalyst over a higher number of reactive trays can lead to improved controllability. More recently, Huang et al.31 report that, by appropriately extending the reactive zone into the stripping section and enriching sections for reactive distillation columns with exothermic and endothermic reactions, respectively, it is possible to reduce the energy costs significantly with the possibility of improved open-loop and closed-loop dynamics. These results, when viewed in totality, suggest that, in spite of simplifying assumptions such as ideal vapor-liquid equilibrium (VLE), high nonlinearity exists in the generic ideal RD column, causing controllability problems. Kienle and Marquardt32 note that, for RD processes in the kinetic regime with significantly different individual component boiling points, the ideal RD system being one such process,3,33 new phenomena are introduced by the interaction of separation and reaction and understanding the same is a “challenging field for future research”. Indeed, in the RD control literature, very few articles explicitly address the implications of nonlinear effects due to reaction-separation interaction on control system performance and design.20,21,34-36 In this context, the hypothetical ideal RD column remains of much relevance. Of particular interest are the implications of nonlinearity on closed-loop control performance and its consideration in control-system design for robustness. To the best of our knowledge, the same has not been addressed in the extant literature, and this article is a contribution in that direction. To limit the scope of the work, only temperature-based inferential control of the ideal RD column is considered, given its relevance in an industrial setting. In the following, a two-point temperature-control structure for the generic ideal RD column is systematically evaluated for nonlinear dynamic phenomena. First, the steady-state open- and closed-loop input-output (IO) relations are presented to show the existence of input multiplicity in the tray temperatures. The severity of the input multiplicity is significantly mitigated if a reactive tray temperature is controlled instead of a rectifying tray temperature. Closed-loop dynamic simulation results are then presented, confirming the occurrence of nonlinear dynamic phenomena predicted by the bifurcation study. The article concludes emphasizing the key role of steady-state multiplicity in designing a robust control system for RD systems. Implications of Multiplicity From the control perspective, steady-state multiplicities32 are classified into output and input multiplicities. The former refers to multiple output values for a given input, while the latter refers to the same output for multiple input values. In the case of output

Figure 3. (a) Steady-state gains of tray temperatures with respect to FA and VS and corresponding (b) SVD results.

2780

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008

Figure 5. Steady-state variation of (a) T2 with respect to FA while T18 is held constant by VS and (b) T18 with respect to VS while T2 is held constant by FA.

Figure 4. Steady-state variation of (a) T2, (b) T12, and (c) T18 with respect to FA, FB, and VS. Table 2. Niederlinski Index (NI) and Relative Gain Array (RGA) for Two Possible Pairings of Control Structure CS pairing

NI

VS-T18; FA-T2

0.8862

VS-T12; FA-T2

0.7345

[ [

RGA

T18 1.128 -0.128 ∧) T 1.128 2 -0.128 T12 1.361 -0.361 ∧) T 1.361 2 -0.361

] ]

multiplicity, a feedback control loop ensures that the process stays at the desired steady state and does not drift to a different steady state. Input multiplicity, on the other hand, results in the possibility of “wrong” control action or steady-state transitions even with feedback control.34,36 Input multiplicity can, thus, severely compromise the robustness of a control system. Consider an IO relation as in Figure 1 that exhibits input multiplicity. The base-case operating condition is marked ‘o’, and the points where the output crosses its base-case value

causing input multiplicity are marked ‘1’ and ‘2’. If the input is used to control the output, the input multiplicity causes ambiguity in the control action to be taken. Around the basecase operating condition, the controller must be direct acting. For an initial steady state at point ‘a’, the controller sees a positive error and so reduces the input to bring the system back to the base-case operating condition. On the other hand, for an initial steady state at point ‘b’, the controller sees a negative error signal and, therefore, increases the input instead of decreasing it. This leads to wrong control action. If the IO relation turns back again, the system would settle at the new steady state corresponding to Point ‘3’. If the IO relation does not turn back, the wrong control action would cause a constraint, such as a saturated valve or column flooding, to be hit. The cause of wrong control action is sign reversal in the error signal into the controller. This is different from process gain sign reversal corresponding to peaks and valleys in the IO relation. Controller error sign reversal occurs at the point of crossover in the IO relation and is marked as ‘*’ in Figure 1. Input multiplicity due to crossover in the IO relation thus leads to the possibility of wrong control action and steady-state transition. The latter is also sometimes referred to as closed-loop multiplicity as the column may end up at a steady state different from the base case even with the control system on. In the above, for simplicity, the possibility of steady-state transition or wrong control action was explained for a feedback loop IO pairing exhibiting input multiplicity. Wrong control action or a steady-state transition may occur even if the feedback loop IO relation is well-behaved, i.e., no multiplicity, but there are other column inputs such as disturbances or manipulated variables for other loops that cause input multiplicity in a controlled output.

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2781

pressure dynamics are much faster than the tray temperature dynamics so that, for a well-tuned pressure controller adjusting the condenser duty, the slow temperature loops “see” a column at constant pressure. To assist in the selection of the control tray locations, sensitivity analysis and singular value decomposition (SVD) analysis, as practiced in ordinary distillation control design,37,38 are used. Figure 3 plots the tray temperature sensitivities (or gain) with respect to FA and VS. The first two left singular vectors are also shown in the figure. The first singular vector shows that FA should control the temperature of tray 2 (T2) in the stripping section. The second singular vector shows two possible tray locations, tray 18 (T18) in the rectifying section or tray 12 (T12) in the reactive section, which may be controlled using VS. The conventional metrics of the Niederlinski index and the relative gain for these two possibilities are tabulated in Table 2, showing that both control tray locations are acceptable. Steady-State IO Relations

Figure 6. Steady-state variation of (a) T2 with respect to FA while T12 is held constant by VS and (b) T12 with respect to VS while T2 is held constant by FA. Table 3. Ultimate Gain and Ultimate Period Control Loops pairing

KU (ultimate gain)

PU (min) (ultimate period)

VS-T18 VS-T12 FA-T12 FA-T2

5.84 13.50 32.60 3.53

4.8 5.7 5.7 8.7

Ideal RD Column Base Design. The generic ideal column design and operating and RD model parameters have been taken from Kaymak and Luyben27 and are reported in Table 1 for ready reference. There are 5 rectifying, 10 reactive, and 5 stripping trays in RD column, and the trays are numbered from bottom to top. The catalyst holdup is 1 kmol per reactive tray. The heavy reactant B is fed immediately above the reactive section, while the light reactant A is fed immediately below the reactive section. The two feeds are fed stoichiometrically at a feed rate of 12.6 mol/s each. The distillate rate is specified to be the same as the fresh feed rate, and the reflux ratio is adjusted to 2.6915 for 95 mol % product purities. Control Structure. The control structure studied in this work is shown in Figure 2. The heavy reactant B feed, FB, acts as the production rate handle. The light reactant A feed, FA, controls a sensitive tray temperature in the stripping section. The vapor boilup, VS, is manipulated to control the temperature of a reactive or rectifying tray. The reflux ratio is held constant, and the distillate and bottoms control the reflux drum and reboiler levels, respectively. The column pressure is assumed to be constant. This is a reasonable assumption because the

In an early paper, Roat et al.34 showed that control structures designed using conventional metrics based on local behavior around the operating steady state, such as open-loop gain, Niederlinski index, and relative gain, may still result in poor performance due to system nonlinearity. An azeotropic column and a reactive distillation column were used to underscore rigorous dynamic simulations as the only true means of validating a control system. In view of the high nonlinearity in RD systems, it is prudent to study the steady-state variation in the candidate control tray temperatures with the column inputs, also referred to as the steady-state input-output (IO) relations, to see if the local point metrics characterize the column behavior for the expected operating range of the inputs. The control structure for the example RD column consists of two temperature loops. The steady-state variation in the control tray temperatures with respect to the column inputs with both the temperature loops open and one of the loops closed is evaluated next. IO Relations with Both Loops Open. The steady-state variations of T2 (stripping tray temperature), T12 (reactive tray temperature), and T18 (rectifying tray temperature) with respect to FA, FB, and VS are plotted in Figure 4. For easy comparison, the base-case temperature is subtracted so that all the curves pass through the origin at the base-case operating condition. The inputs are changed by 35% around the base case in the plots. It is seen in Figure 4a that T2 exhibits gain sign reversal in the direction of excess B (FB increases or FA decreases). A crossover in the IO relation occurs as FB is increased by 27.2%. The IO relation for T12 exhibits gain sign reversal for a decrease in FA or FB (see Figure 4b). The crossover point is slightly beyond -35%. For an increase in FA, the IO relation is flat with only a slight increase in T12. The T18-FB IO relation exhibits crossover for +22.7%, -22.5%, and -30.8% change in FB (see Figure 4c). Negligible change in T18 occurs for an increase in FA up to +35% and for a slight decrease. For a large decrease in FA, T18 increases. The IO relations of the three temperatures are well-behaved with respect to VS in that an increase beyond the base-case value causes an increase in the temperature and vice versa. For the control structure considered, FB is the production rate handle and is, therefore, a disturbance into the column. As seen from Figure 4 IO relations, if T18 is controlled, a steady-state transition is possible for a large decrease in FB (IO relation turns back again), while wrong control action is likely for a large increase (IO relation does not turn back again). Controlling T12,

2782


Figure 7. Closed-loop dynamics of manipulated (FA and VS) and controlled variables (T2 and T18) with product purities for a series of step changes in FB. Control loop pairings: VS-T18 and FA-T2.

Figure 8. Temperature profile of RD column at base case and after steadystate transition.

on the other hand, avoids a crossover in the IO relation with respect to FB for a decrease, while the crossover for an increase occurs for a larger increase in FB compared to T18. Controlling T12 thus provides a larger operating window with no crossover in the IO relation and may, therefore, result in better controllability. IO Relations with One Loop Closed. Holding one tray temperature constant can significantly alter the steady-state IO relation that the other loop “sees”. To evaluate the same, Figure 5 plots the variation in each of the controlled temperatures with respect to their corresponding manipulated variable with the other controlled temperature at its set point (base-case value). These plots are generated by exploiting the flexibility in column specifications of our in-house Naphtali-Sandholm simulator for RD columns.39 To trace the variation in T2 with FA holding T18 (or T12) constant, the two column specifications are the reflux ratio and the tray temperature T18 (or T12). With this specification

set, the reboiler duty necessary to maintain T18 (or T12) at its specified value gets calculated as a dependent variable. The complementary plot of the variation in T18 (or T12) with reboiler duty holding T2 constant uses the reflux ratio and the reboiler duty as the column specifications. The FA for maintaining T2 at its specified value is calculated using an outer iteration loop. The steady-state IO relation of T2 with respect to FA holding T18 constant is plotted in Figure 5a. The complementary IO relation of T18 with respect to VS holding T2 constant is shown in Figure 5b. The two plots show that there are four distinct steady states for which both T2 and T18 equal their base-case values. The base-case steady state is marked as O, while the three other distinct steady states are marked as I, J, and K. The stability of these steady states is determined by the sign of the local slope, i.e., the process gain, with the other temperature loop closed. A steady state would be stable only if the process gain sign for each of the temperature loops is the same as the corresponding open-loop process gain sign at the base-case design. Accordingly, steady states O and K are stable, while steady states I and J are unstable. The latter is unstable since the local slope in the T18-VS IO relation is positive, whereas the corresponding open-loop slope (Figure 4c) is negative. The plots in Figure 5 suggest that the column moving toward I during transients can result in wrong control action. Figure 6 plots the complementary IO relations of T2-FA (T12 constant) and T12-VS (T2 constant). These correspond to the IO relation that each of the temperature controllers sees with the other loop on automatic when T12 (instead of T18) and T2 are controlled. In addition to the base-case steady state marked O, two distinct steady states, Q and R, are seen in plot. Both these steady states are unstable so that wrong control action may occur for large production rate changes in either direction. Dynamic Simulation Results The occurrence of input multiplicity only indicates the possibility of wrong control action or a steady-state transition.

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2783



The largest production rate change, the primary disturbance into the column, that can be handled effectively depends on the process dynamics. Rigorous closed-loop dynamic simulations are performed for this purpose. The two temperature loops are tuned using the ultimate gain and period obtained from a relay

feedback test.40 Two first-order lags of 60 s each are used for every temperature measurement. The Tyreus-Luyben controller (TLC) settings41 are then calculated. The ultimate gain and period of the two temperature loops, T2-FA and T18/T12-VS, are reported in Table 3. When T18 is controlled, the TL controller

2784



tuning parameters in the rectifying temperature control loop have to be detuned by a factor of 5 to suppress sustained closedloop oscillations as the ultimate period of the T18-VS loop is comparable to the T2-FA loop. Detuning is not required when T12 is controlled. Consider the case where T18, a rectifying tray temperature, is controlled using VS. To test for the robustness of this control configuration, the closed-loop response for an input disturbance sequence of a series of large step changes in FB, the production rate handle, is obtained. Figure 7 plots the response for a +20% step change in FB about the base case at 0 and 15 h followed by a large -40% step change at 30 h to bring FB back to its base-case value. The first two step changes are handled effectively with the two temperature loops ensuring that FA and VS increase by ∼20% in response to the 20% step increases in FB. The distillate and bottoms purities are also maintained close to their design values once steady state is reached. For the large step decrease of 40% in FB, however, a steady-state transition occurs. Instead of decreasing, VS increases by a small amount and FA decreases by only ∼7.5%. The distillate purity of component C decreases to 77.3 mol %, while the bottoms purity of component D increases to 97.7%. This steady state corresponds to the stable steady state marked K in parts a and b of Figure 5 with the VS and FA being above the base-case values. The temperature profiles corresponding to this new steady state and the base-case steady state are shown in Figure 8. As expected, the controlled tray temperatures, T2 and T18, are exactly equal in both the temperature profiles. Figure 9 plots the closed-loop response for the complementary disturbance sequence of -15% step changes in FB about the base case at 0 and 15 h followed by a large step change of +30% back to its base-case value at 30 h. The first two step changes are handled effectively with FA and VS decreasing by 15% to match the change in FB and the product purities being maintained close to their design values. The large 30% step

increase in FB back to its base-case value, however, results in wrong control action with both FA and VS decreasing instead of increasing. As seen in the bifurcation diagram in Figure 5a, a stable steady state does not exist for a large increase in FB so that there is no steady state at which the column can settle down. The FA and VS valves eventually end up shutting down. Wrong control action thus causes a constraint to be hit. The closed-loop results show that controlling T18 and T2 results in poor control system robustness. The alternate control tray location of T12, instead of T18, significantly improves control system robustness with the product purities maintained close to the design values for the same two disturbance sequences in FB. The closed-loop responses for step changes of +20%, +20%, and -40% about the base case in FB at times 0, 15, and 30 h, respectively, are shown in Figure 10. For all three step changes, the temperature controllers adjust FA and VS to match the step change in FB. Also, the steady-state product purities are maintained within 1% of their design values. Similarly for a step-input sequence of -15%, -15%, and +30% FB, the closed-loop response in Figure 11 shows that the steadystate purities are maintained within 1% with the FA and VS moving to match the changes in FB. Clearly, altering the control tray location to tray 12 significantly improves the control system robustness. The bifurcation study in Figure 6b corresponding to controlling T12 and T2 suggests that wrong control action may occur for a large production rate change in either direction. To test for the same, larger step changes in FB were input in incremental steps of 5%. Wrong control action occurs for a -70% change in FB, while the control system is able to handle even a 100% increase in FB. Detuning the T12-VS loop by a factor of 5 over the TL controller settings causes wrong control action for a -50% and a +70% step change in the production rate handle, FB. Even as such large step changes would seldom be implemented in practice, the result confirms the possibility of

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2785 Table 4. IAE in Product Purities for CS7 and CS2 For (20% Change in Throughput Manipulation CS7

CS2

throughput change

+20%

-20%

+20%

-20%

xC, D xD, B

1.0921 0.3027

1.3704 0.4665

0.9147 0.2024

0.8971 0.1856

argument holds for a -25% step change in FB. Controlling T12, on the other hand, avoids the initial wrong direction of response in VS. This makes the control system more robust, with much larger step changes being needed for wrong control action. As with distillation systems, the initial direction of response to a disturbance thus plays an important role in determining the overall control system robustness.42 Comparison with Previous Work

Figure 12. Open-loop dynamics of (a) T2, (b) T12, and (c) T18 with respect to different step changes in FB.

wrong control action for a large production rate change in either direction. For the same detuning factor of 5 in the T18-VS loop, a steady-state transition occurs for a -25% step change in FB while the wrong control action occurs for a +30% change. These results quantitatively show that controlling T12 (reactive tray) gives significantly improved control system robustness compared to controlling rectifying T18 (rectifying tray). Proper choice of the control tray location is, thus, crucial in the design of a robust RD control system. The reason for the better performance of T12 over T18 can be traced to the open-loop response to a step change in FB in Figure 12. Responses for different magnitude step changes are plotted in the figure. Notice that T18 and T2 exhibit an inverse response, while T12 does not. Thus, when T18 and T2 are controlled, for a >30% increase in FB, the wrong initial direction of response in VS and FA coupled with input multiplicity causes the control system to succumb to wrong control action. A complementary

The previous articles on RD column control by Kaymak and Luyben27,28 are of particular relevance to this work. The authors recommended a two-temperature control structure where T2 and T12 are maintained using FA and FB, respectively, with throughput manipulation using VS (labeled CS7 in their work) as the best overall. Controlling T18 using FB was not recommended due to an inverse response giving a very large reset time and, hence, a sluggish closed-loop response. In the context of input multiplicity between T18 and FB (Figure 4a), CS7 with T18-FB and T2-FA pairings fails for (35% throughput change. Specifically, wrong control action occurs for a -35% change and a steady-state transition occurs for +35% change using standard TLC settings.41 The control system robustness expectedly improves if T12 is controlled instead of T18, with the control system handling a (40% throughput change. Figure 13 compares the closed-loop response of CS2 and CS7 with T12 and T2 as the controlled variables for a (40% throughput change. For consistent comparison, the standard TLC tuning procedure is applied to all the temperature loops. The closed-loop response for CS7 is highly oscillatory, with the FB valve saturating for the large throughput change in either direction. CS2, on the other hand, gives a relatively smooth response. For a quantitative comparison, Table 4 reports the Integral Absolute Error (IAE) values of the distillate and bottoms product purities. The values indicate that CS2 is superior to CS7 in terms of the tightness of product purity control achieved. In particular, the performance is significantly better for a throughput decrease. Kaymak and Luyben27 also showed that, for CS7 with T12 and T2 controlled, both temperature controllers are direct acting. An increase (decrease) in VS for a throughput change causes the control tray temperatures to increase (decrease) so that the negative gain of the control tray temperatures with respect to the two feeds (direct action) causes both the feeds to increase (decrease) in tandem, maintaining the delicate stoichiometric feed balance in the initial transient. In CS2 considered here, the T12-VS loop is indirect acting. Here, as FB (throughput manipulator) is increased, T2 increases and T12 decreases. To maintain T12, VS must increase (indirect action), and to maintain T2, FA must increase (direct action). The open-loop sensitivities in CS2 are, thus, well-behaved. With respect to the initial direction of response, the inverse response of T2 with respect to FB (see Figure 12a), however, causes FA to initially move in the opposite direction of the throughput change. The overall transient stoichiometric imbalance (FA-FB) in CS2 is, however, less compared to that in CS7, as seen in Figure 14. CS2 thus

2786


Figure 13. Closed-loop responses of CS2 and CS7 control structures with T12 and T2 as the controlled variables for (40% throughput change.

KU ) ultimate gain, dimensionless L/D ) reflux ratio M ) manipulated variable P ) pressure, bar PU ) ultimate period, min R ) ideal gas constant t ) time, h Tn ) temperature of nth tray, K U ) SVD parameters VS ) vapor boilup, mol/s xC ) liquid mole fraction of C xD ) liquid mole fraction of D Figure 14. Transient stoichiometric imbalance (FA-FB) of CS7 and CS2 for (20% change in throughput manipulator.

performs the key regulatory task of stoichiometric feed balancing better than CS7, which explains the tighter product-purity control achieved. Conclusions This article shows that high nonlinearity occurs in the generic ideal RD column, which significantly affects control system robustness. In contrast to conventional wisdom, results show that controlling the most sensitive tray temperature (a rectifying tray) makes the control structure studied susceptible to nonlinear dynamic phenomena due to input multiplicity. A steady-state transition occurs for a large production rate decrease, while wrong control action occurs for a large production rate increase. Controlling a less sensitive reactive tray significantly improves the control system robustness as the severity of input multiplicity is mitigated and also the initial inverse response with respect to the heavy reactant feed (the production rate handle) is avoided. The article highlights the importance of studying steady-state input-output relations for understanding the nonlinear dynamic phenomena as well as for the proper selection of the control tray location for good control system robustness. Acknowledgment The financial support from the Department of Science and Technology, Government of India, is gratefully acknowledged. Nomenclature B ) bottoms flow rate, mol/s D ) distillate flow rate, mol/s FA ) feed flow rate of component A, mol/s FB ) feed flow rate of component B, mol/s KP ) steady-state gain, K/% change in manipulated variable

Greek Symbols Rj ) relative volatility of component j Literature Cited (1) Backhaus, A. A. Continuous Processes for the Manufacture of Esters. U.S. Patent 1,400,849, 1921. (2) DeGarmo, J. L.; Parulekar, V. L.; Pinjala, V. Consider Reactive Distillation. Chem. Eng. Prog. 1992, 88, 43. (3) Luyben, W. L. Economic and Dynamic Impact of use of Excess Reactant in Reactive Distillation Systems. Ind. Eng. Chem. Res. 2000, 39, 2935. (4) Kaymak, D. B.; Luyben, W. L. Quantitative Comparison of Reactive Distillation with Conventional Multiunit Reactor/Column/Recycle Systems for Different Chemical Equilibrium Constants. Ind. Eng. Chem. Res. 2004, 43, 2493. (5) Agreda, V. H.; Partin, L. R.; Heise, W. H. High Purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, 86, 40. (6) Kenig, E. Y.; Ba¨der, H.; Go´rak, A.; Beβling, B.; Adrian, T.; Schoenmakers, H. Investigation of Ethyl Acetate Reactive Distillation Process. Chem. Eng. Sci. 2001, 56, 6185. (7) Leyes, C. E.; Othmer, D. F. Esterification of Butanol and Acetic Acid. Ind. Eng. Chem. 1945, 37, 968. (8) Smith, L. A. Method for the Preparation of Methyl Tertiary Butyl Ether. U.S. Patent 4,978,807, 1990. (9) Matouq, M.; Quitain, A. T.; Takahashi, K.; Goto, S. Reactive Distillation for Synthesizing Ethyl tert-Butyl Ether from Low-Grade Alcohol Catalyzed by Potassium Hydrogen Sulfate. Ind. Eng. Chem. Res. 1996, 35, 982. (10) Hickey, T. P.; Adams, J. R. (Chemical Research & Licencing Company) U.S. Patent 5,321,163, 1994. (11) Sharma, M. M.; Mahajani S. M. Industrial Applications of Reactive Distillation. In ReactiVe DistillationsStatus and Future Directions; Sundmacher, K., Kienle, A., Eds.; Wiley-VCH: Weinheim, Germany, 2003; p 3. (12) Gru¨ner, S.; Mohl, K. D.; Kienle, A.; Gilles, E. D.; Fernholtz, G.; Friedrich, M. Non-Linear Control of a Reactive Distillation Column. Control Eng. Pract. 2003, 11, 915. (13) Mohl, K. D.; Kienle, A.; Gilles, E. D.; Rapmund, P.; Sundmacher, K.; Hoffmann, U. Steady-State Multiplicities in Reactive Distillation Columns for the Production of Fuel Ethers MTBE and TAME: Theoretical Analysis and Experimental Verification. Chem. Eng. Sci. 1999, 54, 1029. (14) Ciric, A. R.; Miao, P. Steady-State Multiplicities in an Ethylene Glycol Reactive Distillation Column. Ind. Eng. Chem. Res. 1994, 33, 2738.

Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2787 (15) Gu¨ttinger, T. E.; Morari, M. Predicting Multiple Steady-states in Distillation: Singularity Analysis and Reactive Systems. Comput. Chem. Eng. 1997, 21 (SUPPL1), S995. (16) Jacobs, R.; Krishna, R. Multiple Solutions in Reactive Distillation for Methyl tert-Butyl Ether Synthesis. Ind. Eng. Chem. Res. 1993, 32, 1706. (17) Hauan, S.; Hertzberg, T.; Lien, K. M. Why Methyl tert-Butyl Ether Production by Reactive Distillation May Yield Multiple Solutions. Ind. Eng. Chem. Res. 1995, 34, 987. (18) Sneesby, M. G.; Tade´, M. O.; Smith, T. N. Steady-State Transitions in the Reactive Distillation of MTBE. Comput. Chem. Eng. 1998, 22, 879. (19) Tian, Y. C.; Zhao, F.; Bisowarno, B. H.; Tade´, M. O. PatternBased Predictive Control for ETBE Reactive Distillation. J. Process Control 2003, 13, 57. (20) Bisowarno, B. H.; Tian, Y. C.; Tade´, M. O. Model Gain Scheduling Control of an Ethyl tert-Butyl Ether Reactive Distillation Column. Ind. Eng. Chem. Res. 2003, 42, 3584. (21) Khaledi, R.; Young, B. R. Modeling and Model Predictive Control of Composition and Conversion in an ETBE Reactive Distillation Column. Ind. Eng. Chem. Res. 2005, 44, 3134. (22) Al-Arfaj, M. A.; Luyben, W. L. Comparison of Alternative Control Structures for an Ideal Two-Product Reactive Distillation Column. Ind. Eng. Chem. Res. 2000, 39, 3298. (23) Al-Arfaj, M. A.; Luyben, W. L. Comparative Control Study of Ideal and Methyl Acetate Reactive Distillation. Chem. Eng. Sci. 2002, 57, 5039. (24) Al-Arfaj, M. A.; Luyben, W. L. Control of Ethylene Glycol Reactive Distillation Column. AIChE J. 2002, 48, 905. (25) Al-Arfaj, M. A.; Luyben, W. L. Design and Control of Olefin Metathesis Reactive Distillation Column. Chem. Eng. Sci. 2002, 57, 715. (26) Al-Arfaj, M. A.; Luyben, W. L. Control study of Ethyl tert-Butyl Ether Reactive Distillation. Ind. Eng. Chem. Res. 2002, 41, 3784. (27) Kaymak, D. B.; Luyben, W. L. Comparison of Two types of Two Temperature Control Structures for Reactive Distillation Columns. Ind. Eng. Chem. Res. 2005, 44, 4625. (28) Kaymak, D. B.; Luyben, W. L. Evaluation of a Two Temperature Control Structure for a Two-Reactant/Two-Product Type of Reactive Distillation Column. Chem. Eng. Sci. 2006, 61, 4432. (29) Hung, S. B.; Lee, M. J.; Tang, Y. T.; Chen, Y. W.; Lai, I. K.; Hung, W. J.; Huang, H. P.; Yu, C. C. Control of Different Reactive Distillation Configurations. AIChE J. 2006, 52, 1423. (30) Olanrewaju, M. J.; Al-Arfaj, M. A. Impact of Disturbance Magnitudes and Directions on the Dynamic Behavior of a Generic Reactive Distillation. Chem. Eng. Process. 2006, 45, 140.

(31) Huang, K.; Nakaiwa, M.; Tsutsumi, A. Towards further internal heat Integration in design of reactive distillation columns. Part II. The process dynamics and operation. Chem. Eng. Sci. 2006, 61, 5377. (32) Kienle, A.; Marquardt, W. Nonlinear Dynamics and Control of Reactive Distillation Processes. In ReactiVe distillationsStatus and future directions; Sundmacher, K., Kienle, A., Eds.; Wiley-VCH: Weinheim, Germany, 2002; p 241. (33) Pavan Kumar, M. V.; Kaistha, N. Decentralized control of a kinetically controlled ideal reactive distillation column. Chem. Eng. Sci. 2008, 63, 228. (34) Roat, S.; Downs, J.; Vogel, E.; Doss, J. Integration of Rigorous Dynamic Modeling and Control System Synthesis for Distillation Columns. In Chemical Process Control, CPC III; Elsevier: Amesterdam, The Netherlands, 1986; p 99. (35) Lin, Y. D.; Huang, H. P.; Yu, C. C. Relay feedback tests for highly nonlinear processes: Reactive distillation. Ind. Eng. Chem. Res. 2006, 45, 4081. (36) Pavan Kumar, M. V.; Kaistha, N. Role of Multiplicity in Reactive Distillation Control Structure Design. J. Process Control 2008, doi: 1016/j.jprocont.2007.12.001. (37) Moore C. F. Selection of Controlled and Manipulated Variables. In Practical Distillation Control; Luyben, W. L., Ed.; Van Nostrand Reinhold: New York, 1992; p 140. (38) Luyben, W. L. Evaluation of Criteria for Selecting Temperature Control Trays in Distillation Columns. J. Process Control 2006, 16, 115. (39) Singh, R.; Pavan Kumar, M. V.; Kaistha, N. Steady State Reactive Distillation Simulation using the Naphtali-Sandholm Method. Can. J. Chem. Eng. 2007, 1, 75. (40) Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645. (41) Tyreus, B. D.; Luyben, W. L. Tuning PI Controllers for Integrator/ Dead Time Processes. Ind. Eng. Chem. Res. 1992, 31, 2625. (42) Skogestad, S.; Morari, M. Understanding the Dynamic Behavior of Distillation Columns. Ind. Eng. Chem. Res. 1988, 27, 1848.

ReceiVed for reView December 17, 2007 Accepted February 1, 2008 IE701720R

Steady-State Multiplicity and Its Implications on the Control of an Ideal

Recommend Documents