Ind. Eng. Chem. Process Des. Dev. 1903, 22, 361-366
considered the optimum pH of the feed is 7.65; (b) the sensitivity of the total operation costs for small variations in the pH (0.1 pH unit) is small; (c) the temperature should preferably lie between 50 and 55 O C , or as close as possible to this range, while avoiding microbiological growth in the system; (d) the catalyst particles should be as small as the pressure drop over the bed allows, when the effectiveness factor is smaller than one; and (e) the total operation costs are sensitive to variations in the catalyst price, rather insensitive to variations in the yearly costs per reactor. Nomenclature De = effective diffusion coefficient in a catalyst particle, mz/s E,, = total weight of catalyst in a reactor, kg of dry cat. G = glucose concentration, kmol/m3 Go = glucose concentration in the feed solution, kmol/m3 G, = glucose concentration in a solution which is in thermodynamic equilibrium, kmol/m3 H = height of the catalyst bed, m K = thermodynamic equilibrium constant of the glucose to fructose isomerization k, = first-order rate constant, m3/s kg of dry cat. L = ratio between volume and external surface of a catalyst particle, m f i t ) = production rate of HFS per kg dry catalyst at operating time t, kg of HFS/s kg of dry cat. R = gas constant, J/K mol
36 1
Th = Thiele modulus (see eq 6) t = operating time, h x = axial coordinate in the catalyst bed, m c
= effectiveness factor
Pk = weight of dry catalyst/m3wet catalyst, kg of dry cat./m3
of wet cat. density of HFS,kg/m3 = flow rate, m3/s = fractional conversion (eq 9) = time after which a charge of catalyst is replaced, h Registry No. Glucose isomerase, 9055-00-9.
p = 9 T
Literature Cited Boersma, J. 0.;Vellenga, K.; de Wllt, H. G. J.; Joosten, G. E. H. Blofechnol. B h n g . 1070, 21, 1711. Colquhoun-Lee. I.; Stepanek, J. Chem. Eng. 1974, 1 , 108. Keulen, M. van; Veilenga, K.; Joosten, 0. E. H. Bbfechnol. B h n g . 1981. 23, 1437. Kikkert. A.; Vellenga, K.; ce ! WIR, H. G. J.; Joosten, G. E. H. Blofechnol. B h n g . 1981, 23, 1007. Roels, J. A,, van filburg, R. Sfaerke 1070, 31. 17. Smith, J. M. “Chemical Engineering Kinetics”, 2nd ed.; McGraw-HiII: New York, 1970; Chapter 11. Straatsma, H.; Vellenga, K.; de Wilt, H. 0. J.; Joosten. G. E. H. Ind. Eng. Chem. Process Des. Dev. 1089, preceding paper in this issue. Vellenga, K. Ph.D. Thesis State Unlversky Groningen. The Netherlands, 1978.
Received for review March 2, 1981 Revised manuscript received July 13, 1982 Accepted September 23, 1982
Control of High-Purity Distillation Columns Carmelo Fuentes and Wllllam L. Luyben’ Department of Chemical Engineedng, Lehigh University, Bethiehem, Pennsylvanla 180 75
The dynamic behavior of distlllation columns with high-purity products (down to 10 ppm) has been studied via digital simulation. The effects of product purity, relative volatility, composition analyzer sampling time, and magnitude of disturbance have been explored. Results show that systems with low relative volatility (a= 2) respond slowly enough so that good control can be achieved at very high purity levels with a 5-min analyzer dead time. However, systems with high relative volatility (a= 4) respond so quickly that large deviations in product purities occur before the analyzer can respond. The dynamic response of the column is very nonlinear. Effective control was obtained by using a composition/temperature cascade system. An intermediatetray temperature was controiled to achieve fast dynamic response to disturbances, and the setpoint of the temperature controller was reset from a product composition controller. The secondary temperature controller gave better control for feed composition disturbances when it was proportional only and loosely tuned. The opposite was true for feed rate disturbances.
Introduction The control of distillation columns is probably one of the most studied areas of process control. Tolliver (1980) has recently compiled a useful comprehensive list of references. Most of the columns studied have had product purities that were low to moderate (0.1 to 5 mol 90impurity). Very few papers consider the dynamics and control of highpurity columns, despite their industrial importance. Boyd (1975) used a double differential control scheme to maintain overhead purities of about 10 ppm in a benzene/ toluene separation. ’&reus (1976) reported difficult control problems for a methanol/water column with product purities in the range of lo00 ppm. A highly nonlinear behavior was reported. The control problems with high-purity columns have been so severe that many process designers try to avoid
making high-purity products out of both ends of a column simultaneously. It is a very common practice in the chemical industry to build two columns instead of one and to provide large intermediate tanks to handle recycle flows between these columns. Pure products are produced out of one end of each column. This practice increases both capital investment and energy costs. Thus, there is considerable economic incentive to improve the control of high-purity columns. The purpose of this paper is to explore the dynamics and controllability of high-purity distillation columns. The dynamic behavior was investigated by linear analysis and by digital simulation of a nonlinear mathematical model. The dynamic responses of the open-loop system for changes in various manipulated and disturbance variables were first studied in order to gain some insight into the dynamic difficulties associated with the control of these
01~~-43051831 I 212 - 0 3 ~ 1 ~ 0 1 . 5 0 1 00 1983 American Chemical Society
Id.Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1083
362
Table I. Design Parameters prod purity, concn of impurity
re1 volatility, Q!
mol %
2 4
IO6
I
PPm 50 000 1000 10 50 000 1000 10
5 0.1 0.001 5 0.1 0.001
50,000 p p m
total trays
feed rate, kg-mol/min
reflux ratio
reboiler heat input, lo6 kcal/h
18 40 60 13 26 40
36.3 31.8 31.8 65.8 61.7 61.7
2.03 2.35 2.37 0.521 0.693 0.700
9.18 8.87 8.92 8.33 8.71 8.74
Table 11. Results of Linear Analysis
re1 volatility, 01
Q 10 Q
prod purity, ppm
steady -state gain (xB/V), mole fraction/ kg-mol/min)
time const, min
50 000 1000 10 50 000 1000 10
-0.0284 -0.0357 -0.0299 -0.0209 -0.0233 -0.0238
30 1680 170 000 10 450 45 400
v
2
m
X
-AZ [I
= 4
I
- 1
4
1000 p p m
I o5
I P;'
50,000
ppm I
E = 4
-
101
50
-
a
CAZ
to4 4
X
IO
0
,=====
Q
-AV
4 I t
100 0 T I M E (minutes)
50
zK;v
100
Figure 1. Open-loop response for feed composition step change.
columns. Then several types of closed-loop control systems were investigated. Systems Studied Distillation columns with product purities ranging from 5 mol % to 10 ppm (molar) impurity in both distillate and bottoms products were studied for two values of relative volatility (a= 2 and a = 4). Table I gives values of design parameters for the six columns explored. Each was assumed to be 12 ft (3.66 m) in diameter. The ratio of actual to minimum reflux ratio was set at 1.2 for the low relative volatility cases and 1.05 for the high relative volatility cases. Other assumptions included constant relative volatility, equimolal overflow, theoretical trays, total condenser, partial reboiler, and saturated liquid feed and reflux. Tray hydraulics were given by the Francis Weir formula with 1.5-in. (3.81 cm) outlet weirs. Open-Loop Dynamic Behavior The nonlinear ordinary differential equations describing the dynamic behavior of these columns were linearized, and the frequency responses were obtained by the stepping technique of Lamb and Rippin. See Luyben (1973). The results of this linear analysis are summarized in Table 11. Dynamics are predicted to speed up somewhat as relative volatility increases. Extremely slow response
a a 100
-
100
; -
IO
0.1
0.1
0
50
100 T I ME
0
50
100
(minuter)
Figure 2. Open-loop response for step change in vapor boil-up.
is predicted for high-purity columns. The explanation of these predicted effects lies in the small concentration changes from tray to tray as purity is increased and as relative volatility is reduced. These small concentration changes from tray to tray make some of the coefficents in the linearized equations very small, giving small eigenvalues. The nonlinear dynamic model was then simulated. The dynamic open-loop responses of product compositions (XB and xD) were obtained for step changes in various input variables for the six columns studied. Figure 1 shows the responses for both positive and negative step changes in feed composition (0.5 to 0.6 mole
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 363 50,000 p p m
a
.
2
P-a
10
Y
1000
1000 I
-
100
X
10
-
10
1000 p p m =
0
-
4
a IO00
10
-A R 50
0
100
loot//
x 0
50
100
T I M E
T I M E
IO
0
100
50
a
( m i n u t e s )
Figure 3. Open-loop response for step change in reflux. a
=
G
2
I
o
'
F
1000 100
-
a
;lo'E = 2
1000
x
- 10
X
1
(minutes)
Figure 5. Open-loop response for various magnitude changes in vapor boil-up.
100 10
L
Y:F; =
4
Y
1000
xm
-
100
a
x
0
0
50
100
T I M E
50
100
I
0
O 50
U
100
10
20
100
T I M E
I
,v
-
-0 1
1000
0
1 0 101
0
L
50
100
( m i n u t e s )
Figure 6. Open-loop response for various magnitude changes in reflux.
( m i n u t e s )
Figure 4. Open-loop response for various magnitude changes in feed composition.
fraction and 0.5 to 0.4 mole fraction). Figure 2 gives results for f10% changes in vapor boilup. Figure 3 shows the open-loop responses for f10% step changes in reflux flow rate. These responses are highly nonlinear. With disturbances of these magnitudes, the response is completely different for a positive change than for a negative change. There is little difference in the dynamic behavior of systems with different relative volatilities when purity levels are low. However, as purity is increased, the dynamic responses (particularly to feed composition disturbances) begin to differ greatly for different relative volatilities. For systems
with high relative volatility the response is quite fast and highly nonlinear. Disturbances in feed composition are felt very quickly in the bottom of the column. Figures 4 to 6 give open-loop responses for input changes of different magnitudes for the 1000 ppm columns with relative volatilities of 2 and 4. These results show that the system responds more slowly as the magnitude of the disturbance is reduced. This nonlinear effect explains why the time constants predicted by linear analysis are so much larger than those observed in the nonlinear simulation. Linear analysis is based on small perturbations. These open-loop responses give us some insight into the control problems to be expected with high-purity columns. The fast dynamic response of high relative volatility columns require a fast control system to avoid large changes
364
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
Table 111. Bottoms Composition Controller Settings anaconre1 lyzer bottoms controller vola- dead compn, troller reset time, tility , time, XB gain,= ~i -Kc rI, min TD,min ppm 2 0 50 000 5.2 1.5 1000 5.8 1.5 10 7 .a 1.5 5 50 000 0.26 15 1000 0.24 15 10 0.21 15 4 0 50 000 2.1 1.5 1000 2.6 1.5 10 4.0 1.5 5 50 000 0.14 15 1000 0.11 15 10 0.10 15
=
TD
0
~~~
a 1 4
0 . 2 50,000
E
-2
ppm
t
s'5
9
Composition transmitter spans of 50 ppm, 5000 ppm, and 20 mol % were assumed for the three purity cases. A valve span (giving vapor boil-up changes for a change in controller output signal) of 100 kg-mol/min was assumed for all cases. a
in product purities in the impure direction. Unfortunately, most of these high-purity columns depend on chromatographs for composition analysis. These devices introduce a significant deadtime into the control loop (2 to 30 min). Thus the basic problem is one of being unable to detect the disturbance fast enough in a highpurity column with fast dynamic response.
Closed-Loop Control Feedback controllers were designed for each end of the column: re& controlling distillate composition and vapor boil-up controlling bottoms composition. The frequency response results from the stepping technique were used to design controllers for two cases: no dead time and a 5-min dead time in the composition analyzer. Table I11 gives controller settings for the bottom composition controller. Since the process transfer functions were almost first-order lags, very tight controller settings could be used when no dead time was present. A fairly fast reset time of 1.5 min was selected. Controller gains were determined by using a root locus plot to find the gain that gave critical damping. With the 5-min dead time in the loop, the Ziegler-Nichols settings are given in Table 111. In several wes,these settings gave too oscillatory behavior when applied to the nonlinear model. The closed-loop response of the nonlinear system with proportional-integral (PI) composition controllers was evaluated for both positive and negative step changes in feed composition from 0.5 to 0.6 and from 0.5 to 0.4 mole fraction. Figure 7 shows that bottoms composition could be controlled very well when no analyzer dead time was present. This was true for any purity level and any relative volatility. The controller settings are so tight that corrective action is made before the column can drift into a nonlinear region. Simulations also showed that distillate composition could be well controlled when no analyzer dead time was present. In fact, simultaneous control of both product compositions was found to be easily achieved when no analyzer dead times were assumed, even with very high purities. However, the introduction of a 5-min dead time into the control loop changed control performance drastically. As shown in Table 111, controller settings had to be loosened considerably for closed-loop stability. The result was much
?
5.0
m 4.8
X
s.0
1
I"
X
4.51
1000 -
p p m
,
c
a
tAZ
v
1 0 0 -
m
!
x
9501
- AZ
x
-
-
$10 5 r
a l
x
93k
E a
I 10 6 1
- AZ
E!oo-
* A- A ZZ
10 0
m
950
;9 . s t i
0
12
6
0
24 T I
M E
12
(minutes)
Figure 7. Closed-loop response with no dead time. To
-
=
5
a = 2 50,000
ppm
E a
a
-
0
v
v
-A Z
4
m X 01
-
E a
1000
ppm
P
m 1000 X
soot
- AZ
10
P
P
In
~
0
0
36
72
T I M E
0
36
72
(minutes)
Figure 8. Closed-loop response with 5 min dead time.
larger errors in product compositions and the tendency for the column to drift into a region of highly nonlinear response.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
A Z = 2 0 % and
Table IV. Temperature Profile
TD = 5
AZ
= 4
I
TD = I
o /'
temp, "C
reboiler
98.89 98.67 98.33 97.52 95.64 91.70 85.01 76.91 70.27 66.37 64.51 63.71 63.38 63.01 62.46 61.71 60.72 55.74 54.67 54.45 54.44
Temperature
3e
72
0
38
= 4,10 ppm Column)
plate
5 6 7 8 9 10 11 12 13 14 15 16 (feed) 17 18 19 20 25 30 35 40
A 2 = I O '10
Xe
(CY
72
K,
Control
Tio
=0.24
T ,
= 5
+AZ
,751 '--
T I M E
365
(minutes)
Figure 9. Effect of magnitude of disturbance and dead time.
Figure 8 gives results for control of XB at different purity levels and different relative volatilities with a 5-min dead time. Adequate control of xB was achieved up to purity levels of 10 ppm for relative volatility of 2. Similar results were obtained for control of X D only and for simultaneous control of both XB and XD when relative volatility was low (a = 2). However, for a relative volatility of 4, very poor control resulted (note changes in scales in Figure 8). Extremely large errors in product purities occurred. For example, in the 10 ppm and a = 4 case with a step change in feed composition from 0.5 to 0.6, bottoms composition rose from 10 ppm to about 100 ppm. Thus our conclusion is that high-purity columns with high relative volatilities cannot be adequately controlled with a simple product compostion controller when a realistic analyzer dead time is present. The smaller the magnitude of the disturbance and/or the smaller the dead time in the analyzer, the better control will be. These effects are shown in Figure 9 for the 10 ppm and a = 4 case. Controller gain can be doubled as dead time is reduced from 5 to 1 min, giving much smaller errors in product purity.
Temperature/Composition Cascade Control In order to overcome the deadtime in the composition analyzer, a temperature/composition cascade control system was studied for the high-purity (10 ppm), high relative volatility (a = 4) case with a 5 min dead time in the composition analyzer. This was the case that was the most difficult to control with just a composition controller. Fortunately, an intermediate temperature control tray can usually be used in this type of column because there is a fairly large change in temperature from the top to the bottom of the column. Since the relative volatility is large, the temperature profile will have a reasonably hefty break somewhere in the column. Very little temperature change from tray to tray occurs in the very top and bottom of the column due to the high purity levels.
Composition/ Temperature Ks = O
Tio
2 4
r.=5
KM=0.6
Control r,,,=I5
la5
I75
I
01
K, = o I
( P - Only)
- AZ Tio
Cascade
i85[
XB
T I M E
"/_
K,=
1.0
T,
=I2
-AZ
IO
(minutes)
Figure 10. Cascade control-feed composition disturbances.
The steady-state temperature profile for this column is given in Table IV. Tray 10 was selected as the intermediate control tray because it was quite responsive to changes in all input variables. A 30-9temperature measurement lag was assumed. Figure 10 (upper curves) shows the response of tray 10 temperature and bottoms composition when only a PI temperature controller was used. A temperature transmitter span of 25 O C and a valve gain of 100 kg-mol/min were assumed to determine the dimensionless controller gain. Positive and negative step disturbances in feed composition were imposed on the system. Good control of tray 10 temperature was achieved, but bottoms composition settled out at values away from the specification of 10 ppm.
368 Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 Temperature
Control
Ks = o . z 4
rs = 5
1751
Composition/Tempera t u r e Ks=024
Cascade K
T,: 5
1931
201
1751
01
K,= 0 . 1
0
33
T I M E
~
K M =1 . 0
( P - Only)
66
M
Control
0
3 3
o
"1 ~5
T m = 12
66
( m i n u t e s )
Figure 11. Cascade control-feed rate disturbances.
A bottoms composition controller was then added to the control system. The output of the composition controller changed the setpoint of the cascade temperature controller. Figure 10 (lower curves) gives the response of the system for feed composition disturbances with several controller settings. The composition controller had to be tuned very loosely in order to avoid oscillatory response. The temperature controller has a tendency to overcompensate and drive bottoms composition in the wrong direction for feed composition disturbances. After some experimentation, we found that using a proportional-only slave temperature controller with a low gain improved the response of the system to feed composition disturbances. However, as shown in Figure 11,the loosely tuned proportional-only temperature controller does not do as good
a job for feed rate disturbances as a more tightly tuned PI controller. This problem could be overcome by using feedforward control to compensate for feed rate changes, i.e., ratio steam and reflux to feed. Conclusions This study indicates that distillation columns producing high-purity products can be effectively controlled despite their highly nonlinear behavior. The control system must be able to detect disturbances quickly and take corrective action to prevent the column from dropping into a nonlinear region where one or both products go far off specification. High-purity columns respond much more quickly than predicted by linear analysis. This fact must be recognized in specifying analyzer cycle times and in designing control systems. Nomenclature F = feed rate, kg-mol/min K, = controller gain, dimensionless KM = master composition controller gain, dimensionless Ks = slave temperature controller gain, dimensionless P = proportional-only controller PI = proportional-integra1 controller TD = composition analyzer dead time, min XB = bottoms compositon, ppm more-volatile component 1 - XD = distillate composition, ppm less component V = vapor boilup, kg-mol/min z = feed composition, mole fraction more-volatile component Greek Letters a = relative volatility 71 = controller reset constant, min T M = Master controller reset constant, min 7s = slave controller reset constant, min Literature Cited Boyd. D. M. chem.Eng. Prop. 1871, 71(6),55. Luyben, W. L. "Procees MndeHng, Simulation and Control for Chemical Engineers"; W a w H I # : New York, 1973; p 8 7 . Tolilver, T. L. P a m presented at the Mey 1980 Dlstllletlon Dynamics and Control Covse at Lehbh Univ.rdty, Rdhkhem, PA. Tyreus. B. D.; Luyben, W. L. Chem. fng. Prog. 1976, 72(9), 59.
Receiued for review March 2, 1981 Revised manuscript received September 17, 1982 Accepted November 20, 1982
