Profile Position Control of Batch Distillation Based on a Nonlinear

A profile position control strategy based on the nonlinear wave model is ... wave propagation model and the quasi-dynamic model by Quintero-Marmol and...
0 downloads 0 Views 117KB Size
Download PDF
Ind. Eng. Chem. Res. 2001, 40, 4111-4120

4111

Profile Position Control of Batch Distillation Based on a Nonlinear Wave Model Myungwan Han† and Sunwon Park*,‡ Chemical Engineering Department, Chungnam National University, Taejon 305-764, Korea, and Chemical Engineering Department, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea

Batch distillation poses challenging operation and control problems caused by a great change in process gain and a very nonlinear response in distillate composition during the operation. A nonlinear wave model captures an essential dynamic characteristic of the batch distillation column. A profile position control strategy based on the nonlinear wave model is proposed for the control of batch distillation columns. The profile position and distillate composition are estimated from selected tray temperature measurements by observers, which are based on the combination of the nonlinear wave propagation model and the quasi-dynamic model by QuinteroMarmol and Luyben (Chem. Eng. Sci. 1992, 47, 887). This control strategy has been applied to binary and multicomponent separation of ideal and nonideal mixtures. The proposed control scheme is shown to deal with severe nonlinear and varying process gains of batch operation and to be robust to model-plant mismatches. Introduction Batch distillation has received increased attention in recent years because it is suitable for production of lowvolume, high-value specialty chemicals. A batch column can handle a wide range of feed compositions, relative volatilities, and product specifications. The same column can be used for a number of different separation tasks. However, the nonlinear and time-varying nature of the process poses a challenging operation and control problems, but little attention has been paid to the problem of control. Regardless of the objective of distillation, the purity of the products should be strictly regulated. An off-specification product means a significant economic loss. A control scheme based on a better understanding of the dynamic characteristic of a batch distillation column is needed for tight composition control in batch distillation. A batch rectifier is commonly operated in two ways: constant reflux ratio and constant distillate composition. Constant reflux ratio operation keeps the reflux ratio constant during the batch. The distillate composition changes continuously with time. The transfer from cut to cut occurs when the average composition in the product receiver meets the product specification purity for the corresponding component. Therefore, it needs online measurements of the product composition or estimator to infer the product composition. However, it is well-known that product compositions cannot be economically measured online because of the delay between obtainment of the sample and the output of the analyzer. Without accurate information about the product composition, the operation is apt to produce offspecification products with a significant economic loss or should be run with some marginal safety. Optimal operation involves the use of a mathematical * To whom correspondence should be addressed. Tel: 0082-42-869-3920. Fax: 00-82-42-869-3910. E-mail: swpark@ convex.kaist.ac.kr. † Chungnam National University. ‡ Korea Advanced Institute of Science and Technology.

technique to determine the optimal reflux policies that maximize a profit function. The policies are intrinsically determined in an open loop; that is, there is no feedback from a real plant. Therefore, the operating mode may be very susceptible to model-plant mismatches, similar to constant reflux ratio operation. Usually the optimal profiles cannot be tracked with conventional linear controllers, because of the nonlinear and time-varying nature of the process. A constant distillate composition policy has been reported in several studies to give results close to the optimal policy, provided that there are an adequate number of stages in the column.1-3 Therefore, a reasonable mode of operation would appear to be the use of a constant distillate composition policy for each product cut. Then, the optimization problem can be reduced to the one determining how to operate during the slop-cut operation. Constant distillate composition operation is a challenging problem because one needs to adjust the reflux rate continuously. Finefrock et al.7 recognized that controlling the distillate purity in a batch distillation operation is very difficult, even when binary mixtures are used. The distillate composition control becomes more difficult with high-purity separation and a mixture of high relative volatilities. Bosley and Edgar3 summarized the difficulties faced with optimization and control of batch distillation: (1) poor agreement of model thermodynamic vapor/liquid equilibrium (VLE) correlation with actual plant data; (2) time-varying process gains and time constants; (3) large open-loop interactions; (4) poor agreement between model and plant data; (5) low process signal-to-noise ratio; (6) online sensors which are unavailable or give delayed results; (7) difficult and computationally expensive state estimation. This work is concerned with the development of a control scheme to deal with these difficulties. QuinteroMarmol et al.4 and Quintero-Marmol and Luyben5 reported that the profile propagates with a little variation in shape to the top of the column, that is, wavelike behavior. It supports that dynamic characteristic of a

10.1021/ie990845n CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001

4112

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

Table 1. Model and System Characteristics

vapor boilup, mol/h tray holdup (st. st.), mol nominal reflux drum holdup, mol total feed charge, mol nominal feed composition (mole fraction) estimated feed composition (mole fraction) tray hydraulic time constant, h number of ideal trays relative volatility nominal composition setpoint

ideal binary system

nonideal binary system

ternary system

100 1 10 400 0.5 0.7 0.001 40 1.5 0.99

100 1 10 800 0.25/0.75 0.35/0.65 0.001 10 nonideal 0.8

100 1 10 1000 0.3/0.4/0.3 0.333/0.333/0.3334 0.001 40 9/3/1 0.99/0.99

batch distillation can be well represented by a nonlinear wave model. Here, the wave is defined as a spatial structure propagating at a constant velocity without changing its shape along a spatial coordinate. We choose the quasi-dynamic (QD) model of Quintero-Marmol and Luyben5 to calculate the composition of each tray in the distillation column so that we can get a column composition profile instantaneously. The nonlinear wave model and QD model are combined to give the wave velocity, column composition profile, and distillate composition. The proposed control scheme is based on the generic model control (GMC) scheme of Lee and Sullivan,6 which incorporates the nonlinear wave model. Three examples involving ideal, nonideal, and ternary systems are chosen to examine the applicability of the control scheme to a real column. Summary of Batch Distillation Control Quintero-Marmo et al.4 and Quintero-Marmol and Luyben5 studied constant reflux ratio operation of batch column and proposed several methods for estimating distillate composition online: for example, the extended Luenberger observer and the QD model. They used the QD method based on the fact that, after the startup period, the composition profile in the upper part of the batch distillation column becomes essentially binary. A temperature is measured in the upper middle section of the column. The composition of the vapor phase of the appropriate binary mixture at that location is calculated from the measured temperature, pressure, and VLE relationships. This calculated binary vapor is fed into a dynamic mathematical model of the section of the column above this temperature measurement tray. The Luenberger observer was found to be slightly better than the QD observer, but it is more difficult to design. They proposed use of the QD observer for most batch distillation applications. Bosley and Edgar3 implemented online the optimal operation trajectory determined a priori by optimizing offline the performance of a batch column; they proved that nonlinear model predictive control is effective in tracking the optimal performance. They showed in simulation experiments that the profit for one set of conditions using a constant composition policy was 43% higher than the profit for operation at a constant reflux ratio. The optimal result showed another 4% profit over constant composition. Finefrock et al.7 analyzed the problem of composition control in a batch column. They indicated that a batch column is an integrating system; the depletion of light components with time during the batch operation takes a role somewhat like a ramp load and produces a control problem. The column experiences a great deal of change from a low-plant-gain composition space (i.e., the steady-

state space) to a high-plant-gain composition space. This means that the control gain should be increased appropriately during the operation. They suggested using a gain-scheduled PI controller to obtain tight control of the distillate composition. Belanger and Luyben8 proposed recently PII control to reject the ramplike disturbances. Nonlinear model predictive control (NMPC) can be selected to overcome the time-varying characteristics. NMPC may be effective for optimal batch distillation if we consider the complexity and computational requirement. Barolo and Berto9,10 studied constant composition control in batch distillation and proposed a control scheme based on nonlinear internal model control.11 They used an extended Luenberger observer proposed by Quintero-Marmol et al.4 for estimating tray compositions and distillate composition. The control scheme uses the estimated tray compositions of vapor or liquid in the upper section of the column as well as the estimated distillate composition to calculate the reflux flow rate for the control of distillate composition. They showed the advantage of their approach over conventional PI control. Model Description In this work, we consider the separation of both binary (ideal and nonideal) and ternary mixtures using a batch rectifier. The assumptions made in the model of the batch rectifier are as follows: (i) perfect mixing and equilibrium on all trays; (ii) negligible vapor holdup; (iii) constant stage pressures and tray efficiencies; (iv) constant vapor flows (the energy balance is neglected); (v) total condensation with no subcooling in the condenser. This model considers varying molar holdups on each tray so that the liquid flow rate from the tray varies with the corresponding molar holdup. The following linearized version of the Francis weir formula is used:

Lm ) Lm,0 +

Hm - H0 τ

(1)

Lm,0 is the reference value of the internal liquid flow rate, Hm and H0 are the actual and reference molar holdups on tray m, and τ is the tray hydraulic time constant. Furthermore, we relax the assumption of constant reflux drum level to build a new operation strategy for batch column. The model parameters and system characteristics are listed in Table 1 for ideal binary, nonideal binary, and ternary mixtures. The following equations are used to calculate the temperature and liquid-phase composition on any tray when the VLE of the mixture to be separated exhibits

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4113 Table 2. Vapor-Pressure Coefficients for the Ideal Binary System (Temperature in K; Vapor Pressure in mmHg) component 1

component 2

A1

B1

A2

B2

-4000.44

12.2410

-4000.44

11.3242

Table 3. Antoine Constants and Wilson Parameters for the Ethanol-Water System Antoine Constants (log P (mmHg) ) A - B/[T (°C) + C] EtOH water

A

B

C

8.04494 7.96681

1554.800 1668.210

222.65 228.000

Wilson Parameters 0.18202 0.78317

Λ12 Λ21

Table 4. Vapor-Pressure Coefficients for the Ternary System [ln Pj (mmHg) ) Aj/T (K) - Bj] component 1

component 2

component 3

A1

B1

A2

B2

A3

B3

-4000.44

12.4233

-4000.44

11.3277

-4000.44

10.2261

constant relative volatility as in the approach of QuinteroMarmol et al.4:

yi Pi° ) P xi yi

Ri

)

xi

NC

∑ Rkxk k)1

ln Pi° )

Ai + Bi T

(2)

where Pi° is the vapor pressure of component i and Ri is the relative volatility. Therefore, the stage temperature can be determined:

Ai

Tm ) ln

RiPtot NC

(3) - Bi

∑ Rkxm,k

k)1

where Ptot is the total pressure and xm,j is the liquidphase composition of component i on stage m. The constants Ai and Bi are given in Tables 2 and 4. For a binary system, we can determine the composition on a tray from the tray temperature as the following:

xm,i )

{

[

}

Ai 1 RiPtot exp - Bi - Ri-1 Ri - Ri-1 Tm

xi-1 ) 1 - xi

for components i - 1 and i

] (4)

Profile Position Control Recently, the dynamics of a continuous distillation column has been well explained as wave propagation theoretically as well as experimentally (Marquardt,13,14 Hwang,15 Hwang et al.,16 Ting et al.,17 and Kienle18). The control of distillation columns based on wave

propagation models has been studied by several researchers. Marquardt and co-workers13,14,23 initiated studies of wave propagation model based control for distillation columns. Han and Park19-22 developed a profile position control strategy based on both a nonlinear wave model and GMC for control of various types of continuous distillation columns and applied the strategy to startup control of distillation columns. Rehm and Allgo¨wer24 designed a robust control strategy for experimental distillation columns from wave models. Balasubrahanya and Doyle25,26 studied nonlinear control of both conventional and reactive distillation columns using a wave propagation model. Shin et al.27 proposed a new nonlinear profile position observer using tray temperatures instead of tray compositions based on a nonlinear wave model. Quintero-Marmol and Luyben5 have found that the dynamics of a batch distillation column is also well described by wave phenomena. After the startup period, during which the column operates at total reflux, a composition front generates in the lower part of the column and propagates to the top of the column during the course of the batch cycle. Only two components constitute the composition front. Then another binary composition front generates and moves to the top of the column and so on. During the propagation of the composition fronts, the column profile maintains nearly constant shape, which is a characteristic of the wave. This behavior can be well explained by nonlinear wave theory. Batch distillation can be thought to occur according to the following sequences. At the time when the first product starts to be withdrawn, a composition front is located in the lower part of the column and separates the lightest and intermediate components: this wave propagates along the column as the light product is removed. When the wave nears the top of the column, the first slop cut, which is essentially a binary mixture of light and intermediate components, begins to be withdrawn. When the purity of the intermediate component reaches its specification, the withdrawal of the intermediate product starts. Now a second wave forms near the base of the column. The front moves up the column when the intermediate component is removed in the distillate. The wave position is directly related to the distillate composition. Therefore, the control of the profile position can be a key to distillate composition control. The wave velocity can be written as15

S˙ )

V∆y/∆x - L 1 + r∆y/∆x

(5)

During the propagation of the column profile, the profile maintains nearly constant shape so that locating the position of the wave at a certain point takes an important role in maintaining a constant distillate composition during the batch operation. Choosing the profile position as a controlled variable makes the proposed control scheme deal with time-varying process gains and time constants. The process gain and time constant varies with time when the controlled variable is distillate composition. The shape of these fronts depends on the reflux ratio and the relative volatilities. For the constant distillate composition policy, the reflux ratio varies continuously during the operation. The profile position should be adjusted a little as the distillation proceeds. The dy-

4114

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

The proposed control scheme needs an observer for estimating the profile position. The observer incorporates eq 5 into the form of a Luenberger type equation:

S˙ )

V∆y/∆x - L 1 + r∆y/∆x

m

+

K1(xˆ j) (xˆ j - xj) ∑ j)l

(8)

where j is the measurement tray number and l and m are the number of the first measurement tray and the number of the last measurement tray in the column, respectively. From the above equation, we get the representative slope of concentration ∆y/∆x.

∆y/∆x )

S˙ + L V - rS˙

(9)

The representative slope, which is the slope of the equilibrium curve at the representative concentration, can be estimated from eq 9. The observer computes the concentration of each measurement tray from the estimated profile position with the assumption that the profile has a linear form in the steepest region as the following:

xˆ j ) K2(Sj - S) + xs

Figure 1. Dynamic column composition and temperature profiles in batch distillation.

namic composition and temperature profiles during constant distillate composition control are shown in Figure 1. The figure shows that the profiles maintain similar forms and that the inflection point of the profiles, which can be regarded as a profile position, remains at almost the same position, although the bottom composition varies continuously. For control of the profile position, GMC8 is employed because the overall structure of GMC allows the incorporation of the nonlinear wave model directly into the framework. The other control framework such as NMPC can be thought to be an alternative for control of the profile position. However, we found that it is hard to incorporate the nonlinear wave model into the framework. We choose the wave position of a batch column as the state vector in a GMC equation.

∫0t(S* - S) dt′

S˙ ) K1(S* - S) + K2

(6)

where S and S* are the profile position and its setpoint, respectively. Combining eqs 5 and 6 gives one equation for profile position control of batch distillation.

V∆y/∆x - L - K1(S* - S) - K2 1 + r∆y/∆x

∫0t(S* - S) dt′ ) 0 (7)

Both the profile position and the slope of the equilibrium curve at the representative concentration can be estimated by the profile position observer. If we determine the boilup rate V, the manipulated variable, reflux rate L, can be calculated.

(10)

where Sj is the normalized distance at which the jth tray is located. The representative concentration, xs, can be determined as the concentration at which the equilibrium curve has the same slope as ∆y/∆x. The K2 term represents the slope of the model equation that calculates the concentration in the steep region. The K1 term can be written as the following form:

K1 ) K0 exp[-b(xˆ j - xs)2]

(11)

where K0 denotes the gain of the weighting function and b adjusts the range of weighting. The details of the observer are found in the work of Han and Park.19 Temperature measurements in the steepest region have advantages of low sensitivity to pressure and noise variation. Based on the measured tray temperatures, the QD model or Luenberger observer proposed by Quintero-Marmol and Luyben5 provides not only the liquid-phase composition of each tray to the profile position observer but also the concentration of the distillate product to the distillate composition controller. The measured tray temperature can be converted to the liquid-phase composition of the tray using eq 4, and the liquid-phase composition is used for the QD or Luenberger observer. Then the profile position observer adjusts the profile position to fit the tray compositions of the model specified in eq 10 with the tray compositions calculated by the QD observer. The profile position controller (GMC) is used for instantaneous control to move the profile position to the desired location (S*) in the column. However, an offset from the setpoint in a product composition may occur when the profile position is fixed at a specific point. The reflux rate varies with time when we choose the constant distillate composition control policy so that the shape of profile also varies to some extent. The offset in distillate composition from the setpoint can be compensated by profile position/composition cascade. The profile position control only leads the distillate composition near its setpoint despite the time-varying

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4115

Figure 2. Schematic diagram of the proposed control scheme.

nature of the process. The composition controller in the cascade scheme eliminates the offset that occurs when maintaining the profile position at a specific point of the column. The velocity form of the PI control scheme is chosen for the composition controller in order to transfer from cut to cut with a reduced bump. The observer, GMC, and distillate composition controller have sampling times of 0.005, 0.01, and 0.03 h. Equations 6 and 7 are implemented as a digital form by replacing the integrals with summations and the derivative terms with first-order backward differences. The schematic diagram of the proposed control scheme is illustrated in Figure 2. Batch Distillation Startup Strategy The batch distillation startup period consists of the following three steps:28 (1) preheating of the still charge to its bubble point, (2) filling the column and the constant holdup, and (3) running without distillate withdrawal and taking the unit to a steady state. The total-reflux startup phase is carried out until the process approaches the steady state or the first product component reaches a desired concentration. The controller is activated to lead the distillate composition to the desired value and maintains it at the setpoint. The distillate is withdrawn as a product until the reflux ratio reaches a prespecified limiting value. This startup strategy has a drawback that the distillate product is usually obtained at a purer level than needed. We modified the startup policy to reduce the startup time and to obtain a distillate product as needed. The profile position control scheme is started when the distillate composition is near the composition specification, and the distillate composition controller is activated after the purity specification is met. The proposed controller does not wait until the product specification is met or steady state is obtained. However, until the product in the reflux drum meets the product specification, the control scheme does not begin product withdrawal. As a result, the level of reflux drum goes higher; i.e., the drum level is not constant. This reduces the sensitivity of the distillate composition to a control action or disturbances and contributes to moving the distillate composition to its setpoint smoothly. We found that the proposed control scheme is robust with the modification. This modification can be applied to the transition from slop cut to product removal. Composition Estimator The proposed control scheme employs a distillate composition controller to provide a setpoint to the profile

Figure 3. Conventional temperature control (Kc ) -8 mol/(h °C), τI ) 0.1 h): (A) 25th tray temperature; (B) distillate composition.

position controller. The distillate composition controller can utilize the measurement of distillate composition using an online measuring device, such as a gas chromatograph. Delayed responses of such devices do not pose a significant problem to the proposed control scheme, because the profile position controller does not need a frequent updating of its setpoint. However, the measuring device requires high investment and maintenance cost. Therefore, practical and economic measurements are still needed. Quintero-Marmol and Luyben5 proposed a QD model for estimating the distillate composition. They found that the composition fronts move through the column during the course of the batch cycle as explained in the previous section. What is in the upper part of the batch distillation column at one time during most of the batch cycle is some binary mixture of components of 1 and 2. Then it becomes components 2 and 3. This continues for as many components as there are in the initial feed to the unit. This binary wavelike behavior of batch distillation columns is the basis for the QD estimator. A temperature is measured in the upper middle section

4116

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

Figure 5. Dynamic profile position and reflux flow rate under the proposed control scheme for an ideal binary mixture. Figure 4. Distillate composition control for an ideal binary mixture: (A) proposed control scheme; (B) PI composition control.

of the column. The composition of the vapor phase of the appropriate binary mixture at that point in time and at that location is calculated from the measured temperature, pressure, and VLE relationship. This calculated binary vapor is fed into a dynamic mathematical model of the column section above this temperature measurement tray. Tray and distillate compositions are estimated by using the following mass balance differential equations:

Hm Hm

dxˆ l ) Lxˆ l+1 + Vyl-1 - Lxˆ l - Vyˆ l dt

dxˆ m ) Lxˆ m+1 + Vyˆ m-1 - Lxˆ m - Vyˆ m, dt m ) l + 1, ..., N HD

dxˆ D ) V(yˆ N - xˆ D) dt

(12)

where the (l - 1)th tray is a measurement tray and yl-1 is updated every sampling time. We allow additional measurement inputs to eq 12 to improve the quality of the estimation. Holdup of each tray is assumed to be constant. The QD model provides column composition measurements (xˆ m, m ) l, ..., N) to the profile position observer for estimating the profile position and distillate composition (xˆ D) to the distillate composition controller. Note that the estimated tray composition (xˆ m) is treated as the measured tray composition (xm) in the profile position observer. The QD model consists of ordinary differential equations so that an initial condition for the state variable is needed. We make a different initial

guess of the composition of the initial charge from the actual composition as shown in Table 1. The distillate composition and tray compositions predicted by the model converge quickly to the actual compositions during the period of total reflux operation as shown in Figures 4, 8, and 9. Temperature Control The tray temperature controller is usually considered in industry in order to keep the distillate composition close to the desired value. Barolo and Berto9,10 reported that the distillate composition drifts away from the specification when a temperature controller is used. Figure 1B shows the variations of temperature profiles during the constant distillate composition control. The tray temperature changes greatly (from 355 to 363 K) during the period. The column profile changes and becomes steeper with an increase of the reflux ratio (that is, with the progress of the operation). Figure 3 shows that keeping the 25th tray temperature at its setpoint does not guarantee that the distillate composition will be closer to the desired value. Ideal Binary System The relative volatility of this ideal binary system is 1.5, and the composition setpoint is 0.99, which requires high-purity separation. The details of the system are given in Table 1. Figure 4 shows the performance comparison between the proposed control scheme and the conventional PI controller. Profile position control leads the distillate composition near its setpoint but produces some offset from the setpoint. We eliminate the offset by employing the profile position/composition cascade. One can notice that the profile position control

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4117

One of the difficulties faced with batch distillation operation is a low process signal-to-noise ratio. The robustness of the proposed control scheme to the temperature measurement noise is tested. Random noise within (0.25 °C is added to the temperature measurement. It is shown in Figure 7 that the proposed control scheme is not greatly affected by the temperature noise and maintains good performances. Barolo and Berto9,10 proposed the control scheme based on the framework of nonlinear internal model control11 (NIMC). The Luenberger observer proposed by Quintero-Marmol and Luyben5 provides the distillate composition and system states. Their control equation can be written as the following for the manipulated variable, i.e., reflux flow rate: Figure 6. Distillate composition control for an ideal binary mixture in the presence of model-plant mismatch (relative volatility R): proposed control scheme.

∂yˆ N 1 V (yˆ N-1 - yˆ N) (xD,sp - yˆ N)  Hm ∂xˆ N R) ∂y ˆ 1 N (xˆ - xˆ N) Hm D ∂xˆ N

(13)

where  is a closed-loop time constant and N indicates the top tray. We have tested the NIMC approach for the case of an ideal binary system. The NIMC appears to be sensitive to temperature noise. This may be due to the fact that the differences between estimated tray compositions in eq 13 tend to amplify the noise. Nonideal Binary System

Figure 7. Distillate composition control for an ideal binary mixture in the presence of temperature measurement noise within (0.25 °C: proposed control scheme.

is the key to tight control of the distillate composition. PI composition control is not able to deal with the varying dynamics of the batch distillation, showing an oscillatory response. The high purity of the system gives a more nonlinear response of the distillate composition, which indicates more difficulty in using conventional PI control. The proposed control scheme is shown to give a superior performance over a conventional PI controller. Figure 5 gives dynamic profiles of the profile position and reflux flow rate when using the proposed control scheme. For the QD observer in the proposed control scheme, the 30th and 40th tray temperatures were used to estimate the tray compositions and distillate purity. The model-plant mismatch in VLE is common in batch operation. We examine the effect of the relative volatility error in the control performance. Figure 6 shows that the proposed control scheme using the measurement of the actual distillate composition can overcome up to 25% error in relative volatility. The error in the relative volatility tends to make oscillations in the distillate composition when the startup phase is switched to the production phase. The profile position control is robust to the model-plant mismatch in VLE, but the QD estimator is susceptible to the error. The proposed scheme using an estimated distillate composition shows an offset from the setpoint. It is recommended to use the measurement of the distillate composition for the primary controller when the modelplant mismatch error is significant.

We consider a mixture of ethanol and water that is highly nonideal. The VLE is represented by the Wilson model, and the model parameters are taken from Hirata et al.29 The relative volatility of the mixture varies from 1 to 11. The variability poses a challenging problem to model-based control. We examine the applicability of the proposed control scheme and compare its performance with that of a conventional PI controller. The proposed control scheme needs a VLE relationship, that is, differential of vapor composition with respect to liquid composition. We obtain the following equations by fitting the differential of the VLE curve and temperature in the range of operation through the use of the Wilson model and Antoine equation.

dy/dx ) 0.39 + 5.02 exp{-15.58x}

(14)

T(x) ) 81.2759 + 16.7435 exp{11.4629x} (15) Equation 14 is used to get a representative concentration, that is, a concentration at the profile position; eq 15 calculates the equilibrium temperature at tray liquid composition x for the range from 0.1 to 0.35. The bottom temperature was chosen to estimate the tray compositions using the observers. Performances of a conventional PI composition controller and the proposed control scheme are shown in Figure 8. The proposed control scheme maintains the distillate composition at the setpoint until 7.3 h. The proposed scheme is shown to deal with the problem caused by nonideality of the mixture. However, the PI controller keeps the distillate composition at its setpoint until 5 h. This is because the PI controller cannot deal with the changing gain of the process. The process suffers a control problem caused by ramplike disturbance when the distillate composition is controlled; that is, the composition tends to change continuously as if the ramp load enters the system.

4118

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

Figure 9. Distillate composition control for a ternary mixture using the proposed control scheme: (A) products; (B) impurities.

Ternary System

Figure 8. Distillate composition control for a nonideal binary mixture: (A) proposed control scheme; (B) PI composition control; (C) PII composition control.

Belanger and Luyben8 proposed a proportional plus double-integral (PII) controller as a low-frequency compensator to deal with ramplike disturbances. The controller has the form

[

CO(t) ) Kc (t) +

∫

∫∫

]

1 1 (t) dt + ( (t) dt) dt τ11 τ12 (16)

The objective of the double integral action is to introduce compensation at low frequencies to asymptotically reject the effects of occurring ramplike disturbances. Quality controllers are often tuned as tightly as possible to prevent the production of off-specification product. The closed-loop time constants of these loops are small. We test the applicability of the PII controller to batch distillation operation. A PII controller is shown in Figure 8C to give a somewhat better response in product composition than PI controller. The PII controller maintains the distillate composition longer at the setpoint than the conventional PI controller does. This indicates that the PII control scheme is effective to some degree in dealing with the changing gain of batch distillation. However, tuning of a second time constant should be done carefully; small time constant for tight control of the distillate composition may often lead to more oscillatory responses.

The proposed control scheme can be extended to multicomponent mixtures. This is based on the fact that after the startup only two components at a time are in the upper part of the column, as indicated by QuinteroMarmol and Luyben.5 It is a necessary condition for the profile position observer and QD observer to be used. To make the derivation and the implementation of the control law easy, it is assumed that during each of the two production phases a binary mixture is found in the upper trays of the column. We consider the case of a constant relative volatility ternary mixture. The specifications of the column are listed in Table 1. The proposed control scheme does not wait until steady state is attained. It is activated when the distillate composition is near its setpoint through total reflux operation. The distillate product is withdrawn at the desired constant purity until the reflux ratio reaches a specified maximum value. After the maximum value is reached, the operation is switched to a slop-cut removal phase. During the period, the reflux rate is decreased to move the profile of the intermediate component to the top of the column. When the concentration of the intermediate component becomes close to its setpoint, the proposed control scheme is reactivated for the withdrawal of a second product. The proposed control scheme gives a good performance, as shown in Figure 9A. When the concentration of the distillate becomes close to its setpoint, there exist two impurities in the distillate product: light and heavy. To meet the product specification, one of the impurities can be reduced. However, reducing the light component and reducing the heavy component have the opposite control direction to each other. To reduce the light

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4119

position and reflux flow rate are given in Figure 10, which show smooth changes of the variables during the operation. Conclusions A batch distillation column shows a wavelike behavior that can be well represented by a nonlinear wave model. A strategy based on the nonlinear wave model has been proposed for the control of constant distillate composition in a batch column. The control scheme uses a profile position in the column as a controlled variable, and a distillate composition controller provides the setpoint of the profile position in a cascade manner. The QD model developed by Quintero-Marmol and Luyben5 is employed for estimating the tray compositions and distillate composition. The proposed control scheme has been applied to three examples: ideal binary, nonideal binary, and ternary systems. The results show that the proposed control scheme provides superior performances over a conventional PI controller. A new startup operation strategy is also proposed to minimize waste and reduce an overshoot in the distillate composition during the startup and cut-over period. The proposed control scheme is shown to be robust with the modification of the startup strategy. Model-plant mismatches such as errors in VLE, common in batch distillation operation, are considered. The proposed scheme is shown to deal with the model-plant mismatches and measurement noises. Acknowledgment Figure 10. Dynamic profile position and reflux flow rate under the proposed control scheme for ternary mixture: (A) profile position; (B) reflux flow rate.

impurity, the reflux flow rate should be decreased; to reduce the heavy impurity, the reflux flow rate should be increased. We employ the method to keep the position of the wave front between the intermediate and heavy component at the specified point to fix the amount of heavy component in the top product. While the amount of heavy component in the distillate product is maintained, the light impurity is eliminated gradually in the reflux drum. Figure 9B shows that the light and heavy impurities coexist in the distillate product and the concentration of the light impurity decreases gradually while that of the heavy impurity is maintained. The constant distillate composition policy is close to optimal. This reduces the search dimension of the optimization problem. Now it is only necessary to determine the optimal reflux policy for the slop cut. If the reflux flow is too low, most of the intermediate component is removed as slop cut without meeting specifications of the product. This means that no intermediate product is obtained from the batch so that it is very important to maintain the reflux flow rate over a certain low limit. If the reflux flow rate is high, it takes much time to withdraw the slop cut. There is a tradeoff in determining the reflux flow rate during the slop-cut period. We used a low reflux flow rate during the slopcut period. This was possible because of the high reflux flow rate in the previous product cut, which increases the holdup of the intermediate component in the column. More research on the reflux policy during the slopcut period is needed. Dynamic profiles of the profile

This work was partially supported by the Brain Korea 21 project. Nomenclature Ai ) vapor-pressure constant for component i Bi ) vapor-pressure constant for component i CO ) controller output H ) holdup L ) reflux rate (mol/h) NC ) number of components K ) gain of the Luenberger observer; also, GMC tuning constant R ) reflux rate (mol/h) r ) molar holdup of vapor to liquid T ) temperature (K or °C) V ) vapor rate (mol/h) x ) mole fraction in the liquid phase y ) mole fraction in the vapor phase Subscripts B ) reboiler D ) distillate Superscripts ∧ ) estimated property • ) time derivative * ) setpoint Greek Letters R ) relative volatility  ) error in the PII control scheme; also tuning constant for NIMC τ ) integral reset time; also tray hydraulic time constant

4120

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001

Λ ) activity coefficient ∆ ) prefix for the difference between the two sides of a self-sharpening wave

Literature Cited (1) Robinson, E. R. The Optimization of Batch Distillation Operations. Chem. Eng. Sci. 1969, 24, 1661. (2) Robinson, E. R. Optimal Reflux Policies for Batch Distillation. Chem. Process Eng. 1971, 52, 47. (3) Bosley, J. R.; Edgar T. F. Application of Nonlinear Model predictive Control to Optimal Batch Distillation. In DYCORD+ ’92; Balchen, J. C., et al., Eds.; Pergamon Press: Amsterdam, The Netherlands, 1993; p 303. (4) Quintero-Marmol, E.; Luyben, W. L.; Georgakis, C. Application of an Extended Luenberger observer to the control of multicomponent batch distillation. Ind. Eng. Chem. Res. 1991, 30, 1870. (5) Quintero-Marmol, E.; Luyben, W. L. Inferential ModelBased Control of Multicomponent Batch Distillation. Chem. Eng. Sci. 1992, 47, 4887. (6) Lee, P. L.; Sullivan, G. R. Generic Model Control. Comput. Chem. Eng. 1988, 12, 573. (7) Finefrock, Q. B.; Bosley, J. R.; Edgar, T. F. Gain Scheduled PID Control of Batch Distillation to Overcome Changing System Dynamics. AIChE Annual Meeting, San Francisco, CA, 1994. (8) Belanger, P. W.; Luyben, W. L. Design of Low-Frequency Compensators for Improvement of Plantwide Regulatory Performance. Ind. Eng. Chem. Res. 1997, 36, 5889. (9) Barolo, M.; Berto, S. Composition Control in Batch Distillation: Binary and Multicomponent Mixtures. Ind. Eng. Chem. Res. 1998, 37, 4689-4698. (10) Barolo, M.; Berto, S. An Advanced Strategy for Composition Control in Batch Distillation. DYCOPS-5, Corfu, Greece, June 8-10, 1998; pp 430-435. (11) Barolo, M. On the Equivalence between the GMC and the GLC Controllers. Comput. Chem. Eng. 1994, 18 (8), 769-772. (12) Henson, M. A., Seborg, D. E., Eds. Nonlinear Process Control; Prentice-Hall PTR: London, 1997. (13) Marquardt, W. Nonlinear Model Reduction for Binary Distillation. IFAC symposium: DYCORD 86sDynamics and Control of Chemical Reactors and Distillation Columns, 8-10, Bournemouth, U.K., 1986; p 123. (14) Marquardt, W. Traveling waves in chemical processes. Int. Chem. Eng. 1990, 30 (4), 585. (15) Hwang, Y.-L. Nonlinear Wave Theory for Dynamics of Binary Distillation Columns. AIChE J. 1991, 37, 705. (16) Hwang, Y.-L.; Graham, G. K.; Keller, G. E., II; Ting, J.;

Helfferich, F. G. Experimental Study of Wave Propagation Dynamics of Binary Distillation Columns. AIChE J. 1996, 42, 10. (17) Ting, J.; Helfferich, F.; Hwang, Y.-L.; Graham, G.; Keller, G. Experimental Study of Wave Propagation Dynamics of Multicomponent Distillation Columns. Ind. Eng. Chem. Res. 1999, 38 (10), 3588. (18) Kienle, A. Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory. Chem. Eng. Sci. 2000, 55, 1817. (19) Han, M.; Park, S. Control of High-Purity Distillation Columns Using a Nonlinear Wave Theory. AIChE J. 1993, 39 (5), 787. (20) Han, M.; Park, S. Control of Complex Distillation Columns. J. Chem. Eng. Jpn. 1995, 28 (1), 71. (21) Han, M.; Park, S. Multivariable Control of Heat-Integrated Double Effect Distillation Configurations. J. Process Control 1996, 6 (4), 247. (22) Han, M.; Park, S. Startup of Distillation Columns Using Profile Position Control Based on a Nonlinear Wave Model. Ind. Eng. Chem. Res. 1999, 38 (4), 1565. (23) Marquardt, W.; Amhrhein, M. Development of a Linear Distillation Model from Design-Data for Process Control. Comput. Chem. Eng. 1994, 18S, 349 (24) Rehm, A.; Allgo¨wer, F. Nonlinear h∞ -control of a highpurity distillation column. UKACC International Conference on CONTROL ’96, Exeter, U.K., 1996; Vol. 2, p 1178. (25) Balasubramhanya, L. S.; Doyle, F. J., III. Nonlinear control of a high-purity distillation column using a traveling wave model. AIChE J. 1997, 43 (3), 703. (26) Balasubramhanya, L. S.; Doyle, F. J., III. Nonlinear modelbased control of a batch reactive distillation column. 5th IFAC Symposium on Dynamics and Control of Process Systems, Corfu, Greece, June 8-10, 1998; p 123. (27) Shin, J.; Seo, H.; Han, M.; Park, S. A Nonlinear Profile Observer Using Tray Temperatures for High-Purity Distillation Column Control. Chem. Eng. Sci. 2000, 55, 807. (28) Diwekar, U. M. Batch Distillation. Simulation, Design and Control; Taylor & Francis: London, 1995. (29) Hirata, M.; Ohe, S.; Nagahama, K. Computer Aided Data Book of Vapor-Liquid Equilibria; Kodansha-Elsevier: New York, 1975.

Received for review November 23, 1999 Revised manuscript received May 20, 2001 Accepted July 2, 2001 IE990845N