Ind. Eng. Chem. Res. 1988, 27, 99-104 r(s) = set point [SI = solvent concentration, mol/L TR = reactor temperature, "C Tw = jacket temperature, "C u(s) = plant input U = overall heat-transfer coefficient, kcal/(m2 h "C) Uo = parameter appearing in the equation of U , eq A-14 X ( t f )= conversion of final polymer product y ( s ) = plant output Greek Symbols = ith moment of the concentration of dead polymer Xi = ith moment of the concentration of polymer radical
K~
Superscripts
= estimated value
* = desired value
Appendix. Mathematical Description of Styrene Suspension Polymerization Batch Reactor The Material Balances:
and
99
DM = 924.0 - 0.981(TR - 273.15)
(A-10)
Dp = 1084.0 - 0.605(TR .- 273.15)
(A-11)
X is the monomer conversion which can be given by X = (NMO - V[MI)/NMO
(A-12)
The Heat Balance: D,c,d(VTR)/dt = VAH(k,[M]Xo) - UA(TR - Tw) (A-13) where D,, cp, AH, U , A, and Tw are density of polymer, heat capacity of reaction mixture, heat of polymerization reaction, overall heat-transfer coefficient, heat-transfer area, and jacket temperature, respectively. The overall heat-transfer coefficient U is assumed to written by U = Uo- 300 exp(1 - 1/X) (A-14)
d(V[I])/dt = kd[I]V
(A-1)
That is, U decreases with monomer conversion X . Numerical values of reaction parameters and conditions used for the simulations are listed in Table 11. Registry No. Styrene, 100-42-5; polystyrene, 9003-53-6.
d ( V[M]) /dt = -k,[M] XoV
(-4-2)
Literature Cited
dpo/dt = k,XozV/2
(A-3)
dpl/dt = k,XoXiV
(A-4)
Arnold, K.; Johnson, A. F.; Ramsay, J . Proc. IFAC PRPP 4 Autom. Ghent, Belg. 1980, 359-367. Couso, D. A.; Alassia, L. M.; Meira, G. R. J . Appl. Polym. Sci. 1985, 30, 3249-3265. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Deu. 1982, 21, 308-323. Hicks, J.; Mohan, A.; Ray, W. H. Can. J. Chem. Eng. 1969, 47, 590-597. Hoffman, R. F.; Schreiber, S.; Rosen, G. Ind. Eng. Chem. 1964,56, 51. Laundau, Y. D. Adaptive Control; Marcel Dekker: New York, 1979. Louie, B. R.; Soong, D. S. J . Appl. Polym. Sci. 1985, 30, 3707-3749. Martin, J. R.; Idunes, R. W.; Johnson, J. F. Polym. Eng. Sci. 1982, 22, 205-228. Nishimura. H.: Yokovama. F. Kaeaku KoPaku 1968. 32. 601-607. Osakada, K.; Fan, L."T. J.'Appl. kolym. f c i . 1970, 1'4, 3065. Sachs, M. E.; Lee, S.; Biesenberger, J. A. Chem. Eng. Sci. 1973,28,
dpz/dt = kk(XoX2
+ X1X1)V
(A-5)
where Xo, X1, and Xz are the zero-, first-, and second-order moments of the growing polymer chains. By use of the quasi-steady-state assumption, these are expressed by = (2fkd[I]/kt,)1'2 = (ktJo Xz =
A1
+ kp[Ml)/k,
+ 2k,[MIX,/kk/XO
(-4-6) (A-7) (-4-8)
The volume of the reactor mixture V [m3] can be written by V = ( N M o / D M )( ~(1- DM/DP)X) (A-9) where N M O [mol] is the amount of monomer initially fed into the reactor, DM and Dp [mol/m3] are densities of monomer and polymer which are given by
241.
Tadmor, Z.; Biesenberger, J. A. Znd. Eng. Chem. Fundam. 1966,5, 336. Takamatsu, T.; Shioya, S.; Okada, Y. Proc. IFAC Workshop Adapt. Control Chem. Proc., Frankfurt, FRG 1985, 120-125. Receiued for review December 31, 1986 Accepted August 24, 1987
Comparison of the Dynamic Performances of Three Heat-Integrated Distillation Configurations Teh-ping Chiangt and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
T h e closed-loop load responses of three types of heat-integrated distillation configurations were compared quantitatively with the response of a conventional single column. The methanol/water separation with low product purities (96/4) was used as the specific distillation system. The "LSR" (light-split/reverse) configuration was found to be dynamically almost as good as the single column while consuming 40% less energy. Conventional multiloop single-input-single-output (SISO) controllers gave effective control of the 3 X 3 multivariable LSR process. There have been many studies of energy-conserving process configurations from the steady-state design point of view. Since most energy-conserving processes involve Present address: Union Chemical Laboratories, Industrial Technology Research Institute, Hsin-chu, Taiwan, Republic of China. 0888-588518812627-0099$01.50/0
more interrelated units and multivariable interacting control loops, it is important that their dynamic operability and transient performance also be considered. There have been a few studies of the dynamics of energy-conserving systems. Tyreus and Luyben (1975,1976) studied the control of side-stream and heat-integrated columns for binary separations. Doukas and Luyben (1978, 0 1988 American Chemical Society
100 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 Table I. Steady-State Specifications: Single-Column Base Case feed rate, mol/min 45000 0.5 feed composition, mole fraction distillate rate, mol/min 22500 distillate composition, mole fraction 0.96 bottoms rate, mol/min 22500 0.04 bottoms composition, mole fraction operating pressure, mmHg 760 re1 volatility, a 7.58-2.45 no. of trays 12 5 feed tray location reflux ratio 0.831 36.9 reboiler heat duty, lo4 kcal/min reboiler temp, "C 94.5 64.9 reflux drum temp, "C column diameter, m 3.2
1981) explored control systems for side-stream columns and for a prefractionator configuration for separating ternary mixtures. Mosler (1979) gave control schemes for heat-integrated, side-stream, and vapor-recompression columns. Ryskamp (1980), Lenhoff and Morari (1982), Frey et al. (1984), and Abu-Eishah and Luyben (1985) looked a t alternative control schemes for heat-integrated columns. Ogunnaike et al. (1983) studied the control of a side-stream column. Elaahi and Luyben (1985) explored the control of a complex configuration of columns for the separation of quaternary mixtures. Alatiqi and Luyben (1986) compared the dynamic response of a conventional two-column configuration with that of a complex side-stream/stripper configuration. Levien and Morari (1987) experimentally studied a side-stream column with a side rectifier, exploring resiliency and internal model control. The purpose of this paper is to present a quantitative comparison of the dynamic performances of a conventional single distillation column with several alternative heatintegrated configurations. Details of the steady-state design procedure for the various configurations are given by Chiang and Luyben (1983).
Systems Studied A. Single Column. A conventional single distillation column was designed on the basis of 1.1times the minimum reflux ratio. Table I gives the steady-state design parameters. The chemical system methanol/water was used a t atmospheric pressure. Low-purity products (96 mol 70methanol distillate and 4 mol % methanol bottoms) were assumed. The energy consumption of the single column was 36.9 X lo4 kcal/min. B. Feed-Split (FS) Heat-Integrated Columns. Figure 1shows the FS configuration. The total feed is split almost equally between the two columns (to satisfy heat balances in both columns). The columns operate a t different pressures. The high-pressure column runs at 3900 mmHg pressure so that its overhead vapor at 112.7 "C can be used t o reboil the low-pressure column whose base temperature is 94.5 "C. Table I1 gives steady-state design parameters for the FS configuration. Specification products are produced at the ends of both columns. Both the high- and low-pressure columns were designed for 1.1 times their respective minimum reflux ratios. The low-pressure column had 12 trays and the high-pressure column had 15 trays. These same number of trays were used in the other heat-integrated configurations. The feed flow rates to the two columns were adjusted to balance the heat removal rate in the high-pressure column with the heat input rate in the low-pressure column. The feed split will be changed by the control system as feed composition changes in order
I
#b / / / / //
Y Figure 1. Feed-split configuration. Table 11. Steady-State Specification: FS column 1 feed rate, mol/min 22506 feed composition, mole fraction 0.5 distillate rate, mol/min 11253 distillate composition, mole fraction 0.96 bottoms rate, mol/min 11253 bottoms composition, mole fraction 0.04 operating pressure, mmHg 3900 re1 volatility, a 6.08-1.92 no. of trays 15 feed tray location 5 reflux ratio 1.173 22.7 reboiler heat duty, lo4 kcal/min reboiler temp, "C 146.7 112.7 reflux drum temp, "C column diameter, m 1.8
column 2 22494 0.5 11247 0.96 11247 0.04 760 7.58-2.45 12 5 0.831 94.5 64.9 2.3
to maintain the energy balances. Note that the energy consumption of the heat-integrated FS configuration is only 22.7 X lo4 kcal/min compared to the 36.9 X lo4 kcal/min needed in the single column. However, also note that the base temperatures in the two configurations are significantly different: 94.5 and 146.7 "C. Thus, higher pressure steam must be used in the FS heat-integrated configuration. C. Light-Split/Forward (LSF) Heat-Integrated Columns. Figure 2 shows the LSF configuration in which approximately half of the light component is removed overhead from the high-pressure column as specification methanol product. The bottoms stream from the highpressure column, which is 35.6 mol % methanol, is fed into the low-pressure column where the rest of the methanol is taken overhead as distillate product. The bottoms stream from the low-pressure column is the water product. The 35.6 mol % bottoms composition from the highpressure column was set by the requirement that the heat removal in the high-pressure column must match the heat addition rate in the low-pressure column. Heat integration is in the same direction as process flows. Table I11 gives steady-state design parameters for the LSF scheme. The energy consumption is slightly higher than in the FS configuration, but the base temperature in the high-pressure column is only 125.6 O C . This means that lower pressure steam can be used in the LSF configuration than in the FS configuration. The two configurations have the same pressures in the high-pressure columns.
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 101
3: '
1
71
H
c-3
.
.
Figure 2. Light-split/forward configuration.
Figure 3. Light-split/reverse configuration.
Table 111. Steady-State Specifications: LSF column 1 feed rate, mol/min 45000 0.5 feed composition, mole fraction 10737 distillate rate, mol/min distillate composition, mole fraction 0.96 bottoms rate, mol/min 34263 bottoms composition, mole fraction 0.356 operating pressure, mmHg 3900 re1 volatility, a 6.08-1.92 no. of trays 15 4 feed tray location 1.1 reflux ratio 24.1 reboiler heat duty, lo4 kcal/min reboiler temp, "C 125.6 112.7 reflux drum temp, "C 2.0 column diameter, m
Table V. Pairings of Controlled and Manipulated Variables for t h e Four Schemes manipulated controlled variable variable single column XD R
~
Table IV. Steads-State SDecifications: LSR column 1 feed rate, mol/min 45000 feed composition, mole fraction 0.5 11612 distillate rate, mol/min distillate composition, mole fraction 0.96 33388 bottoms rate, mol/min 0.34 bottoms composition, mole fraction 760 operating pressure, mmHg 7.58-2.45 re1 volatility, a 12 no. of trays 3 feed tray location 0.772 reflux ratio reboiler heat duty, lo4 kcal/min reboiler temp, "C 76.4 reflux drum temp, "C 64.9 column diameter, m 2.3
column 2 34263 0.356 11763 0.96 22500 0.04 760 7.58-2.45 12 3 1.0
FS
D. Light-Split/Reverse (LSR) Heat-Integrated Columns. Figure 3 shows the LSR configuration. The first column is the low-pressure column, and the second column is the high-pressure column. Table IV gives steady-state design parameters. The bottoms composition in the low-pressure column is 34 mol % methanol as set by the requirement that the heat loads must be balanced. Note that the pressure in the high-pressure column is only 2280 mmHg, compared with 3900 mmHg required in the FS and LSF configurations. This implies that the LSR scheme may be the best for those chemical systems in which relative volatilities increase significantly with de-
QR
XDH
RH QRH RL
XBH XDL XBL
LSF
XDH XDL ~ B L
94.5 64.9 2.4
column 2 33388 0.34 10888 0.96 22500 0.04 2280 6.34-2.2 15 5 1.24 22.6 128.1 96.0 2.0
XB
LSR
FHIFL RH RL QRH
XDH
RL RH
XBH
QRH
XDL
creasing pressure. The energy consumption is only 22.6 X lo4 kcal/min, the lowest of all the heat-integrated systems, for the specific methanol/water system studied. The base temperature in the high-pressure column is 128.1 "C, which is slightly higher than in the LSF case (125.6 "C) but significantly lower than in the FS case (146.7
"C). Control A. Controlled Variables. The four systems studied all represent multivariable systems. Even the single column is a 2 X 2 system since we wish to control both overhead xD and bottoms xB compositions. The FS configuration is a 4 x 4 system since there are two compositions in each column: high-pressure column (xDH and xBH) and low-pressure column (xDLand xBL). The LSF and LSR configurations are 3 x 3 systems: two X D ' S and one xB. Composition analyzer deadtimes of 1.2 and 6 min were used in the study. B. Manipulated Variables. Many possible choices of manipulated variables exist for each scheme. Since the methanol/water system has a low reflux ratio, manipulation of reflux flow rate R and heat input QR was selected on the basis of engineering judgment to control product compositions. Base level was controlled by bottoms flow rate and reflux drum level by distillate flow rate. In all the heat-integrated configurations, there are two reflux flow rates and one heat input to manipulate. In the
102 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 Table VI. Process Transfer Functions A. Conventional Single Column -4.44 1) (15.5s+ 1)(2s + -33.4 1) (23s+ l)(s + 1) J
3.6
L
-
1 +!
L
B. FS
t.45 (14s+ 1)(4s + 1) 17.3e4gs (17s+ 1)(0.5s+ 1) 0.22~' 2s (17.5s+ 1)(4s + 1) 1.82~~ (21s+ l)(s + 1)
-7.4 (16s+ 1)(4s + 1) -41 (21s + l)(s + 1) -4.66 (13s+ 1)(4s + 1) -34.5 (20s + l ) ( s+ 1)
0
0.35 (25.7s+ 1)(2s + 1) 0 9.2e43s 20s + 1 3.6 0.042(78.7s+ 1) (13s+ 1)(4s + 1) (21s+ 1)(11.6s+ 1)(3s + 1) 12.2e499~ -6.92e4 (18.5s+ l)(s + 1) 20s + 1
19.7e433S i25s + l)(s + 1) 0.75e-55 (15.6s+ 1)(2s + 1)) 16.61e4& (25s + 1)(2s +ii
RL
-
C. LSF
4.96 (9.6s+ 1)* 6.39e-'j5 (16.7s+ 1)(6s + l)(s+l) 0.29(-12s+ l)e-5s (13s + 1)(6s + 1)3
-7.5 0 (15.4s+ 1)(3s + 1) -37.4 (17s+ 1)(3s + 1) -10.92 (21s+ 1)(6s + 1)
0.98~~~
5.55
D. LSR L14.3~ s4::': + 5.6e-' 5s (14.5s+ 1)(6s + 0.27e-?' (9.5s+ 1 ) 3
=l
-4.6 1) (14s+ 1)(4s + 1s -36.4 1) (15.4s+ 1)(4s + 1 ) -9 (16s + 1)(4s + 1)
+
2.44~~~ (38.5s+ 1)(5s + 1)*
J
7.56e4.9S (12s+ l)(s + 1) 5.51 (13.5s+ 1)(4s + 1)
r
0.414(-8~+ 1) (6s+ l)3(2s+ 1)
1
(10s-0.25e-"55 + 1)(1Os 1)
+
1
Table VII. RGA and NI Values RGA
XD XB
-10
XDH XBH XDL XBL
-
-LO
-.
..i3
*
7
-
~~-
-1
6
-. 2
-08
-04
3
04
XDH XBL
.cg
F-eqdenc)
XDL
Figure 4. Morari resiliency indices.
FS configuration, the feed split (FH/FL) can also be adjusted to give the required fourth manipulated variable. The pairings of controlled and manipulate variables for the four schemes are summarized in Table V. C. Transfer Functions. Table VI gives the transfer functions for the four systems derived from pulse testing dynamic models of each system. These transfer functions fit the Bode plots fairly well down to phase angles of -180 deg. Table VI1 gives RGA values and Niederlinski indices (NI) for all systems with the variable pairings chosen. The results indicate reasonable pairings. Figure 4 gives the frequency-dependent Morari resiliency indices (MRI's) for the four systems. The MRI is the minimum singular value of the plant open-loop transfer function matrix. As used by Yu and Luyben (1986), the MRI gives a measure of the inherent controllability of the system. The larger the MRI value, the more controllable the system. Surprisingly, the LSR configuration has a somewhat larger MRI value than the single column. The FS has the lowest MRI value, indicating that its control will be the most difficult.
XDL
XBH XDH
R QR RH QRH RL FHIFL A. Conventional Single Column: NI = 0.55 1.8 0.8 0.8 1.8 B. FS: NI = 0.78 2.06 -0.96 -1.0 1.28 0.04 -0.57 -0.1 1.25 C. LSF: NI 1.63 -0.67 0.04
0 0
1.54 -0.54
-0.1 0.72 -0.01 0.39
= 0.38
-0.63 2.6 -0.97
0 -0.93 1.93
D. LSR: NI = 0.49 -0.02 -0.33 1.35 -0.67 2.05 -0.38 0.03 1.69 -0.72
D. Controller Tuning. Two methods were used to tune the multiloop SISO controllers in all systems: BLT tuning and SVT tuning. The BLT method, presented by Luyben (1986), uses the closeness of a multivariable Nyquist plot to the (-1,O) point as a closed-loop criterion and detunes equally all loops from the Ziegler-Nichols settings (both gains and reset times). The SVT method, developed by Chiang (1985) and based on the work of Doyle and Stein (1981) and Tyreus (1984), uses a -12-db (db = decibel) minimum singular value of the closed-loop transfer function matrix [I + (GB)-']. G is the matrix of open-loop plant transfer functions, and B is the diagonal matrix of SISO feedback P I controllers. The singular values of a matrix are measures of the size of the matrix. The smallest singular value of the [I + (GB)-l] matrix is similar to the reciprocal of the maximum closed-loop log modulus of a SISO system. Therefore,
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 103 Table VIII. Tuning Constants: Values of Gain and Reset ( K C / q ) analyzer deadtime, min XDH single column 1.2 ZN BLT SVT 6 ZN BLT SVT 1.2 ZN 1.6/10 FS BLT 0.36/45 SVT 0.34/24 6 ZN 0.4/26 BLT 0.16/66 SVT 0.18/35 1.2 ZN LSF 1.5/12 BLT 0.37/49 SVT 0.62/30 6 ZN 0.35/30 BLT 0.16/64 SVT 0.24/42 1.2 ZN 1.3/11 LSR BLT 0.32/41 SVT 0.66/24 ZN BLT SVT
6
XBH
0.31/26 0.15/53 0.25/35
XDL
XBL
1.8/10 0.73125 0.97 j 2 2
-0.32/6.5 -0.13116 -0.2 2) 15
0.45/26 0.26/45 0.32/33
-0.083 / 21 -0.05/36 -0.064/32
-0.24/6.5 -0.054/29 -0.135/14
1.87/10 0.42/45 0.29/22
-1.17/6 -0.26/ 27 -0.11/12
-0.062/21 -0.024/53 -0.029/32
0.46/26 0.18/66 0.22/35
-0.35/20 -0.14/51 -0.098/28
1.39/10 0.34/43 0.72/24
-0.23/10 -0.06/43 -0.11/25
0.34/26 0.16/ 55 0.27/35
-0.055/27 -0.026/57 -0.04/38
-0.22/11 -0.056/41 -0.10/25
1.8/11 0.46/41 0.86/24
-0.052/26 -0.026/53 -0.038/38
0.43/26 0.21/53 0.31/35
0.043
f Y 0.042
a
-. I
0.041
V 2b
I
-2
. , 6
. . 2
-
I
6
I
I
!
!
0.04
!
- 1 L
Loa i.ea_e-cy
Figure 5. Typical maximum and minimum singular values of SVT.
0’07
0.065
designing for a -12-db minumum singular value in a multivariable system is analogous to designing for a +2-db maximum closed-loop log modulus in a SISO system. Figure 5 shows typical maximum and minimum singular values as functions of frequency. The lowest magnitude of the minimum singular value is about -18 db for the controller settings used in the figure. Chiang’s SVT tuning procedure is outlined below. (1)Use Buckley’s procedure (Luyben, 1973) to calculate gains and reset times that give +2-db maximum closedloop log moduli for each individual loop, using just the diagonal elements of the plant transfer function matrix (GJ. (2) Reduce the gains to yield the desired -12-db minimum singular value. The gains can be reduced by the same detuning factor or by a different detuning factor for each loop. Table VI11 gives BLT and SVT controller settings for all configurations. Both tuning methods gave stable responses, but SVT results were somewhat tighter and less conservative, as illustrated in Figure 6. Product compositions from both columns of the FS configuration are shown for a feed composition disturbance (z changed from 0.50 to 0.55) with a 1.2-min analyzer deadtime. The numbers “1”and “2” refer to the column number in the
I
0.06
0.055 0
x
0.05
0.045
0.04
0.035 0
20
40
60
BO
Figure 6. Comparison of BLT and SVT tuning for FS configuration: (a, top) distillate composition; (b, bottom) bottoms composition.
FS configuration (1= high pressure and 2 = low pressure). Results and Conclusions Figure 7 compares the feed composition load responses of the four configurations with an analyzer sampling deadtime of 6 min. The xD values are the average values of the two overhead distillate products from the two columns in the heat-integrated systems. “FS 1”and “FS 2” on the xB plot refer to the bottoms from the high- and low-pressure columns.
104 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
I
0039
+ 0
,
I
20
I
I
40
,
FH = feed flow rate to high-pressure column feed flow rate to low-pressure column FS = feed split configuration G = open-loop plant transfer function matrix G,, = diagonal element of G K , = feedback controller gain LSF = light-split/forward configuration LSR = light-split/reverse configuration MRI = Morari resiliency index NI = Niederlinski index QRH = reboiler heat input to high-pressure column RGA = relative gain array RH = reflux flow rate in high-pressure column RL = reflux flow rate in low-pressure column SVT = singular value tuning ZB = bottoms composition, mole fraction methanol xBH = bottoms composition in high-pressure column xBL = bottoms composition in low-pressure column XD = distillate composition, mole fraction methanol xDH = distillate composition in high-pressure column xDL = distillate composition in low-pressure column z = feed composition, mole fraction methanol ZN = Ziegler-Nichols settings FL =
n
, 60
,
, r,$g
I
(,,nl,OO
I
1
, 120
I
, 140
,
I
I
160
I 180
Greek Symbol T~ = reset time, min OD3
0 32
I
7
L 0
20
40
60
(m,$OO
120
140
L
160
180
Figure 7. Comparison of alternative configurations: (a, top) distillate composition; (b, bottom) bottoms composition.
The response of the FS configuration is the worst in terms of departures from set points. The response of the single column is the best. The responses of the LSR and LSF schemes are fairly similar and are somewhat worse than the single column but still quite stable and acceptable. Since the LSR uses less energy at steady state, it is the best configuration for this specific system. Chiang (1985) also compared steady-state designs for other chemical systems (benzene/toluene and isobutaneln-butane) over a range of feed compositions. There were only small differences among the heat-integrated configurations in terms of steady-state energy consumptions. We would expect that the dynamic comparisons would &o be similar since the basic structure of the systems remains the same. Low product purities were used in this study to avoid the problems of nonlinearity. Good transfer functions could be derived for all configurations, and fair comparisons could be made. We expect that the comparisons of dynamic responses would not be changed significantly if product purities were increased. Luyben (1987) has recently developed an effective procedure for obtaining transfer functions for quite high-purity columns. We plan to compare high-purity systems in future work.
Nomenclature B = feedback controller matrix BLT = biggest log modulus tuning
Literature Cited Abu-Eishah, S. I.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 132. Alatiqi, I. M.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 762. Chiang, T. P. Ph.D. Thesis, Lehigh University, Bethlehem, PA, 1985. Chiang, T. P . ; Luyben, W. L. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 175. Doukas, N.; Luyben, W. L. Instrum. Technol. 1978,25,43. Doukas, N.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 147. Doyle, J. C.; Stein, G. IEEE Trans. Autom. Control 1981, AC-26, 368. Elaahi, A.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 368. Frey, R. M.; Doherty, M. F; Douglas, J. M.; Malone, M. F. Znd. Eng. Chem. Process Des. Deu. 1984, 23, 483. Lenhoff, A. M.; Morari, M. Chem. Eng. Sei. 1982, 37, 245. Levien, K. L.; Morari, M. AIChE J. 1987,33, 83. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1973. Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 654. Luyben, W. L. "Derivation of Transfer Functions for Highly Nonlinear Distillation Columns". Ind. Eng. Chem. Res. 1987, in press. Mosler, H. A. Proc. Workshop on Indust. Proc. Control, Tampa, FL, 1979, p 76. Ogunnaike, B. A,; Lemaire, J. P.; Morari, M. AIChE J. 1983,29,632. Ryskamp, C. J. Hydrocarbon Process. 1980,59(6), 51. Tyreus, B. D. Paper presented at the 1984 Distillation Control Short Course, Lehigh University, Bethlehem, PA, 1984. Tyreus, B. D.; Luyben, W. L. Hydrocarbon Process. 1975,54(7), 93. Tyreus, B. D.; Luyben, W. L. Chem. Eng. Prog. 1976, 72, 59. Yu, C. C.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 498. Received f o r review January 20, 1987 Revised manuscript received August 11, 1987 Accepted September 24, 1987
