705
Ind. Eng. Chem. Res. 1991,30,705-713 American Institute of Chemical Engineers: New York, 1984. Vaselenak, J. A,; Groesmann,.I.E.; Westerberg, A. W. An Embedding Formulation for the Optimal Scheduling and Design of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 1987,26,139-148. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Paper MD 18.5,presented at the CORS/TIMS/ORSA Meeting, Vancouver, May 1989. Wellons, M. C. Scheduling of Multipurpose Batch Plants. Ph.D. Dissertation, Purdue University, December 1989. Wellons, M. C.,Reklaitis, G.V. Scheduling of Multipurpose Batch Chemical Plants. 1. Formatiom of Single-Product Campaigns.
Ind. Eng. Chem. Res. 1991,preceding paper in this issue. Yu, P. L.;Zeleny, M. The Set of All Nondominatd Sohtions in Linear C a w and a Multicriteria Simplex Method. J. Math. Anal. Appl. 1976,49,430-468. Zadeh, L. A. Optimality and Non-Scalar-Valued Performance Criteria. IEEE Trans. Automatic Control 1963,AC-8, 5 M . Zeleny, M. Linear Multiobjective Programming; Springer-Verlag: Berlin and New York, 1974. Received for review December 10, 1989 Revised manuscript received July 26, 1990 Accepted August 10,1990
Control of a Binary Sidestream Distillation Column Heleni S. Papastathopoulou and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, Pennsylvania 18015
This paper presents results of a comprehensive study of the dynamics and control of a large industrial distillation column that separates a binary mixture of propylene and propane into three products: polymer-grade propylene distillate, chemical-grade propylene liquid sidestream, and propane bottoms. The column is a superfractionator with 190 trays and a high reflux ratio. A two-stage vapor recompression system is used for energy conservation. Plant data, both steady state and dynamic, were used to validate steady-state and dynamic models of the column. Then steady-state rating programs were used to calculate steady-state gains for various alternative control structures, and several steady-state indices were used for initial screening of alternatives. The most promising structures were evaluated on the rigorous dynamic model. Use of sidestream location as a manipulative variable was included in the analysis. This strategy yielded an unusual dynamic response: a naturally occurring derivative action in the blending of liquid sidestreams. Several control objectivea were studied, ranging from control of only one composition to control of all three product compositions plus the sidestream flow rate.
Introduction An increasing number of sidestream columns are being used in industry. They offer an energy-efficient method for producing three products from a single column in some separations. Most of the applications are in multicomponent separations where the sidestream is used to remove an intermediate-boiling component. However, there are important industrial applications where a sidestream column is used in binary separations. The most important example is in the separation of propylene and propane when two propylene products are required at different purity levels. This is the system that was studied in this work. The literature on the control of sidestream distillation columns goes back many years: Luyben (1966),Buckley (19691,Mosler (19741,Tyreus and Luyben (19751,Doukas and Luyben (19801,Ogunnaike et al. (1983),and Alatiqi and Luyben (1986). The only paper dealing with binary sidestream columns is that of Tyreus and Luyben (1975), who studied the methanol-water separation with low-purity products. Both computer simulations and experimental tests on a small pilot-scale distillation column (24 trays) were conducted. Use of the location of the sidestream drawoff tray as an additional manipulated variable was recommended. The work reported in this paper is an extension of studies by Tyreus and Luyben (1975)to a very large industrial-scale column making fairly high-purity products *Towhom correspondence should be addressed.
Table I. Nominal Parameter Values for Sidestream Column F,(lb-mol)/h 1000 S, (lb-mol)/h 196 D,(lb-mol)/h 535 0.70 r,.mole fraction of propylene 0.995 xD, mole fraction of propylene 0.93 x g , mole fraction of propylene 0.0002 xB, mole fraction of propylene NT 200 NS 150 NF 60 110 P,psia 1060 MB,lb-mol 180 M D , lb-mol 29 M,,lb-mol 7450 R, (lb-mol)/h 13.3 RR 10 d, ft 1 wH,in. 6.41 WL, ft
Table 11. Stream Proaerties ~
~
~~~
stream no. 1 2 4 6 I 8 IO 11
~~
~
flow rate, (lb-mol)/h 8913.6 7800.3 1113.3 195.7 917.6 917.6 7800.3 7800.3
~~
~
teme, OF 47.8 89.2 105.8 100.1 100.1
47.8 87.2 47.8
0888-5885/91/2630-0705$02.50/00 1991 American Chemical Society
~~
pressure, Dsia 108.9 190.3 223.9 223.9 223.9 108.9 190.3 108.9
vapor fraction 1.0 1.0 1.0 0.0 0.0 0.248 0.0 0.179
706 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 Table 111. Steady-State Gains ml-ma-mg
kll
kl2
k13
kP1
D-S- V R-S- V R-S-B D-S-BR R-S-BR RR-S-BR RR-S-B RR-S- V DV-S-V DV-S-B RV-S-V D-SV-B D-NS-V R-NS-V R-NS-B D-NS-BR R-NS-B RR-NS-BR RR-NS-B RR-NS- V DV-NS-V DV-NS-B RV-NS-V
-7.33 93.0 0.291 -7.94 3.65 4.82 0.570 12.7 -18.1 -0.765 182 -1.54 -8.11 103 0.288 -8.72 4.00 5.30 0.563 14.1 -20.1 -0.756 200
-2.65 -2.65 0.276 -2.89 0.170 -0.824 0.155 -2.65 -2.64 0.162 -2.65 -0.764 -0.073 -0.073 -0.073 -0.073 -0.073 -0,073 -0.073 -0.073 -0.073 -0.073 -0.073
0.306 -97.5 3.65 0.578 -6.67 -4.32 3.50 -6.55 -6.95 3.51 -1.19 2.90 0.303 -108 4.05 0.571 -7.38 -4.80 3.90 -7.28 -7.73 3.90 -1.35
-49.6 628 -0.208 -49.1 22.6 29.8 -0,407 86.2 -123 0.547 1230 -1.93 -48.7 616 -0.204 -48.2 22.2 29.3 -0.400 84.6 -120 0.537 1200
Table IV. Steady-State Control Indices ml-ma-m3 MRI D-s-v 0.053 R-S-V 0.054 0.056 R-S-B 0.056 S-B-R D-S-BR 0.054 R-S-BR 0.056 S-R-BR 0.056 RR-S-BR 0.058 0.059 RR-S-B RR-B-S 0.059 RR-S-V 0.056 DV-S-V 0.056 DV-S-B 0.059 DV-B-S 0.059 RV-S- V 0.056 0.087 D-SV-B 0.087 D-B-SV 0.046 D-NS-V 0.046 NS-D-V 0.045 R-NS-V 0.046 R-NS-B 0.046 NS-B-R 0.047 D-NS-BR 0.047 NS-D-BR 0.046 R-NS-BR 0.046 NS-R-BR 0.046 RR-NS-BR 0.046 NS-RR-BR 0.046 RR-NS-B 0.046 NS-B-RR 0.046 RR-NS-V 0.046 NS-RR-V 0.046 DV-NS-V 0.046 NS-DV-V 0.047 DV-NS-B 0.047 NS-B-DV 0.046 RV-NS-V 0.046 NS-RV-V
kaz -18.8 -18.8 1.01 -18.6 0.293 -5.86 1.10 -18.8 -18.8 1-09 -18.8 0.547 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079
k2a
k31
-0.219 -662 24.8 -0.413 -45.2 -30.7 24.9 -46.5 -49.3 24.9 -10.3 23.8 -0,215 -648 24.3 -0,406 -44.4 -30.1 24.5 -45.8 -48.4 24.4 -10.1
-0.806
10.3 -0.189 -0.411 0.189 0.250 -0.370 1.40 -1.20 0.496 20.0 0.521 -0.887 11.1 -0.189 -0.485 0.223 0.295 -0.370 1.53 -2.18 0.497 21.8
kn -0.317 -0.317 0.011 -0.159 -0.OOo
-0.052 0.091 -0.317 -0.317 0.086 -0.317 0.496 -0.003 -0.003 -0,003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003
ksa -0.199 -11.0 0.411 -0.375 -0.760 -0,628 0.509
-0.953 -0.997 0.504 -0.363 0.664 -0.199 -11.9 0.448 -0.375 -0.818 -0.674 0.547 -1.02 -1.07 0.541 -0.378
~
CN 1020 17200 447 447 978 920 920 753 430 430 1800 2410 429 429 22200 278 278 1070 1070 20100 538 538 1050 1050 1100 1100 922 922 533 533 2130 2130 2870 2870 531 531 26500 26500
NI 0.042 0.001 -4.48 0.419 0.039 -1.24 0.343 0.074 -3.33 0.827 0.009 0.008 -3.38 0.873 0.023 -6.38 0.196 6.96 1.26 0.116 41.4 1.26 6.48 1.27 2.97 0.585 3.61 0.709 33.9 1.26 1.35 0.246 1.29 0.235 34.3 1.26 3.66 0.664
in a system where the separation is very difficult (propylene-propane). Actual experimental data from an operating column at Exxon Chemical's Baytown, TX, chemical plant were used to validate the steady-state and dynamic models. A binary sidestream distillation column presents a more challenging control problem than a conventional twoproduct column. The process is now a 3 X 3 multivariable system instead of the normal 2 X 2. In addition, there are
JEC 0.771 1.98 2.72 0.589 0.899 2.18 0.932 1.47 1.75 1.23 1.84 1.84 1.80 1.27 1.53 5.76 0.758 2.39 0.498 2.76 3.66 0.498 2.28 0.508 2.67 1.02 2.59 0.913 3.38 0.495 2.68 1.04 2.68 1.05
3.39 0.495 2.54 0.776
A11
ha2
Ass
23.6 -22.5 -0.071 2.34 25.5 -0.885 1.44 7.69 0.909 0.909 20.3 20.1 0.850 0.850 22.9 4.95 4.95 0.150 0.776 26.6 0.075 0.776 0.161 0.776 1.04 0.776 0.753 0.776 0.080 0.776 2.01 0.776 2.12 0.776 0.080 0.776 0.554 0.776
28.1 28.1 -1.51 2.54 27.8 -0.437 2.98 8.74 -1.64 2.55 28.1 28.1 -1.63 2.55 28.1 -0.388 2.44 0.167 0.842 0.167 0.167 0.842 0.167 0.835 0.167 1.06 0.167 0.990 0.167 0.846 0.167 2.86 0.167 2.98 0.167 0.846 0.167 1.28
1.18 65.2 -0.267 1.10 1.05 2.11 2.11 1.76 -0.331 1.33 5.66 5.93 -0.327 1.26 2.16 -0.431 4.33 0.935 0.935 56.1 0.008 0.935 0.939 0.939 2.05 2.05 1.69 1.69 0.010 0.933 4.81 4.81 5.04 5.04 0.010 0.934 1.78 1.78
a variety of possible control objectives for a sidestream column. The normal operation would be control of the compositions of all three products (distillatexD, sideatream xs, and bottoms xB), using the three normal manipulated variables (distillate flow D, sidestream flow S, and vapor boilup V). However, there are several alternative operating objectives that lead to different control structures. (1) Control of the compositions of just the distillate and
Ind. Eng. Chem. Res., Vol. 30, No. 4,1991 707
0
t
Fi
Figure 1. Sidestream propylene-propane column with two-stage vapor recompression.
0.0
25.0
50.0
75.0
100.0
Time [minl
t’ ‘0
5s: 0 2
0 E
I
-
Figure 2. D-NS-V control structure.
sidestream: This occurs when maximum propylene recovery is desired, Le., xB is minimized. This mode occurs frequently in the propylene-propane separation because propylene is much more valuable (going to petrochemical production) than is propane (going to LPG). It is particularly common when the distillation column uses vapor recompresaion with a motor-driven, fixed-speed compressor in which energy costs do not increase very much as xB is decreased. This means that frequently the optimum operating strategy is to minimize xB by fixing vapor boilup at its maximum value (as limited by column flooding, compressor load, etc.). (2) Control of sidestream flow rate and product composition: If the sidestream from the column is a demand flow that is set by some downstream unit or customer, its flow rate must be controlled and cannot be used as a manipulated variable. If the control of the compositions
2
s,
I
I
I
I
I
125.0
708 Ind. Eng. Chem. Res., Vol. 30,No.4, 1991
1
ot
0.0
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9
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9
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Time lminl
Time (minl
Figure 4. Comparison of experimental plant test with model predictions: ATV test of xD-D loop.
0
0.0
Time [min)
Figure 5. Open-loop step response of x s to change in CO.
2.5
5.0
7.5
10.0
12.5
Detuning Factor
Figure 6. Dependence of maximum closed-loop log modulus on BLT detuning factor.
Ind. Eng. Chem. Res., Vol. 30,No. 4,1991 709
4w.o
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2400.
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Figure 7 . Comparison of D-S-V and D-S-B structures for -5% feed composition disturbance (solid l i e , D-S-V; dotted line, D-SB).
1
, I
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zw0.0
2400.
im.0
m . 0
xm.
Time (minl
Time Iminl
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'
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Figure 8. Comparison of D-S-V and D-NS-V structures for -5% feed composition disturbance (solid line, D-S-V; dotted line, DNS-V).
Process Studied
by Finco et al. (1989). Pressure-dependent relative volatilities and equimolal overflow were the main assumptions. The former was shown to be valid at the lower pressures encountered in the vapor recompression column. The latter is valid as long as pressure changes are not large. The tray efficiency in the steady-state model was adjusted until the steady-state model matched the plant operating data. An efficiency of 95% was found. The dynamic model was validated by plant tests. These are discussed later in this paper.
The column studied was based on an actual column at the Exxon Chemical Baytown chemical plant. The column separates a 70 mol % propylene, 30 mol % propane feed into a 99.5 mol % propylene polymer-grade distillate product, a 93 mol 5% propylene chemical-grade liquid side-stream product, and a high-purity propane bottoms product that goes to fuel (LPG). The column has about 200 trays and operates with a reflux ratio of 13.3. Since a vapor recompression system is used, the operating pressure in the column is about 110 psia. This pressure is lower than would be required in a conventional column with a water-cooled condenser. Therefore, the relative volatility is increased, making the separation easier. Table I summarizes the steady-state design and operating parameters for the sidestream column. Figure l shows the system. Table 11gives the properties of the various streams numbered in Figure 1. The rigorous, nonlinear steady-state and dynamic models used in the analysis were the same as those used
State-State Analysis The steady-state model was used to calculate steadystate gains for a large number of alternative control structures for the sidestream column. An efficient procedure was developed to determine these gains for any choice of manipulated variables with only three rune of the rating program (Papastathopoulou and Luyben, 1990a). Table I11 gives steady-state gains for several 3 X 3 multivariable systems when XD, X S , and XB are the controlled variables. These gains are dimensionless (scaled by composition and flow transmitter spans). The basic manipulated variables are D, S, and V or D, NS, and V. Ratio schemes are also included such as boilup ratio (BR = V / B ) , reflux ratio (RR = R/D), and sidestream-to-vapor ratio (SV = S / W . Figure 2 shows the D-NS-V control structure. Three sidestream drawoff trays are used with five trays separating them (NS3 - NS2 = NS2 - NSI = ANS = 5). Liquid
(1)exper'mental autotune test data from a large industrial column, (2) exploration of a variety of control structures and control objectives (from a single composition to three compositions plus sidestream flow rate), and (3) the discovery of a new phenomenon (a naturally occurring open-loop derivative action in the sidestream blending system).
710 Ind. Eng. Chem, Res., Vol. 30, No. 4, 1991 Table V. Controller Tuning Parameters ml-ml-m3
D-S-V
XD
xs
D-S-B
XB XD
xs
D-NS-V
XB XD
D-S
XB XD
xs
D-NS
XS XD XS
0
K"
WU
Kc
71
-53.3 -51.9 -26.8 -53.3 -51.9 720 -53.3 1.96 -26.8 -53.3 -51.9 -53.3 1.96
0.197 0.241 0.142 0.197 0.242 0.142 0.197 0.523 0.142 0.197 0.241 0.197 0.532
-7.15 -6.96 -3.60 -7.12 -6.94 96.3 -12.7 0.836 -6.38 -5.28 -5.14 -13.3 0.836
90 73.4 125 90 74 125 51 3 71 122 100 48 3
streams from two of the three trays are mixed together. The signal from the composition controller (CO)sets this split at any point in time. Table IV gives various steady-state indexes calculated for the alternative control structures when all three compositions are controlled by using various pairings and various choices of manipulated variables. The indexes used are summarized below. (1) The Morari resiliency index (MRI) is the minumum singular value of the steady-state gain matrix. The larger its value, the better the control. (2) The relative gain array (RGA) is the Hadamard product of the steady-state gain matrix and ita inverse. Diagonal elements (Aii) close to unity indicate little interaction. (3) The Niederlinski index (NI) is the determinant of the steady-state gain matrix divided by the product of ita diagonal elements. This index must be positive. (4) The condition number (CN) is the ratio of the maximum singular value of the steady-state gain matrix to ita minimum singular value. The smaller this number, the better the control. (5) The Jacobi eigenvalue criterion (JEC) (Mijares et al., 1986) is the largest eigenvalue of the matrix A, where A satisfies the equation K = D(I - A) (1) where K = steadystate gain matrix,D = a diagonal matrix whose elements are those of K, and I = identity matrix. The values of JEC should be between -1 and 1, and the best pairing is that with the smallest JEC. As expected, Table IV shows that the use of reflux in this high reflux ratio column is not desirable (very high CN, JEC greater than 1,and negative RGA). All schemes with negative NI's or RGA's were eliminated. All schemes with very large C N s were eliminated. This still left a large number of possible schemes. Heuristic dynamic considerations were used to reduce the number of cases. All schemes that use B to control sidestream composition or S (or NS) to control overhead composition were eliminated because these manipulated variables have no direct effect on these controlled variables. Of the remaining schemes in Table 111, the 0-S-V, D-SBR, D-NS-V, and D-NS-BR schemes had the lowest JEC's. Note, however, that some of them are greater than unity, indicating control-loop interaction. The schemes using the boilup ratio gave some unexpected and interesting open-loop dynamic responses (open-loop underdamped behavior). They also required the use of transformations to determine the open-loop plant-transfer functions since some of them could not be determined by conventional identification methods. Since these ratio schemes were discussed in a recent paper (Papastathopoulou and Luyben, 1990b), they will not be included in this paper.
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Figure 9. Comparison of D-S and D-NS structures for -5% feed composition disturbance (solid line, D-S; dotted line, D-NS).
One scheme that cannot be analyzed by using the steady-state indices is the D-S-B structure since the steady- state gains are indeterminant. However, the transfer functions for this structure can be obtained at higher frequencies by suitable transformations. The performance of this structure has been compared with the more conventional structures in recent papers (Skogestad et al., 1990; Papastathopoulou and Luyben, 199Oc), and one example of these results will be included in this paper. Dynamic Analysis A. Mathematical Model. The rigorous nonlinear dynamic model of the column consisted of two ordinary differential equations (totalmass and light component balances) and two algebraic equations (vapor-liquid equilibrium and liquid hydraulics) per tray. The Francis weir formula was used for liquid hydraulia. A composition analyzer dead time of 6 min was used for all composition measurements. B. Model Validation. During the week of December 12-16,1988, plant tests were run on an Exxon Chemical propylene-propane splitter at Baytown, TX. Several typea of tests were conducted. Figure 3 compares the plant experimental data with the model predictions for a dual pulse on distillate flow. Figure 4 shows the same comparison for an autotune (ATV test on the xD-D loop). It should be noted that the switching times in both the model and the plant were determined by XD croesing the set point, i.e., the input to the model was not the sequence of pulses in distillate flow found from the plant data. Thus the ATV
Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 711 9
0
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?
e T
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am.0
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Figure 10. Comparison of direct and indirect schemes for +5% feed composition disturbance (solid line, direct; dotted line, indirect).
test was a particularly sensitive evaluation of model fidelity. These results demonstrated that the dynamic mathematical model was accurate enough to use for comparative studies of alternative control structures. C. Controller Tuning. Linear transfer function models were obtained by using the AW method (Luyben, 1987) for the basic structures (D-S-V and D-NS-V) and suitable transformations to obtain the transfer functions for the D-S-B structures. Details are given in Papastathopoulou (1990). Conventional diagonal controller structures were used, and controllers were tuned by using the BLT procedure (Luyben, 1986). Tuning the schemes where the sidestream drawoff location (NS)was used as a manipulated variable was initially thought to be straightforward because the blending of two streams should be essentially instantaneous except for the 6-min dead time from the composition analyzer. Therefore we assumed that the transfer function relating xs to the sidestream composition controller output signal CO would be just the steady-state gain and the 6-min dead time. However, when the sidestream composition controller was designed with this assumed transfer function, the closed-loop response of this loop ( x g C 0 ) was very oscillatory. After considerable investigation, the source of the problem was discovered. Figure 5 shows the open-loop response of x s to a step change in CO. There is a large
instantaneous jump in xs and a slow decay to the lower steady-state value. This is the type of response associated with a derivative unit. It results from the initial blending of the two streams with different compositions, followed by a slow shift in the compositions on the liquid drawoff trays. Therefore, the open-loop transfer function relating x s and CO was not just a dead time and gain but had to contain a lead/lag element. XQ 0.076(3253 + l)e* -= (2) co 81s + 1 Since the sidestream composition loop was much faster than the distillate and bottoms composition loops, it was tuned independently. A reset time equal to half the dead time was selected (T, = 3 min), and the controller gain was found that gave a +2-dB maximum closed-loop log modulus for this SISO loop (K,= 0.836). With these settings held constant in the sidestream composition loop, the other two controllers were tuned, by using the BLT procedure, considering the entire 3 X 3 multivariable process and designing for +6-dB maximum closed-loop log modulus
L?.
Figure 6 shows the dependence of LF- on the detuning factor FBLPNote the very interesting multiple solutions, i.e., there are three values of the detuning factor that give a +6-dB maximum closed-loop log modulus. This phe-
712 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991
nomenon has not been reported before in the literature. Naturally the desired solution is the one that gives the smallest detuning factor. Table V gives the tuning constanta for various control structures studied. It is interesting ta compare the detuning factors of the 3 X 3 cases (D-S-V and P-NS-V) with the corresponding 2 X 2 m e a (D-S and D - N S ) . We would expect that the 2 X 2 systems would be easier to control and therefore would require smaller detuning factors (higher gains and smaller reset times). This was found to be true in the D-NS-V versus D-NS case, but it was not true in the D-S-V versus D-S case.
Results
A. Three Produat Composition Control. Figures 7 and 8 give the closed-loop response of the nonlinear rigorous model of the sidestream column for a 5 % step decrease in feed composition using three control schemes: 0-S-V, D-S-B, and D-NS-V. Note that the distillate compositions are plotted as ppm deviations. All three schemes do a fairly good job in controlling the column, but the D-NS-V does the best job in holding the sidestream composition constant. This structure also has the ad. vantage of holding the sidestream flow rate constant. Therefore, if tight control of the sidestream is important, the D-NS-V scheme is recommended. Other types of disturbances were introduced into the system (including feed rate and sidestream flow rate), and the responses of the various control schemes were similar. The D-NS-V structure was found to work well for all disturbances. The structure was also found to be robust for changes in the purity level of xD. B. Two-Product Composition Control. Figure 9 compares the performance of the D-S and D-NS schemes for a step change in feed composition. Bottoms composition is uncontrolled since vapor boilup is held constant. The D-NS structure does a better job in holding the sidestream composition constant, and at the same time it also holds the sidestream flow rate constant. This could be a distinct advantage of the D-NS structure in some applications. C. One-Product Composition Control and Sidestream Flow Rate Control. In some sidestream columns it may be desirable to control only the composition and the flow rate of the sidestream. Two methods for accomplishing this were studied (1)the direct method, where S is set on flow control and x s is controlled with D; and (2) the indirect method, where S is manipulated to control x s and D is manipulated to control xD,but another controller is added that changes the setpoint of the 3tD controller to slowly bring the sidestream flow rate back to the desired value (this approach is similar to Shinskey’s “valve position control”). Clearly the controller tuning of the direct approach was much more straightforward than the indirect since this is just a single SISO loop. The indirect approach required simultaneous tuning of the XD and x s loops followed by a SISO analysis of the valve position controller with the other two loops closed. Figure 10 compares the performances of these two schemes. The indirect approach holds x s more constant but a t the price of varying sidestream flow rate. It should be emphasized that sidestream drawoff location cannot be used to control the sidestream composition with both distillate D and sidestream S flow rates held constant. Fixing both of these flows makes it essentially impossible to control any composition in the column because the feed split has been fixed.
Conclusions Several control s t r u m have been evaluated for a large binary sidestream distillation column separating propylene and propane. The most effective stru&ure depends on the control objectives. If dietillata composition control has the highest priority, the D-S-V structure is the best. If sidestream composition control has the highest priority, the D-NS-V structure is best. This Structure also can be used to provide a constant or on-demand flow rate of the sidestream. Control of bottom composition is often not required in these systems, particularly when a motordriven vapor recompression system is used, because the economics favor maximumI propylene recovery. Acknowledgment We thank Ken Emigholz, Dave Hokanson, and Alan Schaefer of h o n Chemical for their invaluable assistance in obtaining the plant data and for their many helpful suggestions and comments during the c o w of this work. Financial assistance for H.S.P.from Exxon Chemical is also gratefully acknowledged. Nomenclature
A = matrix in JEC criterion B = bottoms flow rate ((lb-mol)/h) BR = boil-up ratio = V / B CN = condition number CO = controller output from sideet” composition controller D = diagonal matrix in JEC criterion D = distillate flow rate ((lb-mol)/h) d = column diameter (ft) DV = distillate-to-vapor ratio = D / V F = feed flow rate ((lb-mol)/h) FaLT= detuning factor in BLT procedure I = identity matrix JEC = Jacobi eigenvalue criterion K = matrix of steady-state gains K, = controller gain K, = ultimate gain hi. = elementa of steady-state gain matrix = maximum closed-loop log modulus (dB) mi = manipulated variable MB = holdup in column base (lb-mol) MD = holdup in reflux drum (lb-mol) M, = steady-state holdup on each tray (lb-mol) MRI = Morari resiliency index NF = feed tray number (numbered from base) NI = Niederlinski index NS = sidestream drawoff tray number or location NT = total number of trays P = pressure at top of column (psia) R = reflux flow rate ((lb-mol)/h) RGA = relative gain array RR = reflux ratio = R / D RV = reflux-to-vapor ratio = R / V S = sidestream flow rate ((lb-mol)/h) SV = sidestream-to-vapor ratio = S / V V = vapor boilup ((lb-mol)/h) xB =composition of bottoms product (mole fraction of propylene) xD = composition of distillate product (mole fraction of pro-
Lb
pylene) x s = composition of sidestream product (mole fraction of
propylene) wH = weir height (in.) wL = weir length (ft) z = feed composition (mole fraction of propylene) ANS = number of trays separating the sidestream drawoffs
Ind. Eng. Chem. Res. 1991,30,713-721 Xij = diagonal elements of RGA q = controller integral time (min) w,
= ultimate frequency Registry No. Propylene, 115-07-1;propane, 74-98-6.
Literature Cited Alatiqi, I. M.; Luyben, W. L. Control of a Complex Sidestream Column/Stripper Distillation Configuration. Ind. Eng. Chem. Process Des. Dev. 1986,25,762-767. Buckley, P. S. Controls for Sidestream Drawoff Columns. Chem. Eng. Prog. 1969,65 (5),45-51. Doukas, N.; Luyben, W. L. Control of an Energy-Conserving Prefractionator/Sidestream Column Distillation System. Ind. Eng. Chem. Process Des. Dev. 1980,20,147-153. Finco, M. V.; Luyben, W.L.; Polleck, R.E. Control of Distillation Columns with Low Relative Volatilities. Ind. Eng. Chem. Res. 1989,28,75-83. Luyben, W.L. Ten Schemes to Control Distillation Columns with Sidestream Drawoff. ISA J. 1966,13(7),37-42. Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng.Chem. Process Des. Dev. 1986, 25,654-660. Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987, 26, 2490-2495. Mijares, G.; Cole, J. D.; Naugle, N. W.; Preisig, H. A.; Holland, C. D. A New Criterion for the Pairing of Control and Manipulated Variables. AIChE J. 1986,32, 1439-1449.
713
Mosler, H. A. Control of Sidestream and Energy Conservation Distillation Towers. AIChE Workshop on Industrial Process Control, Tampa, FL, 1974. Ogunnaike, B. A,; Lemaire, J. P.; Morari, M.; Ray, W. H. Advanced Multivariable Control of a Pilot-plant Distillation Column. AIChE J . 1983,29,632-640. Papastathopoulou, H. S. Control of Binary Sidestream Distillation Columns with Varying Control Objectives and Constrainta. Ph.D. Thesis, Lehigh University, qethlehem, PA, 1990. Papastathopoulou, H. S.;Luyhn, W. L. A New Method for the Derivation of Steadystate Gains for Multivariable Processes. I d . Eng. Chem. Res. 1990a,29,366-369. Papastathopoulou, H. S.; Luyben, W. L. Potential Pitfalls in Ratio Control Schemes. Ind. Eng. Chem. Res. 199Ob,29, 2044-2053. Papastathopoulou, H. S.; Luyben, W. L. Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure. Ind. Eng. Chem. Res. 199Oc,29,1859-1868. Skogestad, S.; Jacobsen, E. W.; Morari, M. Inadequacy of SteadyState Analysis for Feedback Control: Distillate-Bottom Control of Distillation Columns. Ind. Eng. Chem. Res. 1990, 29, 2339-2346. Tyreus, B.; Luyben, W. L. Control of a Binary Distillation Column with Sidestream Drawoff. Ind. Eng. Chem. Process Des. Dev. 1975, 14,391-398.
Received for review June 27, 1990 Revised manllscript received September 19, 1990 Accepted October 9,1990
SEPARATIONS Multicomponent Batch Distillation Column Design Urmila M.Diwekar*s+ a n d K.P . Madhavan CAD Centre, IIT Powai, Bombay 400076, India Batch distillation is characterized as a system that is difficult to design because compositions are changing continuously with time. The models reported in the literature and used in the simulators like PROCESS and BATCHFRAC, etc., are too complex to use for obtaining optimal design because of high computational time and large memory requirement. In this work we are presenting a short-cut method for design of multicomponent batch distillation columns, operating under variable reflux and constant reflux conditions. The method is essentially a modification of the short-cut method widely used in the design of continuous multicomponent columns. The technique has been tested for both binary and multicomponent systems, and the results compare favorably with the rigorous methods of design. The method is very efficient and can be used for preliminary design and analysis of batch columns. The model used in this method has a number of tuning parameters that can be used for model adaptation. The main features of the short-cut method include lower computational time (which is independent of the number of plates in the column), lower memory requirements, and its adaptability to design. 1. Introduction
The design of a batch distillation column is much more complex in comparison with that of a continuous distillation column as it requires consideration of unsteady-state behavior. The complexity of the problem increases with the number of components in multicomponent systems. The batch distillation column can be designed by taking into account two possible modes of operation: (1)variable Current address: Department of Engineering and Public Policy, Carnegie Mellon University, Pittsburgh, PA 15213.
reflux and constant product composition of (a) all components and (b) one component; (2) constant reflux and variable product composition. The design methods developed to date have some disadvantages. Most of the approaches are derived from the classical McCabe Thiele’s method, which, for the batch distillation case, calls for an iterative solution procedure. Chao (1954)proposed a short-cut method based on the absorption separation factor method. Rose et al. (1950) and Houtman and Hussain (1956)have used the poleheight concept for design, which depends on the width of the intermediate fraction and is applicable only to sharp separations.
0888-5885/91/2630-07 13$02.50/0 0 1991 American Chemical Society
