Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, p 874, Wiley, Yew York, Y. Y.,
19.59. Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. X., Ind. Eng. Chem. 41, 47.5 (1949). Orve, 1%.V., Prausnitz, J. M.,Trans. Faraday SOC.61, 1338 (1965). Prausnitz, J. M,, “Molecular Thermodynamics of Fluid-Phase Equilibria,” p 214, Prentice-Hall, Englewood Cliffs, N. J., 1969. Prausnitz, J. >I., Chueh, P. L., “Computer Calculations for High-Pressure Vapor-Liquid Equilibria,” p 83, Prentice-Hall, Englewood Cliff?, N. J., 1968. Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. End. Chem. 42, 343 (1950).
Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 43, 976 (1951).
Redlich, O., Kister, A. T., Ind. Eng. Chem. 40, 345 (1948). Thompson, R. E., Edmister, W. C., A.I.Ch.E. J . 1 1 , 457 (1965). Van Ness. H. C.. “Classical Thermodvnamics for Son-Electrolyte Solutions,’; p 107, Pergamon, Eimsford, N. Y., 1964. Van Ness, H. C., A.I.Ch.E. J . 19, 238 (1973). Wichterle, I., Kobayashi, R., J . Chem. Eng. Data 17, 9 (1972). RECEIVED for review March 12, 1973 ACCEPTED June 26, 1973 For financial support the authors are grateful to the Sational Science Foundation, the American Petroletim Institute and the Xatural Gas Producers Association.
Parallel Cascade Control William 1. Luyben Department of Chemical Engineering, Lehigh University,Bethlehem, Pa. 18015
The differences between conventional “series” cascade control systems and configurations that are of a “parallel” type are explored. Quantitative analysis and controller design for each type of system shows the improvements in load responses. Second-order processes, with and without deadtime, are studied over a range of time constants.
C a s c a d e control systems are widely used in industry for improving the dynamic response of systems. Cascade systems are particularly useful in reducing the effects of load disturbances that are introduced into the secondary or slave loop. The majority of these systems are of the series type; Le., the output of the secondary loop process transfer function is the input to t,he primary loop process transfer function. d typical example (Figure 1) is tray temperature control in a distillation column cascaded to reboiler steam flow control. Steam flow rate is the output of the secondary loop process transfer function, and steam flow is the input to the primary loop process transfer function (temperature/steam flow). These series cascade systems were quantitatively studied by Franks and Worley (1956). Cascade control is sometimes used in process systems where the primary and secondary process transfer functions are not in series but are in parallel. Jauffret (1973) cited one example, the temperature control of subcooled reflux by cascade control of exit cooling water temperature in a condenser. The manipulative variable, cooling water flow, affects both exit cooling water temperature and reflux temperature through parallel transfer functions. Ainother example (Figure 1) is the overhead composition control of a distillation column by cascade control of a tray temperature. The manipulative variable, reflux flow, affects overhead compositioii and tray temperature t,lirough two parallel process transfer functions. The purpose of this paper is to explore quant,itatively the differences between the series and parallel configurations. Stability Analysis
Figure 1 shows two cascade control loops on a distillation column. The lower temperature control loop maintains a
temperature on some tray in the stripping section of the column by changing the setpoint of a steam flow controller. The secondary or slave process transfer fuiictiou Gs in this system is the valve transfer function, the relationship between the flow controller output and the steam flow rate. The primary or master process transfer function G ~ is I the relationship between steam flow and t,ray temperature. These two process transfer functions are in series, as illustrated in Figure n
L.
The stability of the slave loop depends on the roots of the closed loop characteristic equation 1
+ BsGs = 0
(1)
The stability of the master loop in this conventional series cascade system, with the slave loop on automatic, depends on the roots of the closed loop characteristic equation
Without cascade control, the closed loop characteristic equation for the system is
1
+ BMGMGS 0 =
(3)
The upper temperature control loop in Figure 1 illustrates a parallel cascade process. Overhead vapor composition is controlled by changing the setpoint of a tray t’emperature controller. The manipulative variable in this system is reflux flow rate. It affects both overhead composition and tray temperature through two distinct transfer functions. Thus the process transfer functions are not connected in series. Reflux has parallel effects on temperature and overhead composition. These effects are, of course, interdependent and interacting. Reflux does not affect temperature first, which then affects Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
463
NO
CASCADE
SERIES
YY
( r M' 2 )
\ 4
-I
I
-I
I
8 .
Steam u
t
Figure 1 . Distillation column examples of series and parallel cascade
SERIES -
1
Figure 3. Root locus plots without cascade
CASCADE -
II
1
SERIES
CASCADE ( K s
= i , T~ = i )
Im
\
PARALLEL
M
Im
CASCADE
I
*I Figure 2. Block diagrams of series and parallel cascade
overhead composition. Figure 2 gives a block diagram representation of the system. The two parallel process transfer functions are Gs and Ghl for temperature and composition, respectively. The slave closedloop characteristic equation is the same as in the series case, eq 1, but the closed loop characteristic equation of the master loop in this parallel cascade system, with the slave loop on automatic, is
(4) This differs from eq 2 by the deletion of the Gs term in the numerator. Without cascade control, the closed loop characteristic equation is 1
+ BMGM= 0
(5)
Without cascade control, the slave process transfer function Gs is not involved a t all in controller design in this parallel 464 Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
-2
I
I
- 2
-I
I
Ra
Figure 4. Root locus plots for series cascade system
system. This was not true in the series system (see eq 3). Thus these two processes are distinctly different, and the equations used to design the master controller B J ~ are different. Two systems were studied. I n the first, both Gar and Gs were second-order lags. 1
GM = ~(7MS
+ I)*
PARALLEL
CASCADE
CASCADE
NO
( K s = l , rs = I )
\L
PARALLEL
CASCADE
KM
rM = 2
,
Im
Figure 6. Block diagrams of dead time system with and without parallel cascade -2
-2
Figure 5. Root locus plots for parallel cascade system
Table 1.
Master Controller Gain Settings
Second-Order Process
I
K x = 0.5,
K ~ I= 0.5, without cascade Series
7 a1
0.7 0.7 0.7 0.7
0.5
1 2 4
with cascade
Parallel
Series
Parallel
3 0.6 3 1 3 2 3 3 Dead Time Process
0.5
2.5 2 4 13
0
Y)
Parallel cascade
Without cascade
0.5 1 2
2.5 1.56 1.136
2.5 1.35 0.862
&i
=
Kar
ru = 2
I
3
I,
'\
0.1
0.5
I
Proportional feedback controllers were used.
Ks
NO
0
a
=
(7)
0
I
2
F R E PUENC Y
The slave coiltroller gain k ' s was set equal to one to give a closed loop damping coefficient in the slave loop of 0.707. (See Figure 3.) Root locus plots are given in Figure 4 for the series cascade between 0.5 and 4 with 7s = 1. T o system for values of conserve space, only the upper half of the s plane is shown. I 0) a t the poles of the overall system The paths start ( K ~ = open loop transfer function (s = - ~ / T A I from the master process transfer function and s = -1 i from the closed loop slave loop). The series system is fourth order, so there are four paths. -1design criterion of a 0.5 closed loop damping coefficient gives the values of master controller gain K ~ I shown in Table I. These gains increase as 711 increases. Root locus plots for the parallel cascade system are given in Figure 5. The numerator of the overall system open loop
*
2
w
Kw far gain margin = 2
D
Bs
I
0
n CASCADE
-----
CLScADE
-
I
2
\ 0
(radians /time)
Figure 7. Closed loop load transfer functions for series cascade
transfer function is a fourt'h-order polynomial in s, so there are four paths (only two shown in the upper half plane). The net order of the system is only 2, so the asymptotes do not intersect the imaginary axis (there is no ultimate gain). Gains for a 0.5 closed loop damping coefficient are shown in Table I. They are, of course, different from those in the series system. The gain a t T l I = 0.5 is higher than the gain a t T M = 1. For 711's greater than 1 , the gain increases rapidly with increasing ThI. It should be remembered that these two systems are two different processes. They are not the same process with two Ind. Eng. Chern. Fundarn., Vol. 12, No.
4, 1973 465
rM= 0.5
TM
i
I
I
NO CASCADE ------CASCADE
a
w n
a 0 In a
-1 W -1 -1
a
0
LT
a
-I W -1
a
O" -
O
0
0
2
4
I
CASCADE NO
-
CASCADE
----
4.
0 I
I
0
2
0 -1
n W v)
0 0 -1 0
w n 3 I-
-
2
I
0
FREQUENCY
(radians/time)
Figure 9. Closed loop load transfer functions for parallel dead time process 0
I
2
FREQUEN CY
0
I
2
(ra dians/time)
Figure 8. Closed loop load transfer functions for parallel cascade
I
NO
CASCADE
I.
--I
M
X M
Table I1 No cascade
Series
+
GLU GM) 1 BMGMGS
+
W i t h cascade
OS
t
.-o.s
+ + BsGs) + BMGM)
G L ( ~ GM 1 BsGs(1
+
PARALLEL
CASCADE
IC
different control systems. Therefore they cannot be directly compared as to which is the more effective. Each cascade system should be compared only with its noncascade equivalent. Figure 3 shows root locus plots for the two processes without cascade control. The series system is fourth-order because the Gs and GM transfer functions are both second order and are in series. The parallel system is second order since only Ghf is involved. The second example considered was the same overhead composition-tray temperature distillation column system but with a dead time inserted in the overhead composition loop. Physically this could correspond to an analysis dead time. Figure 6 presents block diagrams of the second-order system studies with +I, = 7s = 1. X gain margin criterion of 2 was used to determine the value of K M with and without parallel cascade control for values of dead time D from 0.5 to 2. Table I summarizes the results. Gains decrease as dead times increase. The gains with parallel cascade are larger than without cascade for dead times greater than 0.5. l o a d Responses
The closed loop response of a process depends on where the load disturbance enters the process, as well as the control system used and the dynamics of the process itself. Load 466 Ind. Eng. Chem. Fundam., Vol. 1 2 , No. 4, 1973
IC 0
IO
20
TIME
Figure 10. Time responses of dead time process to load disturbance with and without parallel cascade
disturbances can enter in the master or the slave loop. I n this study, the load disturbance L was assumed to affect both the slave variable zs and the master variable 211 through the same load transfer function GL (see Figure 2). The closed loop transfer functions between the load L and the master variable Z M for series and parallel systems ( ~ l f / Lare ) shown in Table 11. Frequency domain plots of these closed loop transfer functions provide a convenient way to compare load responses. Figures 7 , 8, and 9 show series and parallel systems, with and without cascade control, using the controller settings given in Table I. Since a perfect closed loop load transfer function is zero, the smaller the magnitude, the better the load response. The zero frequency intercept reflects the differences in con-
troller gains. The improvements attainable by the use of cascade control can be seen for values of 711 greater than 1 and for dead times greater than 0.5. The parallel system with 711 = 1 produced a n interesting result (see Figure 8). The closed loop load transfer functions are identical with and R ithout cascade control. Two different values of K\f are used, but the presence or absence of the slave loop makes the ( z b f / L ) relationship reduced to eyactly the same ratio of polynomials i n s A time domain load respoiise comparison betmeeii parallel cascade and no cascade for the system with a dead time of 2 is shoa i i in Figure 10. Without cascade, the manipulative variable M does not move until time reaches 2 because of the dead time In the parallel cascade y s t e m , the slave loop begins to mol e the manipulative T. ariable immediately. The higher allowable gain with cascade control iewlts in less steady-state error in zhf. Nomenclature Bhf
=
Bs
=
CC
= = = =
CT D FC
feedback controller transfer function iii primary or master loop feedback controller transfer fuiiction in secondary or slave loop compositioii controller composition transmitter dead time flow controller
FT = flow transmitter GL = load transfer function GM = process t'ransfer function in master loop Gs
= =
Ks
= =
1, JI s TC
= =
TT
=
zM
=
K,
= =
zpt = zs = zpt =
process transfer function in slave loop gain of mast'er controller gain of slave controller ultimate gain input load disturbance manipulative variable Laplace transform variable temperature controller temperature transmitter process out,put variable of master loop set point of master controller process output variable of slave loop set point of slave controller
GREEKLETTERS 7b1
=
'TS
=
{
=
time constant of process transfer function in master loop time constant of process transfer fuiiction in slave loop closed loop damping coefficient
literature Cited
Franks, R. G., Worley, C. W., Ind. Eng.Chem. 48,1074 (1956). Jauffret, J. P., 113.Thesis, Lehigh University, Bethlehem, Pa., 1973.
RECEIVED for review April 13, 1973 ACCEPTEDJuly 24, 1973
Batch Fractionation of Ionic Mixtures by Parametric Pumping Timothy J. ButtsI1 Norman H. Sweed," and Arthur A. Camero Department of Chemical Engineering, Princeton Cniuersity, Princeton, .V. J . 08540
Direct, thermal parametric pumping has been used to fractionate experimentally K+-H+ and K+-Naf-H+ mixtures using Dowex 50x8 as the ion exchanger. The binary exchange equilibrium i s influenced b y temperature so that desorption of K+ and adsorption of H+ occur simultaneously on heating. K+ accumulates in the top reservoir with separation factors exceeding 2000:1, while H+ accumulates in the bottom with separation factors exceeding 2000: 1 in the opposite direction. In a ternary exchange experiment, the K f separation factor was 52,000: 1 , accumulating in the top, H+ was 97,OOO:l in the opposite direction, and No+ almost completely disappeared from both reservoirs. The influence of resin swelling due to temperature i s investigated with an equilibrium theory model.
P a r a m e t r i c pumping is a cyclic process which theory and experiment have shoivii can produce substantial separations of liquid mixtures. The misture to be separated is placed in a chromatographic bed wherein both the temperature aiid fluid flow direction are chaiiged periodically and synchroiiously. This cyclic operatioil interacts constructively with the natural depeiideiice of sorption equilibria on temperature to give separations which build on themselves cycle after cycle. All previous experiments and most theory (Sweed, 1971) have dealt with binary mixtures. However, Butts, et al. 1
Presently at Exxon Chemical Company, Florham, Park, S . J.
(1972) , did esainine theoretically the separation of multisolut'e mixtures. Their analysis was based on the equilibrium theory of the parametric pump, originally proposed by Pigford, et al. (1969), wherein adsorption equilibria are assumed linear aiid noncompetit'ive. 13utts, et nl. (1972), showed that parametric pumps can fractionate multisolute mixtures for such noncompetitive equilibria. However, in most real systems, adsorption is riot a linear phenomenon. Except a t low concentrations, adsorbates typically compete for adsorbent sites, giving rise to nonlinear, Langmiur type isotherms even when only a single adsorbing species is present. I n ion exchange the sorbing (i.e.) eschangInd. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
467
