percentage of success became as low as 19% in example 7.8 and 22%in example 7.9. On the upper side, MMM achieved a percentage of success of 100% in six examples, while SQL achieved a percentage of success of 100% in two examples (one of which is the obvious linear-quadratic example 7.1). For the nine examples, the cumulative percentage of success was 93% for MMM and 74% for SQL. From all these data, the higher robustness of MMM is apparent. (ii) Computer Time per Run. Inspection of the average computer time per run Tavshows that SQL is superior to MMM in five examples and inferior in four examples. However, when one looks at the effective computer time per run Teff, the situation is just the opposite: MMM is superior to SQL in six examples and inferior in three examples. (iii) Relative Efficiency Index. In section 6, the relative efficiency index E was introduced as a way of combining percentage of success with average computer time per run, while arriving at a parameter which is machine independent, to some degree. This efficiency index E was defined as follows
< 1.2 are not significant. In particular, this applies to examples 7.2,7.3, and 7.6. If one excludes these examples, then conclusion (iii) must be modified as follows: E < 0.8 in two examples and E > 1.2 in four examples. Therefore, even accounting for possible imprecision in computer time measurements, MMM compares favorably with SQL. (v) As shown by eq 31 and 32 and Table I, the relative advantage of MMM with respect to SQL should become more apparent for large systems (large n) involving many constraints (large q ) . Example 7.9, which includes n = 10 variables and q = 9 constraints supports this point of view (see Table 111). Acknowledgment This research was supported by the National Science Foundation, Grant No. GP-41158. The authors are indebted to Professor A. Miele for stimulating discussions. This work was done while the first author held a post-doctoral position with the Aero-Astronautics Group of Rice University, Houston, Texas. Literature Cited
E=
(Teff)SQL (Teff)MMM
-
(Tav)SQL (P)MMM
(104)
(Tav)MMM (P)SQL
This relative efficiency index is defined so that E < 1 indicates superiority of SQL with respect to MMM, while E > 1 indicates inferiority of SQL with respect to MMM. Inspection of Table I11 shows that E < 1 in three examples and E > 1 in six examples. Thus, from the examples investigated, we conclude that MMM compares favorably with SQL. (iv) The results of Tables I1 and I11 must be taken with a grain of salt, since computer times are precise only to f20%. This being the case, values of E in the range 0.8 I E
Hestenes, M. R.. J. Optim. Theory Appl., 4 (5),303-320 (1969). Isaacson. E., Keller, H. E., "Analysis of Numerical Methods," pp 34-37, Wiley, New York, N.Y., 1966. Luus. R., Jaakola, H. I., lnd. Eng. Chem., Process Des. Dev., 12, 380-383 (1973). Miele. A., Huang, H. Y.. Heideman, J. L., J. Optim. Theory Appl., 4,(4), 213343 (1969). Miele, A,, Levy, A. V., lyer. R. R., Well, K. H., "Modified Quasilinearization Method for Mathematical Programming Problems and Optimal Control Problems," "Control and Dynamic Systems, Advances in Theory and Applications," Vol. 9, pp 239-307, C. T. Leondes, Ed., Academic Press, New York, N.Y.. 1973. Miele, A., Moseley, P. E., Levy, A. V., Coggins, G. M., J. Optim. Theory Appl., I O , (1). 1-30 (1972).
Received for review August 26, 1974 Accepted May 14,1975
Control of a Binary Distillation Column with Sidestream Drawoff Bjorn Tyreus and W. L. Luyben' Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 180 75
The control of three compositions in a binary distillation column with a liquid sidestream was studied theoretically and experimentally. Different control schemes, all with sidestream flow rate as one manipulative variable, were simulated on a digital computer. None of these schemes gave satisfactory control performance due to limitations in the range of steady-state operability. A unique control system, in which sidestream composition was controlled by manipulating the drawoff location, was tested by simulation and experimentally. Overhead composition was approximately controlled by holding the temperature of a tray near the top of the column with reflux flow. Bottoms composition was similarly controlled via a tray temperature in the lower section of the column. Experiments with this scheme on a 24-tray, 8 in. diameter column showed good performance for load as well as set point disturbances.
Introduction Distillation columns with sidestream drawoffs are frequently used in the chemical and petroleum industries. Three or more products may be produced from a single column, thereby reducing the number of columns required and in some cases also reducing the energy consumption. Most industrial sidestream columns handle multicompo-
nent systems, but sidestream columns are occasionally used in binary systems to produce products of different purities. The use of sidestream columns in binary systems has, in the past, been infrequent because it has been more practical to use one of several alternative processing schemes to end up with two different products. One alternative is to produce an excess of the purest product and then blend this with the feed to obtain an inInd. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
391
n
l l
n
I
I
1
1
I -
i
Figure 1. Distillation column with basic instrumentation.
Table I. Average Operating Conditions; Methanol-Water System System pressure, 1 atm Feed rate, 3.29 l./min (0.87 gal/min) Feed temperature, 62°C Feed composition, 0.32 (mole fraction MeOH) Reflux rate, 2.35 l./min (0.62 gal/min) Reflux temperature, 52°C Distillate rate, 1.06 l./min (0.28 gal/min) Distillate composition, 0.95 (mole fraction MeOH) Sidestream rate, 0.75 l./min (0.20 gal/min) Tray 1 9 .temperature, 69.5”C Sidestream temperature, 71°C (tray 17) Tray 15 temperature, 73.5”C Sidestream composition, 0.71 (mole fraction MeOH) Bottoms rate. 1.51 l./min (0.40 gal/min) Bottoms composition, 0.02 (mole fraction MeOH) Tray efficiencies, 42% vapor phase i n stripping section; 30% i n sidestream section; 35% in rectifying section Reboiler holdup, 11 1. (600 g-mol) Reflux drum holdup, 5 1. (115 g-mol) Average tray holdup, 1 5 g-mol Deadtime i n reflux piping, 0.5 min Steam line pressure, 7 kg/cm2 (100 psig) Calandria steam pressure, 1.47 kg/cm2 (21 psig) Steam flow, 1.47 kg/min (3.3 lb/min) Heat loss, negligible termediate-grade product. Since the energy requirement for separation is a nonlinear function of product purity, the energy requirements for this system would be greater than for the sidestream system. More tankage and blending facilities would also be required. A more widely used alternative is to run a conventional column in a “blocked operation” scheme. The two products are made alternately and sent to storage for use as needed. This processing technique naturally requires more tankage, but it also introduces a significant control problem in swinging the column from one grade to another. The column dynamics can be quite slow. Time constants of 10-20 hr are not unusual in low relative volatility or high-purity systems. During these transitions, product would have to be run into low-purity tankage, thereby giving away product quality. The transition to the high-purity product 392 Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
would be particularly slow. Thus this system would require more energy than the sidestream column. These problems have been long recognized, but with inexpensive fuel, it was more economical to use an inefficient processing scheme than to take on the control difficulties of simultaneous control of three compositions. Energy economics have changed drastically in recent years, and more complex systems are now being used to reduce energy consumption. This paper presents the results of simulation and experimental studies of a binary sidestream column. I t reveals what the control problems are and presents a workable, practical control system. Thus a more energy-efficient alternative to the old brute-force techniques can be effectively employed. Little has appeared in the literature on the control of sidestream columns. Luyben (1966) has qualitatively discussed several control schemes for liquid as well as vapor sidestream columns. Buckley (1969) has similarly discussed some control schemes and the application of override controls. Shinskey (1969) suggested a specific control scheme for a deisobutanizer which has a single sidestream drawoff. Wood and Berry (1973) mentioned the increasing control problems with sidestream columns, although no analysis was made. Aside from the important question of energy saving, there is still another reason for studying binary sidestream columns. Since there are many multicomponent sidestream columns in use, there is a direct need to understand and master the dynamics and control of these highly complex units. By first looking a t the simpler binary case, insight can be gained about sidestream columns in general to facilitate the later study of the multicomponent case. This work is therefore intended as the first part of a broader program to study the dynamics and control of sidestream columns. Although the present work is limited to binary systems, some of the concepts may be applicable to multicomponent systems. For example, a petroleum fractionator producing different products a t various sidestream tray locations could crudely be treated by considering a single physical property (such as mid-boiling point, gravity, or viscosity) which varies continuously up and down the column, much as mole fractions vary in the binary system. The control problems encountered and the control technique developed in our study of binary systems may be directly extendable to these more complex systems. Experimental System A 24-tray, 20 cm i.d. bubble-cap column was used to separate mixtures of methanol-water a t atmospheric pressure. The column had a vertical thermosyphon reboiler, feed preheater, and overhead condenser and accumulator system. Figure 1 shows the basic setup of transmitters and controllers used in this study. Temperatures near the top and bottom of the column were controlled by pneumatic cascade controllers. The temperature sensors were Moore Products Nullmatic temperature transmitters (50 to 100°C range for top loop and 60 to 120° on bottom loop). The sidestream composition was continuously measured by a Princo Densitrol. All flow measurements were made with Foxboro pneumatic integral orifice D/P cells. The control system for the sidestream, to be discussed later, was implemented on a TR 20 analog computer (Electronic Associates, Inc.). Eight variables were recorded together with several tray temperatures, the latter on a Leeds and Northrup Speedomax multipoint temperature recorder. Average operating conditions are given in Table I. Feed and reflux temperatures were not controlled but were fairly
constant during the experiments (both were subcooled liquids). Theoretical Models Four different mathematical models were developed and used on the Lehigh University CDC 6400 digital computer. The models are: (I) a steady-state model for the existing column without a sidestream; (11) a dynamic model without a sidestream using model (I) as an initiator; (111) a steadystate model with one liquid sidestream for any tray; and (IV) a dynamic niodel with a sidestream using model (111) as an initiator. The steady-state models determine steady state conditions for the nonequimolal overflow, nonideal system. The dynamic models solve the nonlinear ordinary differential equations describing the column. Models I and 11 were used in the following ways. (A) Several steady states were obtained with the experimental system for the column without a sidestream. The temperature profiles were compared with results from nlodel I under the same conditions. IJsing tray efficiencies as parameters the model was fitted to the experimental data. Tray efficiencies of 40% in the stripping section and 35%in the rectifying section were obtained. (B) IJsirig the efficiencies from (A) in the dynamic m(~del (II), transient responses were simulated for different disturbances. The same disturbances were introduced on the experimental system, and the results were compared. In order t,o get good agreement between the results, the relationship between liquid flow from a tray and the holdup on the same tray had to be determined. A first-order lag was assumed to apply to each tray. The time constant was assumed to be the same f w all trays and was determined by fitting the experimental dynamic data (see Figures 2 and 3). A deadtime CJf 0.3 min, due to heat transfer delays in the reboiler, was also included in the model. Figure 2 shows the resulting dynamic response curves of the column when subject to a step change in reflux flow from 1.42 l./min to 1.07 l./min. As seen from the figure, the model was able to describe the pinch situation developed at the feed plate by the step change in reflux flow rate. Figure 3 shows the response, for a different case, to a pulse in reflux. The pulse had a magnitude of -50% of reflux flow and a duration of 1.25 min. During these tests only the level controllers were held on automatic while all other controllers were in manual. With the information obtained in (A) and (B), models 111 and IV were used to simulate the column with a sidestream and with various control schemes.
f
1 L
L
2 1 \
Figure 2, DyriamiL response curves to (1.42to 1 0 7 l./min)
T
2
P
step change in reflux flow
L ,
Figure 3. Dynaniic response curves to a pulse in reflux flow (-5096, duration 1 25 min)
No
-S
Dynamic Simulations The dynamic program (IV) was used to determine the nine transfer functions: xr,lR, X D / S ,x D / F s , xslR, xJS, xs/Fs, x B I R , x B / S , xs/Fs: through pulse testing and Fourier analysis. With the aid of these transfer functions, the loops in different control schemes could be tuned. The control schemes were thereafter tested for various disturbances, load as well as setpoint. Figure 4 shows the three schemes that were tested. The other three possibilities of control schemes for the variables considered were not found interesting since two of them involve nested control figurations and the third is a trivial variation of scheme no. 1. The loops were initially tuned according to the Ziegler-Nichols criteria and corrections were made afterwards to decrease interaction. Results All three configurations had one great disadvantage. They could not maintain sidestream composition after a
Figure 4. Differedit control schenies simulated puter.
011 a
digital com-
setpoint change in overhead composition. A small decrease in the overhead composition setpoint resulted in a situation with no sidestream flow at all. A small increase, on the other hand, forced the controllers to take out the maximum flow of sidestream, but they were still not able to hold sidestream composition. Control strategy no. 2 was very interesting and could not handle a load disturbance of 10% change in feed composition. Control strategy no. 3 could Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
393
-
I
0.80
0
erational constraints or if we had pure dynamic instability problems.
t
,
1 02
34
SIDESTREAV
0.G
0.8
FLSWRATE, S (GMOLISEC.)
Figure 5. The composition of a sidestream from tray 17 as a function of the drawoff rate.
0.651
3 2
04
5
(--MOL
08
00 I
SEC I
Figure 6. Comparison of sidestream compositions from different trays for two values of XD.
-
I j N j l 7 , XD=0.95 0.70
NC18, X d 0 . 9 4 0.2
0.4
5
0.0
0.8
(G-MOL.ISEt)
Figure 7. Comparison of sidestream compositions from different trays for two values of XD.
handle a load disturbance and was superior even to strategy no. 1. In strategy no. 1 there was a high degree of interaction between the top loop and the sidestream loop. This interaction was reduced by loosening the top loop and/or by inserting decouplers between the top loop and the sidestream loop. The inability to handle overhead setpoint changes however still persisted. The fact that the suggested control schemes could not handle overhead setpoint changes made them useless for this system since the type of disturbance mentioned would probably be quite common in an actual plant. To find the cause of the failure of all these control systems we checked to see if we had run into steady-state op394
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
A Modified Control Scheme Steady-State Considerations. In order to explore what happened to the column for setpoint changes in XD, sidestream composition was calculated as a function of sidestream flowrate a t steady-state conditions with model 111. Overhead composition XD served as a parameter and x g was kept constant. Results are shown in Figure 5. Figure 5 clearly reveals why it is impossible to hold sidestream composition using sidestream flowrate as a manipulative variable in any configuration. There is simply no overlapping of the curves even for such small changes in XD as f0.01 mole fraction. Suppose, for example, that we were operating with an overhead composition of 0.95 mole fraction and a sidestream flowrate corresponding to point A in Figure 5. The sidestream composition would then be at point B. If we now changed overhead composition to 0.94 mole fraction and maintained the sidestream flowrate, we would be operating at point C. Since any change in sidestream flowrate would move us along the curve D-E, we would never be able to reach point B again. In fact the closest we would get is to point D, but then we have no sidestream flow at all. The problem could, however, be overcome by using liquids from several trays and mixing these into one sidestream. Figure 6 shows that either the liquid from tray 16 or a mixture of liquids from trays 15 and 17 could be used to maintain sidestream composition despite a change in X D from 0.95 to 0.96 mole fraction. Figure 7 similarly shows that either liquid from tray 18 or a mixture of liquids from trays 17 and 19 could be used to hold x s for a change in XD from 0.95 to 0.94 mole fraction. Dynamic Results. The case where a sidestream was drawn from tray 17 and mixed with streams from either tray 15 or 19, when needed, was simulated in model IV. Overhead composition was controlled by reflux flow and bottoms composition by heat input. Feed, with a composition of 0.32 mole fraction, was introduced on tray 11. The simulations showed excellent control for all kinds of disturbances. Since in this scheme one degree of freedom is used to vary sidestream location, sidestream flowrate can be held constant. Just varying the amount of sidestream drawn from two or more trays, when the overall sidestream rate is constant, turns out to have very little effect on overhead and bottoms compositions. This means that no extra interaction is introduced as was the case in the control schemes previously discussed. Physical Realization The existing column had to be modified to withdraw liquid sidestreams from three trays. This was achieved by inserting a %-in. 0.d. copper tube into the column on trays 15, 17, and 19. The open end of the tube was positioned close to the downcomers and submerged in liquid. The pressure in the column was enough to fill the tubes with liquid. The rest of the driving force, to get liquid out of the column, was the liquid head from the tray down to a product tank. On each of the three streams there was a control valve (linear trim, no positioner, air to open, Cv = 0.5). The outlets from these valves were mixed and run through a flow sensor and a Densitrol in series. Figure 8 shows the basic setup of the valves, the flow meter, and the Densitrol. Figure 8 also shows the control scheme used to vary sidestream drawoff location and at the same time hold the sidestream flowrate constant. Four fixed gain relays, three
*-
11 Figure 10. Nyquist plots of G for various values of K,1.
F l
CT t
,
.
1x1
L--~,
~- M U L T I P L I E R ~-F I X E D G A I N R E L A Y
Ls ~- ~-L O W S E L E C T O R Figure 8. Setup of valves, controllers, and computing hardware for sidestream control. 1
O U T P U T F R O M C O M P O S I T I O N CONTROLLER (
Kg/crn2 )
Figure 9. Control signals generated by fixed gain relays. multipliers, and a low selector were used to generate the signals shown in Figure 9. In this study the relays, multipliers, and the two sidestream controllers were implemented on the analog computer. Only one piece of equipment for direct composition measurement was available, namely the Densitrol on the sidestream. Control schemes with intermediate tray temperatures were chosen in order to hold bottoms and distillate compositions near desired values. The reason why base and top tray temperature should not be used directly is that the equilibrium line ( T vs. x ) is very flat a t high purities for methanol-water. The location of the bottom loop sensor was chosen to be on tray 5, where the temperature profile began to break. The location of the top loop sensor was chosen similarly to be tray 20. Tuning of the Controllers. As indicated earlier, it was believed that the sidestream drawoff would not deteriorate the control system significantly. Therefore the tuning procedure was concentrated on the top and bottom loops in order to reduce interaction between these two. The sidestream was not manipulated in this procedure. Through pulse testing and Fourier analysis of the dynamic model (IV) the four transfer functions P11, P I Z ,Pzr, P22 were determined. These are
Figure 11. Nyquist plots of G for various values of
TI^.
Dimensionless gains are given in the transfer functions above; i.e., flow and temperature transmitter spans have been included. When the top and bottom loops were tuned independently, using these transfer functions and ZieglerNichols criteria, the following were obtained: top loop controller (&): Kcl = 62, 711 = 0.5 min; bottom loop controller (Bz):kc2 = 4, 712 = 1 min. For an interacting control system, however, it is generally not safe to tune the loops independently if decoupling has not been provided for. The stability of the total closed-loop system depends on both controllers B1 and & as given by the characteristic equation (1). l+G=O
(1)
where C = BlPll + B2P22 + BlB2(PllP22 - P12P21). In order to explore the stability region for the system, Nyquist plots of G were made. Figure 10 shows these plots for different values of Kcl and Ziegler-Nichols a values for Kc2, 712, and 711.The plots reveal that the system is unstable for Kcl = 45 and that it is stable but has a very small phase margin for Kcl = 20. The system is made more stable with decreasing Kcl. Figure 11 shows how the phase margin could be improved by increasing 711 when Kcl is held constant. The reason for backing off on the top loop settings as opposed to backing off on the bottom loop settings was that sidestream composition was affected more by changes in reflux flow than by changes in vapor boilup. This is easily seen from the steady state gains (SSG) of the transfer functions of the system.
SSG of T2oIR = 16 SSG of T5IFs = 140
("C/g-moleIsec) (OClkgImin)
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
(2)
(3) 395
0.37,
S
15
"B D B L* 5
10
TIME
I3 1
15 20 (MINUTES1
-
4
25
30
15
20
25
(MINIJTESI
Figure 13. Setpoint change on sidestream composition controller (-0.025 mole fraction).
(OC/g-mole/sec)
(4)
(OC/kg/min)
(5)
(4) and (2) give the ratio T17/T20 = 0.9. (5) and (3) give the ratio T17/T5 = 0.08. The ratios show that the reflux change necessary to compensate for a 0.5OC disturbance on tray 20 will bring about a 0.45OC disturbance on tray 17. In order for steam to make the same change in tray 17 temperature we need a 5.5OC change on tray 5 temperature. This indicates which manipulative variable has the stronger effect on sidestream composition. Since we are interested in making as small upsets in sidestream composition as possible it seems best to make changes in reflux more gradual. The final settings were chosen to give a gain margin of 2.4 and a phase margin = 30': top loop: K,1 = 7, 711 = 2 min; bottom loop: K,2 = 4, 7 1 2 = 1 min. The new control system was simulated and tested on model IV. A 0.5-min measurement delay on sidestream composition was included in the model to correspond to the deadtime in the sidestream piping and the density analyzer. The sidestream controllers were empirically tuned to give a stable system with the smallest possible value of the integral of the absolute error in sidestream composition. Initially two PI controllers were used to manipulate sidestream composition and sidestream flowrate. It was, however, found in the simulations that the proportional action was undesirable on the composition controller. Only very small values of K , were allowed for the loop to remain stable. When an integral-only controller was tested, the system showed excellent performance. The flow controller could also be changed to an integral-only controller but this time with no noticeable effect on system stability. The empirically found settings were: composition controller: 71 = 2 min; flow controller: T I = 0.1 min. Experimental Results The experimental system was tested for several setpoint changes and load disturbances. Figure 12 shows the case where a setpoint change in tray 20 temperature was made. 396 Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
10 TIME
troller; corresponding to a 0.01 mole fraction decrease in XD.
SSG of T17IFs = 11
I__-_
L
5
Figure 12. Setpoint change (t0.6OC) on tray 20 temperature con-
SSG of T17lR = 14
::t
Figure 13 shows the response of a setpoint change in sidestream composition. The XD compositions shown on these figures were calculated from tray 24 temperature measurements. A typical operating situation might be that of Figure 14 where a demand was made of increased purity on overhead as well as sidestream compositions. In this case a change in X D from 0.94 to 0.96 and in x s from 0.675 to 0.69 mole fraction was requested. Setpoint changes in tray 5 temperature could also be made as in Figure 15. As pointed out earlier, a setpoint change in tray 5 temperature caused much less upset in sidestream composition than a change in tray 20 temperature. Finally, the response to a change in feed flowrate is shown in Figure 16. This disturbance, like all the other disturbances, was tried for two different feed compositions, namely 0.24 and 0.32 mole fraction. Figure 16 shows the case of a 16%increase in feed flowrate when feed composition was maintained at 0.24 mole fraction. Sidestream flowrate was observed to have sharp spikes whenever a valve opened. The phenomenon is believed to have been caused by "sticking valves" and can probably be avoided if valves with positioners are utilized. The theoretical improvements to be made with valve positioners can be visualized in Figure 17. Here the experiment from Figure 12 is compared with a simulation under the same conditions. The spikes in sidestream flowrate show when valve 19 opened in the experiments. In the simulation, without sticky control valves, compensation for the error in sidestream composition was made much earlier and thus reduced the integral of the absolute error. A few remarks should be made about the design of the sidestream trays. In our case we could most likely not avoid having some vapor bubbles enter the drawoff tubes and condense on their way down to the flowmeter. If this occurred, no problems arose as long as the same fraction of vapor entered at all instances of time. If, however, the balance was disturbed the tray above would experience a change in temperature. For the particular case where the sidestream was drawn from tray 19 alone, an interaction of the sort indicated above took place. Since the top loop sen-
xD R
" " 1 0
0')
,
1
XS
s
S
0.41
,
,
,
,
,
03
I
T5 14
1 3
"E3
D B
;:-------I 5
10
15 20 TIME ( M I N U T E S )
J
Figure 14. Simultaneous changes in tray 20 setpoint and side-
10 15 20 TIME (MINUTES)
5
30
25
30
25
Figure 16. A 16%increase in feed flowrate.
stream composition setpoint. 0
0 0
0.4
S
EXPERIMENT
"
03
0
0
0
0
"
~
0 0
a
-
T5 "
0
0
0
0
2
3
3
0
0
,
~
.
.
vB
B
1.3 1.2
b
P
I 5
10
20 TIME ( M I N U T E S ) 15
D
0.5
point. sor was located on tray 20, only a small change in the conditions on tray 19 would cause a change in reflux rate and consequently on liquid rates in the column. The new reflux rate would not only restore conditions on tray 20 but also change them on tray 19. Therefore a sustained oscillation in reflux and sidestream flows was noted for this particular case. No such oscillation was recorded in the simulations for the same situations but in the simulation a pure liquid drawoff was assumed. I t can therefore be concluded that the sidestream trays should be designed to assure a liquid only sidestream.
n
c
c
c
c
o
5
25
Figure 15. A 1 . 2 O C change in tray 5 temperature controller set-
0
OA
5
15
10 TIME
20
25
30
(MINUTES)
Figure 17. Comparison between experimental and simulated data for a setpoint change on tray 20 temperature controller.
Conclusions Simulation and experimental studies of the dynamics and control of binary distillation columns with sidestreams revealed that manipulation of sidestream flow rate was very ineffective. The reason for this was that sidestream flow rate had very little effect on sidestream composition, causing material balance constraints to be encountered for very small disturbances. Varying the location of the sidestream withdrawal tray was found to give good control of sidestream composition. Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
397
Simultaneous closed loop control of overhead, bottoms, and sidestream compositions was achieved both in simulation studies and experimentally. Extension of these binary results to multicomponent systems may be fairly direct, particularly in petroleum fractionators where a single physical property like mid-boiling point, gravity, or viscosity can be used for each sidestream. Work is currently underway in our laboratory to extend these studies to ternary systems. Nomenclature B = bottoms flow rate, g-mol/sec BL = top loop controller transfer function Bz = bottom loop controller transfer function CC = composition controller CT = composition transmitter D = distillate flow rate, g-mol/sec F = feed flow rate, g-mol/sec F s = steam flow rate, kg/min FC = flow controller FT = flowtransmitter G = system transfer function K,1 = top loop controller gain K,z = bottom loop controller gain
LC = level controller L T = level transmitter N s = sidestream tray location P = process transfer function R = reflux flow rate, g-mol/sec S = sidestream flow rate, g-mol/sec s = Laplace transform variable TN = tray temperature, "C TC = temperature controller TT = temperature transmitter VB = vapor boilup, g-mol/sec XB = bottoms composition, mole fraction XD = distillate composition, mole fraction x s = sidestream composition, mole fraction 71 = controller reset time, min 711 = top loop controller reset time, min 712 = bottom loop controller reset time, min Literature Cited Buckley, P. S., Chem. Eng. Progr., 65 (5),45 (1969). Luyben. W. L., /SA J., 37 (July 1966). Shinskey, F. G., OilGasJ., 67, 147 (28, 1969). Wood, R. K., Berry, M. W., Chem. Eng. Sei., 28, 1707 (1973).
Receiued for review September 9,1974 Accepted May 27,1975
Optimal Tuning of Digital PID Controllers Using Dynamic-Stochastic Models John F. MacGregor, J. D. Wright,*,' and Huynh Man Hong Department of Chemical Engineering, McMaster University, &milton, Ontario, Canada L8S 4L 7 and Systems and Engineering Division, Process Control Department, Alcan, Arvida, Quebec, Canada
A method is presented for tuning digital controllers by modelling the disturbances in the process by an autoregressive-integrated-moving-average type of stochastic model and the process dynamics by a discrete transfer function. The criterion of optimality is minimum variance at the output possibly subject to a constraint on the variance of the manipulated variable. This procedure is intended to cover the situation where one is limited to a specified form of controller such as PID controllers. The method is applied and compared with other methods on two actual processes: a steam-jacketed continuous stirred tank reactor and a system of heat exchangers in series.
1. Introduction
A process sampled a t discrete equispaced intervals of time can often be represented by a discrete linear transfer function model of the form
In this equation the backward shift operator B is defined such that But = ut-1 and Bkut = U t + . The discrete transfer function model V ( B ) relates the dynamic characteristics of the output deviation Yt from equilibrium to the deviation of the input variable ut from its corresponding equilibrium value. It can usually be expressed as the ratio of two finite polynomials in B, w(B) = (COO - wlB - . . . . . . - w,BS) and 6 ( B ) = (1 - blB - . . . . . . - 6,B') plus a delay term Bb where b represents the number of whole periods of Address correspondence to this author at McMaster University.
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delay in the sampled process. The term Nt represents the noise in the process output. It measures the joint effect a t the output of all unobserved disturbances and would represent the deviation from target that would occur in the output at time t if no control actions were taken. I t is the presence of this noise that dictates the need for control action. Discrete time series models capable of representing such correlated noise processes were first investigated in the 1920's by Yule (1927) and by Slutsky (1937). More recently Box and Jenkins (1970) have presented a unified theoretical and practical treatment of these autoregressive-integrated-moving-average (ARIMA) time series models. The general ARIMA model of order (p,cl,q) can be written in the form
