Dual Composition Control in Continuous, Middle-Vessel Distillation

Article pubs.acs.org/IECR

Dual Composition Control in Continuous, Middle-Vessel Distillation Columns, with a Draw Stream in the Middle Vessel Rosendo Monroy-Loperena†,‡,* and José Alvarez-Ramírez§ †

ROMON Paseo de los Pirules 124, 04250 Distrito Federal, México Departamento de Energía Universidad Autónoma MetropolitanaAzcapotzalco Avenida San Pablo Xalpa 180, 02200, Distrito Federal, México § Departamento de Ingeniería de Procesos e Hidráulica Universidad Autónoma MetropolitanaIztapalapa Apartado Postal 55-534, 09340 Distrito Federal, México ‡

ABSTRACT: The dual composition control problem for a class of continuous middle-vessel distillation, which accepts a draw stream in the middle vessel, is studied in this work. This distillation configuration is advantageous when the separation goal is to recover high-purity light and heavy components from a multicomponent mixture with sufficiently large separation between relative volatilities. By using an estimator of the input−output modeling error, a first-order output-feedback compensator is designed which is shown to be equivalent to a multi-input multi-output proportional−integral (MIMO PI) controller. Tuning issues in terms of two time constants are given, and the ability of the controller to regulate the distillate and bottom product compositions in the face of sampled/delayed measurements is shown via numerical simulations.

1. INTRODUCTION Distillation processes have been designed to operate as a batch column, where a still is charged and distilled, or to operate continuously, where a stream is fed to a column and distillate and bottoms products are withdrawn continuously. Batch distillation is usually preferred for low throughputs, when fine or specialty chemicals are produced, or when there is intermittent or seasonal chemical production. In contrast, continuous distillation is favored for large-scale throughputs and continuous upstream feeds. However, an unconventional type of batch distillation column, called “middle-vessel batch distillation”, has been proposed. It consists of a rectifying section and a stripping section with a feed tray in the middle. The liquid feed is charged to an intermediate vessel, and a liquid stream is continuously recycled between the feed/ withdrawal tray and the feed vessel, so that the compositions of the liquid stream in the feed tray and in the feed vessel are kept close. Liquid streams can be continuously withdrawn from the top and bottom of the column. Robinson and Gilliland1 proposed, for the first time, the use of a middle vessel in the distillation process. Subsequently, Bortolini and Guarise2 performed an analysis of the dynamics and operability of the process. Since the work of Takamatsu et al.,3 who apparently rediscovered the process, several papers have been published dealing with middle-vessel batch distillation columns. Barolo and Papini4 studied the dual composition control problem of continuous middle-vessel distillation columns (CMVDC) with different control configurations, which are commonly used in conventional continuous distillation, showing that the middle vessel provides a way to reduce the interaction between the composition loops. As a result, the control performance of middle-vessel columns may be made remarkably superior to that of conventional columns. Phimister and Seader5 expanded the work of Barolo and Papini by showing that an inoperable configuration for dual composition control in conventional © 2012 American Chemical Society

continuous distillation where the distillate and bottom flows are manipulated can be used in CMVDC, with quite good performance. Bezzo and Barolo6 studied the fundamental dynamic behavior of CMVDC, taking into account the middlevessel holdup. Finally, Bezzo et al.7 proposed a control configuration based on model predictive control (MPC). The objective of this work can be described as follows: (i) to address some dynamic issues of CMVDC when a draw stream in the middle vessel is present; (ii) to propose a control design for the dual composition control of a CMVDC, assuming that the nominal plant model is relative degree one, and using an inverse control with an observer that provides estimates of the underlying modeling error signals; and (iii) to show, through numerical simulations, the performance of the proposed controller under expected operating conditions.

2. PROCESS DESCRIPTION In general, a CMVDC is a distillation column with a large vessel between the rectifying and stripping sections. Referring to Figure 1, a liquid stream S from the rectifying section is sent to an external middle vessel T. Variations to this configuration include the use of a partial-liquid side draw, a heat stream added to the middle vessel, and the transmission of a vapor stream from the stripping section of the column to the middle vessel. The middle vessel, denoted by the tank T, is also provided with another three streams: an input feed stream F, and two streams to dump it (one from the middle vessel, E, and a recycle stream from the middle vessel to the column, R). One important feature of this type of distillation process is that, under steady-state conditions, the distillation column has Received: Revised: Accepted: Published: 4624

December 23, 2011 February 13, 2012 February 18, 2012 February 21, 2012 dx.doi.org/10.1021/ie203018k | Ind. Eng. Chem. Res. 2012, 51, 4624−4631

Industrial & Engineering Chemistry Research

Article

Table 1. Case Study Configuration specifications number of stages, including condenser and reboiler number of recycle and side-stream stages feed flow rate, F [mol/min] side-stream rate, S [mol/min] recycle flow rate, R nominal distillate product [mol fraction] component 1, x1,1 component 2, x2,1 component 3, x3,1 nominal bottom product [mol fraction] component 1, x1,N component 2, x2,N component 3, x3,N condenser holdup [mol] reboiler holdup [mol] stage holdup [mol] middle-vessel holdup [mol] nominal reflux, L1 [mol] nominal vapor boilup, V [mol] hydraulic time constant, τh [min]

Figure 1. Schematic diagram of the middle-vessel continuous distillation column.

the constraint (R − S) = (D + B) > 0, implying that the product flow rates can be manipulated by means of the difference of the recycling stream and the side stream. Let us consider a separation of three components, ordered according to their relative volatility. It is expected that, under continuous operation conditions, the composition of the side

value 35 17 10 22 30 0.7640 0.2360 0.0 0.0 0.2622 0.7378 10.0 10.0 1.0 100.0 26.0 30.0 0.0667

stream S, which goes to the middle-vessel tank T, has a larger amount of impurities of components 1 and 3, despite the fact that the operation was performed under close to total reflux operating conditions. Besides, the stream S is mixed with

Figure 2. Step response of the continuous middle-vessel distillation columns (CMVDC) under a ±2% disturbance in the control inputs. 4625

dx.doi.org/10.1021/ie203018k | Ind. Eng. Chem. Res. 2012, 51, 4624−4631


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Figure 3. Response of the CMVDC under the proposed control law for three different values of the estimation time constant τe.

stream F in the middle-vessel tank T, which limits the separation of component 2 as a nearly pure product. Nevertheless, one can expect that continuous operation in a middle-vessel configuration can separate the most-volatile and least-volatile components to nearly pure, given the conditions D < z1F and B < z3F, making this configuration attractive for the separation of two components from a multicomponent mixture to nearly pure using only one distillation column.

understand, and widely accepted among operators. For these reasons, the LV configuration will be considered in detail in this section. However, other control configurations can be derived along the same lines (see, for instance, Bezzo and Barolo6). First, consider a linear dynamic process represented by a general state-space equation as in the following:

ẏ = A 0y + Bu z = Cy

3. A ROBUST FEEDBACK CONTROL DESIGN Consider the continuous operation with dual composition control where the control objective is to track prescribed product compositions via manipulations of the reflux rate and the vapor boilup rate. To achieve the control objective, an LV configuration (also called the “energy-balance configuration”) is considered, where the reflux rate (L1) is used to control the overhead purity (xi,1, where i refers to the component of interest and 1 refers to the condenser stage) and the vapor boilup rate (V) is used to control the bottoms composition (xj,N, where j refers to the component of interest and N refers to the reboiler stage). In practice, however, the heat supply to the reboiler, instead of the vapor boilup, is manipulated. Although the LV configuration may not be the best one from the point of view of coupling between control loops (see, for instance, Shinskey8), it is a commonly used control structure for dual composition control (see, for instance, Häggblom and Waller9), because it is simple to implement, easy to

(1)

where y, u, and z represent state, control, and output, respectively, and the matrices A0, B, and C are of appropriate dimensions. Note that if the available process model is in a transfer function description, it can be directly represented in a state-space model as in (1). Following standard ideas from single-column distillation processes, a simple one-time constant model can be described as follows (see, for instance, Chien et al.10 and Skogestad et al.11): y=

1 Ku 1 + τ0s

(2)

where y = {x1D,x3B}T and u = {L,V′}T in deviation variables. In addition, τ0 is the open-loop dominant time constant and K is 4626



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Figure 4. Response of the CMVDC under the proposed control law for two different operation conditions.

rate τc−1. Since the modeling error signal η(t) is unknown, an approximation of the inverse-dynamics feedback giving the dynamic behavior (6) is

the process gain matrix (K ∈ R2×2). According to system (1), one has that

A0 = −

1 I τ0

u = B−1[(A c‐A 0)y‐η̃ ]

(3)

where η̃(t) is an approximation of the actual signal η(t). Based on some observability properties of the modeling error signal η(t), and on input u and output y measurements, the approximate signal η̃(t) can be obtained via the following first-order filter:

and

B=

1 K τ0

(4)

where I denotes the identity matrix. To compensate for the approximations of using model (1), consider the following representation of the CMVDC: ẏ = A 0y + Bu + η (5)

ẇ = ‐A 0y‐Bu‐Te[w + y]

(9)

where

where η is the modeling error that accounts for model/plant mismatches. Based on the CMVDC representation in (5), a robust feedback controller can be developed along the same lines of modeling error compensation approaches (see, for instance, Alvarez-Ramirez12). Let ẏ = A cy (6) be the desired controlled dynamics, where 1 Ac = − I τc

(8)

Te =

1 I τe

(10)

where τe > 0, is the estimation time-constant. This filter provides an online estimate of the modeling error signal η(t), which is subsequently used by the approximate inversedynamics feedback (8) to counteract its effects and achieve exponential tracking of the desired composition trajectory. Besides, the smaller the estimation time-constant τe, the faster the convergence of the modeling error signal η̃(t) to the actual one η(t). Hence, in the limit as τe → 0, the approximate inverse-dynamics feedback (8) converges to the exact one u = B−1[(Ac − A0)y − η], and the dynamics converges to the prescribed behavior y(t) = exp(tAc)y(0). Of course, the

(7)

and τc > 0 is a prescribed time constant. Notice that y(t) = exp(tAc)y(0), so that y(t) → 0 exponentially with convergence 4627



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Figure 5. Response of the CMVDC under the proposed control law as a diagonal controller for two different operation conditions.

loop response time-constant τc, the MIMO PI control gains depend only on the estimation time-constant τe, which can be reduced to adjust the modeling error estimation response. That is, the directionality of the control parameters τc and τe is well established in the sense that the control response is enhanced as such time-constants are reduced. The control design structure described above suggests the following guidelines on tuning of the proposed controller: (a) In a first step, set the value of the closed-loop time constant τc. Values of τc of the order of 0.5−1.0 of τ0 are suggested. (b) After the value of the closed-loop time-constant has been set, select a suitable value of the estimation time constant τe. In principle, the estimation of the modeling error signal η(t) must be faster than the prescribed closed-loop convergence time τc. However, excessively large values of τe can excite unmodeled (actuator) dynamics and amplify excessive measurement noise. To avoid these undesirable effects, set τe at a value no smaller than the dominant time delay.

minimum allowable value of the estimation time-constant τe is limited by the underlying measurement noise, sampling frequency and unmodeled (actuator) dynamics. Summarizing, based on the model (1), the proposed measurement-driven feedback controller for the CMVDC can be written as follows: u = B−1[(A c‐A 0)y‐η̃ ] ẇ = ‐A 0y‐Bu‐Te[w + y] η̃ = Te[w + y]

It can be shown that this controller is equivalent to a linear multi-input multi-output proportional−integral (MIMO PI) compensator with controller gain and integral reset time constant parametrized by Ac, Te, and B. In fact, by computing the controller transfer function y → u, it is possible to describe the controller as u = udc + Kcy + KI∫ 0t y(σ) dσ, where udc is a dc-like constant input and the controller Kc and integral KI gains are given by K c = B−1[A c − A 0 − Te] −1

KI = B A cTe

4. NUMERICAL SIMULATIONS Consider the problem of the separation of an equimolar mixture of a ternary ideal mixture, where the components are ordered according to their relative volatility, and are referred as component 1 for the most volatile and component 3 for the least volatile. The components have the following relative

(11)

The main advantage of the MIMO PI controller with parametrization (11) over traditional ones is that the tuning can be carried out quite efficiently. Specifically, for a given closed4628



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Figure 6. Performance of the proposed control law under measurement noise.

The plant parameters τ0 and K were estimated from step responses (see Figure 2), from where the following estimates were obtained: τ0 = 5 min and

volatilities: αT = [9, 3, 1]. It is noted that the difference between the relative volatilities for light and heavy components is sufficiently large for performing separation on the middle-vessel distillation configuration of Figure 1. According with the CMVDC column described in Table 1, several numerical simulations using a model based on the usual collection of material balances, vapor−liquid equilibrium relationships, and liquid hydraulics correlations (see, for instance, Monroy-Loperena and Alvarez-Ramirez13), where the assumption of negligible vapor holdup, theoretical trays, perfect mixing on trays, constant operating pressure, total condensation with no subcooling, and adiabatic operation are assumed. An energy balance for the column is not performed. Constant molar overflow assumption is used, but the model takes into account the dynamics of the molar holdups on each tray, and the internal liquid rate on each stage is determined by means of the linearized version of the Francis weir formula: Lk = Lk̃ +

⎡ 0.219031 −0.161634 ⎤ K=⎢ ⎥ ⎣−0.16525 0.216183⎦

Consider the following conditions and parameters to be used in the proposed control scheme to regulate the distillate and bottom compositions: (1) The regulation task is to keep the composition of component 1 in the distillate as x1,1 = 0.9, and the bottom composition for component 3 as x3,N = 0.9. (2) The closed-loop time-constant is taken as τc = 4 min. With this set of control parameters, the following numerical simulations were carried out: (1) Effect of the estimation time-constant τe. Figure 3 shows the time evolution of the controlled distillation for three different values of the estimation time-constant τe. As expected, a better close-loop response is obtained for smaller values of the estimation time, but this induces an oscillatory response. However, in principle, the proposed controller law can achieve the closed-loop performance. (2) Change in the operation conditions. To show the performance of the proposed control law in a more real situation, consider that, after 500 min of normal operation, the composition of the feed is changed from equimolar

Mk − M̃ k τh

where L̃ k and M̃ k are nominal liquid flow rate and holdup in tray k, respectively, and τh is the hydraulic time constant (τh > 0). 4629



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Figure 7. Performance of the proposed control law under sampled and delayed measurements.

to zT = [0.4, 0.2, 0.4]. Referring to Figure 4, note that a smooth behavior in the manipulated variables is obtained. However, note that a long time (nearly 1000 min) is necessary to reach the steady state, because of the cushion effect of the middle vessel. Next, consider that, after 3000 min of operation, the set points of the distillate product and bottom products are changed to 0.95. Note that the proposed controller preserves a smooth behavior. (3) Diagonal MIMO PI. Diagonal controllers are often preferred in practice, because they are robust and relatively simple to understand and tune. The proposed controller law can be easily transformed to decentralized (singleloop controllers), by taking the gain matrix, K, as diagonal. Figure 5 shows the behavior of the MVCDC for the same changes in the operation conditions used in the above case. Nevertheless, the diagonal controller gives an acceptable behavior, one can note that a better behavior is obtained with more information. (4) Effect of measurement noise. Measurement noise is always present in practical situations. Figure 6 presents the response of the proposed controller when measured compositions are subjected to ±0.01 measurement noise. As can be seen in the figure, the controller is able to provide acceptable performance without excessive amplification of measurement noise.

(5) Effects of sample measurement and time delays. Despite advances in online composition analyzers, they introduce time delay into the control loop and also common case is that the measurements are sampled. To analyze this situation, a case with a measurement delay and sampling time equal to 0.0833 min is studied for a change in the setpoints of the composition of the distillate and bottom products from 0.9 to 0.95. Referring to Figure 7, a smooth behavior is obtained, showing that the proposed control law is capable of handling this situation without serious problems.

5. CONCLUSIONS Through the analysis of continuous middle-vessel distillation columns (CMVDC) allowing a draw stream in the middle vessel, one can expect that this configuration is appropriate for the separation of the most-volatile and least-volatile components using only one column. We proposed a multi-input multioutput proportional−integral (MIMO PI) control design for the dual composition control of a CMVDC. Under the assumption that the nominal plant is relative degree one, the idea is to use inverse control with an observer that provides estimates of the underlying modeling error signals. This control design approach leads to classical multivariate proportional− integral (PI) controllers with a novel parametrization of the controller gain and the integral time matrices. Finally, through numerical simulations, we showed that the proposed 4630



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control law exhibits a good performance for common operation conditions.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +525 5 5581 4982. Fax: +525 5 5581 4982. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Robinson, C. S.; Gilliland, E. R. Elements of Fractional Distillation, 4th ed.; McGraw−Hill Book Co.: New York, 1950. (2) Bortolini, P.; Guarise, G. B. A New Practice of Batch Distillation (in Ital.). Quad. Ing. Chim. Ital. 1970, 6, 150−159. (3) Takamatsu, T.; Hashimoto, I.; Hasebe, S. Optimal Design and Operation of a Batch Process with Intermediate Storage Tanks. Ind. Eng. Chem. Process Des. Dev. 1982, 21 (3), 431−440. (4) Barolo, M.; Papini, C. A. Improving Dual Composition Control in Continuous Distillation by a Novel Column Design. AIChE J. 2000, 46 (1), 146−159. (5) Phimister, J. R.; Seider, W. D. Distillate−Bottoms Control of Middle-Vessel Distillation Columns. Ind. Eng. Chem. Res. 2000, 39 (6), 1840−1849. (6) Bezzo, F.; Barolo, M. Understanding the Dynamic Behaviour of Middle-Vessel Continuous Distillation Columns. Chem. Eng. Sci. 2005, 60 (2), 553−563. (7) Bezzo, F.; Micheletti, F.; Muradore, R.; Barolo, M. Using MPC to Control Middle-Vessel Continuous Distillation Columns. J. Process Control 2005, 15 (8), 925−930. (8) Shinskey, F. G. Distillation Control: For Productivity and Energy Conservation, 2nd ed.; McGraw−Hill Book Co.: New York, 1984. (9) Häggblom, K. E.; Waller, K. V. Transformations and Consistency Relations of Distillation Control Structures. AIChE J. 1988, 34 (10), 1634−1648. (10) Chien, I.-L.; Tang, Y. T.; Chang, T.-S. Simple Nonlinear Controller for High-Purity Distillation Columns. AIChE J. 1997, 43 (11), 3111−3116. (11) Skogestad, S.; Morari, M.; Doyle, J. C. Robust Control of IllConditioned Plants: High-Purity Distillation. IEEE Trans. Autom. Control 1988, 33 (12), 1092−1105. (12) Alvarez-Ramírez, J. Adaptive Control of Feedback Linearizable Systems: A Modeling Error Compensation Approach. Int. J. Robust Nonlinear Control 1999, 9 (6), 361−377. (13) Monroy-Loperena, R.; Alvarez-Ramírez, J. Dual Composition Control in a Middle-Vessel Batch Distillation Column. Ind. Eng. Chem. Res. 2001, 40 (20), 4377−4390.

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Dual Composition Control in Continuous, Middle-Vessel Distillation

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