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Ind. Eng. Chem. Res. 2000, 39, 378-386
PROCESS DESIGN AND CONTROL Output-Feedback Control of Reactive Batch Distillation Columns Rosendo Monroy-Loperena† and Jose Alvarez-Ramirez*,‡ Estrategia Sinergica S.A. de C.V., Paseo de los Pirules 124, Col. Paseos de Taxquen˜ a, Mexico D.F., 04250 Mexico, and Departamento de Ingenieria de Procesos, Universidad Autonoma MetropolitanasIztapalapa, Apartado Postal 55-534, Mexico D.F., 09340 Mexico
In this work, an output-feedback control for the regulation of distillate purity via manipulations of the reflux ratio in reactive batch distillation is designed. The approach is based on an approximate model of the composition dynamics and makes use of a reduced-order observer to estimate the modeling error. An input/output linearizing feedback is proposed where the estimated modeling error is included to achieve robust tracking of a composition reference. It is shown that the resulting controller has the structure of a proportional-integral derivative (PID) controller with antireset windup. The controller performance is tested using a simulation example including strong uncertainties in the reaction model. An interesting finding is that the required reflux ratio policy to reach asymptotically a constant reference resembles the reflux ratio policy obtained from posing an optimization technique (Mujtaba, I. M.; Macchietto, S. Ind. Eng. Chem. Res. 1997, 36, 2287-2295). 1. Introduction Batch distillation with or without chemical reaction is used in industry for the production of small amounts of products with high added value and for processes where flexibility is required. Distillation with chemical reaction is well suited for processes where one of the products has a lower boiling point than other products and reactants. The higher volatility of this product induces a decrease of its concentration in the liquid phase, thus leading to higher reactant conversions than with reaction alone. Batch processes, as reactive batch distillation, are inherently dynamic. In a rough sense, they can be seen as composed of a stationary process driven by an integrator. While both the dynamic modeling of reactive batch distillation (RBD) and its optimization have been studied to some extent,1-3 issues related to the feedback regulation of this process have rarely been addressed. A method for computing operational policies for RBD systems using simulation techniques was reported by Albet et al.1 Sorensen and Skogestad4 developed control strategies by repeated simulations with the SPEEDUP package. Some aspects related to the controllability of RBD columns were also addressed. It is clear that a drawback of this approach is that extensive simulations using a simple model are required, thus leading to a high computational burden. In a recent work, Mujtaba and Macchietto3 reported a method for obtaining optimal operational policies for the RBD process. Their approach is presented as a proper dynamic optimization problem incorporating a detailed dynamic model, which * Corresponding author. E-mail:
[email protected]. Fax: +52-5-7244900. † Estrategia Sinergica S.A. de C.V. ‡ Universidad Autonoma MetropolitanasIztapalapa.
resulted in a nonlinear programming problem with the dynamics of the RBD process as constraints. In general, the solution of the resulting optimization problem can be computationally very expensive3 and therefore may not be suitable for on-line implementation. To alleviate this problem, Mujtaba and Macchietto3 used polynomial curve-fitting techniques. As a result, it was shown that by judicious use of repeated solutions of the dynamic optimization problem a priori, an algebraic representation of the optimal solutions can be obtained and very efficient calculation of the optimal reflux ratio can be performed. A criticism to optimal policies approaches, such as that reported by Mujtaba and Macchietto,3 is that the optimal solution (optimal reflux ratio) depends strongly on the model. Because models may present strong uncertainties in the parameters and reaction rates, these may lead to serious robustness problems and performance degradation with respect to the computed optimal profit. In fact, Mujtaba and Macchietto3 have shown that changes in system parameters can significantly change the operating condition (the reflux rate policies) of the plant. During the past decade, the feedback linearization control technique has been successfully used to address some of the practical control problems, such as the control of nonlinear fermentation processes,5 polymerization processes,6,7 and pH neutralization processes.8 Successful applications to nonminimum-phase nonlinear systems have been reported.9-11 The important practical case of output-feedback regulation has also been addressed,11-14 where the idea is to use some kind of state estimation within an input/output feedback linearization framework. In the spirit of these papers, our work addresses several aspects of the output control of RBD processes. Because the profitability of the process is closely related to the distillate composition,3 the
10.1021/ie990382l CCC: $19.00 © 2000 American Chemical Society Published on Web 12/16/1999
Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 379 Table 1. Input Data for Ethanol Esterification Using Conventional Batch Distillation no. of ideal separation stages (including reboiler and total condenser) total fresh feed (kmol) feed composition (acetic acid, ethanol, ethyl acetate, water) (mole fraction) internal plates holdup (kmol) condenser holdup (kmol) condenser vapor load (kmol/h) column pressure (bar)
10 5 0.45, 0.45, 0.0, 0.1 0.1 0.0125 2.5 1.013
Table 2. Vapor-Liquid Equilibrium and Kinetic Data for Ethanol Esterification Vapor-Liquid Equilibrium acetic acid (1) + ethanol (2) a ethyl acetate (3) + water (4) K1 ) (2.25 × 10-2)T - 7.812, T > 347.6 K K1 ) 0.001, T e 347.6 K log K2 ) -2.3 × 103/T + 6.588 log K3 ) -2.3 × 103/T + 6.742 log K4 ) -2.3 × 103/T + 6.484 Kinetic Data rate of reaction, gmol/(L min); r ) k1C1C2 - k2C3C4, where rate constants are k1 ) 4.76 × 10-4 and k2 ) 1.63 × 10-4 and Ci stands for concentration in gmol/L for the ith component
Figure 1. Schematic diagram of the reactive batch reactifier.
control objective is to track a prescribed distillate composition via manipulations of the reflux ratio. It is assumed that the condenser duty is used for pressure control and the distillate flow for condenser level control. The main interests are (i) establishing an outputfeedback strategy with guaranteed tracking properties despite strong uncertainties in the dynamics of the RBD process and (ii) showing via a specific example from Mujtaba and Macchietto3 that the resulting reflux ratio policy approaches that obtained via optimization techniques. The control design is based on an approximate model of the composition dynamics and makes use of a reduced-order observer to estimate the modeling error. An input/output linearizing feedback is proposed where the estimated modeling error is included to achieve robust tracking of a composition reference. It is shown that the resulting controller has the structure of a PID controller with an antireset windup scheme. The controller performance is tested using a simulation model including strong uncertainties in the reaction model. An interesting finding is that the required reflux ratio policy to reach asymptotically a constant reference approximates very closely that obtained from using optimization techniques.3 2. Problem Statement A schematic diagram of a RBD column is presented in Figure 1. A dynamic model for an n-stage RBD process consists of the mass- and energy-balance equations.2,15,16 The model includes column holdup, rigorous phase equilibria, and chemical reaction on the plates. The model is fairly detailed and assumes negligible vapor holdup on plates, perfect mixing on trays, fast energy dynamics, constant operating pressure, and total condensation with no subcooling. The stages are counted from top to bottom. Subindex D is assigned to the
condenser drum, and subindex R is assigned to the reboiler drum. A complete description of the model is made by Cuille and Reklaitis15 and Mujtaba and Macchietto.3 For the sake of completeness, the model and its main variables are described in the appendix. It should be remarked that although formation of azeotropes are quite common in reactive distillation, this situation is not considered in the model, for convenience and simplicity in presentation. The worked example presented by Mujtaba and Macchietto3 considers the esterification of ethanol and acetic acid. The reaction products are ethyl acetate and water. The reversible reaction scheme is the following:
acetic acid (1) + ethanol (2) a ethyl acetate (3) + water (4) The boiling temperatures are respectively 391.1, 351.5, 350.3, and 373.2 K. Ethyl acetate, the main product, has the lowest boiling temperature in the mixture and consequently has the highest volatility. The continuous removal of this product by distillation will shift the chemical equilibrium further to the right and will improve conversion of reactants. The data defining the column configuration, feed, feed composition, etc., for the example are given in Table 1 of Mujtaba and Macchietto.3 Table 2 of the same work presents the vapor-liquid equilibria and kinetic data. Although our results are presented for this example, they are intended for the general case of RBD processes. Let xD,3 be the concentration of ethyl acetate in the distillate flow. In addition, let wref(t) be a reference trajectory. The control problem is to track the wref(t) by manipulations of the reflux ratio rf ) L0/V1. 3. A State Feedback Control Design In this section, we will build a feedback control under the assumption of complete knowledge and state measurement. To this end, we will follow the methodology described by Barolo and Berto17 for nonreactive distillation columns. For simplicity in algebraical manipulations, let us take the reflux flow rate L0 as the manipulated variable.
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Moreover, let us assume that the vapor flow rate in the first plate V1 and the condenser drum holdup hD vary slowly; i.e., dV1/dt ≈ 0 and dhD/dt ≈ 0. The reflux ratio is then given by rf ) L0/V1. The dynamics of the distillate composition xD,3 are given by (see the appendix)
hD
dxD,3 ) V1(y1,3 - xD,3) + hDRD(XD,TD) dt
dx1,3 ) V2y2,3 + L0xD,3 - V1y1,3 - L1x1,3 + h1 dt dh1 h1R1(X1,T1) - x1,3 (2) dt For simplicity in notation, introduce the following functions:
fD,3 ) [V1(y1,3 - xD,3) + hDRD(XD,TD)]/hD
(3a)
f1,3 ) V2y2,3 - V1y1,3 - L1x1,3 + H1R1(X1,T1) - x1,3
]
dh1 /h1 dt (3b)
Because y1,3 ) E(x1,3) (liquid-vapor equilibria relationship), we have that
hD
d2xD,3
) dt2 dE(x1,3) dRD V1 [f1,3 + (x3,D/h1)L0] - fD,3 + hD (4) x1,3 dt
[(
)
]
where
dRD ∂RD dXD ∂RD dTD ) + dt ∂XD dt ∂TD dt and dXD/dt and dTD/dt are given by the mass and energy balances in the condenser. In eq 4, the control input L0 affects directly the dynamics of the controlled concentration xD,3 via the second-order derivative d2xD,3/dt2, and therefore the relative degree of the system is 2.17 Let e ) xD,3 - wref be the tracking error. Suppose the following stable error trajectory description:18-20
de d2e + 2ξcτc-1 + τc-2e ) 0 dt dt2
d2xD,3 dt
2
)
d2wref dt
2
+ 2ξcτc-1
(1)
where XD ) (xD,1, ..., xD,4) is the distillate composition, y1,3 is the ethyl acetate mole fraction in the vapor leaving the first stage (the column stages are counted from top to bottom), RD(XD,TD) is the reaction rate, and TD denotes the temperature in the condenser drum. Following the methodology described by Barolo and Berto,17 the relative degree of the system is not 1 because the dynamics of xD,3 are not directly affected by the control input L0. On the other hand, the dynamics of the ethyl acetate concentration in the first stage are given by
[
for xD,3 are given by
(5)
where ξc and τc are respectively the closed-loop damping factor and time constant. Then, the desired dynamics
dwref + τc-2wref dt dxD,3 2ξcτc-1 - τc-2xD,3 (6) dt
From eqs 4 and 6, we obtain the theoretical control input LT0 , which provides the trajectory tracking error behavior (5) and is given by
(
LT0 ) -Ψ +
d2wref dt
2
+ 2ξcτc-1
dwref + τc-2wref dt
2ξcτc-1
)
dxD,3 - τc-2xD,3 /φ (7) dt
where
[(
Ψ ) (h1hD)-1 V1
)
]
dE(x1,3) dRD f1,3 - fD,3 + hD dx1,3 dt
(8a)
and
φ ) (V1/hDh1)xD,3
dE(x1,3) dx1,3
(8b)
The control input LT0 is well-defined provided that φ(t) * 0, for all t > 0. The main feature of the feedback function (7) is that it leads to asymptotic convergence of the tracking error to zero within a mean operating time τc. In this way, tuning of the controller (7) can be easily made just by choosing the closed-loop time constant τc and damping coefficient ξc. To avoid unrealistic situations due to hard input bounds, a saturated version of eq 7 is proposed: T ) Sat(LT0 ; L0,max, L0,min) L0,sat
(9)
where Sat is a standard saturating function with upper and lower limits L0,max and L0,min, respectively. A. Numerical Simulations with the Theoretical Control Law (9). We have carried out several numerical simulations with the example presented by Mujtaba and Macchietto.3 Figure 2 shows the dynamics of the uncontrolled distillate composition for several values of the reflux ratio rf ) L0/V1. It is noted that the concentration of the reactant acetic acid in the distillate flow goes to zero immediately. However, the concentration of this reactant is very high in the bottom plates (not shown). This behavior is due to the fact that the acetic acid has the higher boiling temperature, so that it is maintained at the bottom plates where the chemical reaction mechanism becomes more important than the distillate separation mechanism. For total reflux operation, the maximum achievable ethyl acetate concentration is 0.964 mole fraction. This value imposes a limit in the achievable product purity under batch operation. In fact, for batch operation (0 < rf < 1), the ethyl acetate mole fraction increases in the first part of the batch time, achieves a maximum value, and then decreases until the end of the operation. The first part of the batch operation where the ethyl acetate mole fraction increases can be called the reaction phase because the chemical reaction is the main drive of ethyl acetate in the distillate product. Compared with nonreactive batch
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Figure 2. Dynamics of the uncontrolled distillate composition for several values of the reflux ratio rf ) L0/V1.
distillation where the mole fraction of the more volatile component decreases along the operation of the uncontrolled batch distillation, the last part of the batch operation can be called the separation phase. In fact, the process is controlled by the separation during the second phase. It is also noted that the smaller the reflux ratio, the smaller the operating time where the maximum ethyl acetate mole fraction is achieved. Hence, inefficient batch operation is obtained with smaller values of the reflux ratio. As in Mujtaba and Macchietto,3 assume that the control objective is to regulate the distillate composition at a given constant product purity wref. We have carried out numerical simulations for two given purities, wref ) 0.7 and 0.8. The reflux ratio level was computed with
the feedback control law (8) for batch time tf ) 40 h, although there is not an inherent restriction to compute the reflux flow rate for any batch time. The reboiler heat duty was kept constant. The control parameters were chosen as ξc ) 1 and τc ) 0.01 h. Besides, L0,max ) V1 and L0,min ) 0.8 V1 (i.e., rf,max ) 1 and rf,min ) 0.8). Figure 3 shows the dynamics of the computed reflux ratio and product purity. After about 2 h, the product purity is maintained at its reference value until the end of the batch time. The reflux ratio attains its upper limit (L0,satT/V1 ) 1) during the first part of the batch operation. This behavior is due to the fact that the column is started up with zero mole fraction of products (ethyl acetate and water) in all trays. During the reaction phase, the reflux ratio decreases to compensate
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Figure 3. Dynamics of the computed reflux ratio and product purity under the theoretical feedback control law (9).
for the high capacity of the process to give product at the desired purity. This is expected because, as the product species are withdrawn by distillation, the reaction goes further to the right. As Mujtaba and Macchietto3 have pointed out, this increase is very sharp at the beginning because it is easier to shift equilibrium by eliminating the plentiful product at the given purity. The curve is flattened near the end of the batch time because it is progressively more difficult to remove the product at the given purity. At a certain batch time (about 6 h), the separation mechanism starts to control the process dynamics and the column behaves like a nonreactive batch column. During the separation phase, the reflux ratio increases to compensate the decrease in the reaction drive to generate ethyl acetate. This behavior is maintained until the end of the batch time. Compare the behavior of the reflux ratio vs batch time in Figure 2 with that in Figure 5 of Mujtaba and Macchietto’s paper.3 Both reflux ratio policies display the same shape with almost the same quantitative evolution. An advantage of the feedback control approach (9) is that this is easy to compute. On the contrary, the computation of the dynamic optimization problem seems to be computer time-consuming. 4. Robust Output Feedback Control Design As in Mujtaba and Macchietto’s approach, given an accurate model of the RBD process, the computed reflux ratio policy can be computed off-line and used in the open loop. A drawback of this approach is that reaction rate parameters may be highly uncertain, which may lead to serious degradation of the optimized performance (e.g., maximum profit or maximum conversion).
Figure 4. Time evolution of the reflux ratio and product purity for wref ) 0.7 and two different values of the estimation time constant τe.
On the other hand, an on-line computation of eq 7 would require measurements of vapor and liquid composition which is expensive. In this section, we will design a feedback control for on-line implementation. To this end, we will assume that the only concentration measurement is the product purity. This measurement can be accomplished through a chromatograph or state estimators via temperature measurements (see the interesting work by Quintero-Marmol and Luyben21 on this topic). Equation 4 can be written as
d2xD,3 dt2
) Ψ + φL0
(10)
It is noted that the function Ψ involves a set of quite complex and uncertain functions, such as the time derivative of the reaction rate dR1/dt. As a worst case control design, assume that the function Ψ is unknown. On the other hand, the function φ involves the gradient of the vapor-liquid equilibrium relationship dE(x1,3)/ dx1,3. In general, it is expected that dE(x3,1)/dx3,1 be positive. Let φ j be an estimate of φ, which can be taken as
φ j ) (V1/hDh1)wref dE(wref)/dx1,3
(11)
where dE(wref)/dx1,3 is the gradient of E(x) evaluated at the reference value wref and V1/hDh1 can be computed from the nominal design values V1, hD, and h1. Introduce the modeling error function
η ) ψ + (φ - φ j )L0
(12)
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Then, eq 10 can be written as
d2xD,3 dt2
)η+φ j L0
(13)
Following the ideas described in section 3, the theoretical feedback control leading to desired closed-loop behavior (5) is
(
LT0 ) -η +
d2wref dt
2
+ 2ξcτc-1
dwref + τc-2wref dt
2ξcτc-1
)
dxD,3 - τc-2xD,3 /φ j (14) dt
This feedback control cannot be implemented just as it is because the modeling error η and the time derivative dxD,3/dt are not available for feedback. An alternative is to use estimates of η and dxD,3/dt. To this end, introduce the variable z ) dxD,3/dt. Then, eq 12 can be written as
dxD,3 )z dt dz )η+φ j L0 dt Let zj and η j be estimates of z and η j , respectively. The following estimator is proposed:
(
)
dxD,3 dzj )η j+φ j L0 + 2ξeτe-1 - zj dt dt
(
)
dxD,3 dη j ) τe-2 - zj dt dt
Figure 5. Time evolution of the reflux ratio and product purity for wref ) 0.8 and two different values of the estimation time constant τe.
(15)
where ξc and τc are respectively the estimation damping factor and time constant. In fact, in the case that η ) constant, the following stable estimation error trajectory description is obtained: 2
d ee dt
2
+ 2ξeτe-1
(
LP0 ) -η j+
+ 2ξcτc-1
dwref + τc-2wref dt
)
such that a saturated version becomes P ) Sat(LP0 ; L0,max,L0,min) L0,sat
where ee ) dxD,3/dt - zj is the estimation error. The estimator (15) plays the role of a reduced-order observer for the “unmeasured” states z and η. To implement (15), j define the variables q1 ) zj - 2ξeτe-1xD,3 and q2 ) η j are computed from τe-2xD,3. Then, the estimates zj and η the following equations:
dq1 )η j+φ j L0 - 2ξeτe-1zj dt (16)
(19)
In this way, the reflux ratio policy is computed online with the feedback function (19) and the estimators (16) and (17). This control law has an interesting structure. It can be interpreted as a PID-like control law with an antireset windup (ARW) scheme.22,23 In fact, after straightforward algebraic manipulations, it can be concluded that the feedback control (16)-(19) can be written as
LP0
where
-1
)φ j
d2wref dt2
- CPID(s) F(s) er P GARW(s) (LP0 - L0,sat ) (20)
where CPID(s) is a classical PID controller with control gain, integral, and derivative time constants given by
zj ) q1 + 2ξcτe-1 dxD,3/dt η j ) q2 + τc-2 dxD,3/dt
dt
2
j (18) 2ξcτc-1zj - τc-2xD,3 /φ
dee + τe-2ee ) 0 dt
dq2 ) -τe-2zj dt
d2wref
(17)
The initial conditions for eq 16 can be chosen as follows. Because the signals z(t) and η(t) are unknown, take q1(0) ) -2ξcτe-1xD,3(0) and q2(0) ) -τe - 2xD,3(0). In this way, the practical feedback control is given by
j -1 Kc ) φ τD )
ξcτc + ξeτe ξeτeτc2 + ξcτcτe2
τc2 + 4ξeξcτeτc + τe2 2(ξcτc + ξeτe)
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τI ) 2(ξcτc + ξeτe)
(21)
F(s) is a first-order filter (i.e., F(s) ) 1/(τfs+1)) with filter time constant given by
τf )
τcτe 2(ξeτc + ξcτe)
(22)
and GARW(s) is the following ARW operator acting on P : the “saturation error” LP0 - L0,sat
GARW(s) )
τc - 2ξcτe2s s(τe2τcs + 2τe[ξeτc + ξcτe])
(23)
Regarding the structure of the PID control configuration (20), the following comments are in order: (a) If φ j ) constant, the feedback function (16)-(19) is a linear controller that can be easily implemented in actual inexpensive technologies (e.g., programmable logic controllers). (b) When the actuator saturates, the feedback signal P P ) tries to drive the error LP0 - L0,sat GARW(s) (LP0 - L0,sat to zero by recomputing the integral action, such that the controller output becomes exactly at the saturation limit. This prevents the controller from windup.22,23 (c) It is noted that the PID control parameters are symmetric functions of the nominal closed-loop parameters {ξc,τc} and the observer parameters {ξe,τe}. In other words, the PID control parameters are invariant under the shifts {ξc,τc} f {ξe,τe} and {ξe,τe} f {ξc,τc}. This means that the reference model (5) and the estimators (16) and (17) have the same effects on the PID performance. (d) Although the PID representation (20) and the control law (16)-(19) are input/output equivalent to each other, probably the key advantage of the proposed PID control configuration (16)-(19) lies in the fact that the controller states are meaningful variables as estimates of the physical plant states and the model/plant mistmaches. It follows that the estimates zj and η j can be used to monitor the performance of the process or detect failures of actuators and sensors. (e) Dead time in the input channel imposes serious limitations in the achievable closed-loop performance.23 Following some internal model control ideas (see Morari and Zafiriou23) and given an upper bound for the dead time, the estimation and closed-loop time constants should be set at values not smaller than the dead-time upper bound. This tuning guideline will be used in numerical simulations below. A. Numerical Simulations with the Practical Control Law (16)-(19). We have carried out several numerical simulations to illustrate the performance of the proposed robust control design. From nominal design parameters and the vapor-liquid equilibrium relationship, we have chosen φ j ) 18.28 (see eq 11). The closed-loop damping factor and time constant, ξc and τc, have been chosen as in the simulation above. The estimation damping factor has been chosen as ξe ) 1. Figure 4 shows the time evolution of the reflux ratio and product purity for wref ) 0.7 and two different values of the estimation time constant τe. For comparison, the ideal behavior under the theoretical feedback control (9) is also shown. It is noted that the smaller the estimation time constant, the closer the behavior to the ideal one. This behavior is also observed for the
Figure 6. Dynamics of the controlled RBD column for τe ) 0.025 h and three different values of the input dead time.
case wref ) 0.8 (see Figure 5). This implies that, in principle, the ideal behavior under the theoretical feedback control (9) could be achieved as τe f 0. Of course, this is not possible in the presence of dead times. Figure 6 shows the dynamics of the controlled RBD column for τc ) τe ) 0.025 h and three different values of the dead time. It is noted in this case that perfect regulation of the product purity is not achieved. After control input saturation, product purity is maintained below the required purity. This behavior is induced by the delayed information used by the control input, which induces smaller reflux ratios. Such a regulation offset cannot be completely removed despite the presence of the integral action in the output feedback control law (see eqs 21 and 22) because the nonregulated RBD dynamics (i.e., the internal dynamics) are nonvanishing. That is, because these uncontrolled dynamics correspond to batch processes, they induce a time-varying behavior in the modeling error function η. Anyway, the maximum deviation from the reference is about 3%. The numerical simulations above have been carried out for the feed composition3 0.45/0.45/0.0/0.1. However, within a batch distillation process, the feed composition can change from one batch to another. If the control of product purity is based on optimization techniques3 and implemented in the open loop, the reflux ratio policy should be computed for every batch to be processed. This is not the case when feedback control is used. Assume that the feed composition is 0.2/0.3/0.0/0.5. Figure 7 shows the dynamics of the controlled RBD column using the same parameters as in Figure 6. Because the feed of acetic acid and ethanol is not stoichiometric, the production of ethyl acetate is less than that in the
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1. The vapor-phase holdup is assumed to be negligible compared to the liquid-phase holdup on each phase. 2. Chemical reactions in the vapor phase are neglected. 3. The initial state of the column is the steady-state total reflux condition with no reactions. 4. The liquid volumetric holdups on the plates will be assumed to be constant. Thus, the model is directed at simulating the dynamics of the main production period during which the hydrodynamic conditions are not widely varying. 5. The pressure drops and the plate efficiencies are constant during the operation. 6. The control of levels in condenser and reboiler drums is perfect. Consider the following notation:
Figure 7. Dynamics of the controlled RBD column for τe ) 0.025 h and three different values of the input deadtime. The feed composition was 0.2/0.3/0.0/0.5.
former case. This is represented by the fact that the reaction phase is shorter than that in the stoichiometric feed case. 5. Conclusions The regulation of product purity in reactive batch distillation columns has been studied in this work. It has been found that a reflux ratio policy computed from a feedback control function resembles the reflux ratio policy obtained from a nonlinear optimization problem. The proposed control policy should be implemented in a closed-loop fashion to avoid loss of performance due to strong uncertainties in the process model. To this end, a robust control design has been proposed, which for implementation only requires measurement or estimation of the product purity. It has been shown that the resulting control law is equivalent to a classical PID controller with an ARW scheme. Several numerical simulations have been presented to illustrate the performance of the controlled column under strong uncertainties in the process dynamics and important dead times due to measurement and estimation. Appendix. Theoretical Model In this appendix we consider the mass- and energybalance equations for a batch rectification column shown in Figure 1. The set of mass-balance differential equations consists of the total mass balance and mass balance on each component.15 The energy-balance differential equation merely consists of the application of the first law of thermodynamics. The model is valid under the following assumptions:
D ) overhead product flow rate Hl ) molar enthalpy of the liquid phase Hv ) molar enthalpy of the vapor phase L ) liquid flow rate M ) molar mass n ) number of plates Q ) rate of heat generation by chemical reaction activity R ) vector of reaction rates T ) thermodynamic temperature t ) time V ) vapor flow rate X ) vector of mole fractions (liquid) Y ) vector of mole fraction (vapor) F ) density Also, consider the following subscripts: D ) condenser 1 ) plate 1 (top) j ) plate j n ) plate n (bottom) B ) reboiler i ) component i
The following differential equations result.
Condenser dhD ) V1 - L0 - D + hD dt
∑RD,i(XD,TD)
(A.1)
d(hDXD) ) V1Y1 - L0XD - DXD + hDRD(XD,TD) (A.2) dt d(hDHl,D) ) V1Hv,1 - (L0 + D)Hv,D + QD(XD,TD) dt (A.3) Plate j, 1 e j e n dhj ) Vj+1 + Lj-1 - Vj - Lj + hj dt
∑Rj,i(Xj,Tj)
(A.4)
d(hjXj) ) Vj+1Yj+1 + Lj-1Xj-1 - VjXj - LjYj + dt hjRj(Xj,Tj) (A.5) d(hjHl,j) ) Vj+1Hv,j+1 + Lj-1Hl,j-1 - VjHv,j - LjHl,j + dt Qj(Xj,Tj) (A.6)
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Reboiler dhB ) Ln - VB + hB dt
∑RB,i(XB,TB)
d(hBXB) ) LnXn - VBYB + hBRB dt
(A.7)
(2) Mujtaba, I. M.; Macchietto, S. Simultaneous optimization of design and operation of multicomponent batch distillation column-single and multiple separation duties. J. Process Control 1996, 6, 27.
(A.8)
(3) Mujtaba, I. M.; Macchietto, S. Efficient optimization of batch distillation with chemical reaction using polynomial curve fitting techniques. Ind. Eng. Chem. Res. 1997, 36, 2287.
d(hBHl,B) ) LnHl,n - VBHv,B + QB(XB,TB) (A.9) dt To obtain a fully determined system, the variables appearing in these balance equations must also satisfy the following equations. Constraint in the Volume. The volume of the liquid phase in the condenser and on each plate is assumed to be constant. Consequently, the corresponding molar holdups are functions only of the temperature, the pressure, and the compositions:
hj ) τjFj/Mj
(A.10)
(4) Sorensen, E.; Skogestad, S. Control strategies for reactive batch distillation. J. Process Control 1994, 4, 205. (5) Henson, M. A.; Seborg, D. E. Nonlinear control strategies for continuous fermenters. Chem. Eng. Sci. 1992, 47, 821. (6) Schork, F. J.; Deshpande, P. B.; Leffew, K. W. Control of Polymerization Reactors; Marcel Dekker: New York, 1993. (7) Soroush, M.; Kravaris, C. Nonlinear control of a polymerization CSTR with singular characteristic matrix. AIChE J. 1994, 40, 6. (8) Henson, M. A.; Seborg, D. E. Adaptive nonlinear control of a pH neutralization process. IEEE Trans. Control Syst. Tech. 1994, 2, 169.
Constraint due to Vapor-Liquid Equilibrium
(9) Kravaris, C.; Daoutidis, P. Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems. Comput. Chem. Eng. 1990, 49, 439.
Yj ) KjXj
(10) Wright, R. A.; Kravaris, C. Nonminimum-phase compensation for nonlinear processes. AIChE J. 1992, 38, 26.
(A.11)
with nc
yj,i ) 1 ∑ i)1
(A12)
YB ) KBXB
(A.13)
for the reboiler,
with nc
(11) Kravaris, C.; Daoutidis, P.; Wright, M. A. Output feedback control of nonminimum-phase nonlinear processes. Chem. Eng. Sci. 1994, 49, 2107. (12) Limquenco, L. C.; Kantor, J. C. Nonlinear output feedback control of an exhotermic reactor. Comput. Chem. Eng. 1990, 14, 427. (13) Wu, W.; Chou, Y. S. Robust output regulation for nonlinear chemical processes with unmesurable disturbances. AIChE J. 1995, 41, 2565. (14) Soroush, M.; Kravaris, C. Nonlinear control of a batch polymerization reactor: an experimental study. AIChE J. 1992, 38, 1429.
yB,i ) 1 ∑ i)1
(A.14)
YD ) KBXB
(A.15)
(16) Bosely, J. R., Jr.; Edgar, T. F. Appropriate modeling assumptions for batch distillation optimization and control. In Proceedings of 5th International Seminar on Process Systems Engineering, Kyongju, Korea, May 30-June 3, 1994; Vol. 1, p 477.
(A.16)
(17) Barolo, M.; Berto, F. Composition control in batch distillation: binary and multicomponent mixtures. Ind. Eng. Chem. Res. 1998, 37, 4689-4698.
Constraint in the Enthalpy. If the temperature and the composition of the liquid phase are known, then the molar enthalpy of this liquid phase is determined. At each time, the molar enthalpy calculated according to the value of the composition (given by the integration of the mass balance equations) and the temperature (given by the liquid-vapor equilibrium constraint equations) must be equal to the molar enthalpy calculated by the integration of the energy-mass balance equations:
(18) Bartusiak, R. D.; Georgakis, C.; Reilly, M. J. Nonlinear feedforward/feedback control structures designed by reference systems synthesis. Chem. Eng. Sci. 1989, 44, 1837-1851.
(15) Cuille, P. E.; Reklaitis, G. V. Dynamic simulation of multicomponent batch rectification with chemical reaction. Comput. Chem. Eng. 1986, 10, 389.
for the condenser,
with nc
yD,i ) 1 ∑ i)1
Hl,j ) Hl(Xj,Tj,Pj)
(A.17)
Total enthalpies are used in this formulation; thus, no heat of reaction terms are required in eqs A.3, A.6, and A.9. Literature Cited (1) Albet, J.; Le Lann, J. M.; Julia, X.; Koehret, B. Rigorous simulation of multicomponent multisequence batch reactive distillation. Proceedings of Computed-Oriented Process Engineering; Elsevier Science Publishers B. V.: Amsterdam, The Netherlands, 1991; p 75.
(19) McLellan, P. J.; Harris, T. J.; Bacon, D. W. Error trajectory descriptions of nonlinear controller designs. Chem. Eng. Sci. 1990, 45, 3017-3034. (20) Lee, P. L.; Sullivan, G. R. Generic Model Control (GMC). Comput. Chem. Eng. 1988, 12, 573-580. (21) Quintero-Marmol, E.; Luyben, W. L. Inferential modelbased control of multicomponent batch distillation. Chem. Eng. Sci. 1992, 47, 887. (22) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, N. N. A unified framework for the study of anti-windup designs. Automatica 1994, 30, 1869. (23) Morari, E.; Zafiriou, E. Robust Process Control; PrenticeHall: New York, 1989.
Received for review June 1, 1999 Revised manuscript received October 11, 1999 Accepted October 21, 1999 IE990382L
