(2) Bain, J. L., Jr., master of science thesis, Univer ity of Texas,
Nomenclature d = tab diameter, inches e = absolute entrainment, lb./hr.
F F, Fh
Austin, Tex., 1957. (3) Rain, J. L., Jr., Van Winkle, M., A.Z.Ch.E. J. 1, No. 3, 363 (1961). (4) Church, W. L., Green, A. C., Lee, D. C., Mayfield, D. F., Rasmussen, B. W., Znd. Eng. Chem. 44, 2238 (1952). (5) Fair, J. R., PetrolChem. Engr. 33 (9), 45 (1961). (6) Forgrieve, John, “Commercial Jet-Tray Fractionators,” Proceedings of Symposium on Distillation, Brighton, England (1960), Institution of Chemical Engineers, 16 Belgrave Square, London, S. W. 1. (7) Jones, J. B., Pyle, C., Chem. Eng. Progr. 51, 424 (1955).
F factor, (vapor velocity) (vapor density)0.5 F factor based on superficial column velocity F factor based on hole velocity flow parameter, (L,/G,) ( ~ J ~ L ) ~ J FLv G gas rate, lb./hr.-sq. ft. column free area G, gas rate, lb./lhr. height of the weir, inches H, L liquid rate, lb./hr. ft. of weir length liquid rate, lb./hr. L, = total tray pressure drop, inches of water AP = superficial column velocity, ft./sec. u, U, = hole velocity, ft./sec. = fractional entrainment, e / ( e Lo) = angle of jet tabs from plate surface e p = density, lb./cu. ft. = = = = = = = = =
Material supplementary to this article has been deposited as Document No. 9135 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D.C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints, or $1.75 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
+
literature Cited (1) Arnold, D. S., Plank, C. A., Schoenborn, E. M., Chem. Eng. Progr. 48, 633 (1952).
RECEIVED for review March 18, 1966 ACCEPTEDSeptember 19, 1966
APPLICATION 0F CONVENT ION A L LOOP TIJNING TO S A M P L E D - D A T A S Y S T E M S H E N R Y A,, M O S L E R , L O W E L L B .
KOPPEL, AND DONALD R. COUGHANOWR
School of Chtmical Engineering, Purdue University, Lafayette, Znd.
For sampled’-data systems, in which sampling periods are of the same order as the process time constants, the response of a conditionally stable loop may or may not b e periodic, depending on whether the dominant roots are real or complex. Experimental determination of the ultimate gain will b e difficult where the response is not periodic. The value of the ultimate gain is not a useful design criterion for sampled-data control systems. The ultimate period, where the conditionally stable response is periodic, is dependent on the samlding period and not on the process dynamics, and hence does not give useful process dynamics informaticin. ‘These conclusions apply in a practical sense only to systems sampled sufficiently slowly to b e samplecl-data in nature. Systems sampled rapidly behave essentially as their continuous-data analogs and may b e anallyzed using continuous-data techniques.
HE method of loop tuning ( 7 , 6 ) is a well known technique Tfor determinir g suitable controller settings for conventional continuous-data systems. I n this method, a process with unknown dynam cs is operated in closed-loop proportional control. The loop is increased until conditional stability is evidenced by continuous, undamped cycling. Hence, there are available two important dynamic parameters of the system in an inexpensive frequency response test: the frequency a t which the process has 180’ of phase lag, and the amplitude ratio of the process at this frequency. These parameters, termed the ultirr ate frequency and the reciprocal ultimate gain, are used to (determine controller settings with reasonably successful results. However, extension of this method to sampled-data sys tems yields considerably less information about the process dynamics, and hence is not as useful a design technique. A sampled-data system results when the controller receives feedback information only at discrete instants of time. Such behavior is typical when, for example, a chromatographic device provides the measurement, or when digital control is being used. This paper shows that the response of a conditionally stable sampled-data system may or may not be periodic, explains the occurrence of the aperiodic response, verifies
these results experimentally, and discusses loop tuning of sampled-data systems in light of this information. This aperiodic phenomenon has been previously noted ( 2 ); however, no explanation was given. Analysis of System
The closed-loop system to be analyzed was examined in a n earlier paper (5). The process is first-order with delay, G p ( s ) = exp ( - u r s ) / ( r s l), and is controlled in a sampleddata feedback loop with a proportional controller, G, = K,. Because of the existence of delay time in the loop, the order of the characteristic equation depends on the ratio of sampling period to delay time. For the general case in which
+
u/(n
+ 1) 6
T/r
< a/n
n = 0, 1, 2,
...
the open-loop 2-transform is (5) d - b 1 - d . d) Z(n+l)(z - b) ’ n = 0 , 1 , 2 , z+-
G,G,H(z) = K(l
-
...
(1)
JANUARY 1 9 6 7
101
where b E e-=/7 VOL. 6
NO. 1
+
Hence, the order of the characteristic equation is n 2, where n 1 of the open-loop poles are located a t the origin in the z-domain, one pole is a t b, and the zero is a t ( b - d ) / (1 d). Since 0 < b < 1 and b < d 6 1, the open-loop zero must always be negative. The stability of the closed-loop system can be found by using the modified Routh-Hurwitz criterion as in an earlier paper ( 5 ) . As an example, for a = 0.2, the ultimate gain, K,, is plotted as a function of sampling period in Figure 1. For a / 3 6 T/r < m-i.e., n = 0, 1, 2-the constraints on the loop gain of the system were evaluated analytically. For smaller values of T/r, the curve was linearly extrapolated to the ultimate loop gain of the analogous continuous-data feedback system. Previous work (5) has shown that if T/r > (TT),,,~~,where ( is the dimensionless sampling period yielding the maximum K , for n = 0, the two roots will be real when the system is conditionally stable-Le., K = K,. However, for a 6 T/r < ( T / T ) ~ ~ the = ,roots are complex a t conditional stability. ( T / T ) was ~ ~ shown ~ to be related to the delay time in the following manner:
+
-
(3)
Root locus diagrams in the z-plane are constructed by graphical techniques identical to those used in the s-plane (4). I n each of the three cases, since n = 0, there are two open-loop poles and one open-loop zero. Because the zero is always negative, and thus to the left of both poles on the diagram, the loci will always be real for small K , become complex as K increases, and eventually return to the real axis. However, the stability boundary in the z-domain is the unit circle (4). For T/T > ( T / T ) ~ the ~ , loci return to the real axis before crossing the stability boundary, and hence the roots are real at conditional stability as shown in Figure 2a. Conditional stability occurs a t the first crossing of the unit circle by a locus. However, for a 6 T/r < ( T / T ) ~ the ~ , loci cross the stability boundary before returning to the real axis, and hence the roots are now complex at conditional stability, as shown in Figure 2c. Figure 26 shows the borderline case for T / r = ( T / T ) ~ = . For T/r < a < (T/r)max-i.e., n > 0-additional, open-loop poles are located at the origin, and some, if not all, of the roots will be complex at conditional stability. Transient Response of Closed-loop System
If the loop gain is increased to the ultimate gain, and the closed-loop system is perturbed in some manner, the characteristics of the resulting response will differ, depending on whether the roots on the unit circle are real or complex. These roots will Le termed the dominant roots. The stable roots which lie within the unit circle will affect only the initial part of the transient, and after a finite period of time, their
I t may be shown from Equations 2 and 3 that (T/r),,,*=is always greater than a. To illustrate how the location of ( T / T ) affects ~ ~ the character of the roots a t conditional stability, consider a = 0.2. Equation 2 yields (T/T),,,~~ = 0.5811. For n = 0, Figure 2 gives examples of the root loci for the three possible cases: T / r > ( T / T ) ~ T~ /,r = (TAT-=, and a 6 T/r < (T/rImax.
f i r
Unit c ircl e
I
5 t
(T
3
0
0.2
1 r’max
0.4 0.6 TI?-
\
0.8
1.0
Figure 1. Sampled-data stability constraint for firstorder process with delay (a = 0.2) as a function of sampling period Extrapolated part of curve to continuous ultimate gain 102
I&EC PROCESS DESIGN A N D DEVELOPMENT
Figure 2. Root locus diagrams for firstorder process with delay (a = 0.2) with different sampling periods = 0.70 > (T/dmal, pl = 0, p2 = 0.497, = 0.280, K, = 5.28 b. T/r = 0.58 = (T/r)max, p i = 0,p2 = 0.560, 11 = -0.392, K, = 8.07 T I r = 0.50 < (r/dmU,pl = 0, pn = 0.607, 21 = -0.517, K. = 7.45 a. r / r 11
effect will decay to a negligible level. The response after this time will be termed the quasi-steady-state response. For T / T > ( T / T ) ~the ~ : dominant ~, root will be real and negative, equal to - 1. Applying the inverse Z-transform (4) immediately shows that this root causes a response which is of the form
c(nT) = k [I
- (-l)n]
6
3 dt)
0
(4)
-3 where k is some constant. Hence, the response oscillates with a period of 2 T . For T / r < ( T / T ) ~ ~ the: (dominant , roots will be complex conjugates ( 4 ) ,with magnitude unity, z = exp ( j 4 ) and z = exp ( - j $ ) , where 0 < 0.5811. For the specific case T / T = 0.7 > ( T / T ) ~ ~ and K = K , = 5.273, the closed-loop response of the system is shown in Figure 3a, which corresponds to the root locus of Figure 2a. The system has been perturbed by letting c ( t = 0) = -1.0, which implies an initial error, e ( t = 0) = r - c = 1.0. After the initial part of the response due to the stable pole has decayed, the response is periodic with a period of 2 T . For this case there would be no problem in experimentally determining the ultimate gain by the method of loop tuning, Now copsider the case T / T = 0.5, which is less than ( T / T ) ~ & for a = 0.2. For this case K , = 7.447, and the dominant roots are complex. If this closed-loop system is perturbed in the same manner as in the previous case, the response will be that shown in Figure 36, \\hich corresponds to the root locus of Figure 2c. The aperiodic nature of the response is evident, and for this case it would be difficult to identify the ultimate gain from the transient response. The case of T / T = ( T / T presents ) ~ ~ ~difficulties of a different nature. The initial psrt of the transient response exists over a much greater period of time. I n addition, the system is much more sensitive to small perturbations. Letting c ( t = 0) = - 1.0, as x e did in the 1 . ~ 0previous cases, results in peak values of c ( as great as 40, implying that saturation would be an additional problem. This is the result of the high value of K , a t ( T / T ) * ~No ~ ~transient . response is shown for the case of T/T = ( T / T ) ~ ~ = . As the sampling period is made small, T / r +-0, the nature of the response a t conditional stability must approach that of the continuous loop. Consideration of the root locus diagram
20
0 I
I
I
I
I
I
I
4 dt) 0
-4 0
(b)
lo t / T
20
Figure 3. Closed-loop response of first order with delay, sampled-data system ( a = 0.2) for different sampling periods at K = K , System subjected to a momentary disturbance a. l / r = 0.7 > (T/r),,,nx, K = K, = 5.279 b. T / T = 0.5 < ( l / ~ )K ~ =*K,~=, 7.447
for T / T + 0 shows that Equation 1 has a large number of poles at z = 0, a pole at z = 1-, and a zero between z = 0 and z = ~ , - m , which zero is therefore insignificant. Conditional stability occurs as the pair of roots, from the breakaway point on the real axis just to the left of the pole at z = 1-, crosses the unit circle. Hence, 4 in Equation 5 is arbitrarily close to zero. This in turn means that the response at sampling instants given in Equation 5 is arbitrarily close to periodic, and as the sampling period is continuously decreased, the periodic conditionally stable response of the continuous loop is approached. ~Of course, this demonstration simply proves continuity of results. For practical interest, cases with very small sampling intervals may be analyzed as if they were continuous. Of primary concern in the present paper are processes sampled sufficiently slowly to be sampled-data in nature. Higher-Order Systems
Linear sampled-data systems whose processes are described by complex transfer functions will exhibit the phenomena previously discussed. These systems may also be analyzed in the z-domain by using the characteristic equation. A stability analysis can again be performed by the modified RouthHurwitz criterion; however, the amount of algebra required for higher-order systems precludes the use of this method. Instead, the root locus approach can be adopted. If the dominant roots are on the real axis in the z-domain when the system is conditionally stable, the quasi-steady-state response will be periodic with a period of 2T. But, if the dominant roots are complex, the aperiodic quasi-steady-state response will again be expected. VOL. 6
NO. 1
JANUARY 1 9 6 7
103
T o illustrate this phenomenon for higher-order systems, a second-order process with delay was simulated on the analog computer. The process transfer function considered was
where a = 0.164 and f = 0.535. These dynamic parameters were chosen to correspond closely to those of the experimental system discussed below. A sample-and-hold device was introduced into the feedback part of the loop. For various sampling rates the closed-loop system was disturbed, and the ultimate gain found by trial-and-error by observing the process output, as would be done in an experimental test for ultimate
gain. Two interesting responses are shown in Figure 4. Figure 4a is for TIT = 8a and K = 4.61. The resulting response is aperiodic, and we can only assume we are close to the ultimate gain because all peaks are similar in magnitude. In Figure 4a, the top channel reports the process output, The middle channel reports the sampled output which is being fed back to the controller, and the bottom channel reports the manipulated variable which is the output of the controller. The sampling period was then increased to T/T = 15a. Figure 46 illustrates the response for K = 1.59. Here the system is easily recognized as being conditionally stable. For this response the dominant root must lie on the real axis, because the sampled output is periodic and has a period of 2T. This illustrates that higher-order systems may be expected to exhibit the same characteristics as the first-order process with delay, characteristics which have been discussed in detail. Experimental Variflcation
Figure 4. Transient response for analog simulation of second-order process with delay a = 0.164, f = 0.535 Top channel. System output
Middle channel. Sampled output Bottom channel. Manipulated variable a. T/r = 0a = 1.31, K = 4.61, P,/r = 3.22 b. T / r = 2.46, K = 1.59, Puli- = 4.95
Figure 5.
Transient response of real system, second-order with delay
Results reported in same manner as Figure 4. 4.73, r/i- = 1 .si 104
T o present further evidence that the aperiodic phenomenon does exist for real systems controlled in a sampled-data manner, a physical system was tested. The equipment used was a water heating system, consisting of two stirred tanks connected by a delay tube. This system is described in an earlier paper ( 5 ) . Water flows through the system at a constant rate, and the control objective is to maintain the temperature of the water in tank 2, which is measured periodically, by manipulating the heat input rate into tank 1, the upstream tank. The ratio of the second tank volume to the first tank volume is 0.535. The ratio of the delay tube volume to the first tank volume is 0.087, and if 5% mixing delay for each of the two tanks is included in the delay term (5), the water process has approximately the same transfer function as Equation 6. For the particular flow rate used, 7 = 2.51 minutes and the process gain, K , = 1.15. The sampling period was set as T = 3.28 minutes, so that T/s= 8a as in Figure 4a. A disturbance was introduced into the system, and the loop gain was increased until the closed-loop system appeared to be conditionally stable. Figure 5 illustrates the response of the system. The results are reported in the same manner as in Figure 4. To eliminate noise disturbances, a 10-second filter was placed on the output of the temperature-monitoring element, and thus the system actually had a third time constant (in addition to side capacitances, measurement lags, etc.) which was relatively small compared to the two tank time constants. Thus, while its response is not identical to Figure 4a, highly similar behavior is exhibited. In particular, the aperiodic nature of the quasi-steady-state response is experimentally verified.
a = 0.1 64, f = 0.535,
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
i- =
2.51 min., T = 3.28 min.,
KI,,, =
Design Considerations
d
A major difficulty to be anticipated in use of loop tuning for sampled-data systems is recognition of the existence of conditional stability. Thus, Figures 4a and 5 show that if the
G,(s)
quasi-steady-state response is aperiodic, it may require considerable effort to determine whether the controller gain is at, above, or below the ultimate gain. This is particularly true if a chromatographic or similar measurement is used, so that only the sampled output, shown on the middle channel, may be observed. Even when conditional stability is recognized, it is difficult to use the information. The effective test input shown by the manipulated variable in the third channel of Figures 4 a and 5 is a sequencie of aperiodic square pulses. Dynamic analysis will require taking the numerical Fourier transforms, instead of the simple sinusoidal result one obtains in the continuous-data system. If such calculations are to be used, it is probably simpler to make a pulse test (7) a t the outset, than to search for the ultimate gain. I n light of these considerations, it is not difficult to see why K / K , (or gain margin) is not a good design criterion, as the duthors observed previously ( 5 ) . I n the case where thc quasi-steady-state response is periodic, as in Figure 4 b , it should not be difficult to find K , experimentally. The period of the quasi-steady-state response is 2T, and thus is not dependent on the process time constant, 7. This is in contrast to the continuous-data case where the period corresponds to the frequency at which the process exhibits 180’ of phase lag. A test a t this crossover frequency is very important for controller design, because of its close relation to system stability. Hence, again the dynamic test is of limited value in the sampled-data case, because the input form is not well suited to gibing dynamic information, and because the observed frequency is more directly related to the sampling period, T , than to the process time constant, T . Nomenclature
delay time of process as a fraction of major time constant = e -T/T = output of system
=
a
b
0)
Gp(5)
H K
= = =
eO-(n+’)T/T
process error ratio of two time constants for second-order process = controller transfer function = process transfer function = transfer function of zero-order hold = loop gain of system =
dz
K7l
=
ultimate gain
m
= manipulated variable
i
T = sampling period T/+ = normalized sampling period ( T / T ) ~=, normalized ~ sampling period for which K , is maximized for T / T 2 a 2 = e-transform variable a = constant 6 = constant 7 = major time constant of process Ac knowledgment
The financial assistance of The Esso Research and Engineering Co., The Monsanto Co., and The Procter and Gamble Co., and the generous donation of computing time by the Purdue University Computer Science Center are gratefully acknowledged.
References
(1) Coughanowr, D. R., Koppel, L. B., “Process Systems Analysis and Control,” McGraw-Hill, New York, 1965. ( 2 ) Hartwigsen, C. C., Mortimer, K., Ruopp, D. E., Zoss, L. M.; “Analysis of Sampling Rates and Controller Settings for Direct Digital Control,” ISA Conference, Preprint 30.1-4-65 (October 1965). (3) Kuo, B. C., “Analysis and Synthesis of Sampled-Data Control Systems,” pp. 164-69, Prentice-Hall, Englewood Cliffs, N. J., 1963. (4) Lindorff, D. P., “Theory of Sampled-Data Control Systems,” ChaD. 4. Wilev. New York. 1965. (5) Mbsler, H. A , Koppel, L. B., Coughanowr, D. R., IND.END. CHEM.PROCESS DESIGN DEVELOP. 5,297 (1966). (6) Ziegler, J. G., Nichols, N. B., Trans. A S M E 6 4 , 759 (1942). RECEIVED for review March 28, 1966 ACCEPTEDSeptember 29, 1966
PLUTONIUM AND URANIUM HEXAFLUORIDE HYDROLYSIS KINETICS ROBERT W. KESSIE Chemical Engineering Division, Argonne National Laboratory, Argonne, Ill.
THE reaction of the highly volatile plutonium hexafluoride (PuFs) with atmo,,pheric moisture provides a basis for secondary containment of the highly toxic plutonium. Plutonium is removed from the air directly on the solid surfaces of reaction or in the case of aerosol formation is removed by high efficiency filters. The product of the hydrolysis reaction has been identified as plutclnyl fluoride (2, 4,7) (Pu02F2) and the over-all reaction is PuFs(g)
+ 2HzO(g)
+
+
PuOZF~(S) 4HF(g)
(1)
Uranium hexafluoride (UF6) gives homologous reaction prod-
ucts with similar reaction rates. Since uranium is much less toxic than plutonium, UF6 was used to check out techniques and procedures with much less stringent safety precautions than would be required with plutonium. The objective of this investigation was to measure the hydrolysis rates of PuFc. These data would be useful in evaluating the safety of containment facilities used for PuF8. The initial experiments were performed with UF6. I n preliminary experiments of the hydrolysis of UF6 on filters, the following observations were made principally from direct examination of the reaction product in an electron V O L 6 NO. 1
JANUARY 1967
105
