Use of IWuMple Temperatures for the Control of Multfcomponent DisflHation Columns Cheng-Chlng Vu and Wllllam L. Luyben’ Depertment of Chemical Enginwing, Leh@
Unlvmity, BetMshem, Pennsylvania 18015
Conventional single-tray temperature control often produces very poor results when variations occur in non-key components in the feed of a multicomponent distillation column. This paper demonstrates that the set point of the temperature controller can be reset by using several secondary temperature measurements to detect changes in non-key and key components in the feed. Both dynamic and steady-state performances are evaluated for two different columns and are shown to be as good as or better than Brosiiow’s more complex inferential control.
Introduction In general, the objective of distillation control is to maintain the desired product quality. Despite advances in on-line composition analyzers, direct composition measurement is expensive, difficult to maintain, and can introduce undesirable time-delay into the control loop. One altemative is to use a number of simple, reliable, and inexpensive secondary measurements for indirect composition control. The conventional temperature control technique uses the temperature of one tray somewhere in the column. An “optimaln tray is selected such that the steady-state composition error in the product is minimized. This type of single temperature control is not always accurate especially in multicomponent systems, since holding a tray temperature constant does not necessarily result in constant product quality. Luyben (1973) analyzed a parallel wcade control scheme in which the temperature controller set point is changed by the output from the composition controller. The parallel cascade control does not have steady-state composition error, but it requires an on-line analyzer. Pressure variations affect temperature measurements in some columns. Several techniques have been proposed for overcoming pressure-effect problems: pressure compensated temperature, differential temperature, and double differential temperature control. Boyd (1975) demonstrated that controlling the difference between two temperature differentials in each section of the column gave improved performance. Luyben (1972) examined the problems associated with control of distillation columns with sharp temperature profiles. He recommended using multiple temperature measurements in order to achieve profile position control. Mosler and Weber (1974) proposed plural temperature control to maintain product qualities in both ends of a distillation column. They used the sum of two temperatures located near opposite ends of the column as an index of the split of products between both ends to adjust one manipulative variable. The difference between the same temperatures was used as an index of both product qualities to adjust the second manipulative variable. Brosilow and co-workers (Weber and Brosilow, 1972; Joseph and Brosilow, 1978) proposed a more complex control scheme called “inferential control”. A linear combination of temperature measurements and steam and reflux flow rates was used to estimate product compositions. Brosilow and Tong (1978) incorporated dynamics into inferential control. When applied to an industrial multicomponent debutanizer column, substantial im0 1964305184f 1123-O59O$Q1S O / 0
provement over single temperature control was reported. Patke et al. (1982) evaluated inferential, parallel cascade, and single temperature control schemes and recommended inferential control. However, their control schemes gave 5% steady-state offset in product composition, which could be a serious problem in many systems. Shah and Luyben (1979) proposed static nonlinear rigorous estimators for binary systems, involving tray to tray material and energy balances to estimate product composition. Rigorous estimators handle column nonlinearities better than linear estimators. One important consideration is the amount of on-line computation required. In the binary case, this is not excessive, but in multicomponent systems it may prove to be beyond the capability of many process control computing systems. In multicomponent distillation syBtems, variations of non-key components in the feed affect the temperature measurements strongly. Therefore, conventional temperature control techniques often infer erroneous product quality. Parallel cascade control eliminates steady-state composition error by changing temperature controller set points. Instead of using the output from a composition controller to reset the temperature controller, it may be possible to use convenient secondary measurements to generate the desired temperature set point. The objective of this study is to design composition control systems by using several temperature measurements in multicomponent distillation. Systems Studied Two columns were chosen for study based on the nature of non-key components in the feed. The first column studied was a three-component, 32-tray deisobutanizer (column l), separating isobutane (LK) and n-butane (HK) with propane (LLK) included in the feed. The control objective was to maintain constant normalized distillate composition X,N by manipulating reflux flow rate (Figure 1). Vapor boil-up was constant. The steady-state operating conditions are summarized in Table I. XDN= XD(LK)/(XD(LK)+ XD(HK)) (1) The second system studied was a five-component,20tray depropanizer (Patke et al., 1982, column 2), separating propane (LK) and isobutane (HK) with propylene (LLK), isobutene (HHK), and n-butane (HHK) present in the feed. The composition of isobutane (HK) in the overhead product was maintained by adjusting reflux flow rate. Steady-state values of parameters are given in Table 11. Steady-State Aspects The steady-state design was based on the Wang-Henke (1966) method. The analysis of column 1 is presented to 0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 591 Table 11. Steady-State Operation Conditions of Depropanizer (Column 2)
A.
Wok H I M
63Scm
Wdr L-h
IS1 m
compositions, mole fraction propylene (LLK) propane (LK) isobutane (HK) isobutene (HHK1) n-butane (HHK2) flow rate, kg mol/h temperature, “C
feed
distillate
bottoms
0.2134 0.1559 0.2614 0.0935 0.2758 189.15 74.06
0.60153 0.39341 0.00453 0.00034 0.00019 61.82 24.62
0.02494 0.04057 0.38163 0.13874 0.40962 127.32 98.95
B.
0
Figure 1. The deisobutanizer column (column 1)with single temperature control. Table I. Steady-State Operation Conditions of Deisobutanizer (Column 1)
item
S-S value
reflux flow rate, kg-mol/h top pressure, kPa bottom pressure, kPa reboiler heat duty, GJ/h total no. of trays feed tray tray efficiency, 5% column diameter, m weir length, m weir height, cm approximate 01 between LLK/LK
190.76 1962.9 1983.6 4.6 20 8 100 1.07 0.91 7.87 1.13
9
?
“1,
3
A. feed distillate bottoms compositions, mole fraction propane (LLK) 0.06 isobutane (LK) 0.42 n-butane (HK) 0.52 720.0 flow rate, kg-mol/h temperature, “C 62.83
0.150 0.806 0.044 288.0 51.55
1x 0.163 0.837 432.0 70.59
B. item reflux flow rate, kg-mol/h top pressure, kPa bottom pressure, kPa reboiler heat duty, GJ/h total no. of trays feed tray tray efficiency, % base holdup, kgmol reflux drum holdup, kg-mol average tray holdup, kg-mol column diameter, m weir length, m weir height, cm approximate 01 between LLK/‘LK
value 25 56 .O 827.4 857.4 47.4 32 14 100 254.4 237.0 6.0 2.30 1.61 6.35 1.90
0.OOo
.om
.oIo
.ow
.om
ZF Q L K l ~ mole frocvion
illustrate the design procedure for the proposed multiple-temperature control system. A. Single Temperature Control (2’). The “optimum” temperature control tray was selected by fiiding that tray which gave minimum steady-state error in distillate composition in face of the “worst” disturbance. In this case the worst disturbance was a change in the feed composition of propane (LLK). Figure 2 shows the steady-state error in distillate composition when a temperature was controlled on a single tray. Five different trays are shown. Tray 22 is the optimum single temperature control tray since it gives the least change in distillate composition. These results are plotted in a different way in Figure 3. B. Temperature/Differentid Temperature Control (TDT).Figure 4 shows the temperature profiles which
.loo
.lco
C
Figure 2. Steady-state distillate composition error for single temperature control for change in feed composition of propane (D1; column 1).
give zero steady-state distillate composition offset for changes in LLK feed composition, Le., changes in ZF(LLK) with the ratio of key components kept constant. This disturbance is called “Dl” hereafter. The temperature controller set point must be changed to keep XDNconstant. Figure 4 shows that the temperature differential between the top of the column and a tray down in the rectifying section changes directly with variations in the amount of LLK in the feed. Therefore, it should be possible to use this AT signal to change the set point of the temperature controller.
592
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
P
ZF(LLK)=
.OO
.oo
1 I
1
1t.000
t8.OOO
1
I
I
18.000
tt.000
t4.000
I
U.00
1
D4
.02
.oe ZFILLK)
.on
10
.I2
mole M i o n in C3
Figure 5. Temperature differentials and desired temperature set point for feed composition change in propane (Dl; column 1).
300.000
T U 1 YO.
I
i
Figure 3. Steady-state distillate composition error for different temperature control tray location in face of feed composition change in propane (Dl; column 1). mLL*
Figure 6. TDT control scheme.
change in temperature differentials. The temperature difference between top tray and tray 22 was chosen. (2) AT1 = T22 - T32 The middle curve in Figure 5 shows that ATl gives a good indication of change in LLK feed composition. (4) Adjust the temperature set point according to the top curve in Figure 5. The temperature set point compensation becomes T228et = a, + alATl (3) mo.om
84.000
u.000
W.00
rw.
@EO
w.000
ro.00
14.000
c)
Figure 4. Desired temperature profile in face of feed composition change in propane (Dl; column 1).
The design procedure for TDT control is as follows: (1) Find the "optimum" single temperature control tray (Figures 2 and 3). (2) Generate the desired temperature profiles (Figure 4) for the worst disturbance (DI). (3) Locate the section of the column where there is the most
The TDT control uses temperature differential to reset temperature controller set point (Figure 6). TDT control works well for LLK feed composition changes. However, for feed composition changes in key-componenta (LK and HK), it may not work as well because big temperature changes in the top tray do not occur. Temperature profiles (Figure 7) were plotted for feed composition disturbance in key components (called D2 hereafter) with ZF(LLK) held constant. The top curve in Figure 9 shows that the TDT temperature set point would move in the wrong
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
63
593
1
'I 6
.36
.40
.30
.42
ZF(LK1
Figure 7. Desired temperature profile for feed composition change in key components (D2; column 1).
direction for D2 disturbance. C. Temperature/Dual Differential Temperature Control ( T D 2 T ) .The bottom curve in Figure 5 shows that another temperature differential (AT2= T18- T2J does not change much as LLK in the feed changes. However, Figure 8 shows that this AT2does change as the LK/HK ratio in the feed varies. Therefore, it should be possible to combine AT1 and ATz to detect either feed composition disturbance. The design procedure for TD2Tcontrol is as follows: (1) Generate the desired temperature profiles (Figure 7) for feed composition changes in the key components (D2). (2) Locate the section of the column in which the temperature differential is almost unchanged for a LLK disturbance but does change for a key component disturbance. The second temperature differential for the column 1case is
AT2 = Ti, - T2z
.44
.48
.46
mole froction in
iC4
Figure 8. Temperature differentials and desired temperature set point for feed composition change in key components (D2; column 1).
63
4
(4)
(3) Find the slope a2such that after adding uz(ATz- ATzk) to a. + alATl the desired temperature set point is obtained (top curve in Figure 9) for D2. ATZhis the AT2 temperature difference with "base case" feed composition. The final form of TD2T control is =
+ alAT1 + aZ(AT2 - ATZbc)
(5)
.36
.3S
.40
I
I
.42
.4 4
2 F(LK1
.46
.48
mole fraction in i C,
Two temperature differentials are used to adjust the set point of the temperature controller in the TD2Tcontrol scheme (Figure 10).
Figure 9. Construction of TD2Tcontrol (the final form of TD2T is T2znet= a. + a,AT, + a2(AT2 - ATzb"))(column 1).
Steady-State Results In order to test the effectiveness of TDT and TD2T control on column 1, additional disturbances were considered. Disturbance D3 is a change in ZF(LK) with constant ZF(LLK)/ZF(HK) ratio. Disturbance D4 is a change in ZF(HK) with constant ZF(LLK)/ZF(LK) ratio. Figure 11 shows the steady-state distillate composition error for different types of control and for the four dis-
turbances. The results show that D1 is the worst disturbance for single temperature control. TDT control reduced the steady-state distillate composition error for disturbance D1, but not for D2, D3, and D4. TD2Tcontrol performs well for all four disturbances. It should be mentioned here that only feed composition changes were considered. Disturbances in feed flow rate should be handled by some ratio system, e.g., steam to feed.
594
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
I
& r?
1
AZF(LLKb 30 %
20 %
I0 %
Figure 10. TD2T control scheme. Feed C o w a i l i o n Change ‘9
In
0 propano ( L W with comlon~ LKlHKrOtio
9
A
+ X
1 butone (UO i butane (LKI
-30%
\-200/e
with constont LKI
1 4
with Oonstont LLKlHK rolio n butam (HKI with Conotant LLKlLK m i 0
TBT CONTROL
I
8.000
11.000
1
11.000
1
I
17.000
16.000
I
18.000
I C1.000
TRAY YO.
z a
x
Figure 12. Steady-state distillate composition error for different temperature control tray location in face of feed composition change in propylene (LLK) (column 2). a
9
TDT CONTROL
1
0
4
INFERENTl4L CONTROL
T CONTROL Tray 2 2
0
::
-.ow
-.w
-.om
0.000
.oto
.ow
T CONTROL (Troy 14)
.ow
DZF
Figure 11. Steady-state distillate composition error for T,TDT, and TD2Tcontrol (column 1).
In the second application, TD2T control was tested on a five-component depropanizer (column 2). Following the proposed design procedure, tray 14 is the optimum temperature control tray for propane (LLK) feed composition disturbance (Figure 12). The temperature control tray, tray 20, selected by Patke et al. (1982) actually is the worst choice since the LLK has the most effect on the top tray. The steady-state distillate composition error for various feed composition disturbances with single temperature control using tray 14 and tray 20 are shown in Figure 13. Single temperature control performs fairly well in this column with the proper choice of the control tray. The TD2T control scheme was design using the procedure discussed previously.
TldEet = a.
+ alATl + a2AT2
(6)
where AT, = T14- Tz0and AT2 = T15- Tisb”(Tisb”is the
Figure 13. Steady-state distillate composition error for T , TD2T, and inferential control (column 2).
base case value of tray 15’s temperature). The results (Figure 13) show that TD? control works better than the inferential control system recommended by Patke et al. (1982). Table III summarizes the results of this study and compares them with the results reported by Patke et al. (1982). The disturbances used in their study were percent change in feed component flow rates. The LLK feed
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 595
I
1 I
GAlNS,'C/(KgmWmiJ 1603
* h2/R G M =E / R Gml
2.210
I
-
3
0
0
-
2
.01
001
FREQUENCY, radians/time
2
5 I
Figure 14. Bode plots. Table IV. Controller Settings
In 9 P-
K,, (kg-mol/)/"C K,, dimensionlessa
2.62 3.62
2.54 0.60 9.35
ri, min Assume temperature transmitter span of 20 "C and valve gain of 85.2 kg-mol/min.
(D
(I
c'?
0 I
l
11.19
composition disturbances have different effects on these two columns. Column 2 handled the LLK disturbance better under single temperature control than column 1. The reason for this is that the relative volatility a between LLK and LK is about 1.9 for column 1and about 1.13 for column 2. The low a in column 2 means that the difference in boiling points between LLK and LK is small. Therefore, the non-key effect is not as serious in column 2 as in column 1. From a steady-state standpoint, TD2Tcontrol appears to be very effective. Next, we will examine the dynamic performance and compare it with single temperature control.
l
I
.
v I
Q
Dynamic Aspects Column 1 was chosen for dynamic study because the steady-state performance of TD2T is much better than conventional T control. The set of nonlinear ordinary differential equations describing the column was solved via digital simulation. Tray hydraulics were approximated by the Francis Weir formula. The energy model proposed by Fuentes and Luyben (1982) was used. Table I gives data on holdups. Bode plots, Figure 14, for transfer functions Gm, and Gm, were generated by pulse testing the rigorous mathematical model. Gm, = T,,/R
(7)
Gm, = E / R
(8)
where E is the error signal between TzZset calculated by TD2T and T2,.
E = a.
+ alTl + a2(T2- Tzbc)- T22
(9)
PI controllers were designed using Ziegler-Nichols settings (see Table IV). B1 is the controller for conventional T
596
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
-
T CONTRCL
A
”1
DI
LZF(LLK1.f
02
-n D2 G Z F ( L K l = t 02
T2z
62
Tsz
R
TI8
C.
T $T
PRECOMPENSATED
D 3 b Z F ( L K ) = * 02
0
FINAL
D.
z
T D T
0 X
__1_1 ~
W AZF(HK):* 02
n
m
0
Ib
20
30
40
50
60
T$me. min
Figure 17. Dynamic responses of TDzTcontrol for change in feed compositions.
Figure 15. Block diagrams.
I1-I
c)
Y)
%9 ,I 0
01 AZFILLIO=02
I
I
1
10
20
30
I
I
40
50
I 60
Tima, min
Figure 16, Dynamic responses of conventional temperature control for feed composition disturbances. control and B2is the controller when TD2Tcontrol was implemented. Block diagrams for T and TD2T control are shown in Figure 15 (A and B). In practical applications, we would like to have the T controller capable of working with or without the TD2Tset-point adjustment. This means the temperature controller should work equally well in either “local automatic” or “remote set”. To achieve this, a precompensator C is used (Figure 5c). Precompensator C is of the form
The final TD2T control block is shown in Figure 15d. The dynamic responses of step changes in feed composition (Dl-D4) of T and TD2Tcontrol are shown in Figures
16 and 17. The dynamic performances are approximately the same. However, the steady-state distillate composition error is much less with TD2Tcontrol than with T control for all disturbances. Conclusions TD2Tcontrol seems to offer an attractive method for indirect composition control of a distillation column. It has several advantages over the conventional technique and more complex “inferential control”. (1)The steady-state distillation composition error is less than with conventional single temperature control. (2) The design procedure is more physically understandable to the user than the design procedure for inferential control. (3) TD2Tworks as a set-point compensator. The control system can be designed to be stable with or without setpoint adjustment of the temperature controller. Nomenclature E = controller transfer function C = precompensator transfer function D1 = feed composition change in LLK with LK/HK ratio constant D2 = feed composition change in LK with LLK constant D3 = feed composition change in LK with LLK/HK ratio constant D4 = feed composition change in HK with LLK/LK ratio constant E = the error between the temperature set point generated by TD2Tand the temperature measurement G m = process transfer function HHK = heavier than heavy key component HK = heavy key component K , = controller gain LK = light key component LLK = lighter than light key component R = reflux flow rate XDN= normalized distillate composition, XDN= XD(LK)/ (XD(LK)+ XD(HK)) ZFG) = feed composition, mole fraction component j Greek Letters a = T,
relative volatility
= integral time constant (minutes)
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 597-602
Literature Cited Boyd, D. M. Chem. f n g . frog. 1975, 77(6), 55. Brosilow, C. B.; Tong, M. AIChE J. 1978, 24, 492. Fuentes, C.; Luyben, W. L. Ind. Eng. Chem. Fundam. 1982, 27, 323. Joseph, B.; Brosilow, C. 8 . AIChE J . 1978, 24, 485. Luyben. W. L. AIChE J . 1972, 78, 238. Luyben. W. L. Ind. Eng. Chem. Fundam. 1973, 72, 463. Mosler, H. A.; Weber, R. U S . Patent 3855074, 1974.
597
Patke, N. G.; Deshpande, P. B.; Chou, A. Ind. Eng. Chem. Process Des. Dev. 1982, 27, 266. Shah, M. K.;Luyben, W. L. Chem. Eng. Symp. Ser. No. 56 1979, 2.6/1. Wang, J. C.; Henke, G. E. Hydrocarbon Process. 1966, 45(8), 155. Weber, R.; Brosllow, C. B. AIChE J . 1972, 78, 614.
Received for review June 2, 1983 Accepted October 14, 1983
Combined Feedforward-Feedback Servo Control Scheme for an Exothermic Batch Reactor Arthur Jutan” and Ashok Uppal Xerox Research Centre of Canada, Mississauga, Ontario, Canada L5K 2L 7
Temperature control of an exothermic batch reaction is studied here. Only the energy balance is used in the derivation of the control algorithm. Coupling to the mass balance is through the heat generation term which can be predicted from the process. The two control variables, steam pressure and coolant flow, are related through a single variable. This variable is manipulated through the feedforward algorithm to compensate for the heat generation. I t is further modified through Dahlin’s feedback algorithm applied to the locally linearized process. Simulations are presented which show the effectiveness of the control scheme and its sensitivity to modeling and measurement errors.
Introduction The control of a batch reactor system provides an interesting challenge in that the system has no steady state. A servo control strategy is, thus, required to move the system dong a predetermined desired temperature vs. time trajectory. This problem has commonly been approached by using open-loop optimal control theory, in which the trajectory was obtained as a solution to, say, a “minimum time of reaction” problem. The difficulties encountered with these optimal servo control strategies was that no allowance was made for modeling errors and there was no feedback from the process through process measurements. The control strategies that were obtained were not robust and often quite sensitive to model differences. Their subsequent implementation on actual processes was often unsuccessful (Fm, 1973). In addition, any nonlinearities in the process model would greatly increase the computational effort required for the solution to the optimal problem and thus limit ita usefulness for on-line control strategies. This paper seeks to establish the feasibility of a new control algorithm for servo control of a batch reactor. This is accomplished through a computer simulation using a mathematical model of a pilot plant batch reactor. The next stage would be the actual on-line implementation, the details of which will be reported in a subsequent paper. Here, we develop a feedforward-feedback control algorithm for an exothermic batch reactor system. Both the feedforward and feedback algorithms can be separately tuned and both algorithms rely on process measurements (with superimposed measurement error) as well as a process model for control calculations. Dahlin’s algorithm is used for the feedback algorithm because of its ability to compensate for process dead time (Dahlin, 1968). The nonlinear heat generation term in the energy balance is treated as an inferential disturbance (predicted from process measurements) and a feedforward algorithm is 0196-4305/84/1123-0597$01.50/0
used to compensate for its effect on the controlled variable. Literature values were used to obtain typical reactor parameters for the model. The reactor dynamics were then simulated on a Data General Eclipse S/230. The control algorithms were tested by using superimposed measurement and modeling errors and results are compared to a standard three-term Proportional Integral Derivative control algorithm (Coughanowr et al., 1965)with derivative action on the process variable to avoid derivative kick due to set point changes. The details of the reaction system are unfortunately proprietory (Xerox Inc.) and discussion of this area will be in general terms only. However, the methodology presented here applies to any exothermic/endothermic batch reaction system; therefore this restriction should not cause any serious difficulties. T h e Batch Reactor Control Problem The need for a servo control system occurs during process startup and shutdown where it may be desirable to follow predetermined trajectories in say temperature vs. time. Usually, this part of the process control is performed manually since most control schemes are tuned for regulatory control of a system around some steady state. The current work was motivated because of the need to collect isothermal kinetic data from a series of batch experiments. In order to minimize data collection time, the reactor temperature had to be brought up to the required temperature as rapidly as possible with minimum overshoot and then this temperature had to be maintained for some period of time during which data were collected. Following this period, the reactor had to be quenched in a controlled manner. These requirements indicated that a time/temperature profile of the shape shown in Figure 1 should be used. Process Description. The reactor model used in this initial simulation study is based on an existing 2-gal batch reactor as shown in Figure 2. The reactor is filled with 0 1984 American Chemical Society
