266
Ind. Eng. Chem. Process Des. Dev.
k = specific velocity per volume, l/min atm" Mo = initial mass of coal, g M = mass of coal at some instance, g P = pressure, atm R = constant of gases, cal/mol K Ro = mean radius of the particles, cm r = reaction rate, mol/min cm2 t = time of reaction, min T = temperature, "C or K U = fraction of moisture of coal (weight) V = fraction of volatile matter (weight) X = conversion of solid 2 = fraction of ash in coal (weight) bs = dry basis T = time for total conversion, min Literature Cited
1982,21, 266-272
Castellan. L. M.S. Thesis, Federal University of Rio de Janeiro, COPE, 1978. Chan, E. M.; Papic, M. M. Can. J . Chem. Eng. 1976, 5 4 , 645. Gas Development Corporation Report, Chicago, Sd.15 p, 1976. Goring, G. E.; Curran, G. P.; Zielke, C. W.; Gorin, E. Ind. Eng. Chem. 1953, 4 5 , 2586. Hunt, B. E.; Mori, S.;Katz, S.; Peck, R. E. Ind. Eng. Chem. 1953, 45, 677. Jensen, 0. A. Ind. Eng. Chem. Frocess Des. Dev. 1975, 14, 308. Johnson, J. L. Adv. Chem. Ser. 1974, 131, 145. Kiei, H. E.; Sahaglan, J.; Sundestrom, D. W. Ind. Eng. Chem. Process Des. Dev. 1975, 14. 470. Levenspiel, 0. "Chemical Reaction Engineering"; Wiley: New York, 1967. May, W. G.; Mueller, R. H.; Sweetser, S. B. Ind. Eng. Chem. 1958, 5 0 , 1289. Piicher, J. M.; Warker, P. L., Jr.; Wright, C. C. Ind. Eng. Chem. 1955, 47, 1742. Ride. B. E.; Havesian, D. Ind. Eng. Chem. Process Des. Dev. 1975, 14. 70. Von Fredersdorf, C. G.; Elliot, M. A. "Chemistry of Coal Utilization"; Wiiey: New York, 1963; p 892. Wen C. Y. Ind. Eng. Chem. 1968, 60.34.
Received for review October 31, 1980 Accepted August 31, 1981
Batchelder, H. R.; Busche, R. M.; Armstrong, W. P. Ind. Eng. Chem. 1953, 4 5 , 1856-78.
Evaluation of Inferential and Parallel Cascade Schemes for Distillation Control Nandklshor 0. Patke' and Pradeep B. Deshpande' Chemical and Environmental Engineering Department, University of Louisviiie, Louisville, Kentucky 40292
Adam C. Chou Mobil Research and Development Corporation, Princeton, New Jersey 08540
This paper presents a comparative assessment of inferential and parallel cascade schemes for controlling the top-product composition of a simulated multicomponent distillation tower. The steady-state data employed in the simulation study pertain to an industrial type column. The paper compares the closed-loop response of the single-temperature feedback scheme with the inferential and parallel cascade schemes. The findings of this work support the implementation of the inferential control scheme for this column.
Introduction Distillation columns are widely used in petroleum and chemical process industries. Improved composition control of these units is important not only from quality control considerations but also because distillation units consume vast amounts of energy. The columns are generally subjected to disturbances in the feed, and the control of product quality is achieved by maintaining a suitable tray temperature near its set point. This type of single-temperature feedback control is not always effective since maintaining a constant tray temperature does not necessarily result in constant product composition (Jones, 1971). Cascade control is used to improve the dynamic response of the column. In this type of cascade scheme, the top product composition is measured and then fed to a composition controller whose output serves as the set point of the tray-temperature controller. This scheme is referred to as the parallel cascade control system (Luyben, 1973) 'Department of Chemical Engineering,Ohio University, Athens, Ohio 45701.
because the manipulated variable, reflux flow, affects the top product composition and the tray temperature through two parallel transfer functions. The parallel cascade control system requires an on-line composition analyzer. The on-line composition analyzers introduce undersirable time delays in the control loops (Meyer et al., 1979). In addition, they are expensive and can be difficult to maintain (Scott, 1968; Hadley, 1969; Painter et al., 1978). The inferential control scheme circumvents many of the problems associated with the composition analyzers. The control scheme, which has been proposed by Brosilow and co-workers (1978), uses the easily available tray temperatures to estimate the product composition and uses the estimated composition to determine the required control effort. Many industrial multicomponent columns similar to the one on which this study is based employ single-temperature feedback control schemes. For the reasons cited earlier, this conventional feedback control scheme is sometimes inadequate. The purpose of this study is to assess through simulation, the benefits of the inferential control scheme and compare it with the parallel cascade
0196-4305/82/1121-0266$01.25/00 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 267
system and the single-temperature feedback system. T h e Inferential Control Technique The inferential control technique uses the secondary process outputs (in this study, stage temperatures) and the manipulated variable to estimate the effect of unknown feed disturbances on product quality. A brief description of this method follows. For details, the reader is referred to the original papers by Joseph and Brosilow (1978), Brosilow and Tong (1978), and Cwiklinski (1978). The static input-output model of the distillation column which relates the dependent variables (stage temperatures, top product composition) to the independent variables (feed disturbances, manipulated variables) is given by
or in matrix form
y = bT u + cm
and
11=[:z“:II!+ ...
ek
aklak2
Pk
y(s) = bT(s) u(s)
+ c(s) m(s)
(7)
O(s) = AT(s) u(s)
+ P(s)m(s)
(8)
and Brosilow and co-workers (1978) have suggested that a suitable lead-lag network should be used for dynamic compensation. Thus, the dynamic estimator is given by
where CY@) is an element of the vector a ( s ) for the measurement i and q ( 0 ) is the static estimator given in eq 4. The lead time constant rli is the average of all the time constants in the matrix A that correspond to the measurement i and the lag time constant rw is the average of all time constants in the matrix b that corresponds to the primary process output y. The dynamic estimate of the top product composition is given by the equation
9(s) = 4s) m(s) + aT(s)(e(s) - P(s)m(s)) (10) Since 9(s) is a deviation variable, the objective of the control effort would be to make j ( s ) zero. Thus, by setting (2)
ak5
or
e=ATu+Pm Note that all the variables in eq 1and 2 are perturbation variables. The variables 0 and m are measurable but the output variable y is assumed to be unavailable. The least-squares estimation theory is applied to the static input-output model of the process to obtain an estimate, 9, of the product composition, y, as a linear combination of the temperature measurements, 8, and the manipulated variable, m. The resulting equation is y = aT ( e - P m ) cm (3)
+
In eq 3 CY is the steady-state estimator which is given by a = (ATA)-lATb (4) Two quantities called projection error, Eo,and condition number, K , measure the accuracy and the sensitivity, respectively, of the estimator. They are given by the equations
Eo = (Ilrll/llbll) X 100
in the Laplace domain. The system equations are
(5)
where
r = residual vector b - Aa 11.11 = norm of the vector (.) K ( A ~ A=) (Xmax/Xmin)l/:! (6) where &,Lh= maximum and minimum eigenvalues of the matrix A TA. In a distillation column having many stages, it is possible to measure as many temperatures. From this total, an adequate number is selected such that the matrix A has a low projection error and a low condition number. The foregoing discussion was based on a static model of the process. It is generally necessary to add some form of dynamic compensation to the estimator in order to form an acceptable control system (Foss, 1973). For small deviations around the steady state, the distillation system can be modeled as a linear dynamic system
9 ( s ) to zero, eq 10 may be solved for m(s) to give m(s) = -c-l(s) aT(s)(e(s) - P(s) m(s)) (11) = G(s) aT(s)(e(s) - P(s)m(s)) where G(s) C(S) = -1 (12) Equation 11 is the controller algorithm which determines the manipulated variable from the measured temperatures. Since 4 s ) is fitted to a first-order transfer function, G(s) would be a lead network which is physically unrealizable. In order to realize this transfer function, a lag time constant which is ten times smaller than the lead-time constant may be used (Coughanowr and Koppel, 1965). Equation 11 can be inverted so as to give the controller equation in the time domain. The stability aspects of the algorithm are discussed by Brosilow (1978). Simulation Studies The schematic of a depropanizer column on which the simulation studies are based is shown in Figure 1. Through steady-state simulation, it has been determined that the operating column may be described as one having 20 ideal stages. The steady-state simulation work was based on a rigorous plate-to-plate calculation procedure known as Sorel’s method (Van Winkle, 1967). The steady-state operating data are summarized in Table I. The dynamic model of the column is developed according to the procedure presented by Luyben (1973). All the simulation work is based on the 20-tray tower and is carried out on a DEC 1090 computer. The primary objective of the control schemes is to maintain the composition of isobutane in the top product at its steady-state level. In simulating the control systems it is assumed that the dynamics of the flow and liquid level loops are much faster than the dynamics of the temperature control loop. The steam flow is assumed to be on flow control. The liquid levels in the column base and in the reflux drum are controlled by manipulating the overhead product and the bottoms product, respectively. Conventional Control Studies. The schematic of the relevent portion of the conventional control system is shown in Figure 2. In this system the temperature on
268
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 Coolant 125.0
0, c
124.4
E
I
I - 4 2 4
?
D3
1246
B
f
1
'K-
1248
D Y
v,
1
1242
tL
D5
DI
0
60
30
90
I
Time, min,
273 psig 2lO.F
D Top Product
Valve Trays Weir Height 3' Weir Length 3 I'
0
60 Time, min.
30
J
90
Figure 3. Open-loop response of the temperature on stage 20 to feed disturbances. Figure 1. Schematic of the depropanizer column.
fi 7
1
,@
i,1"4 P Reflux
I" I
.
60
,007.
D12
i
7
50
40 + P 0
set pt
30
20
-
u Distillate
IO
f
2 Y
Figure 2. Feedback control system.
ideal stage 20 is controlled by manipulating the flow of reflux with a view to maintain the top product composition of isobutane constant. The transient open-loop responses of tray-20 temperature and overhead isobutane composition to various exponential-step feed upsets of the form (1 - e-",025t)are shown in Figures 3 and 4, respectively. These figures show that although the temperature change is less than 1 OF, the composition varies by about 20 to 45% from its steady-state value. The tuning constants for the proportional + integral (PI) temperature controller are determined by the ultimatecycle method (Ziegler and Nichols, 1942). They are: gain, K, = 202.5 (lb-mol/h)/OF and integral time, TI = 0.0694 h. The closed-loop responses of temperature and composition to the various disturbances are shown in Figures 5 and 6, respectively. It may be observed that although the temperature is restored to its original set point within 18 min, with less than 0.5 O F overshoot, the composition of isobutane shows a large deviation ranging from about *9.5% to *48.5%, depending upon the type of upset. Parallel Cascade System. From the closed-loop responses shown in Figures 5 and 6 it is clear that some type of cascaded composition control is required to reduce the steady-state offset in the isobutane concentration. To
..E
".
O
g L
40
E
.20
e
0
+
-30
5
t
003 -40
120
60
0
160
l i m e , min
Figure 4. Open-loop response of isobutane composition to disturbances in the feed.
0
6
12
18
T i m e , min
Figure 5. Closed-loop response of the feedback control system to disturbances in the feed flow rate.
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 269 Table I
A. Steady-State Operating Conditionsa components propylene n-propane isobutane 1-butene n-butane
feed compn, mole frac.
distillate compn, mole frac.
bottoms compn, mole frac.
0.21340 0.15590 0.26140 0.09350 0.27580
0.60579 0.38931 0.0 0454 0.00022 0.00016
0.02287 0.04262 0.38611 0.13879 0.40962
B. Steady-State Column Temperature and Composition Profiles composition, mole fraction stage no. reboiler 1
temp, "F
C3H6
213.58 208.90 203.82 198.53 193.24 188.23 183.71 179.86 176.67 171.55 165.33 158.26 150.93 144.05 138.27 133.81 130.58 128.32 126.73 125.56 124.63 76.00
0.02290 0.03746 0.05560 0.07706 0.10112 0.12666 0.15240 0.17719 0.20021 0.22066 0.24765 0.28109 0.31933 0.35923 0.39730 0.43146 0.46144 0.48845 0.51441 0.54132 0.57103 0.60519
n-C,H,
i-C4Hlo
i-C,H,
n-C,H,,
0.04267 0.38608 0.13877 0.40958 0.06355 0.39444 0.13254 0.37200 2 0.08633 0.39155 0.12546 0.34105 3 0.10969 0.38012 0.11800 0.31511 4 0.13200 0.36307 0.11061 0.29319 5 0.15164 0.34327 0.10368 0.27474 6 0.16739 0.32322 0.09754 0.25944 7 0.17868 0.30472 0.09233 0.24701 8 0.18557 0.28885 0.08813 0.23721 9 0.21230 0.29263 0.07912 0.19526 10 0.24708 0.28428 0.06783 0.15316 11 0.28901 0.26192 0.05488 0.11309 12 0.33477 0.22667 0.04149 0.07775 13 0.37895 0.18325 0.02914 0.04950 14 0.41590 0.13852 0.01905 0.02923 15 0.44208 0.09861 0.01169 0.01616 16 0.45654 0.06675 0.00681 0.00847 17 0.46017 0.04332 0.00380 0.00425 18 0.45444 0.02704 0.00205 0.00206 19 0.44051 0.01616 0.00106 0.00096 20 0.41898 0.00907 0.00051 0.00042 condenser 0.38991 0.00453 0.00022 0.00016 a Feed stage = 8th tray from the reboiler; feed flow rate = 417 lb-mol/h; top product flow rate = 136.3 lb-mol/h; bottom product flow rate = 280.7 lb-mol/h; reflux flow rate = 419.9 lb-mol/h; reboiler heat duty = 3.9 x lo6 Btu/h; number of theoretical trays = 20. ,007
I 4 30 O
,006
3
-
1 ' 0
ns
,005
f f
I I T!w Dt.
F
O E
Distillate
P
Figure 7. Parallel cascade control system.
c c
,004 F
Y
P
,003
O2
4-50
,002 0
60
120
180
Time, min.
Figure 6. Closed-loop response of the feedback control system to disturbances in the feed.
achieve this objective, a parallel cascade control system has been tested (Luyben, 1973). The block diagram of the parallel cascade control scheme is shown in Figure 7. This
strategy requires an on-line composition analyzer. As shown in Figure 7, the composition signal is fed to the composition controller which is the master controller of the cascade system. The output of the composition controller serves as the set point of the slave temperature controller. The output of the temperature controller manipulates the reflux. In the simulation study the composition analyzer is represented as a pure dead-time element having a value of 6 min. Thus the composition measurements are assumed to be taken every 6 min and held constant. The sampling interval is also assumed to be 6 min. The tuning constants of the temperature controller remain the same but the master controller parameters, determined by trial and error, are gain, K, = 100 OF/mole fraction i-C4 and integral time, T~ = 0.1 h. The closed-loop responses of the system to various feed upsets are shown in Figure 8. This figure shows that the
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
270
Table 11. Transfer Functions Relating Overhead Isobutane Composition to t h e Disturbances and t h e Reflux Flow Rate propylene 1
feed components gain k, mole frac./ lb-mol/min time constant, 7 , min transfer functions
n-propane
isobutane
1-butene
n-butane
2
3
4
5
reflux rate
-0.00634
0.00812
-0.00066
-0.00585
-0.00657
-0.00596
19.2
25.2
12.0
21.0
22.8
7.8
-0.00634 (19.2s-t 1 )
-0.00812 (25.2s+ 1)
-0.00066 (12s+ 1 )
-0.00585 (21s+ 1 )
-0.00657 (22.8s + 1)
-0.00596 (7.8s+ 1)
Table 111. Transfer Functions Relating Temperatures on Stages to Disturbances and Reflux Flow Rate stage no. 15 20 7 14 8 1
13
propylene 1
n-propane
is0butane 3
2-butene
n-butane
2
4
5
- 13.685 (7.2s + 1 ) -3.910 (21s + 1) -21.101 -------(7.8s + 1 ) -18.00 (9s + 1) -18.674 (4.5s + 1 ) -14.831 (7.8s + 1) -22.247 (5s + 1 )
-12.185 (9.6s + 1) 1.477 (36s + 1) -24.554 ---_---(4.8s + 1) -18.462 (9.6s + 1 ) -20.585 ( 6 s + 1) -21.508 (7.8s + 1 ) -24.923 (9s + 1 )
-1.431 (7.8s + 1) -0.550 (36s + 1 ) -2.532 (4.8s + 1 ) -1.761 (4.8s + 1 ) -0.55 (9s + 1 ) -8.257 (4.8s + 1 ) -2.037 (10s + 1)
-7.962 (9.6s + 1 ) -1.231 (15s + 1) -4.308 -_-__--(9s + 1 ) -9.692 (9s + 1) -2.154 (12s + 1 ) -7.385 (5.4s 1) -11.385 (10s 1 )
-9.965 (10.8s 1) -1.565 (18s + 1) -5.687 (12.6s 1) -13.096 1) (10.8s -3.339 (18s t 1) -7.461 (9.6s + 1) -15.757 (12s t 1)
0050
f
0046
z
-E
+
+
+
+
;(,)
11
6
+
7
0052
0048
+
-12.118 (6s 1) -2.858 (19.5s + 1 ) -21.564 (4.2s + 1) -16.877 (3.6s + 1) -18.406 (6s + 1) -26.366 (6s 1) -22.55 (5s + 1 )
+--pi-
00%
,."
+
reflux ratio
0044
0
0042
0040 0038
- i -
Control
I
Process
System
0036
6= 30
60
90
120
I50
180
210
T i m e , min
-
r
multiple variables
single variable
Figure 8. Closed-loop response of the parallel cascade control system to disturbances in the feed.
steady-state offset has been eliminated but an overshoot of about *17% has resulted. Inferential Control Studies. The block diagram of the inferential control system is shown in Figure 9. As seen from eq 7 and 8 several open-loop transfer functions are required for implementation of the inferential control scheme. These transfer functions are developed from step tests; the independent variables ( m and u ) are perturbed, one at a time, and the resulting response of the dependent variables (0 and y) is recorded. Then, suitable first-order transfer functions are fitted to these curves. Tables I1 and I11 show these transfer functions relating the overhead
Figure 9. Inferential control system.
isobutane composition and several stage temperatures to the independent variables. The next step in the design of the inferential control system is to select suitable stage temperatures for measurement. The steady-state gains for all the stage temperatures are used to calculate the projection error and the condition number for a given measurement set. Following the procedure suggested by Joseph (19751, five sets of temperature measurements are selected. The projection error is generally a decreasing function of the number of measurements. On the other hand, the condition number
Table IV. Selection of Measurements
Measurement Set I Temperature 14
(projection error) condition x 100 no.
set no.
locations (stage n o . )
1 2 3 4 5
14 14,s 14,8,1 14,8,1, 13 14,8,15, 20,7
17.55 11.19 10.45 6.95 0.16
- 5 0
D4
1.0 4.66 7.38 76.01 326.40
25
02
03
-u
--25
.,0042
60
I
160
I20 T i m e , min.
(20s + 1) (8.64s + 1)
5
-
(7.08s+ 1)
,0044
I (14,8, 1,13)
,0042
(9.9s+ 1) (20s + 1) (7.08s+ l),
1/
04 ~
k r Measurement Temperatures Set 14, 2 E
5 0
D5
I
6.0
I20
Time
180
,min
Figure 10. Closed-loop response of the inferential control system to disturbances in the feed.
is generally an increasing function of the number of measurements. Table IV lists the projection errors and the condition numbers associated with five sets of temperature measurements. The gains of the estimator are evaluated using eq 4. The dynamic compensation in the form of a lead-lag network is added as described earlier. The final form of these estimators is shown in Table V. The controller transfer function as evaluated from eq 12 is (7.8s + 1) G(s) = 167.79 o.78s +
By inverting eq 11into the time domain, the inferential control system is simulated. The closed-loop responses of the control system to various disturbances are shown in
Figures 10 and 11. The condition number associated with five temperature measurements is large and the response of the system which uses this set is highly unstable. This appears to confirm the usefulness of the condition number criterion in the construction of the inferential controller. A comparison of the responses of the three control systems studied reveals that the dynamic response of the inferential control system in all four cases is superior. The settling times with the inferential system are 30 min shorter than those which result when the conventional feedback system is used and about 60 min shorter than the parallel cascade system. In the inferential system, the composition overshoot is less than about &9% from its steady value. The steady-state deviations in i-C4 composition for all the simulation runs are listed in Table VI. The steady-state offset with inferential control, which in most cases is about & 5 % , could be a potential problem. However, in an experimental study of a binary distillation
Table VI. Percent Deviations in Overhead Isobutane Composition disturbances disturbances in % of feed comDonent flow rates open loop feedback control using temperature on stage 20 parallel cascade control inferential control set 1 (stage 14) inferential control set 2 (stages 14,8) inferential control set 3 (stages 14,8, 1) inferential control set 4 (stages 14,8, 1,13)
in propylene
z; E$ f, Ea
D5
t ,0048
+ 10%
f.S
u c
estimator vector, a
'4
z,e z ;c
- 0
I
stage numbers
-
:a
E B
DI
8.64s + 1) (4.37x 10-4( ) (20s+ 1) (8.64s+ 1)
4
7.5
u
06
Table V. Estimators set no.
-
n-propane
D1
-10% in propylene D2
+lo% in
-10% in n-propane
+5% in total feed
-5% in total feed
D3
D4
D5
D6
-20.7 48.46
35.68 -43.61
-19.38 -35.02
38.55 47.80
-31.28 9.69
99.12 -9.25
0.01 1.95
-0.02 -2.10
-0.01 -2.51
0.01 2.59
0.0 -5.39
0.0 6.38
-1.5
1.37
-5.39
5.58
-6.61
8.11
-1.51
1.59
-4.66
5.11
-4.97
6.86
-3.75
3.83
-3.93
3.92
-7.0
8.26
272
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 Measurement Set 3 Temperatures 14.8, I
-
c = steady-state gain relating composition and manipulated 7.5
D6
5 D4
-
25
D2
,0044 I
k
7
1
2
5 D3
D5
0
60
I20
8!
-5 -5
180
Time , min.
, 10
,0050
,0048
,0046
- -
\ ,0044 Measurement kt 4 Temperatures 14,8,1 , I 3
D,
j-25
D3
_ c -_I
I
, E
u n
(a)
DS ,0042
60
I20
IS0
variable D1 = +lo% upset in the feed flow rate of propylene D2 = -10% upset in the feed flow rate of propylene D3 = +lo% upset in the feed flow rate of propane D4 = -10% upset in the feed flow rate of propane D5 = +5% upset in the total feed flow rate D6 = -5% upset in the total feed flow rate D7 = +lo% upset in the total feed flow rate of isobutane D8 = -10% upset in the total feed flow rate of isobutane D9 = +lo% upset in the total feed flow rate of 1-butene D10 = -10% upset in the total feed flow rate of 1-butene D11 = +lo% upset in the total feed flow rate of n-butane D12 = -10% upset in the total feed flow rate of n-butane Eo = projection error associated with a measurement set G = controller transfer function KO = controller gain m = the manipulated variable P = steady-state gain matrix relating manipulated variable to the measurement s = laplace transform variable t = time u = column vector of disturbances in the feed y = controlled variable Greek Letters a = column vector representing steady-state estimator B = column vector of measurements (temperatures) K ( - ) = condition number of matrix X = eigenvalues of a matrix TI = integral time constant of the controller T X = lead time constant in the estimator corresponding to ith measurement T~ = lag time constant in the estimator corresponding to composition, y Special Symbols (-) = denotes the entity, as a matrix or a vector (9 = an estimate of (-)T= transpose of matrix (-) (.)-l = inverse of matrix (-) 11-11 = norm of the matrix or vector (.) 1. = absolute value of (-)
-7 5
Time, min
Figure 11. Closed-loop response of the inferential control system to feed disturbances.
control system (Patke and Deahpande, 1980)the inferential system did not give rise to steady-state offset. While the extrapolation of the results to multicomponent systems may not be warranted, at least there is experimental evidence with a binary system which suggests that steadystate offset may not be a problem. It must also be pointed out that the choice of control tray in parallel cascade and feedback systems and changes in chromatograph cycle time would undoubtedly affect the quality of control. The effect of these variables on control has not been explored in this study. Conclusions This paper has presented a comparative study of inferential, parallel cascade, and conventional control schemes for the top-product composition control of a depropanizer column. The results of this study suggest that the inferential control scheme should be implemented on this column. The work reported in this paper will be useful to those who are involved in the design of distillation control systems. Nomenclature A = a matrix of steady-stategains relating stage temperatures and disturbances a;= ith column vector of matrix A b = column vector of steady-state gains relating composition and disturbances
(s),
(e)
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Received for review May 2, 1980 Revised manuscript received June 6 , 1981 Accepted October 8,1981
