FEEDFORWARD CONTROL OF DISTILLATION COLUMNS W .
L.
L U Y B E N ’ A N D
J . A.
GERSTER
L’nznluerszty of Delaware, .Vewark. Del. The effectiveness of feedforward control for a 1 0-tray and a 40-tray distillation column has been studied. The function of the controller was to prevent the overhead and bottoms product compositions from changing in spite of disturbances in feed rate and in feed composition by manipulating vapor rate and reflux rate to the column. The performance of the controller was determined by analog simulation of a 10-tray and a 40-tray column, and by experimental tests in a 1 0-tray, 2-foot-diameter pilot-plant column.
control has been the traditional method of controlcolumns. I t often provides acceptable control despite scant knowledge of the dynamic character of the column. However, corrective action is taken only after product qualities have deviated from their desired values, and long times are often required to bring the unit hack to its desired steady state, especially in large towers. Also. undesirable interaction of feedback control loops may occur, as most columns are multivariable. Feedforward control in principle eliminates these difficulties by sensing the input disturbances as they enter the column and taking the proper corrective action. Thus. control problems inherent in dead-time and distributed parameter systems are theoretically eliminated. Use of feedforward controllers in practice has its limitations. First, the system may not correspond exactly to the mathematical or empirical model used to predict its dynamic behavior. Second, changes in product compositions caused by unmeasured input disturbances go uncorrected. Third, physical and process limitations may constrain the manipulative inputs so that they cannot be adjusted to values required by the feedforward controller. For these reasons, secondary feedback control would be required in the plant to prevent a drift of product composition. This combined feedforward-feedback control is not considered in this paper. EEDBACK
F ling distillation
Background
Feedforward control is discussed qualitatively by Calvert and Coulman (4) and Dobson (5) Application to general multivariable linear systems is discussed by Bollinger and Lamb ( 3 ) ,who used matrix methods to calculate both feedforward and feedback controllers so as to minimize “penalty functions.” Lamb. Pigford, and Rippin (7) synthesized a feedforward controller using a linear model for a seven-tray column. Transfer functions for the column were computed from a frequency-domain solution of the model ; matrix methods Mere then used to solve for the feedforward controller transfer functions. In the present study these procedures were modified to calculate the feedforward controller transfer functions directly. Lupfer and Parsons (8) describe the synthesis of feedforward controllers for a distillation column from an empiricall? determined tramient response of the column Present address. Enginerring’Department. E. I. du Pont de Kernours & Go., Inc.. Louviers Building. S e w a r k , Del. 374
I&EC PROCESS DESIGN A N D DEVELOPMENT
Pilot-Column Studies
Scope of Work. T h e effectiveness of a feedfor\\.ard controller was investigated experimentally for the same 2-foot, 10-tray distillation column for which intensive transient response data have been determined ( 7 , 2, 70, 72). Eighteen runs were made \vith step changes in feed rate or feed composition to the column. Corrective action was made as prescribed by the simple feedforbvard controller transfer functions obtained from Bode plots. Changes in composition for the tray liquids and for the overhead and bottoms products \vere determined as a function of time after the step change. T h e experimental work \vas carried out \vith the acetonebenzene system at 25 p.s.i.g. Reflux ratios of 1.3 and 4.1 were employed for several values of feed composition between 40 and 60 mole % acetone. Cold feed Lias introduced to the fifth tray from the bottom and averaged 10 gallons per minute. All tests were made at vapor velocities \vhich \\-ere about 60Yc of the flooding rate. T h e tower itself has been fully described (6). Briefl)-, it consists of 10 bubble-cap trays on 18-inch spacing. Outlet weir height is 2 inches. T h e system includes a naturalcirculation, thermosiphon reboiler. condenser, 100-gallon reflux drum. and feed, reflux, and bottoms pumps. Liquid hold-up on each tray is 2.40 gallons; in each downpipe, 1.20 gallons; in the reboiler system, 67 gallons; and in the entire reboiler-column-condenser system. 172 gallons. Transfer Functions. T o achieve feedforward control of the pilot-plant column, it was first necessary to compute the transfer functions between the input disturbances (feed rate: F. and feed composition, z F ) and the manipulative or corrective inputs (reflux rate, R. and vapor boilup rate. V,) such that the control objectives are met: that the perturbations in composition of the overhead and bottoms products be zero-i.e., x D = xI1 = 0. Thus: the feedforward controller (FFC)of the distillation s)-stem, assuming it to be linear. is made up of four separate transfer functions; in matrix notation, these are:
(In this paper. variables without an overhead bar are perturbations from the steady-state values. \vhile barred quantities are the steady-state values-for example. the reflux rate at any instant is 2 R ) Thus if the transfer functions are known and if upsets in feed rate and feed composition are measured, proper values for the corrective inputs are readily determined as follous:
+
T h e transient behavior of the lO-tray pilot-plant column is described by a set of 26 linear differential equations of the
perturbation type (70). Two equations are required for each tray: one to describe ].he composition-time relationship and a second to show the liquid rate-time behavior. Additional equations are needed for the reflux drum, the reboiler, the seal pan below the bottom tray, and the sections of top and feed trays used to preheat cold feed streams to those trays. These 26 equations were Laplace-transformed and put into the frequency domain by substituting I W for s, the Laplace operator; 1 is and w is the frequency in radians per minute. At any particular value of U , a set of 26 algebraic equations results with coefficients that are complex numbers. These equations were soived for the feedforward ccntroller transfer functions by iinposing the two control criteria equations, X D = ytD = 0, ,and solving the simultaneous complex algebraic equations for R(F, zB) and V(F,z F ) .
and
? !dt!
=
(a>
(R
- L10)
In the frequency domain, these equations become (with XD
=
0): 0
47,
J W xi0
=
bixio - 0
bzR - b3Vio f 0
=
(6)
-
f
b4~10
bjxg
(7)
and
JULIO = bsR - b d i o
(8)
where the b constants are the constant coefficients in Equations 3, 4, and 5. The corresponding pair of equations for tray 9 is:
To illustrate the method, the splitting of top and feed trays and the contacting section above the reboiler (70) is omitted below. Consider the equations for the top of the column; for the condenser,
jwxg = b7Lio - baVg
f b,xio - bioxg f biixs
IwLg = b6Lio - bsLg
(9)
(10)
Substituting Equations 7 and 8 into Equation 9, noting that Equation 6 gives x10 = 0, and recalling that V g = Vlo= V, (instantaneous propagation of vapor disturbances) give the following result for any given value of w :
(3)
x8
= (biz
f jbl3)Vr -
f jbl5)R
(bl4
(11)
T h e stepping procedure is continued tray-by-tray downward until the tray above the feed tray is reached, where the result is ~5
Table 1.
(hls
Correctiue Act ion
-
(12)
Values of Corrective Action for Pilot-Plant Column _ _ ~
~~~
Operating Conditions
+ ~ h i ~ ) V-r ( h i s f 3big)R
To attain required Jinal steady state (steady-state gain = K )
L/D
ZF
1.3 1.3 1.3 4.1 4.1 4 1
0.45 0,494 0.55 0.48 0.50 0.53
0.6074 0.6072 0.6070 1.8918 1 ,8923 1 ,8956
1.3 1.3 1.3 4.1 4.1 4.1 4.1 4.1
0.45 0.494 0.55 0.48 0.49 0.50 0.51 0.53
-0,6604 -0.6852 -0.6511 - 0.0498 -0,2549 -0,4400 -0,5915 -0,8048
1.3 1.3 1.3 4.1 4.1 4.1
0.45 0.494 0.55 0 48 0 50 0.53
1.5054 1 ,5052 1 5049 2 9636 2.9642 2.9681
1.3 1.3 1.3 4.1 4.1 4.1 4.1 4.1
0.45 0.494 0.55 0.48 0.49 0.50 0.51 0.53
1.1869 1.1885 1.3885 0.8138 0.5664 0.3476 0.1733 - 0.0590
Dynamic feedforward controller (transfer function)
1/(0.3390s f 1/(0.3571s 1/(0.3745s 1/(0.9259s f 1/(0 9615s 1/(0.9709s f
1)
++ 11 )) 1) + 11 ))
1/10.3704s C 1 P i)(o ,4405s 1j z 1/(0.4545s f 1)2 (0.3636s 1 )/(s 1)(O 2299s 1/(0.5882s 1)2 1/(0.6667s f 1)2 1/(0.7353s i- 1 ) 2 1/(0.6757s
+
+ + + +
1/(0.4219s f 1/(0.4608s f 1/(0.5076s f 1/(0.9901s 1/(1.0640s C 1/(1.0750s
+1
)2
1) 1) 1) 1) 1)
+ + 1)
vs / z F
+ 1)/(0.6250s f 1)[0.40822s2+ (1.6)(0.4082)sf I ] + 1)[0.41672s2f (1.6)(0.4167)s+ l ] + 1)/(0.6993s 1)/(0.4545s f 1)[0.4167*s2+ (1.2)(0.4167)s4- I ] + 1 ) ( 1 . 2 6 6+~ 1)[0.35462~2 f ( 1 . 6 ) ( 0 . 3 5 4 6 )+ ~ I] (3.125s + 1)/(1.235s f 1)/0.4O0O2s2+ (1.6)(0.4000)sf 11 (10s $. 1)/(1 ,111s f 1)[0.43482s2+ (1.4)(0.4348).r+ 11
(1.111s (1.250s f (0.667s (1 ,429s
VOL. 3
NO. 4
OCTOBER 1 9 6 4
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.4t the bottom of the column, where the control objective is to maintain a constant bottoms product composition, yTV = 0. Applying the same stepping procedure from the bottom upwards. the result for the tray bel014 the feed tray becomes
T h e final step in the calculation of the transfer functions is to write perturbation equations similar to Equations 4 and 5 for the feed tray itself; additional terms involving F ahd zF are present. These equations, plus Equations 12 and 13, can be rearranged to give the desired result
R
=
Vs =
+ +jbp)zF (bzs + jb2dF + ( b a +
(h j b d F 4-
(626
jb8Jzp
(14) (1 3
~~
Table II.
Perturbation in Feed Coni#osition
0 +0 02 +0 05
t o .10 +0.15
0 +0.02 +0.05 $0 10 +0.15
Calculations of this type were programmed for a G-15 Bendix computer so that values of the four complex numbers in Equations 14 and 15 could be obtained at various values of 7,and A w. Values of the steady-state parameters f. 5. (required for each tray as coefficients in the equations similar to Equation 4). were also obtained in a separate Bendix program in which the necessary vapor-liquid, enthalpy, and density d a t a for the acetone-benzene system were expressed as polynomials in temperature and composition. \'slues for liquid holdup and for rate of change of liquid holdup with respect to liquid rate were obtained from experimental data. These procedures and tables of the computed values have been described (6>9: 7 7). Computed results were expressed in terms of Bode plots of log modulus and phase angle us. w for each transfer function, R;’F, R,!z,, V , ’F? and Vv,/z,. Results were obtained for reflux ratios of Ll’D = 1.3,and 4.1 and for various feed compositions between 45 and 55 mole yc. ,411 of the Bode plots or frequency response curves are given by Luyben ( 9 ) . T h e curves of log modulus us. w were approximated by ratios of polynomials in s. First- and second-order leads and lags were used to obtain a reasonably simple result; a n attempt was made to obtain a good fit especially in the low frequency part of the curve. No attempt was made to fit the phase angle curves, because this would complicate the transfer function by requiring terms such as ( e - “ ) or (s 1) ‘(s - 1). Numerical values of the steady-state gains and the approximate feedforward controller transfer functions are given in Table I. Inspection of Table I shows that the time constants for the feedforward controller increase as the reflux ratio is increased a t any z F >because of the larger concentration difference on adjacent trays a t the higher reflux ratios. Feed composition has a stronger effect upon the results a t the higher reflux ratio because the composition change over the feed tray is greatest in this instance. Range of Validity of Linear Model. T h e transient and steady-state results for the pilot-plant column given in Table I are based upon the assumption of a linear model. I n fact, the system is actually nonlinear and it is of interest to determine ho\v large the perturbations can become before the linear model becomes invalid. To test this. steady-state values for x I ) and ,yW were computed from the exact steady-state program mentioned above as perturbations in zF were increased from 2 to 20 mole yo. The required changes in R and V, were computed from the steady-state gains of Table I. If the linear model holds, the product compositions should remain constant. As shown in Table I I ? the product compositions deviate by less than 1.3 mole Yc with 1070 changes in z F . At the higher reflux ratio. the column is more nonlinear because of the larger composition changes per tray.
+
376
l&EC PROCESS DESIGN A N D DEVELOPMENT
Validity of linear Model for Pilot-Plant Column
+0 20
niodrl
inodel
calliv
L / D = 1.3; Initial 2 p 0.1035 0.1033 0.1035 0 1060 0 1035 0.1009 0.1035 0 1153 0.1035 0.1189
L I D = 4.1; Initial 2~ 0.0384 0.0384 0.0384 0.041 1 0.0384 0.0449 0.0384 0 0515 0.0384 0.0598 0 0384 0.0690
=
0.494
=
0.8398 0.8398
0 8398
0.8398 0 8398
0.8420
0.8398 0.500 0.8950
0.8410 0.8442 0.8523 0.8950 0,8968
0.8950 0.8950 0.8950 0.8950
0,8992 0.9023 0,9067
0.8950
0.9116
Effectiveness of Control on Pilot Column
Analog Studies. T h e effectiveness of a feedforward controller for the pilot column was studied first by simulating the column on a PACE analog computer. The analog circuit used to simulate the transient behavior of the pilot-plant column is given in earlier studies (70). The approximate transfer functions of Table I ivere developed on other components of the analog computer to simulate the feedfor\z.ard controller of the analog column. Complete analog circuit diagrams are given by Luyben ( 9 , 7 7 ) . Figure 1 sho\\s the modified step disturbance in either F or zF imposed upon the system. T h e disturbance \vas not a sharp step? so that the analog components could readily describe the derivative of the disturbance. Figure 2 shobvs rhe corrective action demanded by the controller in order to inaintain constant product purities a t all times when the feed composition changes as shown in Fjgure 1. The required changes vary \vith time and lvith the steady-state values in I? and in 7, of ‘I) and ZF. Corrective action varies greatly ivith f , a t the higher reflux ratio. T h e dynamic part of the corrective action occurs during the first 3 or 4 minutes follo\\-ing the input
I
2
T I M E , MINUTES
Figure 1. Modified step disturbance in feed rate or feed composition to column
70
Units of disturbance. change in f e e d rate or change in f e e d composition, mole acetone
yo
I
I
-
- 0 2
C
6
4
TIME, MINUTES V s f Z F FOR PILOT COLUMN R / & FOR PIILOT COLUMN
.-&.0
2
4
55
6
TIME, MINUTES
1
-4
2
#
1 I
4
TIME, MINUTES
Figure 2. Changes in reflux rate and vapor boilup rate required for pilot-plant column Feed composition or f e e d r a t e changes as in Figure 1 . Parameters: r t e a d y state reflux ratio, 1/01 steady-state f e e d composition, IF
change. The required corrective action for a feed rate change is also shown in Figure 2. In this case 2, has little effect upon the result. The analog studies showed that control was almost perfect. .4bsolutely perfect control ( x D and xw staying exactly zero) was not expected because of the approximate transfer functions used. Overhead and bottoms compositions of the analog column never deviated more than 0.2 mole % acetone. This maximum deviation occurred 4 to 5 minutes after the input change. T h e analog studies also showed that the deviations in product compositions are held to small values (less than 1 mole yo) when the corrective action was taken instantaneously-i.e., when values of R and V , were immediately changed to the values required for the new, final steady state. This is not surprising for this small 10-tray column, as the dynamic feedforward controller brought these flow rates to their final steady-state values in only 3 to 4 minutes. Experimental Studies. Step changes in feed rate and in feed composition were introduced into the pilot-plant column, and in the first set of experiments the corrective action was is, changes in both reboiler steam taken instantaneously-that flow and reflux flow were made at the same time that feed rate or feed composition was changed. A series of 18 runs of this type was made a t reflux ratios of 1.3 and 4.1. Step changes in feed rate varied from 5 to 107, of the original value, and step changes in feed composition varied from 5 to 10 mole yo acetone. Results for two typical runs are shown in Figures 3 and 4. In Figure 3 where the feed rate was increased by 5%, the corrective action was made instantaneously. The tray compositions did not move appreciably from their original values, and there was no measurable transient period. Figure 4 shows the result for a change in feed composition from 48 to 57 mole yo acetone. The corrective action was made instantaneously. In this instance the tray compositions
6 -2 REBOILER: Rw.0.069
a 0
-2 0
2
4
6
8
T I M E , MINUTES
2
4
6
8
T I M E , MINUTES
Figure 3. Experimental changes in composition of tray liquids and product compositions in pilot-plant column
Figure 4. Experimental changes in composition of tray liquids and product compositions in pilot-plant column Feed composition increased b y 9 mole % a n d cor-
F e e d r a t e increased b y 5% a n d corrective action in reflux r a t e a n d v a p o r bailup r a t e t a k e n instantaneously
rective action in reflux r a t e a n d v a p o r boilup r a t e taken instanta neously
VOL. 3
NO. 4
OCTOBER 1964
377
0
*e
-6
-12
t6
-18
+4 -24
-30
, 005 0 1
05
I O
5
10
I
t2
0
I ,
50
00501
FREQUENCY. RADIANS/MIN
05
I
5
10
50
FREQUENCY. RADIANS/HIN.
+ -I
w
0
3
-6
e
-I2
K
-2 -3
;
-10
-4
9 -24 0
- -30 L
0
p -36 -40 0 0 5 0.1
05
I
5
IO
FREQUENCY. RADIANS/MIN.
50 0 0 5 01
I
1
05
I
’
\ I S
IO
50
FREQUENCY, RADIANS/MIN
Figure 5. Bode plots for feedforward controller transfer functions for lo-, 20-, and 40-tray columns
all change as expected but the overhead and bottoms product compositions remain unchanged. The results of the other 16 runs also indicated that instantaneous corrective action gave essentially perfect control for the pilot-plant column. There was thus no need for ‘experiments in which dynamic corrective action would be taken. T h e effectiveness of instantaneous corrective action was a t least partly due to the presence of small lags in the experimental equipment. A lag of 30 to 60 seconds existed between the time the set point was changed on the steam-rate controller and the time that the vapor rate to the column was increased. Such lags made the “instantaneous” corrective action similar to the dynamic corrective action. Ten-, Twenty-, and Forty-Tray Column Studies
Instantaneous corrective action becomes less attractive as the number of trays increases, because the time required for establishment of a new steady state is increased. Only a few minutes were required for the 10-tray column ‘studied above, but hours may be required for very large columns. When the number of trays is large, the corrective action must be applied a t the proper rate over the transient period if product compositions are to remain constant. T o determine the effect of increasing the number of trays on the form and effectiveness of feedforward controllers: a computer study was made in which feedforward-controller transfer functions were obtained for columns of 10, 20, and 40 trays. Analog studies were made for the 10- and 40-tray cases; both column and computer were simulated to determine effectiveness of control. T h e basis for the computer studies was: Binary system of relative volatility, a = 1.4 Equimolar overflow within column Saturated liquid feed and reflux streams 100yo tray efficiencies Liquid perfectly mixed on each tray Reflux ratio, L I D , of 4.0 Flow rates, in moles per minute, of 100 for the feed and 50 for the distillate Holdups, in moles, of 100 for the reboiler, 100 for the condenser, 30 for the stripping trays, a?d 25 for the rectifying trays. The flow rate chosen would correspond to those employed in a 15-foot-diameter column operating a t 80% of flooding 378
TIME, 5 MINUTES IO
I8
TIME, MINUTES
I
l&EC PROCESS DESIGN A N D DEVELOPMENT
Figure 6. Changes in reflux rate and vapor boilup rate required for 40-tray column subjected to step changes in feed rate or feed composition as in Figure 1 F e e d f o r w a r d controller responses Inrt. Instantaneous corrective action Simple. Simple dynamic corrective actjon Dyn. Dynamic corrective action using transfer Table 111
functions
of
with a liquid density of 50 pounds per cu. foot and a vapor of density of 1.1 pounds per cu. foot. The holdup on each tray would be 3 inches of clear liquid. Results can be used for a column of any size with the same relative holdups and hydraulics by changing the time scale. Controller Transfer Functions. The feedforward controller transfer functions were computed in the same manner as for the pilot-scale column. Typical Bode plots are given as Figure 5. T h e magnitude ratio plots show interesting resonance peaks in the medium to high frequency range. Resonance peaks decrease in size and sharpness as the number of trays is reduced; columns with fewer trays behave less as distributed parameter systems. The Bode plots were approximated with simple transfer functions as described previously (Table 111). The response of the feedforward controller to step, pulse, and ramp changes in F and in z, were determined for three cases: (1) instantaneous corrective action; (2) dynamic corrective action, but with the controller transfer functions approximated by first-order lags (designated as a “simple” dynamic controller) ; and (3) dynamic corrective action using the transfer functions listed in Table 111. Responses for the step input case are shown in Figure 6 for the 40-tray column. The corrective action in the dynamic cases takes place over about a 15-minute period for the 40-tray column. For transfer functions KIF’ and V 8 / F . the simple dynamic controller is more practical and closely approximates the more exact dynamic control. For R/z,, a second-order lag is sufficient for good control (see Table 111). For V s / z F ,combination of a first-order lead with first- and second-order lags is required. The results of Table 111 are more easily interpreted if they are expressed in terms of steady-state gains ( K = value of transfer function at zcro frequency), breakpoint frequencies (w” = value of frequency where magnitude ratio has a value of - 3 decibels), and time constants ( 7 = time constant assuming system behaves as a first-order lag). These quantities are given in Tables I11 and IV, where the effect of the number of trays is made evident.
Table 111.
of Trays
Ll-o.
Values of Corrective Action for Columns Considered in Computer Studies Correctioe .4ction To attain required Dynamic Jeedforward controller Jnal steady state ( Transfer function) (steadystate gain = K )
10 20 40
R/F 10.1429s 4- l ) / ( s C 1110 07692.r2 10.50110 0769lr 11 (0.2632s f lj,/(l .786;'+ 1)[0.166?is2+'(0.30)(0.1667js f 11 (O.jOOs $. 1)/(3.333s I)[0.33332szf ( 0 . 2 0 ) ( 0 . 3 3 3 3 ) ~ 11
+
2.000 2,000 2.000
10 20 40
+
+
+
l/(l.ll:s f 1) 1/[1.4712s2f (2.0)(1,471)s 1/[3.4482s2 (2.0)(3.448)s
-54.32 -57.87 -47.70
++ 1111
+
V/F
10 20 40
2.500 2.500 2,500
(0.10s
(0.25s ( 0 50s
+ l ) / ( s + 1)[0.08333's2 + ( 0 . 6 0 ) ( 0 . 0 8 3 3 3 )+~ 11 + l ) / ( l , 7 8 6 r + 1)[0.14292s2f (0.50)(0.1429)sf + 1)/(3.704s f
v/SF
10 20 40
Transfer Function
+
+
+
+
Steady-State Gains and Breakpoint Frequencies for Columns Considered in Computer Studies Case Considered .Y = 70 = 20 .Y = 40 s = 10 .v = 70 s = 70 cy = 7.J cy = 1.4 a = 7.4 cy = 7.8 cy = 7.4 a = 7.4 L I D = .d.0 L ! D = 4.0 L / D = 4.0 L I D = 4.0 L I D = 2.0 L / D = 4.0 l / a = 78.7 7 / a = 78.7 7 / ' a = 78.7 7 / a = 78.7 l/a = 78.1 7 l a = 70.0 F = 700 F = 700 F = 700 F = 700 F = 200 F = 700 2.00
R,'F
+
(0.8333s 1)/(0.4545s 1)[0.27032s2 f (2.0)(0.2703)sf 11 ( 2 .50s 1)/(1 ,111s 1)[0.86962s2f i l . 8 ) ( 0 . 8 6 9 6 ) ~ I ] (5.00sf 1 ) / ( 2 8 j 7 s f 1)[2.4392s2 (1 .6)(2.439)s f 11
134.5 86.72 76.87
Table IV.
+
+ (0.20)(0.3125)s+ 1111
1)[0.31252s2
2.00
.57,87 2.50 86.72
1.03 0.92
1 03 4.60
0.57 0.445 0.55 1.75
Steady-State Gain, K 2 00 - 47 70 2 50 76.87 Breakpoint Frequency, 0.285 0.194 0.270 0.560
Values of breakpoint frequency for a column containing .V trays can be related to the breakpoint frequency for a
column of 10 trays by the empirical equation
\shere exponent P has a value of 1.00 for transfer functions K F and V,7'F, 1.20 for R;'z,, and 1.65 for Vs/'zF. T h e relationship is, of coursr, valid only when the two columns are identical in all respects except for the value of 5. T h e feedforward controller time constants in Table I11 may be used to estimate the range required for the equipment serving as a feedfonvard controller; the T values range from 0.1 to 5 . 2 minutes. A few other Bode plots were computed to get a qualitative idea of the rffects of varying relative volatility, reflux ratio, and tray hydraulics upon the feedforward controller transfer functions. These Bode plots, along with all of those mentioned previously. are given by Luyben ( 9 ) . Values of K , w'. and T are shown in Tables I11 and I V . Reference to thew tables shows that the T values increase with increasing reflux ratio and Jvitli increasing hydraulic lag. and decrease dightly Lvith relative volatility. For a change in z F , the R values decrease Lvith increasing relative volatility or reflux ratio.
2.00 -27.32 2.50 96.37
1 .00 -222.40 1.50 261.20
2.00 -54 42 2.50 134 4
1.12 1.42 1.08 4.55
1.73 1.46 1 .68 6.90
0.56 0.55 0.54 4.40
WF
Effectiveness of Controller. T h e effectiveness of the feedforward controller was tested for columns of 10 and 40 trays by analog simulation of both the controller and the columncondenser-reboiler system. Step or pulse disturbances in F or t P were introduced, corrective action was taken by the controller, and the compositions of the two products and various tray liquids were recorded as functions of time. A total of 1 7 amplifiers was required to simulate the controller, and 83 amplifiers were required to simulate the 40-tray column. A change in reflux rate introduced to the top of the '40-tray column was not felt by the bottom tray for 3 minutes and took 6 minutes to reach 80% of its final value. T h e corrective action administered by the controller was the dynamic case shown in Figure 6. T h e effectiveness of control is shown in Figure 7 for the 10and 40-tray columns. T h e deviations in product compositions are seen to be very small during the transient period and are eventually reduced to zero. T h e approximations to the true Bode plot relationships (Table 111) are apparently precise enough to permit good feedforward control. However, deviations in product compositions are larger for the 40-tray case. This indicates that control becomes more difficult as N increases, and more dynamically precise feedforward controllers are required. Also, the bottom section of the column is more sensitive to the accuracy of the feedforward controller; this VOL.
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OCTOBER 1 9 6 4
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TIME, MINUTES
TIME, MINUTES
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8
W
O t I
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t v) - I
XD
- I N S T CONTROL
B
I
U L Y CHANGE I N
8
I
~ L - X ~ - D Y NCONTROL
F
X -,
INST CONTROL
40 TRkYS PULSECHANGE I N 2 ,
-2
I 10
20
50
3
0
10
TIME, MINUTES
20
30
TIME, MINUTES
Figure 7. Change in composition of overhead and bottoms products, subjected to change in feed rate or feed composition
XD
and
XW,
for 10- and 40-tray analog column
Results shown for cases of no control, instantaneous corrective action, and dynamic control b y feedforward controller
is due to the instantaneous propagation of changes in vapor rate and its strong effect on the reboiler. If larger reboiler (and condenser) holdups were employed, the system would be less sensitive, and less dynamically precise feedforward controllers would be required.
Much more research is needed in this area. Generalized correlations of the feedforward controller transfer functions are desirable, including effects of multicomponents and unequal molar overflow. Combinations of feedforward and feedback control should be investigated.
Conclusion
Nomenclature
Feedforward controllers should be considerrd for distillation columns Relatively simple controllers appear to be adequate. For small input disturbances. a linear model for the system can be used to determine controller transfer functions. These could be verified, improved. and optimized experimentally in the plant.
a
380
l & E C P R O C E S S DESIGN AND DEVELOPMENT
b
= constant in Equation = constant
D
=
5
distillate rate, l b moles hr.
F = feed rate. lb. moles hr. FFC = feedforward conti oller transfer function = liquid holdup on tra). lb. moles H 7
=a
K L rn .V
P R s
t
V x
y z
steady-state gain liquid rate: Ib. mo1es;hr. slope of equilibrium curve number of trays in column = exponent in Equation 16 = reflux rate. Ib. moles hr. = Laplace operator = time, hours = vapor rate, Ib. moles, hr. = mole fraction of component in liquid = mole fraction of component in vapor = mole fraction of component in feed
literature Cited
= = = =
GREEKLETTERS cy = relative volatility = time constant assuming first-order system T w = frequency. radians, min. W X = breakpoint frequency, radiansimin.
SVBSCRIPTS
D
= distillate
E‘
= = = =
feed rectifying section of column s stripping section of column it’ bottoms product 1. 2. etc. = tra)- number or constant designation A bar over a variable indicates that it is a steady-state quantity: a variable without the bar is a perturbation from ateady state. 7
(1) Baber, M. F., Edwards, L. L., Harper, W. T.. Witte, M. D., Gerster. J. A , , Chem. Eng. Progr. Symp. Se7. 57, No. 36, 148 (1961). (2) Baber. M. F., Gerster. J. A , , A.I.Ch.E. J . 8, 407 (1962). FUND4MENTALS (3) Bollinger, R. E., I,amb, D. E., IND.ENG.CHEM. 1, 245 (1962). (4) Calvert. S., Coulman, G., Chem. Eng. Progr. 57, 45 (1961). (5) Dobson, J . G.. Interkama 1960. Dussrldorf, 1960. (6) ,Grrster, J. A , ? Hill, A . B.. Hochgraf. N. N.. Robinson, D. G., ‘’ rray Efficiencies in Distillation Columns,” Final Report from UniLersity of Delaware to A.1.Ch.E. Research Committee, American Institute of Chemical Engineers, New York, 1958. (7) Lamb, D. E.. Pigford, R. I,., Rippin, D. W., Chem. Eng. Progr. Symp. Ser. 57, No. 36, 132 (1961). (8) Lupfer, D. E., Parsons, J. R., Chem. En
