On the Controllability of Middle-Vessel Continuous Distillation Columns

Ind. Eng. Chem. Res. 2004, 43, 2721-2729

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On the Controllability of Middle-Vessel Continuous Distillation Columns Fabrizio Bezzo, Simona Varrasso, and Massimiliano Barolo* Dipartimento di Principi e Impianti di Ingegneria Chimica (DIPIC), Universita` di Padova, via Marzolo 9, 35131 Padova PD, Italy

Controllability issues for conventional and middle-vessel continuous distillation columns are addressed in this paper by using different controllability and resiliency measures. It is shown that, by using a middle-vessel continuous column (MVCC), the sensitivity of the control system performance to the direction of input disturbances can be significantly reduced with respect to a conventional column, in such a way that effective composition control can be achieved with simple proportional-integral controllers even for highly ill-conditioned columns. The tuning of composition loops in an MVCC is shown to be easier than that in a conventional column. 1. Introduction It is well-known that static and dynamic nonlinearities, control loop interactions, and system directionality can make the dual composition control in distillation columns a challenging problem. These control challenges can be tackled effectively by using advanced control systems. However, advanced controllers are often difficult to derive and to implement, expensive to maintain, and sometimes difficult for plant operators to understand.1 To improve the control performance and reduce the energy consumption, it may instead be convenient to modify the plant design in such a way as to make the column inherently easier to control, even with conventional proportional-integral (PI) controllers. The interaction between column design and column control in continuous distillation has been recently addressed by Barolo and Papini,2 who proposed a novel distillation column configuration called a middle-vessel continuous column (MVCC). In this column (Figure 1a), a total draw-off tray is used to collect all of the liquid coming from the rectifying section and divert it to an intermediate external vessel, to which the process feed is also fed. A P-only level controller is exploited to drive the liquid from the middle vessel to the column at the most convenient rate. The authors showed that, by using this kind of column, the dynamic interaction between composition loops can be almost suppressed, in such a way that simple PI controllers are able to provide tight composition control. Moreover, the controller performance was shown to be almost unaffected by the middle-vessel holdup. Along the same lines, Phimister and Seider3 proved that an MVCC is particularly suitable for implementation of the distillate-bottoms (DB) control structure, which conversely is known to suffer from a lack of integrity when applied to conventional columns. They found that the middle vessel acts like an “inventory regulator” when the DB scheme is employed, thus allowing for temporary accumulation of liquid matter, which (on the other hand) may not be tolerable in the absence of the middle vessel.4 * To whom correspondence should be addressed. Tel.: +39 049.827.5473. Fax: +39 049.827.5461. E-mail: max.barolo@ unipd.it.

The above two studies are the only ones dealing with the dynamics and control of MVCCs that have been published in the open literature so far. Therefore, several issues on these columns are currently open for investigation. This paper aims at providing a contribution to the understanding of the controllability of MVCCs. In particular, we study how the middle vessel affects the sensitivity of the control system response to the direction of the disturbance inputs and to the tuning of the composition loops. The investigation will be carried out with reference to the models of two columns with different steady-state and dynamic responses. 2. Theoretical Background Controllability and resiliency measures can be used as diagnostic tools during the design of a process control system. The objective is to obtain some information suggesting whether the process will be controlled easily or the final design will instead require a sophisticated control system to satisfy the process specifications. Lewin5 points out that the use of simple screening measures is particularly important in the early design stages when competing process flowsheets need evaluation. Typical controllability measures include the process condition number γ, the disturbance condition number γd, and the disturbance cost DC.6,7 These measures can assist the designer in rejecting schemes that may have unacceptable static or dynamic properties. This is particularly important for ill-conditioned processes, i.e., those having a high directionality.8 The directionality in a multivariable system indicates by how much the amplification of a manipulated input vector changes with the input vector direction.9-11 Consider the linearized system of the form

y(s) ) G(s) u(s) + Gd(s) d(s)

(1)

where y is the output vector, u is the vector of manipulated inputs, d is the vector of disturbances, and G and Gd are the process and disturbance transfer function matrixes (respectively). The process condition number is defined as

γ)σ j (G)/σ(G)

10.1021/ie030489f CCC: $27.50 © 2004 American Chemical Society Published on Web 04/28/2004

(2)

2722 Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004

Figure 1. (a) MVCC and (b) conventional columns with LV control structure.

where σ j (G) and σ(G) are respectively the largest and smallest nonzero singular values of G. The condition number provides a direct measure of the directionality of the system. A large condition number indicates that the gain of the plant changes significantly with the input direction. Therefore, from the control standpoint, if γ is large, the use of a simple decentralized control system (such as a conventional multiloop PI control system) leads to performances that strongly depend on the type of perturbation. The minimum singular value (also called the Morari resiliency index, MRI) is itself used as a measure of the invertibility of the system and therefore represents a measure of the potential problems of the system under feedback control, in that the lower the value of G, the harder the control of the system.12 To just quote one recent example of the usefulness of γ and MRI in the field of distillation control, Serra et al.13 have used these indices to assess the controllability of different multicomponent distillation column configurations. The disturbance condition number6 γd is defined as

γd )

||G-1Gdd||2 σ j (G) ||Gdd||2

(3)

where the symbol ||‚||2 indicates the Euclidean norm of a vector. It can be demonstrated that γd depends on the direction of the disturbance but not on its magnitude. In this paper, we will mainly consider a different controllability index, i.e., DC,7 defined as

DC ) ||G-1Gdd||2

(4)

DC provides an indication of the feedback effort required to completely reject the disturbance vector d. As in the formulation of γd, the formulation of DC assumes that perfect control is achieved, and therefore DC defines an upper bound to the performance, which is independent of any particular controller design or tuning. Because DC is a frequency-dependent measure, the effect of 2D disturbances can be analyzed in terms of a 3D plot in which DC is represented as a function of the frequency ω and the direction θ of d. Because γ, γd, and DC are scaling-dependent measures, the input variables (reflux rate and vapor boilup rates, L and V) and output variables (distillate and

Table 1. Column C1: Some Parameters no. of real trays feed tray (numbering from the bottom) top pressure (mmHg) Murphree tray efficiency feed composition (ethanol m.f.) distillate composition (ethanol mole fraction) bottoms composition (ethanol ppm by mole) feed rate (L/h) distillate rate (L/h) bottoms rate (L/h) reflux rate (L/h) steam rate (kg/h) condenser holdup (kg) reboiler holdup (kg) middle-vessel holdup (kg)

12 4 755 0.56 0.1424 0.7696 200 140.0 55.2 91.2 245.2 131.0 18 58 50

bottom compositions, xD and xB) are scaled between zero and twice the nominal value.14 Thus, the nominal-scaled values of these variables are 0, with an allowed variation of (0.5. As a consequence, if one element of G-1Gdd is greater than 0.5 at steady state, actuator saturation will occur for the corresponding input, which will result in offset. At higher frequencies, large DC values identify the disturbance directions for which the high-frequency modes are attenuated with difficulty or not at all. The maximum magnitude of a disturbance is assumed to be 20% of the full-scale disturbance range. 3. Example Columns The controllability of a conventional continuous distillation column (Figure 1b) will be compared to that of an MVCC for two different test columns. Because our purpose is mainly to provide an understanding of the effect of the middle vessel on the controllability of a column rather than to specifically design the “best” control system for that column, only the so-called LV control configuration will be considered (Figure 1). Although this configuration might not be the best from the control standpoint in certain cases,15 it is, nevertheless, the most frequently used in the industry.16 To improve the response to feed flow upsets, the reflux and boilup rates were also ratioed to the feed rate in the composition loops,17 but no major changes in the relative merits of the conventional column and of the MVCC with these ratioed controllers were observed. The first column (C1) separates an ethanol/water mixture. Table 1 summarizes some of the column design

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2723 Table 2. Column C2: Some Parameters. no. of ideal trays relative volatility feed tray (numbering from the bottom) feed composition (light component molar fraction) distillate composition (mole fraction) bottoms composition (mole fraction) feed rate (kmol/min) distillate rate (kmol/min) bottoms rate (kmol/min) reflux rate (kmol/min) boilup rate (kmol/min) condenser holdup (kmol) reboiler holdup (kmol) middle-vessel holdup (kmol)

18 2.5 9 0.5 0.99 0.01 1.9 0.95 0.95 1.9221 2.8721 10.9 15.8 35

and operating parameters. The column is described by a detailed tray-to-tray model taking into account mass and enthalpy balances, with nonlinear tray hydraulics based on the actual tray geometry. The second column (C2) is assumed to separate a constant-relative-volatility binary mixture, with nonlinear tray hydraulics. The basic column parameters are reported in Table 2. 4. Controllability Analysis The controllability indices are obtained from linear models of the form

x3 (t) ) Ax(t) + Bu(t) y(t) ) Cx(t) + Du(t)

(5)

which can be derived from the rigorous models using the procedure described by Skogestad.18 The linear model is used to produce the DC maps following the procedure of Lewin.7

Decentralized PI controllers are used in the composition loops, while P-only controllers are used in the level loops (the tuning of the level loops will not be discussed because it does not affect the composition loop dynamics appreciably with an LV control structure). The (unmeasured) disturbances are represented by variations in the feed flow and in the feed composition. The steady-state value λ11(0) for the diagonal element of the relative gain array matrix is about 85 for column C1 and 20 for column C2. Especially in the C1 case, that indicates a very large steady-state coupling between the quality loops. Both columns have |λ11(ω)| = 1 at high frequencies. However, consistent with what was demonstrated by Barolo and Papini,2 |λ11(ω)| gets closer to 1 in the MVCCs at significantly lower frequencies than in the conventional columns (Figure 2a), so that the MVCCs reduce the dynamic coupling between the composition loops. The steady-state condition number γ(0) is about 9900 and 90 for columns C1 and C2, respectively. Therefore, both columns have a strong steady-state directionality (actually, very strong for column C1). A dynamic measure of γ was also performed: Figure 2b reports the dynamic behavior of γ(ω) for conventional and MVCC configurations in column C1 (column C2 behavior is qualitatively similar). This shows that γ decreases at intermediate frequencies (in the range important for feedback control,18 i.e., ∼10-2-100 rad/min) and that the MVCC provides only a marginal reduction of directionality. Thus, the effect of the middle vessel seems not to be significant from this point of view. Further insight is provided by the DC index. Figure 3 plots the DC index for conventional C1 and MVCC C1 (θ ) 0° refers to a feed flow disturbance only, while θ ) 90° refers to a feed composition disturbance only).

Figure 2. (a) Diagonal element λ11 of the RGA matrix and (b) condition number for column C1 (conventional and MVCC configurations).

Figure 3. Column C1: DC for the (a) conventional configuration and (b) MVCC configuration.


Figure 4. Column C2: DC for the (a) conventional configuration and (b) MVCC configuration. Table 3. Composition Controller Tuning According to the Shen-Yu Relay Feedback Method

conventional middle vessel

conventional middle vessel

distillate

bottoms

Column C1 KD ) 6.989 × 10-2 (m3/min)/(mole fraction) τD ) 19.0 min KD ) 2.215 × 10-1 (m3/min)/(mole fraction) τD ) 12.5 min

KB ) 7.425 × 102 (kg/min)/(mole fraction) τB ) 13.2 min KB ) 3.967 × 103 (kg/min)/(mole fraction) τB ) 5.0 min

Column C2 KD ) 7.95 (kmol/min)/(mole fraction) τD ) 48.0 min KD ) 29.10 (kmol/min)/(mole fraction) τD ) 37.0 min

KB ) 8.90 (kmol/min)/(mole fraction) τB ) 37.0 min KB ) 25.70 (kmol/min)/(mole fraction) τB ) 30.0 min

The DC index for C2 (in both conventional and MVCC configurations) is plotted in Figure 4. The middle-vessel level controller gain was set at 1.0 × 10-4 (m3/min)/kg in column C1 and 0.1 (kmol/min)/kmol in column C2; looser level control makes the middle vessel prone to overfill or drain. The DC surfaces show that no limitations due to actuator constraints are experienced (DC < 0.16 at all frequencies for all disturbances in both columns). Note that a more proper approach to detecting input saturations would be to plot each element of the G-1Gdd vector separately (eq 4). This was actually done, but no major differences either in the general trend of the DCi plots or in the maximum DCi values were observed for both test columns. As expected, the required steady-state control effort (in terms of DC) is quite markedly dependent upon the disturbance direction and is the same for the conventional and MVCC configurations. Regardless of the configuration (conventional or MVCC), for column C1 the worst disturbance direction (i.e., the one that calls for a larger steady-state control effort) is ∼25°, while the best disturbance direction is ∼115°; for column C2, the worst and best disturbance directions are ∼10° and ∼100°, respectively. The effect of the middle vessel on the column controllability is clearly highlighted by the DC plots. In the frequency range important for feedback control, the middle vessel reduces the required control effort and makes it almost independent of the disturbance direction. Thus, the column controllability is improved. It can also be observed that DC is slightly more dependent on the frequency in the MVCCs, but this has a minor impact on the controllability. 5. Performance Evaluation 5.1. Relay Tuning. Simulations were carried out to assess the above findings. First, a tuning procedure that

could actually be implemented in an experimental/ industrial setting was considered in order to tune the composition controllers. For this purpose, the quality loops were tuned according to the relay-feedback procedure suggested by Shen and Yu19 for multivariable systems; to eliminate one possible source of arbitrariness that could bias the tuning results, no further online adjustments of the tuning parameters were attempted. Table 3 lists the tuning parameters that were obtained for conventional and MVCC C1 and C2 columns; in the following, K indicates the controller gain and τ the integral time constant. The control system response to disturbances entering the columns at different directions was evaluated. A 20% step increase was considered for each disturbance; the disturbances entered the column at time t ) 20 and 50 min for column C1 and C2, respectively. The system responses are illustrated in Figures 5-8. As expected, the control performance is dependent on the disturbance direction for the conventional columns (Figures 5 and 7). This is particularly evident for conventional column C1: the worst-direction disturbance causes significant under/overshoots in the controlled variables and calls for very large settling times. In practice, it appears that a single set of tuning parameters is not able to provide effective rejection of all possible disturbances in the conventional columns. Such behavior can be explained by looking at the DC index in Figures 3a and 4a, which shows that the control effort needed to reject a disturbance markedly depends on the disturbance direction in the frequency range important for feedback control. On the other hand, Figures 3b and 4b anticipate that the dynamic control effort is much less affected by the disturbance direction if an MVCC is considered. The simulation results confirm this indication, as illustrated in Figures 6 and 8. Although the steady-state change in the manipulated variables (steady-state DC) is obvi-

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2725

Figure 5. Conventional C1. Responses to best-direction and worst-direction disturbances with Shen-Yu tuning of the composition controllers: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate.

Figure 6. MVCC C1. Responses to best-direction and worst-direction disturbances with Shen-Yu tuning of the composition controllers: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate.

ously the same as that in the relevant conventional column, the dynamic response is substantially improved in the MVCCs: the controllers do a good job in the rejection of both the worst and best disturbances, with fairly small deviations from the composition setpoints, smooth moves of the manipulated variables, and short settling times. Thus, a single set of tuning parameters appears to be enough for rejecting all possible disturbances in an MVCC.

The fact that at higher frequencies DC is smaller for an MVCC than it is for a conventional column (as the DC surfaces indicate) cannot be directly abstracted from the analysis of Figures 5-8. Actually, comparing the manipulated variable responses in a conventional column to those of the relevant MVCC in order to get an indirect comparison of the DC values may be misleading. In fact, the formulation of DC (eq 4) implies that perfect disturbance rejection is achieved, an occurrence


Figure 7. Conventional C2. Responses to best-direction and worst-direction disturbances with Shen-Yu tuning of the composition controllers: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate.

Figure 8. MVCC C2. Responses to best-direction and worst-direction disturbances with Shen-Yu tuning of the composition controllers: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate.

that can be assumed to be “almost” true for the MVCCs (Figures 6a and 8a) but certainly cannot for the conventional columns (Figures 5a and 7a). To enforce the limiting condition of perfect control and allow for a direct comparison of the input profiles, nonlinear model-based controllers were derived for both test columns in both configurations using tools from differential geometry.20 As an example, Figure 9a shows

the controlled variable responses for column C2 (conventional and MVCC) subject to the worst-direction disturbance with nonlinear composition control: it can be seen that the disturbance is perfectly rejected for both column configurations. Figure 9b shows the manipulated variable responses, which are now truly representative of the DC values. Clearly, it “costs” more (in terms of input changes) to reject the disturbance in the


Figure 9. Column C2: (a) controlled variable and (b) manipulated variable responses to the worst-direction disturbance under “perfect” control.

Figure 10. Conventional C1. Responses of the worst-direction tuning set (full line) and of the best-direction tuning set (dashed line) to the worst-direction disturbance: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate.

conventional column than it does to reject that in the MVCC. Similar results were obtained for different disturbance directions and for column C1. 5.2. Optimal Tuning. It has been shown in the previous subsection that a single set of tuning parameters may not be effective to counteract all possible disturbances affecting a conventional column; conversely, this does not seem to be the case for an MVCC. We now further investigate this issue, by obtaining different tuning sets, according to the direction of the disturbance entering the column. The optimal tuning for a given disturbance was accomplished by adjustment of the controller gains and integral times in such a way as to minimize a suitable penalty function embedding information on the control performance in terms of setpoint deviation, settling time, and valve travel.21 After some trials, the following composite objective function was selected:

fob ) R × ITAED + β × ITAEB + φ × CtL + δ × CtV (6)

where

ITAED )

∫tt |xD,sp - xD|t dt u

0

ITAEB )

∫tt |xB,sp - xB|t dt u

0

In the formulas above, t0 is the time at which the step disturbance enters the column, tu is the time at which the simulation is stopped, and xJ,sp - xJ is the difference between the setpoint value and the current value of the controlled variable. n

CtL )

|L(i) - L(i-1)| ∑ i)1

n

CtV )

|V(i) - V(i-1)| ∑ i)1

In the formulas above, i stands for the ith control action, n for the total number of control actions, L for the reflux rate, and V for the boilup rate. The dimensional weighting coefficients R, β, φ, and δ were given values [1, 10, 0.1, 0.5] and [1, 1, 200, 300]


Figure 11. MVCC C1. Responses of the worst-direction tuning set (full line) and of the best-direction tuning set (dashed line) to the worst-direction disturbance: (a) distillate composition; (b) bottoms composition; (c) reflux rate; (d) boilup rate. Table 4. Conventional Column Tuning Optimization [KD in (m3/min)/(mole fraction), KB in (kg/min)/(mole fraction) for C1, KD and KB in (kmol/min)/(mole fraction) for C2; τD and τB in min] initial fob final fob Conventional C1 worst direction θw ) 25° 106.4 48.7

best direction θb ) 115°

12.9

5.1

Conventional C2 worst direction θw ) 10° 887.0 490.2


228.5

69.4

Table 5. . MVCC Tuning Optimization [KD in (m3/min)/ (mole fraction), KB in (kg/min)/(mole fraction) for C1, KD and KB in (kmol/min)/(mole fraction) for C2; τD and τB in min]

optimal tuning KD ) 4.13 × 10-4 τD ) 6.88 × 10-3 KB ) 1.03 × 103 τB ) 11.1 KD ) 1.187 × 10-1 τD ) 3.17 KB ) 8.05 × 102 τB ) 3.51 KD ) 6.5 τD ) 7.5 KB ) 4.5 τB ) 8.5 KD ) 20.1 τD ) 30.0 KB ) 1.3 τB ) 18.0

for columns C1 and C2, respectively (the unit dimension of each weighting coefficient is the inverse of the unit dimension of the penalty term it multiplies, in such a way as to make fob be dimensionless). The disturbances entered the column according to either the best or worst directions, and the objective function was minimized accordingly; therefore, “direction-dependent” optimal tunings were obtained. The optimizations were started from the Shen-Yu tuning sets; other initializations seemed to make the optimizer more prone to getting entrapped in local minima. Tables 4 and 5 summarize the optimization results. As expected, the optimal tuning is very markedly dependent on the disturbance direction, as far as a conventional column is considered. For example, the distillate loop gain changes by ∼280 times with the

initial fob final fob worst direction θw ) 25°


MVCC C1 38.9 18.9

4.3

2.4

worst direction θw ) 10°

MVCC C2 338.1 216.8


103.3

38.0

optimal tuning KD ) 4.659 × 10-1 τD ) 9.79 × 10-1 KB ) 5.972 × 103 τB ) 2.45 KD ) 3.110 × 10-1 τD ) 9.21 × 10-1 KB ) 3.127 × 103 τB ) 5.81 KD ) 10.0 τD ) 3 KB ) 22.0 τB ) 8 KD ) 21.9 τD ) 5 KB ) 36.6 τB ) 33

disturbance direction in conventional C1 and by ∼3 times with that in conventional C2. Conversely, it changes by ∼1.5 times in MVCC C1 and by ∼2 times in MVCC C2. Therefore, because the optimal tuning parameters do not seem to be affected very much by the disturbance direction in the MVCC, we can indeed state that the direction of the disturbance is not very much of an issue for control as far as the MVCC is considered (incidentally, it is interesting to note that the ShenYu method provides a tuning set not too far from the optimal one in the MVCCs). To make our reasoning clearer, a further test was considered. Namely, the worst-direction disturbance was applied to column C1 (at t ) 20 min), and either the optimal controllers (worst-direction tuning; full line) or the unoptimal controllers (best-direction tuning;


broken line) were asked to reject the disturbance. The control responses are reported in Figure 10 for conventional C1 and in Figure 11 for MVCC C1 (results for C2 are qualitatively similar). It can be seen that the tuning severely impacts the controller performance in the conventional column. However, no such problem is encountered in the MVCC: the manipulated variable profiles are smooth, the controlled variables closely track the setpoint, and the settling time is short, regardless of the controller tuning. This is an indication that providing a satisfactory tuning for an MVCC can indeed be much simpler than providing one for a conventional column because the control response itself is not very dependent upon the tuning. 6. Conclusions This paper has presented a contribution on the understanding of the dynamics and control of continuous distillation columns with a middle vessel. It has been shown that the use of an MVCC configuration can produce significant benefits to the column controllability. In particular, the middle vessel reduces the control system sensitivity to the direction of disturbance inputs, so that the MVCC controllers are able to provide effective composition control for a wide spectrum of disturbance inputs with a simple decentralized control system based on multiloop PI controllers. Conversely, the performance of the quality controllers may be heavily dependent on the disturbance direction in a conventional column, particularly if the column condition number is large. It has also been shown that tuning of the composition loops in a MVCC is easier than tuning in a conventional column because the modified column design makes the controller performance much less affected by the tuning. Acknowledgment This research was carried out in the framework of the MIUR-PRIN 2002 project “Operability and controllability of middle-vessel distillation columns” (reference no. 2002095147_002). Literature Cited (1) Riggs, J. B.; Beauford, M.; Watts, J. Using Tray-to-Tray Models for Distillation Control. In Nonlinear Process Control: Applications of Generic Model Control; Lee, P. L., Ed.; SpringerVerlag: London, 1993. (2) Barolo, M.; Papini, C. A. Improving Dual Composition Control in Continuous Distillation by a Novel Column Design. AIChE J. 2000, 46, 146-159.

(3) Phimister, J. R.; Seider, W. D. Distillate-Bottoms Control of Middle-Vessel Distillation Columns. Ind. Eng. Chem. Res. 2000, 39, 1840-1849. (4) Finco, M. V.; Luyben, W. L.; Polleck, R. E. Control of Distillation Columns with Low Relative Volatilities. Ind. Eng. Chem. Res. 1989, 28, 75-83. (5) Lewin, D. R. Interaction of Design and Control. Proceedings of the 7th IEEE Mediterranean Conference on Control and Automation, Haifa, Israel, 1999. (6) Skogestad, S.; Morari, M. Effect of Disturbance Directions on Closed-Loop Performance. Ind. Eng. Chem. Res. 1987, 26, 2029-2035. (7) Lewin, D. R. A Simple Tool for Disturbance Resiliency Diagnosis and Feedforward Control Design. Comput. Chem. Eng. 1996, 20, 13-25. (8) Waller, J. B.; Waller, K. V. Defining Directionality: Use of Directionality Measures with Respect to Scaling. Ind. Eng. Chem. Res. 1995, 34, 1244-1252. (9) Andersen, H. W.; Ku¨mmel, M. Evaluating Estimation of Gain Directionality. Part 1: Methodology. J. Process Control 1992, 2, 59-66. (10) Andersen, H. W.; Ku¨mmel, M. Evaluating Estimation of Gain Directionality. Part 2: A Case Study of Binary Distillation. J. Process Control 1992, 2, 67-86. (11) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley and Sons: New York, 1989. (12) Hernandez, S.; Jimenez, A. Controllability Analysis of Thermally Coupled Distillation Systems. Ind. Eng. Chem. Res. 1999, 38, 3957-3963. (13) Serra, M.; Espun˜a, A.; Puigjaner, A. Controllability of a Different Multicomponent Distillation Arrangements. Ind. Eng. Chem. Res. 2003, 42, 1773-1782. (14) Seider, W. D.; Seader, J. D.; Lewin, D. R. Process Design Principles. Synthesis, Analysis, and Evaluation; John Wiley and Sons: New York, 1999; p 467. (15) Shinskey, F. G. Distillation Control; McGraw-Hill: New York, 1992. (16) Ha¨ggblom K. E.; Waller K. W. Control Structures, Consistency, and Transformations. In Practical Distillation Control; Luyben, W. L., Ed.; Van Nostrand Reinhold: New York, 1992. (17) Hurowitz, S.; Anderson, J.; Duvall, M.; Riggs, J. B. Distillation Control Configuration Selection. J. Process Control 2003, 13, 357-362. (18) Skogestad, S. Dynamics and Control of Distillation Columns: a Tutorial Introduction. Chem. Eng. Res. Des. 1997, 75, 539-562. (19) Shen, S. H.; Yu, C. C. Use of Relay-Feedback Test for Automatic Tuning of Multivariable Systems. AIChE J. 1994, 40, 627-646. (20) Henson, M. A., Seborg, D. E., Eds. Nonlinear Process Control; Prentice-Hall: Upper Saddle River, NJ, 1997. (21) Ramchandran, S.; Rhinehart, R. R. A Very Simple Structure for Neural Network Control of Distillation. J. Process Control 1995, 5, 115-128.

Received for review June 10, 2003 Revised manuscript received February 23, 2004 Accepted March 23, 2004 IE030489F

On the Controllability of Middle-Vessel Continuous Distillation Columns

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