ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT
li
n
THEODORE J. WILLIAMS USAF Institute o f Technology, Wright-Patterson Air Force Base, Ohio
ROBERT T. HARNETT Wright Air Developmenf Center, Wrighf-Patterson Air Force Base, Ohio
ARTHUR ROSE The Pennsylvania State University, University Park, Pa.
P
REVIOUS ~ o r khas shown, through theoretical computer calculations, that very close control of a radically varying
feed composition in continuous distillation is entirely possible if the proper precautions are taken in the choice of column sampling locations, the type of control utilized, and the magnitude of the controller constants employed. This article reports additional studies t o show, first, t h a t the problem of varying feed rates can be treated in much the same manner as those of varying feed compositions if an additional separate controller is used t o maintain the boilup rate as a definite multiple of the feed input rate. Secondly, the effect of an imperfect sampler-that is, one incapable of detecting small variations of the sampled variableon the over-all controllability of the column and on the optimum location of the sampling element has been specifically investigated. I n the previous work, by far the best control was obtained when sampling on the top plate. It was also found that the effect of the larger error signal possible on the intermediate plate was counterbalanced by an instability induced through the fluid flow lags between the plates. It was therefore recommended t h a t top-plate sampling be used. As this recommendation was contrary t o current industrial practice, this intensive investigation of its true applicability was indicated. Thirdly, some effects of the nonidealities in column operation which may arise under transient conditions are discussed; these include (1) a reflux stream whose temperature is different from t h a t of the boiling liquid on the top plate and (2) a feed that is maintained a t a constant quality rather than a constant temperature (varying quality) as was previously considered. Also reported here are the results of a theoretical investigation of a method of column control that utilizes a control of the feed input rate. With this method, as the feed composition varies, the feed rate is controlled so as to keep the rate a t which the more volatile
I
i Control of Varying Feed Rafes and Effect of Column Design on Over-All Controllability
Arthur Rose R. T. Harneft T. J. Williams
Effect of imperfect S a m p h g on Control Response and Optimum Sampler location
T. J. Williams
Effects of Feed ond Reflux Quality and Comparison of Input vs. Output Control
1008
T. J. Williams R. T. Harnett
component enters the column as nearly as possible a constant. T h a t is, the product (Fz,) is maintained constant. A comparison of this method with one employing output sampling is given.
A previous paper (14) presented the equations necessary for carrying out the calculations required in order to follow the progress of the automatic control of a continuous column while it is subject to feed composition changes. Let us n o v regulate the boilup rate (VI)so that it is, as nearly as possible, twice the input feed rate. If the previously described composition controller can still maintain the distillate rate to ensure an adequate separation, then a distillation system which can handle any reasonable variation of input conditions is a t hand. Heat) capacity lags in the steam chest of the column will result in a transient effect when changing the boilup rate. Therefore the boilup rate must be expressed as a function of the steam rate. If the assumption of a first-order exponential lag can again be made, this expression will be
A rate measuring device when placed in the feed input line can generate an error signal by comparing the actual instantaneous feed rate with the mean rate. This error is specified as
e(F) = ( F
-
F,)
(2)
Then, by a relation similar to Equation 10d in reference (I-$),
Because of the nature of a rate-measuring device as compared t o a composition-measuring device no relation similar to Equation 9 in ( 2 4 ) is necessary here. Equation 3 is of course the equation of action of the boilup rate controller. Here k refers to the proportional constant-Le., the gain required for proportional control; k' is the gain for derivative control, and k" the gain for integral control. They correspond to the terms K,K', and K" of the composition controller discussed in ( 2 4 ) . Column parameters used in this study were as follows: 1. Holdup, plate, H , t o H6 2. Holdup, stillpot, HI
3.
Feed, F
4.
Feed quality, q
INDUSTRIAL AND ENGINEERING CHEMISTRY
2.33 1b.-moles/plate 23.3 1b.-moles or 2.33 1b.moles 200 1b.-moles/hour or 3.33 1b.-molesiminute 0.0 t o 1.0,initially 0.5
Vol. 48, No. 6
PROCESS CONTROL
Nomenclature = distillate or tops rate, moles per unit time; a s sub-
D
script refers to distillate E ( x ) = error in distillate composition as transmitted to cone
Fo
= = =
H
=
J
=
K
=
k
=
L
=
P P S
= =
T
=
t V
=
W
=
X
=
Xfo
=
v
=
01
=
F
=
=
a: :b
P
troller base of Naperian logarithms, exponential feed rate, moles per unit time reference feed rate, moles per unit time holdup, amount of material entrapped on any plate, moles constant relating steam condensation rate to boilup rate (all forms) constants in equation of composition controller response (all forms) constants in equation of boilup rate controller response liquid rate, moles per unit time; subscript refers to location of stream differential operator, d/dt quality of feed-Le., fraction as saturated liquid steam condensation rate, pounds temperature time vapor rate, moles per unit time; subscript refers to location of stream bottoms product rate, moles per unit time; subscript refers to bottoms liquid composition, mole fraction of more volatile component; subscript refers to location of stream reference feed composition, used in evaluating Q and feed plate location vapor composition, mole fraction of more volatile component relative volatility
= equilibrium relation, II =
1
+
OiX (a!
- l)z
= function used in evaluating q
A D = required change in distillate rate e(x) = error in distillate composition
e ( F ) = error in feed rate 7
=
1
time constant if part of transfer function, rp 1'* true time delay if part of real translation term, e - T p
+
Subscripts C = condenser f = feed
0.5 f 0.12 or 0.35 f 0.12 Initially 4: 1 2.65 when xf, = 0.5 2.97 when x f o = 0.35 5.0 Relative vdatility, a: Liquid flow exponential lag, T Z 0.1 minute Feed input exponential lag, 7 4 0.1 minute 0.5, 1.0, and 2.0 minutes Condenser delay times, 7 6 Mixing lag in cornposition 0.1 minute samplers, 7 6 2.0 minutes Heat transfer lag, 7 7
5 . Feed composition, z t 6. Reflux ratio, L I D 7 . Quality constant, p 8. 9. 10. 11. 12. 13.
These particular values were chosen since they correspond to those of the plate design problem of Robinson and Gilliland (IO) and provide a convenient basis for the study of column behavior. Figure 1 is a diagrammatical representation of the complete distillation setup as visualized here. The feed inlet temperature control and the condenser water control are assumed to be external t o this problem as previously mentioned. Figure 2 presents the block diagram of the theoretical model-i.e., a physical representation showing each of the terms in the various pertinent equations ( 1 7 ) in their proper physical relatibnship t o each other. Results of Tests. Three methods of column sampling were effective in controlling composition changes-sampling the top plate or an intermediate plate alone or sampling simultaneously from both the top and the bottom plates. I n the latter case the top plate error was used if the top plate composition fell below
June 1956
I
NOT CONSIDERED IN PROOLLY
Icw.Ns,E BETTOMS PRWLICT
Figure 1.
Distillation column automatic control
a certain predetermined value. The bottom plate error was used in case the bottom plate composition rose above another predetermined value. Each of these three methods gave satisfactory control. However, of the three, the control afforded by the top and bottom plate, two-point sampling method was somewhat less satisfactory than the others. Top plate, single-point sampling gave by far the best results. If now the composition controller is to be depended on t o maintain product compositions and if the two-point sampling method proves satisfactory, all three methods will have been proved. As mentioned previously, the boilup rate controller was used merely t o maintain a balance in column flow rates. Let us first consider the effect of the composition controller alone-i.e., without the boilup rate controller. Figure 3 shows the results of such a test. Here the composition controller is able t o compensate for the changing rates but only at a sacrifice in the output compositions. The effect of a changing separating efficiency as the relative magnitude of the feed rate and boilup rate vary is quite evident between the right- and left-hand sides of Figure 3. A constant boilup rate in combination with a decreased feed rate means, in effect, a higher reflux ratio and thus a wider separation between the distillate and bottoms compositions. The left-hand side of Figure 4 shows this effect. On the other hand, an increasing feed rate gives effectively a lower reflux ratio and thus a decreased separation as is shown by the righthand side of Figure 4. The effect of adding the boilup rate controller is shown by comparing Figures 3 and 4. Considerable improvement is shown by the boilup rate controller with a proportional constant of 1.0 and an even greater one when the controller is equipped with a proportional constant of 2.0. The fluctuations present in the plate 5 compositions of this test are signs of instability which with larger proportional constants render the column unstable-i.e., the boilup rate controller causes rate fluctuations beyond the ability of the composition controller to correct, and the distillation system becomes unstable. A small amount of integral control helps smooth out some of this instability. However the addition of a still larger amount of integral control again adds instability t o the system. Since there is no delay between the appearance of a feed rate change a t the point of flow measurement and its detection and use by the controller, derivative control would call for a n infinite correction at the moment of change but no correction thereafter.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1009
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Effect of Column Design on Over-All Controllability
M S S TRPHSFER AND HYDRAULIC LAG
,
FEED f *f
p’@JLCI\>URIRW
LAB
W
Figure 2.
Distillation column automatic control-dynamic relations
It would therefore be completely unsatisfactory as a control medium in this case. When i t was attempted t o control a sinusoidal variation in feed rate, the top plate fluctuations obtained were similar for both types of feed rate variation for similar values of the controller constants. However, in no case that gave a satisfactory top plate response, was satisfactory control of a sinusoidally fluctuating feed rate obtained. That is, the boilup rate was not kept reasonably constant during the period of feed rate change. It must be kept in mind, of course, that the use of top-plate sampling alone Iyould have allowed much larger values of the constants of the boilup rate controller. Thus very possibly a complete control of the boilup rate could have been obtained for a sinusoidally varying input. Figure 5 summarizes the attempts t o provide for boilup control for all types of feed rate changes. The limiting factor is the ability of the composition controller l o handle the rapid flov variations caused by the boilup rate controller and not that of an actual instability in the boilup system. The response of the boilup system alone is very simple and could readily be worked out by classical automatic control analysis methods ( 1 7 ) . However, the response of the rest of the column system to the AOTV variations imposed by the act of correcting the boilup rate is the important factor, and this response must be worked out by the methods explained herein. Figure 5 and each of the summary graphs similar to it shorn a point labeled “Recommended Control Point.” This point represents t h a t combination of controller constants which gives the closest control (minimum error) consistent with rapid response, minimum overshoot, and minimum circuit “noise.” It must be realized t h a t the choice of this point was in many cases somewhat arbitrary since each graph contains many combinations of constants t h a t would give satisfactory response. Some idea of the actual errors involved can be obtained by studying Figures 22, 24, 26, and 27.
1010
Effect of Value of Alpha. The previous investigations of the optimum controller constants to be used by the composition controller for top-plate sampling alone and intermediate-plate sampling alone have been repeated for the condition of a relative volatility ( a )of 3.0 rather than the previously utilized value of 5.0. The results are summarized graphically in Figure 6 for top-plate sampling and in Figure 8 for intermediate-plate sampling. These figures should be compared with Figures 7 and 9, respectively. The results may also be summarized in words by saying that for the top-plate cases the proportional control alone is more stable, while the integral control is less so. On the other hand, for the intermediate-plate case, proportional control is less stable, and the integral control is more so. This apparent contradiction is due to a combination of two factors. First, as was shonm by other work of this investigation ( I S ) , the distillation system response becomes much faster as the relative volatility is reduced. Therefore the column responds more rapidly to the corrections applied by a proportional controller thus helping t o reduce overcorrections and resulting oscillations. I n integral control, the necessity of integrating out a previous correction before applying a new one requires a slow system t o prevent an oscillation. Thus, for this more responsive system, integral control becomes more oscillatory. Secondly, again because the distillation system is more responsive, the fluid flow lag now becomes relatively more important. This results in an opposite effect to t h a t just mentioned for top-plate sampling when one considers intermediateplate sampling and compares Figures 8 and 9. Thus, proportional control is now less stable. Integral control also now shows a n-ider oscillatory region before becoming unstable. This latter analysis n-ould also hold for bottom-plate sampling alone and would result in greater instability for that configuration also. Effect of Magnitude of Condenser True Time Delay. All previous computations of this study have been made using a condenser true time delay ( T ~ of) l/z minute. The effect of delays of 1 and 2 minutes is now considered. Since the top-plate-only sampling arrangement has shown itself able to counteract the effect of condenser delay, the method of sampling the distillate line only was used in these tests. The results for a 1-minute delay turned out to be rather surprising t o these authors. As is s h o r n by Figure 10, a better control response was obtained than that for a l/Z-minute time de1a.i. (Figure 11); this is of course contrary to the normal expectation. Computer graphs of the condition with a 1-minute time delay showed a much sloiver natural frequency of oscillation than that for a 1/2-minute delay with apparently a higher frequency superimposed. Since the faster natural frequency of the system gives a period that approximates the actual time delay of the condenser, the column is apparently deriving its stability by sampling one cycle ahead of that actually being corrected. Thus the 1-minute delay gave a fortuitous condition not usually present with a delay of this type. This fact was exemplified by consideration of the effects of a 2minute true time delay v7here control was completely unsatisfactory. Effect of Magnitude of Still Pot Holdup. While the holdup of the normal-sized still pot is of the order of ten times that of a normal plate, cases arise, particularly with heat sensitive materials, when a much smaller still pot holdup is necessary. It is therefore of value t o consider the effect of such an arrangement on the controllability of the column. The summary graph for top-plate sampling for this still pot holdup arrangement is essentially identical to Figure 7. The slight variation of the upper limit of the proportional constant between the t r o figures is considered insignificant.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL 0.955
0.046
XI
5
0,030
0.m
0.0l.4 0.83
0.195 0.63
P
P
3.33
3.33
5.83
5.83 -4.0
0.955
5
U-V,
0.675
0
t4.0
0.795
Figure 3.
Figure 4.
Response of column to feed rate change
K
Response of column to boilup control = 2.0, K’ = 0,K” = 0.01
Composition controller sampling on top and bottom plates K = 16.0, K’ = 0, K” = 0.16
The effect of bottom plate sampling is considered in Figures 12 and 13. The summary graph for the small still pot (Figure 12) when compared with the summary graph for a normal still pot (Figure 13) shows the present example to be much more susceptible t o instability than the regular holdup case. The normal still pot, because of its large holdup, is very slow to change and thus acts with an important stabilizing effect on the rest of the column. With the loss of this stabilizing effect, the full impact of the plate fluid flow lags is exhibited, thus ensuring a relatively poor control response for the.bottom plate. The response of all the intermediate plates is altered toward instability to a degree between t h a t shown and the negligible change of the top plate.
Effect of Imperfect Sampling on Control Response and Optimum Sampler Location
spectively, of the composition being sampled. These are discussed as 1, 2, and 5% dead space, respectively. In order t o establish an order of magnitude for these dead spaces in terms of temperature if the detecting element were a thermocouple, resistance thermometer, temperature bulb, or other such device; the equation of Rose ( 1 8 ) was employed. By this means the dependence of composition on temperature for any specified relative volatility can be expressed. A base temperature of 100’ C. was used. Figure 15 shows vividly the necessity of using other means than temperature as a detecting medium for low relative-volatility mixtures in order to reduce the dead space to the limits t o be specified later. Similar studies can be made for each of the other methods of sampling and indeed must be made in order to choose the best possible method.
The task of the sampling device of any feed-back type automatic controller is to detect any and all changes in the value of the controlled variable away from the previously set control point. The sampler need not be able t o determine,the absolute magnitude of the variable it is monitoring but must only be able t o detect variations in this quantity. A common error in these devices is the presence of “dead space” or the inability to detect or transmit the smaller variations in the controlled variable away from the set point. This type of imperfection can be represented graphically (Figure 14) and mathematically as follows: E(Z) = (Z
where x
- Zdesired)
- (D.Z.1
(4a)
> Zdeslred, and 00
(Z
- x d e s i r e d ) > (D.Z.) E(z) = 0
where
(Z
- zdesired) 5 E(z) =
1 0
30
40
PROPORTIONAL
(4b)
Figure
60 CONSTANT
60
70
80
5. Relationship of controller constants for boilup rate controller
(D.Z.) (Z
IO
- zdasired)
+ (D.Z.)
(4c)
Composition controller sampling top and bottom plates K = 16.0, K” = 0.16
where z < Zdesired, and (Z -
%desired)
> (D.2.)
I n all these cases z is the controlled variable, x d e s i r e d is the set point, D.Z. is the dead zone and R(z) is the error as finally transmitted to the controller. Of course this transmitted error would also be subjected to any mixing lags or other such effect present in the sampling chamber as discussed in earlier works (14, 19). The calculations of this study were carried out using plate composition as the controlled variable and establishing the dead space as equivalent to 0.01, 0.02, and 0.05 mole fraction, re-
June 1956
It is also interesting t o consider what values of dead space present-day instrument manufacturers specify for their products. Dead space is usually spoken of in the advertising literature as sensitivity and, as expected, varies widely depending on the instrument chosen. For temperature-measuring instruments the variation is from 1” t o 3” C. for the simpler relay-operated types to 0.015” C. or even lower for the more complex electronic and pneumatic types available today (2, 7 , 16). On-stream, infrared analyzers are now being developed with a sensitivity of (8).
INDUSTRIAL AND ENGINEERING CHEMISTRY
1011
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Calculation Procedure. The previous computation work Thowed t h a t if a particular sampling arrangement and set of controller constants could adequately control a variation in feed composition it could also handle, either separately or simultaneously, a variation in feed rate. I n other words the maintenance of a definite plate composition was the more difficult task. Also, i t was established that a step variation (sudden change) in feed composition was more likely t o cause instability for a, particular control setup than was the application of a sinusoidal or continuously varying input. Finally, derivative or rate control was shown t o be inapplicable. With these facts in mind a considerable portion of the previous work v a s repeated for each of the dead zones mentioned in order to establish the regions of stability, oscillation, and finally of instability for various combinations of controller constants and t o determine the dependence of final output composition variation on the sampler location involved and the dead space postulated. Effect on Column Controllability. Figures 16, 17, and 18, give the response diagrams, respectively, for top-plate sampling only and for 1, 2, and 5% dead space. These should be compared with Figure 7 which also gives the results obtainable with no dead space in the sampler. Several facts immediately present themselves to the reader on studying these figures.
1."
PROPORTIONAL
Figure 8.
1012
CONSTANT
1. The presence of any amount of dead space results in an oscillatory region a t high values of the proportional constant which is not present for the example of no dead space. 2. I n addition, for small dead spaces instability occurs a t much smaller values of the proportional constant. As the size of dead space increases the point of onset of instability returns to a value which is even larger than that for the perfect sampler. 3. Perhaps the most startling result evident from these figures is the effect of sampler dead space on the stability with integral or reset control. First, the oscillatory region noted in the upper
Table I. Effect of Dead Space on Point of Onset of Instability for Large Values of Integral Constant Dead Space, Mole Fraction
Point of Oscillation
Point of
Instability
Top-Plate Sampling 0 0.01 0.02 0.05
0.3 '
0.6
... ... ...
0.7 3.7 18.5
Intermediate-Plate Sampling 0 0.01 0.02
0.05
0.0 0.0
0.03 0.04 0.05
0.0 0.0
.. PROPORTIONAL
0.07
.. CONSTANT
..
Figure 9. Relative volatility = 5.0 Relative volatility = 3.0 Relationship of controller constants for intermediate-plate sampling INDUSTRIAL AND ENGINEERING CHEMISTRY
voi. 48, N ~ 6.
PROCESS CONTROL
*
0.i2
-
c
I P
0
J
a
“ %
:
0.01
5
0.0 1.0 PROPORTIONAL
0.0 PROPORTIONAL
Figure 10.
CONSTANT
Condenser time delay, 1 minute
Figure 1 1 .
P.0 CONSTANT
Condenser time delay, Y2 minute
Relationship of controller constants for distillate sampling only
left-hand portion of Figure 7 did not occur with dead space present. I n addition, as dead space increased the point of onset of instability with increasing integral constants was greatly retracted (Table I). This point, however, is not contrary t o the findings of other investigators since Truxal has reported a similar occurrence (18). 4. These effects did not alter the recommended control point, however, and the most satisfactory control is still obtained by using proportional control only a n d a proportional constant of 16. 5. It is interesting t o note the existence of a singular instability point at a proportional constant of 22 for 2% dead space and t o compare this with the shape of the unstable region of Figures 16 and 18.
not materially affect the recommended control point. The recommended points for 1 and 2% dead space are the same as for no dead space. The use of a proportional constant of 1.4 alone is recommended for higher dead spaces. The reasons for this are that a large integral constant used without a correspondingly large proportional constant results in overcontrol and a “wandering” control point especially with dead space present. Integral control finds its main use in the elimination of the residual error always present to some degree with proportional control and especially so for small values of the proportional constant. Integral control used alone is seldom if ever satisfactory.
Optimum Sampler Location Figures 9, 19, 20, and 21 similarly show the effect of dead space on the controllability of the column with sampling on an intermediate plate-plate 4. As before, several salient points can be noted:
1. Oscillation occurs with slightly smaller values of the proportional constant-that is, 1.5 instead of 2.0. This is true for all values of dead s ace. 2. A similar e&ct of increasing stability with integral control t o t h a t already described is present here but t o a considerably less degree as is shown in Table I and in Figures 19, 20 and 21. 3. The effect on instability with high proportionai constants here is as equally unexpected as were the findings concerning integral control for top-plate sampling. The oscillatory region expands t o a proportional constant of 9.0 for a 1% dead space and apparent1 without limit for larger values of dead space. 4. Again t%e greatly expanded region of controllability does
If calculations are run for each of the chosen dead spaces using both top-plate-only and intermediate-plate-only sampling and the resulting output compositions compared Figure 22 is obtained. Here, the error in output composition is plotted versus dead space for each sampling location. For dead spaces of 1.5y0 or less, topplate sampling gives the best control (the least error). However, because of the much wider variation of compositions on intermediate plates, the effect of a given dead space on the output composition is correspondingly reduced. Thus for large values of dead space the output error resulting from intermediateplate sampling becomes less than for top-plate sampling. The error in output composition with intermediate plate sampling when no dead space is present is due to the fluid flow lag between the plates. If, therefore, we attribute a similar fluid
0.10
008
c 4 L 0 0
z
0.08
a
=Y 0.04
z 0.02
0.0 .. PROPORTIONAL
Figure 12.
PROPORTIONAL
OONSTANT
Still pot holdup equal to holdup of other plates
Figure 13.
..
OONSTANT
Still pot holdup equal to ten times that of other plates
Relationship of controller constants for bottom-plate sampling only
June 1956
INDUSTRIAL AND ENGINEERING CHEMISTRY
1013
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT flow lag error t o the examples with dead space present, a better comparison between the effect of dead space for top-plate and intermediate-plate sampling is obtained (Figure 23). Postulating further from these results one may predict the curve that would be obtained for sampling on a plate which is situated tn.0 plates below the top plate. The larger error with no dead space is of course due t o the presence of two fluid flow lags between sampling point and output. However, we may a t the same time predict an even smaller effect of dead space as is shown in Figure 24.
Effects of Feed and Reflux Quality and Comparison of Input vs. Output Control While practically all theoretical work in distillation is carried out under the assumption that the reflux enters the top plate a t the boiling point of the material on the plate, i t is well kno-m 1
Figure 14.
required for the reflux material to flow through the condenser and associated lines. As an example of this, consider a column which is operating in the steady state according t o Equation 6. Let us now allow a sudden increase to occur in the vapor rate from the top plate, for a short period of time, due to some as yet unspecified external factor. If D is held constant, there mill be an increase in L and a corresponding decrease in 8’ a t the end of the condenser delay period. The decrease in V’ will in turn be propagated to L causing an increase in T i ’ a t the end of the second condenser delay period. Thus for valucs off 2 1.0, any variations in V are self-propagating For values off S 1.0, the resulting oscillation will die out with a speed inversely proportional to f. Besides a rate fluctuation, the presence of cold reflux can also cause a composition fluctuation since the reflux ratio of the column is now (1 f f ) L / D instead of LID. Therefore the value of D must be varied during the transient fluctuation discussed previously if the heads composition is t o be maintained reasonably constant. With this in mind, let us now proceed to investigate the effect which cold reflux will have on the controllability and control response of a column subjected to various methods of automatic control. For this purpose the five-plate column simulated here and the earlier work of the study ( 1 4 ) will again be utilized. Both top-plate and intermediate-plate (plate 4) sampling will be investigated. The optimum controller setting for each case as previously developed mill be used. I t is assumed that the external influence which causes a change in the vapor rate is a composition change of 1 0 . 1 2 mole fraction in the feed t o the column. Since the feed is assumed to be a t constant temperature, this corresponds also to a vapor rate change in the column-i.e., p varies between 0 and 1.0 for a = 5.0 as assumed here. Both a step variation and a sinusoidal variation with a frequency of 1 cycle in 10 minutes are used
Graphical representation of dead space
I
\
r
\
i
I
l
that the majority of operating columns must actually use a reflux which is much colder than the top-plate boiling temperature. A complete study of the transient and automatic control aspects of continuous distillation should therefore determine what effect such a condition may have on the response of the column. I n the ideal, steady-state example, the top-plate vapor rate, the reflux rate, and the distillate take-off rate, are related by the equation: V=T,+D (5)
If the reflux is cold a certain fraction of original vapor flov is condensed in heating the reflux to the plate temperature. Therefore, again in the steady state
V‘= V
-fL
=
L 4- D
(6)
where fLis the amount of vapor condensed in heating the reflux, andf may be termed the reflux condensation factor. This factor is a function of the heat capacity, heat of vaporization, and boiling point of the reflux material as well as its actual temperature prior to entrance onto the top plate.
where AHb is the heat of vaporization, Tb is the boiling point> T Eis the reflux temperature, and C, is the heat capacity. Figure 25 relates f t o Ieflux temperature for several typical hydrocarbon materials at 1 atmosphere ambient pressure (4, 1 6 ) . Similar graphs can readily be constructed for other materials and conditions. Equation 6 xi11 not be correct during a period of transient operation since the values of V , L, and D cannot be related directly in algebraic equation form because of the finite period of time 1014
20
30 RELATIVE
Figure 15.
I
40
50
V O L A T I L I T Y , CL
Relationship of temperature dead space to composition dead space
Results. Figures 26 and 27 show the results obtained. The total variation of top-plate composition (z6)was chosen as the parameter which gave the best comparison between step and sinusoidal inputs. It is to be remembered, however, that the output composition variation caused by a step input quickly assumes a much smaller steady-state value after an initial large deviation ( 1 9 ) . It is the initial large deviation that is piotted here. The increased stability and smaller variation present with the top-plate sampling example is due to the fact that it can immediately detect the effect of a rate change through its accompanying
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL 10.0
8.0
6.0
4.0
2.0
0.0 0.0
PROPORTIONAL CONSTANT
Figure 16.
10.0
80.0
30.0
40.0
PROPORTIONAL
Figure 17.
Dead space, 0.01 mole fraction
50.0
60.0
70.0
80.0
CONSTANT
Dead space, 0.02 mole fraction
Relationship of controller constants for top-pla'te sampling
composition change and can then make the proper adjustment in the distillate takeoff rate t o compensate. The fourth plate sampler on the other hand is handicapped by the presence of the exponential time delay of the top plate (Figure 2) which hinders the detection of the composition changes t h a t are occurring and the application of the proper corrections. It is thus much easier for an oscillation to be set u p and finally for a n unstable condition t o occur with plate-four sampling than with top-plate sampling. The slight decrease in the output variation for the step input of Figure 27 when the value off is near 0.5 or 0.6 is probably due t o a reduction of the initial deviation of the developing oscillation since the natural frequency of the column is longer than the time required for the material to flow through the condenser ( '/z minute i n this case). Thus with high values off the increased L begins to decrease Ti appreciably before i t can reach its highest value. Similarly the decrease in the output variation for the sinusoidal case is probably due t o the phase relationship between V and L. T h u s L is large when an increasing $1 makes V large and thus effectively limits its possible variation. Possible Remedies. Beyond the obvious remedies of a combination of top-plate sampling and a well insulated reflux return line, the effect of a n accumulator in the reflux line deserves attention. An accumulator will always place a long time constant in the reflux return line. However, if the reflux stream rate is determined from a level control in the accumulator, fluctuations in vapor rate will still be propagated to the reflux stream and thus can allow a n oscillation or instability to occur. If, on the other hand, the rate of the reflux stream is controlled from the
plate composition controller and the distillate take-off comes from the level control of the accumulator, vapor rate fluctuations cannot b e propagated and the described events cannot occur. Thus a control scheme similar to t h a t of Robinson and Gilliland (11) will effectively prevent these fluctuations from occurring while a scheme such as t h a t described by Kern (6) will not. T h e latter however seems t o be the more prevalent system (6).
Effect of Feed Quality As mentioned previously, the earlier work of this investigation was carried out assuming a constant temperature feed and thus a feed whose quality varied widely as its composition varied. However, since this stipulation can impose a severely fluctuating load on the feed heat exchange, i t is difficult t o attain in actual practice. It is therefore of interest t o determine the actual effect on column controllability of the use of a constant quality feed rather than one with a constant temperature. Again the effect on column controllability with both top-plate sampling and intermediate-plate sampling is determined by using the simulated column previously described. It has been shown t h a t derivative control is ineffective in controlling this column (14). It has also been shown t h a t when various combinations of proportional and integral or reset constants (gains) were applied in controlling the column, a plot of the proportional constant used versus the integral constant used would effectively map our regions of satisfactory control, oscillatory response, and
20.0
16.0
c z U
,"
12.0
0 0 -I
,K 4
8.0 w
z
-
I+
4.0
.._
*O
10.0
ZOO
30.0
40.0
PROPORTIONAL
Figure 18.
50.0
60.0
70.0
80.0
CONSTANT
Top-plate sampling; dead space, mole fraction
0.05
Figure 19.
Intermediate-plate sampling; mole fraction
dead space 0.01
Relationship of controller constants for top-plate sampling
June 19%
INDUSTRIAL AND ENGINEERING CHEMISTRY
1015
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Figure 20.
Dead space, 0.02 mole fraction
Figure 21.
Dead space, 0.05 mole fraction
Relationship of controller constants for intermediate-plate sampling
oscillatory region for large values of the integral constant is larger for constant quality feed, there is little over-all difference between the two response patterns obtained. Either will give quite satisfactory control. With intermediate-plate sampling, on the other hand, quite different results were obtained as is shown by Figures 9 and 29 For constant temperature feed (Figure 9 ) the boundaries between the various regions were quite sharp, the response of the column being definitely entirely satisfactory, oscillatory, or completely unstable as the various control combinations were tried on the computer. However, for the constant quality feed there was a definite region where an unpredictable response was obtained. Figure 29 shows this under “Conditionally Stable Region.” Figures 30 and 31 show one aspect of the unpredictability. Figure 30 gives a typical response nThich was obtained in the top-plate composition for a set of controller constants from within this region. An entirely different transient was obtained with each cycle of the sinusoidal feed composition change imposed on the column. Repeated solutions on the computer failed in every case to give duplicate results. For some runs a satisfactory stable response was obtained, for others a completely unstable condition, and for the majority a response similar to that shown in Figure 30 was the result. Figure 31 by way of contrast shows a comparative rksponse obtained with the constant temperature feed. Notice the exact duplication of the transient response with each cycle of the sinusoidal input n-hich is now obtained.
of instability obtainable n-ith the various possible control combinations. I n addition that combination which gave the best over-all control could be plotted as a point in the resulting field. This method has been used here. Results. Figures 7 and 28 present such a plot of the results of the calculations for top-plate sampling using constant temperature and constant quality feeds, respectively. While the
0.0
0.01
0.02 DEAD
Figure 22.
0.03 SPACE
0.04
0.05
Effect of dead space and sampling location on final output error
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A D D I T I O N A L CONTRIBUTION OF D E A D SPbCE T O O U T P U T ERROR
052-
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Figure 23.
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Sampling on plate 4 only
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Figure 24.
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0.02 DEAD
0.03
SPhCE
0.04
0.05
Sampling on plate 2 below top plate
Probable cause of final output error with dead space present
1016
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL Theoretical Discussion. It is interesting at this point to postulate a reason for this behavior. Why should the column response be unpredictable for certain combinations of operating and control conditions? Consider what occurs in the column when the composition of a constant temperature feed increases. Since a greater percentage of the more volatile component is present, the feed boils a t a lower temperature resulting in a greater percentage of vapor in the feed. This larger vapor stream increases the boilup from the feed plate thus helping to carry the greater amount of more volatile component present toward the top of the column. Thus the composition of the feed plate liquid need not increase t o as great a n extent t o maintain the material balance relationships in the column. Similarly a decrease in feed composition increases q and aids the transfer of the increased amount of less volatile component t o the bottom of the column. Depending on the reflux ratio employed, these effects may be nearly sufficient t o compensate for the feed composition change regardless of the plate composition prevalent on the feed plate a t the moment the change in feed composition occurs. With a constant quality feed the column flow rates cannot change and the effect of a feed composition change must be compensated for entirely by a change in feed plate composition. Now the response of the column is very definitely dependent on the conditions existent on the feed plate a t the moment the feed composition change occurs instead of nearly independent as before. Thus, since it is very difficult to duplicate internal conditions of the column prior to application of an input change, different response patterns are obtained. With top-plate sampling the sampler is close enough t o the control point t o effectively compensate for the response fluctuations. However, when plate-four sampling is used, the existence of the resulting time delays between the sampling and control points permits the resulting different transients to develop and may be too late in applying a correction t o prevent instability, if they are severe.
IS0 140
130 120 110 IL
100
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i
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70 ! I .
00
w
5
60
tI
40
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30 20 10
0
01
02
03
04
05
06
07
08
09
IO
f Figure 25. Values of reflux condensation factor for various reflux temperatures and typical heads products
pertinent. However, in distillation the input stream has two pertinent variables since i t possesses both a rate and a composition, either of which will affect the performance of the column. While the composition of the feed stream is fixed by the performance of prior units of the plant, the feed rate can be varied through judicious use of a tankage system. Therefore the possibility exists of maintaining a n adequate control of the column output by the proper regulation of the input feed rate t o compensate for feed composition changes. Inspection of this material balance equation for the more volatile component in the column
+
Fx! = D X D WX,
(8)
gives a clue t o a method for accomplishing this. If the value of the quantity F x j can be maintained constant and if the values of D and W are independently regulated constant, then X D and xw should be constant. It must be borne in mind, however, t h a t Equation 8 is a steady-state equation, and i t may not necessarily apply in the transient condition. Results. Calculations were made to determine the results obtainable with the method of sampling when the quantity F x j was maintained exactly constant. This, of course, assumes a perfect controller. For a constant q and a sinusoidal variation
Input vs. Output Control Ordinary servomechanism theory as used in automatic control studies demands the application of the principles of “feedback”t h a t is, the output of the system is sampled and the proper corrections are then applied to the system operating conditions when and where necessary t o maintain the output at some definite state ( I , 17). I n the general automatic control case studied there is usually only one variable of the input stream which is
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0.08
n 0.05
F a
5
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0
y.
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2
003
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B
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0
f
Figure
June 1956
f
26. Sampling on an intermediate plate Figure 27. Sampling on top plate Effect of cold reflux on response and stability of control INDUSTRIAL AND ENGINEERING CHEMISTRY
1017
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
2C PROPORTIONAL CONSTANT
Figure 28.
Figure 29.
Top-plate sampling
Relationship of controller constants-constant
of feed composition the response \-,-as as good or better than that possible with the best type of output control using top-plate sampling, and, contrary t o the prcviously discussed example, better than that obtainable with a constant temperature feed. Similar calculations with step changes in feed composition showed a slightly better response for the output control than for input control although the difference was again small. The relation between constant q and constant temperature feed was again maintained. The reason for the pooier response in the bottom-plate composition for the constant temperature feed is due to the fact that the variation of F can effectively compensate for the variation of z j and maintain the output compositions constant. Therefore, the additional correction given by the variation of p for a constant temperature feed is actually an overcorrection and causes a n error in the opposite sense.
30
43
50
6C
70
80
PROPORTIONAL CONSTPlNT
Intermediate-plate sampling
quality feed
of operation of the column. Thus if the column is the majoi piece of apparatus in the process and the rate of production of the column feed stream is not significant, then input control will prove satisfactory. If, however, the column is only one unit in a long process and if the rate of production of the column feed stream is variable, then output control will be more satisfactory.
r 0.m
0.W 0.555
4 0.875
0.155
Figure 31.
Figure 30. Representative response obtained using control constants from conditionally stable region of Figure 29 and constant quality feed
Because the column is noT effectively operating as an “open loop”-that is, there is no correction possible if an error occurs in the output--two conditions must be met. First, the control of F in relation to 2, must be as nearly exact as possible since any error in the constant value of F z , will cause a drift in the output which cannot be rectified. Secondly, the values of D and W or more important D and V must be independently controlled as constants since the output compositions are also dependent on these quantities. The Place of Input Control. The question of the use of input control versus the use of output control raises a question in the philosophy of plant-wide process control. il column operating on output control can handle a wide range of feed rates and compositions dependent only on the ability of the particular plate or packing design used t o handle the range of throughputs entailed without a significant change in efficiency. On the other hand input control effectively enslaves the rest of the plant t o the rate
1018
Corresponding response to that of Figure 30 if constant temperature feed is utilized
Another point worth considering in this discussion is that of the completely automated plant of the future. Here when plant production rates are determined and varied by a master controller, the output controlled column n i t h its wider range of operation will be more compatible than one with input control. The question of stream analysis is a point in favor of input control, however. Column output streams are usually very difficult t o analyze since they are so close to pure material in composition and an analysis error is therefore much more significant. Feed streanis on the other hand usually have an intermediate composition and thus lend themselves more readily to analysis. Conclusions I n addition t o the conclusions of the previous paper (14), the following ronclusions may be drawn concerning the column simulated by the computer; these should be capable of extrapolation to larger columns. 1. When an adequate composition controller-i.e., a method of controlling the effect of feed composition changes-is available, a satisfactory control of radical feed rate change can be provided by supplying an additional controller to maintain the column boilup rate as a given ratio of the feed rate. Care must be talrcn, hoyever, t o ensure that corrections to the boilup rate are not so violent as t o render the composition control system unstable. 2. As the relative volatility increases, the action of the distillation process speeds u p making the effect of any exponential or true time lags more important.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL 3. Condenser holdup is in general destabilizing. However in certain cases the delay may equal the time of one cycle of the natural frequency of the system, and a false stability may be shown at this point. Still pot holdup on the other hand exerts a stabilizing influence on the control of the column since a large capacity here gives a “smoothing out” effect on composition and rate changes in the column. 4. Every effort should be made to obtain a sampling device giving as small a dead space as possible in order to keep the final output error as low as possible. If a device with a dead space of 1.5% or less can be used, the recommendations previously made should be followed. If, however, necessity requires the use of a larger dead space, sampling on an intermediate plate should be used in order t o reduce output error to a minimum. 5 . Simultaneous sampling on both top and bottom plates would be subject to the same errors a t high dead spaces as would top-plate sampling alone and is therefore not recommended when dead space is present. 6. A reflux temperature different from the top-plate boiling temperature is a significant factor only if the changes in vapor rate caused by i t amount t o 50qlCor more of the original liquid rate for intermediate-plate sampling or 80% or more for topplate sampling. However variations sufficient t o cause liquid rate deviations of this magnitude or greater can cause oscillations and even instability in the control of the column. The distillation of compounde with properties similar to the higher straight-chain paraffins would be most critical in this respect. 7 . Where output control is being utilized, the maintenance of a constant feed temperature is very important, especially if intermediate plate sampling is in use. However, for the example of input control described herein, a constant quality feed is indicated. 8. Input control has proved capable of quite close control of column output compositions if the column’s internal flow rates can be adequately controlled. I t s use bears investigation when the distillation column can be the flow rate determining element in the process. However, where this condition may prove a handicap the use of output control will be more satisfactory. Ease of stream analysis on the other hand will favor input control.
Acknowledgment
the time and computer facilities to make this study possible. The assistance of Clarence L. Johnson, John E. Cornett, James McKenna, and John Sansome in carrying out the computations is gratefully acknowledged. The authors are also especially grateful to Noland Poffenberger of the Dow Chemical Co. who originally suggested the input control scheme based on some work done a t Dow Chemical.
References (1) Brown, G. S., Campbell, D. P., “Principles of Servomechanisms,’’Wiley, Sew York, 1948. ( 2 ) Foxboro Co., Foxboro, Mass., Bull. 461-2, Model 40 controller, 1953. (3) Gardner, M. F., Barnes, J. L., “Transients in Linear Systems,” Wiley, Sew York, 1942. Handbook of Chemistry and Physics (C. D. Hodgman, editorin-chief), 29th ed., Chemical Rubber Publ. Co., Cleveland, Ohio, 1945. Kern, D. Q., “Process Heat Transfer,” chap. 21, pp. 765-89, McGraw-Hill, S e w York, 1950.
O’Connor, Ward, Philadelphia-Wilmington Section, Am. Inst. Chem. Engrs., April 27,1954. Partlow Carp., S e w Hartford, S . Y., Bull. 101, Industrial Temperature Measurement and Control, 1950. Perkin-Elmer Carp., Norwalk, Conn., “Tri Nan Analyzer,” 1954.
Poffenberger, Noland, private communication, Aug. 31, 1955. Robinson, C. S., Gilliland, E. R., “Elements of Fractional Distillation,” 4th ed., pp. 433-7, hlcGraw-Hill, Division of Industrial and Engineering Chemistry, Sew York, 1950. Ibid., p. 473.
Rose, Arthur, IND.ENG.CHEM.,33, 596 (1941). Rose, Arthur, Johnson, C. L., Williams, T. J., Division of Indus-
trial and Engineering Chemistry, 128th Meeting, ACS, Minneapolis, Minn., September 1955. Rose, Arthur, Williams, T. J., 1 r D . ENG.CHEM.47, 2284-9 (1955).
Rossini, F. D., others, “Selected Values of Properties of Hydrocarbons,” Circ. C 461, National Bureau of Standards, Washington, D. C., 1947. Taylor Instrument Co., Rochester, N. Y . , Bull. 98097, Transet control system, 1954. Thaler, G. J., Brown, R. G., “Servomechanism Analysis,” McGraw-Hill, New York, 1953. Truxal, John G., “Automatic Feed-Back Control System Synthesis,” McGraw-Hill, New York, 1955. Williams, T. J., Ph.D. thesis, The Pennsylvania State University, 1955.
The authors wish to express their thanks t o the USAF Institute of Technology and Wright Air Development Center for providing
June 1956
R E C E I V Ereview: D ~ ~ ~ (1) September 27, 1956; (2) January 20, 1956; (3) February 20, 1956.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED April 24, 1956.
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